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94de20e7266a43b76375f83492d1971f63d6e300be70f68838def84c1a9da7a8 | from __future__ import print_function, division
from sympy.core.add import Add
from sympy.core.assumptions import check_assumptions
from sympy.core.containers import Tuple
from sympy.core.compatibility import as_int, is_sequence, ordered
from sympy.core.exprtools import factor_terms
from sympy.core.function import _mexpand
from sympy.core.mul import Mul
from sympy.core.numbers import Rational
from sympy.core.numbers import igcdex, ilcm, igcd
from sympy.core.power import integer_nthroot, isqrt
from sympy.core.relational import Eq
from sympy.core.singleton import S
from sympy.core.symbol import Symbol, symbols
from sympy.core.sympify import _sympify
from sympy.functions.elementary.complexes import sign
from sympy.functions.elementary.integers import floor
from sympy.functions.elementary.miscellaneous import sqrt
from sympy.matrices.dense import MutableDenseMatrix as Matrix
from sympy.ntheory.factor_ import (
divisors, factorint, multiplicity, perfect_power)
from sympy.ntheory.generate import nextprime
from sympy.ntheory.primetest import is_square, isprime
from sympy.ntheory.residue_ntheory import sqrt_mod
from sympy.polys.polyerrors import GeneratorsNeeded
from sympy.polys.polytools import Poly, factor_list
from sympy.simplify.simplify import signsimp
from sympy.solvers.solveset import solveset_real
from sympy.utilities import default_sort_key, numbered_symbols
from sympy.utilities.misc import filldedent
# these are imported with 'from sympy.solvers.diophantine import *
__all__ = ['diophantine', 'classify_diop']
class DiophantineSolutionSet(set):
"""
Container for a set of solutions to a particular diophantine equation.
The base representation is a set of tuples representing each of the solutions.
Parameters
==========
symbols : list
List of free symbols in the original equation.
parameters: list (optional)
List of parameters to be used in the solution.
Examples
========
Adding solutions:
>>> from sympy.solvers.diophantine.diophantine import DiophantineSolutionSet
>>> from sympy.abc import x, y, t, u
>>> s1 = DiophantineSolutionSet([x, y], [t, u])
>>> s1
set()
>>> s1.add((2, 3))
>>> s1.add((-1, u))
>>> s1
{(-1, u), (2, 3)}
>>> s2 = DiophantineSolutionSet([x, y], [t, u])
>>> s2.add((3, 4))
>>> s1.update(*s2)
>>> s1
{(-1, u), (2, 3), (3, 4)}
Conversion of solutions into dicts:
>>> list(s1.dict_iterator())
[{x: -1, y: u}, {x: 2, y: 3}, {x: 3, y: 4}]
Substituting values:
>>> s3 = DiophantineSolutionSet([x, y], [t, u])
>>> s3.add((t**2, t + u))
>>> s3
{(t**2, t + u)}
>>> s3.subs({t: 2, u: 3})
{(4, 5)}
>>> s3.subs(t, -1)
{(1, u - 1)}
>>> s3.subs(t, 3)
{(9, u + 3)}
Evaluation at specific values. Positional arguments are given in the same order as the parameters:
>>> s3(-2, 3)
{(4, 1)}
>>> s3(5)
{(25, u + 5)}
>>> s3(None, 2)
{(t**2, t + 2)}
"""
def __init__(self, symbols_seq, parameters=None):
super().__init__()
if not is_sequence(symbols_seq):
raise ValueError("Symbols must be given as a sequence.")
self.symbols = tuple(symbols_seq)
if parameters is None:
self.parameters = symbols('%s1:%i' % ('t', len(self.symbols) + 1), integer=True)
else:
self.parameters = tuple(parameters)
def add(self, solution):
if len(solution) != len(self.symbols):
raise ValueError("Solution should have a length of %s, not %s" % (len(self.symbols), len(solution)))
super(DiophantineSolutionSet, self).add(Tuple(*solution))
def update(self, *solutions):
for solution in solutions:
self.add(solution)
def dict_iterator(self):
for solution in ordered(self):
yield dict(zip(self.symbols, solution))
def subs(self, *args, **kwargs):
result = DiophantineSolutionSet(self.symbols, self.parameters)
for solution in self:
result.add(solution.subs(*args, **kwargs))
return result
def __call__(self, *args):
if len(args) > len(self.parameters):
raise ValueError("Evaluation should have at most %s values, not %s" % (len(self.parameters), len(args)))
return self.subs(zip(self.parameters, args))
class DiophantineEquationType:
"""
Internal representation of a particular diophantine equation type.
Parameters
==========
equation
The diophantine equation that is being solved.
free_symbols: list (optional)
The symbols being solved for.
Attributes
==========
total_degree
The maximum of the degrees of all terms in the equation
homogeneous
Does the equation contain a term of degree 0
homogeneous_order
Does the equation contain any coefficient that is in the symbols being solved for
dimension
The number of symbols being solved for
"""
name = None
def __init__(self, equation, free_symbols=None):
self.equation = _sympify(equation).expand(force=True)
if free_symbols is not None:
self.free_symbols = free_symbols
else:
self.free_symbols = list(self.equation.free_symbols)
if not self.free_symbols:
raise ValueError('equation should have 1 or more free symbols')
self.free_symbols.sort(key=default_sort_key)
self.coeff = self.equation.as_coefficients_dict()
if not all(_is_int(c) for c in self.coeff.values()):
raise TypeError("Coefficients should be Integers")
self.total_degree = Poly(self.equation).total_degree()
self.homogeneous = 1 not in self.coeff
self.homogeneous_order = not (set(self.coeff) & set(self.free_symbols))
self.dimension = len(self.free_symbols)
def matches(self):
"""
Determine whether the given equation can be matched to the particular equation type.
"""
return False
class Univariate(DiophantineEquationType):
name = 'univariate'
def matches(self):
return self.dimension == 1
class Linear(DiophantineEquationType):
name = 'linear'
def matches(self):
return self.total_degree == 1
class BinaryQuadratic(DiophantineEquationType):
name = 'binary_quadratic'
def matches(self):
return self.total_degree == 2 and self.dimension == 2
class InhomogeneousTernaryQuadratic(DiophantineEquationType):
name = 'inhomogeneous_ternary_quadratic'
def matches(self):
if not (self.total_degree == 2 and self.dimension == 3):
return False
if not self.homogeneous:
return False
return not self.homogeneous_order
class HomogeneousTernaryQuadraticNormal(DiophantineEquationType):
name = 'homogeneous_ternary_quadratic_normal'
def matches(self):
if not (self.total_degree == 2 and self.dimension == 3):
return False
if not self.homogeneous:
return False
if not self.homogeneous_order:
return False
nonzero = [k for k in self.coeff if self.coeff[k]]
return len(nonzero) == 3 and all(i**2 in nonzero for i in self.free_symbols)
class HomogeneousTernaryQuadratic(DiophantineEquationType):
name = 'homogeneous_ternary_quadratic'
def matches(self):
if not (self.total_degree == 2 and self.dimension == 3):
return False
if not self.homogeneous:
return False
if not self.homogeneous_order:
return False
nonzero = [k for k in self.coeff if self.coeff[k]]
return not (len(nonzero) == 3 and all(i**2 in nonzero for i in self.free_symbols))
class InhomogeneousGeneralQuadratic(DiophantineEquationType):
name = 'inhomogeneous_general_quadratic'
def matches(self):
if not (self.total_degree == 2 and self.dimension >= 3):
return False
if not self.homogeneous_order:
return True
else:
# there may be Pow keys like x**2 or Mul keys like x*y
if any(k.is_Mul for k in self.coeff): # cross terms
return not self.homogeneous
return False
class HomogeneousGeneralQuadratic(DiophantineEquationType):
name = 'homogeneous_general_quadratic'
def matches(self):
if not (self.total_degree == 2 and self.dimension >= 3):
return False
if not self.homogeneous_order:
return False
else:
# there may be Pow keys like x**2 or Mul keys like x*y
if any(k.is_Mul for k in self.coeff): # cross terms
return self.homogeneous
return False
class GeneralSumOfSquares(DiophantineEquationType):
name = 'general_sum_of_squares'
def matches(self):
if not (self.total_degree == 2 and self.dimension >= 3):
return False
if not self.homogeneous_order:
return False
if any(k.is_Mul for k in self.coeff):
return False
return all(self.coeff[k] == 1 for k in self.coeff if k != 1)
class GeneralPythagorean(DiophantineEquationType):
name = 'general_pythagorean'
def matches(self):
if not (self.total_degree == 2 and self.dimension >= 3):
return False
if not self.homogeneous_order:
return False
if any(k.is_Mul for k in self.coeff):
return False
if all(self.coeff[k] == 1 for k in self.coeff if k != 1):
return False
if not all(is_square(abs(self.coeff[k])) for k in self.coeff):
return False
# all but one has the same sign
# e.g. 4*x**2 + y**2 - 4*z**2
return abs(sum(sign(self.coeff[k]) for k in self.coeff)) == self.dimension - 2
class CubicThue(DiophantineEquationType):
name = 'cubic_thue'
def matches(self):
return self.total_degree == 3 and self.dimension == 2
class GeneralSumOfEvenPowers(DiophantineEquationType):
name = 'general_sum_of_even_powers'
def matches(self):
if not self.total_degree > 3:
return False
if self.total_degree % 2 != 0:
return False
if not all(k.is_Pow and k.exp == self.total_degree for k in self.coeff if k != 1):
return False
return all(self.coeff[k] == 1 for k in self.coeff if k != 1)
# these types are known (but not necessarily handled)
# note that order is important here (in the current solver state)
all_diop_classes = [
Linear,
Univariate,
BinaryQuadratic,
InhomogeneousTernaryQuadratic,
HomogeneousTernaryQuadraticNormal,
HomogeneousTernaryQuadratic,
InhomogeneousGeneralQuadratic,
HomogeneousGeneralQuadratic,
GeneralSumOfSquares,
GeneralPythagorean,
CubicThue,
GeneralSumOfEvenPowers,
]
diop_known = set([diop_class.name for diop_class in all_diop_classes])
def _is_int(i):
try:
as_int(i)
return True
except ValueError:
pass
def _sorted_tuple(*i):
return tuple(sorted(i))
def _remove_gcd(*x):
try:
g = igcd(*x)
except ValueError:
fx = list(filter(None, x))
if len(fx) < 2:
return x
g = igcd(*[i.as_content_primitive()[0] for i in fx])
except TypeError:
raise TypeError('_remove_gcd(a,b,c) or _remove_gcd(*container)')
if g == 1:
return x
return tuple([i//g for i in x])
def _rational_pq(a, b):
# return `(numer, denom)` for a/b; sign in numer and gcd removed
return _remove_gcd(sign(b)*a, abs(b))
def _nint_or_floor(p, q):
# return nearest int to p/q; in case of tie return floor(p/q)
w, r = divmod(p, q)
if abs(r) <= abs(q)//2:
return w
return w + 1
def _odd(i):
return i % 2 != 0
def _even(i):
return i % 2 == 0
def diophantine(eq, param=symbols("t", integer=True), syms=None,
permute=False):
"""
Simplify the solution procedure of diophantine equation ``eq`` by
converting it into a product of terms which should equal zero.
For example, when solving, `x^2 - y^2 = 0` this is treated as
`(x + y)(x - y) = 0` and `x + y = 0` and `x - y = 0` are solved
independently and combined. Each term is solved by calling
``diop_solve()``. (Although it is possible to call ``diop_solve()``
directly, one must be careful to pass an equation in the correct
form and to interpret the output correctly; ``diophantine()`` is
the public-facing function to use in general.)
Output of ``diophantine()`` is a set of tuples. The elements of the
tuple are the solutions for each variable in the equation and
are arranged according to the alphabetic ordering of the variables.
e.g. For an equation with two variables, `a` and `b`, the first
element of the tuple is the solution for `a` and the second for `b`.
Usage
=====
``diophantine(eq, t, syms)``: Solve the diophantine
equation ``eq``.
``t`` is the optional parameter to be used by ``diop_solve()``.
``syms`` is an optional list of symbols which determines the
order of the elements in the returned tuple.
By default, only the base solution is returned. If ``permute`` is set to
True then permutations of the base solution and/or permutations of the
signs of the values will be returned when applicable.
>>> from sympy.solvers.diophantine import diophantine
>>> from sympy.abc import a, b
>>> eq = a**4 + b**4 - (2**4 + 3**4)
>>> diophantine(eq)
{(2, 3)}
>>> diophantine(eq, permute=True)
{(-3, -2), (-3, 2), (-2, -3), (-2, 3), (2, -3), (2, 3), (3, -2), (3, 2)}
Details
=======
``eq`` should be an expression which is assumed to be zero.
``t`` is the parameter to be used in the solution.
Examples
========
>>> from sympy.abc import x, y, z
>>> diophantine(x**2 - y**2)
{(t_0, -t_0), (t_0, t_0)}
>>> diophantine(x*(2*x + 3*y - z))
{(0, n1, n2), (t_0, t_1, 2*t_0 + 3*t_1)}
>>> diophantine(x**2 + 3*x*y + 4*x)
{(0, n1), (3*t_0 - 4, -t_0)}
See Also
========
diop_solve()
sympy.utilities.iterables.permute_signs
sympy.utilities.iterables.signed_permutations
"""
from sympy.utilities.iterables import (
subsets, permute_signs, signed_permutations)
eq = _sympify(eq)
if isinstance(eq, Eq):
eq = eq.lhs - eq.rhs
try:
var = list(eq.expand(force=True).free_symbols)
var.sort(key=default_sort_key)
if syms:
if not is_sequence(syms):
raise TypeError(
'syms should be given as a sequence, e.g. a list')
syms = [i for i in syms if i in var]
if syms != var:
dict_sym_index = dict(zip(syms, range(len(syms))))
return {tuple([t[dict_sym_index[i]] for i in var])
for t in diophantine(eq, param, permute=permute)}
n, d = eq.as_numer_denom()
if n.is_number:
return set()
if not d.is_number:
dsol = diophantine(d)
good = diophantine(n) - dsol
return {s for s in good if _mexpand(d.subs(zip(var, s)))}
else:
eq = n
eq = factor_terms(eq)
assert not eq.is_number
eq = eq.as_independent(*var, as_Add=False)[1]
p = Poly(eq)
assert not any(g.is_number for g in p.gens)
eq = p.as_expr()
assert eq.is_polynomial()
except (GeneratorsNeeded, AssertionError):
raise TypeError(filldedent('''
Equation should be a polynomial with Rational coefficients.'''))
# permute only sign
do_permute_signs = False
# permute sign and values
do_permute_signs_var = False
# permute few signs
permute_few_signs = False
try:
# if we know that factoring should not be attempted, skip
# the factoring step
v, c, t = classify_diop(eq)
# check for permute sign
if permute:
len_var = len(v)
permute_signs_for = [
GeneralSumOfSquares.name,
GeneralSumOfEvenPowers.name]
permute_signs_check = [
HomogeneousTernaryQuadratic.name,
HomogeneousTernaryQuadraticNormal.name,
BinaryQuadratic.name]
if t in permute_signs_for:
do_permute_signs_var = True
elif t in permute_signs_check:
# if all the variables in eq have even powers
# then do_permute_sign = True
if len_var == 3:
var_mul = list(subsets(v, 2))
# here var_mul is like [(x, y), (x, z), (y, z)]
xy_coeff = True
x_coeff = True
var1_mul_var2 = map(lambda a: a[0]*a[1], var_mul)
# if coeff(y*z), coeff(y*x), coeff(x*z) is not 0 then
# `xy_coeff` => True and do_permute_sign => False.
# Means no permuted solution.
for v1_mul_v2 in var1_mul_var2:
try:
coeff = c[v1_mul_v2]
except KeyError:
coeff = 0
xy_coeff = bool(xy_coeff) and bool(coeff)
var_mul = list(subsets(v, 1))
# here var_mul is like [(x,), (y, )]
for v1 in var_mul:
try:
coeff = c[v1[0]]
except KeyError:
coeff = 0
x_coeff = bool(x_coeff) and bool(coeff)
if not any([xy_coeff, x_coeff]):
# means only x**2, y**2, z**2, const is present
do_permute_signs = True
elif not x_coeff:
permute_few_signs = True
elif len_var == 2:
var_mul = list(subsets(v, 2))
# here var_mul is like [(x, y)]
xy_coeff = True
x_coeff = True
var1_mul_var2 = map(lambda x: x[0]*x[1], var_mul)
for v1_mul_v2 in var1_mul_var2:
try:
coeff = c[v1_mul_v2]
except KeyError:
coeff = 0
xy_coeff = bool(xy_coeff) and bool(coeff)
var_mul = list(subsets(v, 1))
# here var_mul is like [(x,), (y, )]
for v1 in var_mul:
try:
coeff = c[v1[0]]
except KeyError:
coeff = 0
x_coeff = bool(x_coeff) and bool(coeff)
if not any([xy_coeff, x_coeff]):
# means only x**2, y**2 and const is present
# so we can get more soln by permuting this soln.
do_permute_signs = True
elif not x_coeff:
# when coeff(x), coeff(y) is not present then signs of
# x, y can be permuted such that their sign are same
# as sign of x*y.
# e.g 1. (x_val,y_val)=> (x_val,y_val), (-x_val,-y_val)
# 2. (-x_vall, y_val)=> (-x_val,y_val), (x_val,-y_val)
permute_few_signs = True
if t == 'general_sum_of_squares':
# trying to factor such expressions will sometimes hang
terms = [(eq, 1)]
else:
raise TypeError
except (TypeError, NotImplementedError):
fl = factor_list(eq)
if fl[0].is_Rational and fl[0] != 1:
return diophantine(eq/fl[0], param=param, syms=syms, permute=permute)
terms = fl[1]
sols = set([])
for term in terms:
base, _ = term
var_t, _, eq_type = classify_diop(base, _dict=False)
_, base = signsimp(base, evaluate=False).as_coeff_Mul()
solution = diop_solve(base, param)
if eq_type in [
Linear.name,
HomogeneousTernaryQuadratic.name,
HomogeneousTernaryQuadraticNormal.name,
GeneralPythagorean.name]:
sols.add(merge_solution(var, var_t, solution))
elif eq_type in [
BinaryQuadratic.name,
GeneralSumOfSquares.name,
GeneralSumOfEvenPowers.name,
Univariate.name]:
for sol in solution:
sols.add(merge_solution(var, var_t, sol))
else:
raise NotImplementedError('unhandled type: %s' % eq_type)
# remove null merge results
if () in sols:
sols.remove(())
null = tuple([0]*len(var))
# if there is no solution, return trivial solution
if not sols and eq.subs(zip(var, null)).is_zero:
sols.add(null)
final_soln = set([])
for sol in sols:
if all(_is_int(s) for s in sol):
if do_permute_signs:
permuted_sign = set(permute_signs(sol))
final_soln.update(permuted_sign)
elif permute_few_signs:
lst = list(permute_signs(sol))
lst = list(filter(lambda x: x[0]*x[1] == sol[1]*sol[0], lst))
permuted_sign = set(lst)
final_soln.update(permuted_sign)
elif do_permute_signs_var:
permuted_sign_var = set(signed_permutations(sol))
final_soln.update(permuted_sign_var)
else:
final_soln.add(sol)
else:
final_soln.add(sol)
return final_soln
def merge_solution(var, var_t, solution):
"""
This is used to construct the full solution from the solutions of sub
equations.
For example when solving the equation `(x - y)(x^2 + y^2 - z^2) = 0`,
solutions for each of the equations `x - y = 0` and `x^2 + y^2 - z^2` are
found independently. Solutions for `x - y = 0` are `(x, y) = (t, t)`. But
we should introduce a value for z when we output the solution for the
original equation. This function converts `(t, t)` into `(t, t, n_{1})`
where `n_{1}` is an integer parameter.
"""
sol = []
if None in solution:
return ()
solution = iter(solution)
params = numbered_symbols("n", integer=True, start=1)
for v in var:
if v in var_t:
sol.append(next(solution))
else:
sol.append(next(params))
for val, symb in zip(sol, var):
if check_assumptions(val, **symb.assumptions0) is False:
return tuple()
return tuple(sol)
def diop_solve(eq, param=symbols("t", integer=True)):
"""
Solves the diophantine equation ``eq``.
Unlike ``diophantine()``, factoring of ``eq`` is not attempted. Uses
``classify_diop()`` to determine the type of the equation and calls
the appropriate solver function.
Use of ``diophantine()`` is recommended over other helper functions.
``diop_solve()`` can return either a set or a tuple depending on the
nature of the equation.
Usage
=====
``diop_solve(eq, t)``: Solve diophantine equation, ``eq`` using ``t``
as a parameter if needed.
Details
=======
``eq`` should be an expression which is assumed to be zero.
``t`` is a parameter to be used in the solution.
Examples
========
>>> from sympy.solvers.diophantine.diophantine import diop_solve
>>> from sympy.abc import x, y, z, w
>>> diop_solve(2*x + 3*y - 5)
(3*t_0 - 5, 5 - 2*t_0)
>>> diop_solve(4*x + 3*y - 4*z + 5)
(t_0, 8*t_0 + 4*t_1 + 5, 7*t_0 + 3*t_1 + 5)
>>> diop_solve(x + 3*y - 4*z + w - 6)
(t_0, t_0 + t_1, 6*t_0 + 5*t_1 + 4*t_2 - 6, 5*t_0 + 4*t_1 + 3*t_2 - 6)
>>> diop_solve(x**2 + y**2 - 5)
{(-2, -1), (-2, 1), (-1, -2), (-1, 2), (1, -2), (1, 2), (2, -1), (2, 1)}
See Also
========
diophantine()
"""
var, coeff, eq_type = classify_diop(eq, _dict=False)
if eq_type == Linear.name:
return diop_linear(eq, param)
elif eq_type == BinaryQuadratic.name:
return diop_quadratic(eq, param)
elif eq_type == HomogeneousTernaryQuadratic.name:
return diop_ternary_quadratic(eq, parameterize=True)
elif eq_type == HomogeneousTernaryQuadraticNormal.name:
return diop_ternary_quadratic_normal(eq, parameterize=True)
elif eq_type == GeneralPythagorean.name:
return diop_general_pythagorean(eq, param)
elif eq_type == Univariate.name:
return diop_univariate(eq)
elif eq_type == GeneralSumOfSquares.name:
return diop_general_sum_of_squares(eq, limit=S.Infinity)
elif eq_type == GeneralSumOfEvenPowers.name:
return diop_general_sum_of_even_powers(eq, limit=S.Infinity)
if eq_type is not None and eq_type not in diop_known:
raise ValueError(filldedent('''
Alhough this type of equation was identified, it is not yet
handled. It should, however, be listed in `diop_known` at the
top of this file. Developers should see comments at the end of
`classify_diop`.
''')) # pragma: no cover
else:
raise NotImplementedError(
'No solver has been written for %s.' % eq_type)
def classify_diop(eq, _dict=True):
# docstring supplied externally
matched = False
diop_type = None
for diop_class in all_diop_classes:
diop_type = diop_class(eq)
if diop_type.matches():
matched = True
break
if matched:
return diop_type.free_symbols, dict(diop_type.coeff) if _dict else diop_type.coeff, diop_type.name
# new diop type instructions
# --------------------------
# if this error raises and the equation *can* be classified,
# * it should be identified in the if-block above
# * the type should be added to the diop_known
# if a solver can be written for it,
# * a dedicated handler should be written (e.g. diop_linear)
# * it should be passed to that handler in diop_solve
raise NotImplementedError(filldedent('''
This equation is not yet recognized or else has not been
simplified sufficiently to put it in a form recognized by
diop_classify().'''))
classify_diop.func_doc = ( # type: ignore
'''
Helper routine used by diop_solve() to find information about ``eq``.
Returns a tuple containing the type of the diophantine equation
along with the variables (free symbols) and their coefficients.
Variables are returned as a list and coefficients are returned
as a dict with the key being the respective term and the constant
term is keyed to 1. The type is one of the following:
* %s
Usage
=====
``classify_diop(eq)``: Return variables, coefficients and type of the
``eq``.
Details
=======
``eq`` should be an expression which is assumed to be zero.
``_dict`` is for internal use: when True (default) a dict is returned,
otherwise a defaultdict which supplies 0 for missing keys is returned.
Examples
========
>>> from sympy.solvers.diophantine import classify_diop
>>> from sympy.abc import x, y, z, w, t
>>> classify_diop(4*x + 6*y - 4)
([x, y], {1: -4, x: 4, y: 6}, 'linear')
>>> classify_diop(x + 3*y -4*z + 5)
([x, y, z], {1: 5, x: 1, y: 3, z: -4}, 'linear')
>>> classify_diop(x**2 + y**2 - x*y + x + 5)
([x, y], {1: 5, x: 1, x**2: 1, y**2: 1, x*y: -1}, 'binary_quadratic')
''' % ('\n * '.join(sorted(diop_known))))
def diop_linear(eq, param=symbols("t", integer=True)):
"""
Solves linear diophantine equations.
A linear diophantine equation is an equation of the form `a_{1}x_{1} +
a_{2}x_{2} + .. + a_{n}x_{n} = 0` where `a_{1}, a_{2}, ..a_{n}` are
integer constants and `x_{1}, x_{2}, ..x_{n}` are integer variables.
Usage
=====
``diop_linear(eq)``: Returns a tuple containing solutions to the
diophantine equation ``eq``. Values in the tuple is arranged in the same
order as the sorted variables.
Details
=======
``eq`` is a linear diophantine equation which is assumed to be zero.
``param`` is the parameter to be used in the solution.
Examples
========
>>> from sympy.solvers.diophantine.diophantine import diop_linear
>>> from sympy.abc import x, y, z
>>> diop_linear(2*x - 3*y - 5) # solves equation 2*x - 3*y - 5 == 0
(3*t_0 - 5, 2*t_0 - 5)
Here x = -3*t_0 - 5 and y = -2*t_0 - 5
>>> diop_linear(2*x - 3*y - 4*z -3)
(t_0, 2*t_0 + 4*t_1 + 3, -t_0 - 3*t_1 - 3)
See Also
========
diop_quadratic(), diop_ternary_quadratic(), diop_general_pythagorean(),
diop_general_sum_of_squares()
"""
var, coeff, diop_type = classify_diop(eq, _dict=False)
if diop_type == Linear.name:
result = _diop_linear(var, coeff, param)
if param is None:
result = result(*[0]*len(result.parameters))
if len(result) > 0:
return list(result)[0]
else:
return tuple([None] * len(result.parameters))
def _diop_linear(var, coeff, param):
"""
Solves diophantine equations of the form:
a_0*x_0 + a_1*x_1 + ... + a_n*x_n == c
Note that no solution exists if gcd(a_0, ..., a_n) doesn't divide c.
"""
if 1 in coeff:
# negate coeff[] because input is of the form: ax + by + c == 0
# but is used as: ax + by == -c
c = -coeff[1]
else:
c = 0
# Some solutions will have multiple free variables in their solutions.
if param is None:
params = [symbols('t')]*len(var)
else:
temp = str(param) + "_%i"
params = [symbols(temp % i, integer=True) for i in range(len(var))]
result = DiophantineSolutionSet(var, params)
if len(var) == 1:
q, r = divmod(c, coeff[var[0]])
if not r:
result.add((q,))
return result
else:
return result
'''
base_solution_linear() can solve diophantine equations of the form:
a*x + b*y == c
We break down multivariate linear diophantine equations into a
series of bivariate linear diophantine equations which can then
be solved individually by base_solution_linear().
Consider the following:
a_0*x_0 + a_1*x_1 + a_2*x_2 == c
which can be re-written as:
a_0*x_0 + g_0*y_0 == c
where
g_0 == gcd(a_1, a_2)
and
y == (a_1*x_1)/g_0 + (a_2*x_2)/g_0
This leaves us with two binary linear diophantine equations.
For the first equation:
a == a_0
b == g_0
c == c
For the second:
a == a_1/g_0
b == a_2/g_0
c == the solution we find for y_0 in the first equation.
The arrays A and B are the arrays of integers used for
'a' and 'b' in each of the n-1 bivariate equations we solve.
'''
A = [coeff[v] for v in var]
B = []
if len(var) > 2:
B.append(igcd(A[-2], A[-1]))
A[-2] = A[-2] // B[0]
A[-1] = A[-1] // B[0]
for i in range(len(A) - 3, 0, -1):
gcd = igcd(B[0], A[i])
B[0] = B[0] // gcd
A[i] = A[i] // gcd
B.insert(0, gcd)
B.append(A[-1])
'''
Consider the trivariate linear equation:
4*x_0 + 6*x_1 + 3*x_2 == 2
This can be re-written as:
4*x_0 + 3*y_0 == 2
where
y_0 == 2*x_1 + x_2
(Note that gcd(3, 6) == 3)
The complete integral solution to this equation is:
x_0 == 2 + 3*t_0
y_0 == -2 - 4*t_0
where 't_0' is any integer.
Now that we have a solution for 'x_0', find 'x_1' and 'x_2':
2*x_1 + x_2 == -2 - 4*t_0
We can then solve for '-2' and '-4' independently,
and combine the results:
2*x_1a + x_2a == -2
x_1a == 0 + t_0
x_2a == -2 - 2*t_0
2*x_1b + x_2b == -4*t_0
x_1b == 0*t_0 + t_1
x_2b == -4*t_0 - 2*t_1
==>
x_1 == t_0 + t_1
x_2 == -2 - 6*t_0 - 2*t_1
where 't_0' and 't_1' are any integers.
Note that:
4*(2 + 3*t_0) + 6*(t_0 + t_1) + 3*(-2 - 6*t_0 - 2*t_1) == 2
for any integral values of 't_0', 't_1'; as required.
This method is generalised for many variables, below.
'''
solutions = []
for i in range(len(B)):
tot_x, tot_y = [], []
for j, arg in enumerate(Add.make_args(c)):
if arg.is_Integer:
# example: 5 -> k = 5
k, p = arg, S.One
pnew = params[0]
else: # arg is a Mul or Symbol
# example: 3*t_1 -> k = 3
# example: t_0 -> k = 1
k, p = arg.as_coeff_Mul()
pnew = params[params.index(p) + 1]
sol = sol_x, sol_y = base_solution_linear(k, A[i], B[i], pnew)
if p is S.One:
if None in sol:
return result
else:
# convert a + b*pnew -> a*p + b*pnew
if isinstance(sol_x, Add):
sol_x = sol_x.args[0]*p + sol_x.args[1]
if isinstance(sol_y, Add):
sol_y = sol_y.args[0]*p + sol_y.args[1]
tot_x.append(sol_x)
tot_y.append(sol_y)
solutions.append(Add(*tot_x))
c = Add(*tot_y)
solutions.append(c)
if param is None:
# just keep the additive constant (i.e. replace t with 0)
solutions = [i.as_coeff_Add()[0] for i in solutions]
result.add(solutions)
return result
def base_solution_linear(c, a, b, t=None):
"""
Return the base solution for the linear equation, `ax + by = c`.
Used by ``diop_linear()`` to find the base solution of a linear
Diophantine equation. If ``t`` is given then the parametrized solution is
returned.
Usage
=====
``base_solution_linear(c, a, b, t)``: ``a``, ``b``, ``c`` are coefficients
in `ax + by = c` and ``t`` is the parameter to be used in the solution.
Examples
========
>>> from sympy.solvers.diophantine.diophantine import base_solution_linear
>>> from sympy.abc import t
>>> base_solution_linear(5, 2, 3) # equation 2*x + 3*y = 5
(-5, 5)
>>> base_solution_linear(0, 5, 7) # equation 5*x + 7*y = 0
(0, 0)
>>> base_solution_linear(5, 2, 3, t) # equation 2*x + 3*y = 5
(3*t - 5, 5 - 2*t)
>>> base_solution_linear(0, 5, 7, t) # equation 5*x + 7*y = 0
(7*t, -5*t)
"""
a, b, c = _remove_gcd(a, b, c)
if c == 0:
if t is not None:
if b < 0:
t = -t
return (b*t , -a*t)
else:
return (0, 0)
else:
x0, y0, d = igcdex(abs(a), abs(b))
x0 *= sign(a)
y0 *= sign(b)
if divisible(c, d):
if t is not None:
if b < 0:
t = -t
return (c*x0 + b*t, c*y0 - a*t)
else:
return (c*x0, c*y0)
else:
return (None, None)
def diop_univariate(eq):
"""
Solves a univariate diophantine equations.
A univariate diophantine equation is an equation of the form
`a_{0} + a_{1}x + a_{2}x^2 + .. + a_{n}x^n = 0` where `a_{1}, a_{2}, ..a_{n}` are
integer constants and `x` is an integer variable.
Usage
=====
``diop_univariate(eq)``: Returns a set containing solutions to the
diophantine equation ``eq``.
Details
=======
``eq`` is a univariate diophantine equation which is assumed to be zero.
Examples
========
>>> from sympy.solvers.diophantine.diophantine import diop_univariate
>>> from sympy.abc import x
>>> diop_univariate((x - 2)*(x - 3)**2) # solves equation (x - 2)*(x - 3)**2 == 0
{(2,), (3,)}
"""
var, coeff, diop_type = classify_diop(eq, _dict=False)
if diop_type == Univariate.name:
return set([(int(i),) for i in solveset_real(
eq, var[0]).intersect(S.Integers)])
def divisible(a, b):
"""
Returns `True` if ``a`` is divisible by ``b`` and `False` otherwise.
"""
return not a % b
def diop_quadratic(eq, param=symbols("t", integer=True)):
"""
Solves quadratic diophantine equations.
i.e. equations of the form `Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0`. Returns a
set containing the tuples `(x, y)` which contains the solutions. If there
are no solutions then `(None, None)` is returned.
Usage
=====
``diop_quadratic(eq, param)``: ``eq`` is a quadratic binary diophantine
equation. ``param`` is used to indicate the parameter to be used in the
solution.
Details
=======
``eq`` should be an expression which is assumed to be zero.
``param`` is a parameter to be used in the solution.
Examples
========
>>> from sympy.abc import x, y, t
>>> from sympy.solvers.diophantine.diophantine import diop_quadratic
>>> diop_quadratic(x**2 + y**2 + 2*x + 2*y + 2, t)
{(-1, -1)}
References
==========
.. [1] Methods to solve Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0, [online],
Available: http://www.alpertron.com.ar/METHODS.HTM
.. [2] Solving the equation ax^2+ bxy + cy^2 + dx + ey + f= 0, [online],
Available: http://www.jpr2718.org/ax2p.pdf
See Also
========
diop_linear(), diop_ternary_quadratic(), diop_general_sum_of_squares(),
diop_general_pythagorean()
"""
var, coeff, diop_type = classify_diop(eq, _dict=False)
if diop_type == BinaryQuadratic.name:
return set(_diop_quadratic(var, coeff, param))
def _diop_quadratic(var, coeff, t):
u = Symbol('u', integer=True)
x, y = var
A = coeff[x**2]
B = coeff[x*y]
C = coeff[y**2]
D = coeff[x]
E = coeff[y]
F = coeff[S.One]
A, B, C, D, E, F = [as_int(i) for i in _remove_gcd(A, B, C, D, E, F)]
# (1) Simple-Hyperbolic case: A = C = 0, B != 0
# In this case equation can be converted to (Bx + E)(By + D) = DE - BF
# We consider two cases; DE - BF = 0 and DE - BF != 0
# More details, http://www.alpertron.com.ar/METHODS.HTM#SHyperb
result = DiophantineSolutionSet(var, [t, u])
discr = B**2 - 4*A*C
if A == 0 and C == 0 and B != 0:
if D*E - B*F == 0:
q, r = divmod(E, B)
if not r:
result.add((-q, t))
q, r = divmod(D, B)
if not r:
result.add((t, -q))
else:
div = divisors(D*E - B*F)
div = div + [-term for term in div]
for d in div:
x0, r = divmod(d - E, B)
if not r:
q, r = divmod(D*E - B*F, d)
if not r:
y0, r = divmod(q - D, B)
if not r:
result.add((x0, y0))
# (2) Parabolic case: B**2 - 4*A*C = 0
# There are two subcases to be considered in this case.
# sqrt(c)D - sqrt(a)E = 0 and sqrt(c)D - sqrt(a)E != 0
# More Details, http://www.alpertron.com.ar/METHODS.HTM#Parabol
elif discr == 0:
if A == 0:
s = _diop_quadratic([y, x], coeff, t)
for soln in s:
result.add((soln[1], soln[0]))
else:
g = sign(A)*igcd(A, C)
a = A // g
c = C // g
e = sign(B/A)
sqa = isqrt(a)
sqc = isqrt(c)
_c = e*sqc*D - sqa*E
if not _c:
z = symbols("z", real=True)
eq = sqa*g*z**2 + D*z + sqa*F
roots = solveset_real(eq, z).intersect(S.Integers)
for root in roots:
ans = diop_solve(sqa*x + e*sqc*y - root)
result.add((ans[0], ans[1]))
elif _is_int(c):
solve_x = lambda u: -e*sqc*g*_c*t**2 - (E + 2*e*sqc*g*u)*t\
- (e*sqc*g*u**2 + E*u + e*sqc*F) // _c
solve_y = lambda u: sqa*g*_c*t**2 + (D + 2*sqa*g*u)*t \
+ (sqa*g*u**2 + D*u + sqa*F) // _c
for z0 in range(0, abs(_c)):
# Check if the coefficients of y and x obtained are integers or not
if (divisible(sqa*g*z0**2 + D*z0 + sqa*F, _c) and
divisible(e*sqc*g*z0**2 + E*z0 + e*sqc*F, _c)):
result.add((solve_x(z0), solve_y(z0)))
# (3) Method used when B**2 - 4*A*C is a square, is described in p. 6 of the below paper
# by John P. Robertson.
# http://www.jpr2718.org/ax2p.pdf
elif is_square(discr):
if A != 0:
r = sqrt(discr)
u, v = symbols("u, v", integer=True)
eq = _mexpand(
4*A*r*u*v + 4*A*D*(B*v + r*u + r*v - B*u) +
2*A*4*A*E*(u - v) + 4*A*r*4*A*F)
solution = diop_solve(eq, t)
for s0, t0 in solution:
num = B*t0 + r*s0 + r*t0 - B*s0
x_0 = S(num)/(4*A*r)
y_0 = S(s0 - t0)/(2*r)
if isinstance(s0, Symbol) or isinstance(t0, Symbol):
if check_param(x_0, y_0, 4*A*r, t) != (None, None):
ans = check_param(x_0, y_0, 4*A*r, t)
result.add((ans[0], ans[1]))
elif x_0.is_Integer and y_0.is_Integer:
if is_solution_quad(var, coeff, x_0, y_0):
result.add((x_0, y_0))
else:
s = _diop_quadratic(var[::-1], coeff, t) # Interchange x and y
while s:
result.add(s.pop()[::-1]) # and solution <--------+
# (4) B**2 - 4*A*C > 0 and B**2 - 4*A*C not a square or B**2 - 4*A*C < 0
else:
P, Q = _transformation_to_DN(var, coeff)
D, N = _find_DN(var, coeff)
solns_pell = diop_DN(D, N)
if D < 0:
for x0, y0 in solns_pell:
for x in [-x0, x0]:
for y in [-y0, y0]:
s = P*Matrix([x, y]) + Q
try:
result.add([as_int(_) for _ in s])
except ValueError:
pass
else:
# In this case equation can be transformed into a Pell equation
solns_pell = set(solns_pell)
for X, Y in list(solns_pell):
solns_pell.add((-X, -Y))
a = diop_DN(D, 1)
T = a[0][0]
U = a[0][1]
if all(_is_int(_) for _ in P[:4] + Q[:2]):
for r, s in solns_pell:
_a = (r + s*sqrt(D))*(T + U*sqrt(D))**t
_b = (r - s*sqrt(D))*(T - U*sqrt(D))**t
x_n = _mexpand(S(_a + _b)/2)
y_n = _mexpand(S(_a - _b)/(2*sqrt(D)))
s = P*Matrix([x_n, y_n]) + Q
result.add(s)
else:
L = ilcm(*[_.q for _ in P[:4] + Q[:2]])
k = 1
T_k = T
U_k = U
while (T_k - 1) % L != 0 or U_k % L != 0:
T_k, U_k = T_k*T + D*U_k*U, T_k*U + U_k*T
k += 1
for X, Y in solns_pell:
for i in range(k):
if all(_is_int(_) for _ in P*Matrix([X, Y]) + Q):
_a = (X + sqrt(D)*Y)*(T_k + sqrt(D)*U_k)**t
_b = (X - sqrt(D)*Y)*(T_k - sqrt(D)*U_k)**t
Xt = S(_a + _b)/2
Yt = S(_a - _b)/(2*sqrt(D))
s = P*Matrix([Xt, Yt]) + Q
result.add(s)
X, Y = X*T + D*U*Y, X*U + Y*T
return result
def is_solution_quad(var, coeff, u, v):
"""
Check whether `(u, v)` is solution to the quadratic binary diophantine
equation with the variable list ``var`` and coefficient dictionary
``coeff``.
Not intended for use by normal users.
"""
reps = dict(zip(var, (u, v)))
eq = Add(*[j*i.xreplace(reps) for i, j in coeff.items()])
return _mexpand(eq) == 0
def diop_DN(D, N, t=symbols("t", integer=True)):
"""
Solves the equation `x^2 - Dy^2 = N`.
Mainly concerned with the case `D > 0, D` is not a perfect square,
which is the same as the generalized Pell equation. The LMM
algorithm [1]_ is used to solve this equation.
Returns one solution tuple, (`x, y)` for each class of the solutions.
Other solutions of the class can be constructed according to the
values of ``D`` and ``N``.
Usage
=====
``diop_DN(D, N, t)``: D and N are integers as in `x^2 - Dy^2 = N` and
``t`` is the parameter to be used in the solutions.
Details
=======
``D`` and ``N`` correspond to D and N in the equation.
``t`` is the parameter to be used in the solutions.
Examples
========
>>> from sympy.solvers.diophantine.diophantine import diop_DN
>>> diop_DN(13, -4) # Solves equation x**2 - 13*y**2 = -4
[(3, 1), (393, 109), (36, 10)]
The output can be interpreted as follows: There are three fundamental
solutions to the equation `x^2 - 13y^2 = -4` given by (3, 1), (393, 109)
and (36, 10). Each tuple is in the form (x, y), i.e. solution (3, 1) means
that `x = 3` and `y = 1`.
>>> diop_DN(986, 1) # Solves equation x**2 - 986*y**2 = 1
[(49299, 1570)]
See Also
========
find_DN(), diop_bf_DN()
References
==========
.. [1] Solving the generalized Pell equation x**2 - D*y**2 = N, John P.
Robertson, July 31, 2004, Pages 16 - 17. [online], Available:
http://www.jpr2718.org/pell.pdf
"""
if D < 0:
if N == 0:
return [(0, 0)]
elif N < 0:
return []
elif N > 0:
sol = []
for d in divisors(square_factor(N)):
sols = cornacchia(1, -D, N // d**2)
if sols:
for x, y in sols:
sol.append((d*x, d*y))
if D == -1:
sol.append((d*y, d*x))
return sol
elif D == 0:
if N < 0:
return []
if N == 0:
return [(0, t)]
sN, _exact = integer_nthroot(N, 2)
if _exact:
return [(sN, t)]
else:
return []
else: # D > 0
sD, _exact = integer_nthroot(D, 2)
if _exact:
if N == 0:
return [(sD*t, t)]
else:
sol = []
for y in range(floor(sign(N)*(N - 1)/(2*sD)) + 1):
try:
sq, _exact = integer_nthroot(D*y**2 + N, 2)
except ValueError:
_exact = False
if _exact:
sol.append((sq, y))
return sol
elif 1 < N**2 < D:
# It is much faster to call `_special_diop_DN`.
return _special_diop_DN(D, N)
else:
if N == 0:
return [(0, 0)]
elif abs(N) == 1:
pqa = PQa(0, 1, D)
j = 0
G = []
B = []
for i in pqa:
a = i[2]
G.append(i[5])
B.append(i[4])
if j != 0 and a == 2*sD:
break
j = j + 1
if _odd(j):
if N == -1:
x = G[j - 1]
y = B[j - 1]
else:
count = j
while count < 2*j - 1:
i = next(pqa)
G.append(i[5])
B.append(i[4])
count += 1
x = G[count]
y = B[count]
else:
if N == 1:
x = G[j - 1]
y = B[j - 1]
else:
return []
return [(x, y)]
else:
fs = []
sol = []
div = divisors(N)
for d in div:
if divisible(N, d**2):
fs.append(d)
for f in fs:
m = N // f**2
zs = sqrt_mod(D, abs(m), all_roots=True)
zs = [i for i in zs if i <= abs(m) // 2 ]
if abs(m) != 2:
zs = zs + [-i for i in zs if i] # omit dupl 0
for z in zs:
pqa = PQa(z, abs(m), D)
j = 0
G = []
B = []
for i in pqa:
G.append(i[5])
B.append(i[4])
if j != 0 and abs(i[1]) == 1:
r = G[j-1]
s = B[j-1]
if r**2 - D*s**2 == m:
sol.append((f*r, f*s))
elif diop_DN(D, -1) != []:
a = diop_DN(D, -1)
sol.append((f*(r*a[0][0] + a[0][1]*s*D), f*(r*a[0][1] + s*a[0][0])))
break
j = j + 1
if j == length(z, abs(m), D):
break
return sol
def _special_diop_DN(D, N):
"""
Solves the equation `x^2 - Dy^2 = N` for the special case where
`1 < N**2 < D` and `D` is not a perfect square.
It is better to call `diop_DN` rather than this function, as
the former checks the condition `1 < N**2 < D`, and calls the latter only
if appropriate.
Usage
=====
WARNING: Internal method. Do not call directly!
``_special_diop_DN(D, N)``: D and N are integers as in `x^2 - Dy^2 = N`.
Details
=======
``D`` and ``N`` correspond to D and N in the equation.
Examples
========
>>> from sympy.solvers.diophantine.diophantine import _special_diop_DN
>>> _special_diop_DN(13, -3) # Solves equation x**2 - 13*y**2 = -3
[(7, 2), (137, 38)]
The output can be interpreted as follows: There are two fundamental
solutions to the equation `x^2 - 13y^2 = -3` given by (7, 2) and
(137, 38). Each tuple is in the form (x, y), i.e. solution (7, 2) means
that `x = 7` and `y = 2`.
>>> _special_diop_DN(2445, -20) # Solves equation x**2 - 2445*y**2 = -20
[(445, 9), (17625560, 356454), (698095554475, 14118073569)]
See Also
========
diop_DN()
References
==========
.. [1] Section 4.4.4 of the following book:
Quadratic Diophantine Equations, T. Andreescu and D. Andrica,
Springer, 2015.
"""
# The following assertion was removed for efficiency, with the understanding
# that this method is not called directly. The parent method, `diop_DN`
# is responsible for performing the appropriate checks.
#
# assert (1 < N**2 < D) and (not integer_nthroot(D, 2)[1])
sqrt_D = sqrt(D)
F = [(N, 1)]
f = 2
while True:
f2 = f**2
if f2 > abs(N):
break
n, r = divmod(N, f2)
if r == 0:
F.append((n, f))
f += 1
P = 0
Q = 1
G0, G1 = 0, 1
B0, B1 = 1, 0
solutions = []
i = 0
while True:
a = floor((P + sqrt_D) / Q)
P = a*Q - P
Q = (D - P**2) // Q
G2 = a*G1 + G0
B2 = a*B1 + B0
for n, f in F:
if G2**2 - D*B2**2 == n:
solutions.append((f*G2, f*B2))
i += 1
if Q == 1 and i % 2 == 0:
break
G0, G1 = G1, G2
B0, B1 = B1, B2
return solutions
def cornacchia(a, b, m):
r"""
Solves `ax^2 + by^2 = m` where `\gcd(a, b) = 1 = gcd(a, m)` and `a, b > 0`.
Uses the algorithm due to Cornacchia. The method only finds primitive
solutions, i.e. ones with `\gcd(x, y) = 1`. So this method can't be used to
find the solutions of `x^2 + y^2 = 20` since the only solution to former is
`(x, y) = (4, 2)` and it is not primitive. When `a = b`, only the
solutions with `x \leq y` are found. For more details, see the References.
Examples
========
>>> from sympy.solvers.diophantine.diophantine import cornacchia
>>> cornacchia(2, 3, 35) # equation 2x**2 + 3y**2 = 35
{(2, 3), (4, 1)}
>>> cornacchia(1, 1, 25) # equation x**2 + y**2 = 25
{(4, 3)}
References
===========
.. [1] A. Nitaj, "L'algorithme de Cornacchia"
.. [2] Solving the diophantine equation ax**2 + by**2 = m by Cornacchia's
method, [online], Available:
http://www.numbertheory.org/php/cornacchia.html
See Also
========
sympy.utilities.iterables.signed_permutations
"""
sols = set()
a1 = igcdex(a, m)[0]
v = sqrt_mod(-b*a1, m, all_roots=True)
if not v:
return None
for t in v:
if t < m // 2:
continue
u, r = t, m
while True:
u, r = r, u % r
if a*r**2 < m:
break
m1 = m - a*r**2
if m1 % b == 0:
m1 = m1 // b
s, _exact = integer_nthroot(m1, 2)
if _exact:
if a == b and r < s:
r, s = s, r
sols.add((int(r), int(s)))
return sols
def PQa(P_0, Q_0, D):
r"""
Returns useful information needed to solve the Pell equation.
There are six sequences of integers defined related to the continued
fraction representation of `\\frac{P + \sqrt{D}}{Q}`, namely {`P_{i}`},
{`Q_{i}`}, {`a_{i}`},{`A_{i}`}, {`B_{i}`}, {`G_{i}`}. ``PQa()`` Returns
these values as a 6-tuple in the same order as mentioned above. Refer [1]_
for more detailed information.
Usage
=====
``PQa(P_0, Q_0, D)``: ``P_0``, ``Q_0`` and ``D`` are integers corresponding
to `P_{0}`, `Q_{0}` and `D` in the continued fraction
`\\frac{P_{0} + \sqrt{D}}{Q_{0}}`.
Also it's assumed that `P_{0}^2 == D mod(|Q_{0}|)` and `D` is square free.
Examples
========
>>> from sympy.solvers.diophantine.diophantine import PQa
>>> pqa = PQa(13, 4, 5) # (13 + sqrt(5))/4
>>> next(pqa) # (P_0, Q_0, a_0, A_0, B_0, G_0)
(13, 4, 3, 3, 1, -1)
>>> next(pqa) # (P_1, Q_1, a_1, A_1, B_1, G_1)
(-1, 1, 1, 4, 1, 3)
References
==========
.. [1] Solving the generalized Pell equation x^2 - Dy^2 = N, John P.
Robertson, July 31, 2004, Pages 4 - 8. http://www.jpr2718.org/pell.pdf
"""
A_i_2 = B_i_1 = 0
A_i_1 = B_i_2 = 1
G_i_2 = -P_0
G_i_1 = Q_0
P_i = P_0
Q_i = Q_0
while True:
a_i = floor((P_i + sqrt(D))/Q_i)
A_i = a_i*A_i_1 + A_i_2
B_i = a_i*B_i_1 + B_i_2
G_i = a_i*G_i_1 + G_i_2
yield P_i, Q_i, a_i, A_i, B_i, G_i
A_i_1, A_i_2 = A_i, A_i_1
B_i_1, B_i_2 = B_i, B_i_1
G_i_1, G_i_2 = G_i, G_i_1
P_i = a_i*Q_i - P_i
Q_i = (D - P_i**2)/Q_i
def diop_bf_DN(D, N, t=symbols("t", integer=True)):
r"""
Uses brute force to solve the equation, `x^2 - Dy^2 = N`.
Mainly concerned with the generalized Pell equation which is the case when
`D > 0, D` is not a perfect square. For more information on the case refer
[1]_. Let `(t, u)` be the minimal positive solution of the equation
`x^2 - Dy^2 = 1`. Then this method requires
`\sqrt{\\frac{\mid N \mid (t \pm 1)}{2D}}` to be small.
Usage
=====
``diop_bf_DN(D, N, t)``: ``D`` and ``N`` are coefficients in
`x^2 - Dy^2 = N` and ``t`` is the parameter to be used in the solutions.
Details
=======
``D`` and ``N`` correspond to D and N in the equation.
``t`` is the parameter to be used in the solutions.
Examples
========
>>> from sympy.solvers.diophantine.diophantine import diop_bf_DN
>>> diop_bf_DN(13, -4)
[(3, 1), (-3, 1), (36, 10)]
>>> diop_bf_DN(986, 1)
[(49299, 1570)]
See Also
========
diop_DN()
References
==========
.. [1] Solving the generalized Pell equation x**2 - D*y**2 = N, John P.
Robertson, July 31, 2004, Page 15. http://www.jpr2718.org/pell.pdf
"""
D = as_int(D)
N = as_int(N)
sol = []
a = diop_DN(D, 1)
u = a[0][0]
if abs(N) == 1:
return diop_DN(D, N)
elif N > 1:
L1 = 0
L2 = integer_nthroot(int(N*(u - 1)/(2*D)), 2)[0] + 1
elif N < -1:
L1, _exact = integer_nthroot(-int(N/D), 2)
if not _exact:
L1 += 1
L2 = integer_nthroot(-int(N*(u + 1)/(2*D)), 2)[0] + 1
else: # N = 0
if D < 0:
return [(0, 0)]
elif D == 0:
return [(0, t)]
else:
sD, _exact = integer_nthroot(D, 2)
if _exact:
return [(sD*t, t), (-sD*t, t)]
else:
return [(0, 0)]
for y in range(L1, L2):
try:
x, _exact = integer_nthroot(N + D*y**2, 2)
except ValueError:
_exact = False
if _exact:
sol.append((x, y))
if not equivalent(x, y, -x, y, D, N):
sol.append((-x, y))
return sol
def equivalent(u, v, r, s, D, N):
"""
Returns True if two solutions `(u, v)` and `(r, s)` of `x^2 - Dy^2 = N`
belongs to the same equivalence class and False otherwise.
Two solutions `(u, v)` and `(r, s)` to the above equation fall to the same
equivalence class iff both `(ur - Dvs)` and `(us - vr)` are divisible by
`N`. See reference [1]_. No test is performed to test whether `(u, v)` and
`(r, s)` are actually solutions to the equation. User should take care of
this.
Usage
=====
``equivalent(u, v, r, s, D, N)``: `(u, v)` and `(r, s)` are two solutions
of the equation `x^2 - Dy^2 = N` and all parameters involved are integers.
Examples
========
>>> from sympy.solvers.diophantine.diophantine import equivalent
>>> equivalent(18, 5, -18, -5, 13, -1)
True
>>> equivalent(3, 1, -18, 393, 109, -4)
False
References
==========
.. [1] Solving the generalized Pell equation x**2 - D*y**2 = N, John P.
Robertson, July 31, 2004, Page 12. http://www.jpr2718.org/pell.pdf
"""
return divisible(u*r - D*v*s, N) and divisible(u*s - v*r, N)
def length(P, Q, D):
r"""
Returns the (length of aperiodic part + length of periodic part) of
continued fraction representation of `\\frac{P + \sqrt{D}}{Q}`.
It is important to remember that this does NOT return the length of the
periodic part but the sum of the lengths of the two parts as mentioned
above.
Usage
=====
``length(P, Q, D)``: ``P``, ``Q`` and ``D`` are integers corresponding to
the continued fraction `\\frac{P + \sqrt{D}}{Q}`.
Details
=======
``P``, ``D`` and ``Q`` corresponds to P, D and Q in the continued fraction,
`\\frac{P + \sqrt{D}}{Q}`.
Examples
========
>>> from sympy.solvers.diophantine.diophantine import length
>>> length(-2 , 4, 5) # (-2 + sqrt(5))/4
3
>>> length(-5, 4, 17) # (-5 + sqrt(17))/4
4
See Also
========
sympy.ntheory.continued_fraction.continued_fraction_periodic
"""
from sympy.ntheory.continued_fraction import continued_fraction_periodic
v = continued_fraction_periodic(P, Q, D)
if type(v[-1]) is list:
rpt = len(v[-1])
nonrpt = len(v) - 1
else:
rpt = 0
nonrpt = len(v)
return rpt + nonrpt
def transformation_to_DN(eq):
"""
This function transforms general quadratic,
`ax^2 + bxy + cy^2 + dx + ey + f = 0`
to more easy to deal with `X^2 - DY^2 = N` form.
This is used to solve the general quadratic equation by transforming it to
the latter form. Refer [1]_ for more detailed information on the
transformation. This function returns a tuple (A, B) where A is a 2 X 2
matrix and B is a 2 X 1 matrix such that,
Transpose([x y]) = A * Transpose([X Y]) + B
Usage
=====
``transformation_to_DN(eq)``: where ``eq`` is the quadratic to be
transformed.
Examples
========
>>> from sympy.abc import x, y
>>> from sympy.solvers.diophantine.diophantine import transformation_to_DN
>>> A, B = transformation_to_DN(x**2 - 3*x*y - y**2 - 2*y + 1)
>>> A
Matrix([
[1/26, 3/26],
[ 0, 1/13]])
>>> B
Matrix([
[-6/13],
[-4/13]])
A, B returned are such that Transpose((x y)) = A * Transpose((X Y)) + B.
Substituting these values for `x` and `y` and a bit of simplifying work
will give an equation of the form `x^2 - Dy^2 = N`.
>>> from sympy.abc import X, Y
>>> from sympy import Matrix, simplify
>>> u = (A*Matrix([X, Y]) + B)[0] # Transformation for x
>>> u
X/26 + 3*Y/26 - 6/13
>>> v = (A*Matrix([X, Y]) + B)[1] # Transformation for y
>>> v
Y/13 - 4/13
Next we will substitute these formulas for `x` and `y` and do
``simplify()``.
>>> eq = simplify((x**2 - 3*x*y - y**2 - 2*y + 1).subs(zip((x, y), (u, v))))
>>> eq
X**2/676 - Y**2/52 + 17/13
By multiplying the denominator appropriately, we can get a Pell equation
in the standard form.
>>> eq * 676
X**2 - 13*Y**2 + 884
If only the final equation is needed, ``find_DN()`` can be used.
See Also
========
find_DN()
References
==========
.. [1] Solving the equation ax^2 + bxy + cy^2 + dx + ey + f = 0,
John P.Robertson, May 8, 2003, Page 7 - 11.
http://www.jpr2718.org/ax2p.pdf
"""
var, coeff, diop_type = classify_diop(eq, _dict=False)
if diop_type == BinaryQuadratic.name:
return _transformation_to_DN(var, coeff)
def _transformation_to_DN(var, coeff):
x, y = var
a = coeff[x**2]
b = coeff[x*y]
c = coeff[y**2]
d = coeff[x]
e = coeff[y]
f = coeff[1]
a, b, c, d, e, f = [as_int(i) for i in _remove_gcd(a, b, c, d, e, f)]
X, Y = symbols("X, Y", integer=True)
if b:
B, C = _rational_pq(2*a, b)
A, T = _rational_pq(a, B**2)
# eq_1 = A*B*X**2 + B*(c*T - A*C**2)*Y**2 + d*T*X + (B*e*T - d*T*C)*Y + f*T*B
coeff = {X**2: A*B, X*Y: 0, Y**2: B*(c*T - A*C**2), X: d*T, Y: B*e*T - d*T*C, 1: f*T*B}
A_0, B_0 = _transformation_to_DN([X, Y], coeff)
return Matrix(2, 2, [S.One/B, -S(C)/B, 0, 1])*A_0, Matrix(2, 2, [S.One/B, -S(C)/B, 0, 1])*B_0
else:
if d:
B, C = _rational_pq(2*a, d)
A, T = _rational_pq(a, B**2)
# eq_2 = A*X**2 + c*T*Y**2 + e*T*Y + f*T - A*C**2
coeff = {X**2: A, X*Y: 0, Y**2: c*T, X: 0, Y: e*T, 1: f*T - A*C**2}
A_0, B_0 = _transformation_to_DN([X, Y], coeff)
return Matrix(2, 2, [S.One/B, 0, 0, 1])*A_0, Matrix(2, 2, [S.One/B, 0, 0, 1])*B_0 + Matrix([-S(C)/B, 0])
else:
if e:
B, C = _rational_pq(2*c, e)
A, T = _rational_pq(c, B**2)
# eq_3 = a*T*X**2 + A*Y**2 + f*T - A*C**2
coeff = {X**2: a*T, X*Y: 0, Y**2: A, X: 0, Y: 0, 1: f*T - A*C**2}
A_0, B_0 = _transformation_to_DN([X, Y], coeff)
return Matrix(2, 2, [1, 0, 0, S.One/B])*A_0, Matrix(2, 2, [1, 0, 0, S.One/B])*B_0 + Matrix([0, -S(C)/B])
else:
# TODO: pre-simplification: Not necessary but may simplify
# the equation.
return Matrix(2, 2, [S.One/a, 0, 0, 1]), Matrix([0, 0])
def find_DN(eq):
"""
This function returns a tuple, `(D, N)` of the simplified form,
`x^2 - Dy^2 = N`, corresponding to the general quadratic,
`ax^2 + bxy + cy^2 + dx + ey + f = 0`.
Solving the general quadratic is then equivalent to solving the equation
`X^2 - DY^2 = N` and transforming the solutions by using the transformation
matrices returned by ``transformation_to_DN()``.
Usage
=====
``find_DN(eq)``: where ``eq`` is the quadratic to be transformed.
Examples
========
>>> from sympy.abc import x, y
>>> from sympy.solvers.diophantine.diophantine import find_DN
>>> find_DN(x**2 - 3*x*y - y**2 - 2*y + 1)
(13, -884)
Interpretation of the output is that we get `X^2 -13Y^2 = -884` after
transforming `x^2 - 3xy - y^2 - 2y + 1` using the transformation returned
by ``transformation_to_DN()``.
See Also
========
transformation_to_DN()
References
==========
.. [1] Solving the equation ax^2 + bxy + cy^2 + dx + ey + f = 0,
John P.Robertson, May 8, 2003, Page 7 - 11.
http://www.jpr2718.org/ax2p.pdf
"""
var, coeff, diop_type = classify_diop(eq, _dict=False)
if diop_type == BinaryQuadratic.name:
return _find_DN(var, coeff)
def _find_DN(var, coeff):
x, y = var
X, Y = symbols("X, Y", integer=True)
A, B = _transformation_to_DN(var, coeff)
u = (A*Matrix([X, Y]) + B)[0]
v = (A*Matrix([X, Y]) + B)[1]
eq = x**2*coeff[x**2] + x*y*coeff[x*y] + y**2*coeff[y**2] + x*coeff[x] + y*coeff[y] + coeff[1]
simplified = _mexpand(eq.subs(zip((x, y), (u, v))))
coeff = simplified.as_coefficients_dict()
return -coeff[Y**2]/coeff[X**2], -coeff[1]/coeff[X**2]
def check_param(x, y, a, t):
"""
If there is a number modulo ``a`` such that ``x`` and ``y`` are both
integers, then return a parametric representation for ``x`` and ``y``
else return (None, None).
Here ``x`` and ``y`` are functions of ``t``.
"""
from sympy.simplify.simplify import clear_coefficients
if x.is_number and not x.is_Integer:
return (None, None)
if y.is_number and not y.is_Integer:
return (None, None)
m, n = symbols("m, n", integer=True)
c, p = (m*x + n*y).as_content_primitive()
if a % c.q:
return (None, None)
# clear_coefficients(mx + b, R)[1] -> (R - b)/m
eq = clear_coefficients(x, m)[1] - clear_coefficients(y, n)[1]
junk, eq = eq.as_content_primitive()
return diop_solve(eq, t)
def diop_ternary_quadratic(eq, parameterize=False):
"""
Solves the general quadratic ternary form,
`ax^2 + by^2 + cz^2 + fxy + gyz + hxz = 0`.
Returns a tuple `(x, y, z)` which is a base solution for the above
equation. If there are no solutions, `(None, None, None)` is returned.
Usage
=====
``diop_ternary_quadratic(eq)``: Return a tuple containing a basic solution
to ``eq``.
Details
=======
``eq`` should be an homogeneous expression of degree two in three variables
and it is assumed to be zero.
Examples
========
>>> from sympy.abc import x, y, z
>>> from sympy.solvers.diophantine.diophantine import diop_ternary_quadratic
>>> diop_ternary_quadratic(x**2 + 3*y**2 - z**2)
(1, 0, 1)
>>> diop_ternary_quadratic(4*x**2 + 5*y**2 - z**2)
(1, 0, 2)
>>> diop_ternary_quadratic(45*x**2 - 7*y**2 - 8*x*y - z**2)
(28, 45, 105)
>>> diop_ternary_quadratic(x**2 - 49*y**2 - z**2 + 13*z*y -8*x*y)
(9, 1, 5)
"""
var, coeff, diop_type = classify_diop(eq, _dict=False)
if diop_type in (
"homogeneous_ternary_quadratic",
"homogeneous_ternary_quadratic_normal"):
sol = _diop_ternary_quadratic(var, coeff)
if len(sol) > 0:
x_0, y_0, z_0 = list(sol)[0]
else:
x_0, y_0, z_0 = None, None, None
if parameterize:
return _parametrize_ternary_quadratic(
(x_0, y_0, z_0), var, coeff)
return x_0, y_0, z_0
def _diop_ternary_quadratic(_var, coeff):
x, y, z = _var
var = [x, y, z]
# Equations of the form B*x*y + C*z*x + E*y*z = 0 and At least two of the
# coefficients A, B, C are non-zero.
# There are infinitely many solutions for the equation.
# Ex: (0, 0, t), (0, t, 0), (t, 0, 0)
# Equation can be re-written as y*(B*x + E*z) = -C*x*z and we can find rather
# unobvious solutions. Set y = -C and B*x + E*z = x*z. The latter can be solved by
# using methods for binary quadratic diophantine equations. Let's select the
# solution which minimizes |x| + |z|
result = DiophantineSolutionSet(var)
def unpack_sol(sol):
if len(sol) > 0:
return list(sol)[0]
return None, None, None
if not any(coeff[i**2] for i in var):
if coeff[x*z]:
sols = diophantine(coeff[x*y]*x + coeff[y*z]*z - x*z)
s = sols.pop()
min_sum = abs(s[0]) + abs(s[1])
for r in sols:
m = abs(r[0]) + abs(r[1])
if m < min_sum:
s = r
min_sum = m
result.add(_remove_gcd(s[0], -coeff[x*z], s[1]))
return result
else:
var[0], var[1] = _var[1], _var[0]
y_0, x_0, z_0 = unpack_sol(_diop_ternary_quadratic(var, coeff))
if x_0 is not None:
result.add((x_0, y_0, z_0))
return result
if coeff[x**2] == 0:
# If the coefficient of x is zero change the variables
if coeff[y**2] == 0:
var[0], var[2] = _var[2], _var[0]
z_0, y_0, x_0 = unpack_sol(_diop_ternary_quadratic(var, coeff))
else:
var[0], var[1] = _var[1], _var[0]
y_0, x_0, z_0 = unpack_sol(_diop_ternary_quadratic(var, coeff))
else:
if coeff[x*y] or coeff[x*z]:
# Apply the transformation x --> X - (B*y + C*z)/(2*A)
A = coeff[x**2]
B = coeff[x*y]
C = coeff[x*z]
D = coeff[y**2]
E = coeff[y*z]
F = coeff[z**2]
_coeff = dict()
_coeff[x**2] = 4*A**2
_coeff[y**2] = 4*A*D - B**2
_coeff[z**2] = 4*A*F - C**2
_coeff[y*z] = 4*A*E - 2*B*C
_coeff[x*y] = 0
_coeff[x*z] = 0
x_0, y_0, z_0 = unpack_sol(_diop_ternary_quadratic(var, _coeff))
if x_0 is None:
return result
p, q = _rational_pq(B*y_0 + C*z_0, 2*A)
x_0, y_0, z_0 = x_0*q - p, y_0*q, z_0*q
elif coeff[z*y] != 0:
if coeff[y**2] == 0:
if coeff[z**2] == 0:
# Equations of the form A*x**2 + E*yz = 0.
A = coeff[x**2]
E = coeff[y*z]
b, a = _rational_pq(-E, A)
x_0, y_0, z_0 = b, a, b
else:
# Ax**2 + E*y*z + F*z**2 = 0
var[0], var[2] = _var[2], _var[0]
z_0, y_0, x_0 = unpack_sol(_diop_ternary_quadratic(var, coeff))
else:
# A*x**2 + D*y**2 + E*y*z + F*z**2 = 0, C may be zero
var[0], var[1] = _var[1], _var[0]
y_0, x_0, z_0 = unpack_sol(_diop_ternary_quadratic(var, coeff))
else:
# Ax**2 + D*y**2 + F*z**2 = 0, C may be zero
x_0, y_0, z_0 = unpack_sol(_diop_ternary_quadratic_normal(var, coeff))
if x_0 is None:
return result
result.add(_remove_gcd(x_0, y_0, z_0))
return result
def transformation_to_normal(eq):
"""
Returns the transformation Matrix that converts a general ternary
quadratic equation ``eq`` (`ax^2 + by^2 + cz^2 + dxy + eyz + fxz`)
to a form without cross terms: `ax^2 + by^2 + cz^2 = 0`. This is
not used in solving ternary quadratics; it is only implemented for
the sake of completeness.
"""
var, coeff, diop_type = classify_diop(eq, _dict=False)
if diop_type in (
"homogeneous_ternary_quadratic",
"homogeneous_ternary_quadratic_normal"):
return _transformation_to_normal(var, coeff)
def _transformation_to_normal(var, coeff):
_var = list(var) # copy
x, y, z = var
if not any(coeff[i**2] for i in var):
# https://math.stackexchange.com/questions/448051/transform-quadratic-ternary-form-to-normal-form/448065#448065
a = coeff[x*y]
b = coeff[y*z]
c = coeff[x*z]
swap = False
if not a: # b can't be 0 or else there aren't 3 vars
swap = True
a, b = b, a
T = Matrix(((1, 1, -b/a), (1, -1, -c/a), (0, 0, 1)))
if swap:
T.row_swap(0, 1)
T.col_swap(0, 1)
return T
if coeff[x**2] == 0:
# If the coefficient of x is zero change the variables
if coeff[y**2] == 0:
_var[0], _var[2] = var[2], var[0]
T = _transformation_to_normal(_var, coeff)
T.row_swap(0, 2)
T.col_swap(0, 2)
return T
else:
_var[0], _var[1] = var[1], var[0]
T = _transformation_to_normal(_var, coeff)
T.row_swap(0, 1)
T.col_swap(0, 1)
return T
# Apply the transformation x --> X - (B*Y + C*Z)/(2*A)
if coeff[x*y] != 0 or coeff[x*z] != 0:
A = coeff[x**2]
B = coeff[x*y]
C = coeff[x*z]
D = coeff[y**2]
E = coeff[y*z]
F = coeff[z**2]
_coeff = dict()
_coeff[x**2] = 4*A**2
_coeff[y**2] = 4*A*D - B**2
_coeff[z**2] = 4*A*F - C**2
_coeff[y*z] = 4*A*E - 2*B*C
_coeff[x*y] = 0
_coeff[x*z] = 0
T_0 = _transformation_to_normal(_var, _coeff)
return Matrix(3, 3, [1, S(-B)/(2*A), S(-C)/(2*A), 0, 1, 0, 0, 0, 1])*T_0
elif coeff[y*z] != 0:
if coeff[y**2] == 0:
if coeff[z**2] == 0:
# Equations of the form A*x**2 + E*yz = 0.
# Apply transformation y -> Y + Z ans z -> Y - Z
return Matrix(3, 3, [1, 0, 0, 0, 1, 1, 0, 1, -1])
else:
# Ax**2 + E*y*z + F*z**2 = 0
_var[0], _var[2] = var[2], var[0]
T = _transformation_to_normal(_var, coeff)
T.row_swap(0, 2)
T.col_swap(0, 2)
return T
else:
# A*x**2 + D*y**2 + E*y*z + F*z**2 = 0, F may be zero
_var[0], _var[1] = var[1], var[0]
T = _transformation_to_normal(_var, coeff)
T.row_swap(0, 1)
T.col_swap(0, 1)
return T
else:
return Matrix.eye(3)
def parametrize_ternary_quadratic(eq):
"""
Returns the parametrized general solution for the ternary quadratic
equation ``eq`` which has the form
`ax^2 + by^2 + cz^2 + fxy + gyz + hxz = 0`.
Examples
========
>>> from sympy import Tuple, ordered
>>> from sympy.abc import x, y, z
>>> from sympy.solvers.diophantine.diophantine import parametrize_ternary_quadratic
The parametrized solution may be returned with three parameters:
>>> parametrize_ternary_quadratic(2*x**2 + y**2 - 2*z**2)
(p**2 - 2*q**2, -2*p**2 + 4*p*q - 4*p*r - 4*q**2, p**2 - 4*p*q + 2*q**2 - 4*q*r)
There might also be only two parameters:
>>> parametrize_ternary_quadratic(4*x**2 + 2*y**2 - 3*z**2)
(2*p**2 - 3*q**2, -4*p**2 + 12*p*q - 6*q**2, 4*p**2 - 8*p*q + 6*q**2)
Notes
=====
Consider ``p`` and ``q`` in the previous 2-parameter
solution and observe that more than one solution can be represented
by a given pair of parameters. If `p` and ``q`` are not coprime, this is
trivially true since the common factor will also be a common factor of the
solution values. But it may also be true even when ``p`` and
``q`` are coprime:
>>> sol = Tuple(*_)
>>> p, q = ordered(sol.free_symbols)
>>> sol.subs([(p, 3), (q, 2)])
(6, 12, 12)
>>> sol.subs([(q, 1), (p, 1)])
(-1, 2, 2)
>>> sol.subs([(q, 0), (p, 1)])
(2, -4, 4)
>>> sol.subs([(q, 1), (p, 0)])
(-3, -6, 6)
Except for sign and a common factor, these are equivalent to
the solution of (1, 2, 2).
References
==========
.. [1] The algorithmic resolution of Diophantine equations, Nigel P. Smart,
London Mathematical Society Student Texts 41, Cambridge University
Press, Cambridge, 1998.
"""
var, coeff, diop_type = classify_diop(eq, _dict=False)
if diop_type in (
"homogeneous_ternary_quadratic",
"homogeneous_ternary_quadratic_normal"):
x_0, y_0, z_0 = list(_diop_ternary_quadratic(var, coeff))[0]
return _parametrize_ternary_quadratic(
(x_0, y_0, z_0), var, coeff)
def _parametrize_ternary_quadratic(solution, _var, coeff):
# called for a*x**2 + b*y**2 + c*z**2 + d*x*y + e*y*z + f*x*z = 0
assert 1 not in coeff
x_0, y_0, z_0 = solution
v = list(_var) # copy
if x_0 is None:
return (None, None, None)
if solution.count(0) >= 2:
# if there are 2 zeros the equation reduces
# to k*X**2 == 0 where X is x, y, or z so X must
# be zero, too. So there is only the trivial
# solution.
return (None, None, None)
if x_0 == 0:
v[0], v[1] = v[1], v[0]
y_p, x_p, z_p = _parametrize_ternary_quadratic(
(y_0, x_0, z_0), v, coeff)
return x_p, y_p, z_p
x, y, z = v
r, p, q = symbols("r, p, q", integer=True)
eq = sum(k*v for k, v in coeff.items())
eq_1 = _mexpand(eq.subs(zip(
(x, y, z), (r*x_0, r*y_0 + p, r*z_0 + q))))
A, B = eq_1.as_independent(r, as_Add=True)
x = A*x_0
y = (A*y_0 - _mexpand(B/r*p))
z = (A*z_0 - _mexpand(B/r*q))
return _remove_gcd(x, y, z)
def diop_ternary_quadratic_normal(eq, parameterize=False):
"""
Solves the quadratic ternary diophantine equation,
`ax^2 + by^2 + cz^2 = 0`.
Here the coefficients `a`, `b`, and `c` should be non zero. Otherwise the
equation will be a quadratic binary or univariate equation. If solvable,
returns a tuple `(x, y, z)` that satisfies the given equation. If the
equation does not have integer solutions, `(None, None, None)` is returned.
Usage
=====
``diop_ternary_quadratic_normal(eq)``: where ``eq`` is an equation of the form
`ax^2 + by^2 + cz^2 = 0`.
Examples
========
>>> from sympy.abc import x, y, z
>>> from sympy.solvers.diophantine.diophantine import diop_ternary_quadratic_normal
>>> diop_ternary_quadratic_normal(x**2 + 3*y**2 - z**2)
(1, 0, 1)
>>> diop_ternary_quadratic_normal(4*x**2 + 5*y**2 - z**2)
(1, 0, 2)
>>> diop_ternary_quadratic_normal(34*x**2 - 3*y**2 - 301*z**2)
(4, 9, 1)
"""
var, coeff, diop_type = classify_diop(eq, _dict=False)
if diop_type == HomogeneousTernaryQuadraticNormal.name:
sol = _diop_ternary_quadratic_normal(var, coeff)
if len(sol) > 0:
x_0, y_0, z_0 = list(sol)[0]
else:
x_0, y_0, z_0 = None, None, None
if parameterize:
return _parametrize_ternary_quadratic(
(x_0, y_0, z_0), var, coeff)
return x_0, y_0, z_0
def _diop_ternary_quadratic_normal(var, coeff):
x, y, z = var
a = coeff[x**2]
b = coeff[y**2]
c = coeff[z**2]
try:
assert len([k for k in coeff if coeff[k]]) == 3
assert all(coeff[i**2] for i in var)
except AssertionError:
raise ValueError(filldedent('''
coeff dict is not consistent with assumption of this routine:
coefficients should be those of an expression in the form
a*x**2 + b*y**2 + c*z**2 where a*b*c != 0.'''))
(sqf_of_a, sqf_of_b, sqf_of_c), (a_1, b_1, c_1), (a_2, b_2, c_2) = \
sqf_normal(a, b, c, steps=True)
A = -a_2*c_2
B = -b_2*c_2
result = DiophantineSolutionSet(var)
# If following two conditions are satisfied then there are no solutions
if A < 0 and B < 0:
return result
if (
sqrt_mod(-b_2*c_2, a_2) is None or
sqrt_mod(-c_2*a_2, b_2) is None or
sqrt_mod(-a_2*b_2, c_2) is None):
return result
z_0, x_0, y_0 = descent(A, B)
z_0, q = _rational_pq(z_0, abs(c_2))
x_0 *= q
y_0 *= q
x_0, y_0, z_0 = _remove_gcd(x_0, y_0, z_0)
# Holzer reduction
if sign(a) == sign(b):
x_0, y_0, z_0 = holzer(x_0, y_0, z_0, abs(a_2), abs(b_2), abs(c_2))
elif sign(a) == sign(c):
x_0, z_0, y_0 = holzer(x_0, z_0, y_0, abs(a_2), abs(c_2), abs(b_2))
else:
y_0, z_0, x_0 = holzer(y_0, z_0, x_0, abs(b_2), abs(c_2), abs(a_2))
x_0 = reconstruct(b_1, c_1, x_0)
y_0 = reconstruct(a_1, c_1, y_0)
z_0 = reconstruct(a_1, b_1, z_0)
sq_lcm = ilcm(sqf_of_a, sqf_of_b, sqf_of_c)
x_0 = abs(x_0*sq_lcm//sqf_of_a)
y_0 = abs(y_0*sq_lcm//sqf_of_b)
z_0 = abs(z_0*sq_lcm//sqf_of_c)
result.add(_remove_gcd(x_0, y_0, z_0))
return result
def sqf_normal(a, b, c, steps=False):
"""
Return `a', b', c'`, the coefficients of the square-free normal
form of `ax^2 + by^2 + cz^2 = 0`, where `a', b', c'` are pairwise
prime. If `steps` is True then also return three tuples:
`sq`, `sqf`, and `(a', b', c')` where `sq` contains the square
factors of `a`, `b` and `c` after removing the `gcd(a, b, c)`;
`sqf` contains the values of `a`, `b` and `c` after removing
both the `gcd(a, b, c)` and the square factors.
The solutions for `ax^2 + by^2 + cz^2 = 0` can be
recovered from the solutions of `a'x^2 + b'y^2 + c'z^2 = 0`.
Examples
========
>>> from sympy.solvers.diophantine.diophantine import sqf_normal
>>> sqf_normal(2 * 3**2 * 5, 2 * 5 * 11, 2 * 7**2 * 11)
(11, 1, 5)
>>> sqf_normal(2 * 3**2 * 5, 2 * 5 * 11, 2 * 7**2 * 11, True)
((3, 1, 7), (5, 55, 11), (11, 1, 5))
References
==========
.. [1] Legendre's Theorem, Legrange's Descent,
http://public.csusm.edu/aitken_html/notes/legendre.pdf
See Also
========
reconstruct()
"""
ABC = _remove_gcd(a, b, c)
sq = tuple(square_factor(i) for i in ABC)
sqf = A, B, C = tuple([i//j**2 for i,j in zip(ABC, sq)])
pc = igcd(A, B)
A /= pc
B /= pc
pa = igcd(B, C)
B /= pa
C /= pa
pb = igcd(A, C)
A /= pb
B /= pb
A *= pa
B *= pb
C *= pc
if steps:
return (sq, sqf, (A, B, C))
else:
return A, B, C
def square_factor(a):
r"""
Returns an integer `c` s.t. `a = c^2k, \ c,k \in Z`. Here `k` is square
free. `a` can be given as an integer or a dictionary of factors.
Examples
========
>>> from sympy.solvers.diophantine.diophantine import square_factor
>>> square_factor(24)
2
>>> square_factor(-36*3)
6
>>> square_factor(1)
1
>>> square_factor({3: 2, 2: 1, -1: 1}) # -18
3
See Also
========
sympy.ntheory.factor_.core
"""
f = a if isinstance(a, dict) else factorint(a)
return Mul(*[p**(e//2) for p, e in f.items()])
def reconstruct(A, B, z):
"""
Reconstruct the `z` value of an equivalent solution of `ax^2 + by^2 + cz^2`
from the `z` value of a solution of the square-free normal form of the
equation, `a'*x^2 + b'*y^2 + c'*z^2`, where `a'`, `b'` and `c'` are square
free and `gcd(a', b', c') == 1`.
"""
f = factorint(igcd(A, B))
for p, e in f.items():
if e != 1:
raise ValueError('a and b should be square-free')
z *= p
return z
def ldescent(A, B):
"""
Return a non-trivial solution to `w^2 = Ax^2 + By^2` using
Lagrange's method; return None if there is no such solution.
.
Here, `A \\neq 0` and `B \\neq 0` and `A` and `B` are square free. Output a
tuple `(w_0, x_0, y_0)` which is a solution to the above equation.
Examples
========
>>> from sympy.solvers.diophantine.diophantine import ldescent
>>> ldescent(1, 1) # w^2 = x^2 + y^2
(1, 1, 0)
>>> ldescent(4, -7) # w^2 = 4x^2 - 7y^2
(2, -1, 0)
This means that `x = -1, y = 0` and `w = 2` is a solution to the equation
`w^2 = 4x^2 - 7y^2`
>>> ldescent(5, -1) # w^2 = 5x^2 - y^2
(2, 1, -1)
References
==========
.. [1] The algorithmic resolution of Diophantine equations, Nigel P. Smart,
London Mathematical Society Student Texts 41, Cambridge University
Press, Cambridge, 1998.
.. [2] Efficient Solution of Rational Conices, J. E. Cremona and D. Rusin,
[online], Available:
http://eprints.nottingham.ac.uk/60/1/kvxefz87.pdf
"""
if abs(A) > abs(B):
w, y, x = ldescent(B, A)
return w, x, y
if A == 1:
return (1, 1, 0)
if B == 1:
return (1, 0, 1)
if B == -1: # and A == -1
return
r = sqrt_mod(A, B)
Q = (r**2 - A) // B
if Q == 0:
B_0 = 1
d = 0
else:
div = divisors(Q)
B_0 = None
for i in div:
sQ, _exact = integer_nthroot(abs(Q) // i, 2)
if _exact:
B_0, d = sign(Q)*i, sQ
break
if B_0 is not None:
W, X, Y = ldescent(A, B_0)
return _remove_gcd((-A*X + r*W), (r*X - W), Y*(B_0*d))
def descent(A, B):
"""
Returns a non-trivial solution, (x, y, z), to `x^2 = Ay^2 + Bz^2`
using Lagrange's descent method with lattice-reduction. `A` and `B`
are assumed to be valid for such a solution to exist.
This is faster than the normal Lagrange's descent algorithm because
the Gaussian reduction is used.
Examples
========
>>> from sympy.solvers.diophantine.diophantine import descent
>>> descent(3, 1) # x**2 = 3*y**2 + z**2
(1, 0, 1)
`(x, y, z) = (1, 0, 1)` is a solution to the above equation.
>>> descent(41, -113)
(-16, -3, 1)
References
==========
.. [1] Efficient Solution of Rational Conices, J. E. Cremona and D. Rusin,
Mathematics of Computation, Volume 00, Number 0.
"""
if abs(A) > abs(B):
x, y, z = descent(B, A)
return x, z, y
if B == 1:
return (1, 0, 1)
if A == 1:
return (1, 1, 0)
if B == -A:
return (0, 1, 1)
if B == A:
x, z, y = descent(-1, A)
return (A*y, z, x)
w = sqrt_mod(A, B)
x_0, z_0 = gaussian_reduce(w, A, B)
t = (x_0**2 - A*z_0**2) // B
t_2 = square_factor(t)
t_1 = t // t_2**2
x_1, z_1, y_1 = descent(A, t_1)
return _remove_gcd(x_0*x_1 + A*z_0*z_1, z_0*x_1 + x_0*z_1, t_1*t_2*y_1)
def gaussian_reduce(w, a, b):
r"""
Returns a reduced solution `(x, z)` to the congruence
`X^2 - aZ^2 \equiv 0 \ (mod \ b)` so that `x^2 + |a|z^2` is minimal.
Details
=======
Here ``w`` is a solution of the congruence `x^2 \equiv a \ (mod \ b)`
References
==========
.. [1] Gaussian lattice Reduction [online]. Available:
http://home.ie.cuhk.edu.hk/~wkshum/wordpress/?p=404
.. [2] Efficient Solution of Rational Conices, J. E. Cremona and D. Rusin,
Mathematics of Computation, Volume 00, Number 0.
"""
u = (0, 1)
v = (1, 0)
if dot(u, v, w, a, b) < 0:
v = (-v[0], -v[1])
if norm(u, w, a, b) < norm(v, w, a, b):
u, v = v, u
while norm(u, w, a, b) > norm(v, w, a, b):
k = dot(u, v, w, a, b) // dot(v, v, w, a, b)
u, v = v, (u[0]- k*v[0], u[1]- k*v[1])
u, v = v, u
if dot(u, v, w, a, b) < dot(v, v, w, a, b)/2 or norm((u[0]-v[0], u[1]-v[1]), w, a, b) > norm(v, w, a, b):
c = v
else:
c = (u[0] - v[0], u[1] - v[1])
return c[0]*w + b*c[1], c[0]
def dot(u, v, w, a, b):
r"""
Returns a special dot product of the vectors `u = (u_{1}, u_{2})` and
`v = (v_{1}, v_{2})` which is defined in order to reduce solution of
the congruence equation `X^2 - aZ^2 \equiv 0 \ (mod \ b)`.
"""
u_1, u_2 = u
v_1, v_2 = v
return (w*u_1 + b*u_2)*(w*v_1 + b*v_2) + abs(a)*u_1*v_1
def norm(u, w, a, b):
r"""
Returns the norm of the vector `u = (u_{1}, u_{2})` under the dot product
defined by `u \cdot v = (wu_{1} + bu_{2})(w*v_{1} + bv_{2}) + |a|*u_{1}*v_{1}`
where `u = (u_{1}, u_{2})` and `v = (v_{1}, v_{2})`.
"""
u_1, u_2 = u
return sqrt(dot((u_1, u_2), (u_1, u_2), w, a, b))
def holzer(x, y, z, a, b, c):
r"""
Simplify the solution `(x, y, z)` of the equation
`ax^2 + by^2 = cz^2` with `a, b, c > 0` and `z^2 \geq \mid ab \mid` to
a new reduced solution `(x', y', z')` such that `z'^2 \leq \mid ab \mid`.
The algorithm is an interpretation of Mordell's reduction as described
on page 8 of Cremona and Rusin's paper [1]_ and the work of Mordell in
reference [2]_.
References
==========
.. [1] Efficient Solution of Rational Conices, J. E. Cremona and D. Rusin,
Mathematics of Computation, Volume 00, Number 0.
.. [2] Diophantine Equations, L. J. Mordell, page 48.
"""
if _odd(c):
k = 2*c
else:
k = c//2
small = a*b*c
step = 0
while True:
t1, t2, t3 = a*x**2, b*y**2, c*z**2
# check that it's a solution
if t1 + t2 != t3:
if step == 0:
raise ValueError('bad starting solution')
break
x_0, y_0, z_0 = x, y, z
if max(t1, t2, t3) <= small:
# Holzer condition
break
uv = u, v = base_solution_linear(k, y_0, -x_0)
if None in uv:
break
p, q = -(a*u*x_0 + b*v*y_0), c*z_0
r = Rational(p, q)
if _even(c):
w = _nint_or_floor(p, q)
assert abs(w - r) <= S.Half
else:
w = p//q # floor
if _odd(a*u + b*v + c*w):
w += 1
assert abs(w - r) <= S.One
A = (a*u**2 + b*v**2 + c*w**2)
B = (a*u*x_0 + b*v*y_0 + c*w*z_0)
x = Rational(x_0*A - 2*u*B, k)
y = Rational(y_0*A - 2*v*B, k)
z = Rational(z_0*A - 2*w*B, k)
assert all(i.is_Integer for i in (x, y, z))
step += 1
return tuple([int(i) for i in (x_0, y_0, z_0)])
def diop_general_pythagorean(eq, param=symbols("m", integer=True)):
"""
Solves the general pythagorean equation,
`a_{1}^2x_{1}^2 + a_{2}^2x_{2}^2 + . . . + a_{n}^2x_{n}^2 - a_{n + 1}^2x_{n + 1}^2 = 0`.
Returns a tuple which contains a parametrized solution to the equation,
sorted in the same order as the input variables.
Usage
=====
``diop_general_pythagorean(eq, param)``: where ``eq`` is a general
pythagorean equation which is assumed to be zero and ``param`` is the base
parameter used to construct other parameters by subscripting.
Examples
========
>>> from sympy.solvers.diophantine.diophantine import diop_general_pythagorean
>>> from sympy.abc import a, b, c, d, e
>>> diop_general_pythagorean(a**2 + b**2 + c**2 - d**2)
(m1**2 + m2**2 - m3**2, 2*m1*m3, 2*m2*m3, m1**2 + m2**2 + m3**2)
>>> diop_general_pythagorean(9*a**2 - 4*b**2 + 16*c**2 + 25*d**2 + e**2)
(10*m1**2 + 10*m2**2 + 10*m3**2 - 10*m4**2, 15*m1**2 + 15*m2**2 + 15*m3**2 + 15*m4**2, 15*m1*m4, 12*m2*m4, 60*m3*m4)
"""
var, coeff, diop_type = classify_diop(eq, _dict=False)
if diop_type == GeneralPythagorean.name:
return list(_diop_general_pythagorean(var, coeff, param))[0]
def _diop_general_pythagorean(var, coeff, t):
if sign(coeff[var[0]**2]) + sign(coeff[var[1]**2]) + sign(coeff[var[2]**2]) < 0:
for key in coeff.keys():
coeff[key] = -coeff[key]
n = len(var)
index = 0
for i, v in enumerate(var):
if sign(coeff[v**2]) == -1:
index = i
m = symbols('%s1:%i' % (t, n), integer=True)
ith = sum(m_i**2 for m_i in m)
L = [ith - 2*m[n - 2]**2]
L.extend([2*m[i]*m[n-2] for i in range(n - 2)])
sol = L[:index] + [ith] + L[index:]
lcm = 1
for i, v in enumerate(var):
if i == index or (index > 0 and i == 0) or (index == 0 and i == 1):
lcm = ilcm(lcm, sqrt(abs(coeff[v**2])))
else:
s = sqrt(coeff[v**2])
lcm = ilcm(lcm, s if _odd(s) else s//2)
for i, v in enumerate(var):
sol[i] = (lcm*sol[i]) / sqrt(abs(coeff[v**2]))
result = DiophantineSolutionSet(var)
result.add(sol)
return result
def diop_general_sum_of_squares(eq, limit=1):
r"""
Solves the equation `x_{1}^2 + x_{2}^2 + . . . + x_{n}^2 - k = 0`.
Returns at most ``limit`` number of solutions.
Usage
=====
``general_sum_of_squares(eq, limit)`` : Here ``eq`` is an expression which
is assumed to be zero. Also, ``eq`` should be in the form,
`x_{1}^2 + x_{2}^2 + . . . + x_{n}^2 - k = 0`.
Details
=======
When `n = 3` if `k = 4^a(8m + 7)` for some `a, m \in Z` then there will be
no solutions. Refer [1]_ for more details.
Examples
========
>>> from sympy.solvers.diophantine.diophantine import diop_general_sum_of_squares
>>> from sympy.abc import a, b, c, d, e
>>> diop_general_sum_of_squares(a**2 + b**2 + c**2 + d**2 + e**2 - 2345)
{(15, 22, 22, 24, 24)}
Reference
=========
.. [1] Representing an integer as a sum of three squares, [online],
Available:
http://www.proofwiki.org/wiki/Integer_as_Sum_of_Three_Squares
"""
var, coeff, diop_type = classify_diop(eq, _dict=False)
if diop_type == GeneralSumOfSquares.name:
return set(_diop_general_sum_of_squares(var, -int(coeff[1]), limit))
def _diop_general_sum_of_squares(var, k, limit=1):
# solves Eq(sum(i**2 for i in var), k)
n = len(var)
if n < 3:
raise ValueError('n must be greater than 2')
result = DiophantineSolutionSet(var)
if k < 0 or limit < 1:
return result
sign = [-1 if x.is_nonpositive else 1 for x in var]
negs = sign.count(-1) != 0
took = 0
for t in sum_of_squares(k, n, zeros=True):
if negs:
result.add([sign[i]*j for i, j in enumerate(t)])
else:
result.add(t)
took += 1
if took == limit:
break
return result
def diop_general_sum_of_even_powers(eq, limit=1):
"""
Solves the equation `x_{1}^e + x_{2}^e + . . . + x_{n}^e - k = 0`
where `e` is an even, integer power.
Returns at most ``limit`` number of solutions.
Usage
=====
``general_sum_of_even_powers(eq, limit)`` : Here ``eq`` is an expression which
is assumed to be zero. Also, ``eq`` should be in the form,
`x_{1}^e + x_{2}^e + . . . + x_{n}^e - k = 0`.
Examples
========
>>> from sympy.solvers.diophantine.diophantine import diop_general_sum_of_even_powers
>>> from sympy.abc import a, b
>>> diop_general_sum_of_even_powers(a**4 + b**4 - (2**4 + 3**4))
{(2, 3)}
See Also
========
power_representation
"""
var, coeff, diop_type = classify_diop(eq, _dict=False)
if diop_type == GeneralSumOfEvenPowers.name:
for k in coeff.keys():
if k.is_Pow and coeff[k]:
p = k.exp
return set(_diop_general_sum_of_even_powers(var, p, -coeff[1], limit))
def _diop_general_sum_of_even_powers(var, p, n, limit=1):
# solves Eq(sum(i**2 for i in var), n)
k = len(var)
result = DiophantineSolutionSet(var)
if n < 0 or limit < 1:
return result
sign = [-1 if x.is_nonpositive else 1 for x in var]
negs = sign.count(-1) != 0
took = 0
for t in power_representation(n, p, k):
if negs:
result.add([sign[i]*j for i, j in enumerate(t)])
else:
result.add(t)
took += 1
if took == limit:
break
return result
## Functions below this comment can be more suitably grouped under
## an Additive number theory module rather than the Diophantine
## equation module.
def partition(n, k=None, zeros=False):
"""
Returns a generator that can be used to generate partitions of an integer
`n`.
A partition of `n` is a set of positive integers which add up to `n`. For
example, partitions of 3 are 3, 1 + 2, 1 + 1 + 1. A partition is returned
as a tuple. If ``k`` equals None, then all possible partitions are returned
irrespective of their size, otherwise only the partitions of size ``k`` are
returned. If the ``zero`` parameter is set to True then a suitable
number of zeros are added at the end of every partition of size less than
``k``.
``zero`` parameter is considered only if ``k`` is not None. When the
partitions are over, the last `next()` call throws the ``StopIteration``
exception, so this function should always be used inside a try - except
block.
Details
=======
``partition(n, k)``: Here ``n`` is a positive integer and ``k`` is the size
of the partition which is also positive integer.
Examples
========
>>> from sympy.solvers.diophantine.diophantine import partition
>>> f = partition(5)
>>> next(f)
(1, 1, 1, 1, 1)
>>> next(f)
(1, 1, 1, 2)
>>> g = partition(5, 3)
>>> next(g)
(1, 1, 3)
>>> next(g)
(1, 2, 2)
>>> g = partition(5, 3, zeros=True)
>>> next(g)
(0, 0, 5)
"""
from sympy.utilities.iterables import ordered_partitions
if not zeros or k is None:
for i in ordered_partitions(n, k):
yield tuple(i)
else:
for m in range(1, k + 1):
for i in ordered_partitions(n, m):
i = tuple(i)
yield (0,)*(k - len(i)) + i
def prime_as_sum_of_two_squares(p):
"""
Represent a prime `p` as a unique sum of two squares; this can
only be done if the prime is congruent to 1 mod 4.
Examples
========
>>> from sympy.solvers.diophantine.diophantine import prime_as_sum_of_two_squares
>>> prime_as_sum_of_two_squares(7) # can't be done
>>> prime_as_sum_of_two_squares(5)
(1, 2)
Reference
=========
.. [1] Representing a number as a sum of four squares, [online],
Available: http://schorn.ch/lagrange.html
See Also
========
sum_of_squares()
"""
if not p % 4 == 1:
return
if p % 8 == 5:
b = 2
else:
b = 3
while pow(b, (p - 1) // 2, p) == 1:
b = nextprime(b)
b = pow(b, (p - 1) // 4, p)
a = p
while b**2 > p:
a, b = b, a % b
return (int(a % b), int(b)) # convert from long
def sum_of_three_squares(n):
r"""
Returns a 3-tuple `(a, b, c)` such that `a^2 + b^2 + c^2 = n` and
`a, b, c \geq 0`.
Returns None if `n = 4^a(8m + 7)` for some `a, m \in Z`. See
[1]_ for more details.
Usage
=====
``sum_of_three_squares(n)``: Here ``n`` is a non-negative integer.
Examples
========
>>> from sympy.solvers.diophantine.diophantine import sum_of_three_squares
>>> sum_of_three_squares(44542)
(18, 37, 207)
References
==========
.. [1] Representing a number as a sum of three squares, [online],
Available: http://schorn.ch/lagrange.html
See Also
========
sum_of_squares()
"""
special = {1:(1, 0, 0), 2:(1, 1, 0), 3:(1, 1, 1), 10: (1, 3, 0), 34: (3, 3, 4), 58:(3, 7, 0),
85:(6, 7, 0), 130:(3, 11, 0), 214:(3, 6, 13), 226:(8, 9, 9), 370:(8, 9, 15),
526:(6, 7, 21), 706:(15, 15, 16), 730:(1, 27, 0), 1414:(6, 17, 33), 1906:(13, 21, 36),
2986: (21, 32, 39), 9634: (56, 57, 57)}
v = 0
if n == 0:
return (0, 0, 0)
v = multiplicity(4, n)
n //= 4**v
if n % 8 == 7:
return
if n in special.keys():
x, y, z = special[n]
return _sorted_tuple(2**v*x, 2**v*y, 2**v*z)
s, _exact = integer_nthroot(n, 2)
if _exact:
return (2**v*s, 0, 0)
x = None
if n % 8 == 3:
s = s if _odd(s) else s - 1
for x in range(s, -1, -2):
N = (n - x**2) // 2
if isprime(N):
y, z = prime_as_sum_of_two_squares(N)
return _sorted_tuple(2**v*x, 2**v*(y + z), 2**v*abs(y - z))
return
if n % 8 == 2 or n % 8 == 6:
s = s if _odd(s) else s - 1
else:
s = s - 1 if _odd(s) else s
for x in range(s, -1, -2):
N = n - x**2
if isprime(N):
y, z = prime_as_sum_of_two_squares(N)
return _sorted_tuple(2**v*x, 2**v*y, 2**v*z)
def sum_of_four_squares(n):
r"""
Returns a 4-tuple `(a, b, c, d)` such that `a^2 + b^2 + c^2 + d^2 = n`.
Here `a, b, c, d \geq 0`.
Usage
=====
``sum_of_four_squares(n)``: Here ``n`` is a non-negative integer.
Examples
========
>>> from sympy.solvers.diophantine.diophantine import sum_of_four_squares
>>> sum_of_four_squares(3456)
(8, 8, 32, 48)
>>> sum_of_four_squares(1294585930293)
(0, 1234, 2161, 1137796)
References
==========
.. [1] Representing a number as a sum of four squares, [online],
Available: http://schorn.ch/lagrange.html
See Also
========
sum_of_squares()
"""
if n == 0:
return (0, 0, 0, 0)
v = multiplicity(4, n)
n //= 4**v
if n % 8 == 7:
d = 2
n = n - 4
elif n % 8 == 6 or n % 8 == 2:
d = 1
n = n - 1
else:
d = 0
x, y, z = sum_of_three_squares(n)
return _sorted_tuple(2**v*d, 2**v*x, 2**v*y, 2**v*z)
def power_representation(n, p, k, zeros=False):
r"""
Returns a generator for finding k-tuples of integers,
`(n_{1}, n_{2}, . . . n_{k})`, such that
`n = n_{1}^p + n_{2}^p + . . . n_{k}^p`.
Usage
=====
``power_representation(n, p, k, zeros)``: Represent non-negative number
``n`` as a sum of ``k`` ``p``\ th powers. If ``zeros`` is true, then the
solutions is allowed to contain zeros.
Examples
========
>>> from sympy.solvers.diophantine.diophantine import power_representation
Represent 1729 as a sum of two cubes:
>>> f = power_representation(1729, 3, 2)
>>> next(f)
(9, 10)
>>> next(f)
(1, 12)
If the flag `zeros` is True, the solution may contain tuples with
zeros; any such solutions will be generated after the solutions
without zeros:
>>> list(power_representation(125, 2, 3, zeros=True))
[(5, 6, 8), (3, 4, 10), (0, 5, 10), (0, 2, 11)]
For even `p` the `permute_sign` function can be used to get all
signed values:
>>> from sympy.utilities.iterables import permute_signs
>>> list(permute_signs((1, 12)))
[(1, 12), (-1, 12), (1, -12), (-1, -12)]
All possible signed permutations can also be obtained:
>>> from sympy.utilities.iterables import signed_permutations
>>> list(signed_permutations((1, 12)))
[(1, 12), (-1, 12), (1, -12), (-1, -12), (12, 1), (-12, 1), (12, -1), (-12, -1)]
"""
n, p, k = [as_int(i) for i in (n, p, k)]
if n < 0:
if p % 2:
for t in power_representation(-n, p, k, zeros):
yield tuple(-i for i in t)
return
if p < 1 or k < 1:
raise ValueError(filldedent('''
Expecting positive integers for `(p, k)`, but got `(%s, %s)`'''
% (p, k)))
if n == 0:
if zeros:
yield (0,)*k
return
if k == 1:
if p == 1:
yield (n,)
else:
be = perfect_power(n)
if be:
b, e = be
d, r = divmod(e, p)
if not r:
yield (b**d,)
return
if p == 1:
for t in partition(n, k, zeros=zeros):
yield t
return
if p == 2:
feasible = _can_do_sum_of_squares(n, k)
if not feasible:
return
if not zeros and n > 33 and k >= 5 and k <= n and n - k in (
13, 10, 7, 5, 4, 2, 1):
'''Todd G. Will, "When Is n^2 a Sum of k Squares?", [online].
Available: https://www.maa.org/sites/default/files/Will-MMz-201037918.pdf'''
return
if feasible is not True: # it's prime and k == 2
yield prime_as_sum_of_two_squares(n)
return
if k == 2 and p > 2:
be = perfect_power(n)
if be and be[1] % p == 0:
return # Fermat: a**n + b**n = c**n has no solution for n > 2
if n >= k:
a = integer_nthroot(n - (k - 1), p)[0]
for t in pow_rep_recursive(a, k, n, [], p):
yield tuple(reversed(t))
if zeros:
a = integer_nthroot(n, p)[0]
for i in range(1, k):
for t in pow_rep_recursive(a, i, n, [], p):
yield tuple(reversed(t + (0,) * (k - i)))
sum_of_powers = power_representation
def pow_rep_recursive(n_i, k, n_remaining, terms, p):
if k == 0 and n_remaining == 0:
yield tuple(terms)
else:
if n_i >= 1 and k > 0:
for t in pow_rep_recursive(n_i - 1, k, n_remaining, terms, p):
yield t
residual = n_remaining - pow(n_i, p)
if residual >= 0:
for t in pow_rep_recursive(n_i, k - 1, residual, terms + [n_i], p):
yield t
def sum_of_squares(n, k, zeros=False):
"""Return a generator that yields the k-tuples of nonnegative
values, the squares of which sum to n. If zeros is False (default)
then the solution will not contain zeros. The nonnegative
elements of a tuple are sorted.
* If k == 1 and n is square, (n,) is returned.
* If k == 2 then n can only be written as a sum of squares if
every prime in the factorization of n that has the form
4*k + 3 has an even multiplicity. If n is prime then
it can only be written as a sum of two squares if it is
in the form 4*k + 1.
* if k == 3 then n can be written as a sum of squares if it does
not have the form 4**m*(8*k + 7).
* all integers can be written as the sum of 4 squares.
* if k > 4 then n can be partitioned and each partition can
be written as a sum of 4 squares; if n is not evenly divisible
by 4 then n can be written as a sum of squares only if the
an additional partition can be written as sum of squares.
For example, if k = 6 then n is partitioned into two parts,
the first being written as a sum of 4 squares and the second
being written as a sum of 2 squares -- which can only be
done if the condition above for k = 2 can be met, so this will
automatically reject certain partitions of n.
Examples
========
>>> from sympy.solvers.diophantine.diophantine import sum_of_squares
>>> list(sum_of_squares(25, 2))
[(3, 4)]
>>> list(sum_of_squares(25, 2, True))
[(3, 4), (0, 5)]
>>> list(sum_of_squares(25, 4))
[(1, 2, 2, 4)]
See Also
========
sympy.utilities.iterables.signed_permutations
"""
for t in power_representation(n, 2, k, zeros):
yield t
def _can_do_sum_of_squares(n, k):
"""Return True if n can be written as the sum of k squares,
False if it can't, or 1 if k == 2 and n is prime (in which
case it *can* be written as a sum of two squares). A False
is returned only if it can't be written as k-squares, even
if 0s are allowed.
"""
if k < 1:
return False
if n < 0:
return False
if n == 0:
return True
if k == 1:
return is_square(n)
if k == 2:
if n in (1, 2):
return True
if isprime(n):
if n % 4 == 1:
return 1 # signal that it was prime
return False
else:
f = factorint(n)
for p, m in f.items():
# we can proceed iff no prime factor in the form 4*k + 3
# has an odd multiplicity
if (p % 4 == 3) and m % 2:
return False
return True
if k == 3:
if (n//4**multiplicity(4, n)) % 8 == 7:
return False
# every number can be written as a sum of 4 squares; for k > 4 partitions
# can be 0
return True
|
e86e3693cd1718524e19c726786200353290089a373085d66a1003b7b7d4f906 | r"""
This module contains :py:meth:`~sympy.solvers.ode.dsolve` and different helper
functions that it uses.
:py:meth:`~sympy.solvers.ode.dsolve` solves ordinary differential equations.
See the docstring on the various functions for their uses. Note that partial
differential equations support is in ``pde.py``. Note that hint functions
have docstrings describing their various methods, but they are intended for
internal use. Use ``dsolve(ode, func, hint=hint)`` to solve an ODE using a
specific hint. See also the docstring on
:py:meth:`~sympy.solvers.ode.dsolve`.
**Functions in this module**
These are the user functions in this module:
- :py:meth:`~sympy.solvers.ode.dsolve` - Solves ODEs.
- :py:meth:`~sympy.solvers.ode.classify_ode` - Classifies ODEs into
possible hints for :py:meth:`~sympy.solvers.ode.dsolve`.
- :py:meth:`~sympy.solvers.ode.checkodesol` - Checks if an equation is the
solution to an ODE.
- :py:meth:`~sympy.solvers.ode.homogeneous_order` - Returns the
homogeneous order of an expression.
- :py:meth:`~sympy.solvers.ode.infinitesimals` - Returns the infinitesimals
of the Lie group of point transformations of an ODE, such that it is
invariant.
- :py:meth:`~sympy.solvers.ode.checkinfsol` - Checks if the given infinitesimals
are the actual infinitesimals of a first order ODE.
These are the non-solver helper functions that are for internal use. The
user should use the various options to
:py:meth:`~sympy.solvers.ode.dsolve` to obtain the functionality provided
by these functions:
- :py:meth:`~sympy.solvers.ode.ode.odesimp` - Does all forms of ODE
simplification.
- :py:meth:`~sympy.solvers.ode.ode.ode_sol_simplicity` - A key function for
comparing solutions by simplicity.
- :py:meth:`~sympy.solvers.ode.constantsimp` - Simplifies arbitrary
constants.
- :py:meth:`~sympy.solvers.ode.ode.constant_renumber` - Renumber arbitrary
constants.
- :py:meth:`~sympy.solvers.ode.ode._handle_Integral` - Evaluate unevaluated
Integrals.
See also the docstrings of these functions.
**Currently implemented solver methods**
The following methods are implemented for solving ordinary differential
equations. See the docstrings of the various hint functions for more
information on each (run ``help(ode)``):
- 1st order separable differential equations.
- 1st order differential equations whose coefficients or `dx` and `dy` are
functions homogeneous of the same order.
- 1st order exact differential equations.
- 1st order linear differential equations.
- 1st order Bernoulli differential equations.
- Power series solutions for first order differential equations.
- Lie Group method of solving first order differential equations.
- 2nd order Liouville differential equations.
- Power series solutions for second order differential equations
at ordinary and regular singular points.
- `n`\th order differential equation that can be solved with algebraic
rearrangement and integration.
- `n`\th order linear homogeneous differential equation with constant
coefficients.
- `n`\th order linear inhomogeneous differential equation with constant
coefficients using the method of undetermined coefficients.
- `n`\th order linear inhomogeneous differential equation with constant
coefficients using the method of variation of parameters.
**Philosophy behind this module**
This module is designed to make it easy to add new ODE solving methods without
having to mess with the solving code for other methods. The idea is that
there is a :py:meth:`~sympy.solvers.ode.classify_ode` function, which takes in
an ODE and tells you what hints, if any, will solve the ODE. It does this
without attempting to solve the ODE, so it is fast. Each solving method is a
hint, and it has its own function, named ``ode_<hint>``. That function takes
in the ODE and any match expression gathered by
:py:meth:`~sympy.solvers.ode.classify_ode` and returns a solved result. If
this result has any integrals in it, the hint function will return an
unevaluated :py:class:`~sympy.integrals.integrals.Integral` class.
:py:meth:`~sympy.solvers.ode.dsolve`, which is the user wrapper function
around all of this, will then call :py:meth:`~sympy.solvers.ode.ode.odesimp` on
the result, which, among other things, will attempt to solve the equation for
the dependent variable (the function we are solving for), simplify the
arbitrary constants in the expression, and evaluate any integrals, if the hint
allows it.
**How to add new solution methods**
If you have an ODE that you want :py:meth:`~sympy.solvers.ode.dsolve` to be
able to solve, try to avoid adding special case code here. Instead, try
finding a general method that will solve your ODE, as well as others. This
way, the :py:mod:`~sympy.solvers.ode` module will become more robust, and
unhindered by special case hacks. WolphramAlpha and Maple's
DETools[odeadvisor] function are two resources you can use to classify a
specific ODE. It is also better for a method to work with an `n`\th order ODE
instead of only with specific orders, if possible.
To add a new method, there are a few things that you need to do. First, you
need a hint name for your method. Try to name your hint so that it is
unambiguous with all other methods, including ones that may not be implemented
yet. If your method uses integrals, also include a ``hint_Integral`` hint.
If there is more than one way to solve ODEs with your method, include a hint
for each one, as well as a ``<hint>_best`` hint. Your ``ode_<hint>_best()``
function should choose the best using min with ``ode_sol_simplicity`` as the
key argument. See
:py:meth:`~sympy.solvers.ode.ode.ode_1st_homogeneous_coeff_best`, for example.
The function that uses your method will be called ``ode_<hint>()``, so the
hint must only use characters that are allowed in a Python function name
(alphanumeric characters and the underscore '``_``' character). Include a
function for every hint, except for ``_Integral`` hints
(:py:meth:`~sympy.solvers.ode.dsolve` takes care of those automatically).
Hint names should be all lowercase, unless a word is commonly capitalized
(such as Integral or Bernoulli). If you have a hint that you do not want to
run with ``all_Integral`` that doesn't have an ``_Integral`` counterpart (such
as a best hint that would defeat the purpose of ``all_Integral``), you will
need to remove it manually in the :py:meth:`~sympy.solvers.ode.dsolve` code.
See also the :py:meth:`~sympy.solvers.ode.classify_ode` docstring for
guidelines on writing a hint name.
Determine *in general* how the solutions returned by your method compare with
other methods that can potentially solve the same ODEs. Then, put your hints
in the :py:data:`~sympy.solvers.ode.allhints` tuple in the order that they
should be called. The ordering of this tuple determines which hints are
default. Note that exceptions are ok, because it is easy for the user to
choose individual hints with :py:meth:`~sympy.solvers.ode.dsolve`. In
general, ``_Integral`` variants should go at the end of the list, and
``_best`` variants should go before the various hints they apply to. For
example, the ``undetermined_coefficients`` hint comes before the
``variation_of_parameters`` hint because, even though variation of parameters
is more general than undetermined coefficients, undetermined coefficients
generally returns cleaner results for the ODEs that it can solve than
variation of parameters does, and it does not require integration, so it is
much faster.
Next, you need to have a match expression or a function that matches the type
of the ODE, which you should put in :py:meth:`~sympy.solvers.ode.classify_ode`
(if the match function is more than just a few lines, like
:py:meth:`~sympy.solvers.ode.ode._undetermined_coefficients_match`, it should go
outside of :py:meth:`~sympy.solvers.ode.classify_ode`). It should match the
ODE without solving for it as much as possible, so that
:py:meth:`~sympy.solvers.ode.classify_ode` remains fast and is not hindered by
bugs in solving code. Be sure to consider corner cases. For example, if your
solution method involves dividing by something, make sure you exclude the case
where that division will be 0.
In most cases, the matching of the ODE will also give you the various parts
that you need to solve it. You should put that in a dictionary (``.match()``
will do this for you), and add that as ``matching_hints['hint'] = matchdict``
in the relevant part of :py:meth:`~sympy.solvers.ode.classify_ode`.
:py:meth:`~sympy.solvers.ode.classify_ode` will then send this to
:py:meth:`~sympy.solvers.ode.dsolve`, which will send it to your function as
the ``match`` argument. Your function should be named ``ode_<hint>(eq, func,
order, match)`. If you need to send more information, put it in the ``match``
dictionary. For example, if you had to substitute in a dummy variable in
:py:meth:`~sympy.solvers.ode.classify_ode` to match the ODE, you will need to
pass it to your function using the `match` dict to access it. You can access
the independent variable using ``func.args[0]``, and the dependent variable
(the function you are trying to solve for) as ``func.func``. If, while trying
to solve the ODE, you find that you cannot, raise ``NotImplementedError``.
:py:meth:`~sympy.solvers.ode.dsolve` will catch this error with the ``all``
meta-hint, rather than causing the whole routine to fail.
Add a docstring to your function that describes the method employed. Like
with anything else in SymPy, you will need to add a doctest to the docstring,
in addition to real tests in ``test_ode.py``. Try to maintain consistency
with the other hint functions' docstrings. Add your method to the list at the
top of this docstring. Also, add your method to ``ode.rst`` in the
``docs/src`` directory, so that the Sphinx docs will pull its docstring into
the main SymPy documentation. Be sure to make the Sphinx documentation by
running ``make html`` from within the doc directory to verify that the
docstring formats correctly.
If your solution method involves integrating, use :py:obj:`~.Integral` instead of
:py:meth:`~sympy.core.expr.Expr.integrate`. This allows the user to bypass
hard/slow integration by using the ``_Integral`` variant of your hint. In
most cases, calling :py:meth:`sympy.core.basic.Basic.doit` will integrate your
solution. If this is not the case, you will need to write special code in
:py:meth:`~sympy.solvers.ode.ode._handle_Integral`. Arbitrary constants should be
symbols named ``C1``, ``C2``, and so on. All solution methods should return
an equality instance. If you need an arbitrary number of arbitrary constants,
you can use ``constants = numbered_symbols(prefix='C', cls=Symbol, start=1)``.
If it is possible to solve for the dependent function in a general way, do so.
Otherwise, do as best as you can, but do not call solve in your
``ode_<hint>()`` function. :py:meth:`~sympy.solvers.ode.ode.odesimp` will attempt
to solve the solution for you, so you do not need to do that. Lastly, if your
ODE has a common simplification that can be applied to your solutions, you can
add a special case in :py:meth:`~sympy.solvers.ode.ode.odesimp` for it. For
example, solutions returned from the ``1st_homogeneous_coeff`` hints often
have many :obj:`~sympy.functions.elementary.exponential.log` terms, so
:py:meth:`~sympy.solvers.ode.ode.odesimp` calls
:py:meth:`~sympy.simplify.simplify.logcombine` on them (it also helps to write
the arbitrary constant as ``log(C1)`` instead of ``C1`` in this case). Also
consider common ways that you can rearrange your solution to have
:py:meth:`~sympy.solvers.ode.constantsimp` take better advantage of it. It is
better to put simplification in :py:meth:`~sympy.solvers.ode.ode.odesimp` than in
your method, because it can then be turned off with the simplify flag in
:py:meth:`~sympy.solvers.ode.dsolve`. If you have any extraneous
simplification in your function, be sure to only run it using ``if
match.get('simplify', True):``, especially if it can be slow or if it can
reduce the domain of the solution.
Finally, as with every contribution to SymPy, your method will need to be
tested. Add a test for each method in ``test_ode.py``. Follow the
conventions there, i.e., test the solver using ``dsolve(eq, f(x),
hint=your_hint)``, and also test the solution using
:py:meth:`~sympy.solvers.ode.checkodesol` (you can put these in a separate
tests and skip/XFAIL if it runs too slow/doesn't work). Be sure to call your
hint specifically in :py:meth:`~sympy.solvers.ode.dsolve`, that way the test
won't be broken simply by the introduction of another matching hint. If your
method works for higher order (>1) ODEs, you will need to run ``sol =
constant_renumber(sol, 'C', 1, order)`` for each solution, where ``order`` is
the order of the ODE. This is because ``constant_renumber`` renumbers the
arbitrary constants by printing order, which is platform dependent. Try to
test every corner case of your solver, including a range of orders if it is a
`n`\th order solver, but if your solver is slow, such as if it involves hard
integration, try to keep the test run time down.
Feel free to refactor existing hints to avoid duplicating code or creating
inconsistencies. If you can show that your method exactly duplicates an
existing method, including in the simplicity and speed of obtaining the
solutions, then you can remove the old, less general method. The existing
code is tested extensively in ``test_ode.py``, so if anything is broken, one
of those tests will surely fail.
"""
from __future__ import print_function, division
from collections import defaultdict
from itertools import islice
from sympy.functions import hyper
from sympy.core import Add, S, Mul, Pow, oo, Rational
from sympy.core.compatibility import ordered, iterable
from sympy.core.containers import Tuple
from sympy.core.exprtools import factor_terms
from sympy.core.expr import AtomicExpr, Expr
from sympy.core.function import (Function, Derivative, AppliedUndef, diff,
expand, expand_mul, Subs, _mexpand)
from sympy.core.multidimensional import vectorize
from sympy.core.numbers import NaN, zoo, I, Number
from sympy.core.relational import Equality, Eq
from sympy.core.symbol import Symbol, Wild, Dummy, symbols
from sympy.core.sympify import sympify
from sympy.logic.boolalg import (BooleanAtom, And, Not, BooleanTrue,
BooleanFalse)
from sympy.functions import cos, cosh, exp, im, log, re, sin, sinh, sqrt, \
atan2, conjugate, Piecewise, cbrt, besselj, bessely, airyai, airybi
from sympy.functions.combinatorial.factorials import factorial
from sympy.integrals.integrals import Integral, integrate
from sympy.matrices import wronskian, Matrix
from sympy.polys import (Poly, RootOf, rootof, terms_gcd,
PolynomialError, lcm, roots, gcd)
from sympy.polys.polyroots import roots_quartic
from sympy.polys.polytools import cancel, degree, div
from sympy.series import Order
from sympy.series.series import series
from sympy.simplify import (collect, logcombine, powsimp, # type: ignore
separatevars, simplify, trigsimp, posify, cse)
from sympy.simplify.powsimp import powdenest
from sympy.simplify.radsimp import collect_const
from sympy.solvers import checksol, solve
from sympy.solvers.pde import pdsolve
from sympy.utilities import numbered_symbols, default_sort_key, sift
from sympy.solvers.deutils import _preprocess, ode_order, _desolve
from .subscheck import sub_func_doit
#: This is a list of hints in the order that they should be preferred by
#: :py:meth:`~sympy.solvers.ode.classify_ode`. In general, hints earlier in the
#: list should produce simpler solutions than those later in the list (for
#: ODEs that fit both). For now, the order of this list is based on empirical
#: observations by the developers of SymPy.
#:
#: The hint used by :py:meth:`~sympy.solvers.ode.dsolve` for a specific ODE
#: can be overridden (see the docstring).
#:
#: In general, ``_Integral`` hints are grouped at the end of the list, unless
#: there is a method that returns an unevaluable integral most of the time
#: (which go near the end of the list anyway). ``default``, ``all``,
#: ``best``, and ``all_Integral`` meta-hints should not be included in this
#: list, but ``_best`` and ``_Integral`` hints should be included.
allhints = (
"factorable",
"nth_algebraic",
"separable",
"1st_exact",
"1st_linear",
"Bernoulli",
"Riccati_special_minus2",
"1st_homogeneous_coeff_best",
"1st_homogeneous_coeff_subs_indep_div_dep",
"1st_homogeneous_coeff_subs_dep_div_indep",
"almost_linear",
"linear_coefficients",
"separable_reduced",
"1st_power_series",
"lie_group",
"nth_linear_constant_coeff_homogeneous",
"nth_linear_euler_eq_homogeneous",
"nth_linear_constant_coeff_undetermined_coefficients",
"nth_linear_euler_eq_nonhomogeneous_undetermined_coefficients",
"nth_linear_constant_coeff_variation_of_parameters",
"nth_linear_euler_eq_nonhomogeneous_variation_of_parameters",
"Liouville",
"2nd_linear_airy",
"2nd_linear_bessel",
"2nd_hypergeometric",
"2nd_hypergeometric_Integral",
"nth_order_reducible",
"2nd_power_series_ordinary",
"2nd_power_series_regular",
"nth_algebraic_Integral",
"separable_Integral",
"1st_exact_Integral",
"1st_linear_Integral",
"Bernoulli_Integral",
"1st_homogeneous_coeff_subs_indep_div_dep_Integral",
"1st_homogeneous_coeff_subs_dep_div_indep_Integral",
"almost_linear_Integral",
"linear_coefficients_Integral",
"separable_reduced_Integral",
"nth_linear_constant_coeff_variation_of_parameters_Integral",
"nth_linear_euler_eq_nonhomogeneous_variation_of_parameters_Integral",
"Liouville_Integral",
)
lie_heuristics = (
"abaco1_simple",
"abaco1_product",
"abaco2_similar",
"abaco2_unique_unknown",
"abaco2_unique_general",
"linear",
"function_sum",
"bivariate",
"chi"
)
def get_numbered_constants(eq, num=1, start=1, prefix='C'):
"""
Returns a list of constants that do not occur
in eq already.
"""
ncs = iter_numbered_constants(eq, start, prefix)
Cs = [next(ncs) for i in range(num)]
return (Cs[0] if num == 1 else tuple(Cs))
def iter_numbered_constants(eq, start=1, prefix='C'):
"""
Returns an iterator of constants that do not occur
in eq already.
"""
if isinstance(eq, (Expr, Eq)):
eq = [eq]
elif not iterable(eq):
raise ValueError("Expected Expr or iterable but got %s" % eq)
atom_set = set().union(*[i.free_symbols for i in eq])
func_set = set().union(*[i.atoms(Function) for i in eq])
if func_set:
atom_set |= {Symbol(str(f.func)) for f in func_set}
return numbered_symbols(start=start, prefix=prefix, exclude=atom_set)
def dsolve(eq, func=None, hint="default", simplify=True,
ics= None, xi=None, eta=None, x0=0, n=6, **kwargs):
r"""
Solves any (supported) kind of ordinary differential equation and
system of ordinary differential equations.
For single ordinary differential equation
=========================================
It is classified under this when number of equation in ``eq`` is one.
**Usage**
``dsolve(eq, f(x), hint)`` -> Solve ordinary differential equation
``eq`` for function ``f(x)``, using method ``hint``.
**Details**
``eq`` can be any supported ordinary differential equation (see the
:py:mod:`~sympy.solvers.ode` docstring for supported methods).
This can either be an :py:class:`~sympy.core.relational.Equality`,
or an expression, which is assumed to be equal to ``0``.
``f(x)`` is a function of one variable whose derivatives in that
variable make up the ordinary differential equation ``eq``. In
many cases it is not necessary to provide this; it will be
autodetected (and an error raised if it couldn't be detected).
``hint`` is the solving method that you want dsolve to use. Use
``classify_ode(eq, f(x))`` to get all of the possible hints for an
ODE. The default hint, ``default``, will use whatever hint is
returned first by :py:meth:`~sympy.solvers.ode.classify_ode`. See
Hints below for more options that you can use for hint.
``simplify`` enables simplification by
:py:meth:`~sympy.solvers.ode.ode.odesimp`. See its docstring for more
information. Turn this off, for example, to disable solving of
solutions for ``func`` or simplification of arbitrary constants.
It will still integrate with this hint. Note that the solution may
contain more arbitrary constants than the order of the ODE with
this option enabled.
``xi`` and ``eta`` are the infinitesimal functions of an ordinary
differential equation. They are the infinitesimals of the Lie group
of point transformations for which the differential equation is
invariant. The user can specify values for the infinitesimals. If
nothing is specified, ``xi`` and ``eta`` are calculated using
:py:meth:`~sympy.solvers.ode.infinitesimals` with the help of various
heuristics.
``ics`` is the set of initial/boundary conditions for the differential equation.
It should be given in the form of ``{f(x0): x1, f(x).diff(x).subs(x, x2):
x3}`` and so on. For power series solutions, if no initial
conditions are specified ``f(0)`` is assumed to be ``C0`` and the power
series solution is calculated about 0.
``x0`` is the point about which the power series solution of a differential
equation is to be evaluated.
``n`` gives the exponent of the dependent variable up to which the power series
solution of a differential equation is to be evaluated.
**Hints**
Aside from the various solving methods, there are also some meta-hints
that you can pass to :py:meth:`~sympy.solvers.ode.dsolve`:
``default``:
This uses whatever hint is returned first by
:py:meth:`~sympy.solvers.ode.classify_ode`. This is the
default argument to :py:meth:`~sympy.solvers.ode.dsolve`.
``all``:
To make :py:meth:`~sympy.solvers.ode.dsolve` apply all
relevant classification hints, use ``dsolve(ODE, func,
hint="all")``. This will return a dictionary of
``hint:solution`` terms. If a hint causes dsolve to raise the
``NotImplementedError``, value of that hint's key will be the
exception object raised. The dictionary will also include
some special keys:
- ``order``: The order of the ODE. See also
:py:meth:`~sympy.solvers.deutils.ode_order` in
``deutils.py``.
- ``best``: The simplest hint; what would be returned by
``best`` below.
- ``best_hint``: The hint that would produce the solution
given by ``best``. If more than one hint produces the best
solution, the first one in the tuple returned by
:py:meth:`~sympy.solvers.ode.classify_ode` is chosen.
- ``default``: The solution that would be returned by default.
This is the one produced by the hint that appears first in
the tuple returned by
:py:meth:`~sympy.solvers.ode.classify_ode`.
``all_Integral``:
This is the same as ``all``, except if a hint also has a
corresponding ``_Integral`` hint, it only returns the
``_Integral`` hint. This is useful if ``all`` causes
:py:meth:`~sympy.solvers.ode.dsolve` to hang because of a
difficult or impossible integral. This meta-hint will also be
much faster than ``all``, because
:py:meth:`~sympy.core.expr.Expr.integrate` is an expensive
routine.
``best``:
To have :py:meth:`~sympy.solvers.ode.dsolve` try all methods
and return the simplest one. This takes into account whether
the solution is solvable in the function, whether it contains
any Integral classes (i.e. unevaluatable integrals), and
which one is the shortest in size.
See also the :py:meth:`~sympy.solvers.ode.classify_ode` docstring for
more info on hints, and the :py:mod:`~sympy.solvers.ode` docstring for
a list of all supported hints.
**Tips**
- You can declare the derivative of an unknown function this way:
>>> from sympy import Function, Derivative
>>> from sympy.abc import x # x is the independent variable
>>> f = Function("f")(x) # f is a function of x
>>> # f_ will be the derivative of f with respect to x
>>> f_ = Derivative(f, x)
- See ``test_ode.py`` for many tests, which serves also as a set of
examples for how to use :py:meth:`~sympy.solvers.ode.dsolve`.
- :py:meth:`~sympy.solvers.ode.dsolve` always returns an
:py:class:`~sympy.core.relational.Equality` class (except for the
case when the hint is ``all`` or ``all_Integral``). If possible, it
solves the solution explicitly for the function being solved for.
Otherwise, it returns an implicit solution.
- Arbitrary constants are symbols named ``C1``, ``C2``, and so on.
- Because all solutions should be mathematically equivalent, some
hints may return the exact same result for an ODE. Often, though,
two different hints will return the same solution formatted
differently. The two should be equivalent. Also note that sometimes
the values of the arbitrary constants in two different solutions may
not be the same, because one constant may have "absorbed" other
constants into it.
- Do ``help(ode.ode_<hintname>)`` to get help more information on a
specific hint, where ``<hintname>`` is the name of a hint without
``_Integral``.
For system of ordinary differential equations
=============================================
**Usage**
``dsolve(eq, func)`` -> Solve a system of ordinary differential
equations ``eq`` for ``func`` being list of functions including
`x(t)`, `y(t)`, `z(t)` where number of functions in the list depends
upon the number of equations provided in ``eq``.
**Details**
``eq`` can be any supported system of ordinary differential equations
This can either be an :py:class:`~sympy.core.relational.Equality`,
or an expression, which is assumed to be equal to ``0``.
``func`` holds ``x(t)`` and ``y(t)`` being functions of one variable which
together with some of their derivatives make up the system of ordinary
differential equation ``eq``. It is not necessary to provide this; it
will be autodetected (and an error raised if it couldn't be detected).
**Hints**
The hints are formed by parameters returned by classify_sysode, combining
them give hints name used later for forming method name.
Examples
========
>>> from sympy import Function, dsolve, Eq, Derivative, sin, cos, symbols
>>> from sympy.abc import x
>>> f = Function('f')
>>> dsolve(Derivative(f(x), x, x) + 9*f(x), f(x))
Eq(f(x), C1*sin(3*x) + C2*cos(3*x))
>>> eq = sin(x)*cos(f(x)) + cos(x)*sin(f(x))*f(x).diff(x)
>>> dsolve(eq, hint='1st_exact')
[Eq(f(x), -acos(C1/cos(x)) + 2*pi), Eq(f(x), acos(C1/cos(x)))]
>>> dsolve(eq, hint='almost_linear')
[Eq(f(x), -acos(C1/cos(x)) + 2*pi), Eq(f(x), acos(C1/cos(x)))]
>>> t = symbols('t')
>>> x, y = symbols('x, y', cls=Function)
>>> eq = (Eq(Derivative(x(t),t), 12*t*x(t) + 8*y(t)), Eq(Derivative(y(t),t), 21*x(t) + 7*t*y(t)))
>>> dsolve(eq)
[Eq(x(t), C1*x0(t) + C2*x0(t)*Integral(8*exp(Integral(7*t, t))*exp(Integral(12*t, t))/x0(t)**2, t)),
Eq(y(t), C1*y0(t) + C2*(y0(t)*Integral(8*exp(Integral(7*t, t))*exp(Integral(12*t, t))/x0(t)**2, t) +
exp(Integral(7*t, t))*exp(Integral(12*t, t))/x0(t)))]
>>> eq = (Eq(Derivative(x(t),t),x(t)*y(t)*sin(t)), Eq(Derivative(y(t),t),y(t)**2*sin(t)))
>>> dsolve(eq)
{Eq(x(t), -exp(C1)/(C2*exp(C1) - cos(t))), Eq(y(t), -1/(C1 - cos(t)))}
"""
if iterable(eq):
match = classify_sysode(eq, func)
eq = match['eq']
order = match['order']
func = match['func']
t = list(list(eq[0].atoms(Derivative))[0].atoms(Symbol))[0]
# keep highest order term coefficient positive
for i in range(len(eq)):
for func_ in func:
if isinstance(func_, list):
pass
else:
if eq[i].coeff(diff(func[i],t,ode_order(eq[i], func[i]))).is_negative:
eq[i] = -eq[i]
match['eq'] = eq
if len(set(order.values()))!=1:
raise ValueError("It solves only those systems of equations whose orders are equal")
match['order'] = list(order.values())[0]
def recur_len(l):
return sum(recur_len(item) if isinstance(item,list) else 1 for item in l)
if recur_len(func) != len(eq):
raise ValueError("dsolve() and classify_sysode() work with "
"number of functions being equal to number of equations")
if match['type_of_equation'] is None:
raise NotImplementedError
else:
if match['is_linear'] == True:
# These conditions have to be improved upon in future for the new solvers
# added in systems.py
if match.get('is_general', False):
solvefunc = globals()['sysode_linear_neq_order%(order)s' % match]
else:
solvefunc = globals()['sysode_linear_%(no_of_equation)seq_order%(order)s' % match]
else:
solvefunc = globals()['sysode_nonlinear_%(no_of_equation)seq_order%(order)s' % match]
sols = solvefunc(match)
if ics:
constants = Tuple(*sols).free_symbols - Tuple(*eq).free_symbols
solved_constants = solve_ics(sols, func, constants, ics)
return [sol.subs(solved_constants) for sol in sols]
return sols
else:
given_hint = hint # hint given by the user
# See the docstring of _desolve for more details.
hints = _desolve(eq, func=func,
hint=hint, simplify=True, xi=xi, eta=eta, type='ode', ics=ics,
x0=x0, n=n, **kwargs)
eq = hints.pop('eq', eq)
all_ = hints.pop('all', False)
if all_:
retdict = {}
failed_hints = {}
gethints = classify_ode(eq, dict=True)
orderedhints = gethints['ordered_hints']
for hint in hints:
try:
rv = _helper_simplify(eq, hint, hints[hint], simplify)
except NotImplementedError as detail:
failed_hints[hint] = detail
else:
retdict[hint] = rv
func = hints[hint]['func']
retdict['best'] = min(list(retdict.values()), key=lambda x:
ode_sol_simplicity(x, func, trysolving=not simplify))
if given_hint == 'best':
return retdict['best']
for i in orderedhints:
if retdict['best'] == retdict.get(i, None):
retdict['best_hint'] = i
break
retdict['default'] = gethints['default']
retdict['order'] = gethints['order']
retdict.update(failed_hints)
return retdict
else:
# The key 'hint' stores the hint needed to be solved for.
hint = hints['hint']
return _helper_simplify(eq, hint, hints, simplify, ics=ics)
def _helper_simplify(eq, hint, match, simplify=True, ics=None, **kwargs):
r"""
Helper function of dsolve that calls the respective
:py:mod:`~sympy.solvers.ode` functions to solve for the ordinary
differential equations. This minimizes the computation in calling
:py:meth:`~sympy.solvers.deutils._desolve` multiple times.
"""
r = match
func = r['func']
order = r['order']
match = r[hint]
if isinstance(match, SingleODESolver):
solvefunc = match
elif hint.endswith('_Integral'):
solvefunc = globals()['ode_' + hint[:-len('_Integral')]]
else:
solvefunc = globals()['ode_' + hint]
free = eq.free_symbols
cons = lambda s: s.free_symbols.difference(free)
if simplify:
# odesimp() will attempt to integrate, if necessary, apply constantsimp(),
# attempt to solve for func, and apply any other hint specific
# simplifications
if isinstance(solvefunc, SingleODESolver):
sols = solvefunc.get_general_solution()
else:
sols = solvefunc(eq, func, order, match)
if iterable(sols):
rv = [odesimp(eq, s, func, hint) for s in sols]
else:
rv = odesimp(eq, sols, func, hint)
else:
# We still want to integrate (you can disable it separately with the hint)
if isinstance(solvefunc, SingleODESolver):
exprs = solvefunc.get_general_solution(simplify=False)
else:
match['simplify'] = False # Some hints can take advantage of this option
exprs = solvefunc(eq, func, order, match)
if isinstance(exprs, list):
rv = [_handle_Integral(expr, func, hint) for expr in exprs]
else:
rv = _handle_Integral(exprs, func, hint)
if isinstance(rv, list):
rv = _remove_redundant_solutions(eq, rv, order, func.args[0])
if len(rv) == 1:
rv = rv[0]
if ics and not 'power_series' in hint:
if isinstance(rv, (Expr, Eq)):
solved_constants = solve_ics([rv], [r['func']], cons(rv), ics)
rv = rv.subs(solved_constants)
else:
rv1 = []
for s in rv:
try:
solved_constants = solve_ics([s], [r['func']], cons(s), ics)
except ValueError:
continue
rv1.append(s.subs(solved_constants))
if len(rv1) == 1:
return rv1[0]
rv = rv1
return rv
def solve_ics(sols, funcs, constants, ics):
"""
Solve for the constants given initial conditions
``sols`` is a list of solutions.
``funcs`` is a list of functions.
``constants`` is a list of constants.
``ics`` is the set of initial/boundary conditions for the differential
equation. It should be given in the form of ``{f(x0): x1,
f(x).diff(x).subs(x, x2): x3}`` and so on.
Returns a dictionary mapping constants to values.
``solution.subs(constants)`` will replace the constants in ``solution``.
Example
=======
>>> # From dsolve(f(x).diff(x) - f(x), f(x))
>>> from sympy import symbols, Eq, exp, Function
>>> from sympy.solvers.ode.ode import solve_ics
>>> f = Function('f')
>>> x, C1 = symbols('x C1')
>>> sols = [Eq(f(x), C1*exp(x))]
>>> funcs = [f(x)]
>>> constants = [C1]
>>> ics = {f(0): 2}
>>> solved_constants = solve_ics(sols, funcs, constants, ics)
>>> solved_constants
{C1: 2}
>>> sols[0].subs(solved_constants)
Eq(f(x), 2*exp(x))
"""
# Assume ics are of the form f(x0): value or Subs(diff(f(x), x, n), (x,
# x0)): value (currently checked by classify_ode). To solve, replace x
# with x0, f(x0) with value, then solve for constants. For f^(n)(x0),
# differentiate the solution n times, so that f^(n)(x) appears.
x = funcs[0].args[0]
diff_sols = []
subs_sols = []
diff_variables = set()
for funcarg, value in ics.items():
if isinstance(funcarg, AppliedUndef):
x0 = funcarg.args[0]
matching_func = [f for f in funcs if f.func == funcarg.func][0]
S = sols
elif isinstance(funcarg, (Subs, Derivative)):
if isinstance(funcarg, Subs):
# Make sure it stays a subs. Otherwise subs below will produce
# a different looking term.
funcarg = funcarg.doit()
if isinstance(funcarg, Subs):
deriv = funcarg.expr
x0 = funcarg.point[0]
variables = funcarg.expr.variables
matching_func = deriv
elif isinstance(funcarg, Derivative):
deriv = funcarg
x0 = funcarg.variables[0]
variables = (x,)*len(funcarg.variables)
matching_func = deriv.subs(x0, x)
if variables not in diff_variables:
for sol in sols:
if sol.has(deriv.expr.func):
diff_sols.append(Eq(sol.lhs.diff(*variables), sol.rhs.diff(*variables)))
diff_variables.add(variables)
S = diff_sols
else:
raise NotImplementedError("Unrecognized initial condition")
for sol in S:
if sol.has(matching_func):
sol2 = sol
sol2 = sol2.subs(x, x0)
sol2 = sol2.subs(funcarg, value)
# This check is necessary because of issue #15724
if not isinstance(sol2, BooleanAtom) or not subs_sols:
subs_sols = [s for s in subs_sols if not isinstance(s, BooleanAtom)]
subs_sols.append(sol2)
# TODO: Use solveset here
try:
solved_constants = solve(subs_sols, constants, dict=True)
except NotImplementedError:
solved_constants = []
# XXX: We can't differentiate between the solution not existing because of
# invalid initial conditions, and not existing because solve is not smart
# enough. If we could use solveset, this might be improvable, but for now,
# we use NotImplementedError in this case.
if not solved_constants:
raise ValueError("Couldn't solve for initial conditions")
if solved_constants == True:
raise ValueError("Initial conditions did not produce any solutions for constants. Perhaps they are degenerate.")
if len(solved_constants) > 1:
raise NotImplementedError("Initial conditions produced too many solutions for constants")
return solved_constants[0]
def classify_ode(eq, func=None, dict=False, ics=None, **kwargs):
r"""
Returns a tuple of possible :py:meth:`~sympy.solvers.ode.dsolve`
classifications for an ODE.
The tuple is ordered so that first item is the classification that
:py:meth:`~sympy.solvers.ode.dsolve` uses to solve the ODE by default. In
general, classifications at the near the beginning of the list will
produce better solutions faster than those near the end, thought there are
always exceptions. To make :py:meth:`~sympy.solvers.ode.dsolve` use a
different classification, use ``dsolve(ODE, func,
hint=<classification>)``. See also the
:py:meth:`~sympy.solvers.ode.dsolve` docstring for different meta-hints
you can use.
If ``dict`` is true, :py:meth:`~sympy.solvers.ode.classify_ode` will
return a dictionary of ``hint:match`` expression terms. This is intended
for internal use by :py:meth:`~sympy.solvers.ode.dsolve`. Note that
because dictionaries are ordered arbitrarily, this will most likely not be
in the same order as the tuple.
You can get help on different hints by executing
``help(ode.ode_hintname)``, where ``hintname`` is the name of the hint
without ``_Integral``.
See :py:data:`~sympy.solvers.ode.allhints` or the
:py:mod:`~sympy.solvers.ode` docstring for a list of all supported hints
that can be returned from :py:meth:`~sympy.solvers.ode.classify_ode`.
Notes
=====
These are remarks on hint names.
``_Integral``
If a classification has ``_Integral`` at the end, it will return the
expression with an unevaluated :py:class:`~.Integral`
class in it. Note that a hint may do this anyway if
:py:meth:`~sympy.core.expr.Expr.integrate` cannot do the integral,
though just using an ``_Integral`` will do so much faster. Indeed, an
``_Integral`` hint will always be faster than its corresponding hint
without ``_Integral`` because
:py:meth:`~sympy.core.expr.Expr.integrate` is an expensive routine.
If :py:meth:`~sympy.solvers.ode.dsolve` hangs, it is probably because
:py:meth:`~sympy.core.expr.Expr.integrate` is hanging on a tough or
impossible integral. Try using an ``_Integral`` hint or
``all_Integral`` to get it return something.
Note that some hints do not have ``_Integral`` counterparts. This is
because :py:func:`~sympy.integrals.integrals.integrate` is not used in
solving the ODE for those method. For example, `n`\th order linear
homogeneous ODEs with constant coefficients do not require integration
to solve, so there is no
``nth_linear_homogeneous_constant_coeff_Integrate`` hint. You can
easily evaluate any unevaluated
:py:class:`~sympy.integrals.integrals.Integral`\s in an expression by
doing ``expr.doit()``.
Ordinals
Some hints contain an ordinal such as ``1st_linear``. This is to help
differentiate them from other hints, as well as from other methods
that may not be implemented yet. If a hint has ``nth`` in it, such as
the ``nth_linear`` hints, this means that the method used to applies
to ODEs of any order.
``indep`` and ``dep``
Some hints contain the words ``indep`` or ``dep``. These reference
the independent variable and the dependent function, respectively. For
example, if an ODE is in terms of `f(x)`, then ``indep`` will refer to
`x` and ``dep`` will refer to `f`.
``subs``
If a hints has the word ``subs`` in it, it means the the ODE is solved
by substituting the expression given after the word ``subs`` for a
single dummy variable. This is usually in terms of ``indep`` and
``dep`` as above. The substituted expression will be written only in
characters allowed for names of Python objects, meaning operators will
be spelled out. For example, ``indep``/``dep`` will be written as
``indep_div_dep``.
``coeff``
The word ``coeff`` in a hint refers to the coefficients of something
in the ODE, usually of the derivative terms. See the docstring for
the individual methods for more info (``help(ode)``). This is
contrast to ``coefficients``, as in ``undetermined_coefficients``,
which refers to the common name of a method.
``_best``
Methods that have more than one fundamental way to solve will have a
hint for each sub-method and a ``_best`` meta-classification. This
will evaluate all hints and return the best, using the same
considerations as the normal ``best`` meta-hint.
Examples
========
>>> from sympy import Function, classify_ode, Eq
>>> from sympy.abc import x
>>> f = Function('f')
>>> classify_ode(Eq(f(x).diff(x), 0), f(x))
('nth_algebraic',
'separable',
'1st_linear',
'Bernoulli',
'1st_homogeneous_coeff_best',
'1st_homogeneous_coeff_subs_indep_div_dep',
'1st_homogeneous_coeff_subs_dep_div_indep',
'1st_power_series', 'lie_group', 'nth_linear_constant_coeff_homogeneous',
'nth_linear_euler_eq_homogeneous',
'nth_algebraic_Integral', 'separable_Integral',
'1st_linear_Integral', 'Bernoulli_Integral',
'1st_homogeneous_coeff_subs_indep_div_dep_Integral',
'1st_homogeneous_coeff_subs_dep_div_indep_Integral')
>>> classify_ode(f(x).diff(x, 2) + 3*f(x).diff(x) + 2*f(x) - 4)
('nth_linear_constant_coeff_undetermined_coefficients',
'nth_linear_constant_coeff_variation_of_parameters',
'nth_linear_constant_coeff_variation_of_parameters_Integral')
"""
ics = sympify(ics)
prep = kwargs.pop('prep', True)
if func and len(func.args) != 1:
raise ValueError("dsolve() and classify_ode() only "
"work with functions of one variable, not %s" % func)
if isinstance(eq, Equality):
eq = eq.lhs - eq.rhs
# Some methods want the unprocessed equation
eq_orig = eq
if prep or func is None:
eq, func_ = _preprocess(eq, func)
if func is None:
func = func_
x = func.args[0]
f = func.func
y = Dummy('y')
xi = kwargs.get('xi')
eta = kwargs.get('eta')
terms = kwargs.get('n')
order = ode_order(eq, f(x))
# hint:matchdict or hint:(tuple of matchdicts)
# Also will contain "default":<default hint> and "order":order items.
matching_hints = {"order": order}
df = f(x).diff(x)
a = Wild('a', exclude=[f(x)])
d = Wild('d', exclude=[df, f(x).diff(x, 2)])
e = Wild('e', exclude=[df])
k = Wild('k', exclude=[df])
n = Wild('n', exclude=[x, f(x), df])
c1 = Wild('c1', exclude=[x])
a3 = Wild('a3', exclude=[f(x), df, f(x).diff(x, 2)])
b3 = Wild('b3', exclude=[f(x), df, f(x).diff(x, 2)])
c3 = Wild('c3', exclude=[f(x), df, f(x).diff(x, 2)])
r3 = {'xi': xi, 'eta': eta} # Used for the lie_group hint
boundary = {} # Used to extract initial conditions
C1 = Symbol("C1")
# Preprocessing to get the initial conditions out
if ics is not None:
for funcarg in ics:
# Separating derivatives
if isinstance(funcarg, (Subs, Derivative)):
# f(x).diff(x).subs(x, 0) is a Subs, but f(x).diff(x).subs(x,
# y) is a Derivative
if isinstance(funcarg, Subs):
deriv = funcarg.expr
old = funcarg.variables[0]
new = funcarg.point[0]
elif isinstance(funcarg, Derivative):
deriv = funcarg
# No information on this. Just assume it was x
old = x
new = funcarg.variables[0]
if (isinstance(deriv, Derivative) and isinstance(deriv.args[0],
AppliedUndef) and deriv.args[0].func == f and
len(deriv.args[0].args) == 1 and old == x and not
new.has(x) and all(i == deriv.variables[0] for i in
deriv.variables) and not ics[funcarg].has(f)):
dorder = ode_order(deriv, x)
temp = 'f' + str(dorder)
boundary.update({temp: new, temp + 'val': ics[funcarg]})
else:
raise ValueError("Enter valid boundary conditions for Derivatives")
# Separating functions
elif isinstance(funcarg, AppliedUndef):
if (funcarg.func == f and len(funcarg.args) == 1 and
not funcarg.args[0].has(x) and not ics[funcarg].has(f)):
boundary.update({'f0': funcarg.args[0], 'f0val': ics[funcarg]})
else:
raise ValueError("Enter valid boundary conditions for Function")
else:
raise ValueError("Enter boundary conditions of the form ics={f(point}: value, f(x).diff(x, order).subs(x, point): value}")
# Any ODE that can be solved with a combination of algebra and
# integrals e.g.:
# d^3/dx^3(x y) = F(x)
ode = SingleODEProblem(eq_orig, func, x, prep=prep)
solvers = {
NthAlgebraic: ('nth_algebraic',),
FirstLinear: ('1st_linear',),
AlmostLinear: ('almost_linear',),
Bernoulli: ('Bernoulli',),
Factorable: ('factorable',),
RiccatiSpecial: ('Riccati_special_minus2',),
}
for solvercls in solvers:
solver = solvercls(ode)
if solver.matches():
for hints in solvers[solvercls]:
matching_hints[hints] = solver
if solvercls.has_integral:
matching_hints[hints + "_Integral"] = solver
eq = expand(eq)
# Precondition to try remove f(x) from highest order derivative
reduced_eq = None
if eq.is_Add:
deriv_coef = eq.coeff(f(x).diff(x, order))
if deriv_coef not in (1, 0):
r = deriv_coef.match(a*f(x)**c1)
if r and r[c1]:
den = f(x)**r[c1]
reduced_eq = Add(*[arg/den for arg in eq.args])
if not reduced_eq:
reduced_eq = eq
if order == 1:
# NON-REDUCED FORM OF EQUATION matches
r = collect(eq, df, exact=True).match(d + e * df)
if r:
r['d'] = d
r['e'] = e
r['y'] = y
r[d] = r[d].subs(f(x), y)
r[e] = r[e].subs(f(x), y)
# FIRST ORDER POWER SERIES WHICH NEEDS INITIAL CONDITIONS
# TODO: Hint first order series should match only if d/e is analytic.
# For now, only d/e and (d/e).diff(arg) is checked for existence at
# at a given point.
# This is currently done internally in ode_1st_power_series.
point = boundary.get('f0', 0)
value = boundary.get('f0val', C1)
check = cancel(r[d]/r[e])
check1 = check.subs({x: point, y: value})
if not check1.has(oo) and not check1.has(zoo) and \
not check1.has(NaN) and not check1.has(-oo):
check2 = (check1.diff(x)).subs({x: point, y: value})
if not check2.has(oo) and not check2.has(zoo) and \
not check2.has(NaN) and not check2.has(-oo):
rseries = r.copy()
rseries.update({'terms': terms, 'f0': point, 'f0val': value})
matching_hints["1st_power_series"] = rseries
r3.update(r)
## Exact Differential Equation: P(x, y) + Q(x, y)*y' = 0 where
# dP/dy == dQ/dx
try:
if r[d] != 0:
numerator = simplify(r[d].diff(y) - r[e].diff(x))
# The following few conditions try to convert a non-exact
# differential equation into an exact one.
# References : Differential equations with applications
# and historical notes - George E. Simmons
if numerator:
# If (dP/dy - dQ/dx) / Q = f(x)
# then exp(integral(f(x))*equation becomes exact
factor = simplify(numerator/r[e])
variables = factor.free_symbols
if len(variables) == 1 and x == variables.pop():
factor = exp(Integral(factor).doit())
r[d] *= factor
r[e] *= factor
matching_hints["1st_exact"] = r
matching_hints["1st_exact_Integral"] = r
else:
# If (dP/dy - dQ/dx) / -P = f(y)
# then exp(integral(f(y))*equation becomes exact
factor = simplify(-numerator/r[d])
variables = factor.free_symbols
if len(variables) == 1 and y == variables.pop():
factor = exp(Integral(factor).doit())
r[d] *= factor
r[e] *= factor
matching_hints["1st_exact"] = r
matching_hints["1st_exact_Integral"] = r
else:
matching_hints["1st_exact"] = r
matching_hints["1st_exact_Integral"] = r
except NotImplementedError:
# Differentiating the coefficients might fail because of things
# like f(2*x).diff(x). See issue 4624 and issue 4719.
pass
# Any first order ODE can be ideally solved by the Lie Group
# method
matching_hints["lie_group"] = r3
# This match is used for several cases below; we now collect on
# f(x) so the matching works.
r = collect(reduced_eq, df, exact=True).match(d + e*df)
if r:
# Using r[d] and r[e] without any modification for hints
# linear-coefficients and separable-reduced.
num, den = r[d], r[e] # ODE = d/e + df
r['d'] = d
r['e'] = e
r['y'] = y
r[d] = num.subs(f(x), y)
r[e] = den.subs(f(x), y)
## Separable Case: y' == P(y)*Q(x)
r[d] = separatevars(r[d])
r[e] = separatevars(r[e])
# m1[coeff]*m1[x]*m1[y] + m2[coeff]*m2[x]*m2[y]*y'
m1 = separatevars(r[d], dict=True, symbols=(x, y))
m2 = separatevars(r[e], dict=True, symbols=(x, y))
if m1 and m2:
r1 = {'m1': m1, 'm2': m2, 'y': y}
matching_hints["separable"] = r1
matching_hints["separable_Integral"] = r1
## First order equation with homogeneous coefficients:
# dy/dx == F(y/x) or dy/dx == F(x/y)
ordera = homogeneous_order(r[d], x, y)
if ordera is not None:
orderb = homogeneous_order(r[e], x, y)
if ordera == orderb:
# u1=y/x and u2=x/y
u1 = Dummy('u1')
u2 = Dummy('u2')
s = "1st_homogeneous_coeff_subs"
s1 = s + "_dep_div_indep"
s2 = s + "_indep_div_dep"
if simplify((r[d] + u1*r[e]).subs({x: 1, y: u1})) != 0:
matching_hints[s1] = r
matching_hints[s1 + "_Integral"] = r
if simplify((r[e] + u2*r[d]).subs({x: u2, y: 1})) != 0:
matching_hints[s2] = r
matching_hints[s2 + "_Integral"] = r
if s1 in matching_hints and s2 in matching_hints:
matching_hints["1st_homogeneous_coeff_best"] = r
## Linear coefficients of the form
# y'+ F((a*x + b*y + c)/(a'*x + b'y + c')) = 0
# that can be reduced to homogeneous form.
F = num/den
params = _linear_coeff_match(F, func)
if params:
xarg, yarg = params
u = Dummy('u')
t = Dummy('t')
# Dummy substitution for df and f(x).
dummy_eq = reduced_eq.subs(((df, t), (f(x), u)))
reps = ((x, x + xarg), (u, u + yarg), (t, df), (u, f(x)))
dummy_eq = simplify(dummy_eq.subs(reps))
# get the re-cast values for e and d
r2 = collect(expand(dummy_eq), [df, f(x)]).match(e*df + d)
if r2:
orderd = homogeneous_order(r2[d], x, f(x))
if orderd is not None:
ordere = homogeneous_order(r2[e], x, f(x))
if orderd == ordere:
# Match arguments are passed in such a way that it
# is coherent with the already existing homogeneous
# functions.
r2[d] = r2[d].subs(f(x), y)
r2[e] = r2[e].subs(f(x), y)
r2.update({'xarg': xarg, 'yarg': yarg,
'd': d, 'e': e, 'y': y})
matching_hints["linear_coefficients"] = r2
matching_hints["linear_coefficients_Integral"] = r2
## Equation of the form y' + (y/x)*H(x^n*y) = 0
# that can be reduced to separable form
factor = simplify(x/f(x)*num/den)
# Try representing factor in terms of x^n*y
# where n is lowest power of x in factor;
# first remove terms like sqrt(2)*3 from factor.atoms(Mul)
num, dem = factor.as_numer_denom()
num = expand(num)
dem = expand(dem)
def _degree(expr, x):
# Made this function to calculate the degree of
# x in an expression. If expr will be of form
# x**p*y, (wheare p can be variables/rationals) then it
# will return p.
for val in expr:
if val.has(x):
if isinstance(val, Pow) and val.as_base_exp()[0] == x:
return (val.as_base_exp()[1])
elif val == x:
return (val.as_base_exp()[1])
else:
return _degree(val.args, x)
return 0
def _powers(expr):
# this function will return all the different relative power of x w.r.t f(x).
# expr = x**p * f(x)**q then it will return {p/q}.
pows = set()
if isinstance(expr, Add):
exprs = expr.atoms(Add)
elif isinstance(expr, Mul):
exprs = expr.atoms(Mul)
elif isinstance(expr, Pow):
exprs = expr.atoms(Pow)
else:
exprs = {expr}
for arg in exprs:
if arg.has(x):
_, u = arg.as_independent(x, f(x))
pow = _degree((u.subs(f(x), y), ), x)/_degree((u.subs(f(x), y), ), y)
pows.add(pow)
return pows
pows = _powers(num)
pows.update(_powers(dem))
pows = list(pows)
if(len(pows)==1) and pows[0]!=zoo:
t = Dummy('t')
r2 = {'t': t}
num = num.subs(x**pows[0]*f(x), t)
dem = dem.subs(x**pows[0]*f(x), t)
test = num/dem
free = test.free_symbols
if len(free) == 1 and free.pop() == t:
r2.update({'power' : pows[0], 'u' : test})
matching_hints['separable_reduced'] = r2
matching_hints["separable_reduced_Integral"] = r2
elif order == 2:
# Liouville ODE in the form
# f(x).diff(x, 2) + g(f(x))*(f(x).diff(x))**2 + h(x)*f(x).diff(x)
# See Goldstein and Braun, "Advanced Methods for the Solution of
# Differential Equations", pg. 98
s = d*f(x).diff(x, 2) + e*df**2 + k*df
r = reduced_eq.match(s)
if r and r[d] != 0:
y = Dummy('y')
g = simplify(r[e]/r[d]).subs(f(x), y)
h = simplify(r[k]/r[d]).subs(f(x), y)
if y in h.free_symbols or x in g.free_symbols:
pass
else:
r = {'g': g, 'h': h, 'y': y}
matching_hints["Liouville"] = r
matching_hints["Liouville_Integral"] = r
# Homogeneous second order differential equation of the form
# a3*f(x).diff(x, 2) + b3*f(x).diff(x) + c3
# It has a definite power series solution at point x0 if, b3/a3 and c3/a3
# are analytic at x0.
deq = a3*(f(x).diff(x, 2)) + b3*df + c3*f(x)
r = collect(reduced_eq,
[f(x).diff(x, 2), f(x).diff(x), f(x)]).match(deq)
ordinary = False
if r:
if not all([r[key].is_polynomial() for key in r]):
n, d = reduced_eq.as_numer_denom()
reduced_eq = expand(n)
r = collect(reduced_eq,
[f(x).diff(x, 2), f(x).diff(x), f(x)]).match(deq)
if r and r[a3] != 0:
p = cancel(r[b3]/r[a3]) # Used below
q = cancel(r[c3]/r[a3]) # Used below
point = kwargs.get('x0', 0)
check = p.subs(x, point)
if not check.has(oo, NaN, zoo, -oo):
check = q.subs(x, point)
if not check.has(oo, NaN, zoo, -oo):
ordinary = True
r.update({'a3': a3, 'b3': b3, 'c3': c3, 'x0': point, 'terms': terms})
matching_hints["2nd_power_series_ordinary"] = r
# Checking if the differential equation has a regular singular point
# at x0. It has a regular singular point at x0, if (b3/a3)*(x - x0)
# and (c3/a3)*((x - x0)**2) are analytic at x0.
if not ordinary:
p = cancel((x - point)*p)
check = p.subs(x, point)
if not check.has(oo, NaN, zoo, -oo):
q = cancel(((x - point)**2)*q)
check = q.subs(x, point)
if not check.has(oo, NaN, zoo, -oo):
coeff_dict = {'p': p, 'q': q, 'x0': point, 'terms': terms}
matching_hints["2nd_power_series_regular"] = coeff_dict
# For Hypergeometric solutions.
_r = {}
_r.update(r)
rn = match_2nd_hypergeometric(_r, func)
if rn:
matching_hints["2nd_hypergeometric"] = rn
matching_hints["2nd_hypergeometric_Integral"] = rn
# If the ODE has regular singular point at x0 and is of the form
# Eq((x)**2*Derivative(y(x), x, x) + x*Derivative(y(x), x) +
# (a4**2*x**(2*p)-n**2)*y(x) thus Bessel's equation
rn = match_2nd_linear_bessel(r, f(x))
if rn:
matching_hints["2nd_linear_bessel"] = rn
# If the ODE is ordinary and is of the form of Airy's Equation
# Eq(x**2*Derivative(y(x),x,x)-(ax+b)*y(x))
if p.is_zero:
a4 = Wild('a4', exclude=[x,f(x),df])
b4 = Wild('b4', exclude=[x,f(x),df])
rn = q.match(a4+b4*x)
if rn and rn[b4] != 0:
rn = {'b':rn[a4],'m':rn[b4]}
matching_hints["2nd_linear_airy"] = rn
if order > 0:
# Any ODE that can be solved with a substitution and
# repeated integration e.g.:
# `d^2/dx^2(y) + x*d/dx(y) = constant
#f'(x) must be finite for this to work
r = _nth_order_reducible_match(reduced_eq, func)
if r:
matching_hints['nth_order_reducible'] = r
# nth order linear ODE
# a_n(x)y^(n) + ... + a_1(x)y' + a_0(x)y = F(x) = b
r = _nth_linear_match(reduced_eq, func, order)
# Constant coefficient case (a_i is constant for all i)
if r and not any(r[i].has(x) for i in r if i >= 0):
# Inhomogeneous case: F(x) is not identically 0
if r[-1]:
eq_homogeneous = Add(eq,-r[-1])
undetcoeff = _undetermined_coefficients_match(r[-1], x, func, eq_homogeneous)
s = "nth_linear_constant_coeff_variation_of_parameters"
matching_hints[s] = r
matching_hints[s + "_Integral"] = r
if undetcoeff['test']:
r['trialset'] = undetcoeff['trialset']
matching_hints[
"nth_linear_constant_coeff_undetermined_coefficients"
] = r
# Homogeneous case: F(x) is identically 0
else:
matching_hints["nth_linear_constant_coeff_homogeneous"] = r
# nth order Euler equation a_n*x**n*y^(n) + ... + a_1*x*y' + a_0*y = F(x)
#In case of Homogeneous euler equation F(x) = 0
def _test_term(coeff, order):
r"""
Linear Euler ODEs have the form K*x**order*diff(y(x),x,order) = F(x),
where K is independent of x and y(x), order>= 0.
So we need to check that for each term, coeff == K*x**order from
some K. We have a few cases, since coeff may have several
different types.
"""
if order < 0:
raise ValueError("order should be greater than 0")
if coeff == 0:
return True
if order == 0:
if x in coeff.free_symbols:
return False
return True
if coeff.is_Mul:
if coeff.has(f(x)):
return False
return x**order in coeff.args
elif coeff.is_Pow:
return coeff.as_base_exp() == (x, order)
elif order == 1:
return x == coeff
return False
# Find coefficient for highest derivative, multiply coefficients to
# bring the equation into Euler form if possible
r_rescaled = None
if r is not None:
coeff = r[order]
factor = x**order / coeff
r_rescaled = {i: factor*r[i] for i in r if i != 'trialset'}
# XXX: Mixing up the trialset with the coefficients is error-prone.
# These should be separated as something like r['coeffs'] and
# r['trialset']
if r_rescaled and not any(not _test_term(r_rescaled[i], i) for i in
r_rescaled if i != 'trialset' and i >= 0):
if not r_rescaled[-1]:
matching_hints["nth_linear_euler_eq_homogeneous"] = r_rescaled
else:
matching_hints["nth_linear_euler_eq_nonhomogeneous_variation_of_parameters"] = r_rescaled
matching_hints["nth_linear_euler_eq_nonhomogeneous_variation_of_parameters_Integral"] = r_rescaled
e, re = posify(r_rescaled[-1].subs(x, exp(x)))
undetcoeff = _undetermined_coefficients_match(e.subs(re), x)
if undetcoeff['test']:
r_rescaled['trialset'] = undetcoeff['trialset']
matching_hints["nth_linear_euler_eq_nonhomogeneous_undetermined_coefficients"] = r_rescaled
# Order keys based on allhints.
retlist = [i for i in allhints if i in matching_hints]
if dict:
# Dictionaries are ordered arbitrarily, so make note of which
# hint would come first for dsolve(). Use an ordered dict in Py 3.
matching_hints["default"] = retlist[0] if retlist else None
matching_hints["ordered_hints"] = tuple(retlist)
return matching_hints
else:
return tuple(retlist)
def equivalence(max_num_pow, dem_pow):
# this function is made for checking the equivalence with 2F1 type of equation.
# max_num_pow is the value of maximum power of x in numerator
# and dem_pow is list of powers of different factor of form (a*x b).
# reference from table 1 in paper - "Non-Liouvillian solutions for second order
# linear ODEs" by L. Chan, E.S. Cheb-Terrab.
# We can extend it for 1F1 and 0F1 type also.
if max_num_pow == 2:
if dem_pow in [[2, 2], [2, 2, 2]]:
return "2F1"
elif max_num_pow == 1:
if dem_pow in [[1, 2, 2], [2, 2, 2], [1, 2], [2, 2]]:
return "2F1"
elif max_num_pow == 0:
if dem_pow in [[1, 1, 2], [2, 2], [1 ,2, 2], [1, 1], [2], [1, 2], [2, 2]]:
return "2F1"
return None
def equivalence_hypergeometric(A, B, func):
from sympy import factor
# This method for finding the equivalence is only for 2F1 type.
# We can extend it for 1F1 and 0F1 type also.
x = func.args[0]
# making given equation in normal form
I1 = factor(cancel(A.diff(x)/2 + A**2/4 - B))
# computing shifted invariant(J1) of the equation
J1 = factor(cancel(x**2*I1 + S(1)/4))
num, dem = J1.as_numer_denom()
num = powdenest(expand(num))
dem = powdenest(expand(dem))
pow_num = set()
pow_dem = set()
# this function will compute the different powers of variable(x) in J1.
# then it will help in finding value of k. k is power of x such that we can express
# J1 = x**k * J0(x**k) then all the powers in J0 become integers.
def _power_counting(num):
_pow = {0}
for val in num:
if val.has(x):
if isinstance(val, Pow) and val.as_base_exp()[0] == x:
_pow.add(val.as_base_exp()[1])
elif val == x:
_pow.add(val.as_base_exp()[1])
else:
_pow.update(_power_counting(val.args))
return _pow
pow_num = _power_counting((num, ))
pow_dem = _power_counting((dem, ))
pow_dem.update(pow_num)
_pow = pow_dem
k = gcd(_pow)
# computing I0 of the given equation
I0 = powdenest(simplify(factor(((J1/k**2) - S(1)/4)/((x**k)**2))), force=True)
I0 = factor(cancel(powdenest(I0.subs(x, x**(S(1)/k)), force=True)))
num, dem = I0.as_numer_denom()
max_num_pow = max(_power_counting((num, )))
dem_args = dem.args
sing_point = []
dem_pow = []
# calculating singular point of I0.
for arg in dem_args:
if arg.has(x):
if isinstance(arg, Pow):
# (x-a)**n
dem_pow.append(arg.as_base_exp()[1])
sing_point.append(list(roots(arg.as_base_exp()[0], x).keys())[0])
else:
# (x-a) type
dem_pow.append(arg.as_base_exp()[1])
sing_point.append(list(roots(arg, x).keys())[0])
dem_pow.sort()
# checking if equivalence is exists or not.
if equivalence(max_num_pow, dem_pow) == "2F1":
return {'I0':I0, 'k':k, 'sing_point':sing_point, 'type':"2F1"}
else:
return None
def ode_2nd_hypergeometric(eq, func, order, match):
from sympy.simplify.hyperexpand import hyperexpand
from sympy import factor
x = func.args[0]
C0, C1 = get_numbered_constants(eq, num=2)
a = match['a']
b = match['b']
c = match['c']
A = match['A']
# B = match['B']
sol = None
if match['type'] == "2F1":
if c.is_integer == False:
sol = C0*hyper([a, b], [c], x) + C1*hyper([a-c+1, b-c+1], [2-c], x)*x**(1-c)
elif c == 1:
y2 = Integral(exp(Integral((-(a+b+1)*x + c)/(x**2-x), x))/(hyperexpand(hyper([a, b], [c], x))**2), x)*hyper([a, b], [c], x)
sol = C0*hyper([a, b], [c], x) + C1*y2
elif (c-a-b).is_integer == False:
sol = C0*hyper([a, b], [1+a+b-c], 1-x) + C1*hyper([c-a, c-b], [1+c-a-b], 1-x)*(1-x)**(c-a-b)
if sol is None:
raise NotImplementedError("The given ODE " + str(eq) + " cannot be solved by"
+ " the hypergeometric method")
# applying transformation in the solution
subs = match['mobius']
dtdx = simplify(1/(subs.diff(x)))
_B = ((a + b + 1)*x - c).subs(x, subs)*dtdx
_B = factor(_B + ((x**2 -x).subs(x, subs))*(dtdx.diff(x)*dtdx))
_A = factor((x**2 - x).subs(x, subs)*(dtdx**2))
e = exp(logcombine(Integral(cancel(_B/(2*_A)), x), force=True))
sol = sol.subs(x, match['mobius'])
sol = sol.subs(x, x**match['k'])
e = e.subs(x, x**match['k'])
if not A.is_zero:
e1 = Integral(A/2, x)
e1 = exp(logcombine(e1, force=True))
sol = cancel((e/e1)*x**((-match['k']+1)/2))*sol
sol = Eq(func, sol)
return sol
sol = cancel((e)*x**((-match['k']+1)/2))*sol
sol = Eq(func, sol)
return sol
def match_2nd_2F1_hypergeometric(I, k, sing_point, func):
from sympy import factor
x = func.args[0]
a = Wild("a")
b = Wild("b")
c = Wild("c")
t = Wild("t")
s = Wild("s")
r = Wild("r")
alpha = Wild("alpha")
beta = Wild("beta")
gamma = Wild("gamma")
delta = Wild("delta")
rn = {'type':None}
# I0 of the standerd 2F1 equation.
I0 = ((a-b+1)*(a-b-1)*x**2 + 2*((1-a-b)*c + 2*a*b)*x + c*(c-2))/(4*x**2*(x-1)**2)
if sing_point != [0, 1]:
# If singular point is [0, 1] then we have standerd equation.
eqs = []
sing_eqs = [-beta/alpha, -delta/gamma, (delta-beta)/(alpha-gamma)]
# making equations for the finding the mobius transformation
for i in range(3):
if i<len(sing_point):
eqs.append(Eq(sing_eqs[i], sing_point[i]))
else:
eqs.append(Eq(1/sing_eqs[i], 0))
# solving above equations for the mobius transformation
_beta = -alpha*sing_point[0]
_delta = -gamma*sing_point[1]
_gamma = alpha
if len(sing_point) == 3:
_gamma = (_beta + sing_point[2]*alpha)/(sing_point[2] - sing_point[1])
mob = (alpha*x + beta)/(gamma*x + delta)
mob = mob.subs(beta, _beta)
mob = mob.subs(delta, _delta)
mob = mob.subs(gamma, _gamma)
mob = cancel(mob)
t = (beta - delta*x)/(gamma*x - alpha)
t = cancel(((t.subs(beta, _beta)).subs(delta, _delta)).subs(gamma, _gamma))
else:
mob = x
t = x
# applying mobius transformation in I to make it into I0.
I = I.subs(x, t)
I = I*(t.diff(x))**2
I = factor(I)
dict_I = {x**2:0, x:0, 1:0}
I0_num, I0_dem = I0.as_numer_denom()
# collecting coeff of (x**2, x), of the standerd equation.
# substituting (a-b) = s, (a+b) = r
dict_I0 = {x**2:s**2 - 1, x:(2*(1-r)*c + (r+s)*(r-s)), 1:c*(c-2)}
# collecting coeff of (x**2, x) from I0 of the given equation.
dict_I.update(collect(expand(cancel(I*I0_dem)), [x**2, x], evaluate=False))
eqs = []
# We are comparing the coeff of powers of different x, for finding the values of
# parameters of standerd equation.
for key in [x**2, x, 1]:
eqs.append(Eq(dict_I[key], dict_I0[key]))
# We can have many possible roots for the equation.
# I am selecting the root on the basis that when we have
# standard equation eq = x*(x-1)*f(x).diff(x, 2) + ((a+b+1)*x-c)*f(x).diff(x) + a*b*f(x)
# then root should be a, b, c.
_c = 1 - factor(sqrt(1+eqs[2].lhs))
if not _c.has(Symbol):
_c = min(list(roots(eqs[2], c)))
_s = factor(sqrt(eqs[0].lhs + 1))
_r = _c - factor(sqrt(_c**2 + _s**2 + eqs[1].lhs - 2*_c))
_a = (_r + _s)/2
_b = (_r - _s)/2
rn = {'a':simplify(_a), 'b':simplify(_b), 'c':simplify(_c), 'k':k, 'mobius':mob, 'type':"2F1"}
return rn
def match_2nd_hypergeometric(r, func):
x = func.args[0]
a3 = Wild('a3', exclude=[func, func.diff(x), func.diff(x, 2)])
b3 = Wild('b3', exclude=[func, func.diff(x), func.diff(x, 2)])
c3 = Wild('c3', exclude=[func, func.diff(x), func.diff(x, 2)])
A = cancel(r[b3]/r[a3])
B = cancel(r[c3]/r[a3])
d = equivalence_hypergeometric(A, B, func)
rn = None
if d:
if d['type'] == "2F1":
rn = match_2nd_2F1_hypergeometric(d['I0'], d['k'], d['sing_point'], func)
if rn is not None:
rn.update({'A':A, 'B':B})
# We can extend it for 1F1 and 0F1 type also.
return rn
def match_2nd_linear_bessel(r, func):
from sympy.polys.polytools import factor
# eq = a3*f(x).diff(x, 2) + b3*f(x).diff(x) + c3*f(x)
f = func
x = func.args[0]
df = f.diff(x)
a = Wild('a', exclude=[f,df])
b = Wild('b', exclude=[x, f,df])
a4 = Wild('a4', exclude=[x,f,df])
b4 = Wild('b4', exclude=[x,f,df])
c4 = Wild('c4', exclude=[x,f,df])
d4 = Wild('d4', exclude=[x,f,df])
a3 = Wild('a3', exclude=[f, df, f.diff(x, 2)])
b3 = Wild('b3', exclude=[f, df, f.diff(x, 2)])
c3 = Wild('c3', exclude=[f, df, f.diff(x, 2)])
# leading coeff of f(x).diff(x, 2)
coeff = factor(r[a3]).match(a4*(x-b)**b4)
if coeff:
# if coeff[b4] = 0 means constant coefficient
if coeff[b4] == 0:
return None
point = coeff[b]
else:
return None
if point:
r[a3] = simplify(r[a3].subs(x, x+point))
r[b3] = simplify(r[b3].subs(x, x+point))
r[c3] = simplify(r[c3].subs(x, x+point))
# making a3 in the form of x**2
r[a3] = cancel(r[a3]/(coeff[a4]*(x)**(-2+coeff[b4])))
r[b3] = cancel(r[b3]/(coeff[a4]*(x)**(-2+coeff[b4])))
r[c3] = cancel(r[c3]/(coeff[a4]*(x)**(-2+coeff[b4])))
# checking if b3 is of form c*(x-b)
coeff1 = factor(r[b3]).match(a4*(x))
if coeff1 is None:
return None
# c3 maybe of very complex form so I am simply checking (a - b) form
# if yes later I will match with the standerd form of bessel in a and b
# a, b are wild variable defined above.
_coeff2 = r[c3].match(a - b)
if _coeff2 is None:
return None
# matching with standerd form for c3
coeff2 = factor(_coeff2[a]).match(c4**2*(x)**(2*a4))
if coeff2 is None:
return None
if _coeff2[b] == 0:
coeff2[d4] = 0
else:
coeff2[d4] = factor(_coeff2[b]).match(d4**2)[d4]
rn = {'n':coeff2[d4], 'a4':coeff2[c4], 'd4':coeff2[a4]}
rn['c4'] = coeff1[a4]
rn['b4'] = point
return rn
def classify_sysode(eq, funcs=None, **kwargs):
r"""
Returns a dictionary of parameter names and values that define the system
of ordinary differential equations in ``eq``.
The parameters are further used in
:py:meth:`~sympy.solvers.ode.dsolve` for solving that system.
Some parameter names and values are:
'is_linear' (boolean), which tells whether the given system is linear.
Note that "linear" here refers to the operator: terms such as ``x*diff(x,t)`` are
nonlinear, whereas terms like ``sin(t)*diff(x,t)`` are still linear operators.
'func' (list) contains the :py:class:`~sympy.core.function.Function`s that
appear with a derivative in the ODE, i.e. those that we are trying to solve
the ODE for.
'order' (dict) with the maximum derivative for each element of the 'func'
parameter.
'func_coeff' (dict or Matrix) with the coefficient for each triple ``(equation number,
function, order)```. The coefficients are those subexpressions that do not
appear in 'func', and hence can be considered constant for purposes of ODE
solving. The value of this parameter can also be a Matrix if the system of ODEs are
linear first order of the form X' = AX where X is the vector of dependent variables.
Here, this function returns the coefficient matrix A.
'eq' (list) with the equations from ``eq``, sympified and transformed into
expressions (we are solving for these expressions to be zero).
'no_of_equations' (int) is the number of equations (same as ``len(eq)``).
'type_of_equation' (string) is an internal classification of the type of
ODE.
'is_constant' (boolean), which tells if the system of ODEs is constant coefficient
or not. This key is temporary addition for now and is in the match dict only when
the system of ODEs is linear first order constant coefficient homogeneous. So, this
key's value is True for now if it is available else it doesn't exist.
'is_homogeneous' (boolean), which tells if the system of ODEs is homogeneous. Like the
key 'is_constant', this key is a temporary addition and it is True since this key value
is available only when the system is linear first order constant coefficient homogeneous.
References
==========
-http://eqworld.ipmnet.ru/en/solutions/sysode/sode-toc1.htm
-A. D. Polyanin and A. V. Manzhirov, Handbook of Mathematics for Engineers and Scientists
Examples
========
>>> from sympy import Function, Eq, symbols, diff
>>> from sympy.solvers.ode.ode import classify_sysode
>>> from sympy.abc import t
>>> f, x, y = symbols('f, x, y', cls=Function)
>>> k, l, m, n = symbols('k, l, m, n', Integer=True)
>>> x1 = diff(x(t), t) ; y1 = diff(y(t), t)
>>> x2 = diff(x(t), t, t) ; y2 = diff(y(t), t, t)
>>> eq = (Eq(x1, 12*x(t) - 6*y(t)), Eq(y1, 11*x(t) + 3*y(t)))
>>> classify_sysode(eq)
{'eq': [-12*x(t) + 6*y(t) + Derivative(x(t), t), -11*x(t) - 3*y(t) + Derivative(y(t), t)], 'func': [x(t), y(t)], 'func_coeff': Matrix([
[-12, 6],
[-11, -3]]), 'is_constant': True, 'is_general': True, 'is_homogeneous': True, 'is_linear': True, 'no_of_equation': 2, 'order': {x(t): 1, y(t): 1}, 'type_of_equation': 'type1'}
>>> eq = (Eq(diff(x(t),t), 5*t*x(t) + t**2*y(t) + 2), Eq(diff(y(t),t), -t**2*x(t) + 5*t*y(t)))
>>> classify_sysode(eq)
{'eq': [-t**2*y(t) - 5*t*x(t) + Derivative(x(t), t) - 2, t**2*x(t) - 5*t*y(t) + Derivative(y(t), t)],
'func': [x(t), y(t)], 'func_coeff': {(0, x(t), 0): -5*t, (0, x(t), 1): 1, (0, y(t), 0): -t**2,
(0, y(t), 1): 0, (1, x(t), 0): t**2, (1, x(t), 1): 0, (1, y(t), 0): -5*t, (1, y(t), 1): 1},
'is_linear': True, 'no_of_equation': 2, 'order': {x(t): 1, y(t): 1}, 'type_of_equation': None}
"""
from sympy.solvers.ode.systems import neq_nth_linear_constant_coeff_match
# Sympify equations and convert iterables of equations into
# a list of equations
def _sympify(eq):
return list(map(sympify, eq if iterable(eq) else [eq]))
eq, funcs = (_sympify(w) for w in [eq, funcs])
for i, fi in enumerate(eq):
if isinstance(fi, Equality):
eq[i] = fi.lhs - fi.rhs
t = list(list(eq[0].atoms(Derivative))[0].atoms(Symbol))[0]
matching_hints = {"no_of_equation":i+1}
matching_hints['eq'] = eq
if i==0:
raise ValueError("classify_sysode() works for systems of ODEs. "
"For scalar ODEs, classify_ode should be used")
# find all the functions if not given
order = dict()
if funcs==[None]:
funcs = []
for eqs in eq:
derivs = eqs.atoms(Derivative)
func = set().union(*[d.atoms(AppliedUndef) for d in derivs])
for func_ in func:
funcs.append(func_)
temp_eqs = eq
match = neq_nth_linear_constant_coeff_match(temp_eqs, funcs, t)
if match is not None:
return match
funcs = list(set(funcs))
if len(funcs) != len(eq):
raise ValueError("Number of functions given is not equal to the number of equations %s" % funcs)
func_dict = dict()
for func in funcs:
if not order.get(func, False):
max_order = 0
for i, eqs_ in enumerate(eq):
order_ = ode_order(eqs_,func)
if max_order < order_:
max_order = order_
eq_no = i
if eq_no in func_dict:
func_dict[eq_no] = [func_dict[eq_no], func]
else:
func_dict[eq_no] = func
order[func] = max_order
funcs = [func_dict[i] for i in range(len(func_dict))]
matching_hints['func'] = funcs
for func in funcs:
if isinstance(func, list):
for func_elem in func:
if len(func_elem.args) != 1:
raise ValueError("dsolve() and classify_sysode() work with "
"functions of one variable only, not %s" % func)
else:
if func and len(func.args) != 1:
raise ValueError("dsolve() and classify_sysode() work with "
"functions of one variable only, not %s" % func)
# find the order of all equation in system of odes
matching_hints["order"] = order
# find coefficients of terms f(t), diff(f(t),t) and higher derivatives
# and similarly for other functions g(t), diff(g(t),t) in all equations.
# Here j denotes the equation number, funcs[l] denotes the function about
# which we are talking about and k denotes the order of function funcs[l]
# whose coefficient we are calculating.
def linearity_check(eqs, j, func, is_linear_):
for k in range(order[func] + 1):
func_coef[j, func, k] = collect(eqs.expand(), [diff(func, t, k)]).coeff(diff(func, t, k))
if is_linear_ == True:
if func_coef[j, func, k] == 0:
if k == 0:
coef = eqs.as_independent(func, as_Add=True)[1]
for xr in range(1, ode_order(eqs,func) + 1):
coef -= eqs.as_independent(diff(func, t, xr), as_Add=True)[1]
if coef != 0:
is_linear_ = False
else:
if eqs.as_independent(diff(func, t, k), as_Add=True)[1]:
is_linear_ = False
else:
for func_ in funcs:
if isinstance(func_, list):
for elem_func_ in func_:
dep = func_coef[j, func, k].as_independent(elem_func_, as_Add=True)[1]
if dep != 0:
is_linear_ = False
else:
dep = func_coef[j, func, k].as_independent(func_, as_Add=True)[1]
if dep != 0:
is_linear_ = False
return is_linear_
func_coef = {}
is_linear = True
for j, eqs in enumerate(eq):
for func in funcs:
if isinstance(func, list):
for func_elem in func:
is_linear = linearity_check(eqs, j, func_elem, is_linear)
else:
is_linear = linearity_check(eqs, j, func, is_linear)
matching_hints['func_coeff'] = func_coef
matching_hints['is_linear'] = is_linear
if len(set(order.values())) == 1:
order_eq = list(matching_hints['order'].values())[0]
if matching_hints['is_linear'] == True:
if matching_hints['no_of_equation'] == 2:
if order_eq == 1:
type_of_equation = check_linear_2eq_order1(eq, funcs, func_coef)
elif order_eq == 2:
type_of_equation = check_linear_2eq_order2(eq, funcs, func_coef)
# If the equation doesn't match up with any of the
# general case solvers in systems.py and the number
# of equations is greater than 2, then NotImplementedError
# should be raised.
else:
type_of_equation = None
else:
if matching_hints['no_of_equation'] == 2:
if order_eq == 1:
type_of_equation = check_nonlinear_2eq_order1(eq, funcs, func_coef)
else:
type_of_equation = None
elif matching_hints['no_of_equation'] == 3:
if order_eq == 1:
type_of_equation = check_nonlinear_3eq_order1(eq, funcs, func_coef)
else:
type_of_equation = None
else:
type_of_equation = None
else:
type_of_equation = None
matching_hints['type_of_equation'] = type_of_equation
return matching_hints
def check_linear_2eq_order1(eq, func, func_coef):
x = func[0].func
y = func[1].func
fc = func_coef
t = list(list(eq[0].atoms(Derivative))[0].atoms(Symbol))[0]
r = dict()
# for equations Eq(a1*diff(x(t),t), b1*x(t) + c1*y(t) + d1)
# and Eq(a2*diff(y(t),t), b2*x(t) + c2*y(t) + d2)
r['a1'] = fc[0,x(t),1] ; r['a2'] = fc[1,y(t),1]
r['b1'] = -fc[0,x(t),0]/fc[0,x(t),1] ; r['b2'] = -fc[1,x(t),0]/fc[1,y(t),1]
r['c1'] = -fc[0,y(t),0]/fc[0,x(t),1] ; r['c2'] = -fc[1,y(t),0]/fc[1,y(t),1]
forcing = [S.Zero,S.Zero]
for i in range(2):
for j in Add.make_args(eq[i]):
if not j.has(x(t), y(t)):
forcing[i] += j
if not (forcing[0].has(t) or forcing[1].has(t)):
# We can handle homogeneous case and simple constant forcings
r['d1'] = forcing[0]
r['d2'] = forcing[1]
else:
# Issue #9244: nonhomogeneous linear systems are not supported
return None
# Conditions to check for type 6 whose equations are Eq(diff(x(t),t), f(t)*x(t) + g(t)*y(t)) and
# Eq(diff(y(t),t), a*[f(t) + a*h(t)]x(t) + a*[g(t) - h(t)]*y(t))
p = 0
q = 0
p1 = cancel(r['b2']/(cancel(r['b2']/r['c2']).as_numer_denom()[0]))
p2 = cancel(r['b1']/(cancel(r['b1']/r['c1']).as_numer_denom()[0]))
for n, i in enumerate([p1, p2]):
for j in Mul.make_args(collect_const(i)):
if not j.has(t):
q = j
if q and n==0:
if ((r['b2']/j - r['b1'])/(r['c1'] - r['c2']/j)) == j:
p = 1
elif q and n==1:
if ((r['b1']/j - r['b2'])/(r['c2'] - r['c1']/j)) == j:
p = 2
# End of condition for type 6
if r['d1']!=0 or r['d2']!=0:
if not r['d1'].has(t) and not r['d2'].has(t):
if all(not r[k].has(t) for k in 'a1 a2 b1 b2 c1 c2'.split()):
# Equations for type 2 are Eq(a1*diff(x(t),t),b1*x(t)+c1*y(t)+d1) and Eq(a2*diff(y(t),t),b2*x(t)+c2*y(t)+d2)
return "type2"
else:
return None
else:
if all(not r[k].has(t) for k in 'a1 a2 b1 b2 c1 c2'.split()):
# Equations for type 1 are Eq(a1*diff(x(t),t),b1*x(t)+c1*y(t)) and Eq(a2*diff(y(t),t),b2*x(t)+c2*y(t))
return "type1"
else:
r['b1'] = r['b1']/r['a1'] ; r['b2'] = r['b2']/r['a2']
r['c1'] = r['c1']/r['a1'] ; r['c2'] = r['c2']/r['a2']
if p:
return "type6"
else:
# Equations for type 7 are Eq(diff(x(t),t), f(t)*x(t) + g(t)*y(t)) and Eq(diff(y(t),t), h(t)*x(t) + p(t)*y(t))
return "type7"
def check_linear_2eq_order2(eq, func, func_coef):
x = func[0].func
y = func[1].func
fc = func_coef
t = list(list(eq[0].atoms(Derivative))[0].atoms(Symbol))[0]
r = dict()
a = Wild('a', exclude=[1/t])
b = Wild('b', exclude=[1/t**2])
u = Wild('u', exclude=[t, t**2])
v = Wild('v', exclude=[t, t**2])
w = Wild('w', exclude=[t, t**2])
p = Wild('p', exclude=[t, t**2])
r['a1'] = fc[0,x(t),2] ; r['a2'] = fc[1,y(t),2]
r['b1'] = fc[0,x(t),1] ; r['b2'] = fc[1,x(t),1]
r['c1'] = fc[0,y(t),1] ; r['c2'] = fc[1,y(t),1]
r['d1'] = fc[0,x(t),0] ; r['d2'] = fc[1,x(t),0]
r['e1'] = fc[0,y(t),0] ; r['e2'] = fc[1,y(t),0]
const = [S.Zero, S.Zero]
for i in range(2):
for j in Add.make_args(eq[i]):
if not (j.has(x(t)) or j.has(y(t))):
const[i] += j
r['f1'] = const[0]
r['f2'] = const[1]
if r['f1']!=0 or r['f2']!=0:
if all(not r[k].has(t) for k in 'a1 a2 d1 d2 e1 e2 f1 f2'.split()) \
and r['b1']==r['c1']==r['b2']==r['c2']==0:
return "type2"
elif all(not r[k].has(t) for k in 'a1 a2 b1 b2 c1 c2 d1 d2 e1 e1'.split()):
p = [S.Zero, S.Zero] ; q = [S.Zero, S.Zero]
for n, e in enumerate([r['f1'], r['f2']]):
if e.has(t):
tpart = e.as_independent(t, Mul)[1]
for i in Mul.make_args(tpart):
if i.has(exp):
b, e = i.as_base_exp()
co = e.coeff(t)
if co and not co.has(t) and co.has(I):
p[n] = 1
else:
q[n] = 1
else:
q[n] = 1
else:
q[n] = 1
if p[0]==1 and p[1]==1 and q[0]==0 and q[1]==0:
return "type4"
else:
return None
else:
return None
else:
if r['b1']==r['b2']==r['c1']==r['c2']==0 and all(not r[k].has(t) \
for k in 'a1 a2 d1 d2 e1 e2'.split()):
return "type1"
elif r['b1']==r['e1']==r['c2']==r['d2']==0 and all(not r[k].has(t) \
for k in 'a1 a2 b2 c1 d1 e2'.split()) and r['c1'] == -r['b2'] and \
r['d1'] == r['e2']:
return "type3"
elif cancel(-r['b2']/r['d2'])==t and cancel(-r['c1']/r['e1'])==t and not \
(r['d2']/r['a2']).has(t) and not (r['e1']/r['a1']).has(t) and \
r['b1']==r['d1']==r['c2']==r['e2']==0:
return "type5"
elif ((r['a1']/r['d1']).expand()).match((p*(u*t**2+v*t+w)**2).expand()) and not \
(cancel(r['a1']*r['d2']/(r['a2']*r['d1']))).has(t) and not (r['d1']/r['e1']).has(t) and not \
(r['d2']/r['e2']).has(t) and r['b1'] == r['b2'] == r['c1'] == r['c2'] == 0:
return "type10"
elif not cancel(r['d1']/r['e1']).has(t) and not cancel(r['d2']/r['e2']).has(t) and not \
cancel(r['d1']*r['a2']/(r['d2']*r['a1'])).has(t) and r['b1']==r['b2']==r['c1']==r['c2']==0:
return "type6"
elif not cancel(r['b1']/r['c1']).has(t) and not cancel(r['b2']/r['c2']).has(t) and not \
cancel(r['b1']*r['a2']/(r['b2']*r['a1'])).has(t) and r['d1']==r['d2']==r['e1']==r['e2']==0:
return "type7"
elif cancel(-r['b2']/r['d2'])==t and cancel(-r['c1']/r['e1'])==t and not \
cancel(r['e1']*r['a2']/(r['d2']*r['a1'])).has(t) and r['e1'].has(t) \
and r['b1']==r['d1']==r['c2']==r['e2']==0:
return "type8"
elif (r['b1']/r['a1']).match(a/t) and (r['b2']/r['a2']).match(a/t) and not \
(r['b1']/r['c1']).has(t) and not (r['b2']/r['c2']).has(t) and \
(r['d1']/r['a1']).match(b/t**2) and (r['d2']/r['a2']).match(b/t**2) \
and not (r['d1']/r['e1']).has(t) and not (r['d2']/r['e2']).has(t):
return "type9"
elif -r['b1']/r['d1']==-r['c1']/r['e1']==-r['b2']/r['d2']==-r['c2']/r['e2']==t:
return "type11"
else:
return None
def check_nonlinear_2eq_order1(eq, func, func_coef):
t = list(list(eq[0].atoms(Derivative))[0].atoms(Symbol))[0]
f = Wild('f')
g = Wild('g')
u, v = symbols('u, v', cls=Dummy)
def check_type(x, y):
r1 = eq[0].match(t*diff(x(t),t) - x(t) + f)
r2 = eq[1].match(t*diff(y(t),t) - y(t) + g)
if not (r1 and r2):
r1 = eq[0].match(diff(x(t),t) - x(t)/t + f/t)
r2 = eq[1].match(diff(y(t),t) - y(t)/t + g/t)
if not (r1 and r2):
r1 = (-eq[0]).match(t*diff(x(t),t) - x(t) + f)
r2 = (-eq[1]).match(t*diff(y(t),t) - y(t) + g)
if not (r1 and r2):
r1 = (-eq[0]).match(diff(x(t),t) - x(t)/t + f/t)
r2 = (-eq[1]).match(diff(y(t),t) - y(t)/t + g/t)
if r1 and r2 and not (r1[f].subs(diff(x(t),t),u).subs(diff(y(t),t),v).has(t) \
or r2[g].subs(diff(x(t),t),u).subs(diff(y(t),t),v).has(t)):
return 'type5'
else:
return None
for func_ in func:
if isinstance(func_, list):
x = func[0][0].func
y = func[0][1].func
eq_type = check_type(x, y)
if not eq_type:
eq_type = check_type(y, x)
return eq_type
x = func[0].func
y = func[1].func
fc = func_coef
n = Wild('n', exclude=[x(t),y(t)])
f1 = Wild('f1', exclude=[v,t])
f2 = Wild('f2', exclude=[v,t])
g1 = Wild('g1', exclude=[u,t])
g2 = Wild('g2', exclude=[u,t])
for i in range(2):
eqs = 0
for terms in Add.make_args(eq[i]):
eqs += terms/fc[i,func[i],1]
eq[i] = eqs
r = eq[0].match(diff(x(t),t) - x(t)**n*f)
if r:
g = (diff(y(t),t) - eq[1])/r[f]
if r and not (g.has(x(t)) or g.subs(y(t),v).has(t) or r[f].subs(x(t),u).subs(y(t),v).has(t)):
return 'type1'
r = eq[0].match(diff(x(t),t) - exp(n*x(t))*f)
if r:
g = (diff(y(t),t) - eq[1])/r[f]
if r and not (g.has(x(t)) or g.subs(y(t),v).has(t) or r[f].subs(x(t),u).subs(y(t),v).has(t)):
return 'type2'
g = Wild('g')
r1 = eq[0].match(diff(x(t),t) - f)
r2 = eq[1].match(diff(y(t),t) - g)
if r1 and r2 and not (r1[f].subs(x(t),u).subs(y(t),v).has(t) or \
r2[g].subs(x(t),u).subs(y(t),v).has(t)):
return 'type3'
r1 = eq[0].match(diff(x(t),t) - f)
r2 = eq[1].match(diff(y(t),t) - g)
num, den = (
(r1[f].subs(x(t),u).subs(y(t),v))/
(r2[g].subs(x(t),u).subs(y(t),v))).as_numer_denom()
R1 = num.match(f1*g1)
R2 = den.match(f2*g2)
# phi = (r1[f].subs(x(t),u).subs(y(t),v))/num
if R1 and R2:
return 'type4'
return None
def check_nonlinear_2eq_order2(eq, func, func_coef):
return None
def check_nonlinear_3eq_order1(eq, func, func_coef):
x = func[0].func
y = func[1].func
z = func[2].func
fc = func_coef
t = list(list(eq[0].atoms(Derivative))[0].atoms(Symbol))[0]
u, v, w = symbols('u, v, w', cls=Dummy)
a = Wild('a', exclude=[x(t), y(t), z(t), t])
b = Wild('b', exclude=[x(t), y(t), z(t), t])
c = Wild('c', exclude=[x(t), y(t), z(t), t])
f = Wild('f')
F1 = Wild('F1')
F2 = Wild('F2')
F3 = Wild('F3')
for i in range(3):
eqs = 0
for terms in Add.make_args(eq[i]):
eqs += terms/fc[i,func[i],1]
eq[i] = eqs
r1 = eq[0].match(diff(x(t),t) - a*y(t)*z(t))
r2 = eq[1].match(diff(y(t),t) - b*z(t)*x(t))
r3 = eq[2].match(diff(z(t),t) - c*x(t)*y(t))
if r1 and r2 and r3:
num1, den1 = r1[a].as_numer_denom()
num2, den2 = r2[b].as_numer_denom()
num3, den3 = r3[c].as_numer_denom()
if solve([num1*u-den1*(v-w), num2*v-den2*(w-u), num3*w-den3*(u-v)],[u, v]):
return 'type1'
r = eq[0].match(diff(x(t),t) - y(t)*z(t)*f)
if r:
r1 = collect_const(r[f]).match(a*f)
r2 = ((diff(y(t),t) - eq[1])/r1[f]).match(b*z(t)*x(t))
r3 = ((diff(z(t),t) - eq[2])/r1[f]).match(c*x(t)*y(t))
if r1 and r2 and r3:
num1, den1 = r1[a].as_numer_denom()
num2, den2 = r2[b].as_numer_denom()
num3, den3 = r3[c].as_numer_denom()
if solve([num1*u-den1*(v-w), num2*v-den2*(w-u), num3*w-den3*(u-v)],[u, v]):
return 'type2'
r = eq[0].match(diff(x(t),t) - (F2-F3))
if r:
r1 = collect_const(r[F2]).match(c*F2)
r1.update(collect_const(r[F3]).match(b*F3))
if r1:
if eq[1].has(r1[F2]) and not eq[1].has(r1[F3]):
r1[F2], r1[F3] = r1[F3], r1[F2]
r1[c], r1[b] = -r1[b], -r1[c]
r2 = eq[1].match(diff(y(t),t) - a*r1[F3] + r1[c]*F1)
if r2:
r3 = (eq[2] == diff(z(t),t) - r1[b]*r2[F1] + r2[a]*r1[F2])
if r1 and r2 and r3:
return 'type3'
r = eq[0].match(diff(x(t),t) - z(t)*F2 + y(t)*F3)
if r:
r1 = collect_const(r[F2]).match(c*F2)
r1.update(collect_const(r[F3]).match(b*F3))
if r1:
if eq[1].has(r1[F2]) and not eq[1].has(r1[F3]):
r1[F2], r1[F3] = r1[F3], r1[F2]
r1[c], r1[b] = -r1[b], -r1[c]
r2 = (diff(y(t),t) - eq[1]).match(a*x(t)*r1[F3] - r1[c]*z(t)*F1)
if r2:
r3 = (diff(z(t),t) - eq[2] == r1[b]*y(t)*r2[F1] - r2[a]*x(t)*r1[F2])
if r1 and r2 and r3:
return 'type4'
r = (diff(x(t),t) - eq[0]).match(x(t)*(F2 - F3))
if r:
r1 = collect_const(r[F2]).match(c*F2)
r1.update(collect_const(r[F3]).match(b*F3))
if r1:
if eq[1].has(r1[F2]) and not eq[1].has(r1[F3]):
r1[F2], r1[F3] = r1[F3], r1[F2]
r1[c], r1[b] = -r1[b], -r1[c]
r2 = (diff(y(t),t) - eq[1]).match(y(t)*(a*r1[F3] - r1[c]*F1))
if r2:
r3 = (diff(z(t),t) - eq[2] == z(t)*(r1[b]*r2[F1] - r2[a]*r1[F2]))
if r1 and r2 and r3:
return 'type5'
return None
def check_nonlinear_3eq_order2(eq, func, func_coef):
return None
@vectorize(0)
def odesimp(ode, eq, func, hint):
r"""
Simplifies solutions of ODEs, including trying to solve for ``func`` and
running :py:meth:`~sympy.solvers.ode.constantsimp`.
It may use knowledge of the type of solution that the hint returns to
apply additional simplifications.
It also attempts to integrate any :py:class:`~sympy.integrals.integrals.Integral`\s
in the expression, if the hint is not an ``_Integral`` hint.
This function should have no effect on expressions returned by
:py:meth:`~sympy.solvers.ode.dsolve`, as
:py:meth:`~sympy.solvers.ode.dsolve` already calls
:py:meth:`~sympy.solvers.ode.ode.odesimp`, but the individual hint functions
do not call :py:meth:`~sympy.solvers.ode.ode.odesimp` (because the
:py:meth:`~sympy.solvers.ode.dsolve` wrapper does). Therefore, this
function is designed for mainly internal use.
Examples
========
>>> from sympy import sin, symbols, dsolve, pprint, Function
>>> from sympy.solvers.ode.ode import odesimp
>>> x , u2, C1= symbols('x,u2,C1')
>>> f = Function('f')
>>> eq = dsolve(x*f(x).diff(x) - f(x) - x*sin(f(x)/x), f(x),
... hint='1st_homogeneous_coeff_subs_indep_div_dep_Integral',
... simplify=False)
>>> pprint(eq, wrap_line=False)
x
----
f(x)
/
|
| / 1 \
| -|u2 + -------|
| | /1 \|
| | sin|--||
| \ \u2//
log(f(x)) = log(C1) + | ---------------- d(u2)
| 2
| u2
|
/
>>> pprint(odesimp(eq, f(x), 1, {C1},
... hint='1st_homogeneous_coeff_subs_indep_div_dep'
... )) #doctest: +SKIP
x
--------- = C1
/f(x)\
tan|----|
\2*x /
"""
x = func.args[0]
f = func.func
C1 = get_numbered_constants(eq, num=1)
constants = eq.free_symbols - ode.free_symbols
# First, integrate if the hint allows it.
eq = _handle_Integral(eq, func, hint)
if hint.startswith("nth_linear_euler_eq_nonhomogeneous"):
eq = simplify(eq)
if not isinstance(eq, Equality):
raise TypeError("eq should be an instance of Equality")
# Second, clean up the arbitrary constants.
# Right now, nth linear hints can put as many as 2*order constants in an
# expression. If that number grows with another hint, the third argument
# here should be raised accordingly, or constantsimp() rewritten to handle
# an arbitrary number of constants.
eq = constantsimp(eq, constants)
# Lastly, now that we have cleaned up the expression, try solving for func.
# When CRootOf is implemented in solve(), we will want to return a CRootOf
# every time instead of an Equality.
# Get the f(x) on the left if possible.
if eq.rhs == func and not eq.lhs.has(func):
eq = [Eq(eq.rhs, eq.lhs)]
# make sure we are working with lists of solutions in simplified form.
if eq.lhs == func and not eq.rhs.has(func):
# The solution is already solved
eq = [eq]
# special simplification of the rhs
if hint.startswith("nth_linear_constant_coeff"):
# Collect terms to make the solution look nice.
# This is also necessary for constantsimp to remove unnecessary
# terms from the particular solution from variation of parameters
#
# Collect is not behaving reliably here. The results for
# some linear constant-coefficient equations with repeated
# roots do not properly simplify all constants sometimes.
# 'collectterms' gives different orders sometimes, and results
# differ in collect based on that order. The
# sort-reverse trick fixes things, but may fail in the
# future. In addition, collect is splitting exponentials with
# rational powers for no reason. We have to do a match
# to fix this using Wilds.
#
# XXX: This global collectterms hack should be removed.
global collectterms
collectterms.sort(key=default_sort_key)
collectterms.reverse()
assert len(eq) == 1 and eq[0].lhs == f(x)
sol = eq[0].rhs
sol = expand_mul(sol)
for i, reroot, imroot in collectterms:
sol = collect(sol, x**i*exp(reroot*x)*sin(abs(imroot)*x))
sol = collect(sol, x**i*exp(reroot*x)*cos(imroot*x))
for i, reroot, imroot in collectterms:
sol = collect(sol, x**i*exp(reroot*x))
del collectterms
# Collect is splitting exponentials with rational powers for
# no reason. We call powsimp to fix.
sol = powsimp(sol)
eq[0] = Eq(f(x), sol)
else:
# The solution is not solved, so try to solve it
try:
floats = any(i.is_Float for i in eq.atoms(Number))
eqsol = solve(eq, func, force=True, rational=False if floats else None)
if not eqsol:
raise NotImplementedError
except (NotImplementedError, PolynomialError):
eq = [eq]
else:
def _expand(expr):
numer, denom = expr.as_numer_denom()
if denom.is_Add:
return expr
else:
return powsimp(expr.expand(), combine='exp', deep=True)
# XXX: the rest of odesimp() expects each ``t`` to be in a
# specific normal form: rational expression with numerator
# expanded, but with combined exponential functions (at
# least in this setup all tests pass).
eq = [Eq(f(x), _expand(t)) for t in eqsol]
# special simplification of the lhs.
if hint.startswith("1st_homogeneous_coeff"):
for j, eqi in enumerate(eq):
newi = logcombine(eqi, force=True)
if isinstance(newi.lhs, log) and newi.rhs == 0:
newi = Eq(newi.lhs.args[0]/C1, C1)
eq[j] = newi
# We cleaned up the constants before solving to help the solve engine with
# a simpler expression, but the solved expression could have introduced
# things like -C1, so rerun constantsimp() one last time before returning.
for i, eqi in enumerate(eq):
eq[i] = constantsimp(eqi, constants)
eq[i] = constant_renumber(eq[i], ode.free_symbols)
# If there is only 1 solution, return it;
# otherwise return the list of solutions.
if len(eq) == 1:
eq = eq[0]
return eq
def ode_sol_simplicity(sol, func, trysolving=True):
r"""
Returns an extended integer representing how simple a solution to an ODE
is.
The following things are considered, in order from most simple to least:
- ``sol`` is solved for ``func``.
- ``sol`` is not solved for ``func``, but can be if passed to solve (e.g.,
a solution returned by ``dsolve(ode, func, simplify=False``).
- If ``sol`` is not solved for ``func``, then base the result on the
length of ``sol``, as computed by ``len(str(sol))``.
- If ``sol`` has any unevaluated :py:class:`~sympy.integrals.integrals.Integral`\s,
this will automatically be considered less simple than any of the above.
This function returns an integer such that if solution A is simpler than
solution B by above metric, then ``ode_sol_simplicity(sola, func) <
ode_sol_simplicity(solb, func)``.
Currently, the following are the numbers returned, but if the heuristic is
ever improved, this may change. Only the ordering is guaranteed.
+----------------------------------------------+-------------------+
| Simplicity | Return |
+==============================================+===================+
| ``sol`` solved for ``func`` | ``-2`` |
+----------------------------------------------+-------------------+
| ``sol`` not solved for ``func`` but can be | ``-1`` |
+----------------------------------------------+-------------------+
| ``sol`` is not solved nor solvable for | ``len(str(sol))`` |
| ``func`` | |
+----------------------------------------------+-------------------+
| ``sol`` contains an | ``oo`` |
| :obj:`~sympy.integrals.integrals.Integral` | |
+----------------------------------------------+-------------------+
``oo`` here means the SymPy infinity, which should compare greater than
any integer.
If you already know :py:meth:`~sympy.solvers.solvers.solve` cannot solve
``sol``, you can use ``trysolving=False`` to skip that step, which is the
only potentially slow step. For example,
:py:meth:`~sympy.solvers.ode.dsolve` with the ``simplify=False`` flag
should do this.
If ``sol`` is a list of solutions, if the worst solution in the list
returns ``oo`` it returns that, otherwise it returns ``len(str(sol))``,
that is, the length of the string representation of the whole list.
Examples
========
This function is designed to be passed to ``min`` as the key argument,
such as ``min(listofsolutions, key=lambda i: ode_sol_simplicity(i,
f(x)))``.
>>> from sympy import symbols, Function, Eq, tan, Integral
>>> from sympy.solvers.ode.ode import ode_sol_simplicity
>>> x, C1, C2 = symbols('x, C1, C2')
>>> f = Function('f')
>>> ode_sol_simplicity(Eq(f(x), C1*x**2), f(x))
-2
>>> ode_sol_simplicity(Eq(x**2 + f(x), C1), f(x))
-1
>>> ode_sol_simplicity(Eq(f(x), C1*Integral(2*x, x)), f(x))
oo
>>> eq1 = Eq(f(x)/tan(f(x)/(2*x)), C1)
>>> eq2 = Eq(f(x)/tan(f(x)/(2*x) + f(x)), C2)
>>> [ode_sol_simplicity(eq, f(x)) for eq in [eq1, eq2]]
[28, 35]
>>> min([eq1, eq2], key=lambda i: ode_sol_simplicity(i, f(x)))
Eq(f(x)/tan(f(x)/(2*x)), C1)
"""
# TODO: if two solutions are solved for f(x), we still want to be
# able to get the simpler of the two
# See the docstring for the coercion rules. We check easier (faster)
# things here first, to save time.
if iterable(sol):
# See if there are Integrals
for i in sol:
if ode_sol_simplicity(i, func, trysolving=trysolving) == oo:
return oo
return len(str(sol))
if sol.has(Integral):
return oo
# Next, try to solve for func. This code will change slightly when CRootOf
# is implemented in solve(). Probably a CRootOf solution should fall
# somewhere between a normal solution and an unsolvable expression.
# First, see if they are already solved
if sol.lhs == func and not sol.rhs.has(func) or \
sol.rhs == func and not sol.lhs.has(func):
return -2
# We are not so lucky, try solving manually
if trysolving:
try:
sols = solve(sol, func)
if not sols:
raise NotImplementedError
except NotImplementedError:
pass
else:
return -1
# Finally, a naive computation based on the length of the string version
# of the expression. This may favor combined fractions because they
# will not have duplicate denominators, and may slightly favor expressions
# with fewer additions and subtractions, as those are separated by spaces
# by the printer.
# Additional ideas for simplicity heuristics are welcome, like maybe
# checking if a equation has a larger domain, or if constantsimp has
# introduced arbitrary constants numbered higher than the order of a
# given ODE that sol is a solution of.
return len(str(sol))
def _get_constant_subexpressions(expr, Cs):
Cs = set(Cs)
Ces = []
def _recursive_walk(expr):
expr_syms = expr.free_symbols
if expr_syms and expr_syms.issubset(Cs):
Ces.append(expr)
else:
if expr.func == exp:
expr = expr.expand(mul=True)
if expr.func in (Add, Mul):
d = sift(expr.args, lambda i : i.free_symbols.issubset(Cs))
if len(d[True]) > 1:
x = expr.func(*d[True])
if not x.is_number:
Ces.append(x)
elif isinstance(expr, Integral):
if expr.free_symbols.issubset(Cs) and \
all(len(x) == 3 for x in expr.limits):
Ces.append(expr)
for i in expr.args:
_recursive_walk(i)
return
_recursive_walk(expr)
return Ces
def __remove_linear_redundancies(expr, Cs):
cnts = {i: expr.count(i) for i in Cs}
Cs = [i for i in Cs if cnts[i] > 0]
def _linear(expr):
if isinstance(expr, Add):
xs = [i for i in Cs if expr.count(i)==cnts[i] \
and 0 == expr.diff(i, 2)]
d = {}
for x in xs:
y = expr.diff(x)
if y not in d:
d[y]=[]
d[y].append(x)
for y in d:
if len(d[y]) > 1:
d[y].sort(key=str)
for x in d[y][1:]:
expr = expr.subs(x, 0)
return expr
def _recursive_walk(expr):
if len(expr.args) != 0:
expr = expr.func(*[_recursive_walk(i) for i in expr.args])
expr = _linear(expr)
return expr
if isinstance(expr, Equality):
lhs, rhs = [_recursive_walk(i) for i in expr.args]
f = lambda i: isinstance(i, Number) or i in Cs
if isinstance(lhs, Symbol) and lhs in Cs:
rhs, lhs = lhs, rhs
if lhs.func in (Add, Symbol) and rhs.func in (Add, Symbol):
dlhs = sift([lhs] if isinstance(lhs, AtomicExpr) else lhs.args, f)
drhs = sift([rhs] if isinstance(rhs, AtomicExpr) else rhs.args, f)
for i in [True, False]:
for hs in [dlhs, drhs]:
if i not in hs:
hs[i] = [0]
# this calculation can be simplified
lhs = Add(*dlhs[False]) - Add(*drhs[False])
rhs = Add(*drhs[True]) - Add(*dlhs[True])
elif lhs.func in (Mul, Symbol) and rhs.func in (Mul, Symbol):
dlhs = sift([lhs] if isinstance(lhs, AtomicExpr) else lhs.args, f)
if True in dlhs:
if False not in dlhs:
dlhs[False] = [1]
lhs = Mul(*dlhs[False])
rhs = rhs/Mul(*dlhs[True])
return Eq(lhs, rhs)
else:
return _recursive_walk(expr)
@vectorize(0)
def constantsimp(expr, constants):
r"""
Simplifies an expression with arbitrary constants in it.
This function is written specifically to work with
:py:meth:`~sympy.solvers.ode.dsolve`, and is not intended for general use.
Simplification is done by "absorbing" the arbitrary constants into other
arbitrary constants, numbers, and symbols that they are not independent
of.
The symbols must all have the same name with numbers after it, for
example, ``C1``, ``C2``, ``C3``. The ``symbolname`` here would be
'``C``', the ``startnumber`` would be 1, and the ``endnumber`` would be 3.
If the arbitrary constants are independent of the variable ``x``, then the
independent symbol would be ``x``. There is no need to specify the
dependent function, such as ``f(x)``, because it already has the
independent symbol, ``x``, in it.
Because terms are "absorbed" into arbitrary constants and because
constants are renumbered after simplifying, the arbitrary constants in
expr are not necessarily equal to the ones of the same name in the
returned result.
If two or more arbitrary constants are added, multiplied, or raised to the
power of each other, they are first absorbed together into a single
arbitrary constant. Then the new constant is combined into other terms if
necessary.
Absorption of constants is done with limited assistance:
1. terms of :py:class:`~sympy.core.add.Add`\s are collected to try join
constants so `e^x (C_1 \cos(x) + C_2 \cos(x))` will simplify to `e^x
C_1 \cos(x)`;
2. powers with exponents that are :py:class:`~sympy.core.add.Add`\s are
expanded so `e^{C_1 + x}` will be simplified to `C_1 e^x`.
Use :py:meth:`~sympy.solvers.ode.ode.constant_renumber` to renumber constants
after simplification or else arbitrary numbers on constants may appear,
e.g. `C_1 + C_3 x`.
In rare cases, a single constant can be "simplified" into two constants.
Every differential equation solution should have as many arbitrary
constants as the order of the differential equation. The result here will
be technically correct, but it may, for example, have `C_1` and `C_2` in
an expression, when `C_1` is actually equal to `C_2`. Use your discretion
in such situations, and also take advantage of the ability to use hints in
:py:meth:`~sympy.solvers.ode.dsolve`.
Examples
========
>>> from sympy import symbols
>>> from sympy.solvers.ode.ode import constantsimp
>>> C1, C2, C3, x, y = symbols('C1, C2, C3, x, y')
>>> constantsimp(2*C1*x, {C1, C2, C3})
C1*x
>>> constantsimp(C1 + 2 + x, {C1, C2, C3})
C1 + x
>>> constantsimp(C1*C2 + 2 + C2 + C3*x, {C1, C2, C3})
C1 + C3*x
"""
# This function works recursively. The idea is that, for Mul,
# Add, Pow, and Function, if the class has a constant in it, then
# we can simplify it, which we do by recursing down and
# simplifying up. Otherwise, we can skip that part of the
# expression.
Cs = constants
orig_expr = expr
constant_subexprs = _get_constant_subexpressions(expr, Cs)
for xe in constant_subexprs:
xes = list(xe.free_symbols)
if not xes:
continue
if all([expr.count(c) == xe.count(c) for c in xes]):
xes.sort(key=str)
expr = expr.subs(xe, xes[0])
# try to perform common sub-expression elimination of constant terms
try:
commons, rexpr = cse(expr)
commons.reverse()
rexpr = rexpr[0]
for s in commons:
cs = list(s[1].atoms(Symbol))
if len(cs) == 1 and cs[0] in Cs and \
cs[0] not in rexpr.atoms(Symbol) and \
not any(cs[0] in ex for ex in commons if ex != s):
rexpr = rexpr.subs(s[0], cs[0])
else:
rexpr = rexpr.subs(*s)
expr = rexpr
except IndexError:
pass
expr = __remove_linear_redundancies(expr, Cs)
def _conditional_term_factoring(expr):
new_expr = terms_gcd(expr, clear=False, deep=True, expand=False)
# we do not want to factor exponentials, so handle this separately
if new_expr.is_Mul:
infac = False
asfac = False
for m in new_expr.args:
if isinstance(m, exp):
asfac = True
elif m.is_Add:
infac = any(isinstance(fi, exp) for t in m.args
for fi in Mul.make_args(t))
if asfac and infac:
new_expr = expr
break
return new_expr
expr = _conditional_term_factoring(expr)
# call recursively if more simplification is possible
if orig_expr != expr:
return constantsimp(expr, Cs)
return expr
def constant_renumber(expr, variables=None, newconstants=None):
r"""
Renumber arbitrary constants in ``expr`` to use the symbol names as given
in ``newconstants``. In the process, this reorders expression terms in a
standard way.
If ``newconstants`` is not provided then the new constant names will be
``C1``, ``C2`` etc. Otherwise ``newconstants`` should be an iterable
giving the new symbols to use for the constants in order.
The ``variables`` argument is a list of non-constant symbols. All other
free symbols found in ``expr`` are assumed to be constants and will be
renumbered. If ``variables`` is not given then any numbered symbol
beginning with ``C`` (e.g. ``C1``) is assumed to be a constant.
Symbols are renumbered based on ``.sort_key()``, so they should be
numbered roughly in the order that they appear in the final, printed
expression. Note that this ordering is based in part on hashes, so it can
produce different results on different machines.
The structure of this function is very similar to that of
:py:meth:`~sympy.solvers.ode.constantsimp`.
Examples
========
>>> from sympy import symbols
>>> from sympy.solvers.ode.ode import constant_renumber
>>> x, C1, C2, C3 = symbols('x,C1:4')
>>> expr = C3 + C2*x + C1*x**2
>>> expr
C1*x**2 + C2*x + C3
>>> constant_renumber(expr)
C1 + C2*x + C3*x**2
The ``variables`` argument specifies which are constants so that the
other symbols will not be renumbered:
>>> constant_renumber(expr, [C1, x])
C1*x**2 + C2 + C3*x
The ``newconstants`` argument is used to specify what symbols to use when
replacing the constants:
>>> constant_renumber(expr, [x], newconstants=symbols('E1:4'))
E1 + E2*x + E3*x**2
"""
if type(expr) in (set, list, tuple):
renumbered = [constant_renumber(e, variables, newconstants) for e in expr]
return type(expr)(renumbered)
# Symbols in solution but not ODE are constants
if variables is not None:
variables = set(variables)
constantsymbols = list(expr.free_symbols - variables)
# Any Cn is a constant...
else:
variables = set()
isconstant = lambda s: s.startswith('C') and s[1:].isdigit()
constantsymbols = [sym for sym in expr.free_symbols if isconstant(sym.name)]
# Find new constants checking that they aren't already in the ODE
if newconstants is None:
iter_constants = numbered_symbols(start=1, prefix='C', exclude=variables)
else:
iter_constants = (sym for sym in newconstants if sym not in variables)
# XXX: This global newstartnumber hack should be removed
global newstartnumber
newstartnumber = 1
endnumber = len(constantsymbols)
constants_found = [None]*(endnumber + 2)
# make a mapping to send all constantsymbols to S.One and use
# that to make sure that term ordering is not dependent on
# the indexed value of C
C_1 = [(ci, S.One) for ci in constantsymbols]
sort_key=lambda arg: default_sort_key(arg.subs(C_1))
def _constant_renumber(expr):
r"""
We need to have an internal recursive function so that
newstartnumber maintains its values throughout recursive calls.
"""
# FIXME: Use nonlocal here when support for Py2 is dropped:
global newstartnumber
if isinstance(expr, Equality):
return Eq(
_constant_renumber(expr.lhs),
_constant_renumber(expr.rhs))
if type(expr) not in (Mul, Add, Pow) and not expr.is_Function and \
not expr.has(*constantsymbols):
# Base case, as above. Hope there aren't constants inside
# of some other class, because they won't be renumbered.
return expr
elif expr.is_Piecewise:
return expr
elif expr in constantsymbols:
if expr not in constants_found:
constants_found[newstartnumber] = expr
newstartnumber += 1
return expr
elif expr.is_Function or expr.is_Pow or isinstance(expr, Tuple):
return expr.func(
*[_constant_renumber(x) for x in expr.args])
else:
sortedargs = list(expr.args)
sortedargs.sort(key=sort_key)
return expr.func(*[_constant_renumber(x) for x in sortedargs])
expr = _constant_renumber(expr)
# Don't renumber symbols present in the ODE.
constants_found = [c for c in constants_found if c not in variables]
# Renumbering happens here
expr = expr.subs(zip(constants_found[1:], iter_constants), simultaneous=True)
return expr
def _handle_Integral(expr, func, hint):
r"""
Converts a solution with Integrals in it into an actual solution.
For most hints, this simply runs ``expr.doit()``.
"""
# XXX: This global y hack should be removed
global y
x = func.args[0]
f = func.func
if hint == "1st_exact":
sol = (expr.doit()).subs(y, f(x))
del y
elif hint == "1st_exact_Integral":
sol = Eq(Subs(expr.lhs, y, f(x)), expr.rhs)
del y
elif hint == "nth_linear_constant_coeff_homogeneous":
sol = expr
elif not hint.endswith("_Integral"):
sol = expr.doit()
else:
sol = expr
return sol
# FIXME: replace the general solution in the docstring with
# dsolve(equation, hint='1st_exact_Integral'). You will need to be able
# to have assumptions on P and Q that dP/dy = dQ/dx.
def ode_1st_exact(eq, func, order, match):
r"""
Solves 1st order exact ordinary differential equations.
A 1st order differential equation is called exact if it is the total
differential of a function. That is, the differential equation
.. math:: P(x, y) \,\partial{}x + Q(x, y) \,\partial{}y = 0
is exact if there is some function `F(x, y)` such that `P(x, y) =
\partial{}F/\partial{}x` and `Q(x, y) = \partial{}F/\partial{}y`. It can
be shown that a necessary and sufficient condition for a first order ODE
to be exact is that `\partial{}P/\partial{}y = \partial{}Q/\partial{}x`.
Then, the solution will be as given below::
>>> from sympy import Function, Eq, Integral, symbols, pprint
>>> x, y, t, x0, y0, C1= symbols('x,y,t,x0,y0,C1')
>>> P, Q, F= map(Function, ['P', 'Q', 'F'])
>>> pprint(Eq(Eq(F(x, y), Integral(P(t, y), (t, x0, x)) +
... Integral(Q(x0, t), (t, y0, y))), C1))
x y
/ /
| |
F(x, y) = | P(t, y) dt + | Q(x0, t) dt = C1
| |
/ /
x0 y0
Where the first partials of `P` and `Q` exist and are continuous in a
simply connected region.
A note: SymPy currently has no way to represent inert substitution on an
expression, so the hint ``1st_exact_Integral`` will return an integral
with `dy`. This is supposed to represent the function that you are
solving for.
Examples
========
>>> from sympy import Function, dsolve, cos, sin
>>> from sympy.abc import x
>>> f = Function('f')
>>> dsolve(cos(f(x)) - (x*sin(f(x)) - f(x)**2)*f(x).diff(x),
... f(x), hint='1st_exact')
Eq(x*cos(f(x)) + f(x)**3/3, C1)
References
==========
- https://en.wikipedia.org/wiki/Exact_differential_equation
- M. Tenenbaum & H. Pollard, "Ordinary Differential Equations",
Dover 1963, pp. 73
# indirect doctest
"""
x = func.args[0]
r = match # d+e*diff(f(x),x)
e = r[r['e']]
d = r[r['d']]
# XXX: This global y hack should be removed
global y # This is the only way to pass dummy y to _handle_Integral
y = r['y']
C1 = get_numbered_constants(eq, num=1)
# Refer Joel Moses, "Symbolic Integration - The Stormy Decade",
# Communications of the ACM, Volume 14, Number 8, August 1971, pp. 558
# which gives the method to solve an exact differential equation.
sol = Integral(d, x) + Integral((e - (Integral(d, x).diff(y))), y)
return Eq(sol, C1)
def ode_1st_homogeneous_coeff_best(eq, func, order, match):
r"""
Returns the best solution to an ODE from the two hints
``1st_homogeneous_coeff_subs_dep_div_indep`` and
``1st_homogeneous_coeff_subs_indep_div_dep``.
This is as determined by :py:meth:`~sympy.solvers.ode.ode.ode_sol_simplicity`.
See the
:py:meth:`~sympy.solvers.ode.ode.ode_1st_homogeneous_coeff_subs_indep_div_dep`
and
:py:meth:`~sympy.solvers.ode.ode.ode_1st_homogeneous_coeff_subs_dep_div_indep`
docstrings for more information on these hints. Note that there is no
``ode_1st_homogeneous_coeff_best_Integral`` hint.
Examples
========
>>> from sympy import Function, dsolve, pprint
>>> from sympy.abc import x
>>> f = Function('f')
>>> pprint(dsolve(2*x*f(x) + (x**2 + f(x)**2)*f(x).diff(x), f(x),
... hint='1st_homogeneous_coeff_best', simplify=False))
/ 2 \
| 3*x |
log|----- + 1|
| 2 |
\f (x) /
log(f(x)) = log(C1) - --------------
3
References
==========
- https://en.wikipedia.org/wiki/Homogeneous_differential_equation
- M. Tenenbaum & H. Pollard, "Ordinary Differential Equations",
Dover 1963, pp. 59
# indirect doctest
"""
# There are two substitutions that solve the equation, u1=y/x and u2=x/y
# They produce different integrals, so try them both and see which
# one is easier.
sol1 = ode_1st_homogeneous_coeff_subs_indep_div_dep(eq,
func, order, match)
sol2 = ode_1st_homogeneous_coeff_subs_dep_div_indep(eq,
func, order, match)
simplify = match.get('simplify', True)
if simplify:
# why is odesimp called here? Should it be at the usual spot?
sol1 = odesimp(eq, sol1, func, "1st_homogeneous_coeff_subs_indep_div_dep")
sol2 = odesimp(eq, sol2, func, "1st_homogeneous_coeff_subs_dep_div_indep")
return min([sol1, sol2], key=lambda x: ode_sol_simplicity(x, func,
trysolving=not simplify))
def ode_1st_homogeneous_coeff_subs_dep_div_indep(eq, func, order, match):
r"""
Solves a 1st order differential equation with homogeneous coefficients
using the substitution `u_1 = \frac{\text{<dependent
variable>}}{\text{<independent variable>}}`.
This is a differential equation
.. math:: P(x, y) + Q(x, y) dy/dx = 0
such that `P` and `Q` are homogeneous and of the same order. A function
`F(x, y)` is homogeneous of order `n` if `F(x t, y t) = t^n F(x, y)`.
Equivalently, `F(x, y)` can be rewritten as `G(y/x)` or `H(x/y)`. See
also the docstring of :py:meth:`~sympy.solvers.ode.homogeneous_order`.
If the coefficients `P` and `Q` in the differential equation above are
homogeneous functions of the same order, then it can be shown that the
substitution `y = u_1 x` (i.e. `u_1 = y/x`) will turn the differential
equation into an equation separable in the variables `x` and `u`. If
`h(u_1)` is the function that results from making the substitution `u_1 =
f(x)/x` on `P(x, f(x))` and `g(u_2)` is the function that results from the
substitution on `Q(x, f(x))` in the differential equation `P(x, f(x)) +
Q(x, f(x)) f'(x) = 0`, then the general solution is::
>>> from sympy import Function, dsolve, pprint
>>> from sympy.abc import x
>>> f, g, h = map(Function, ['f', 'g', 'h'])
>>> genform = g(f(x)/x) + h(f(x)/x)*f(x).diff(x)
>>> pprint(genform)
/f(x)\ /f(x)\ d
g|----| + h|----|*--(f(x))
\ x / \ x / dx
>>> pprint(dsolve(genform, f(x),
... hint='1st_homogeneous_coeff_subs_dep_div_indep_Integral'))
f(x)
----
x
/
|
| -h(u1)
log(x) = C1 + | ---------------- d(u1)
| u1*h(u1) + g(u1)
|
/
Where `u_1 h(u_1) + g(u_1) \ne 0` and `x \ne 0`.
See also the docstrings of
:py:meth:`~sympy.solvers.ode.ode.ode_1st_homogeneous_coeff_best` and
:py:meth:`~sympy.solvers.ode.ode.ode_1st_homogeneous_coeff_subs_indep_div_dep`.
Examples
========
>>> from sympy import Function, dsolve
>>> from sympy.abc import x
>>> f = Function('f')
>>> pprint(dsolve(2*x*f(x) + (x**2 + f(x)**2)*f(x).diff(x), f(x),
... hint='1st_homogeneous_coeff_subs_dep_div_indep', simplify=False))
/ 3 \
|3*f(x) f (x)|
log|------ + -----|
| x 3 |
\ x /
log(x) = log(C1) - -------------------
3
References
==========
- https://en.wikipedia.org/wiki/Homogeneous_differential_equation
- M. Tenenbaum & H. Pollard, "Ordinary Differential Equations",
Dover 1963, pp. 59
# indirect doctest
"""
x = func.args[0]
f = func.func
u = Dummy('u')
u1 = Dummy('u1') # u1 == f(x)/x
r = match # d+e*diff(f(x),x)
C1 = get_numbered_constants(eq, num=1)
xarg = match.get('xarg', 0)
yarg = match.get('yarg', 0)
int = Integral(
(-r[r['e']]/(r[r['d']] + u1*r[r['e']])).subs({x: 1, r['y']: u1}),
(u1, None, f(x)/x))
sol = logcombine(Eq(log(x), int + log(C1)), force=True)
sol = sol.subs(f(x), u).subs(((u, u - yarg), (x, x - xarg), (u, f(x))))
return sol
def ode_1st_homogeneous_coeff_subs_indep_div_dep(eq, func, order, match):
r"""
Solves a 1st order differential equation with homogeneous coefficients
using the substitution `u_2 = \frac{\text{<independent
variable>}}{\text{<dependent variable>}}`.
This is a differential equation
.. math:: P(x, y) + Q(x, y) dy/dx = 0
such that `P` and `Q` are homogeneous and of the same order. A function
`F(x, y)` is homogeneous of order `n` if `F(x t, y t) = t^n F(x, y)`.
Equivalently, `F(x, y)` can be rewritten as `G(y/x)` or `H(x/y)`. See
also the docstring of :py:meth:`~sympy.solvers.ode.homogeneous_order`.
If the coefficients `P` and `Q` in the differential equation above are
homogeneous functions of the same order, then it can be shown that the
substitution `x = u_2 y` (i.e. `u_2 = x/y`) will turn the differential
equation into an equation separable in the variables `y` and `u_2`. If
`h(u_2)` is the function that results from making the substitution `u_2 =
x/f(x)` on `P(x, f(x))` and `g(u_2)` is the function that results from the
substitution on `Q(x, f(x))` in the differential equation `P(x, f(x)) +
Q(x, f(x)) f'(x) = 0`, then the general solution is:
>>> from sympy import Function, dsolve, pprint
>>> from sympy.abc import x
>>> f, g, h = map(Function, ['f', 'g', 'h'])
>>> genform = g(x/f(x)) + h(x/f(x))*f(x).diff(x)
>>> pprint(genform)
/ x \ / x \ d
g|----| + h|----|*--(f(x))
\f(x)/ \f(x)/ dx
>>> pprint(dsolve(genform, f(x),
... hint='1st_homogeneous_coeff_subs_indep_div_dep_Integral'))
x
----
f(x)
/
|
| -g(u2)
| ---------------- d(u2)
| u2*g(u2) + h(u2)
|
/
<BLANKLINE>
f(x) = C1*e
Where `u_2 g(u_2) + h(u_2) \ne 0` and `f(x) \ne 0`.
See also the docstrings of
:py:meth:`~sympy.solvers.ode.ode.ode_1st_homogeneous_coeff_best` and
:py:meth:`~sympy.solvers.ode.ode.ode_1st_homogeneous_coeff_subs_dep_div_indep`.
Examples
========
>>> from sympy import Function, pprint, dsolve
>>> from sympy.abc import x
>>> f = Function('f')
>>> pprint(dsolve(2*x*f(x) + (x**2 + f(x)**2)*f(x).diff(x), f(x),
... hint='1st_homogeneous_coeff_subs_indep_div_dep',
... simplify=False))
/ 2 \
| 3*x |
log|----- + 1|
| 2 |
\f (x) /
log(f(x)) = log(C1) - --------------
3
References
==========
- https://en.wikipedia.org/wiki/Homogeneous_differential_equation
- M. Tenenbaum & H. Pollard, "Ordinary Differential Equations",
Dover 1963, pp. 59
# indirect doctest
"""
x = func.args[0]
f = func.func
u = Dummy('u')
u2 = Dummy('u2') # u2 == x/f(x)
r = match # d+e*diff(f(x),x)
C1 = get_numbered_constants(eq, num=1)
xarg = match.get('xarg', 0) # If xarg present take xarg, else zero
yarg = match.get('yarg', 0) # If yarg present take yarg, else zero
int = Integral(
simplify(
(-r[r['d']]/(r[r['e']] + u2*r[r['d']])).subs({x: u2, r['y']: 1})),
(u2, None, x/f(x)))
sol = logcombine(Eq(log(f(x)), int + log(C1)), force=True)
sol = sol.subs(f(x), u).subs(((u, u - yarg), (x, x - xarg), (u, f(x))))
return sol
# XXX: Should this function maybe go somewhere else?
def homogeneous_order(eq, *symbols):
r"""
Returns the order `n` if `g` is homogeneous and ``None`` if it is not
homogeneous.
Determines if a function is homogeneous and if so of what order. A
function `f(x, y, \cdots)` is homogeneous of order `n` if `f(t x, t y,
\cdots) = t^n f(x, y, \cdots)`.
If the function is of two variables, `F(x, y)`, then `f` being homogeneous
of any order is equivalent to being able to rewrite `F(x, y)` as `G(x/y)`
or `H(y/x)`. This fact is used to solve 1st order ordinary differential
equations whose coefficients are homogeneous of the same order (see the
docstrings of
:py:meth:`~sympy.solvers.ode.ode.ode_1st_homogeneous_coeff_subs_dep_div_indep` and
:py:meth:`~sympy.solvers.ode.ode.ode_1st_homogeneous_coeff_subs_indep_div_dep`).
Symbols can be functions, but every argument of the function must be a
symbol, and the arguments of the function that appear in the expression
must match those given in the list of symbols. If a declared function
appears with different arguments than given in the list of symbols,
``None`` is returned.
Examples
========
>>> from sympy import Function, homogeneous_order, sqrt
>>> from sympy.abc import x, y
>>> f = Function('f')
>>> homogeneous_order(f(x), f(x)) is None
True
>>> homogeneous_order(f(x,y), f(y, x), x, y) is None
True
>>> homogeneous_order(f(x), f(x), x)
1
>>> homogeneous_order(x**2*f(x)/sqrt(x**2+f(x)**2), x, f(x))
2
>>> homogeneous_order(x**2+f(x), x, f(x)) is None
True
"""
if not symbols:
raise ValueError("homogeneous_order: no symbols were given.")
symset = set(symbols)
eq = sympify(eq)
# The following are not supported
if eq.has(Order, Derivative):
return None
# These are all constants
if (eq.is_Number or
eq.is_NumberSymbol or
eq.is_number
):
return S.Zero
# Replace all functions with dummy variables
dum = numbered_symbols(prefix='d', cls=Dummy)
newsyms = set()
for i in [j for j in symset if getattr(j, 'is_Function')]:
iargs = set(i.args)
if iargs.difference(symset):
return None
else:
dummyvar = next(dum)
eq = eq.subs(i, dummyvar)
symset.remove(i)
newsyms.add(dummyvar)
symset.update(newsyms)
if not eq.free_symbols & symset:
return None
# assuming order of a nested function can only be equal to zero
if isinstance(eq, Function):
return None if homogeneous_order(
eq.args[0], *tuple(symset)) != 0 else S.Zero
# make the replacement of x with x*t and see if t can be factored out
t = Dummy('t', positive=True) # It is sufficient that t > 0
eqs = separatevars(eq.subs([(i, t*i) for i in symset]), [t], dict=True)[t]
if eqs is S.One:
return S.Zero # there was no term with only t
i, d = eqs.as_independent(t, as_Add=False)
b, e = d.as_base_exp()
if b == t:
return e
def ode_Liouville(eq, func, order, match):
r"""
Solves 2nd order Liouville differential equations.
The general form of a Liouville ODE is
.. math:: \frac{d^2 y}{dx^2} + g(y) \left(\!
\frac{dy}{dx}\!\right)^2 + h(x)
\frac{dy}{dx}\text{.}
The general solution is:
>>> from sympy import Function, dsolve, Eq, pprint, diff
>>> from sympy.abc import x
>>> f, g, h = map(Function, ['f', 'g', 'h'])
>>> genform = Eq(diff(f(x),x,x) + g(f(x))*diff(f(x),x)**2 +
... h(x)*diff(f(x),x), 0)
>>> pprint(genform)
2 2
/d \ d d
g(f(x))*|--(f(x))| + h(x)*--(f(x)) + ---(f(x)) = 0
\dx / dx 2
dx
>>> pprint(dsolve(genform, f(x), hint='Liouville_Integral'))
f(x)
/ /
| |
| / | /
| | | |
| - | h(x) dx | | g(y) dy
| | | |
| / | /
C1 + C2* | e dx + | e dy = 0
| |
/ /
Examples
========
>>> from sympy import Function, dsolve, Eq, pprint
>>> from sympy.abc import x
>>> f = Function('f')
>>> pprint(dsolve(diff(f(x), x, x) + diff(f(x), x)**2/f(x) +
... diff(f(x), x)/x, f(x), hint='Liouville'))
________________ ________________
[f(x) = -\/ C1 + C2*log(x) , f(x) = \/ C1 + C2*log(x) ]
References
==========
- Goldstein and Braun, "Advanced Methods for the Solution of Differential
Equations", pp. 98
- http://www.maplesoft.com/support/help/Maple/view.aspx?path=odeadvisor/Liouville
# indirect doctest
"""
# Liouville ODE:
# f(x).diff(x, 2) + g(f(x))*(f(x).diff(x, 2))**2 + h(x)*f(x).diff(x)
# See Goldstein and Braun, "Advanced Methods for the Solution of
# Differential Equations", pg. 98, as well as
# http://www.maplesoft.com/support/help/view.aspx?path=odeadvisor/Liouville
x = func.args[0]
f = func.func
r = match # f(x).diff(x, 2) + g*f(x).diff(x)**2 + h*f(x).diff(x)
y = r['y']
C1, C2 = get_numbered_constants(eq, num=2)
int = Integral(exp(Integral(r['g'], y)), (y, None, f(x)))
sol = Eq(int + C1*Integral(exp(-Integral(r['h'], x)), x) + C2, 0)
return sol
def ode_2nd_power_series_ordinary(eq, func, order, match):
r"""
Gives a power series solution to a second order homogeneous differential
equation with polynomial coefficients at an ordinary point. A homogeneous
differential equation is of the form
.. math :: P(x)\frac{d^2y}{dx^2} + Q(x)\frac{dy}{dx} + R(x) = 0
For simplicity it is assumed that `P(x)`, `Q(x)` and `R(x)` are polynomials,
it is sufficient that `\frac{Q(x)}{P(x)}` and `\frac{R(x)}{P(x)}` exists at
`x_{0}`. A recurrence relation is obtained by substituting `y` as `\sum_{n=0}^\infty a_{n}x^{n}`,
in the differential equation, and equating the nth term. Using this relation
various terms can be generated.
Examples
========
>>> from sympy import dsolve, Function, pprint
>>> from sympy.abc import x
>>> f = Function("f")
>>> eq = f(x).diff(x, 2) + f(x)
>>> pprint(dsolve(eq, hint='2nd_power_series_ordinary'))
/ 4 2 \ / 2\
|x x | | x | / 6\
f(x) = C2*|-- - -- + 1| + C1*x*|1 - --| + O\x /
\24 2 / \ 6 /
References
==========
- http://tutorial.math.lamar.edu/Classes/DE/SeriesSolutions.aspx
- George E. Simmons, "Differential Equations with Applications and
Historical Notes", p.p 176 - 184
"""
x = func.args[0]
f = func.func
C0, C1 = get_numbered_constants(eq, num=2)
n = Dummy("n", integer=True)
s = Wild("s")
k = Wild("k", exclude=[x])
x0 = match.get('x0')
terms = match.get('terms', 5)
p = match[match['a3']]
q = match[match['b3']]
r = match[match['c3']]
seriesdict = {}
recurr = Function("r")
# Generating the recurrence relation which works this way:
# for the second order term the summation begins at n = 2. The coefficients
# p is multiplied with an*(n - 1)*(n - 2)*x**n-2 and a substitution is made such that
# the exponent of x becomes n.
# For example, if p is x, then the second degree recurrence term is
# an*(n - 1)*(n - 2)*x**n-1, substituting (n - 1) as n, it transforms to
# an+1*n*(n - 1)*x**n.
# A similar process is done with the first order and zeroth order term.
coefflist = [(recurr(n), r), (n*recurr(n), q), (n*(n - 1)*recurr(n), p)]
for index, coeff in enumerate(coefflist):
if coeff[1]:
f2 = powsimp(expand((coeff[1]*(x - x0)**(n - index)).subs(x, x + x0)))
if f2.is_Add:
addargs = f2.args
else:
addargs = [f2]
for arg in addargs:
powm = arg.match(s*x**k)
term = coeff[0]*powm[s]
if not powm[k].is_Symbol:
term = term.subs(n, n - powm[k].as_independent(n)[0])
startind = powm[k].subs(n, index)
# Seeing if the startterm can be reduced further.
# If it vanishes for n lesser than startind, it is
# equal to summation from n.
if startind:
for i in reversed(range(startind)):
if not term.subs(n, i):
seriesdict[term] = i
else:
seriesdict[term] = i + 1
break
else:
seriesdict[term] = S.Zero
# Stripping of terms so that the sum starts with the same number.
teq = S.Zero
suminit = seriesdict.values()
rkeys = seriesdict.keys()
req = Add(*rkeys)
if any(suminit):
maxval = max(suminit)
for term in seriesdict:
val = seriesdict[term]
if val != maxval:
for i in range(val, maxval):
teq += term.subs(n, val)
finaldict = {}
if teq:
fargs = teq.atoms(AppliedUndef)
if len(fargs) == 1:
finaldict[fargs.pop()] = 0
else:
maxf = max(fargs, key = lambda x: x.args[0])
sol = solve(teq, maxf)
if isinstance(sol, list):
sol = sol[0]
finaldict[maxf] = sol
# Finding the recurrence relation in terms of the largest term.
fargs = req.atoms(AppliedUndef)
maxf = max(fargs, key = lambda x: x.args[0])
minf = min(fargs, key = lambda x: x.args[0])
if minf.args[0].is_Symbol:
startiter = 0
else:
startiter = -minf.args[0].as_independent(n)[0]
lhs = maxf
rhs = solve(req, maxf)
if isinstance(rhs, list):
rhs = rhs[0]
# Checking how many values are already present
tcounter = len([t for t in finaldict.values() if t])
for _ in range(tcounter, terms - 3): # Assuming c0 and c1 to be arbitrary
check = rhs.subs(n, startiter)
nlhs = lhs.subs(n, startiter)
nrhs = check.subs(finaldict)
finaldict[nlhs] = nrhs
startiter += 1
# Post processing
series = C0 + C1*(x - x0)
for term in finaldict:
if finaldict[term]:
fact = term.args[0]
series += (finaldict[term].subs([(recurr(0), C0), (recurr(1), C1)])*(
x - x0)**fact)
series = collect(expand_mul(series), [C0, C1]) + Order(x**terms)
return Eq(f(x), series)
def ode_2nd_linear_airy(eq, func, order, match):
r"""
Gives solution of the Airy differential equation
.. math :: \frac{d^2y}{dx^2} + (a + b x) y(x) = 0
in terms of Airy special functions airyai and airybi.
Examples
========
>>> from sympy import dsolve, Function
>>> from sympy.abc import x
>>> f = Function("f")
>>> eq = f(x).diff(x, 2) - x*f(x)
>>> dsolve(eq)
Eq(f(x), C1*airyai(x) + C2*airybi(x))
"""
x = func.args[0]
f = func.func
C0, C1 = get_numbered_constants(eq, num=2)
b = match['b']
m = match['m']
if m.is_positive:
arg = - b/cbrt(m)**2 - cbrt(m)*x
elif m.is_negative:
arg = - b/cbrt(-m)**2 + cbrt(-m)*x
else:
arg = - b/cbrt(-m)**2 + cbrt(-m)*x
return Eq(f(x), C0*airyai(arg) + C1*airybi(arg))
def ode_2nd_power_series_regular(eq, func, order, match):
r"""
Gives a power series solution to a second order homogeneous differential
equation with polynomial coefficients at a regular point. A second order
homogeneous differential equation is of the form
.. math :: P(x)\frac{d^2y}{dx^2} + Q(x)\frac{dy}{dx} + R(x) = 0
A point is said to regular singular at `x0` if `x - x0\frac{Q(x)}{P(x)}`
and `(x - x0)^{2}\frac{R(x)}{P(x)}` are analytic at `x0`. For simplicity
`P(x)`, `Q(x)` and `R(x)` are assumed to be polynomials. The algorithm for
finding the power series solutions is:
1. Try expressing `(x - x0)P(x)` and `((x - x0)^{2})Q(x)` as power series
solutions about x0. Find `p0` and `q0` which are the constants of the
power series expansions.
2. Solve the indicial equation `f(m) = m(m - 1) + m*p0 + q0`, to obtain the
roots `m1` and `m2` of the indicial equation.
3. If `m1 - m2` is a non integer there exists two series solutions. If
`m1 = m2`, there exists only one solution. If `m1 - m2` is an integer,
then the existence of one solution is confirmed. The other solution may
or may not exist.
The power series solution is of the form `x^{m}\sum_{n=0}^\infty a_{n}x^{n}`. The
coefficients are determined by the following recurrence relation.
`a_{n} = -\frac{\sum_{k=0}^{n-1} q_{n-k} + (m + k)p_{n-k}}{f(m + n)}`. For the case
in which `m1 - m2` is an integer, it can be seen from the recurrence relation
that for the lower root `m`, when `n` equals the difference of both the
roots, the denominator becomes zero. So if the numerator is not equal to zero,
a second series solution exists.
Examples
========
>>> from sympy import dsolve, Function, pprint
>>> from sympy.abc import x
>>> f = Function("f")
>>> eq = x*(f(x).diff(x, 2)) + 2*(f(x).diff(x)) + x*f(x)
>>> pprint(dsolve(eq, hint='2nd_power_series_regular'))
/ 6 4 2 \
| x x x |
/ 4 2 \ C1*|- --- + -- - -- + 1|
| x x | \ 720 24 2 / / 6\
f(x) = C2*|--- - -- + 1| + ------------------------ + O\x /
\120 6 / x
References
==========
- George E. Simmons, "Differential Equations with Applications and
Historical Notes", p.p 176 - 184
"""
x = func.args[0]
f = func.func
C0, C1 = get_numbered_constants(eq, num=2)
m = Dummy("m") # for solving the indicial equation
x0 = match.get('x0')
terms = match.get('terms', 5)
p = match['p']
q = match['q']
# Generating the indicial equation
indicial = []
for term in [p, q]:
if not term.has(x):
indicial.append(term)
else:
term = series(term, n=1, x0=x0)
if isinstance(term, Order):
indicial.append(S.Zero)
else:
for arg in term.args:
if not arg.has(x):
indicial.append(arg)
break
p0, q0 = indicial
sollist = solve(m*(m - 1) + m*p0 + q0, m)
if sollist and isinstance(sollist, list) and all(
[sol.is_real for sol in sollist]):
serdict1 = {}
serdict2 = {}
if len(sollist) == 1:
# Only one series solution exists in this case.
m1 = m2 = sollist.pop()
if terms-m1-1 <= 0:
return Eq(f(x), Order(terms))
serdict1 = _frobenius(terms-m1-1, m1, p0, q0, p, q, x0, x, C0)
else:
m1 = sollist[0]
m2 = sollist[1]
if m1 < m2:
m1, m2 = m2, m1
# Irrespective of whether m1 - m2 is an integer or not, one
# Frobenius series solution exists.
serdict1 = _frobenius(terms-m1-1, m1, p0, q0, p, q, x0, x, C0)
if not (m1 - m2).is_integer:
# Second frobenius series solution exists.
serdict2 = _frobenius(terms-m2-1, m2, p0, q0, p, q, x0, x, C1)
else:
# Check if second frobenius series solution exists.
serdict2 = _frobenius(terms-m2-1, m2, p0, q0, p, q, x0, x, C1, check=m1)
if serdict1:
finalseries1 = C0
for key in serdict1:
power = int(key.name[1:])
finalseries1 += serdict1[key]*(x - x0)**power
finalseries1 = (x - x0)**m1*finalseries1
finalseries2 = S.Zero
if serdict2:
for key in serdict2:
power = int(key.name[1:])
finalseries2 += serdict2[key]*(x - x0)**power
finalseries2 += C1
finalseries2 = (x - x0)**m2*finalseries2
return Eq(f(x), collect(finalseries1 + finalseries2,
[C0, C1]) + Order(x**terms))
def ode_2nd_linear_bessel(eq, func, order, match):
r"""
Gives solution of the Bessel differential equation
.. math :: x^2 \frac{d^2y}{dx^2} + x \frac{dy}{dx} y(x) + (x^2-n^2) y(x)
if n is integer then the solution is of the form Eq(f(x), C0 besselj(n,x)
+ C1 bessely(n,x)) as both the solutions are linearly independent else if
n is a fraction then the solution is of the form Eq(f(x), C0 besselj(n,x)
+ C1 besselj(-n,x)) which can also transform into Eq(f(x), C0 besselj(n,x)
+ C1 bessely(n,x)).
Examples
========
>>> from sympy.abc import x
>>> from sympy import Symbol
>>> v = Symbol('v', positive=True)
>>> from sympy.solvers.ode import dsolve
>>> from sympy import Function
>>> f = Function('f')
>>> y = f(x)
>>> genform = x**2*y.diff(x, 2) + x*y.diff(x) + (x**2 - v**2)*y
>>> dsolve(genform)
Eq(f(x), C1*besselj(v, x) + C2*bessely(v, x))
References
==========
https://www.math24.net/bessel-differential-equation/
"""
x = func.args[0]
f = func.func
C0, C1 = get_numbered_constants(eq, num=2)
n = match['n']
a4 = match['a4']
c4 = match['c4']
d4 = match['d4']
b4 = match['b4']
n = sqrt(n**2 + Rational(1, 4)*(c4 - 1)**2)
return Eq(f(x), ((x**(Rational(1-c4,2)))*(C0*besselj(n/d4,a4*x**d4/d4)
+ C1*bessely(n/d4,a4*x**d4/d4))).subs(x, x-b4))
def _frobenius(n, m, p0, q0, p, q, x0, x, c, check=None):
r"""
Returns a dict with keys as coefficients and values as their values in terms of C0
"""
n = int(n)
# In cases where m1 - m2 is not an integer
m2 = check
d = Dummy("d")
numsyms = numbered_symbols("C", start=0)
numsyms = [next(numsyms) for i in range(n + 1)]
serlist = []
for ser in [p, q]:
# Order term not present
if ser.is_polynomial(x) and Poly(ser, x).degree() <= n:
if x0:
ser = ser.subs(x, x + x0)
dict_ = Poly(ser, x).as_dict()
# Order term present
else:
tseries = series(ser, x=x0, n=n+1)
# Removing order
dict_ = Poly(list(ordered(tseries.args))[: -1], x).as_dict()
# Fill in with zeros, if coefficients are zero.
for i in range(n + 1):
if (i,) not in dict_:
dict_[(i,)] = S.Zero
serlist.append(dict_)
pseries = serlist[0]
qseries = serlist[1]
indicial = d*(d - 1) + d*p0 + q0
frobdict = {}
for i in range(1, n + 1):
num = c*(m*pseries[(i,)] + qseries[(i,)])
for j in range(1, i):
sym = Symbol("C" + str(j))
num += frobdict[sym]*((m + j)*pseries[(i - j,)] + qseries[(i - j,)])
# Checking for cases when m1 - m2 is an integer. If num equals zero
# then a second Frobenius series solution cannot be found. If num is not zero
# then set constant as zero and proceed.
if m2 is not None and i == m2 - m:
if num:
return False
else:
frobdict[numsyms[i]] = S.Zero
else:
frobdict[numsyms[i]] = -num/(indicial.subs(d, m+i))
return frobdict
def _nth_order_reducible_match(eq, func):
r"""
Matches any differential equation that can be rewritten with a smaller
order. Only derivatives of ``func`` alone, wrt a single variable,
are considered, and only in them should ``func`` appear.
"""
# ODE only handles functions of 1 variable so this affirms that state
assert len(func.args) == 1
x = func.args[0]
vc = [d.variable_count[0] for d in eq.atoms(Derivative)
if d.expr == func and len(d.variable_count) == 1]
ords = [c for v, c in vc if v == x]
if len(ords) < 2:
return
smallest = min(ords)
# make sure func does not appear outside of derivatives
D = Dummy()
if eq.subs(func.diff(x, smallest), D).has(func):
return
return {'n': smallest}
def ode_nth_order_reducible(eq, func, order, match):
r"""
Solves ODEs that only involve derivatives of the dependent variable using
a substitution of the form `f^n(x) = g(x)`.
For example any second order ODE of the form `f''(x) = h(f'(x), x)` can be
transformed into a pair of 1st order ODEs `g'(x) = h(g(x), x)` and
`f'(x) = g(x)`. Usually the 1st order ODE for `g` is easier to solve. If
that gives an explicit solution for `g` then `f` is found simply by
integration.
Examples
========
>>> from sympy import Function, dsolve, Eq
>>> from sympy.abc import x
>>> f = Function('f')
>>> eq = Eq(x*f(x).diff(x)**2 + f(x).diff(x, 2), 0)
>>> dsolve(eq, f(x), hint='nth_order_reducible')
... # doctest: +NORMALIZE_WHITESPACE
Eq(f(x), C1 - sqrt(-1/C2)*log(-C2*sqrt(-1/C2) + x) + sqrt(-1/C2)*log(C2*sqrt(-1/C2) + x))
"""
x = func.args[0]
f = func.func
n = match['n']
# get a unique function name for g
names = [a.name for a in eq.atoms(AppliedUndef)]
while True:
name = Dummy().name
if name not in names:
g = Function(name)
break
w = f(x).diff(x, n)
geq = eq.subs(w, g(x))
gsol = dsolve(geq, g(x))
if not isinstance(gsol, list):
gsol = [gsol]
# Might be multiple solutions to the reduced ODE:
fsol = []
for gsoli in gsol:
fsoli = dsolve(gsoli.subs(g(x), w), f(x)) # or do integration n times
fsol.append(fsoli)
if len(fsol) == 1:
fsol = fsol[0]
return fsol
def _remove_redundant_solutions(eq, solns, order, var):
r"""
Remove redundant solutions from the set of solutions.
This function is needed because otherwise dsolve can return
redundant solutions. As an example consider:
eq = Eq((f(x).diff(x, 2))*f(x).diff(x), 0)
There are two ways to find solutions to eq. The first is to solve f(x).diff(x, 2) = 0
leading to solution f(x)=C1 + C2*x. The second is to solve the equation f(x).diff(x) = 0
leading to the solution f(x) = C1. In this particular case we then see
that the second solution is a special case of the first and we don't
want to return it.
This does not always happen. If we have
eq = Eq((f(x)**2-4)*(f(x).diff(x)-4), 0)
then we get the algebraic solution f(x) = [-2, 2] and the integral solution
f(x) = x + C1 and in this case the two solutions are not equivalent wrt
initial conditions so both should be returned.
"""
def is_special_case_of(soln1, soln2):
return _is_special_case_of(soln1, soln2, eq, order, var)
unique_solns = []
for soln1 in solns:
for soln2 in unique_solns[:]:
if is_special_case_of(soln1, soln2):
break
elif is_special_case_of(soln2, soln1):
unique_solns.remove(soln2)
else:
unique_solns.append(soln1)
return unique_solns
def _is_special_case_of(soln1, soln2, eq, order, var):
r"""
True if soln1 is found to be a special case of soln2 wrt some value of the
constants that appear in soln2. False otherwise.
"""
# The solutions returned by dsolve may be given explicitly or implicitly.
# We will equate the sol1=(soln1.rhs - soln1.lhs), sol2=(soln2.rhs - soln2.lhs)
# of the two solutions.
#
# Since this is supposed to hold for all x it also holds for derivatives.
# For an order n ode we should be able to differentiate
# each solution n times to get n+1 equations.
#
# We then try to solve those n+1 equations for the integrations constants
# in sol2. If we can find a solution that doesn't depend on x then it
# means that some value of the constants in sol1 is a special case of
# sol2 corresponding to a particular choice of the integration constants.
# In case the solution is in implicit form we subtract the sides
soln1 = soln1.rhs - soln1.lhs
soln2 = soln2.rhs - soln2.lhs
# Work for the series solution
if soln1.has(Order) and soln2.has(Order):
if soln1.getO() == soln2.getO():
soln1 = soln1.removeO()
soln2 = soln2.removeO()
else:
return False
elif soln1.has(Order) or soln2.has(Order):
return False
constants1 = soln1.free_symbols.difference(eq.free_symbols)
constants2 = soln2.free_symbols.difference(eq.free_symbols)
constants1_new = get_numbered_constants(Tuple(soln1, soln2), len(constants1))
if len(constants1) == 1:
constants1_new = {constants1_new}
for c_old, c_new in zip(constants1, constants1_new):
soln1 = soln1.subs(c_old, c_new)
# n equations for sol1 = sol2, sol1'=sol2', ...
lhs = soln1
rhs = soln2
eqns = [Eq(lhs, rhs)]
for n in range(1, order):
lhs = lhs.diff(var)
rhs = rhs.diff(var)
eq = Eq(lhs, rhs)
eqns.append(eq)
# BooleanTrue/False awkwardly show up for trivial equations
if any(isinstance(eq, BooleanFalse) for eq in eqns):
return False
eqns = [eq for eq in eqns if not isinstance(eq, BooleanTrue)]
try:
constant_solns = solve(eqns, constants2)
except NotImplementedError:
return False
# Sometimes returns a dict and sometimes a list of dicts
if isinstance(constant_solns, dict):
constant_solns = [constant_solns]
# after solving the issue 17418, maybe we don't need the following checksol code.
for constant_soln in constant_solns:
for eq in eqns:
eq=eq.rhs-eq.lhs
if checksol(eq, constant_soln) is not True:
return False
# If any solution gives all constants as expressions that don't depend on
# x then there exists constants for soln2 that give soln1
for constant_soln in constant_solns:
if not any(c.has(var) for c in constant_soln.values()):
return True
return False
def _nth_linear_match(eq, func, order):
r"""
Matches a differential equation to the linear form:
.. math:: a_n(x) y^{(n)} + \cdots + a_1(x)y' + a_0(x) y + B(x) = 0
Returns a dict of order:coeff terms, where order is the order of the
derivative on each term, and coeff is the coefficient of that derivative.
The key ``-1`` holds the function `B(x)`. Returns ``None`` if the ODE is
not linear. This function assumes that ``func`` has already been checked
to be good.
Examples
========
>>> from sympy import Function, cos, sin
>>> from sympy.abc import x
>>> from sympy.solvers.ode.ode import _nth_linear_match
>>> f = Function('f')
>>> _nth_linear_match(f(x).diff(x, 3) + 2*f(x).diff(x) +
... x*f(x).diff(x, 2) + cos(x)*f(x).diff(x) + x - f(x) -
... sin(x), f(x), 3)
{-1: x - sin(x), 0: -1, 1: cos(x) + 2, 2: x, 3: 1}
>>> _nth_linear_match(f(x).diff(x, 3) + 2*f(x).diff(x) +
... x*f(x).diff(x, 2) + cos(x)*f(x).diff(x) + x - f(x) -
... sin(f(x)), f(x), 3) == None
True
"""
x = func.args[0]
one_x = {x}
terms = {i: S.Zero for i in range(-1, order + 1)}
for i in Add.make_args(eq):
if not i.has(func):
terms[-1] += i
else:
c, f = i.as_independent(func)
if (isinstance(f, Derivative)
and set(f.variables) == one_x
and f.args[0] == func):
terms[f.derivative_count] += c
elif f == func:
terms[len(f.args[1:])] += c
else:
return None
return terms
def ode_nth_linear_euler_eq_homogeneous(eq, func, order, match, returns='sol'):
r"""
Solves an `n`\th order linear homogeneous variable-coefficient
Cauchy-Euler equidimensional ordinary differential equation.
This is an equation with form `0 = a_0 f(x) + a_1 x f'(x) + a_2 x^2 f''(x)
\cdots`.
These equations can be solved in a general manner, by substituting
solutions of the form `f(x) = x^r`, and deriving a characteristic equation
for `r`. When there are repeated roots, we include extra terms of the
form `C_{r k} \ln^k(x) x^r`, where `C_{r k}` is an arbitrary integration
constant, `r` is a root of the characteristic equation, and `k` ranges
over the multiplicity of `r`. In the cases where the roots are complex,
solutions of the form `C_1 x^a \sin(b \log(x)) + C_2 x^a \cos(b \log(x))`
are returned, based on expansions with Euler's formula. The general
solution is the sum of the terms found. If SymPy cannot find exact roots
to the characteristic equation, a
:py:obj:`~.ComplexRootOf` instance will be returned
instead.
>>> from sympy import Function, dsolve
>>> from sympy.abc import x
>>> f = Function('f')
>>> dsolve(4*x**2*f(x).diff(x, 2) + f(x), f(x),
... hint='nth_linear_euler_eq_homogeneous')
... # doctest: +NORMALIZE_WHITESPACE
Eq(f(x), sqrt(x)*(C1 + C2*log(x)))
Note that because this method does not involve integration, there is no
``nth_linear_euler_eq_homogeneous_Integral`` hint.
The following is for internal use:
- ``returns = 'sol'`` returns the solution to the ODE.
- ``returns = 'list'`` returns a list of linearly independent solutions,
corresponding to the fundamental solution set, for use with non
homogeneous solution methods like variation of parameters and
undetermined coefficients. Note that, though the solutions should be
linearly independent, this function does not explicitly check that. You
can do ``assert simplify(wronskian(sollist)) != 0`` to check for linear
independence. Also, ``assert len(sollist) == order`` will need to pass.
- ``returns = 'both'``, return a dictionary ``{'sol': <solution to ODE>,
'list': <list of linearly independent solutions>}``.
Examples
========
>>> from sympy import Function, dsolve, pprint
>>> from sympy.abc import x
>>> f = Function('f')
>>> eq = f(x).diff(x, 2)*x**2 - 4*f(x).diff(x)*x + 6*f(x)
>>> pprint(dsolve(eq, f(x),
... hint='nth_linear_euler_eq_homogeneous'))
2
f(x) = x *(C1 + C2*x)
References
==========
- https://en.wikipedia.org/wiki/Cauchy%E2%80%93Euler_equation
- C. Bender & S. Orszag, "Advanced Mathematical Methods for Scientists and
Engineers", Springer 1999, pp. 12
# indirect doctest
"""
# XXX: This global collectterms hack should be removed.
global collectterms
collectterms = []
x = func.args[0]
f = func.func
r = match
# First, set up characteristic equation.
chareq, symbol = S.Zero, Dummy('x')
for i in r.keys():
if not isinstance(i, str) and i >= 0:
chareq += (r[i]*diff(x**symbol, x, i)*x**-symbol).expand()
chareq = Poly(chareq, symbol)
chareqroots = [rootof(chareq, k) for k in range(chareq.degree())]
# A generator of constants
constants = list(get_numbered_constants(eq, num=chareq.degree()*2))
constants.reverse()
# Create a dict root: multiplicity or charroots
charroots = defaultdict(int)
for root in chareqroots:
charroots[root] += 1
gsol = S.Zero
# We need keep track of terms so we can run collect() at the end.
# This is necessary for constantsimp to work properly.
ln = log
for root, multiplicity in charroots.items():
for i in range(multiplicity):
if isinstance(root, RootOf):
gsol += (x**root) * constants.pop()
if multiplicity != 1:
raise ValueError("Value should be 1")
collectterms = [(0, root, 0)] + collectterms
elif root.is_real:
gsol += ln(x)**i*(x**root) * constants.pop()
collectterms = [(i, root, 0)] + collectterms
else:
reroot = re(root)
imroot = im(root)
gsol += ln(x)**i * (x**reroot) * (
constants.pop() * sin(abs(imroot)*ln(x))
+ constants.pop() * cos(imroot*ln(x)))
# Preserve ordering (multiplicity, real part, imaginary part)
# It will be assumed implicitly when constructing
# fundamental solution sets.
collectterms = [(i, reroot, imroot)] + collectterms
if returns == 'sol':
return Eq(f(x), gsol)
elif returns in ('list' 'both'):
# HOW TO TEST THIS CODE? (dsolve does not pass 'returns' through)
# Create a list of (hopefully) linearly independent solutions
gensols = []
# Keep track of when to use sin or cos for nonzero imroot
for i, reroot, imroot in collectterms:
if imroot == 0:
gensols.append(ln(x)**i*x**reroot)
else:
sin_form = ln(x)**i*x**reroot*sin(abs(imroot)*ln(x))
if sin_form in gensols:
cos_form = ln(x)**i*x**reroot*cos(imroot*ln(x))
gensols.append(cos_form)
else:
gensols.append(sin_form)
if returns == 'list':
return gensols
else:
return {'sol': Eq(f(x), gsol), 'list': gensols}
else:
raise ValueError('Unknown value for key "returns".')
def ode_nth_linear_euler_eq_nonhomogeneous_undetermined_coefficients(eq, func, order, match, returns='sol'):
r"""
Solves an `n`\th order linear non homogeneous Cauchy-Euler equidimensional
ordinary differential equation using undetermined coefficients.
This is an equation with form `g(x) = a_0 f(x) + a_1 x f'(x) + a_2 x^2 f''(x)
\cdots`.
These equations can be solved in a general manner, by substituting
solutions of the form `x = exp(t)`, and deriving a characteristic equation
of form `g(exp(t)) = b_0 f(t) + b_1 f'(t) + b_2 f''(t) \cdots` which can
be then solved by nth_linear_constant_coeff_undetermined_coefficients if
g(exp(t)) has finite number of linearly independent derivatives.
Functions that fit this requirement are finite sums functions of the form
`a x^i e^{b x} \sin(c x + d)` or `a x^i e^{b x} \cos(c x + d)`, where `i`
is a non-negative integer and `a`, `b`, `c`, and `d` are constants. For
example any polynomial in `x`, functions like `x^2 e^{2 x}`, `x \sin(x)`,
and `e^x \cos(x)` can all be used. Products of `\sin`'s and `\cos`'s have
a finite number of derivatives, because they can be expanded into `\sin(a
x)` and `\cos(b x)` terms. However, SymPy currently cannot do that
expansion, so you will need to manually rewrite the expression in terms of
the above to use this method. So, for example, you will need to manually
convert `\sin^2(x)` into `(1 + \cos(2 x))/2` to properly apply the method
of undetermined coefficients on it.
After replacement of x by exp(t), this method works by creating a trial function
from the expression and all of its linear independent derivatives and
substituting them into the original ODE. The coefficients for each term
will be a system of linear equations, which are be solved for and
substituted, giving the solution. If any of the trial functions are linearly
dependent on the solution to the homogeneous equation, they are multiplied
by sufficient `x` to make them linearly independent.
Examples
========
>>> from sympy import dsolve, Function, Derivative, log
>>> from sympy.abc import x
>>> f = Function('f')
>>> eq = x**2*Derivative(f(x), x, x) - 2*x*Derivative(f(x), x) + 2*f(x) - log(x)
>>> dsolve(eq, f(x),
... hint='nth_linear_euler_eq_nonhomogeneous_undetermined_coefficients').expand()
Eq(f(x), C1*x + C2*x**2 + log(x)/2 + 3/4)
"""
x = func.args[0]
f = func.func
r = match
chareq, eq, symbol = S.Zero, S.Zero, Dummy('x')
for i in r.keys():
if not isinstance(i, str) and i >= 0:
chareq += (r[i]*diff(x**symbol, x, i)*x**-symbol).expand()
for i in range(1,degree(Poly(chareq, symbol))+1):
eq += chareq.coeff(symbol**i)*diff(f(x), x, i)
if chareq.as_coeff_add(symbol)[0]:
eq += chareq.as_coeff_add(symbol)[0]*f(x)
e, re = posify(r[-1].subs(x, exp(x)))
eq += e.subs(re)
match = _nth_linear_match(eq, f(x), ode_order(eq, f(x)))
eq_homogeneous = Add(eq,-match[-1])
match['trialset'] = _undetermined_coefficients_match(match[-1], x, func, eq_homogeneous)['trialset']
return ode_nth_linear_constant_coeff_undetermined_coefficients(eq, func, order, match).subs(x, log(x)).subs(f(log(x)), f(x)).expand()
def ode_nth_linear_euler_eq_nonhomogeneous_variation_of_parameters(eq, func, order, match, returns='sol'):
r"""
Solves an `n`\th order linear non homogeneous Cauchy-Euler equidimensional
ordinary differential equation using variation of parameters.
This is an equation with form `g(x) = a_0 f(x) + a_1 x f'(x) + a_2 x^2 f''(x)
\cdots`.
This method works by assuming that the particular solution takes the form
.. math:: \sum_{x=1}^{n} c_i(x) y_i(x) {a_n} {x^n} \text{,}
where `y_i` is the `i`\th solution to the homogeneous equation. The
solution is then solved using Wronskian's and Cramer's Rule. The
particular solution is given by multiplying eq given below with `a_n x^{n}`
.. math:: \sum_{x=1}^n \left( \int \frac{W_i(x)}{W(x)} \,dx
\right) y_i(x) \text{,}
where `W(x)` is the Wronskian of the fundamental system (the system of `n`
linearly independent solutions to the homogeneous equation), and `W_i(x)`
is the Wronskian of the fundamental system with the `i`\th column replaced
with `[0, 0, \cdots, 0, \frac{x^{- n}}{a_n} g{\left(x \right)}]`.
This method is general enough to solve any `n`\th order inhomogeneous
linear differential equation, but sometimes SymPy cannot simplify the
Wronskian well enough to integrate it. If this method hangs, try using the
``nth_linear_constant_coeff_variation_of_parameters_Integral`` hint and
simplifying the integrals manually. Also, prefer using
``nth_linear_constant_coeff_undetermined_coefficients`` when it
applies, because it doesn't use integration, making it faster and more
reliable.
Warning, using simplify=False with
'nth_linear_constant_coeff_variation_of_parameters' in
:py:meth:`~sympy.solvers.ode.dsolve` may cause it to hang, because it will
not attempt to simplify the Wronskian before integrating. It is
recommended that you only use simplify=False with
'nth_linear_constant_coeff_variation_of_parameters_Integral' for this
method, especially if the solution to the homogeneous equation has
trigonometric functions in it.
Examples
========
>>> from sympy import Function, dsolve, Derivative
>>> from sympy.abc import x
>>> f = Function('f')
>>> eq = x**2*Derivative(f(x), x, x) - 2*x*Derivative(f(x), x) + 2*f(x) - x**4
>>> dsolve(eq, f(x),
... hint='nth_linear_euler_eq_nonhomogeneous_variation_of_parameters').expand()
Eq(f(x), C1*x + C2*x**2 + x**4/6)
"""
x = func.args[0]
f = func.func
r = match
gensol = ode_nth_linear_euler_eq_homogeneous(eq, func, order, match, returns='both')
match.update(gensol)
r[-1] = r[-1]/r[ode_order(eq, f(x))]
sol = _solve_variation_of_parameters(eq, func, order, match)
return Eq(f(x), r['sol'].rhs + (sol.rhs - r['sol'].rhs)*r[ode_order(eq, f(x))])
def _linear_coeff_match(expr, func):
r"""
Helper function to match hint ``linear_coefficients``.
Matches the expression to the form `(a_1 x + b_1 f(x) + c_1)/(a_2 x + b_2
f(x) + c_2)` where the following conditions hold:
1. `a_1`, `b_1`, `c_1`, `a_2`, `b_2`, `c_2` are Rationals;
2. `c_1` or `c_2` are not equal to zero;
3. `a_2 b_1 - a_1 b_2` is not equal to zero.
Return ``xarg``, ``yarg`` where
1. ``xarg`` = `(b_2 c_1 - b_1 c_2)/(a_2 b_1 - a_1 b_2)`
2. ``yarg`` = `(a_1 c_2 - a_2 c_1)/(a_2 b_1 - a_1 b_2)`
Examples
========
>>> from sympy import Function
>>> from sympy.abc import x
>>> from sympy.solvers.ode.ode import _linear_coeff_match
>>> from sympy.functions.elementary.trigonometric import sin
>>> f = Function('f')
>>> _linear_coeff_match((
... (-25*f(x) - 8*x + 62)/(4*f(x) + 11*x - 11)), f(x))
(1/9, 22/9)
>>> _linear_coeff_match(
... sin((-5*f(x) - 8*x + 6)/(4*f(x) + x - 1)), f(x))
(19/27, 2/27)
>>> _linear_coeff_match(sin(f(x)/x), f(x))
"""
f = func.func
x = func.args[0]
def abc(eq):
r'''
Internal function of _linear_coeff_match
that returns Rationals a, b, c
if eq is a*x + b*f(x) + c, else None.
'''
eq = _mexpand(eq)
c = eq.as_independent(x, f(x), as_Add=True)[0]
if not c.is_Rational:
return
a = eq.coeff(x)
if not a.is_Rational:
return
b = eq.coeff(f(x))
if not b.is_Rational:
return
if eq == a*x + b*f(x) + c:
return a, b, c
def match(arg):
r'''
Internal function of _linear_coeff_match that returns Rationals a1,
b1, c1, a2, b2, c2 and a2*b1 - a1*b2 of the expression (a1*x + b1*f(x)
+ c1)/(a2*x + b2*f(x) + c2) if one of c1 or c2 and a2*b1 - a1*b2 is
non-zero, else None.
'''
n, d = arg.together().as_numer_denom()
m = abc(n)
if m is not None:
a1, b1, c1 = m
m = abc(d)
if m is not None:
a2, b2, c2 = m
d = a2*b1 - a1*b2
if (c1 or c2) and d:
return a1, b1, c1, a2, b2, c2, d
m = [fi.args[0] for fi in expr.atoms(Function) if fi.func != f and
len(fi.args) == 1 and not fi.args[0].is_Function] or {expr}
m1 = match(m.pop())
if m1 and all(match(mi) == m1 for mi in m):
a1, b1, c1, a2, b2, c2, denom = m1
return (b2*c1 - b1*c2)/denom, (a1*c2 - a2*c1)/denom
def ode_linear_coefficients(eq, func, order, match):
r"""
Solves a differential equation with linear coefficients.
The general form of a differential equation with linear coefficients is
.. math:: y' + F\left(\!\frac{a_1 x + b_1 y + c_1}{a_2 x + b_2 y +
c_2}\!\right) = 0\text{,}
where `a_1`, `b_1`, `c_1`, `a_2`, `b_2`, `c_2` are constants and `a_1 b_2
- a_2 b_1 \ne 0`.
This can be solved by substituting:
.. math:: x = x' + \frac{b_2 c_1 - b_1 c_2}{a_2 b_1 - a_1 b_2}
y = y' + \frac{a_1 c_2 - a_2 c_1}{a_2 b_1 - a_1
b_2}\text{.}
This substitution reduces the equation to a homogeneous differential
equation.
See Also
========
:meth:`sympy.solvers.ode.ode.ode_1st_homogeneous_coeff_best`
:meth:`sympy.solvers.ode.ode.ode_1st_homogeneous_coeff_subs_indep_div_dep`
:meth:`sympy.solvers.ode.ode.ode_1st_homogeneous_coeff_subs_dep_div_indep`
Examples
========
>>> from sympy import Function, pprint
>>> from sympy.solvers.ode.ode import dsolve
>>> from sympy.abc import x
>>> f = Function('f')
>>> df = f(x).diff(x)
>>> eq = (x + f(x) + 1)*df + (f(x) - 6*x + 1)
>>> dsolve(eq, hint='linear_coefficients')
[Eq(f(x), -x - sqrt(C1 + 7*x**2) - 1), Eq(f(x), -x + sqrt(C1 + 7*x**2) - 1)]
>>> pprint(dsolve(eq, hint='linear_coefficients'))
___________ ___________
/ 2 / 2
[f(x) = -x - \/ C1 + 7*x - 1, f(x) = -x + \/ C1 + 7*x - 1]
References
==========
- Joel Moses, "Symbolic Integration - The Stormy Decade", Communications
of the ACM, Volume 14, Number 8, August 1971, pp. 558
"""
return ode_1st_homogeneous_coeff_best(eq, func, order, match)
def ode_separable_reduced(eq, func, order, match):
r"""
Solves a differential equation that can be reduced to the separable form.
The general form of this equation is
.. math:: y' + (y/x) H(x^n y) = 0\text{}.
This can be solved by substituting `u(y) = x^n y`. The equation then
reduces to the separable form `\frac{u'}{u (\mathrm{power} - H(u))} -
\frac{1}{x} = 0`.
The general solution is:
>>> from sympy import Function, dsolve, pprint
>>> from sympy.abc import x, n
>>> f, g = map(Function, ['f', 'g'])
>>> genform = f(x).diff(x) + (f(x)/x)*g(x**n*f(x))
>>> pprint(genform)
/ n \
d f(x)*g\x *f(x)/
--(f(x)) + ---------------
dx x
>>> pprint(dsolve(genform, hint='separable_reduced'))
n
x *f(x)
/
|
| 1
| ------------ dy = C1 + log(x)
| y*(n - g(y))
|
/
See Also
========
:meth:`sympy.solvers.ode.ode.ode_separable`
Examples
========
>>> from sympy import Function, pprint
>>> from sympy.solvers.ode.ode import dsolve
>>> from sympy.abc import x
>>> f = Function('f')
>>> d = f(x).diff(x)
>>> eq = (x - x**2*f(x))*d - f(x)
>>> dsolve(eq, hint='separable_reduced')
[Eq(f(x), (1 - sqrt(C1*x**2 + 1))/x), Eq(f(x), (sqrt(C1*x**2 + 1) + 1)/x)]
>>> pprint(dsolve(eq, hint='separable_reduced'))
___________ ___________
/ 2 / 2
1 - \/ C1*x + 1 \/ C1*x + 1 + 1
[f(x) = ------------------, f(x) = ------------------]
x x
References
==========
- Joel Moses, "Symbolic Integration - The Stormy Decade", Communications
of the ACM, Volume 14, Number 8, August 1971, pp. 558
"""
# Arguments are passed in a way so that they are coherent with the
# ode_separable function
x = func.args[0]
f = func.func
y = Dummy('y')
u = match['u'].subs(match['t'], y)
ycoeff = 1/(y*(match['power'] - u))
m1 = {y: 1, x: -1/x, 'coeff': 1}
m2 = {y: ycoeff, x: 1, 'coeff': 1}
r = {'m1': m1, 'm2': m2, 'y': y, 'hint': x**match['power']*f(x)}
return ode_separable(eq, func, order, r)
def ode_1st_power_series(eq, func, order, match):
r"""
The power series solution is a method which gives the Taylor series expansion
to the solution of a differential equation.
For a first order differential equation `\frac{dy}{dx} = h(x, y)`, a power
series solution exists at a point `x = x_{0}` if `h(x, y)` is analytic at `x_{0}`.
The solution is given by
.. math:: y(x) = y(x_{0}) + \sum_{n = 1}^{\infty} \frac{F_{n}(x_{0},b)(x - x_{0})^n}{n!},
where `y(x_{0}) = b` is the value of y at the initial value of `x_{0}`.
To compute the values of the `F_{n}(x_{0},b)` the following algorithm is
followed, until the required number of terms are generated.
1. `F_1 = h(x_{0}, b)`
2. `F_{n+1} = \frac{\partial F_{n}}{\partial x} + \frac{\partial F_{n}}{\partial y}F_{1}`
Examples
========
>>> from sympy import Function, pprint, exp
>>> from sympy.solvers.ode.ode import dsolve
>>> from sympy.abc import x
>>> f = Function('f')
>>> eq = exp(x)*(f(x).diff(x)) - f(x)
>>> pprint(dsolve(eq, hint='1st_power_series'))
3 4 5
C1*x C1*x C1*x / 6\
f(x) = C1 + C1*x - ----- + ----- + ----- + O\x /
6 24 60
References
==========
- Travis W. Walker, Analytic power series technique for solving first-order
differential equations, p.p 17, 18
"""
x = func.args[0]
y = match['y']
f = func.func
h = -match[match['d']]/match[match['e']]
point = match.get('f0')
value = match.get('f0val')
terms = match.get('terms')
# First term
F = h
if not h:
return Eq(f(x), value)
# Initialization
series = value
if terms > 1:
hc = h.subs({x: point, y: value})
if hc.has(oo) or hc.has(NaN) or hc.has(zoo):
# Derivative does not exist, not analytic
return Eq(f(x), oo)
elif hc:
series += hc*(x - point)
for factcount in range(2, terms):
Fnew = F.diff(x) + F.diff(y)*h
Fnewc = Fnew.subs({x: point, y: value})
# Same logic as above
if Fnewc.has(oo) or Fnewc.has(NaN) or Fnewc.has(-oo) or Fnewc.has(zoo):
return Eq(f(x), oo)
series += Fnewc*((x - point)**factcount)/factorial(factcount)
F = Fnew
series += Order(x**terms)
return Eq(f(x), series)
def ode_nth_linear_constant_coeff_homogeneous(eq, func, order, match,
returns='sol'):
r"""
Solves an `n`\th order linear homogeneous differential equation with
constant coefficients.
This is an equation of the form
.. math:: a_n f^{(n)}(x) + a_{n-1} f^{(n-1)}(x) + \cdots + a_1 f'(x)
+ a_0 f(x) = 0\text{.}
These equations can be solved in a general manner, by taking the roots of
the characteristic equation `a_n m^n + a_{n-1} m^{n-1} + \cdots + a_1 m +
a_0 = 0`. The solution will then be the sum of `C_n x^i e^{r x}` terms,
for each where `C_n` is an arbitrary constant, `r` is a root of the
characteristic equation and `i` is one of each from 0 to the multiplicity
of the root - 1 (for example, a root 3 of multiplicity 2 would create the
terms `C_1 e^{3 x} + C_2 x e^{3 x}`). The exponential is usually expanded
for complex roots using Euler's equation `e^{I x} = \cos(x) + I \sin(x)`.
Complex roots always come in conjugate pairs in polynomials with real
coefficients, so the two roots will be represented (after simplifying the
constants) as `e^{a x} \left(C_1 \cos(b x) + C_2 \sin(b x)\right)`.
If SymPy cannot find exact roots to the characteristic equation, a
:py:class:`~sympy.polys.rootoftools.ComplexRootOf` instance will be return
instead.
>>> from sympy import Function, dsolve
>>> from sympy.abc import x
>>> f = Function('f')
>>> dsolve(f(x).diff(x, 5) + 10*f(x).diff(x) - 2*f(x), f(x),
... hint='nth_linear_constant_coeff_homogeneous')
... # doctest: +NORMALIZE_WHITESPACE
Eq(f(x), C5*exp(x*CRootOf(_x**5 + 10*_x - 2, 0))
+ (C1*sin(x*im(CRootOf(_x**5 + 10*_x - 2, 1)))
+ C2*cos(x*im(CRootOf(_x**5 + 10*_x - 2, 1))))*exp(x*re(CRootOf(_x**5 + 10*_x - 2, 1)))
+ (C3*sin(x*im(CRootOf(_x**5 + 10*_x - 2, 3)))
+ C4*cos(x*im(CRootOf(_x**5 + 10*_x - 2, 3))))*exp(x*re(CRootOf(_x**5 + 10*_x - 2, 3))))
Note that because this method does not involve integration, there is no
``nth_linear_constant_coeff_homogeneous_Integral`` hint.
The following is for internal use:
- ``returns = 'sol'`` returns the solution to the ODE.
- ``returns = 'list'`` returns a list of linearly independent solutions,
for use with non homogeneous solution methods like variation of
parameters and undetermined coefficients. Note that, though the
solutions should be linearly independent, this function does not
explicitly check that. You can do ``assert simplify(wronskian(sollist))
!= 0`` to check for linear independence. Also, ``assert len(sollist) ==
order`` will need to pass.
- ``returns = 'both'``, return a dictionary ``{'sol': <solution to ODE>,
'list': <list of linearly independent solutions>}``.
Examples
========
>>> from sympy import Function, dsolve, pprint
>>> from sympy.abc import x
>>> f = Function('f')
>>> pprint(dsolve(f(x).diff(x, 4) + 2*f(x).diff(x, 3) -
... 2*f(x).diff(x, 2) - 6*f(x).diff(x) + 5*f(x), f(x),
... hint='nth_linear_constant_coeff_homogeneous'))
x -2*x
f(x) = (C1 + C2*x)*e + (C3*sin(x) + C4*cos(x))*e
References
==========
- https://en.wikipedia.org/wiki/Linear_differential_equation section:
Nonhomogeneous_equation_with_constant_coefficients
- M. Tenenbaum & H. Pollard, "Ordinary Differential Equations",
Dover 1963, pp. 211
# indirect doctest
"""
x = func.args[0]
f = func.func
r = match
# First, set up characteristic equation.
chareq, symbol = S.Zero, Dummy('x')
for i in r.keys():
if type(i) == str or i < 0:
pass
else:
chareq += r[i]*symbol**i
chareq = Poly(chareq, symbol)
# Can't just call roots because it doesn't return rootof for unsolveable
# polynomials.
chareqroots = roots(chareq, multiple=True)
if len(chareqroots) != order:
chareqroots = [rootof(chareq, k) for k in range(chareq.degree())]
chareq_is_complex = not all([i.is_real for i in chareq.all_coeffs()])
# A generator of constants
constants = list(get_numbered_constants(eq, num=chareq.degree()*2))
# Create a dict root: multiplicity or charroots
charroots = defaultdict(int)
for root in chareqroots:
charroots[root] += 1
# We need to keep track of terms so we can run collect() at the end.
# This is necessary for constantsimp to work properly.
#
# XXX: This global collectterms hack should be removed.
global collectterms
collectterms = []
gensols = []
conjugate_roots = [] # used to prevent double-use of conjugate roots
# Loop over roots in theorder provided by roots/rootof...
for root in chareqroots:
# but don't repoeat multiple roots.
if root not in charroots:
continue
multiplicity = charroots.pop(root)
for i in range(multiplicity):
if chareq_is_complex:
gensols.append(x**i*exp(root*x))
collectterms = [(i, root, 0)] + collectterms
continue
reroot = re(root)
imroot = im(root)
if imroot.has(atan2) and reroot.has(atan2):
# Remove this condition when re and im stop returning
# circular atan2 usages.
gensols.append(x**i*exp(root*x))
collectterms = [(i, root, 0)] + collectterms
else:
if root in conjugate_roots:
collectterms = [(i, reroot, imroot)] + collectterms
continue
if imroot == 0:
gensols.append(x**i*exp(reroot*x))
collectterms = [(i, reroot, 0)] + collectterms
continue
conjugate_roots.append(conjugate(root))
gensols.append(x**i*exp(reroot*x) * sin(abs(imroot) * x))
gensols.append(x**i*exp(reroot*x) * cos( imroot * x))
# This ordering is important
collectterms = [(i, reroot, imroot)] + collectterms
if returns == 'list':
return gensols
elif returns in ('sol' 'both'):
gsol = Add(*[i*j for (i, j) in zip(constants, gensols)])
if returns == 'sol':
return Eq(f(x), gsol)
else:
return {'sol': Eq(f(x), gsol), 'list': gensols}
else:
raise ValueError('Unknown value for key "returns".')
def ode_nth_linear_constant_coeff_undetermined_coefficients(eq, func, order, match):
r"""
Solves an `n`\th order linear differential equation with constant
coefficients using the method of undetermined coefficients.
This method works on differential equations of the form
.. math:: a_n f^{(n)}(x) + a_{n-1} f^{(n-1)}(x) + \cdots + a_1 f'(x)
+ a_0 f(x) = P(x)\text{,}
where `P(x)` is a function that has a finite number of linearly
independent derivatives.
Functions that fit this requirement are finite sums functions of the form
`a x^i e^{b x} \sin(c x + d)` or `a x^i e^{b x} \cos(c x + d)`, where `i`
is a non-negative integer and `a`, `b`, `c`, and `d` are constants. For
example any polynomial in `x`, functions like `x^2 e^{2 x}`, `x \sin(x)`,
and `e^x \cos(x)` can all be used. Products of `\sin`'s and `\cos`'s have
a finite number of derivatives, because they can be expanded into `\sin(a
x)` and `\cos(b x)` terms. However, SymPy currently cannot do that
expansion, so you will need to manually rewrite the expression in terms of
the above to use this method. So, for example, you will need to manually
convert `\sin^2(x)` into `(1 + \cos(2 x))/2` to properly apply the method
of undetermined coefficients on it.
This method works by creating a trial function from the expression and all
of its linear independent derivatives and substituting them into the
original ODE. The coefficients for each term will be a system of linear
equations, which are be solved for and substituted, giving the solution.
If any of the trial functions are linearly dependent on the solution to
the homogeneous equation, they are multiplied by sufficient `x` to make
them linearly independent.
Examples
========
>>> from sympy import Function, dsolve, pprint, exp, cos
>>> from sympy.abc import x
>>> f = Function('f')
>>> pprint(dsolve(f(x).diff(x, 2) + 2*f(x).diff(x) + f(x) -
... 4*exp(-x)*x**2 + cos(2*x), f(x),
... hint='nth_linear_constant_coeff_undetermined_coefficients'))
/ / 3\\
| | x || -x 4*sin(2*x) 3*cos(2*x)
f(x) = |C1 + x*|C2 + --||*e - ---------- + ----------
\ \ 3 // 25 25
References
==========
- https://en.wikipedia.org/wiki/Method_of_undetermined_coefficients
- M. Tenenbaum & H. Pollard, "Ordinary Differential Equations",
Dover 1963, pp. 221
# indirect doctest
"""
gensol = ode_nth_linear_constant_coeff_homogeneous(eq, func, order, match,
returns='both')
match.update(gensol)
return _solve_undetermined_coefficients(eq, func, order, match)
def _solve_undetermined_coefficients(eq, func, order, match):
r"""
Helper function for the method of undetermined coefficients.
See the
:py:meth:`~sympy.solvers.ode.ode.ode_nth_linear_constant_coeff_undetermined_coefficients`
docstring for more information on this method.
The parameter ``match`` should be a dictionary that has the following
keys:
``list``
A list of solutions to the homogeneous equation, such as the list
returned by
``ode_nth_linear_constant_coeff_homogeneous(returns='list')``.
``sol``
The general solution, such as the solution returned by
``ode_nth_linear_constant_coeff_homogeneous(returns='sol')``.
``trialset``
The set of trial functions as returned by
``_undetermined_coefficients_match()['trialset']``.
"""
x = func.args[0]
f = func.func
r = match
coeffs = numbered_symbols('a', cls=Dummy)
coefflist = []
gensols = r['list']
gsol = r['sol']
trialset = r['trialset']
if len(gensols) != order:
raise NotImplementedError("Cannot find " + str(order) +
" solutions to the homogeneous equation necessary to apply" +
" undetermined coefficients to " + str(eq) +
" (number of terms != order)")
trialfunc = 0
for i in trialset:
c = next(coeffs)
coefflist.append(c)
trialfunc += c*i
eqs = sub_func_doit(eq, f(x), trialfunc)
coeffsdict = dict(list(zip(trialset, [0]*(len(trialset) + 1))))
eqs = _mexpand(eqs)
for i in Add.make_args(eqs):
s = separatevars(i, dict=True, symbols=[x])
if coeffsdict.get(s[x]):
coeffsdict[s[x]] += s['coeff']
else:
coeffsdict[s[x]] = s['coeff']
coeffvals = solve(list(coeffsdict.values()), coefflist)
if not coeffvals:
raise NotImplementedError(
"Could not solve `%s` using the "
"method of undetermined coefficients "
"(unable to solve for coefficients)." % eq)
psol = trialfunc.subs(coeffvals)
return Eq(f(x), gsol.rhs + psol)
def _undetermined_coefficients_match(expr, x, func=None, eq_homogeneous=S.Zero):
r"""
Returns a trial function match if undetermined coefficients can be applied
to ``expr``, and ``None`` otherwise.
A trial expression can be found for an expression for use with the method
of undetermined coefficients if the expression is an
additive/multiplicative combination of constants, polynomials in `x` (the
independent variable of expr), `\sin(a x + b)`, `\cos(a x + b)`, and
`e^{a x}` terms (in other words, it has a finite number of linearly
independent derivatives).
Note that you may still need to multiply each term returned here by
sufficient `x` to make it linearly independent with the solutions to the
homogeneous equation.
This is intended for internal use by ``undetermined_coefficients`` hints.
SymPy currently has no way to convert `\sin^n(x) \cos^m(y)` into a sum of
only `\sin(a x)` and `\cos(b x)` terms, so these are not implemented. So,
for example, you will need to manually convert `\sin^2(x)` into `[1 +
\cos(2 x)]/2` to properly apply the method of undetermined coefficients on
it.
Examples
========
>>> from sympy import log, exp
>>> from sympy.solvers.ode.ode import _undetermined_coefficients_match
>>> from sympy.abc import x
>>> _undetermined_coefficients_match(9*x*exp(x) + exp(-x), x)
{'test': True, 'trialset': {x*exp(x), exp(-x), exp(x)}}
>>> _undetermined_coefficients_match(log(x), x)
{'test': False}
"""
a = Wild('a', exclude=[x])
b = Wild('b', exclude=[x])
expr = powsimp(expr, combine='exp') # exp(x)*exp(2*x + 1) => exp(3*x + 1)
retdict = {}
def _test_term(expr, x):
r"""
Test if ``expr`` fits the proper form for undetermined coefficients.
"""
if not expr.has(x):
return True
elif expr.is_Add:
return all(_test_term(i, x) for i in expr.args)
elif expr.is_Mul:
if expr.has(sin, cos):
foundtrig = False
# Make sure that there is only one trig function in the args.
# See the docstring.
for i in expr.args:
if i.has(sin, cos):
if foundtrig:
return False
else:
foundtrig = True
return all(_test_term(i, x) for i in expr.args)
elif expr.is_Function:
if expr.func in (sin, cos, exp, sinh, cosh):
if expr.args[0].match(a*x + b):
return True
else:
return False
else:
return False
elif expr.is_Pow and expr.base.is_Symbol and expr.exp.is_Integer and \
expr.exp >= 0:
return True
elif expr.is_Pow and expr.base.is_number:
if expr.exp.match(a*x + b):
return True
else:
return False
elif expr.is_Symbol or expr.is_number:
return True
else:
return False
def _get_trial_set(expr, x, exprs=set([])):
r"""
Returns a set of trial terms for undetermined coefficients.
The idea behind undetermined coefficients is that the terms expression
repeat themselves after a finite number of derivatives, except for the
coefficients (they are linearly dependent). So if we collect these,
we should have the terms of our trial function.
"""
def _remove_coefficient(expr, x):
r"""
Returns the expression without a coefficient.
Similar to expr.as_independent(x)[1], except it only works
multiplicatively.
"""
term = S.One
if expr.is_Mul:
for i in expr.args:
if i.has(x):
term *= i
elif expr.has(x):
term = expr
return term
expr = expand_mul(expr)
if expr.is_Add:
for term in expr.args:
if _remove_coefficient(term, x) in exprs:
pass
else:
exprs.add(_remove_coefficient(term, x))
exprs = exprs.union(_get_trial_set(term, x, exprs))
else:
term = _remove_coefficient(expr, x)
tmpset = exprs.union({term})
oldset = set([])
while tmpset != oldset:
# If you get stuck in this loop, then _test_term is probably
# broken
oldset = tmpset.copy()
expr = expr.diff(x)
term = _remove_coefficient(expr, x)
if term.is_Add:
tmpset = tmpset.union(_get_trial_set(term, x, tmpset))
else:
tmpset.add(term)
exprs = tmpset
return exprs
def is_homogeneous_solution(term):
r""" This function checks whether the given trialset contains any root
of homogenous equation"""
return expand(sub_func_doit(eq_homogeneous, func, term)).is_zero
retdict['test'] = _test_term(expr, x)
if retdict['test']:
# Try to generate a list of trial solutions that will have the
# undetermined coefficients. Note that if any of these are not linearly
# independent with any of the solutions to the homogeneous equation,
# then they will need to be multiplied by sufficient x to make them so.
# This function DOES NOT do that (it doesn't even look at the
# homogeneous equation).
temp_set = set([])
for i in Add.make_args(expr):
act = _get_trial_set(i,x)
if eq_homogeneous is not S.Zero:
while any(is_homogeneous_solution(ts) for ts in act):
act = {x*ts for ts in act}
temp_set = temp_set.union(act)
retdict['trialset'] = temp_set
return retdict
def ode_nth_linear_constant_coeff_variation_of_parameters(eq, func, order, match):
r"""
Solves an `n`\th order linear differential equation with constant
coefficients using the method of variation of parameters.
This method works on any differential equations of the form
.. math:: f^{(n)}(x) + a_{n-1} f^{(n-1)}(x) + \cdots + a_1 f'(x) + a_0
f(x) = P(x)\text{.}
This method works by assuming that the particular solution takes the form
.. math:: \sum_{x=1}^{n} c_i(x) y_i(x)\text{,}
where `y_i` is the `i`\th solution to the homogeneous equation. The
solution is then solved using Wronskian's and Cramer's Rule. The
particular solution is given by
.. math:: \sum_{x=1}^n \left( \int \frac{W_i(x)}{W(x)} \,dx
\right) y_i(x) \text{,}
where `W(x)` is the Wronskian of the fundamental system (the system of `n`
linearly independent solutions to the homogeneous equation), and `W_i(x)`
is the Wronskian of the fundamental system with the `i`\th column replaced
with `[0, 0, \cdots, 0, P(x)]`.
This method is general enough to solve any `n`\th order inhomogeneous
linear differential equation with constant coefficients, but sometimes
SymPy cannot simplify the Wronskian well enough to integrate it. If this
method hangs, try using the
``nth_linear_constant_coeff_variation_of_parameters_Integral`` hint and
simplifying the integrals manually. Also, prefer using
``nth_linear_constant_coeff_undetermined_coefficients`` when it
applies, because it doesn't use integration, making it faster and more
reliable.
Warning, using simplify=False with
'nth_linear_constant_coeff_variation_of_parameters' in
:py:meth:`~sympy.solvers.ode.dsolve` may cause it to hang, because it will
not attempt to simplify the Wronskian before integrating. It is
recommended that you only use simplify=False with
'nth_linear_constant_coeff_variation_of_parameters_Integral' for this
method, especially if the solution to the homogeneous equation has
trigonometric functions in it.
Examples
========
>>> from sympy import Function, dsolve, pprint, exp, log
>>> from sympy.abc import x
>>> f = Function('f')
>>> pprint(dsolve(f(x).diff(x, 3) - 3*f(x).diff(x, 2) +
... 3*f(x).diff(x) - f(x) - exp(x)*log(x), f(x),
... hint='nth_linear_constant_coeff_variation_of_parameters'))
/ / / x*log(x) 11*x\\\ x
f(x) = |C1 + x*|C2 + x*|C3 + -------- - ----|||*e
\ \ \ 6 36 ///
References
==========
- https://en.wikipedia.org/wiki/Variation_of_parameters
- http://planetmath.org/VariationOfParameters
- M. Tenenbaum & H. Pollard, "Ordinary Differential Equations",
Dover 1963, pp. 233
# indirect doctest
"""
gensol = ode_nth_linear_constant_coeff_homogeneous(eq, func, order, match,
returns='both')
match.update(gensol)
return _solve_variation_of_parameters(eq, func, order, match)
def _solve_variation_of_parameters(eq, func, order, match):
r"""
Helper function for the method of variation of parameters and nonhomogeneous euler eq.
See the
:py:meth:`~sympy.solvers.ode.ode.ode_nth_linear_constant_coeff_variation_of_parameters`
docstring for more information on this method.
The parameter ``match`` should be a dictionary that has the following
keys:
``list``
A list of solutions to the homogeneous equation, such as the list
returned by
``ode_nth_linear_constant_coeff_homogeneous(returns='list')``.
``sol``
The general solution, such as the solution returned by
``ode_nth_linear_constant_coeff_homogeneous(returns='sol')``.
"""
x = func.args[0]
f = func.func
r = match
psol = 0
gensols = r['list']
gsol = r['sol']
wr = wronskian(gensols, x)
if r.get('simplify', True):
wr = simplify(wr) # We need much better simplification for
# some ODEs. See issue 4662, for example.
# To reduce commonly occurring sin(x)**2 + cos(x)**2 to 1
wr = trigsimp(wr, deep=True, recursive=True)
if not wr:
# The wronskian will be 0 iff the solutions are not linearly
# independent.
raise NotImplementedError("Cannot find " + str(order) +
" solutions to the homogeneous equation necessary to apply " +
"variation of parameters to " + str(eq) + " (Wronskian == 0)")
if len(gensols) != order:
raise NotImplementedError("Cannot find " + str(order) +
" solutions to the homogeneous equation necessary to apply " +
"variation of parameters to " +
str(eq) + " (number of terms != order)")
negoneterm = (-1)**(order)
for i in gensols:
psol += negoneterm*Integral(wronskian([sol for sol in gensols if sol != i], x)*r[-1]/wr, x)*i/r[order]
negoneterm *= -1
if r.get('simplify', True):
psol = simplify(psol)
psol = trigsimp(psol, deep=True)
return Eq(f(x), gsol.rhs + psol)
def ode_separable(eq, func, order, match):
r"""
Solves separable 1st order differential equations.
This is any differential equation that can be written as `P(y)
\tfrac{dy}{dx} = Q(x)`. The solution can then just be found by
rearranging terms and integrating: `\int P(y) \,dy = \int Q(x) \,dx`.
This hint uses :py:meth:`sympy.simplify.simplify.separatevars` as its back
end, so if a separable equation is not caught by this solver, it is most
likely the fault of that function.
:py:meth:`~sympy.simplify.simplify.separatevars` is
smart enough to do most expansion and factoring necessary to convert a
separable equation `F(x, y)` into the proper form `P(x)\cdot{}Q(y)`. The
general solution is::
>>> from sympy import Function, dsolve, Eq, pprint
>>> from sympy.abc import x
>>> a, b, c, d, f = map(Function, ['a', 'b', 'c', 'd', 'f'])
>>> genform = Eq(a(x)*b(f(x))*f(x).diff(x), c(x)*d(f(x)))
>>> pprint(genform)
d
a(x)*b(f(x))*--(f(x)) = c(x)*d(f(x))
dx
>>> pprint(dsolve(genform, f(x), hint='separable_Integral'))
f(x)
/ /
| |
| b(y) | c(x)
| ---- dy = C1 + | ---- dx
| d(y) | a(x)
| |
/ /
Examples
========
>>> from sympy import Function, dsolve, Eq
>>> from sympy.abc import x
>>> f = Function('f')
>>> pprint(dsolve(Eq(f(x)*f(x).diff(x) + x, 3*x*f(x)**2), f(x),
... hint='separable', simplify=False))
/ 2 \ 2
log\3*f (x) - 1/ x
---------------- = C1 + --
6 2
References
==========
- M. Tenenbaum & H. Pollard, "Ordinary Differential Equations",
Dover 1963, pp. 52
# indirect doctest
"""
x = func.args[0]
f = func.func
C1 = get_numbered_constants(eq, num=1)
r = match # {'m1':m1, 'm2':m2, 'y':y}
u = r.get('hint', f(x)) # get u from separable_reduced else get f(x)
return Eq(Integral(r['m2']['coeff']*r['m2'][r['y']]/r['m1'][r['y']],
(r['y'], None, u)), Integral(-r['m1']['coeff']*r['m1'][x]/
r['m2'][x], x) + C1)
def checkinfsol(eq, infinitesimals, func=None, order=None):
r"""
This function is used to check if the given infinitesimals are the
actual infinitesimals of the given first order differential equation.
This method is specific to the Lie Group Solver of ODEs.
As of now, it simply checks, by substituting the infinitesimals in the
partial differential equation.
.. math:: \frac{\partial \eta}{\partial x} + \left(\frac{\partial \eta}{\partial y}
- \frac{\partial \xi}{\partial x}\right)*h
- \frac{\partial \xi}{\partial y}*h^{2}
- \xi\frac{\partial h}{\partial x} - \eta\frac{\partial h}{\partial y} = 0
where `\eta`, and `\xi` are the infinitesimals and `h(x,y) = \frac{dy}{dx}`
The infinitesimals should be given in the form of a list of dicts
``[{xi(x, y): inf, eta(x, y): inf}]``, corresponding to the
output of the function infinitesimals. It returns a list
of values of the form ``[(True/False, sol)]`` where ``sol`` is the value
obtained after substituting the infinitesimals in the PDE. If it
is ``True``, then ``sol`` would be 0.
"""
if isinstance(eq, Equality):
eq = eq.lhs - eq.rhs
if not func:
eq, func = _preprocess(eq)
variables = func.args
if len(variables) != 1:
raise ValueError("ODE's have only one independent variable")
else:
x = variables[0]
if not order:
order = ode_order(eq, func)
if order != 1:
raise NotImplementedError("Lie groups solver has been implemented "
"only for first order differential equations")
else:
df = func.diff(x)
a = Wild('a', exclude = [df])
b = Wild('b', exclude = [df])
match = collect(expand(eq), df).match(a*df + b)
if match:
h = -simplify(match[b]/match[a])
else:
try:
sol = solve(eq, df)
except NotImplementedError:
raise NotImplementedError("Infinitesimals for the "
"first order ODE could not be found")
else:
h = sol[0] # Find infinitesimals for one solution
y = Dummy('y')
h = h.subs(func, y)
xi = Function('xi')(x, y)
eta = Function('eta')(x, y)
dxi = Function('xi')(x, func)
deta = Function('eta')(x, func)
pde = (eta.diff(x) + (eta.diff(y) - xi.diff(x))*h -
(xi.diff(y))*h**2 - xi*(h.diff(x)) - eta*(h.diff(y)))
soltup = []
for sol in infinitesimals:
tsol = {xi: S(sol[dxi]).subs(func, y),
eta: S(sol[deta]).subs(func, y)}
sol = simplify(pde.subs(tsol).doit())
if sol:
soltup.append((False, sol.subs(y, func)))
else:
soltup.append((True, 0))
return soltup
def _ode_lie_group_try_heuristic(eq, heuristic, func, match, inf):
xi = Function("xi")
eta = Function("eta")
f = func.func
x = func.args[0]
y = match['y']
h = match['h']
tempsol = []
if not inf:
try:
inf = infinitesimals(eq, hint=heuristic, func=func, order=1, match=match)
except ValueError:
return None
for infsim in inf:
xiinf = (infsim[xi(x, func)]).subs(func, y)
etainf = (infsim[eta(x, func)]).subs(func, y)
# This condition creates recursion while using pdsolve.
# Since the first step while solving a PDE of form
# a*(f(x, y).diff(x)) + b*(f(x, y).diff(y)) + c = 0
# is to solve the ODE dy/dx = b/a
if simplify(etainf/xiinf) == h:
continue
rpde = f(x, y).diff(x)*xiinf + f(x, y).diff(y)*etainf
r = pdsolve(rpde, func=f(x, y)).rhs
s = pdsolve(rpde - 1, func=f(x, y)).rhs
newcoord = [_lie_group_remove(coord) for coord in [r, s]]
r = Dummy("r")
s = Dummy("s")
C1 = Symbol("C1")
rcoord = newcoord[0]
scoord = newcoord[-1]
try:
sol = solve([r - rcoord, s - scoord], x, y, dict=True)
if sol == []:
continue
except NotImplementedError:
continue
else:
sol = sol[0]
xsub = sol[x]
ysub = sol[y]
num = simplify(scoord.diff(x) + scoord.diff(y)*h)
denom = simplify(rcoord.diff(x) + rcoord.diff(y)*h)
if num and denom:
diffeq = simplify((num/denom).subs([(x, xsub), (y, ysub)]))
sep = separatevars(diffeq, symbols=[r, s], dict=True)
if sep:
# Trying to separate, r and s coordinates
deq = integrate((1/sep[s]), s) + C1 - integrate(sep['coeff']*sep[r], r)
# Substituting and reverting back to original coordinates
deq = deq.subs([(r, rcoord), (s, scoord)])
try:
sdeq = solve(deq, y)
except NotImplementedError:
tempsol.append(deq)
else:
return [Eq(f(x), sol) for sol in sdeq]
elif denom: # (ds/dr) is zero which means s is constant
return [Eq(f(x), solve(scoord - C1, y)[0])]
elif num: # (dr/ds) is zero which means r is constant
return [Eq(f(x), solve(rcoord - C1, y)[0])]
# If nothing works, return solution as it is, without solving for y
if tempsol:
return [Eq(sol.subs(y, f(x)), 0) for sol in tempsol]
return None
def _ode_lie_group( s, func, order, match):
heuristics = lie_heuristics
inf = {}
f = func.func
x = func.args[0]
df = func.diff(x)
xi = Function("xi")
eta = Function("eta")
xis = match['xi']
etas = match['eta']
y = match.pop('y', None)
if y:
h = -simplify(match[match['d']]/match[match['e']])
y = y
else:
y = Dummy("y")
h = s.subs(func, y)
if xis is not None and etas is not None:
inf = [{xi(x, f(x)): S(xis), eta(x, f(x)): S(etas)}]
if checkinfsol(Eq(df, s), inf, func=f(x), order=1)[0][0]:
heuristics = ["user_defined"] + list(heuristics)
match = {'h': h, 'y': y}
# This is done so that if any heuristic raises a ValueError
# another heuristic can be used.
sol = None
for heuristic in heuristics:
sol = _ode_lie_group_try_heuristic(Eq(df, s), heuristic, func, match, inf)
if sol:
return sol
return sol
def ode_lie_group(eq, func, order, match):
r"""
This hint implements the Lie group method of solving first order differential
equations. The aim is to convert the given differential equation from the
given coordinate system into another coordinate system where it becomes
invariant under the one-parameter Lie group of translations. The converted
ODE can be easily solved by quadrature. It makes use of the
:py:meth:`sympy.solvers.ode.infinitesimals` function which returns the
infinitesimals of the transformation.
The coordinates `r` and `s` can be found by solving the following Partial
Differential Equations.
.. math :: \xi\frac{\partial r}{\partial x} + \eta\frac{\partial r}{\partial y}
= 0
.. math :: \xi\frac{\partial s}{\partial x} + \eta\frac{\partial s}{\partial y}
= 1
The differential equation becomes separable in the new coordinate system
.. math :: \frac{ds}{dr} = \frac{\frac{\partial s}{\partial x} +
h(x, y)\frac{\partial s}{\partial y}}{
\frac{\partial r}{\partial x} + h(x, y)\frac{\partial r}{\partial y}}
After finding the solution by integration, it is then converted back to the original
coordinate system by substituting `r` and `s` in terms of `x` and `y` again.
Examples
========
>>> from sympy import Function, dsolve, exp, pprint
>>> from sympy.abc import x
>>> f = Function('f')
>>> pprint(dsolve(f(x).diff(x) + 2*x*f(x) - x*exp(-x**2), f(x),
... hint='lie_group'))
/ 2\ 2
| x | -x
f(x) = |C1 + --|*e
\ 2 /
References
==========
- Solving differential equations by Symmetry Groups,
John Starrett, pp. 1 - pp. 14
"""
x = func.args[0]
df = func.diff(x)
try:
eqsol = solve(eq, df)
except NotImplementedError:
eqsol = []
desols = []
for s in eqsol:
sol = _ode_lie_group(s, func, order, match=match)
if sol:
desols.extend(sol)
if desols == []:
raise NotImplementedError("The given ODE " + str(eq) + " cannot be solved by"
+ " the lie group method")
return desols
def _lie_group_remove(coords):
r"""
This function is strictly meant for internal use by the Lie group ODE solving
method. It replaces arbitrary functions returned by pdsolve as follows:
1] If coords is an arbitrary function, then its argument is returned.
2] An arbitrary function in an Add object is replaced by zero.
3] An arbitrary function in a Mul object is replaced by one.
4] If there is no arbitrary function coords is returned unchanged.
Examples
========
>>> from sympy.solvers.ode.ode import _lie_group_remove
>>> from sympy import Function
>>> from sympy.abc import x, y
>>> F = Function("F")
>>> eq = x**2*y
>>> _lie_group_remove(eq)
x**2*y
>>> eq = F(x**2*y)
>>> _lie_group_remove(eq)
x**2*y
>>> eq = x*y**2 + F(x**3)
>>> _lie_group_remove(eq)
x*y**2
>>> eq = (F(x**3) + y)*x**4
>>> _lie_group_remove(eq)
x**4*y
"""
if isinstance(coords, AppliedUndef):
return coords.args[0]
elif coords.is_Add:
subfunc = coords.atoms(AppliedUndef)
if subfunc:
for func in subfunc:
coords = coords.subs(func, 0)
return coords
elif coords.is_Pow:
base, expr = coords.as_base_exp()
base = _lie_group_remove(base)
expr = _lie_group_remove(expr)
return base**expr
elif coords.is_Mul:
mulargs = []
coordargs = coords.args
for arg in coordargs:
if not isinstance(coords, AppliedUndef):
mulargs.append(_lie_group_remove(arg))
return Mul(*mulargs)
return coords
def infinitesimals(eq, func=None, order=None, hint='default', match=None):
r"""
The infinitesimal functions of an ordinary differential equation, `\xi(x,y)`
and `\eta(x,y)`, are the infinitesimals of the Lie group of point transformations
for which the differential equation is invariant. So, the ODE `y'=f(x,y)`
would admit a Lie group `x^*=X(x,y;\varepsilon)=x+\varepsilon\xi(x,y)`,
`y^*=Y(x,y;\varepsilon)=y+\varepsilon\eta(x,y)` such that `(y^*)'=f(x^*, y^*)`.
A change of coordinates, to `r(x,y)` and `s(x,y)`, can be performed so this Lie group
becomes the translation group, `r^*=r` and `s^*=s+\varepsilon`.
They are tangents to the coordinate curves of the new system.
Consider the transformation `(x, y) \to (X, Y)` such that the
differential equation remains invariant. `\xi` and `\eta` are the tangents to
the transformed coordinates `X` and `Y`, at `\varepsilon=0`.
.. math:: \left(\frac{\partial X(x,y;\varepsilon)}{\partial\varepsilon
}\right)|_{\varepsilon=0} = \xi,
\left(\frac{\partial Y(x,y;\varepsilon)}{\partial\varepsilon
}\right)|_{\varepsilon=0} = \eta,
The infinitesimals can be found by solving the following PDE:
>>> from sympy import Function, Eq, pprint
>>> from sympy.abc import x, y
>>> xi, eta, h = map(Function, ['xi', 'eta', 'h'])
>>> h = h(x, y) # dy/dx = h
>>> eta = eta(x, y)
>>> xi = xi(x, y)
>>> genform = Eq(eta.diff(x) + (eta.diff(y) - xi.diff(x))*h
... - (xi.diff(y))*h**2 - xi*(h.diff(x)) - eta*(h.diff(y)), 0)
>>> pprint(genform)
/d d \ d 2 d
|--(eta(x, y)) - --(xi(x, y))|*h(x, y) - eta(x, y)*--(h(x, y)) - h (x, y)*--(x
\dy dx / dy dy
<BLANKLINE>
d d
i(x, y)) - xi(x, y)*--(h(x, y)) + --(eta(x, y)) = 0
dx dx
Solving the above mentioned PDE is not trivial, and can be solved only by
making intelligent assumptions for `\xi` and `\eta` (heuristics). Once an
infinitesimal is found, the attempt to find more heuristics stops. This is done to
optimise the speed of solving the differential equation. If a list of all the
infinitesimals is needed, ``hint`` should be flagged as ``all``, which gives
the complete list of infinitesimals. If the infinitesimals for a particular
heuristic needs to be found, it can be passed as a flag to ``hint``.
Examples
========
>>> from sympy import Function
>>> from sympy.solvers.ode.ode import infinitesimals
>>> from sympy.abc import x
>>> f = Function('f')
>>> eq = f(x).diff(x) - x**2*f(x)
>>> infinitesimals(eq)
[{eta(x, f(x)): exp(x**3/3), xi(x, f(x)): 0}]
References
==========
- Solving differential equations by Symmetry Groups,
John Starrett, pp. 1 - pp. 14
"""
if isinstance(eq, Equality):
eq = eq.lhs - eq.rhs
if not func:
eq, func = _preprocess(eq)
variables = func.args
if len(variables) != 1:
raise ValueError("ODE's have only one independent variable")
else:
x = variables[0]
if not order:
order = ode_order(eq, func)
if order != 1:
raise NotImplementedError("Infinitesimals for only "
"first order ODE's have been implemented")
else:
df = func.diff(x)
# Matching differential equation of the form a*df + b
a = Wild('a', exclude = [df])
b = Wild('b', exclude = [df])
if match: # Used by lie_group hint
h = match['h']
y = match['y']
else:
match = collect(expand(eq), df).match(a*df + b)
if match:
h = -simplify(match[b]/match[a])
else:
try:
sol = solve(eq, df)
except NotImplementedError:
raise NotImplementedError("Infinitesimals for the "
"first order ODE could not be found")
else:
h = sol[0] # Find infinitesimals for one solution
y = Dummy("y")
h = h.subs(func, y)
u = Dummy("u")
hx = h.diff(x)
hy = h.diff(y)
hinv = ((1/h).subs([(x, u), (y, x)])).subs(u, y) # Inverse ODE
match = {'h': h, 'func': func, 'hx': hx, 'hy': hy, 'y': y, 'hinv': hinv}
if hint == 'all':
xieta = []
for heuristic in lie_heuristics:
function = globals()['lie_heuristic_' + heuristic]
inflist = function(match, comp=True)
if inflist:
xieta.extend([inf for inf in inflist if inf not in xieta])
if xieta:
return xieta
else:
raise NotImplementedError("Infinitesimals could not be found for "
"the given ODE")
elif hint == 'default':
for heuristic in lie_heuristics:
function = globals()['lie_heuristic_' + heuristic]
xieta = function(match, comp=False)
if xieta:
return xieta
raise NotImplementedError("Infinitesimals could not be found for"
" the given ODE")
elif hint not in lie_heuristics:
raise ValueError("Heuristic not recognized: " + hint)
else:
function = globals()['lie_heuristic_' + hint]
xieta = function(match, comp=True)
if xieta:
return xieta
else:
raise ValueError("Infinitesimals could not be found using the"
" given heuristic")
def lie_heuristic_abaco1_simple(match, comp=False):
r"""
The first heuristic uses the following four sets of
assumptions on `\xi` and `\eta`
.. math:: \xi = 0, \eta = f(x)
.. math:: \xi = 0, \eta = f(y)
.. math:: \xi = f(x), \eta = 0
.. math:: \xi = f(y), \eta = 0
The success of this heuristic is determined by algebraic factorisation.
For the first assumption `\xi = 0` and `\eta` to be a function of `x`, the PDE
.. math:: \frac{\partial \eta}{\partial x} + (\frac{\partial \eta}{\partial y}
- \frac{\partial \xi}{\partial x})*h
- \frac{\partial \xi}{\partial y}*h^{2}
- \xi*\frac{\partial h}{\partial x} - \eta*\frac{\partial h}{\partial y} = 0
reduces to `f'(x) - f\frac{\partial h}{\partial y} = 0`
If `\frac{\partial h}{\partial y}` is a function of `x`, then this can usually
be integrated easily. A similar idea is applied to the other 3 assumptions as well.
References
==========
- E.S Cheb-Terrab, L.G.S Duarte and L.A,C.P da Mota, Computer Algebra
Solving of First Order ODEs Using Symmetry Methods, pp. 8
"""
xieta = []
y = match['y']
h = match['h']
func = match['func']
x = func.args[0]
hx = match['hx']
hy = match['hy']
xi = Function('xi')(x, func)
eta = Function('eta')(x, func)
hysym = hy.free_symbols
if y not in hysym:
try:
fx = exp(integrate(hy, x))
except NotImplementedError:
pass
else:
inf = {xi: S.Zero, eta: fx}
if not comp:
return [inf]
if comp and inf not in xieta:
xieta.append(inf)
factor = hy/h
facsym = factor.free_symbols
if x not in facsym:
try:
fy = exp(integrate(factor, y))
except NotImplementedError:
pass
else:
inf = {xi: S.Zero, eta: fy.subs(y, func)}
if not comp:
return [inf]
if comp and inf not in xieta:
xieta.append(inf)
factor = -hx/h
facsym = factor.free_symbols
if y not in facsym:
try:
fx = exp(integrate(factor, x))
except NotImplementedError:
pass
else:
inf = {xi: fx, eta: S.Zero}
if not comp:
return [inf]
if comp and inf not in xieta:
xieta.append(inf)
factor = -hx/(h**2)
facsym = factor.free_symbols
if x not in facsym:
try:
fy = exp(integrate(factor, y))
except NotImplementedError:
pass
else:
inf = {xi: fy.subs(y, func), eta: S.Zero}
if not comp:
return [inf]
if comp and inf not in xieta:
xieta.append(inf)
if xieta:
return xieta
def lie_heuristic_abaco1_product(match, comp=False):
r"""
The second heuristic uses the following two assumptions on `\xi` and `\eta`
.. math:: \eta = 0, \xi = f(x)*g(y)
.. math:: \eta = f(x)*g(y), \xi = 0
The first assumption of this heuristic holds good if
`\frac{1}{h^{2}}\frac{\partial^2}{\partial x \partial y}\log(h)` is
separable in `x` and `y`, then the separated factors containing `x`
is `f(x)`, and `g(y)` is obtained by
.. math:: e^{\int f\frac{\partial}{\partial x}\left(\frac{1}{f*h}\right)\,dy}
provided `f\frac{\partial}{\partial x}\left(\frac{1}{f*h}\right)` is a function
of `y` only.
The second assumption holds good if `\frac{dy}{dx} = h(x, y)` is rewritten as
`\frac{dy}{dx} = \frac{1}{h(y, x)}` and the same properties of the first assumption
satisfies. After obtaining `f(x)` and `g(y)`, the coordinates are again
interchanged, to get `\eta` as `f(x)*g(y)`
References
==========
- E.S. Cheb-Terrab, A.D. Roche, Symmetries and First Order
ODE Patterns, pp. 7 - pp. 8
"""
xieta = []
y = match['y']
h = match['h']
hinv = match['hinv']
func = match['func']
x = func.args[0]
xi = Function('xi')(x, func)
eta = Function('eta')(x, func)
inf = separatevars(((log(h).diff(y)).diff(x))/h**2, dict=True, symbols=[x, y])
if inf and inf['coeff']:
fx = inf[x]
gy = simplify(fx*((1/(fx*h)).diff(x)))
gysyms = gy.free_symbols
if x not in gysyms:
gy = exp(integrate(gy, y))
inf = {eta: S.Zero, xi: (fx*gy).subs(y, func)}
if not comp:
return [inf]
if comp and inf not in xieta:
xieta.append(inf)
u1 = Dummy("u1")
inf = separatevars(((log(hinv).diff(y)).diff(x))/hinv**2, dict=True, symbols=[x, y])
if inf and inf['coeff']:
fx = inf[x]
gy = simplify(fx*((1/(fx*hinv)).diff(x)))
gysyms = gy.free_symbols
if x not in gysyms:
gy = exp(integrate(gy, y))
etaval = fx*gy
etaval = (etaval.subs([(x, u1), (y, x)])).subs(u1, y)
inf = {eta: etaval.subs(y, func), xi: S.Zero}
if not comp:
return [inf]
if comp and inf not in xieta:
xieta.append(inf)
if xieta:
return xieta
def lie_heuristic_bivariate(match, comp=False):
r"""
The third heuristic assumes the infinitesimals `\xi` and `\eta`
to be bi-variate polynomials in `x` and `y`. The assumption made here
for the logic below is that `h` is a rational function in `x` and `y`
though that may not be necessary for the infinitesimals to be
bivariate polynomials. The coefficients of the infinitesimals
are found out by substituting them in the PDE and grouping similar terms
that are polynomials and since they form a linear system, solve and check
for non trivial solutions. The degree of the assumed bivariates
are increased till a certain maximum value.
References
==========
- Lie Groups and Differential Equations
pp. 327 - pp. 329
"""
h = match['h']
hx = match['hx']
hy = match['hy']
func = match['func']
x = func.args[0]
y = match['y']
xi = Function('xi')(x, func)
eta = Function('eta')(x, func)
if h.is_rational_function():
# The maximum degree that the infinitesimals can take is
# calculated by this technique.
etax, etay, etad, xix, xiy, xid = symbols("etax etay etad xix xiy xid")
ipde = etax + (etay - xix)*h - xiy*h**2 - xid*hx - etad*hy
num, denom = cancel(ipde).as_numer_denom()
deg = Poly(num, x, y).total_degree()
deta = Function('deta')(x, y)
dxi = Function('dxi')(x, y)
ipde = (deta.diff(x) + (deta.diff(y) - dxi.diff(x))*h - (dxi.diff(y))*h**2
- dxi*hx - deta*hy)
xieq = Symbol("xi0")
etaeq = Symbol("eta0")
for i in range(deg + 1):
if i:
xieq += Add(*[
Symbol("xi_" + str(power) + "_" + str(i - power))*x**power*y**(i - power)
for power in range(i + 1)])
etaeq += Add(*[
Symbol("eta_" + str(power) + "_" + str(i - power))*x**power*y**(i - power)
for power in range(i + 1)])
pden, denom = (ipde.subs({dxi: xieq, deta: etaeq}).doit()).as_numer_denom()
pden = expand(pden)
# If the individual terms are monomials, the coefficients
# are grouped
if pden.is_polynomial(x, y) and pden.is_Add:
polyy = Poly(pden, x, y).as_dict()
if polyy:
symset = xieq.free_symbols.union(etaeq.free_symbols) - {x, y}
soldict = solve(polyy.values(), *symset)
if isinstance(soldict, list):
soldict = soldict[0]
if any(soldict.values()):
xired = xieq.subs(soldict)
etared = etaeq.subs(soldict)
# Scaling is done by substituting one for the parameters
# This can be any number except zero.
dict_ = dict((sym, 1) for sym in symset)
inf = {eta: etared.subs(dict_).subs(y, func),
xi: xired.subs(dict_).subs(y, func)}
return [inf]
def lie_heuristic_chi(match, comp=False):
r"""
The aim of the fourth heuristic is to find the function `\chi(x, y)`
that satisfies the PDE `\frac{d\chi}{dx} + h\frac{d\chi}{dx}
- \frac{\partial h}{\partial y}\chi = 0`.
This assumes `\chi` to be a bivariate polynomial in `x` and `y`. By intuition,
`h` should be a rational function in `x` and `y`. The method used here is
to substitute a general binomial for `\chi` up to a certain maximum degree
is reached. The coefficients of the polynomials, are calculated by by collecting
terms of the same order in `x` and `y`.
After finding `\chi`, the next step is to use `\eta = \xi*h + \chi`, to
determine `\xi` and `\eta`. This can be done by dividing `\chi` by `h`
which would give `-\xi` as the quotient and `\eta` as the remainder.
References
==========
- E.S Cheb-Terrab, L.G.S Duarte and L.A,C.P da Mota, Computer Algebra
Solving of First Order ODEs Using Symmetry Methods, pp. 8
"""
h = match['h']
hy = match['hy']
func = match['func']
x = func.args[0]
y = match['y']
xi = Function('xi')(x, func)
eta = Function('eta')(x, func)
if h.is_rational_function():
schi, schix, schiy = symbols("schi, schix, schiy")
cpde = schix + h*schiy - hy*schi
num, denom = cancel(cpde).as_numer_denom()
deg = Poly(num, x, y).total_degree()
chi = Function('chi')(x, y)
chix = chi.diff(x)
chiy = chi.diff(y)
cpde = chix + h*chiy - hy*chi
chieq = Symbol("chi")
for i in range(1, deg + 1):
chieq += Add(*[
Symbol("chi_" + str(power) + "_" + str(i - power))*x**power*y**(i - power)
for power in range(i + 1)])
cnum, cden = cancel(cpde.subs({chi : chieq}).doit()).as_numer_denom()
cnum = expand(cnum)
if cnum.is_polynomial(x, y) and cnum.is_Add:
cpoly = Poly(cnum, x, y).as_dict()
if cpoly:
solsyms = chieq.free_symbols - {x, y}
soldict = solve(cpoly.values(), *solsyms)
if isinstance(soldict, list):
soldict = soldict[0]
if any(soldict.values()):
chieq = chieq.subs(soldict)
dict_ = dict((sym, 1) for sym in solsyms)
chieq = chieq.subs(dict_)
# After finding chi, the main aim is to find out
# eta, xi by the equation eta = xi*h + chi
# One method to set xi, would be rearranging it to
# (eta/h) - xi = (chi/h). This would mean dividing
# chi by h would give -xi as the quotient and eta
# as the remainder. Thanks to Sean Vig for suggesting
# this method.
xic, etac = div(chieq, h)
inf = {eta: etac.subs(y, func), xi: -xic.subs(y, func)}
return [inf]
def lie_heuristic_function_sum(match, comp=False):
r"""
This heuristic uses the following two assumptions on `\xi` and `\eta`
.. math:: \eta = 0, \xi = f(x) + g(y)
.. math:: \eta = f(x) + g(y), \xi = 0
The first assumption of this heuristic holds good if
.. math:: \frac{\partial}{\partial y}[(h\frac{\partial^{2}}{
\partial x^{2}}(h^{-1}))^{-1}]
is separable in `x` and `y`,
1. The separated factors containing `y` is `\frac{\partial g}{\partial y}`.
From this `g(y)` can be determined.
2. The separated factors containing `x` is `f''(x)`.
3. `h\frac{\partial^{2}}{\partial x^{2}}(h^{-1})` equals
`\frac{f''(x)}{f(x) + g(y)}`. From this `f(x)` can be determined.
The second assumption holds good if `\frac{dy}{dx} = h(x, y)` is rewritten as
`\frac{dy}{dx} = \frac{1}{h(y, x)}` and the same properties of the first
assumption satisfies. After obtaining `f(x)` and `g(y)`, the coordinates
are again interchanged, to get `\eta` as `f(x) + g(y)`.
For both assumptions, the constant factors are separated among `g(y)`
and `f''(x)`, such that `f''(x)` obtained from 3] is the same as that
obtained from 2]. If not possible, then this heuristic fails.
References
==========
- E.S. Cheb-Terrab, A.D. Roche, Symmetries and First Order
ODE Patterns, pp. 7 - pp. 8
"""
xieta = []
h = match['h']
func = match['func']
hinv = match['hinv']
x = func.args[0]
y = match['y']
xi = Function('xi')(x, func)
eta = Function('eta')(x, func)
for odefac in [h, hinv]:
factor = odefac*((1/odefac).diff(x, 2))
sep = separatevars((1/factor).diff(y), dict=True, symbols=[x, y])
if sep and sep['coeff'] and sep[x].has(x) and sep[y].has(y):
k = Dummy("k")
try:
gy = k*integrate(sep[y], y)
except NotImplementedError:
pass
else:
fdd = 1/(k*sep[x]*sep['coeff'])
fx = simplify(fdd/factor - gy)
check = simplify(fx.diff(x, 2) - fdd)
if fx:
if not check:
fx = fx.subs(k, 1)
gy = (gy/k)
else:
sol = solve(check, k)
if sol:
sol = sol[0]
fx = fx.subs(k, sol)
gy = (gy/k)*sol
else:
continue
if odefac == hinv: # Inverse ODE
fx = fx.subs(x, y)
gy = gy.subs(y, x)
etaval = factor_terms(fx + gy)
if etaval.is_Mul:
etaval = Mul(*[arg for arg in etaval.args if arg.has(x, y)])
if odefac == hinv: # Inverse ODE
inf = {eta: etaval.subs(y, func), xi : S.Zero}
else:
inf = {xi: etaval.subs(y, func), eta : S.Zero}
if not comp:
return [inf]
else:
xieta.append(inf)
if xieta:
return xieta
def lie_heuristic_abaco2_similar(match, comp=False):
r"""
This heuristic uses the following two assumptions on `\xi` and `\eta`
.. math:: \eta = g(x), \xi = f(x)
.. math:: \eta = f(y), \xi = g(y)
For the first assumption,
1. First `\frac{\frac{\partial h}{\partial y}}{\frac{\partial^{2} h}{
\partial yy}}` is calculated. Let us say this value is A
2. If this is constant, then `h` is matched to the form `A(x) + B(x)e^{
\frac{y}{C}}` then, `\frac{e^{\int \frac{A(x)}{C} \,dx}}{B(x)}` gives `f(x)`
and `A(x)*f(x)` gives `g(x)`
3. Otherwise `\frac{\frac{\partial A}{\partial X}}{\frac{\partial A}{
\partial Y}} = \gamma` is calculated. If
a] `\gamma` is a function of `x` alone
b] `\frac{\gamma\frac{\partial h}{\partial y} - \gamma'(x) - \frac{
\partial h}{\partial x}}{h + \gamma} = G` is a function of `x` alone.
then, `e^{\int G \,dx}` gives `f(x)` and `-\gamma*f(x)` gives `g(x)`
The second assumption holds good if `\frac{dy}{dx} = h(x, y)` is rewritten as
`\frac{dy}{dx} = \frac{1}{h(y, x)}` and the same properties of the first assumption
satisfies. After obtaining `f(x)` and `g(x)`, the coordinates are again
interchanged, to get `\xi` as `f(x^*)` and `\eta` as `g(y^*)`
References
==========
- E.S. Cheb-Terrab, A.D. Roche, Symmetries and First Order
ODE Patterns, pp. 10 - pp. 12
"""
h = match['h']
hx = match['hx']
hy = match['hy']
func = match['func']
hinv = match['hinv']
x = func.args[0]
y = match['y']
xi = Function('xi')(x, func)
eta = Function('eta')(x, func)
factor = cancel(h.diff(y)/h.diff(y, 2))
factorx = factor.diff(x)
factory = factor.diff(y)
if not factor.has(x) and not factor.has(y):
A = Wild('A', exclude=[y])
B = Wild('B', exclude=[y])
C = Wild('C', exclude=[x, y])
match = h.match(A + B*exp(y/C))
try:
tau = exp(-integrate(match[A]/match[C]), x)/match[B]
except NotImplementedError:
pass
else:
gx = match[A]*tau
return [{xi: tau, eta: gx}]
else:
gamma = cancel(factorx/factory)
if not gamma.has(y):
tauint = cancel((gamma*hy - gamma.diff(x) - hx)/(h + gamma))
if not tauint.has(y):
try:
tau = exp(integrate(tauint, x))
except NotImplementedError:
pass
else:
gx = -tau*gamma
return [{xi: tau, eta: gx}]
factor = cancel(hinv.diff(y)/hinv.diff(y, 2))
factorx = factor.diff(x)
factory = factor.diff(y)
if not factor.has(x) and not factor.has(y):
A = Wild('A', exclude=[y])
B = Wild('B', exclude=[y])
C = Wild('C', exclude=[x, y])
match = h.match(A + B*exp(y/C))
try:
tau = exp(-integrate(match[A]/match[C]), x)/match[B]
except NotImplementedError:
pass
else:
gx = match[A]*tau
return [{eta: tau.subs(x, func), xi: gx.subs(x, func)}]
else:
gamma = cancel(factorx/factory)
if not gamma.has(y):
tauint = cancel((gamma*hinv.diff(y) - gamma.diff(x) - hinv.diff(x))/(
hinv + gamma))
if not tauint.has(y):
try:
tau = exp(integrate(tauint, x))
except NotImplementedError:
pass
else:
gx = -tau*gamma
return [{eta: tau.subs(x, func), xi: gx.subs(x, func)}]
def lie_heuristic_abaco2_unique_unknown(match, comp=False):
r"""
This heuristic assumes the presence of unknown functions or known functions
with non-integer powers.
1. A list of all functions and non-integer powers containing x and y
2. Loop over each element `f` in the list, find `\frac{\frac{\partial f}{\partial x}}{
\frac{\partial f}{\partial x}} = R`
If it is separable in `x` and `y`, let `X` be the factors containing `x`. Then
a] Check if `\xi = X` and `\eta = -\frac{X}{R}` satisfy the PDE. If yes, then return
`\xi` and `\eta`
b] Check if `\xi = \frac{-R}{X}` and `\eta = -\frac{1}{X}` satisfy the PDE.
If yes, then return `\xi` and `\eta`
If not, then check if
a] :math:`\xi = -R,\eta = 1`
b] :math:`\xi = 1, \eta = -\frac{1}{R}`
are solutions.
References
==========
- E.S. Cheb-Terrab, A.D. Roche, Symmetries and First Order
ODE Patterns, pp. 10 - pp. 12
"""
h = match['h']
hx = match['hx']
hy = match['hy']
func = match['func']
x = func.args[0]
y = match['y']
xi = Function('xi')(x, func)
eta = Function('eta')(x, func)
funclist = []
for atom in h.atoms(Pow):
base, exp = atom.as_base_exp()
if base.has(x) and base.has(y):
if not exp.is_Integer:
funclist.append(atom)
for function in h.atoms(AppliedUndef):
syms = function.free_symbols
if x in syms and y in syms:
funclist.append(function)
for f in funclist:
frac = cancel(f.diff(y)/f.diff(x))
sep = separatevars(frac, dict=True, symbols=[x, y])
if sep and sep['coeff']:
xitry1 = sep[x]
etatry1 = -1/(sep[y]*sep['coeff'])
pde1 = etatry1.diff(y)*h - xitry1.diff(x)*h - xitry1*hx - etatry1*hy
if not simplify(pde1):
return [{xi: xitry1, eta: etatry1.subs(y, func)}]
xitry2 = 1/etatry1
etatry2 = 1/xitry1
pde2 = etatry2.diff(x) - (xitry2.diff(y))*h**2 - xitry2*hx - etatry2*hy
if not simplify(expand(pde2)):
return [{xi: xitry2.subs(y, func), eta: etatry2}]
else:
etatry = -1/frac
pde = etatry.diff(x) + etatry.diff(y)*h - hx - etatry*hy
if not simplify(pde):
return [{xi: S.One, eta: etatry.subs(y, func)}]
xitry = -frac
pde = -xitry.diff(x)*h -xitry.diff(y)*h**2 - xitry*hx -hy
if not simplify(expand(pde)):
return [{xi: xitry.subs(y, func), eta: S.One}]
def lie_heuristic_abaco2_unique_general(match, comp=False):
r"""
This heuristic finds if infinitesimals of the form `\eta = f(x)`, `\xi = g(y)`
without making any assumptions on `h`.
The complete sequence of steps is given in the paper mentioned below.
References
==========
- E.S. Cheb-Terrab, A.D. Roche, Symmetries and First Order
ODE Patterns, pp. 10 - pp. 12
"""
hx = match['hx']
hy = match['hy']
func = match['func']
x = func.args[0]
y = match['y']
xi = Function('xi')(x, func)
eta = Function('eta')(x, func)
A = hx.diff(y)
B = hy.diff(y) + hy**2
C = hx.diff(x) - hx**2
if not (A and B and C):
return
Ax = A.diff(x)
Ay = A.diff(y)
Axy = Ax.diff(y)
Axx = Ax.diff(x)
Ayy = Ay.diff(y)
D = simplify(2*Axy + hx*Ay - Ax*hy + (hx*hy + 2*A)*A)*A - 3*Ax*Ay
if not D:
E1 = simplify(3*Ax**2 + ((hx**2 + 2*C)*A - 2*Axx)*A)
if E1:
E2 = simplify((2*Ayy + (2*B - hy**2)*A)*A - 3*Ay**2)
if not E2:
E3 = simplify(
E1*((28*Ax + 4*hx*A)*A**3 - E1*(hy*A + Ay)) - E1.diff(x)*8*A**4)
if not E3:
etaval = cancel((4*A**3*(Ax - hx*A) + E1*(hy*A - Ay))/(S(2)*A*E1))
if x not in etaval:
try:
etaval = exp(integrate(etaval, y))
except NotImplementedError:
pass
else:
xival = -4*A**3*etaval/E1
if y not in xival:
return [{xi: xival, eta: etaval.subs(y, func)}]
else:
E1 = simplify((2*Ayy + (2*B - hy**2)*A)*A - 3*Ay**2)
if E1:
E2 = simplify(
4*A**3*D - D**2 + E1*((2*Axx - (hx**2 + 2*C)*A)*A - 3*Ax**2))
if not E2:
E3 = simplify(
-(A*D)*E1.diff(y) + ((E1.diff(x) - hy*D)*A + 3*Ay*D +
(A*hx - 3*Ax)*E1)*E1)
if not E3:
etaval = cancel(((A*hx - Ax)*E1 - (Ay + A*hy)*D)/(S(2)*A*D))
if x not in etaval:
try:
etaval = exp(integrate(etaval, y))
except NotImplementedError:
pass
else:
xival = -E1*etaval/D
if y not in xival:
return [{xi: xival, eta: etaval.subs(y, func)}]
def lie_heuristic_linear(match, comp=False):
r"""
This heuristic assumes
1. `\xi = ax + by + c` and
2. `\eta = fx + gy + h`
After substituting the following assumptions in the determining PDE, it
reduces to
.. math:: f + (g - a)h - bh^{2} - (ax + by + c)\frac{\partial h}{\partial x}
- (fx + gy + c)\frac{\partial h}{\partial y}
Solving the reduced PDE obtained, using the method of characteristics, becomes
impractical. The method followed is grouping similar terms and solving the system
of linear equations obtained. The difference between the bivariate heuristic is that
`h` need not be a rational function in this case.
References
==========
- E.S. Cheb-Terrab, A.D. Roche, Symmetries and First Order
ODE Patterns, pp. 10 - pp. 12
"""
h = match['h']
hx = match['hx']
hy = match['hy']
func = match['func']
x = func.args[0]
y = match['y']
xi = Function('xi')(x, func)
eta = Function('eta')(x, func)
coeffdict = {}
symbols = numbered_symbols("c", cls=Dummy)
symlist = [next(symbols) for _ in islice(symbols, 6)]
C0, C1, C2, C3, C4, C5 = symlist
pde = C3 + (C4 - C0)*h - (C0*x + C1*y + C2)*hx - (C3*x + C4*y + C5)*hy - C1*h**2
pde, denom = pde.as_numer_denom()
pde = powsimp(expand(pde))
if pde.is_Add:
terms = pde.args
for term in terms:
if term.is_Mul:
rem = Mul(*[m for m in term.args if not m.has(x, y)])
xypart = term/rem
if xypart not in coeffdict:
coeffdict[xypart] = rem
else:
coeffdict[xypart] += rem
else:
if term not in coeffdict:
coeffdict[term] = S.One
else:
coeffdict[term] += S.One
sollist = coeffdict.values()
soldict = solve(sollist, symlist)
if soldict:
if isinstance(soldict, list):
soldict = soldict[0]
subval = soldict.values()
if any(t for t in subval):
onedict = dict(zip(symlist, [1]*6))
xival = C0*x + C1*func + C2
etaval = C3*x + C4*func + C5
xival = xival.subs(soldict)
etaval = etaval.subs(soldict)
xival = xival.subs(onedict)
etaval = etaval.subs(onedict)
return [{xi: xival, eta: etaval}]
def sysode_linear_2eq_order1(match_):
x = match_['func'][0].func
y = match_['func'][1].func
func = match_['func']
fc = match_['func_coeff']
eq = match_['eq']
r = dict()
t = list(list(eq[0].atoms(Derivative))[0].atoms(Symbol))[0]
for i in range(2):
eqs = 0
for terms in Add.make_args(eq[i]):
eqs += terms/fc[i,func[i],1]
eq[i] = eqs
# for equations Eq(a1*diff(x(t),t), a*x(t) + b*y(t) + k1)
# and Eq(a2*diff(x(t),t), c*x(t) + d*y(t) + k2)
r['a'] = -fc[0,x(t),0]/fc[0,x(t),1]
r['c'] = -fc[1,x(t),0]/fc[1,y(t),1]
r['b'] = -fc[0,y(t),0]/fc[0,x(t),1]
r['d'] = -fc[1,y(t),0]/fc[1,y(t),1]
forcing = [S.Zero,S.Zero]
for i in range(2):
for j in Add.make_args(eq[i]):
if not j.has(x(t), y(t)):
forcing[i] += j
if not (forcing[0].has(t) or forcing[1].has(t)):
r['k1'] = forcing[0]
r['k2'] = forcing[1]
else:
raise NotImplementedError("Only homogeneous problems are supported" +
" (and constant inhomogeneity)")
if match_['type_of_equation'] == 'type2':
gsol = _linear_2eq_order1_type1(x, y, t, r, eq)
psol = _linear_2eq_order1_type2(x, y, t, r, eq)
sol = [Eq(x(t), gsol[0].rhs+psol[0]), Eq(y(t), gsol[1].rhs+psol[1])]
if match_['type_of_equation'] == 'type6':
sol = _linear_2eq_order1_type6(x, y, t, r, eq)
if match_['type_of_equation'] == 'type7':
sol = _linear_2eq_order1_type7(x, y, t, r, eq)
return sol
# To remove Linear 2 Eq, Order 1, Type 1 when
# Linear 2 Eq, Order 1, Type 2 is removed.
def _linear_2eq_order1_type1(x, y, t, r, eq):
r"""
It is classified under system of two linear homogeneous first-order constant-coefficient
ordinary differential equations.
The equations which come under this type are
.. math:: x' = ax + by,
.. math:: y' = cx + dy
The characteristics equation is written as
.. math:: \lambda^{2} + (a+d) \lambda + ad - bc = 0
and its discriminant is `D = (a-d)^{2} + 4bc`. There are several cases
1. Case when `ad - bc \neq 0`. The origin of coordinates, `x = y = 0`,
is the only stationary point; it is
- a node if `D = 0`
- a node if `D > 0` and `ad - bc > 0`
- a saddle if `D > 0` and `ad - bc < 0`
- a focus if `D < 0` and `a + d \neq 0`
- a centre if `D < 0` and `a + d \neq 0`.
1.1. If `D > 0`. The characteristic equation has two distinct real roots
`\lambda_1` and `\lambda_ 2` . The general solution of the system in question is expressed as
.. math:: x = C_1 b e^{\lambda_1 t} + C_2 b e^{\lambda_2 t}
.. math:: y = C_1 (\lambda_1 - a) e^{\lambda_1 t} + C_2 (\lambda_2 - a) e^{\lambda_2 t}
where `C_1` and `C_2` being arbitrary constants
1.2. If `D < 0`. The characteristics equation has two conjugate
roots, `\lambda_1 = \sigma + i \beta` and `\lambda_2 = \sigma - i \beta`.
The general solution of the system is given by
.. math:: x = b e^{\sigma t} (C_1 \sin(\beta t) + C_2 \cos(\beta t))
.. math:: y = e^{\sigma t} ([(\sigma - a) C_1 - \beta C_2] \sin(\beta t) + [\beta C_1 + (\sigma - a) C_2 \cos(\beta t)])
1.3. If `D = 0` and `a \neq d`. The characteristic equation has
two equal roots, `\lambda_1 = \lambda_2`. The general solution of the system is written as
.. math:: x = 2b (C_1 + \frac{C_2}{a-d} + C_2 t) e^{\frac{a+d}{2} t}
.. math:: y = [(d - a) C_1 + C_2 + (d - a) C_2 t] e^{\frac{a+d}{2} t}
1.4. If `D = 0` and `a = d \neq 0` and `b = 0`
.. math:: x = C_1 e^{a t} , y = (c C_1 t + C_2) e^{a t}
1.5. If `D = 0` and `a = d \neq 0` and `c = 0`
.. math:: x = (b C_1 t + C_2) e^{a t} , y = C_1 e^{a t}
2. Case when `ad - bc = 0` and `a^{2} + b^{2} > 0`. The whole straight
line `ax + by = 0` consists of singular points. The original system of differential
equations can be rewritten as
.. math:: x' = ax + by , y' = k (ax + by)
2.1 If `a + bk \neq 0`, solution will be
.. math:: x = b C_1 + C_2 e^{(a + bk) t} , y = -a C_1 + k C_2 e^{(a + bk) t}
2.2 If `a + bk = 0`, solution will be
.. math:: x = C_1 (bk t - 1) + b C_2 t , y = k^{2} b C_1 t + (b k^{2} t + 1) C_2
"""
C1, C2 = get_numbered_constants(eq, num=2)
a, b, c, d = r['a'], r['b'], r['c'], r['d']
real_coeff = all(v.is_real for v in (a, b, c, d))
D = (a - d)**2 + 4*b*c
l1 = (a + d + sqrt(D))/2
l2 = (a + d - sqrt(D))/2
equal_roots = Eq(D, 0).expand()
gsol1, gsol2 = [], []
# Solutions have exponential form if either D > 0 with real coefficients
# or D != 0 with complex coefficients. Eigenvalues are distinct.
# For each eigenvalue lam, pick an eigenvector, making sure we don't get (0, 0)
# The candidates are (b, lam-a) and (lam-d, c).
exponential_form = D > 0 if real_coeff else Not(equal_roots)
bad_ab_vector1 = And(Eq(b, 0), Eq(l1, a))
bad_ab_vector2 = And(Eq(b, 0), Eq(l2, a))
vector1 = Matrix((Piecewise((l1 - d, bad_ab_vector1), (b, True)),
Piecewise((c, bad_ab_vector1), (l1 - a, True))))
vector2 = Matrix((Piecewise((l2 - d, bad_ab_vector2), (b, True)),
Piecewise((c, bad_ab_vector2), (l2 - a, True))))
sol_vector = C1*exp(l1*t)*vector1 + C2*exp(l2*t)*vector2
gsol1.append((sol_vector[0], exponential_form))
gsol2.append((sol_vector[1], exponential_form))
# Solutions have trigonometric form for real coefficients with D < 0
# Both b and c are nonzero in this case, so (b, lam-a) is an eigenvector
# It splits into real/imag parts as (b, sigma-a) and (0, beta). Then
# multiply it by C1(cos(beta*t) + I*C2*sin(beta*t)) and separate real/imag
trigonometric_form = D < 0 if real_coeff else False
sigma = re(l1)
if im(l1).is_positive:
beta = im(l1)
else:
beta = im(l2)
vector1 = Matrix((b, sigma - a))
vector2 = Matrix((0, beta))
sol_vector = exp(sigma*t) * (C1*(cos(beta*t)*vector1 - sin(beta*t)*vector2) + \
C2*(sin(beta*t)*vector1 + cos(beta*t)*vector2))
gsol1.append((sol_vector[0], trigonometric_form))
gsol2.append((sol_vector[1], trigonometric_form))
# Final case is D == 0, a single eigenvalue. If the eigenspace is 2-dimensional
# then we have a scalar matrix, deal with this case first.
scalar_matrix = And(Eq(a, d), Eq(b, 0), Eq(c, 0))
vector1 = Matrix((S.One, S.Zero))
vector2 = Matrix((S.Zero, S.One))
sol_vector = exp(l1*t) * (C1*vector1 + C2*vector2)
gsol1.append((sol_vector[0], scalar_matrix))
gsol2.append((sol_vector[1], scalar_matrix))
# Have one eigenvector. Get a generalized eigenvector from (A-lam)*vector2 = vector1
vector1 = Matrix((Piecewise((l1 - d, bad_ab_vector1), (b, True)),
Piecewise((c, bad_ab_vector1), (l1 - a, True))))
vector2 = Matrix((Piecewise((S.One, bad_ab_vector1), (S.Zero, Eq(a, l1)),
(b/(a - l1), True)),
Piecewise((S.Zero, bad_ab_vector1), (S.One, Eq(a, l1)),
(S.Zero, True))))
sol_vector = exp(l1*t) * (C1*vector1 + C2*(vector2 + t*vector1))
gsol1.append((sol_vector[0], equal_roots))
gsol2.append((sol_vector[1], equal_roots))
return [Eq(x(t), Piecewise(*gsol1)), Eq(y(t), Piecewise(*gsol2))]
def _linear_2eq_order1_type2(x, y, t, r, eq):
r"""
The equations of this type are
.. math:: x' = ax + by + k1 , y' = cx + dy + k2
The general solution of this system is given by sum of its particular solution and the
general solution of the corresponding homogeneous system is obtained from type1.
1. When `ad - bc \neq 0`. The particular solution will be
`x = x_0` and `y = y_0` where `x_0` and `y_0` are determined by solving linear system of equations
.. math:: a x_0 + b y_0 + k1 = 0 , c x_0 + d y_0 + k2 = 0
2. When `ad - bc = 0` and `a^{2} + b^{2} > 0`. In this case, the system of equation becomes
.. math:: x' = ax + by + k_1 , y' = k (ax + by) + k_2
2.1 If `\sigma = a + bk \neq 0`, particular solution is given by
.. math:: x = b \sigma^{-1} (c_1 k - c_2) t - \sigma^{-2} (a c_1 + b c_2)
.. math:: y = kx + (c_2 - c_1 k) t
2.2 If `\sigma = a + bk = 0`, particular solution is given by
.. math:: x = \frac{1}{2} b (c_2 - c_1 k) t^{2} + c_1 t
.. math:: y = kx + (c_2 - c_1 k) t
"""
r['k1'] = -r['k1']; r['k2'] = -r['k2']
if (r['a']*r['d'] - r['b']*r['c']) != 0:
x0, y0 = symbols('x0, y0', cls=Dummy)
sol = solve((r['a']*x0+r['b']*y0+r['k1'], r['c']*x0+r['d']*y0+r['k2']), x0, y0)
psol = [sol[x0], sol[y0]]
elif (r['a']*r['d'] - r['b']*r['c']) == 0 and (r['a']**2+r['b']**2) > 0:
k = r['c']/r['a']
sigma = r['a'] + r['b']*k
if sigma != 0:
sol1 = r['b']*sigma**-1*(r['k1']*k-r['k2'])*t - sigma**-2*(r['a']*r['k1']+r['b']*r['k2'])
sol2 = k*sol1 + (r['k2']-r['k1']*k)*t
else:
# FIXME: a previous typo fix shows this is not covered by tests
sol1 = r['b']*(r['k2']-r['k1']*k)*t**2 + r['k1']*t
sol2 = k*sol1 + (r['k2']-r['k1']*k)*t
psol = [sol1, sol2]
return psol
def _linear_2eq_order1_type6(x, y, t, r, eq):
r"""
The equations of this type of ode are .
.. math:: x' = f(t) x + g(t) y
.. math:: y' = a [f(t) + a h(t)] x + a [g(t) - h(t)] y
This is solved by first multiplying the first equation by `-a` and adding
it to the second equation to obtain
.. math:: y' - a x' = -a h(t) (y - a x)
Setting `U = y - ax` and integrating the equation we arrive at
.. math:: y - ax = C_1 e^{-a \int h(t) \,dt}
and on substituting the value of y in first equation give rise to first order ODEs. After solving for
`x`, we can obtain `y` by substituting the value of `x` in second equation.
"""
C1, C2, C3, C4 = get_numbered_constants(eq, num=4)
p = 0
q = 0
p1 = cancel(r['c']/cancel(r['c']/r['d']).as_numer_denom()[0])
p2 = cancel(r['a']/cancel(r['a']/r['b']).as_numer_denom()[0])
for n, i in enumerate([p1, p2]):
for j in Mul.make_args(collect_const(i)):
if not j.has(t):
q = j
if q!=0 and n==0:
if ((r['c']/j - r['a'])/(r['b'] - r['d']/j)) == j:
p = 1
s = j
break
if q!=0 and n==1:
if ((r['a']/j - r['c'])/(r['d'] - r['b']/j)) == j:
p = 2
s = j
break
if p == 1:
equ = diff(x(t),t) - r['a']*x(t) - r['b']*(s*x(t) + C1*exp(-s*Integral(r['b'] - r['d']/s, t)))
hint1 = classify_ode(equ)[1]
sol1 = dsolve(equ, hint=hint1+'_Integral').rhs
sol2 = s*sol1 + C1*exp(-s*Integral(r['b'] - r['d']/s, t))
elif p ==2:
equ = diff(y(t),t) - r['c']*y(t) - r['d']*s*y(t) + C1*exp(-s*Integral(r['d'] - r['b']/s, t))
hint1 = classify_ode(equ)[1]
sol2 = dsolve(equ, hint=hint1+'_Integral').rhs
sol1 = s*sol2 + C1*exp(-s*Integral(r['d'] - r['b']/s, t))
return [Eq(x(t), sol1), Eq(y(t), sol2)]
def _linear_2eq_order1_type7(x, y, t, r, eq):
r"""
The equations of this type of ode are .
.. math:: x' = f(t) x + g(t) y
.. math:: y' = h(t) x + p(t) y
Differentiating the first equation and substituting the value of `y`
from second equation will give a second-order linear equation
.. math:: g x'' - (fg + gp + g') x' + (fgp - g^{2} h + f g' - f' g) x = 0
This above equation can be easily integrated if following conditions are satisfied.
1. `fgp - g^{2} h + f g' - f' g = 0`
2. `fgp - g^{2} h + f g' - f' g = ag, fg + gp + g' = bg`
If first condition is satisfied then it is solved by current dsolve solver and in second case it becomes
a constant coefficient differential equation which is also solved by current solver.
Otherwise if the above condition fails then,
a particular solution is assumed as `x = x_0(t)` and `y = y_0(t)`
Then the general solution is expressed as
.. math:: x = C_1 x_0(t) + C_2 x_0(t) \int \frac{g(t) F(t) P(t)}{x_0^{2}(t)} \,dt
.. math:: y = C_1 y_0(t) + C_2 [\frac{F(t) P(t)}{x_0(t)} + y_0(t) \int \frac{g(t) F(t) P(t)}{x_0^{2}(t)} \,dt]
where C1 and C2 are arbitrary constants and
.. math:: F(t) = e^{\int f(t) \,dt} , P(t) = e^{\int p(t) \,dt}
"""
C1, C2, C3, C4 = get_numbered_constants(eq, num=4)
e1 = r['a']*r['b']*r['c'] - r['b']**2*r['c'] + r['a']*diff(r['b'],t) - diff(r['a'],t)*r['b']
e2 = r['a']*r['c']*r['d'] - r['b']*r['c']**2 + diff(r['c'],t)*r['d'] - r['c']*diff(r['d'],t)
m1 = r['a']*r['b'] + r['b']*r['d'] + diff(r['b'],t)
m2 = r['a']*r['c'] + r['c']*r['d'] + diff(r['c'],t)
if e1 == 0:
sol1 = dsolve(r['b']*diff(x(t),t,t) - m1*diff(x(t),t)).rhs
sol2 = dsolve(diff(y(t),t) - r['c']*sol1 - r['d']*y(t)).rhs
elif e2 == 0:
sol2 = dsolve(r['c']*diff(y(t),t,t) - m2*diff(y(t),t)).rhs
sol1 = dsolve(diff(x(t),t) - r['a']*x(t) - r['b']*sol2).rhs
elif not (e1/r['b']).has(t) and not (m1/r['b']).has(t):
sol1 = dsolve(diff(x(t),t,t) - (m1/r['b'])*diff(x(t),t) - (e1/r['b'])*x(t)).rhs
sol2 = dsolve(diff(y(t),t) - r['c']*sol1 - r['d']*y(t)).rhs
elif not (e2/r['c']).has(t) and not (m2/r['c']).has(t):
sol2 = dsolve(diff(y(t),t,t) - (m2/r['c'])*diff(y(t),t) - (e2/r['c'])*y(t)).rhs
sol1 = dsolve(diff(x(t),t) - r['a']*x(t) - r['b']*sol2).rhs
else:
x0 = Function('x0')(t) # x0 and y0 being particular solutions
y0 = Function('y0')(t)
F = exp(Integral(r['a'],t))
P = exp(Integral(r['d'],t))
sol1 = C1*x0 + C2*x0*Integral(r['b']*F*P/x0**2, t)
sol2 = C1*y0 + C2*(F*P/x0 + y0*Integral(r['b']*F*P/x0**2, t))
return [Eq(x(t), sol1), Eq(y(t), sol2)]
def sysode_linear_2eq_order2(match_):
x = match_['func'][0].func
y = match_['func'][1].func
func = match_['func']
fc = match_['func_coeff']
eq = match_['eq']
r = dict()
t = list(list(eq[0].atoms(Derivative))[0].atoms(Symbol))[0]
for i in range(2):
eqs = []
for terms in Add.make_args(eq[i]):
eqs.append(terms/fc[i,func[i],2])
eq[i] = Add(*eqs)
# for equations Eq(diff(x(t),t,t), a1*diff(x(t),t)+b1*diff(y(t),t)+c1*x(t)+d1*y(t)+e1)
# and Eq(a2*diff(y(t),t,t), a2*diff(x(t),t)+b2*diff(y(t),t)+c2*x(t)+d2*y(t)+e2)
r['a1'] = -fc[0,x(t),1]/fc[0,x(t),2] ; r['a2'] = -fc[1,x(t),1]/fc[1,y(t),2]
r['b1'] = -fc[0,y(t),1]/fc[0,x(t),2] ; r['b2'] = -fc[1,y(t),1]/fc[1,y(t),2]
r['c1'] = -fc[0,x(t),0]/fc[0,x(t),2] ; r['c2'] = -fc[1,x(t),0]/fc[1,y(t),2]
r['d1'] = -fc[0,y(t),0]/fc[0,x(t),2] ; r['d2'] = -fc[1,y(t),0]/fc[1,y(t),2]
const = [S.Zero, S.Zero]
for i in range(2):
for j in Add.make_args(eq[i]):
if not (j.has(x(t)) or j.has(y(t))):
const[i] += j
r['e1'] = -const[0]
r['e2'] = -const[1]
if match_['type_of_equation'] == 'type1':
sol = _linear_2eq_order2_type1(x, y, t, r, eq)
elif match_['type_of_equation'] == 'type2':
gsol = _linear_2eq_order2_type1(x, y, t, r, eq)
psol = _linear_2eq_order2_type2(x, y, t, r, eq)
sol = [Eq(x(t), gsol[0].rhs+psol[0]), Eq(y(t), gsol[1].rhs+psol[1])]
elif match_['type_of_equation'] == 'type3':
sol = _linear_2eq_order2_type3(x, y, t, r, eq)
elif match_['type_of_equation'] == 'type4':
sol = _linear_2eq_order2_type4(x, y, t, r, eq)
elif match_['type_of_equation'] == 'type5':
sol = _linear_2eq_order2_type5(x, y, t, r, eq)
elif match_['type_of_equation'] == 'type6':
sol = _linear_2eq_order2_type6(x, y, t, r, eq)
elif match_['type_of_equation'] == 'type7':
sol = _linear_2eq_order2_type7(x, y, t, r, eq)
elif match_['type_of_equation'] == 'type8':
sol = _linear_2eq_order2_type8(x, y, t, r, eq)
elif match_['type_of_equation'] == 'type9':
sol = _linear_2eq_order2_type9(x, y, t, r, eq)
elif match_['type_of_equation'] == 'type10':
sol = _linear_2eq_order2_type10(x, y, t, r, eq)
elif match_['type_of_equation'] == 'type11':
sol = _linear_2eq_order2_type11(x, y, t, r, eq)
return sol
def _linear_2eq_order2_type1(x, y, t, r, eq):
r"""
System of two constant-coefficient second-order linear homogeneous differential equations
.. math:: x'' = ax + by
.. math:: y'' = cx + dy
The characteristic equation for above equations
.. math:: \lambda^4 - (a + d) \lambda^2 + ad - bc = 0
whose discriminant is `D = (a - d)^2 + 4bc \neq 0`
1. When `ad - bc \neq 0`
1.1. If `D \neq 0`. The characteristic equation has four distinct roots, `\lambda_1, \lambda_2, \lambda_3, \lambda_4`.
The general solution of the system is
.. math:: x = C_1 b e^{\lambda_1 t} + C_2 b e^{\lambda_2 t} + C_3 b e^{\lambda_3 t} + C_4 b e^{\lambda_4 t}
.. math:: y = C_1 (\lambda_1^{2} - a) e^{\lambda_1 t} + C_2 (\lambda_2^{2} - a) e^{\lambda_2 t} + C_3 (\lambda_3^{2} - a) e^{\lambda_3 t} + C_4 (\lambda_4^{2} - a) e^{\lambda_4 t}
where `C_1,..., C_4` are arbitrary constants.
1.2. If `D = 0` and `a \neq d`:
.. math:: x = 2 C_1 (bt + \frac{2bk}{a - d}) e^{\frac{kt}{2}} + 2 C_2 (bt + \frac{2bk}{a - d}) e^{\frac{-kt}{2}} + 2b C_3 t e^{\frac{kt}{2}} + 2b C_4 t e^{\frac{-kt}{2}}
.. math:: y = C_1 (d - a) t e^{\frac{kt}{2}} + C_2 (d - a) t e^{\frac{-kt}{2}} + C_3 [(d - a) t + 2k] e^{\frac{kt}{2}} + C_4 [(d - a) t - 2k] e^{\frac{-kt}{2}}
where `C_1,..., C_4` are arbitrary constants and `k = \sqrt{2 (a + d)}`
1.3. If `D = 0` and `a = d \neq 0` and `b = 0`:
.. math:: x = 2 \sqrt{a} C_1 e^{\sqrt{a} t} + 2 \sqrt{a} C_2 e^{-\sqrt{a} t}
.. math:: y = c C_1 t e^{\sqrt{a} t} - c C_2 t e^{-\sqrt{a} t} + C_3 e^{\sqrt{a} t} + C_4 e^{-\sqrt{a} t}
1.4. If `D = 0` and `a = d \neq 0` and `c = 0`:
.. math:: x = b C_1 t e^{\sqrt{a} t} - b C_2 t e^{-\sqrt{a} t} + C_3 e^{\sqrt{a} t} + C_4 e^{-\sqrt{a} t}
.. math:: y = 2 \sqrt{a} C_1 e^{\sqrt{a} t} + 2 \sqrt{a} C_2 e^{-\sqrt{a} t}
2. When `ad - bc = 0` and `a^2 + b^2 > 0`. Then the original system becomes
.. math:: x'' = ax + by
.. math:: y'' = k (ax + by)
2.1. If `a + bk \neq 0`:
.. math:: x = C_1 e^{t \sqrt{a + bk}} + C_2 e^{-t \sqrt{a + bk}} + C_3 bt + C_4 b
.. math:: y = C_1 k e^{t \sqrt{a + bk}} + C_2 k e^{-t \sqrt{a + bk}} - C_3 at - C_4 a
2.2. If `a + bk = 0`:
.. math:: x = C_1 b t^3 + C_2 b t^2 + C_3 t + C_4
.. math:: y = kx + 6 C_1 t + 2 C_2
"""
r['a'] = r['c1']
r['b'] = r['d1']
r['c'] = r['c2']
r['d'] = r['d2']
l = Symbol('l')
C1, C2, C3, C4 = get_numbered_constants(eq, num=4)
chara_eq = l**4 - (r['a']+r['d'])*l**2 + r['a']*r['d'] - r['b']*r['c']
l1 = rootof(chara_eq, 0)
l2 = rootof(chara_eq, 1)
l3 = rootof(chara_eq, 2)
l4 = rootof(chara_eq, 3)
D = (r['a'] - r['d'])**2 + 4*r['b']*r['c']
if (r['a']*r['d'] - r['b']*r['c']) != 0:
if D != 0:
gsol1 = C1*r['b']*exp(l1*t) + C2*r['b']*exp(l2*t) + C3*r['b']*exp(l3*t) \
+ C4*r['b']*exp(l4*t)
gsol2 = C1*(l1**2-r['a'])*exp(l1*t) + C2*(l2**2-r['a'])*exp(l2*t) + \
C3*(l3**2-r['a'])*exp(l3*t) + C4*(l4**2-r['a'])*exp(l4*t)
else:
if r['a'] != r['d']:
k = sqrt(2*(r['a']+r['d']))
mid = r['b']*t+2*r['b']*k/(r['a']-r['d'])
gsol1 = 2*C1*mid*exp(k*t/2) + 2*C2*mid*exp(-k*t/2) + \
2*r['b']*C3*t*exp(k*t/2) + 2*r['b']*C4*t*exp(-k*t/2)
gsol2 = C1*(r['d']-r['a'])*t*exp(k*t/2) + C2*(r['d']-r['a'])*t*exp(-k*t/2) + \
C3*((r['d']-r['a'])*t+2*k)*exp(k*t/2) + C4*((r['d']-r['a'])*t-2*k)*exp(-k*t/2)
elif r['a'] == r['d'] != 0 and r['b'] == 0:
sa = sqrt(r['a'])
gsol1 = 2*sa*C1*exp(sa*t) + 2*sa*C2*exp(-sa*t)
gsol2 = r['c']*C1*t*exp(sa*t)-r['c']*C2*t*exp(-sa*t)+C3*exp(sa*t)+C4*exp(-sa*t)
elif r['a'] == r['d'] != 0 and r['c'] == 0:
sa = sqrt(r['a'])
gsol1 = r['b']*C1*t*exp(sa*t)-r['b']*C2*t*exp(-sa*t)+C3*exp(sa*t)+C4*exp(-sa*t)
gsol2 = 2*sa*C1*exp(sa*t) + 2*sa*C2*exp(-sa*t)
elif (r['a']*r['d'] - r['b']*r['c']) == 0 and (r['a']**2 + r['b']**2) > 0:
k = r['c']/r['a']
if r['a'] + r['b']*k != 0:
mid = sqrt(r['a'] + r['b']*k)
gsol1 = C1*exp(mid*t) + C2*exp(-mid*t) + C3*r['b']*t + C4*r['b']
gsol2 = C1*k*exp(mid*t) + C2*k*exp(-mid*t) - C3*r['a']*t - C4*r['a']
else:
gsol1 = C1*r['b']*t**3 + C2*r['b']*t**2 + C3*t + C4
gsol2 = k*gsol1 + 6*C1*t + 2*C2
return [Eq(x(t), gsol1), Eq(y(t), gsol2)]
def _linear_2eq_order2_type2(x, y, t, r, eq):
r"""
The equations in this type are
.. math:: x'' = a_1 x + b_1 y + c_1
.. math:: y'' = a_2 x + b_2 y + c_2
The general solution of this system is given by the sum of its particular solution
and the general solution of the homogeneous system. The general solution is given
by the linear system of 2 equation of order 2 and type 1
1. If `a_1 b_2 - a_2 b_1 \neq 0`. A particular solution will be `x = x_0` and `y = y_0`
where the constants `x_0` and `y_0` are determined by solving the linear algebraic system
.. math:: a_1 x_0 + b_1 y_0 + c_1 = 0, a_2 x_0 + b_2 y_0 + c_2 = 0
2. If `a_1 b_2 - a_2 b_1 = 0` and `a_1^2 + b_1^2 > 0`. In this case, the system in question becomes
.. math:: x'' = ax + by + c_1, y'' = k (ax + by) + c_2
2.1. If `\sigma = a + bk \neq 0`, the particular solution will be
.. math:: x = \frac{1}{2} b \sigma^{-1} (c_1 k - c_2) t^2 - \sigma^{-2} (a c_1 + b c_2)
.. math:: y = kx + \frac{1}{2} (c_2 - c_1 k) t^2
2.2. If `\sigma = a + bk = 0`, the particular solution will be
.. math:: x = \frac{1}{24} b (c_2 - c_1 k) t^4 + \frac{1}{2} c_1 t^2
.. math:: y = kx + \frac{1}{2} (c_2 - c_1 k) t^2
"""
x0, y0 = symbols('x0, y0')
if r['c1']*r['d2'] - r['c2']*r['d1'] != 0:
sol = solve((r['c1']*x0+r['d1']*y0+r['e1'], r['c2']*x0+r['d2']*y0+r['e2']), x0, y0)
psol = [sol[x0], sol[y0]]
elif r['c1']*r['d2'] - r['c2']*r['d1'] == 0 and (r['c1']**2 + r['d1']**2) > 0:
k = r['c2']/r['c1']
sig = r['c1'] + r['d1']*k
if sig != 0:
psol1 = r['d1']*sig**-1*(r['e1']*k-r['e2'])*t**2/2 - \
sig**-2*(r['c1']*r['e1']+r['d1']*r['e2'])
psol2 = k*psol1 + (r['e2'] - r['e1']*k)*t**2/2
psol = [psol1, psol2]
else:
psol1 = r['d1']*(r['e2']-r['e1']*k)*t**4/24 + r['e1']*t**2/2
psol2 = k*psol1 + (r['e2']-r['e1']*k)*t**2/2
psol = [psol1, psol2]
return psol
def _linear_2eq_order2_type3(x, y, t, r, eq):
r"""
These type of equation is used for describing the horizontal motion of a pendulum
taking into account the Earth rotation.
The solution is given with `a^2 + 4b > 0`:
.. math:: x = C_1 \cos(\alpha t) + C_2 \sin(\alpha t) + C_3 \cos(\beta t) + C_4 \sin(\beta t)
.. math:: y = -C_1 \sin(\alpha t) + C_2 \cos(\alpha t) - C_3 \sin(\beta t) + C_4 \cos(\beta t)
where `C_1,...,C_4` and
.. math:: \alpha = \frac{1}{2} a + \frac{1}{2} \sqrt{a^2 + 4b}, \beta = \frac{1}{2} a - \frac{1}{2} \sqrt{a^2 + 4b}
"""
C1, C2, C3, C4 = get_numbered_constants(eq, num=4)
if r['b1']**2 - 4*r['c1'] > 0:
r['a'] = r['b1'] ; r['b'] = -r['c1']
alpha = r['a']/2 + sqrt(r['a']**2 + 4*r['b'])/2
beta = r['a']/2 - sqrt(r['a']**2 + 4*r['b'])/2
sol1 = C1*cos(alpha*t) + C2*sin(alpha*t) + C3*cos(beta*t) + C4*sin(beta*t)
sol2 = -C1*sin(alpha*t) + C2*cos(alpha*t) - C3*sin(beta*t) + C4*cos(beta*t)
return [Eq(x(t), sol1), Eq(y(t), sol2)]
def _linear_2eq_order2_type4(x, y, t, r, eq):
r"""
These equations are found in the theory of oscillations
.. math:: x'' + a_1 x' + b_1 y' + c_1 x + d_1 y = k_1 e^{i \omega t}
.. math:: y'' + a_2 x' + b_2 y' + c_2 x + d_2 y = k_2 e^{i \omega t}
The general solution of this linear nonhomogeneous system of constant-coefficient
differential equations is given by the sum of its particular solution and the
general solution of the corresponding homogeneous system (with `k_1 = k_2 = 0`)
1. A particular solution is obtained by the method of undetermined coefficients:
.. math:: x = A_* e^{i \omega t}, y = B_* e^{i \omega t}
On substituting these expressions into the original system of differential equations,
one arrive at a linear nonhomogeneous system of algebraic equations for the
coefficients `A` and `B`.
2. The general solution of the homogeneous system of differential equations is determined
by a linear combination of linearly independent particular solutions determined by
the method of undetermined coefficients in the form of exponentials:
.. math:: x = A e^{\lambda t}, y = B e^{\lambda t}
On substituting these expressions into the original system and collecting the
coefficients of the unknown `A` and `B`, one obtains
.. math:: (\lambda^{2} + a_1 \lambda + c_1) A + (b_1 \lambda + d_1) B = 0
.. math:: (a_2 \lambda + c_2) A + (\lambda^{2} + b_2 \lambda + d_2) B = 0
The determinant of this system must vanish for nontrivial solutions A, B to exist.
This requirement results in the following characteristic equation for `\lambda`
.. math:: (\lambda^2 + a_1 \lambda + c_1) (\lambda^2 + b_2 \lambda + d_2) - (b_1 \lambda + d_1) (a_2 \lambda + c_2) = 0
If all roots `k_1,...,k_4` of this equation are distinct, the general solution of the original
system of the differential equations has the form
.. math:: x = C_1 (b_1 \lambda_1 + d_1) e^{\lambda_1 t} - C_2 (b_1 \lambda_2 + d_1) e^{\lambda_2 t} - C_3 (b_1 \lambda_3 + d_1) e^{\lambda_3 t} - C_4 (b_1 \lambda_4 + d_1) e^{\lambda_4 t}
.. math:: y = C_1 (\lambda_1^{2} + a_1 \lambda_1 + c_1) e^{\lambda_1 t} + C_2 (\lambda_2^{2} + a_1 \lambda_2 + c_1) e^{\lambda_2 t} + C_3 (\lambda_3^{2} + a_1 \lambda_3 + c_1) e^{\lambda_3 t} + C_4 (\lambda_4^{2} + a_1 \lambda_4 + c_1) e^{\lambda_4 t}
"""
C1, C2, C3, C4 = get_numbered_constants(eq, num=4)
k = Symbol('k')
Ra, Ca, Rb, Cb = symbols('Ra, Ca, Rb, Cb')
a1 = r['a1'] ; a2 = r['a2']
b1 = r['b1'] ; b2 = r['b2']
c1 = r['c1'] ; c2 = r['c2']
d1 = r['d1'] ; d2 = r['d2']
k1 = r['e1'].expand().as_independent(t)[0]
k2 = r['e2'].expand().as_independent(t)[0]
ew1 = r['e1'].expand().as_independent(t)[1]
ew2 = powdenest(ew1).as_base_exp()[1]
ew3 = collect(ew2, t).coeff(t)
w = cancel(ew3/I)
# The particular solution is assumed to be (Ra+I*Ca)*exp(I*w*t) and
# (Rb+I*Cb)*exp(I*w*t) for x(t) and y(t) respectively
# peq1, peq2, peq3, peq4 unused
# peq1 = (-w**2+c1)*Ra - a1*w*Ca + d1*Rb - b1*w*Cb - k1
# peq2 = a1*w*Ra + (-w**2+c1)*Ca + b1*w*Rb + d1*Cb
# peq3 = c2*Ra - a2*w*Ca + (-w**2+d2)*Rb - b2*w*Cb - k2
# peq4 = a2*w*Ra + c2*Ca + b2*w*Rb + (-w**2+d2)*Cb
# FIXME: solve for what in what? Ra, Rb, etc I guess
# but then psol not used for anything?
# psol = solve([peq1, peq2, peq3, peq4])
chareq = (k**2+a1*k+c1)*(k**2+b2*k+d2) - (b1*k+d1)*(a2*k+c2)
[k1, k2, k3, k4] = roots_quartic(Poly(chareq))
sol1 = -C1*(b1*k1+d1)*exp(k1*t) - C2*(b1*k2+d1)*exp(k2*t) - \
C3*(b1*k3+d1)*exp(k3*t) - C4*(b1*k4+d1)*exp(k4*t) + (Ra+I*Ca)*exp(I*w*t)
a1_ = (a1-1)
sol2 = C1*(k1**2+a1_*k1+c1)*exp(k1*t) + C2*(k2**2+a1_*k2+c1)*exp(k2*t) + \
C3*(k3**2+a1_*k3+c1)*exp(k3*t) + C4*(k4**2+a1_*k4+c1)*exp(k4*t) + (Rb+I*Cb)*exp(I*w*t)
return [Eq(x(t), sol1), Eq(y(t), sol2)]
def _linear_2eq_order2_type5(x, y, t, r, eq):
r"""
The equation which come under this category are
.. math:: x'' = a (t y' - y)
.. math:: y'' = b (t x' - x)
The transformation
.. math:: u = t x' - x, b = t y' - y
leads to the first-order system
.. math:: u' = atv, v' = btu
The general solution of this system is given by
If `ab > 0`:
.. math:: u = C_1 a e^{\frac{1}{2} \sqrt{ab} t^2} + C_2 a e^{-\frac{1}{2} \sqrt{ab} t^2}
.. math:: v = C_1 \sqrt{ab} e^{\frac{1}{2} \sqrt{ab} t^2} - C_2 \sqrt{ab} e^{-\frac{1}{2} \sqrt{ab} t^2}
If `ab < 0`:
.. math:: u = C_1 a \cos(\frac{1}{2} \sqrt{\left|ab\right|} t^2) + C_2 a \sin(-\frac{1}{2} \sqrt{\left|ab\right|} t^2)
.. math:: v = C_1 \sqrt{\left|ab\right|} \sin(\frac{1}{2} \sqrt{\left|ab\right|} t^2) + C_2 \sqrt{\left|ab\right|} \cos(-\frac{1}{2} \sqrt{\left|ab\right|} t^2)
where `C_1` and `C_2` are arbitrary constants. On substituting the value of `u` and `v`
in above equations and integrating the resulting expressions, the general solution will become
.. math:: x = C_3 t + t \int \frac{u}{t^2} \,dt, y = C_4 t + t \int \frac{u}{t^2} \,dt
where `C_3` and `C_4` are arbitrary constants.
"""
C1, C2, C3, C4 = get_numbered_constants(eq, num=4)
r['a'] = -r['d1'] ; r['b'] = -r['c2']
mul = sqrt(abs(r['a']*r['b']))
if r['a']*r['b'] > 0:
u = C1*r['a']*exp(mul*t**2/2) + C2*r['a']*exp(-mul*t**2/2)
v = C1*mul*exp(mul*t**2/2) - C2*mul*exp(-mul*t**2/2)
else:
u = C1*r['a']*cos(mul*t**2/2) + C2*r['a']*sin(mul*t**2/2)
v = -C1*mul*sin(mul*t**2/2) + C2*mul*cos(mul*t**2/2)
sol1 = C3*t + t*Integral(u/t**2, t)
sol2 = C4*t + t*Integral(v/t**2, t)
return [Eq(x(t), sol1), Eq(y(t), sol2)]
def _linear_2eq_order2_type6(x, y, t, r, eq):
r"""
The equations are
.. math:: x'' = f(t) (a_1 x + b_1 y)
.. math:: y'' = f(t) (a_2 x + b_2 y)
If `k_1` and `k_2` are roots of the quadratic equation
.. math:: k^2 - (a_1 + b_2) k + a_1 b_2 - a_2 b_1 = 0
Then by multiplying appropriate constants and adding together original equations
we obtain two independent equations:
.. math:: z_1'' = k_1 f(t) z_1, z_1 = a_2 x + (k_1 - a_1) y
.. math:: z_2'' = k_2 f(t) z_2, z_2 = a_2 x + (k_2 - a_1) y
Solving the equations will give the values of `x` and `y` after obtaining the value
of `z_1` and `z_2` by solving the differential equation and substituting the result.
"""
k = Symbol('k')
z = Function('z')
num, den = cancel(
(r['c1']*x(t) + r['d1']*y(t))/
(r['c2']*x(t) + r['d2']*y(t))).as_numer_denom()
f = r['c1']/num.coeff(x(t))
a1 = num.coeff(x(t))
b1 = num.coeff(y(t))
a2 = den.coeff(x(t))
b2 = den.coeff(y(t))
chareq = k**2 - (a1 + b2)*k + a1*b2 - a2*b1
k1, k2 = [rootof(chareq, k) for k in range(Poly(chareq).degree())]
z1 = dsolve(diff(z(t),t,t) - k1*f*z(t)).rhs
z2 = dsolve(diff(z(t),t,t) - k2*f*z(t)).rhs
sol1 = (k1*z2 - k2*z1 + a1*(z1 - z2))/(a2*(k1-k2))
sol2 = (z1 - z2)/(k1 - k2)
return [Eq(x(t), sol1), Eq(y(t), sol2)]
def _linear_2eq_order2_type7(x, y, t, r, eq):
r"""
The equations are given as
.. math:: x'' = f(t) (a_1 x' + b_1 y')
.. math:: y'' = f(t) (a_2 x' + b_2 y')
If `k_1` and 'k_2` are roots of the quadratic equation
.. math:: k^2 - (a_1 + b_2) k + a_1 b_2 - a_2 b_1 = 0
Then the system can be reduced by adding together the two equations multiplied
by appropriate constants give following two independent equations:
.. math:: z_1'' = k_1 f(t) z_1', z_1 = a_2 x + (k_1 - a_1) y
.. math:: z_2'' = k_2 f(t) z_2', z_2 = a_2 x + (k_2 - a_1) y
Integrating these and returning to the original variables, one arrives at a linear
algebraic system for the unknowns `x` and `y`:
.. math:: a_2 x + (k_1 - a_1) y = C_1 \int e^{k_1 F(t)} \,dt + C_2
.. math:: a_2 x + (k_2 - a_1) y = C_3 \int e^{k_2 F(t)} \,dt + C_4
where `C_1,...,C_4` are arbitrary constants and `F(t) = \int f(t) \,dt`
"""
C1, C2, C3, C4 = get_numbered_constants(eq, num=4)
k = Symbol('k')
num, den = cancel(
(r['a1']*x(t) + r['b1']*y(t))/
(r['a2']*x(t) + r['b2']*y(t))).as_numer_denom()
f = r['a1']/num.coeff(x(t))
a1 = num.coeff(x(t))
b1 = num.coeff(y(t))
a2 = den.coeff(x(t))
b2 = den.coeff(y(t))
chareq = k**2 - (a1 + b2)*k + a1*b2 - a2*b1
[k1, k2] = [rootof(chareq, k) for k in range(Poly(chareq).degree())]
F = Integral(f, t)
z1 = C1*Integral(exp(k1*F), t) + C2
z2 = C3*Integral(exp(k2*F), t) + C4
sol1 = (k1*z2 - k2*z1 + a1*(z1 - z2))/(a2*(k1-k2))
sol2 = (z1 - z2)/(k1 - k2)
return [Eq(x(t), sol1), Eq(y(t), sol2)]
def _linear_2eq_order2_type8(x, y, t, r, eq):
r"""
The equation of this category are
.. math:: x'' = a f(t) (t y' - y)
.. math:: y'' = b f(t) (t x' - x)
The transformation
.. math:: u = t x' - x, v = t y' - y
leads to the system of first-order equations
.. math:: u' = a t f(t) v, v' = b t f(t) u
The general solution of this system has the form
If `ab > 0`:
.. math:: u = C_1 a e^{\sqrt{ab} \int t f(t) \,dt} + C_2 a e^{-\sqrt{ab} \int t f(t) \,dt}
.. math:: v = C_1 \sqrt{ab} e^{\sqrt{ab} \int t f(t) \,dt} - C_2 \sqrt{ab} e^{-\sqrt{ab} \int t f(t) \,dt}
If `ab < 0`:
.. math:: u = C_1 a \cos(\sqrt{\left|ab\right|} \int t f(t) \,dt) + C_2 a \sin(-\sqrt{\left|ab\right|} \int t f(t) \,dt)
.. math:: v = C_1 \sqrt{\left|ab\right|} \sin(\sqrt{\left|ab\right|} \int t f(t) \,dt) + C_2 \sqrt{\left|ab\right|} \cos(-\sqrt{\left|ab\right|} \int t f(t) \,dt)
where `C_1` and `C_2` are arbitrary constants. On substituting the value of `u` and `v`
in above equations and integrating the resulting expressions, the general solution will become
.. math:: x = C_3 t + t \int \frac{u}{t^2} \,dt, y = C_4 t + t \int \frac{u}{t^2} \,dt
where `C_3` and `C_4` are arbitrary constants.
"""
C1, C2, C3, C4 = get_numbered_constants(eq, num=4)
num, den = cancel(r['d1']/r['c2']).as_numer_denom()
f = -r['d1']/num
a = num
b = den
mul = sqrt(abs(a*b))
Igral = Integral(t*f, t)
if a*b > 0:
u = C1*a*exp(mul*Igral) + C2*a*exp(-mul*Igral)
v = C1*mul*exp(mul*Igral) - C2*mul*exp(-mul*Igral)
else:
u = C1*a*cos(mul*Igral) + C2*a*sin(mul*Igral)
v = -C1*mul*sin(mul*Igral) + C2*mul*cos(mul*Igral)
sol1 = C3*t + t*Integral(u/t**2, t)
sol2 = C4*t + t*Integral(v/t**2, t)
return [Eq(x(t), sol1), Eq(y(t), sol2)]
def _linear_2eq_order2_type9(x, y, t, r, eq):
r"""
.. math:: t^2 x'' + a_1 t x' + b_1 t y' + c_1 x + d_1 y = 0
.. math:: t^2 y'' + a_2 t x' + b_2 t y' + c_2 x + d_2 y = 0
These system of equations are euler type.
The substitution of `t = \sigma e^{\tau} (\sigma \neq 0)` leads to the system of constant
coefficient linear differential equations
.. math:: x'' + (a_1 - 1) x' + b_1 y' + c_1 x + d_1 y = 0
.. math:: y'' + a_2 x' + (b_2 - 1) y' + c_2 x + d_2 y = 0
The general solution of the homogeneous system of differential equations is determined
by a linear combination of linearly independent particular solutions determined by
the method of undetermined coefficients in the form of exponentials
.. math:: x = A e^{\lambda t}, y = B e^{\lambda t}
On substituting these expressions into the original system and collecting the
coefficients of the unknown `A` and `B`, one obtains
.. math:: (\lambda^{2} + (a_1 - 1) \lambda + c_1) A + (b_1 \lambda + d_1) B = 0
.. math:: (a_2 \lambda + c_2) A + (\lambda^{2} + (b_2 - 1) \lambda + d_2) B = 0
The determinant of this system must vanish for nontrivial solutions A, B to exist.
This requirement results in the following characteristic equation for `\lambda`
.. math:: (\lambda^2 + (a_1 - 1) \lambda + c_1) (\lambda^2 + (b_2 - 1) \lambda + d_2) - (b_1 \lambda + d_1) (a_2 \lambda + c_2) = 0
If all roots `k_1,...,k_4` of this equation are distinct, the general solution of the original
system of the differential equations has the form
.. math:: x = C_1 (b_1 \lambda_1 + d_1) e^{\lambda_1 t} - C_2 (b_1 \lambda_2 + d_1) e^{\lambda_2 t} - C_3 (b_1 \lambda_3 + d_1) e^{\lambda_3 t} - C_4 (b_1 \lambda_4 + d_1) e^{\lambda_4 t}
.. math:: y = C_1 (\lambda_1^{2} + (a_1 - 1) \lambda_1 + c_1) e^{\lambda_1 t} + C_2 (\lambda_2^{2} + (a_1 - 1) \lambda_2 + c_1) e^{\lambda_2 t} + C_3 (\lambda_3^{2} + (a_1 - 1) \lambda_3 + c_1) e^{\lambda_3 t} + C_4 (\lambda_4^{2} + (a_1 - 1) \lambda_4 + c_1) e^{\lambda_4 t}
"""
C1, C2, C3, C4 = get_numbered_constants(eq, num=4)
k = Symbol('k')
a1 = -r['a1']*t; a2 = -r['a2']*t
b1 = -r['b1']*t; b2 = -r['b2']*t
c1 = -r['c1']*t**2; c2 = -r['c2']*t**2
d1 = -r['d1']*t**2; d2 = -r['d2']*t**2
eq = (k**2+(a1-1)*k+c1)*(k**2+(b2-1)*k+d2)-(b1*k+d1)*(a2*k+c2)
[k1, k2, k3, k4] = roots_quartic(Poly(eq))
sol1 = -C1*(b1*k1+d1)*exp(k1*log(t)) - C2*(b1*k2+d1)*exp(k2*log(t)) - \
C3*(b1*k3+d1)*exp(k3*log(t)) - C4*(b1*k4+d1)*exp(k4*log(t))
a1_ = (a1-1)
sol2 = C1*(k1**2+a1_*k1+c1)*exp(k1*log(t)) + C2*(k2**2+a1_*k2+c1)*exp(k2*log(t)) \
+ C3*(k3**2+a1_*k3+c1)*exp(k3*log(t)) + C4*(k4**2+a1_*k4+c1)*exp(k4*log(t))
return [Eq(x(t), sol1), Eq(y(t), sol2)]
def _linear_2eq_order2_type10(x, y, t, r, eq):
r"""
The equation of this category are
.. math:: (\alpha t^2 + \beta t + \gamma)^{2} x'' = ax + by
.. math:: (\alpha t^2 + \beta t + \gamma)^{2} y'' = cx + dy
The transformation
.. math:: \tau = \int \frac{1}{\alpha t^2 + \beta t + \gamma} \,dt , u = \frac{x}{\sqrt{\left|\alpha t^2 + \beta t + \gamma\right|}} , v = \frac{y}{\sqrt{\left|\alpha t^2 + \beta t + \gamma\right|}}
leads to a constant coefficient linear system of equations
.. math:: u'' = (a - \alpha \gamma + \frac{1}{4} \beta^{2}) u + b v
.. math:: v'' = c u + (d - \alpha \gamma + \frac{1}{4} \beta^{2}) v
These system of equations obtained can be solved by type1 of System of two
constant-coefficient second-order linear homogeneous differential equations.
"""
# FIXME: This function is equivalent to type6 (and broken). Should be removed...
C1, C2, C3, C4 = get_numbered_constants(eq, num=4)
u, v = symbols('u, v', cls=Function)
assert False
p = Wild('p', exclude=[t, t**2])
q = Wild('q', exclude=[t, t**2])
s = Wild('s', exclude=[t, t**2])
n = Wild('n', exclude=[t, t**2])
num, den = r['c1'].as_numer_denom()
dic = den.match((n*(p*t**2+q*t+s)**2).expand())
eqz = dic[p]*t**2 + dic[q]*t + dic[s]
a = num/dic[n]
b = cancel(r['d1']*eqz**2)
c = cancel(r['c2']*eqz**2)
d = cancel(r['d2']*eqz**2)
[msol1, msol2] = dsolve([Eq(diff(u(t), t, t), (a - dic[p]*dic[s] + dic[q]**2/4)*u(t) \
+ b*v(t)), Eq(diff(v(t),t,t), c*u(t) + (d - dic[p]*dic[s] + dic[q]**2/4)*v(t))])
sol1 = (msol1.rhs*sqrt(abs(eqz))).subs(t, Integral(1/eqz, t))
sol2 = (msol2.rhs*sqrt(abs(eqz))).subs(t, Integral(1/eqz, t))
return [Eq(x(t), sol1), Eq(y(t), sol2)]
def _linear_2eq_order2_type11(x, y, t, r, eq):
r"""
The equations which comes under this type are
.. math:: x'' = f(t) (t x' - x) + g(t) (t y' - y)
.. math:: y'' = h(t) (t x' - x) + p(t) (t y' - y)
The transformation
.. math:: u = t x' - x, v = t y' - y
leads to the linear system of first-order equations
.. math:: u' = t f(t) u + t g(t) v, v' = t h(t) u + t p(t) v
On substituting the value of `u` and `v` in transformed equation gives value of `x` and `y` as
.. math:: x = C_3 t + t \int \frac{u}{t^2} \,dt , y = C_4 t + t \int \frac{v}{t^2} \,dt.
where `C_3` and `C_4` are arbitrary constants.
"""
C1, C2, C3, C4 = get_numbered_constants(eq, num=4)
u, v = symbols('u, v', cls=Function)
f = -r['c1'] ; g = -r['d1']
h = -r['c2'] ; p = -r['d2']
[msol1, msol2] = dsolve([Eq(diff(u(t),t), t*f*u(t) + t*g*v(t)), Eq(diff(v(t),t), t*h*u(t) + t*p*v(t))])
sol1 = C3*t + t*Integral(msol1.rhs/t**2, t)
sol2 = C4*t + t*Integral(msol2.rhs/t**2, t)
return [Eq(x(t), sol1), Eq(y(t), sol2)]
def sysode_linear_neq_order1(match):
from sympy.solvers.ode.systems import (_linear_neq_order1_type1,
_linear_neq_order1_type3)
if match['type_of_equation'] == 'type1':
sol = _linear_neq_order1_type1(match)
elif match['type_of_equation'] == 'type3':
sol = _linear_neq_order1_type3(match)
return sol
def sysode_nonlinear_2eq_order1(match_):
func = match_['func']
eq = match_['eq']
fc = match_['func_coeff']
t = list(list(eq[0].atoms(Derivative))[0].atoms(Symbol))[0]
if match_['type_of_equation'] == 'type5':
sol = _nonlinear_2eq_order1_type5(func, t, eq)
return sol
x = func[0].func
y = func[1].func
for i in range(2):
eqs = 0
for terms in Add.make_args(eq[i]):
eqs += terms/fc[i,func[i],1]
eq[i] = eqs
if match_['type_of_equation'] == 'type1':
sol = _nonlinear_2eq_order1_type1(x, y, t, eq)
elif match_['type_of_equation'] == 'type2':
sol = _nonlinear_2eq_order1_type2(x, y, t, eq)
elif match_['type_of_equation'] == 'type3':
sol = _nonlinear_2eq_order1_type3(x, y, t, eq)
elif match_['type_of_equation'] == 'type4':
sol = _nonlinear_2eq_order1_type4(x, y, t, eq)
return sol
def _nonlinear_2eq_order1_type1(x, y, t, eq):
r"""
Equations:
.. math:: x' = x^n F(x,y)
.. math:: y' = g(y) F(x,y)
Solution:
.. math:: x = \varphi(y), \int \frac{1}{g(y) F(\varphi(y),y)} \,dy = t + C_2
where
if `n \neq 1`
.. math:: \varphi = [C_1 + (1-n) \int \frac{1}{g(y)} \,dy]^{\frac{1}{1-n}}
if `n = 1`
.. math:: \varphi = C_1 e^{\int \frac{1}{g(y)} \,dy}
where `C_1` and `C_2` are arbitrary constants.
"""
C1, C2 = get_numbered_constants(eq, num=2)
n = Wild('n', exclude=[x(t),y(t)])
f = Wild('f')
u, v = symbols('u, v')
r = eq[0].match(diff(x(t),t) - x(t)**n*f)
g = ((diff(y(t),t) - eq[1])/r[f]).subs(y(t),v)
F = r[f].subs(x(t),u).subs(y(t),v)
n = r[n]
if n!=1:
phi = (C1 + (1-n)*Integral(1/g, v))**(1/(1-n))
else:
phi = C1*exp(Integral(1/g, v))
phi = phi.doit()
sol2 = solve(Integral(1/(g*F.subs(u,phi)), v).doit() - t - C2, v)
sol = []
for sols in sol2:
sol.append(Eq(x(t),phi.subs(v, sols)))
sol.append(Eq(y(t), sols))
return sol
def _nonlinear_2eq_order1_type2(x, y, t, eq):
r"""
Equations:
.. math:: x' = e^{\lambda x} F(x,y)
.. math:: y' = g(y) F(x,y)
Solution:
.. math:: x = \varphi(y), \int \frac{1}{g(y) F(\varphi(y),y)} \,dy = t + C_2
where
if `\lambda \neq 0`
.. math:: \varphi = -\frac{1}{\lambda} log(C_1 - \lambda \int \frac{1}{g(y)} \,dy)
if `\lambda = 0`
.. math:: \varphi = C_1 + \int \frac{1}{g(y)} \,dy
where `C_1` and `C_2` are arbitrary constants.
"""
C1, C2 = get_numbered_constants(eq, num=2)
n = Wild('n', exclude=[x(t),y(t)])
f = Wild('f')
u, v = symbols('u, v')
r = eq[0].match(diff(x(t),t) - exp(n*x(t))*f)
g = ((diff(y(t),t) - eq[1])/r[f]).subs(y(t),v)
F = r[f].subs(x(t),u).subs(y(t),v)
n = r[n]
if n:
phi = -1/n*log(C1 - n*Integral(1/g, v))
else:
phi = C1 + Integral(1/g, v)
phi = phi.doit()
sol2 = solve(Integral(1/(g*F.subs(u,phi)), v).doit() - t - C2, v)
sol = []
for sols in sol2:
sol.append(Eq(x(t),phi.subs(v, sols)))
sol.append(Eq(y(t), sols))
return sol
def _nonlinear_2eq_order1_type3(x, y, t, eq):
r"""
Autonomous system of general form
.. math:: x' = F(x,y)
.. math:: y' = G(x,y)
Assuming `y = y(x, C_1)` where `C_1` is an arbitrary constant is the general
solution of the first-order equation
.. math:: F(x,y) y'_x = G(x,y)
Then the general solution of the original system of equations has the form
.. math:: \int \frac{1}{F(x,y(x,C_1))} \,dx = t + C_1
"""
C1, C2, C3, C4 = get_numbered_constants(eq, num=4)
v = Function('v')
u = Symbol('u')
f = Wild('f')
g = Wild('g')
r1 = eq[0].match(diff(x(t),t) - f)
r2 = eq[1].match(diff(y(t),t) - g)
F = r1[f].subs(x(t), u).subs(y(t), v(u))
G = r2[g].subs(x(t), u).subs(y(t), v(u))
sol2r = dsolve(Eq(diff(v(u), u), G/F))
if isinstance(sol2r, Expr):
sol2r = [sol2r]
for sol2s in sol2r:
sol1 = solve(Integral(1/F.subs(v(u), sol2s.rhs), u).doit() - t - C2, u)
sol = []
for sols in sol1:
sol.append(Eq(x(t), sols))
sol.append(Eq(y(t), (sol2s.rhs).subs(u, sols)))
return sol
def _nonlinear_2eq_order1_type4(x, y, t, eq):
r"""
Equation:
.. math:: x' = f_1(x) g_1(y) \phi(x,y,t)
.. math:: y' = f_2(x) g_2(y) \phi(x,y,t)
First integral:
.. math:: \int \frac{f_2(x)}{f_1(x)} \,dx - \int \frac{g_1(y)}{g_2(y)} \,dy = C
where `C` is an arbitrary constant.
On solving the first integral for `x` (resp., `y` ) and on substituting the
resulting expression into either equation of the original solution, one
arrives at a first-order equation for determining `y` (resp., `x` ).
"""
C1, C2 = get_numbered_constants(eq, num=2)
u, v = symbols('u, v')
U, V = symbols('U, V', cls=Function)
f = Wild('f')
g = Wild('g')
f1 = Wild('f1', exclude=[v,t])
f2 = Wild('f2', exclude=[v,t])
g1 = Wild('g1', exclude=[u,t])
g2 = Wild('g2', exclude=[u,t])
r1 = eq[0].match(diff(x(t),t) - f)
r2 = eq[1].match(diff(y(t),t) - g)
num, den = (
(r1[f].subs(x(t),u).subs(y(t),v))/
(r2[g].subs(x(t),u).subs(y(t),v))).as_numer_denom()
R1 = num.match(f1*g1)
R2 = den.match(f2*g2)
phi = (r1[f].subs(x(t),u).subs(y(t),v))/num
F1 = R1[f1]; F2 = R2[f2]
G1 = R1[g1]; G2 = R2[g2]
sol1r = solve(Integral(F2/F1, u).doit() - Integral(G1/G2,v).doit() - C1, u)
sol2r = solve(Integral(F2/F1, u).doit() - Integral(G1/G2,v).doit() - C1, v)
sol = []
for sols in sol1r:
sol.append(Eq(y(t), dsolve(diff(V(t),t) - F2.subs(u,sols).subs(v,V(t))*G2.subs(v,V(t))*phi.subs(u,sols).subs(v,V(t))).rhs))
for sols in sol2r:
sol.append(Eq(x(t), dsolve(diff(U(t),t) - F1.subs(u,U(t))*G1.subs(v,sols).subs(u,U(t))*phi.subs(v,sols).subs(u,U(t))).rhs))
return set(sol)
def _nonlinear_2eq_order1_type5(func, t, eq):
r"""
Clairaut system of ODEs
.. math:: x = t x' + F(x',y')
.. math:: y = t y' + G(x',y')
The following are solutions of the system
`(i)` straight lines:
.. math:: x = C_1 t + F(C_1, C_2), y = C_2 t + G(C_1, C_2)
where `C_1` and `C_2` are arbitrary constants;
`(ii)` envelopes of the above lines;
`(iii)` continuously differentiable lines made up from segments of the lines
`(i)` and `(ii)`.
"""
C1, C2 = get_numbered_constants(eq, num=2)
f = Wild('f')
g = Wild('g')
def check_type(x, y):
r1 = eq[0].match(t*diff(x(t),t) - x(t) + f)
r2 = eq[1].match(t*diff(y(t),t) - y(t) + g)
if not (r1 and r2):
r1 = eq[0].match(diff(x(t),t) - x(t)/t + f/t)
r2 = eq[1].match(diff(y(t),t) - y(t)/t + g/t)
if not (r1 and r2):
r1 = (-eq[0]).match(t*diff(x(t),t) - x(t) + f)
r2 = (-eq[1]).match(t*diff(y(t),t) - y(t) + g)
if not (r1 and r2):
r1 = (-eq[0]).match(diff(x(t),t) - x(t)/t + f/t)
r2 = (-eq[1]).match(diff(y(t),t) - y(t)/t + g/t)
return [r1, r2]
for func_ in func:
if isinstance(func_, list):
x = func[0][0].func
y = func[0][1].func
[r1, r2] = check_type(x, y)
if not (r1 and r2):
[r1, r2] = check_type(y, x)
x, y = y, x
x1 = diff(x(t),t); y1 = diff(y(t),t)
return {Eq(x(t), C1*t + r1[f].subs(x1,C1).subs(y1,C2)), Eq(y(t), C2*t + r2[g].subs(x1,C1).subs(y1,C2))}
def sysode_nonlinear_3eq_order1(match_):
x = match_['func'][0].func
y = match_['func'][1].func
z = match_['func'][2].func
eq = match_['eq']
t = list(list(eq[0].atoms(Derivative))[0].atoms(Symbol))[0]
if match_['type_of_equation'] == 'type1':
sol = _nonlinear_3eq_order1_type1(x, y, z, t, eq)
if match_['type_of_equation'] == 'type2':
sol = _nonlinear_3eq_order1_type2(x, y, z, t, eq)
if match_['type_of_equation'] == 'type3':
sol = _nonlinear_3eq_order1_type3(x, y, z, t, eq)
if match_['type_of_equation'] == 'type4':
sol = _nonlinear_3eq_order1_type4(x, y, z, t, eq)
if match_['type_of_equation'] == 'type5':
sol = _nonlinear_3eq_order1_type5(x, y, z, t, eq)
return sol
def _nonlinear_3eq_order1_type1(x, y, z, t, eq):
r"""
Equations:
.. math:: a x' = (b - c) y z, \enspace b y' = (c - a) z x, \enspace c z' = (a - b) x y
First Integrals:
.. math:: a x^{2} + b y^{2} + c z^{2} = C_1
.. math:: a^{2} x^{2} + b^{2} y^{2} + c^{2} z^{2} = C_2
where `C_1` and `C_2` are arbitrary constants. On solving the integrals for `y` and
`z` and on substituting the resulting expressions into the first equation of the
system, we arrives at a separable first-order equation on `x`. Similarly doing that
for other two equations, we will arrive at first order equation on `y` and `z` too.
References
==========
-http://eqworld.ipmnet.ru/en/solutions/sysode/sode0401.pdf
"""
C1, C2 = get_numbered_constants(eq, num=2)
u, v, w = symbols('u, v, w')
p = Wild('p', exclude=[x(t), y(t), z(t), t])
q = Wild('q', exclude=[x(t), y(t), z(t), t])
s = Wild('s', exclude=[x(t), y(t), z(t), t])
r = (diff(x(t),t) - eq[0]).match(p*y(t)*z(t))
r.update((diff(y(t),t) - eq[1]).match(q*z(t)*x(t)))
r.update((diff(z(t),t) - eq[2]).match(s*x(t)*y(t)))
n1, d1 = r[p].as_numer_denom()
n2, d2 = r[q].as_numer_denom()
n3, d3 = r[s].as_numer_denom()
val = solve([n1*u-d1*v+d1*w, d2*u+n2*v-d2*w, d3*u-d3*v-n3*w],[u,v])
vals = [val[v], val[u]]
c = lcm(vals[0].as_numer_denom()[1], vals[1].as_numer_denom()[1])
b = vals[0].subs(w, c)
a = vals[1].subs(w, c)
y_x = sqrt(((c*C1-C2) - a*(c-a)*x(t)**2)/(b*(c-b)))
z_x = sqrt(((b*C1-C2) - a*(b-a)*x(t)**2)/(c*(b-c)))
z_y = sqrt(((a*C1-C2) - b*(a-b)*y(t)**2)/(c*(a-c)))
x_y = sqrt(((c*C1-C2) - b*(c-b)*y(t)**2)/(a*(c-a)))
x_z = sqrt(((b*C1-C2) - c*(b-c)*z(t)**2)/(a*(b-a)))
y_z = sqrt(((a*C1-C2) - c*(a-c)*z(t)**2)/(b*(a-b)))
sol1 = dsolve(a*diff(x(t),t) - (b-c)*y_x*z_x)
sol2 = dsolve(b*diff(y(t),t) - (c-a)*z_y*x_y)
sol3 = dsolve(c*diff(z(t),t) - (a-b)*x_z*y_z)
return [sol1, sol2, sol3]
def _nonlinear_3eq_order1_type2(x, y, z, t, eq):
r"""
Equations:
.. math:: a x' = (b - c) y z f(x, y, z, t)
.. math:: b y' = (c - a) z x f(x, y, z, t)
.. math:: c z' = (a - b) x y f(x, y, z, t)
First Integrals:
.. math:: a x^{2} + b y^{2} + c z^{2} = C_1
.. math:: a^{2} x^{2} + b^{2} y^{2} + c^{2} z^{2} = C_2
where `C_1` and `C_2` are arbitrary constants. On solving the integrals for `y` and
`z` and on substituting the resulting expressions into the first equation of the
system, we arrives at a first-order differential equations on `x`. Similarly doing
that for other two equations we will arrive at first order equation on `y` and `z`.
References
==========
-http://eqworld.ipmnet.ru/en/solutions/sysode/sode0402.pdf
"""
C1, C2 = get_numbered_constants(eq, num=2)
u, v, w = symbols('u, v, w')
p = Wild('p', exclude=[x(t), y(t), z(t), t])
q = Wild('q', exclude=[x(t), y(t), z(t), t])
s = Wild('s', exclude=[x(t), y(t), z(t), t])
f = Wild('f')
r1 = (diff(x(t),t) - eq[0]).match(y(t)*z(t)*f)
r = collect_const(r1[f]).match(p*f)
r.update(((diff(y(t),t) - eq[1])/r[f]).match(q*z(t)*x(t)))
r.update(((diff(z(t),t) - eq[2])/r[f]).match(s*x(t)*y(t)))
n1, d1 = r[p].as_numer_denom()
n2, d2 = r[q].as_numer_denom()
n3, d3 = r[s].as_numer_denom()
val = solve([n1*u-d1*v+d1*w, d2*u+n2*v-d2*w, -d3*u+d3*v+n3*w],[u,v])
vals = [val[v], val[u]]
c = lcm(vals[0].as_numer_denom()[1], vals[1].as_numer_denom()[1])
a = vals[0].subs(w, c)
b = vals[1].subs(w, c)
y_x = sqrt(((c*C1-C2) - a*(c-a)*x(t)**2)/(b*(c-b)))
z_x = sqrt(((b*C1-C2) - a*(b-a)*x(t)**2)/(c*(b-c)))
z_y = sqrt(((a*C1-C2) - b*(a-b)*y(t)**2)/(c*(a-c)))
x_y = sqrt(((c*C1-C2) - b*(c-b)*y(t)**2)/(a*(c-a)))
x_z = sqrt(((b*C1-C2) - c*(b-c)*z(t)**2)/(a*(b-a)))
y_z = sqrt(((a*C1-C2) - c*(a-c)*z(t)**2)/(b*(a-b)))
sol1 = dsolve(a*diff(x(t),t) - (b-c)*y_x*z_x*r[f])
sol2 = dsolve(b*diff(y(t),t) - (c-a)*z_y*x_y*r[f])
sol3 = dsolve(c*diff(z(t),t) - (a-b)*x_z*y_z*r[f])
return [sol1, sol2, sol3]
def _nonlinear_3eq_order1_type3(x, y, z, t, eq):
r"""
Equations:
.. math:: x' = c F_2 - b F_3, \enspace y' = a F_3 - c F_1, \enspace z' = b F_1 - a F_2
where `F_n = F_n(x, y, z, t)`.
1. First Integral:
.. math:: a x + b y + c z = C_1,
where C is an arbitrary constant.
2. If we assume function `F_n` to be independent of `t`,i.e, `F_n` = `F_n (x, y, z)`
Then, on eliminating `t` and `z` from the first two equation of the system, one
arrives at the first-order equation
.. math:: \frac{dy}{dx} = \frac{a F_3 (x, y, z) - c F_1 (x, y, z)}{c F_2 (x, y, z) -
b F_3 (x, y, z)}
where `z = \frac{1}{c} (C_1 - a x - b y)`
References
==========
-http://eqworld.ipmnet.ru/en/solutions/sysode/sode0404.pdf
"""
C1 = get_numbered_constants(eq, num=1)
u, v, w = symbols('u, v, w')
fu, fv, fw = symbols('u, v, w', cls=Function)
p = Wild('p', exclude=[x(t), y(t), z(t), t])
q = Wild('q', exclude=[x(t), y(t), z(t), t])
s = Wild('s', exclude=[x(t), y(t), z(t), t])
F1, F2, F3 = symbols('F1, F2, F3', cls=Wild)
r1 = (diff(x(t), t) - eq[0]).match(F2-F3)
r = collect_const(r1[F2]).match(s*F2)
r.update(collect_const(r1[F3]).match(q*F3))
if eq[1].has(r[F2]) and not eq[1].has(r[F3]):
r[F2], r[F3] = r[F3], r[F2]
r[s], r[q] = -r[q], -r[s]
r.update((diff(y(t), t) - eq[1]).match(p*r[F3] - r[s]*F1))
a = r[p]; b = r[q]; c = r[s]
F1 = r[F1].subs(x(t), u).subs(y(t),v).subs(z(t), w)
F2 = r[F2].subs(x(t), u).subs(y(t),v).subs(z(t), w)
F3 = r[F3].subs(x(t), u).subs(y(t),v).subs(z(t), w)
z_xy = (C1-a*u-b*v)/c
y_zx = (C1-a*u-c*w)/b
x_yz = (C1-b*v-c*w)/a
y_x = dsolve(diff(fv(u),u) - ((a*F3-c*F1)/(c*F2-b*F3)).subs(w,z_xy).subs(v,fv(u))).rhs
z_x = dsolve(diff(fw(u),u) - ((b*F1-a*F2)/(c*F2-b*F3)).subs(v,y_zx).subs(w,fw(u))).rhs
z_y = dsolve(diff(fw(v),v) - ((b*F1-a*F2)/(a*F3-c*F1)).subs(u,x_yz).subs(w,fw(v))).rhs
x_y = dsolve(diff(fu(v),v) - ((c*F2-b*F3)/(a*F3-c*F1)).subs(w,z_xy).subs(u,fu(v))).rhs
y_z = dsolve(diff(fv(w),w) - ((a*F3-c*F1)/(b*F1-a*F2)).subs(u,x_yz).subs(v,fv(w))).rhs
x_z = dsolve(diff(fu(w),w) - ((c*F2-b*F3)/(b*F1-a*F2)).subs(v,y_zx).subs(u,fu(w))).rhs
sol1 = dsolve(diff(fu(t),t) - (c*F2 - b*F3).subs(v,y_x).subs(w,z_x).subs(u,fu(t))).rhs
sol2 = dsolve(diff(fv(t),t) - (a*F3 - c*F1).subs(u,x_y).subs(w,z_y).subs(v,fv(t))).rhs
sol3 = dsolve(diff(fw(t),t) - (b*F1 - a*F2).subs(u,x_z).subs(v,y_z).subs(w,fw(t))).rhs
return [sol1, sol2, sol3]
def _nonlinear_3eq_order1_type4(x, y, z, t, eq):
r"""
Equations:
.. math:: x' = c z F_2 - b y F_3, \enspace y' = a x F_3 - c z F_1, \enspace z' = b y F_1 - a x F_2
where `F_n = F_n (x, y, z, t)`
1. First integral:
.. math:: a x^{2} + b y^{2} + c z^{2} = C_1
where `C` is an arbitrary constant.
2. Assuming the function `F_n` is independent of `t`: `F_n = F_n (x, y, z)`. Then on
eliminating `t` and `z` from the first two equations of the system, one arrives at
the first-order equation
.. math:: \frac{dy}{dx} = \frac{a x F_3 (x, y, z) - c z F_1 (x, y, z)}
{c z F_2 (x, y, z) - b y F_3 (x, y, z)}
where `z = \pm \sqrt{\frac{1}{c} (C_1 - a x^{2} - b y^{2})}`
References
==========
-http://eqworld.ipmnet.ru/en/solutions/sysode/sode0405.pdf
"""
C1 = get_numbered_constants(eq, num=1)
u, v, w = symbols('u, v, w')
p = Wild('p', exclude=[x(t), y(t), z(t), t])
q = Wild('q', exclude=[x(t), y(t), z(t), t])
s = Wild('s', exclude=[x(t), y(t), z(t), t])
F1, F2, F3 = symbols('F1, F2, F3', cls=Wild)
r1 = eq[0].match(diff(x(t),t) - z(t)*F2 + y(t)*F3)
r = collect_const(r1[F2]).match(s*F2)
r.update(collect_const(r1[F3]).match(q*F3))
if eq[1].has(r[F2]) and not eq[1].has(r[F3]):
r[F2], r[F3] = r[F3], r[F2]
r[s], r[q] = -r[q], -r[s]
r.update((diff(y(t),t) - eq[1]).match(p*x(t)*r[F3] - r[s]*z(t)*F1))
a = r[p]; b = r[q]; c = r[s]
F1 = r[F1].subs(x(t),u).subs(y(t),v).subs(z(t),w)
F2 = r[F2].subs(x(t),u).subs(y(t),v).subs(z(t),w)
F3 = r[F3].subs(x(t),u).subs(y(t),v).subs(z(t),w)
x_yz = sqrt((C1 - b*v**2 - c*w**2)/a)
y_zx = sqrt((C1 - c*w**2 - a*u**2)/b)
z_xy = sqrt((C1 - a*u**2 - b*v**2)/c)
y_x = dsolve(diff(v(u),u) - ((a*u*F3-c*w*F1)/(c*w*F2-b*v*F3)).subs(w,z_xy).subs(v,v(u))).rhs
z_x = dsolve(diff(w(u),u) - ((b*v*F1-a*u*F2)/(c*w*F2-b*v*F3)).subs(v,y_zx).subs(w,w(u))).rhs
z_y = dsolve(diff(w(v),v) - ((b*v*F1-a*u*F2)/(a*u*F3-c*w*F1)).subs(u,x_yz).subs(w,w(v))).rhs
x_y = dsolve(diff(u(v),v) - ((c*w*F2-b*v*F3)/(a*u*F3-c*w*F1)).subs(w,z_xy).subs(u,u(v))).rhs
y_z = dsolve(diff(v(w),w) - ((a*u*F3-c*w*F1)/(b*v*F1-a*u*F2)).subs(u,x_yz).subs(v,v(w))).rhs
x_z = dsolve(diff(u(w),w) - ((c*w*F2-b*v*F3)/(b*v*F1-a*u*F2)).subs(v,y_zx).subs(u,u(w))).rhs
sol1 = dsolve(diff(u(t),t) - (c*w*F2 - b*v*F3).subs(v,y_x).subs(w,z_x).subs(u,u(t))).rhs
sol2 = dsolve(diff(v(t),t) - (a*u*F3 - c*w*F1).subs(u,x_y).subs(w,z_y).subs(v,v(t))).rhs
sol3 = dsolve(diff(w(t),t) - (b*v*F1 - a*u*F2).subs(u,x_z).subs(v,y_z).subs(w,w(t))).rhs
return [sol1, sol2, sol3]
def _nonlinear_3eq_order1_type5(x, y, z, t, eq):
r"""
.. math:: x' = x (c F_2 - b F_3), \enspace y' = y (a F_3 - c F_1), \enspace z' = z (b F_1 - a F_2)
where `F_n = F_n (x, y, z, t)` and are arbitrary functions.
First Integral:
.. math:: \left|x\right|^{a} \left|y\right|^{b} \left|z\right|^{c} = C_1
where `C` is an arbitrary constant. If the function `F_n` is independent of `t`,
then, by eliminating `t` and `z` from the first two equations of the system, one
arrives at a first-order equation.
References
==========
-http://eqworld.ipmnet.ru/en/solutions/sysode/sode0406.pdf
"""
C1 = get_numbered_constants(eq, num=1)
u, v, w = symbols('u, v, w')
fu, fv, fw = symbols('u, v, w', cls=Function)
p = Wild('p', exclude=[x(t), y(t), z(t), t])
q = Wild('q', exclude=[x(t), y(t), z(t), t])
s = Wild('s', exclude=[x(t), y(t), z(t), t])
F1, F2, F3 = symbols('F1, F2, F3', cls=Wild)
r1 = eq[0].match(diff(x(t), t) - x(t)*F2 + x(t)*F3)
r = collect_const(r1[F2]).match(s*F2)
r.update(collect_const(r1[F3]).match(q*F3))
if eq[1].has(r[F2]) and not eq[1].has(r[F3]):
r[F2], r[F3] = r[F3], r[F2]
r[s], r[q] = -r[q], -r[s]
r.update((diff(y(t), t) - eq[1]).match(y(t)*(p*r[F3] - r[s]*F1)))
a = r[p]; b = r[q]; c = r[s]
F1 = r[F1].subs(x(t), u).subs(y(t), v).subs(z(t), w)
F2 = r[F2].subs(x(t), u).subs(y(t), v).subs(z(t), w)
F3 = r[F3].subs(x(t), u).subs(y(t), v).subs(z(t), w)
x_yz = (C1*v**-b*w**-c)**-a
y_zx = (C1*w**-c*u**-a)**-b
z_xy = (C1*u**-a*v**-b)**-c
y_x = dsolve(diff(fv(u), u) - ((v*(a*F3 - c*F1))/(u*(c*F2 - b*F3))).subs(w, z_xy).subs(v, fv(u))).rhs
z_x = dsolve(diff(fw(u), u) - ((w*(b*F1 - a*F2))/(u*(c*F2 - b*F3))).subs(v, y_zx).subs(w, fw(u))).rhs
z_y = dsolve(diff(fw(v), v) - ((w*(b*F1 - a*F2))/(v*(a*F3 - c*F1))).subs(u, x_yz).subs(w, fw(v))).rhs
x_y = dsolve(diff(fu(v), v) - ((u*(c*F2 - b*F3))/(v*(a*F3 - c*F1))).subs(w, z_xy).subs(u, fu(v))).rhs
y_z = dsolve(diff(fv(w), w) - ((v*(a*F3 - c*F1))/(w*(b*F1 - a*F2))).subs(u, x_yz).subs(v, fv(w))).rhs
x_z = dsolve(diff(fu(w), w) - ((u*(c*F2 - b*F3))/(w*(b*F1 - a*F2))).subs(v, y_zx).subs(u, fu(w))).rhs
sol1 = dsolve(diff(fu(t), t) - (u*(c*F2 - b*F3)).subs(v, y_x).subs(w, z_x).subs(u, fu(t))).rhs
sol2 = dsolve(diff(fv(t), t) - (v*(a*F3 - c*F1)).subs(u, x_y).subs(w, z_y).subs(v, fv(t))).rhs
sol3 = dsolve(diff(fw(t), t) - (w*(b*F1 - a*F2)).subs(u, x_z).subs(v, y_z).subs(w, fw(t))).rhs
return [sol1, sol2, sol3]
#This import is written at the bottom to avoid circular imports.
from .single import (NthAlgebraic, Factorable, FirstLinear, AlmostLinear,
Bernoulli, SingleODEProblem, SingleODESolver, RiccatiSpecial)
|
de5ab563d36dfd4a6955a8efc1dd42a66b7f071326eea2e2123a0bdbdcff9db1 | #
# This is the module for ODE solver classes for single ODEs.
#
import typing
if typing.TYPE_CHECKING:
from typing import ClassVar
from typing import Dict, Iterable, List, Optional, Type
from sympy.core import S
from sympy.core.exprtools import factor_terms
from sympy.core.expr import Expr
from sympy.core.function import AppliedUndef, Derivative, Function, expand
from sympy.core.numbers import Float
from sympy.core.relational import Equality, Eq
from sympy.core.symbol import Symbol, Dummy, Wild
from sympy.functions import exp, sqrt, tan, log
from sympy.integrals import Integral
from sympy.polys.polytools import cancel, factor, factor_list
from sympy.simplify import simplify
from sympy.simplify.radsimp import fraction
from sympy.utilities import numbered_symbols
from sympy.solvers.solvers import solve
from sympy.solvers.deutils import ode_order, _preprocess
class ODEMatchError(NotImplementedError):
"""Raised if a SingleODESolver is asked to solve an ODE it does not match"""
pass
def cached_property(func):
'''Decorator to cache property method'''
attrname = '_' + func.__name__
def propfunc(self):
val = getattr(self, attrname, None)
if val is None:
val = func(self)
setattr(self, attrname, val)
return val
return property(propfunc)
class SingleODEProblem:
"""Represents an ordinary differential equation (ODE)
This class is used internally in the by dsolve and related
functions/classes so that properties of an ODE can be computed
efficiently.
Examples
========
This class is used internally by dsolve. To instantiate an instance
directly first define an ODE problem:
>>> from sympy import Function, Symbol
>>> x = Symbol('x')
>>> f = Function('f')
>>> eq = f(x).diff(x, 2)
Now you can create a SingleODEProblem instance and query its properties:
>>> from sympy.solvers.ode.single import SingleODEProblem
>>> problem = SingleODEProblem(f(x).diff(x), f(x), x)
>>> problem.eq
Derivative(f(x), x)
>>> problem.func
f(x)
>>> problem.sym
x
"""
# Instance attributes:
eq = None # type: Expr
func = None # type: AppliedUndef
sym = None # type: Symbol
_order = None # type: int
_eq_expanded = None # type: Expr
_eq_preprocessed = None # type: Expr
def __init__(self, eq, func, sym, prep=True):
assert isinstance(eq, Expr)
assert isinstance(func, AppliedUndef)
assert isinstance(sym, Symbol)
assert isinstance(prep, bool)
self.eq = eq
self.func = func
self.sym = sym
self.prep = prep
@cached_property
def order(self) -> int:
return ode_order(self.eq, self.func)
@cached_property
def eq_preprocessed(self) -> Expr:
return self._get_eq_preprocessed()
@cached_property
def eq_expanded(self) -> Expr:
return expand(self.eq_preprocessed)
def _get_eq_preprocessed(self) -> Expr:
if self.prep:
process_eq, process_func = _preprocess(self.eq, self.func)
if process_func != self.func:
raise ValueError
else:
process_eq = self.eq
return process_eq
def get_numbered_constants(self, num=1, start=1, prefix='C') -> List[Symbol]:
"""
Returns a list of constants that do not occur
in eq already.
"""
ncs = self.iter_numbered_constants(start, prefix)
Cs = [next(ncs) for i in range(num)]
return Cs
def iter_numbered_constants(self, start=1, prefix='C') -> Iterable[Symbol]:
"""
Returns an iterator of constants that do not occur
in eq already.
"""
atom_set = self.eq.free_symbols
func_set = self.eq.atoms(Function)
if func_set:
atom_set |= {Symbol(str(f.func)) for f in func_set}
return numbered_symbols(start=start, prefix=prefix, exclude=atom_set)
# TODO: Add methods that can be used by many ODE solvers:
# order
# is_linear()
# get_linear_coefficients()
# eq_prepared (the ODE in prepared form)
class SingleODESolver:
"""
Base class for Single ODE solvers.
Subclasses should implement the _matches and _get_general_solution
methods. This class is not intended to be instantiated directly but its
subclasses are as part of dsolve.
Examples
========
You can use a subclass of SingleODEProblem to solve a particular type of
ODE. We first define a particular ODE problem:
>>> from sympy import Function, Symbol
>>> x = Symbol('x')
>>> f = Function('f')
>>> eq = f(x).diff(x, 2)
Now we solve this problem using the NthAlgebraic solver which is a
subclass of SingleODESolver:
>>> from sympy.solvers.ode.single import NthAlgebraic, SingleODEProblem
>>> problem = SingleODEProblem(eq, f(x), x)
>>> solver = NthAlgebraic(problem)
>>> solver.get_general_solution()
[Eq(f(x), _C*x + _C)]
The normal way to solve an ODE is to use dsolve (which would use
NthAlgebraic and other solvers internally). When using dsolve a number of
other things are done such as evaluating integrals, simplifying the
solution and renumbering the constants:
>>> from sympy import dsolve
>>> dsolve(eq, hint='nth_algebraic')
Eq(f(x), C1 + C2*x)
"""
# Subclasses should store the hint name (the argument to dsolve) in this
# attribute
hint = None # type: ClassVar[str]
# Subclasses should define this to indicate if they support an _Integral
# hint.
has_integral = None # type: ClassVar[bool]
# The ODE to be solved
ode_problem = None # type: SingleODEProblem
# Cache whether or not the equation has matched the method
_matched = None # type: Optional[bool]
# Subclasses should store in this attribute the list of order(s) of ODE
# that subclass can solve or leave it to None if not specific to any order
order = None # type: None or list
def __init__(self, ode_problem):
self.ode_problem = ode_problem
def matches(self) -> bool:
if self.order is not None and self.ode_problem.order not in self.order:
self._matched = False
return self._matched
if self._matched is None:
self._matched = self._matches()
return self._matched
def get_general_solution(self, *, simplify: bool = True) -> List[Equality]:
if not self.matches():
msg = "%s solver can not solve:\n%s"
raise ODEMatchError(msg % (self.hint, self.ode_problem.eq))
return self._get_general_solution()
def _matches(self) -> bool:
msg = "Subclasses of SingleODESolver should implement matches."
raise NotImplementedError(msg)
def _get_general_solution(self, *, simplify: bool = True) -> List[Equality]:
msg = "Subclasses of SingleODESolver should implement get_general_solution."
raise NotImplementedError(msg)
class SinglePatternODESolver(SingleODESolver):
'''Superclass for ODE solvers based on pattern matching'''
def wilds(self):
prob = self.ode_problem
f = prob.func.func
x = prob.sym
order = prob.order
return self._wilds(f, x, order)
def wilds_match(self):
match = self._wilds_match
return [match.get(w, S.Zero) for w in self.wilds()]
def _matches(self):
eq = self.ode_problem.eq_expanded
f = self.ode_problem.func.func
x = self.ode_problem.sym
order = self.ode_problem.order
df = f(x).diff(x)
if order != 1:
return False
pattern = self._equation(f(x), x, 1)
if not pattern.coeff(df).has(Wild):
eq = expand(eq / eq.coeff(df))
eq = eq.collect(f(x), func = cancel)
self._wilds_match = match = eq.match(pattern)
if match is not None:
return self._verify(f(x))
return False
def _verify(self, fx) -> bool:
return True
def _wilds(self, f, x, order):
msg = "Subclasses of SingleODESolver should implement _wilds"
raise NotImplementedError(msg)
def _equation(self, fx, x, order):
msg = "Subclasses of SingleODESolver should implement _equation"
raise NotImplementedError(msg)
class NthAlgebraic(SingleODESolver):
r"""
Solves an `n`\th order ordinary differential equation using algebra and
integrals.
There is no general form for the kind of equation that this can solve. The
the equation is solved algebraically treating differentiation as an
invertible algebraic function.
Examples
========
>>> from sympy import Function, dsolve, Eq
>>> from sympy.abc import x
>>> f = Function('f')
>>> eq = Eq(f(x) * (f(x).diff(x)**2 - 1), 0)
>>> dsolve(eq, f(x), hint='nth_algebraic')
[Eq(f(x), 0), Eq(f(x), C1 - x), Eq(f(x), C1 + x)]
Note that this solver can return algebraic solutions that do not have any
integration constants (f(x) = 0 in the above example).
"""
hint = 'nth_algebraic'
has_integral = True # nth_algebraic_Integral hint
def _matches(self):
r"""
Matches any differential equation that nth_algebraic can solve. Uses
`sympy.solve` but teaches it how to integrate derivatives.
This involves calling `sympy.solve` and does most of the work of finding a
solution (apart from evaluating the integrals).
"""
eq = self.ode_problem.eq
func = self.ode_problem.func
var = self.ode_problem.sym
# Derivative that solve can handle:
diffx = self._get_diffx(var)
# Replace derivatives wrt the independent variable with diffx
def replace(eq, var):
def expand_diffx(*args):
differand, diffs = args[0], args[1:]
toreplace = differand
for v, n in diffs:
for _ in range(n):
if v == var:
toreplace = diffx(toreplace)
else:
toreplace = Derivative(toreplace, v)
return toreplace
return eq.replace(Derivative, expand_diffx)
# Restore derivatives in solution afterwards
def unreplace(eq, var):
return eq.replace(diffx, lambda e: Derivative(e, var))
subs_eqn = replace(eq, var)
try:
# turn off simplification to protect Integrals that have
# _t instead of fx in them and would otherwise factor
# as t_*Integral(1, x)
solns = solve(subs_eqn, func, simplify=False)
except NotImplementedError:
solns = []
solns = [simplify(unreplace(soln, var)) for soln in solns]
solns = [Equality(func, soln) for soln in solns]
self.solutions = solns
return len(solns) != 0
def _get_general_solution(self, *, simplify: bool = True):
return self.solutions
# This needs to produce an invertible function but the inverse depends
# which variable we are integrating with respect to. Since the class can
# be stored in cached results we need to ensure that we always get the
# same class back for each particular integration variable so we store these
# classes in a global dict:
_diffx_stored = {} # type: Dict[Symbol, Type[Function]]
@staticmethod
def _get_diffx(var):
diffcls = NthAlgebraic._diffx_stored.get(var, None)
if diffcls is None:
# A class that behaves like Derivative wrt var but is "invertible".
class diffx(Function):
def inverse(self):
# don't use integrate here because fx has been replaced by _t
# in the equation; integrals will not be correct while solve
# is at work.
return lambda expr: Integral(expr, var) + Dummy('C')
diffcls = NthAlgebraic._diffx_stored.setdefault(var, diffx)
return diffcls
class FirstLinear(SinglePatternODESolver):
r"""
Solves 1st order linear differential equations.
These are differential equations of the form
.. math:: dy/dx + P(x) y = Q(x)\text{.}
These kinds of differential equations can be solved in a general way. The
integrating factor `e^{\int P(x) \,dx}` will turn the equation into a
separable equation. The general solution is::
>>> from sympy import Function, dsolve, Eq, pprint, diff, sin
>>> from sympy.abc import x
>>> f, P, Q = map(Function, ['f', 'P', 'Q'])
>>> genform = Eq(f(x).diff(x) + P(x)*f(x), Q(x))
>>> pprint(genform)
d
P(x)*f(x) + --(f(x)) = Q(x)
dx
>>> pprint(dsolve(genform, f(x), hint='1st_linear_Integral'))
/ / \
| | |
| | / | /
| | | | |
| | | P(x) dx | - | P(x) dx
| | | | |
| | / | /
f(x) = |C1 + | Q(x)*e dx|*e
| | |
\ / /
Examples
========
>>> f = Function('f')
>>> pprint(dsolve(Eq(x*diff(f(x), x) - f(x), x**2*sin(x)),
... f(x), '1st_linear'))
f(x) = x*(C1 - cos(x))
References
==========
- https://en.wikipedia.org/wiki/Linear_differential_equation#First_order_equation
- M. Tenenbaum & H. Pollard, "Ordinary Differential Equations",
Dover 1963, pp. 92
# indirect doctest
"""
hint = '1st_linear'
has_integral = True
order = [1]
def _wilds(self, f, x, order):
P = Wild('P', exclude=[f(x)])
Q = Wild('Q', exclude=[f(x), f(x).diff(x)])
return P, Q
def _equation(self, fx, x, order):
P, Q = self.wilds()
return fx.diff(x) + P*fx - Q
def _get_general_solution(self, *, simplify: bool = True):
P, Q = self.wilds_match()
fx = self.ode_problem.func
x = self.ode_problem.sym
(C1,) = self.ode_problem.get_numbered_constants(num=1)
gensol = Eq(fx, (((C1 + Integral(Q*exp(Integral(P, x)),x))
* exp(-Integral(P, x)))))
return [gensol]
class AlmostLinear(SinglePatternODESolver):
r"""
Solves an almost-linear differential equation.
The general form of an almost linear differential equation is
.. math:: a(x) g'(f(x)) f'(x) + b(x) g(f(x)) + c(x)
Here `f(x)` is the function to be solved for (the dependent variable).
The substitution `g(f(x)) = u(x)` leads to a linear differential equation
for `u(x)` of the form `a(x) u' + b(x) u + c(x) = 0`. This can be solved
for `u(x)` by the `first_linear` hint and then `f(x)` is found by solving
`g(f(x)) = u(x)`.
See Also
========
:meth:`sympy.solvers.ode.single.FirstLinear`
Examples
========
>>> from sympy import Function, pprint, sin, cos
>>> from sympy.solvers.ode import dsolve
>>> from sympy.abc import x
>>> f = Function('f')
>>> d = f(x).diff(x)
>>> eq = x*d + x*f(x) + 1
>>> dsolve(eq, f(x), hint='almost_linear')
Eq(f(x), (C1 - Ei(x))*exp(-x))
>>> pprint(dsolve(eq, f(x), hint='almost_linear'))
-x
f(x) = (C1 - Ei(x))*e
>>> example = cos(f(x))*f(x).diff(x) + sin(f(x)) + 1
>>> pprint(example)
d
sin(f(x)) + cos(f(x))*--(f(x)) + 1
dx
>>> pprint(dsolve(example, f(x), hint='almost_linear'))
/ -x \ / -x \
[f(x) = pi - asin\C1*e - 1/, f(x) = asin\C1*e - 1/]
References
==========
- Joel Moses, "Symbolic Integration - The Stormy Decade", Communications
of the ACM, Volume 14, Number 8, August 1971, pp. 558
"""
hint = "almost_linear"
has_integral = True
order = [1]
def _wilds(self, f, x, order):
P = Wild('P', exclude=[f(x).diff(x)])
Q = Wild('Q', exclude=[f(x).diff(x)])
return P, Q
def _equation(self, fx, x, order):
P, Q = self.wilds()
return P*fx.diff(x) + Q
def _verify(self, fx):
a, b = self.wilds_match()
c, b = b.as_independent(fx) if b.is_Add else (S.Zero, b)
# a, b and c are the function a(x), b(x) and c(x) respectively.
# c(x) is obtained by separating out b as terms with and without fx i.e, l(y)
# The following conditions checks if the given equation is an almost-linear differential equation using the fact that
# a(x)*(l(y))' / l(y)' is independent of l(y)
if b.diff(fx) != 0 and not simplify(b.diff(fx)/a).has(fx):
self.ly = factor_terms(b).as_independent(fx, as_Add=False)[1] # Gives the term containing fx i.e., l(y)
self.ax = a / self.ly.diff(fx)
self.cx = -c # cx is taken as -c(x) to simplify expression in the solution integral
self.bx = factor_terms(b) / self.ly
return True
return False
def _get_general_solution(self, *, simplify: bool = True):
x = self.ode_problem.sym
(C1,) = self.ode_problem.get_numbered_constants(num=1)
gensol = Eq(self.ly, (((C1 + Integral((self.cx/self.ax)*exp(Integral(self.bx/self.ax, x)),x))
* exp(-Integral(self.bx/self.ax, x)))))
return [gensol]
class Bernoulli(SinglePatternODESolver):
r"""
Solves Bernoulli differential equations.
These are equations of the form
.. math:: dy/dx + P(x) y = Q(x) y^n\text{, }n \ne 1`\text{.}
The substitution `w = 1/y^{1-n}` will transform an equation of this form
into one that is linear (see the docstring of
:py:meth:`~sympy.solvers.ode.single.FirstLinear`). The general solution is::
>>> from sympy import Function, dsolve, Eq, pprint
>>> from sympy.abc import x, n
>>> f, P, Q = map(Function, ['f', 'P', 'Q'])
>>> genform = Eq(f(x).diff(x) + P(x)*f(x), Q(x)*f(x)**n)
>>> pprint(genform)
d n
P(x)*f(x) + --(f(x)) = Q(x)*f (x)
dx
>>> pprint(dsolve(genform, f(x), hint='Bernoulli_Integral'), num_columns=110)
-1
-----
n - 1
// / / \ \
|| | | | |
|| | / | / | / |
|| | | | | | | |
|| | (1 - n)* | P(x) dx | (1 - n)* | P(x) dx | (n - 1)* | P(x) dx|
|| | | | | | | |
|| | / | / | / |
f(x) = ||C1 - n* | Q(x)*e dx + | Q(x)*e dx|*e |
|| | | | |
\\ / / / /
Note that the equation is separable when `n = 1` (see the docstring of
:py:meth:`~sympy.solvers.ode.ode.ode_separable`).
>>> pprint(dsolve(Eq(f(x).diff(x) + P(x)*f(x), Q(x)*f(x)), f(x),
... hint='separable_Integral'))
f(x)
/
| /
| 1 |
| - dy = C1 + | (-P(x) + Q(x)) dx
| y |
| /
/
Examples
========
>>> from sympy import Function, dsolve, Eq, pprint, log
>>> from sympy.abc import x
>>> f = Function('f')
>>> pprint(dsolve(Eq(x*f(x).diff(x) + f(x), log(x)*f(x)**2),
... f(x), hint='Bernoulli'))
1
f(x) = -----------------
C1*x + log(x) + 1
References
==========
- https://en.wikipedia.org/wiki/Bernoulli_differential_equation
- M. Tenenbaum & H. Pollard, "Ordinary Differential Equations",
Dover 1963, pp. 95
# indirect doctest
"""
hint = "Bernoulli"
has_integral = True
order = [1]
def _wilds(self, f, x, order):
P = Wild('P', exclude=[f(x)])
Q = Wild('Q', exclude=[f(x)])
n = Wild('n', exclude=[x, f(x), f(x).diff(x)])
return P, Q, n
def _equation(self, fx, x, order):
P, Q, n = self.wilds()
return fx.diff(x) + P*fx - Q*fx**n
def _get_general_solution(self, *, simplify: bool = True):
P, Q, n = self.wilds_match()
fx = self.ode_problem.func
x = self.ode_problem.sym
(C1,) = self.ode_problem.get_numbered_constants(num=1)
if n==1:
gensol = Eq(log(fx), (
(C1 + Integral((-P + Q),x)
)))
else:
gensol = Eq(fx**(1-n), (
(C1 - (n - 1) * Integral(Q*exp(-n*Integral(P, x))
* exp(Integral(P, x)), x)
) * exp(-(1 - n)*Integral(P, x)))
)
return [gensol]
class Factorable(SingleODESolver):
r"""
Solves equations having a solvable factor.
This function is used to solve the equation having factors. Factors may be of type algebraic or ode. It
will try to solve each factor independently. Factors will be solved by calling dsolve. We will return the
list of solutions.
Examples
========
>>> from sympy import Function, dsolve, pprint
>>> from sympy.abc import x
>>> f = Function('f')
>>> eq = (f(x)**2-4)*(f(x).diff(x)+f(x))
>>> pprint(dsolve(eq, f(x)))
-x
[f(x) = 2, f(x) = -2, f(x) = C1*e ]
"""
hint = "factorable"
has_integral = False
def _matches(self):
eq = self.ode_problem.eq
f = self.ode_problem.func.func
x = self.ode_problem.sym
order =self.ode_problem.order
df = f(x).diff(x)
self.eqs = []
eq = eq.collect(f(x), func = cancel)
eq = fraction(factor(eq))[0]
roots = factor_list(eq)[1]
if len(roots)>1 or roots[0][1]>1:
for base,expo in roots:
if base.has(f(x)):
self.eqs.append(base)
if len(self.eqs)>0:
return True
roots = solve(eq, df)
if len(roots)>0:
self.eqs = [(df - root) for root in roots]
if len(self.eqs)==1:
if order>1:
return False
if self.eqs[0].has(Float):
return False
return fraction(factor(self.eqs[0]))[0]-eq!=0
return True
return False
def _get_general_solution(self, *, simplify: bool = True):
func = self.ode_problem.func.func
x = self.ode_problem.sym
eqns = self.eqs
sols = []
for eq in eqns:
try:
sol = dsolve(eq, func(x))
except NotImplementedError:
continue
else:
if isinstance(sol, list):
sols.extend(sol)
else:
sols.append(sol)
if sols == []:
raise NotImplementedError("The given ODE " + str(eq) + " cannot be solved by"
+ " the factorable group method")
return sols
class RiccatiSpecial(SinglePatternODESolver):
r"""
The general Riccati equation has the form
.. math:: dy/dx = f(x) y^2 + g(x) y + h(x)\text{.}
While it does not have a general solution [1], the "special" form, `dy/dx
= a y^2 - b x^c`, does have solutions in many cases [2]. This routine
returns a solution for `a(dy/dx) = b y^2 + c y/x + d/x^2` that is obtained
by using a suitable change of variables to reduce it to the special form
and is valid when neither `a` nor `b` are zero and either `c` or `d` is
zero.
>>> from sympy.abc import x, a, b, c, d
>>> from sympy.solvers.ode import dsolve, checkodesol
>>> from sympy import pprint, Function
>>> f = Function('f')
>>> y = f(x)
>>> genform = a*y.diff(x) - (b*y**2 + c*y/x + d/x**2)
>>> sol = dsolve(genform, y)
>>> pprint(sol, wrap_line=False)
/ / __________________ \\
| __________________ | / 2 ||
| / 2 | \/ 4*b*d - (a + c) *log(x)||
-|a + c - \/ 4*b*d - (a + c) *tan|C1 + ----------------------------||
\ \ 2*a //
f(x) = ------------------------------------------------------------------------
2*b*x
>>> checkodesol(genform, sol, order=1)[0]
True
References
==========
1. http://www.maplesoft.com/support/help/Maple/view.aspx?path=odeadvisor/Riccati
2. http://eqworld.ipmnet.ru/en/solutions/ode/ode0106.pdf -
http://eqworld.ipmnet.ru/en/solutions/ode/ode0123.pdf
"""
hint = "Riccati_special_minus2"
has_integral = False
order = [1]
def _wilds(self, f, x, order):
a = Wild('a', exclude=[x, f(x), f(x).diff(x), 0])
b = Wild('b', exclude=[x, f(x), f(x).diff(x), 0])
c = Wild('c', exclude=[x, f(x), f(x).diff(x)])
d = Wild('d', exclude=[x, f(x), f(x).diff(x)])
return a, b, c, d
def _equation(self, fx, x, order):
a, b, c, d = self.wilds()
return a*fx.diff(x) + b*fx**2 + c*fx/x + d/x**2
def _get_general_solution(self, *, simplify: bool = True):
a, b, c, d = self.wilds_match()
fx = self.ode_problem.func
x = self.ode_problem.sym
(C1,) = self.ode_problem.get_numbered_constants(num=1)
mu = sqrt(4*d*b - (a - c)**2)
gensol = Eq(fx, (a - c - mu*tan(mu/(2*a)*log(x) + C1))/(2*b*x))
return [gensol]
# Avoid circular import:
from .ode import dsolve
|
0dbe66354caf809f71d223db796aeb5a9fa233d9ec764ff4d65176cf14a19650 | from sympy.core.containers import Tuple
from sympy.core.function import (Function, Lambda, nfloat, diff)
from sympy.core.mod import Mod
from sympy.core.numbers import (E, I, Rational, oo, pi)
from sympy.core.relational import (Eq, Gt,
Ne)
from sympy.core.singleton import S
from sympy.core.symbol import (Dummy, Symbol, symbols)
from sympy.functions.elementary.complexes import (Abs, arg, im, re, sign)
from sympy.functions.elementary.exponential import (LambertW, exp, log)
from sympy.functions.elementary.hyperbolic import (HyperbolicFunction,
sinh, tanh, cosh, sech, coth)
from sympy.functions.elementary.miscellaneous import sqrt, Min, Max
from sympy.functions.elementary.piecewise import Piecewise
from sympy.functions.elementary.trigonometric import (
TrigonometricFunction, acos, acot, acsc, asec, asin, atan, atan2,
cos, cot, csc, sec, sin, tan)
from sympy.functions.special.error_functions import (erf, erfc,
erfcinv, erfinv)
from sympy.logic.boolalg import And
from sympy.matrices.dense import MutableDenseMatrix as Matrix
from sympy.matrices.immutable import ImmutableDenseMatrix
from sympy.polys.polytools import Poly
from sympy.polys.rootoftools import CRootOf
from sympy.sets.contains import Contains
from sympy.sets.conditionset import ConditionSet
from sympy.sets.fancysets import ImageSet
from sympy.sets.sets import (Complement, EmptySet, FiniteSet,
Intersection, Interval, Union, imageset, ProductSet)
from sympy.tensor.indexed import Indexed
from sympy.utilities.iterables import numbered_symbols
from sympy.testing.pytest import (XFAIL, raises, skip, slow, SKIP)
from sympy.testing.randtest import verify_numerically as tn
from sympy.physics.units import cm
from sympy.solvers.solveset import (
solveset_real, domain_check, solveset_complex, linear_eq_to_matrix,
linsolve, _is_function_class_equation, invert_real, invert_complex,
solveset, solve_decomposition, substitution, nonlinsolve, solvify,
_is_finite_with_finite_vars, _transolve, _is_exponential,
_solve_exponential, _is_logarithmic,
_solve_logarithm, _term_factors, _is_modular, NonlinearError)
from sympy.abc import (a, b, c, d, e, f, g, h, i, j, k, l, m, n, q, r,
t, w, x, y, z)
def dumeq(i, j):
if type(i) in (list, tuple):
return all(dumeq(i, j) for i, j in zip(i, j))
return i == j or i.dummy_eq(j)
def test_invert_real():
x = Symbol('x', real=True)
def ireal(x, s=S.Reals):
return Intersection(s, x)
# issue 14223
assert invert_real(x, 0, x, Interval(1, 2)) == (x, S.EmptySet)
assert invert_real(exp(x), z, x) == (x, ireal(FiniteSet(log(z))))
y = Symbol('y', positive=True)
n = Symbol('n', real=True)
assert invert_real(x + 3, y, x) == (x, FiniteSet(y - 3))
assert invert_real(x*3, y, x) == (x, FiniteSet(y / 3))
assert invert_real(exp(x), y, x) == (x, FiniteSet(log(y)))
assert invert_real(exp(3*x), y, x) == (x, FiniteSet(log(y) / 3))
assert invert_real(exp(x + 3), y, x) == (x, FiniteSet(log(y) - 3))
assert invert_real(exp(x) + 3, y, x) == (x, ireal(FiniteSet(log(y - 3))))
assert invert_real(exp(x)*3, y, x) == (x, FiniteSet(log(y / 3)))
assert invert_real(log(x), y, x) == (x, FiniteSet(exp(y)))
assert invert_real(log(3*x), y, x) == (x, FiniteSet(exp(y) / 3))
assert invert_real(log(x + 3), y, x) == (x, FiniteSet(exp(y) - 3))
assert invert_real(Abs(x), y, x) == (x, FiniteSet(y, -y))
assert invert_real(2**x, y, x) == (x, FiniteSet(log(y)/log(2)))
assert invert_real(2**exp(x), y, x) == (x, ireal(FiniteSet(log(log(y)/log(2)))))
assert invert_real(x**2, y, x) == (x, FiniteSet(sqrt(y), -sqrt(y)))
assert invert_real(x**S.Half, y, x) == (x, FiniteSet(y**2))
raises(ValueError, lambda: invert_real(x, x, x))
raises(ValueError, lambda: invert_real(x**pi, y, x))
raises(ValueError, lambda: invert_real(S.One, y, x))
assert invert_real(x**31 + x, y, x) == (x**31 + x, FiniteSet(y))
lhs = x**31 + x
base_values = FiniteSet(y - 1, -y - 1)
assert invert_real(Abs(x**31 + x + 1), y, x) == (lhs, base_values)
assert dumeq(invert_real(sin(x), y, x),
(x, imageset(Lambda(n, n*pi + (-1)**n*asin(y)), S.Integers)))
assert dumeq(invert_real(sin(exp(x)), y, x),
(x, imageset(Lambda(n, log((-1)**n*asin(y) + n*pi)), S.Integers)))
assert dumeq(invert_real(csc(x), y, x),
(x, imageset(Lambda(n, n*pi + (-1)**n*acsc(y)), S.Integers)))
assert dumeq(invert_real(csc(exp(x)), y, x),
(x, imageset(Lambda(n, log((-1)**n*acsc(y) + n*pi)), S.Integers)))
assert dumeq(invert_real(cos(x), y, x),
(x, Union(imageset(Lambda(n, 2*n*pi + acos(y)), S.Integers), \
imageset(Lambda(n, 2*n*pi - acos(y)), S.Integers))))
assert dumeq(invert_real(cos(exp(x)), y, x),
(x, Union(imageset(Lambda(n, log(2*n*pi + acos(y))), S.Integers), \
imageset(Lambda(n, log(2*n*pi - acos(y))), S.Integers))))
assert dumeq(invert_real(sec(x), y, x),
(x, Union(imageset(Lambda(n, 2*n*pi + asec(y)), S.Integers), \
imageset(Lambda(n, 2*n*pi - asec(y)), S.Integers))))
assert dumeq(invert_real(sec(exp(x)), y, x),
(x, Union(imageset(Lambda(n, log(2*n*pi + asec(y))), S.Integers), \
imageset(Lambda(n, log(2*n*pi - asec(y))), S.Integers))))
assert dumeq(invert_real(tan(x), y, x),
(x, imageset(Lambda(n, n*pi + atan(y)), S.Integers)))
assert dumeq(invert_real(tan(exp(x)), y, x),
(x, imageset(Lambda(n, log(n*pi + atan(y))), S.Integers)))
assert dumeq(invert_real(cot(x), y, x),
(x, imageset(Lambda(n, n*pi + acot(y)), S.Integers)))
assert dumeq(invert_real(cot(exp(x)), y, x),
(x, imageset(Lambda(n, log(n*pi + acot(y))), S.Integers)))
assert dumeq(invert_real(tan(tan(x)), y, x),
(tan(x), imageset(Lambda(n, n*pi + atan(y)), S.Integers)))
x = Symbol('x', positive=True)
assert invert_real(x**pi, y, x) == (x, FiniteSet(y**(1/pi)))
def test_invert_complex():
assert invert_complex(x + 3, y, x) == (x, FiniteSet(y - 3))
assert invert_complex(x*3, y, x) == (x, FiniteSet(y / 3))
assert dumeq(invert_complex(exp(x), y, x),
(x, imageset(Lambda(n, I*(2*pi*n + arg(y)) + log(Abs(y))), S.Integers)))
assert invert_complex(log(x), y, x) == (x, FiniteSet(exp(y)))
raises(ValueError, lambda: invert_real(1, y, x))
raises(ValueError, lambda: invert_complex(x, x, x))
raises(ValueError, lambda: invert_complex(x, x, 1))
# https://github.com/skirpichev/omg/issues/16
assert invert_complex(sinh(x), 0, x) != (x, FiniteSet(0))
def test_domain_check():
assert domain_check(1/(1 + (1/(x+1))**2), x, -1) is False
assert domain_check(x**2, x, 0) is True
assert domain_check(x, x, oo) is False
assert domain_check(0, x, oo) is False
def test_issue_11536():
assert solveset(0**x - 100, x, S.Reals) == S.EmptySet
assert solveset(0**x - 1, x, S.Reals) == FiniteSet(0)
def test_issue_17479():
from sympy.solvers.solveset import nonlinsolve
f = (x**2 + y**2)**2 + (x**2 + z**2)**2 - 2*(2*x**2 + y**2 + z**2)
fx = f.diff(x)
fy = f.diff(y)
fz = f.diff(z)
sol = nonlinsolve([fx, fy, fz], [x, y, z])
assert len(sol) >= 4 and len(sol) <= 20
# nonlinsolve has been giving a varying number of solutions
# (originally 18, then 20, now 19) due to various internal changes.
# Unfortunately not all the solutions are actually valid and some are
# redundant. Since the original issue was that an exception was raised,
# this first test only checks that nonlinsolve returns a "plausible"
# solution set. The next test checks the result for correctness.
@XFAIL
def test_issue_18449():
x, y, z = symbols("x, y, z")
f = (x**2 + y**2)**2 + (x**2 + z**2)**2 - 2*(2*x**2 + y**2 + z**2)
fx = diff(f, x)
fy = diff(f, y)
fz = diff(f, z)
sol = nonlinsolve([fx, fy, fz], [x, y, z])
for (xs, ys, zs) in sol:
d = {x: xs, y: ys, z: zs}
assert tuple(_.subs(d).simplify() for _ in (fx, fy, fz)) == (0, 0, 0)
# After simplification and removal of duplicate elements, there should
# only be 4 parametric solutions left:
# simplifiedsolutions = FiniteSet((sqrt(1 - z**2), z, z),
# (-sqrt(1 - z**2), z, z),
# (sqrt(1 - z**2), -z, z),
# (-sqrt(1 - z**2), -z, z))
# TODO: Is the above solution set definitely complete?
def test_is_function_class_equation():
from sympy.abc import x, a
assert _is_function_class_equation(TrigonometricFunction,
tan(x), x) is True
assert _is_function_class_equation(TrigonometricFunction,
tan(x) - 1, x) is True
assert _is_function_class_equation(TrigonometricFunction,
tan(x) + sin(x), x) is True
assert _is_function_class_equation(TrigonometricFunction,
tan(x) + sin(x) - a, x) is True
assert _is_function_class_equation(TrigonometricFunction,
sin(x)*tan(x) + sin(x), x) is True
assert _is_function_class_equation(TrigonometricFunction,
sin(x)*tan(x + a) + sin(x), x) is True
assert _is_function_class_equation(TrigonometricFunction,
sin(x)*tan(x*a) + sin(x), x) is True
assert _is_function_class_equation(TrigonometricFunction,
a*tan(x) - 1, x) is True
assert _is_function_class_equation(TrigonometricFunction,
tan(x)**2 + sin(x) - 1, x) is True
assert _is_function_class_equation(TrigonometricFunction,
tan(x) + x, x) is False
assert _is_function_class_equation(TrigonometricFunction,
tan(x**2), x) is False
assert _is_function_class_equation(TrigonometricFunction,
tan(x**2) + sin(x), x) is False
assert _is_function_class_equation(TrigonometricFunction,
tan(x)**sin(x), x) is False
assert _is_function_class_equation(TrigonometricFunction,
tan(sin(x)) + sin(x), x) is False
assert _is_function_class_equation(HyperbolicFunction,
tanh(x), x) is True
assert _is_function_class_equation(HyperbolicFunction,
tanh(x) - 1, x) is True
assert _is_function_class_equation(HyperbolicFunction,
tanh(x) + sinh(x), x) is True
assert _is_function_class_equation(HyperbolicFunction,
tanh(x) + sinh(x) - a, x) is True
assert _is_function_class_equation(HyperbolicFunction,
sinh(x)*tanh(x) + sinh(x), x) is True
assert _is_function_class_equation(HyperbolicFunction,
sinh(x)*tanh(x + a) + sinh(x), x) is True
assert _is_function_class_equation(HyperbolicFunction,
sinh(x)*tanh(x*a) + sinh(x), x) is True
assert _is_function_class_equation(HyperbolicFunction,
a*tanh(x) - 1, x) is True
assert _is_function_class_equation(HyperbolicFunction,
tanh(x)**2 + sinh(x) - 1, x) is True
assert _is_function_class_equation(HyperbolicFunction,
tanh(x) + x, x) is False
assert _is_function_class_equation(HyperbolicFunction,
tanh(x**2), x) is False
assert _is_function_class_equation(HyperbolicFunction,
tanh(x**2) + sinh(x), x) is False
assert _is_function_class_equation(HyperbolicFunction,
tanh(x)**sinh(x), x) is False
assert _is_function_class_equation(HyperbolicFunction,
tanh(sinh(x)) + sinh(x), x) is False
def test_garbage_input():
raises(ValueError, lambda: solveset_real([y], y))
x = Symbol('x', real=True)
assert solveset_real(x, 1) == S.EmptySet
assert solveset_real(x - 1, 1) == FiniteSet(x)
assert solveset_real(x, pi) == S.EmptySet
assert solveset_real(x, x**2) == S.EmptySet
raises(ValueError, lambda: solveset_complex([x], x))
assert solveset_complex(x, pi) == S.EmptySet
raises(ValueError, lambda: solveset((x, y), x))
raises(ValueError, lambda: solveset(x + 1, S.Reals))
raises(ValueError, lambda: solveset(x + 1, x, 2))
def test_solve_mul():
assert solveset_real((a*x + b)*(exp(x) - 3), x) == \
Union({log(3)}, Intersection({-b/a}, S.Reals))
anz = Symbol('anz', nonzero=True)
bb = Symbol('bb', real=True)
assert solveset_real((anz*x + bb)*(exp(x) - 3), x) == \
FiniteSet(-bb/anz, log(3))
assert solveset_real((2*x + 8)*(8 + exp(x)), x) == FiniteSet(S(-4))
assert solveset_real(x/log(x), x) == EmptySet()
def test_solve_invert():
assert solveset_real(exp(x) - 3, x) == FiniteSet(log(3))
assert solveset_real(log(x) - 3, x) == FiniteSet(exp(3))
assert solveset_real(3**(x + 2), x) == FiniteSet()
assert solveset_real(3**(2 - x), x) == FiniteSet()
assert solveset_real(y - b*exp(a/x), x) == Intersection(
S.Reals, FiniteSet(a/log(y/b)))
# issue 4504
assert solveset_real(2**x - 10, x) == FiniteSet(1 + log(5)/log(2))
def test_errorinverses():
assert solveset_real(erf(x) - S.Half, x) == \
FiniteSet(erfinv(S.Half))
assert solveset_real(erfinv(x) - 2, x) == \
FiniteSet(erf(2))
assert solveset_real(erfc(x) - S.One, x) == \
FiniteSet(erfcinv(S.One))
assert solveset_real(erfcinv(x) - 2, x) == FiniteSet(erfc(2))
def test_solve_polynomial():
x = Symbol('x', real=True)
y = Symbol('y', real=True)
assert solveset_real(3*x - 2, x) == FiniteSet(Rational(2, 3))
assert solveset_real(x**2 - 1, x) == FiniteSet(-S.One, S.One)
assert solveset_real(x - y**3, x) == FiniteSet(y ** 3)
a11, a12, a21, a22, b1, b2 = symbols('a11, a12, a21, a22, b1, b2')
assert solveset_real(x**3 - 15*x - 4, x) == FiniteSet(
-2 + 3 ** S.Half,
S(4),
-2 - 3 ** S.Half)
assert solveset_real(sqrt(x) - 1, x) == FiniteSet(1)
assert solveset_real(sqrt(x) - 2, x) == FiniteSet(4)
assert solveset_real(x**Rational(1, 4) - 2, x) == FiniteSet(16)
assert solveset_real(x**Rational(1, 3) - 3, x) == FiniteSet(27)
assert len(solveset_real(x**5 + x**3 + 1, x)) == 1
assert len(solveset_real(-2*x**3 + 4*x**2 - 2*x + 6, x)) > 0
assert solveset_real(x**6 + x**4 + I, x) == ConditionSet(x,
Eq(x**6 + x**4 + I, 0), S.Reals)
def test_return_root_of():
f = x**5 - 15*x**3 - 5*x**2 + 10*x + 20
s = list(solveset_complex(f, x))
for root in s:
assert root.func == CRootOf
# if one uses solve to get the roots of a polynomial that has a CRootOf
# solution, make sure that the use of nfloat during the solve process
# doesn't fail. Note: if you want numerical solutions to a polynomial
# it is *much* faster to use nroots to get them than to solve the
# equation only to get CRootOf solutions which are then numerically
# evaluated. So for eq = x**5 + 3*x + 7 do Poly(eq).nroots() rather
# than [i.n() for i in solve(eq)] to get the numerical roots of eq.
assert nfloat(list(solveset_complex(x**5 + 3*x**3 + 7, x))[0],
exponent=False) == CRootOf(x**5 + 3*x**3 + 7, 0).n()
sol = list(solveset_complex(x**6 - 2*x + 2, x))
assert all(isinstance(i, CRootOf) for i in sol) and len(sol) == 6
f = x**5 - 15*x**3 - 5*x**2 + 10*x + 20
s = list(solveset_complex(f, x))
for root in s:
assert root.func == CRootOf
s = x**5 + 4*x**3 + 3*x**2 + Rational(7, 4)
assert solveset_complex(s, x) == \
FiniteSet(*Poly(s*4, domain='ZZ').all_roots())
# Refer issue #7876
eq = x*(x - 1)**2*(x + 1)*(x**6 - x + 1)
assert solveset_complex(eq, x) == \
FiniteSet(-1, 0, 1, CRootOf(x**6 - x + 1, 0),
CRootOf(x**6 - x + 1, 1),
CRootOf(x**6 - x + 1, 2),
CRootOf(x**6 - x + 1, 3),
CRootOf(x**6 - x + 1, 4),
CRootOf(x**6 - x + 1, 5))
def test__has_rational_power():
from sympy.solvers.solveset import _has_rational_power
assert _has_rational_power(sqrt(2), x)[0] is False
assert _has_rational_power(x*sqrt(2), x)[0] is False
assert _has_rational_power(x**2*sqrt(x), x) == (True, 2)
assert _has_rational_power(sqrt(2)*x**Rational(1, 3), x) == (True, 3)
assert _has_rational_power(sqrt(x)*x**Rational(1, 3), x) == (True, 6)
def test_solveset_sqrt_1():
assert solveset_real(sqrt(5*x + 6) - 2 - x, x) == \
FiniteSet(-S.One, S(2))
assert solveset_real(sqrt(x - 1) - x + 7, x) == FiniteSet(10)
assert solveset_real(sqrt(x - 2) - 5, x) == FiniteSet(27)
assert solveset_real(sqrt(x) - 2 - 5, x) == FiniteSet(49)
assert solveset_real(sqrt(x**3), x) == FiniteSet(0)
assert solveset_real(sqrt(x - 1), x) == FiniteSet(1)
def test_solveset_sqrt_2():
x = Symbol('x', real=True)
y = Symbol('y', real=True)
# http://tutorial.math.lamar.edu/Classes/Alg/SolveRadicalEqns.aspx#Solve_Rad_Ex2_a
assert solveset_real(sqrt(2*x - 1) - sqrt(x - 4) - 2, x) == \
FiniteSet(S(5), S(13))
assert solveset_real(sqrt(x + 7) + 2 - sqrt(3 - x), x) == \
FiniteSet(-6)
# http://www.purplemath.com/modules/solverad.htm
assert solveset_real(sqrt(17*x - sqrt(x**2 - 5)) - 7, x) == \
FiniteSet(3)
eq = x + 1 - (x**4 + 4*x**3 - x)**Rational(1, 4)
assert solveset_real(eq, x) == FiniteSet(Rational(-1, 2), Rational(-1, 3))
eq = sqrt(2*x + 9) - sqrt(x + 1) - sqrt(x + 4)
assert solveset_real(eq, x) == FiniteSet(0)
eq = sqrt(x + 4) + sqrt(2*x - 1) - 3*sqrt(x - 1)
assert solveset_real(eq, x) == FiniteSet(5)
eq = sqrt(x)*sqrt(x - 7) - 12
assert solveset_real(eq, x) == FiniteSet(16)
eq = sqrt(x - 3) + sqrt(x) - 3
assert solveset_real(eq, x) == FiniteSet(4)
eq = sqrt(2*x**2 - 7) - (3 - x)
assert solveset_real(eq, x) == FiniteSet(-S(8), S(2))
# others
eq = sqrt(9*x**2 + 4) - (3*x + 2)
assert solveset_real(eq, x) == FiniteSet(0)
assert solveset_real(sqrt(x - 3) - sqrt(x) - 3, x) == FiniteSet()
eq = (2*x - 5)**Rational(1, 3) - 3
assert solveset_real(eq, x) == FiniteSet(16)
assert solveset_real(sqrt(x) + sqrt(sqrt(x)) - 4, x) == \
FiniteSet((Rational(-1, 2) + sqrt(17)/2)**4)
eq = sqrt(x) - sqrt(x - 1) + sqrt(sqrt(x))
assert solveset_real(eq, x) == FiniteSet()
eq = (sqrt(x) + sqrt(x + 1) + sqrt(1 - x) - 6*sqrt(5)/5)
ans = solveset_real(eq, x)
ra = S('''-1484/375 - 4*(-1/2 + sqrt(3)*I/2)*(-12459439/52734375 +
114*sqrt(12657)/78125)**(1/3) - 172564/(140625*(-1/2 +
sqrt(3)*I/2)*(-12459439/52734375 + 114*sqrt(12657)/78125)**(1/3))''')
rb = Rational(4, 5)
assert all(abs(eq.subs(x, i).n()) < 1e-10 for i in (ra, rb)) and \
len(ans) == 2 and \
set([i.n(chop=True) for i in ans]) == \
set([i.n(chop=True) for i in (ra, rb)])
assert solveset_real(sqrt(x) + x**Rational(1, 3) +
x**Rational(1, 4), x) == FiniteSet(0)
assert solveset_real(x/sqrt(x**2 + 1), x) == FiniteSet(0)
eq = (x - y**3)/((y**2)*sqrt(1 - y**2))
assert solveset_real(eq, x) == FiniteSet(y**3)
# issue 4497
assert solveset_real(1/(5 + x)**Rational(1, 5) - 9, x) == \
FiniteSet(Rational(-295244, 59049))
@XFAIL
def test_solve_sqrt_fail():
# this only works if we check real_root(eq.subs(x, Rational(1, 3)))
# but checksol doesn't work like that
eq = (x**3 - 3*x**2)**Rational(1, 3) + 1 - x
assert solveset_real(eq, x) == FiniteSet(Rational(1, 3))
@slow
def test_solve_sqrt_3():
R = Symbol('R')
eq = sqrt(2)*R*sqrt(1/(R + 1)) + (R + 1)*(sqrt(2)*sqrt(1/(R + 1)) - 1)
sol = solveset_complex(eq, R)
fset = [Rational(5, 3) + 4*sqrt(10)*cos(atan(3*sqrt(111)/251)/3)/3,
-sqrt(10)*cos(atan(3*sqrt(111)/251)/3)/3 +
40*re(1/((Rational(-1, 2) - sqrt(3)*I/2)*(Rational(251, 27) + sqrt(111)*I/9)**Rational(1, 3)))/9 +
sqrt(30)*sin(atan(3*sqrt(111)/251)/3)/3 + Rational(5, 3) +
I*(-sqrt(30)*cos(atan(3*sqrt(111)/251)/3)/3 -
sqrt(10)*sin(atan(3*sqrt(111)/251)/3)/3 +
40*im(1/((Rational(-1, 2) - sqrt(3)*I/2)*(Rational(251, 27) + sqrt(111)*I/9)**Rational(1, 3)))/9)]
cset = [40*re(1/((Rational(-1, 2) + sqrt(3)*I/2)*(Rational(251, 27) + sqrt(111)*I/9)**Rational(1, 3)))/9 -
sqrt(10)*cos(atan(3*sqrt(111)/251)/3)/3 - sqrt(30)*sin(atan(3*sqrt(111)/251)/3)/3 +
Rational(5, 3) +
I*(40*im(1/((Rational(-1, 2) + sqrt(3)*I/2)*(Rational(251, 27) + sqrt(111)*I/9)**Rational(1, 3)))/9 -
sqrt(10)*sin(atan(3*sqrt(111)/251)/3)/3 +
sqrt(30)*cos(atan(3*sqrt(111)/251)/3)/3)]
assert sol._args[0] == FiniteSet(*fset)
assert sol._args[1] == ConditionSet(
R,
Eq(sqrt(2)*R*sqrt(1/(R + 1)) + (R + 1)*(sqrt(2)*sqrt(1/(R + 1)) - 1), 0),
FiniteSet(*cset))
# the number of real roots will depend on the value of m: for m=1 there are 4
# and for m=-1 there are none.
eq = -sqrt((m - q)**2 + (-m/(2*q) + S.Half)**2) + sqrt((-m**2/2 - sqrt(
4*m**4 - 4*m**2 + 8*m + 1)/4 - Rational(1, 4))**2 + (m**2/2 - m - sqrt(
4*m**4 - 4*m**2 + 8*m + 1)/4 - Rational(1, 4))**2)
unsolved_object = ConditionSet(q, Eq(sqrt((m - q)**2 + (-m/(2*q) + S.Half)**2) -
sqrt((-m**2/2 - sqrt(4*m**4 - 4*m**2 + 8*m + 1)/4 - Rational(1, 4))**2 + (m**2/2 - m -
sqrt(4*m**4 - 4*m**2 + 8*m + 1)/4 - Rational(1, 4))**2), 0), S.Reals)
assert solveset_real(eq, q) == unsolved_object
def test_solve_polynomial_symbolic_param():
assert solveset_complex((x**2 - 1)**2 - a, x) == \
FiniteSet(sqrt(1 + sqrt(a)), -sqrt(1 + sqrt(a)),
sqrt(1 - sqrt(a)), -sqrt(1 - sqrt(a)))
# issue 4507
assert solveset_complex(y - b/(1 + a*x), x) == \
FiniteSet((b/y - 1)/a) - FiniteSet(-1/a)
# issue 4508
assert solveset_complex(y - b*x/(a + x), x) == \
FiniteSet(-a*y/(y - b)) - FiniteSet(-a)
def test_solve_rational():
assert solveset_real(1/x + 1, x) == FiniteSet(-S.One)
assert solveset_real(1/exp(x) - 1, x) == FiniteSet(0)
assert solveset_real(x*(1 - 5/x), x) == FiniteSet(5)
assert solveset_real(2*x/(x + 2) - 1, x) == FiniteSet(2)
assert solveset_real((x**2/(7 - x)).diff(x), x) == \
FiniteSet(S.Zero, S(14))
def test_solveset_real_gen_is_pow():
assert solveset_real(sqrt(1) + 1, x) == EmptySet()
def test_no_sol():
assert solveset(1 - oo*x) == EmptySet()
assert solveset(oo*x, x) == EmptySet()
assert solveset(oo*x - oo, x) == EmptySet()
assert solveset_real(4, x) == EmptySet()
assert solveset_real(exp(x), x) == EmptySet()
assert solveset_real(x**2 + 1, x) == EmptySet()
assert solveset_real(-3*a/sqrt(x), x) == EmptySet()
assert solveset_real(1/x, x) == EmptySet()
assert solveset_real(-(1 + x)/(2 + x)**2 + 1/(2 + x), x) == \
EmptySet()
def test_sol_zero_real():
assert solveset_real(0, x) == S.Reals
assert solveset(0, x, Interval(1, 2)) == Interval(1, 2)
assert solveset_real(-x**2 - 2*x + (x + 1)**2 - 1, x) == S.Reals
def test_no_sol_rational_extragenous():
assert solveset_real((x/(x + 1) + 3)**(-2), x) == EmptySet()
assert solveset_real((x - 1)/(1 + 1/(x - 1)), x) == EmptySet()
def test_solve_polynomial_cv_1a():
"""
Test for solving on equations that can be converted to
a polynomial equation using the change of variable y -> x**Rational(p, q)
"""
assert solveset_real(sqrt(x) - 1, x) == FiniteSet(1)
assert solveset_real(sqrt(x) - 2, x) == FiniteSet(4)
assert solveset_real(x**Rational(1, 4) - 2, x) == FiniteSet(16)
assert solveset_real(x**Rational(1, 3) - 3, x) == FiniteSet(27)
assert solveset_real(x*(x**(S.One / 3) - 3), x) == \
FiniteSet(S.Zero, S(27))
def test_solveset_real_rational():
"""Test solveset_real for rational functions"""
x = Symbol('x', real=True)
y = Symbol('y', real=True)
assert solveset_real((x - y**3) / ((y**2)*sqrt(1 - y**2)), x) \
== FiniteSet(y**3)
# issue 4486
assert solveset_real(2*x/(x + 2) - 1, x) == FiniteSet(2)
def test_solveset_real_log():
assert solveset_real(log((x-1)*(x+1)), x) == \
FiniteSet(sqrt(2), -sqrt(2))
def test_poly_gens():
assert solveset_real(4**(2*(x**2) + 2*x) - 8, x) == \
FiniteSet(Rational(-3, 2), S.Half)
def test_solve_abs():
n = Dummy('n')
raises(ValueError, lambda: solveset(Abs(x) - 1, x))
assert solveset(Abs(x) - n, x, S.Reals) == ConditionSet(x, Contains(n, Interval(0, oo)), {-n, n})
assert solveset_real(Abs(x) - 2, x) == FiniteSet(-2, 2)
assert solveset_real(Abs(x) + 2, x) is S.EmptySet
assert solveset_real(Abs(x + 3) - 2*Abs(x - 3), x) == \
FiniteSet(1, 9)
assert solveset_real(2*Abs(x) - Abs(x - 1), x) == \
FiniteSet(-1, Rational(1, 3))
sol = ConditionSet(
x,
And(
Contains(b, Interval(0, oo)),
Contains(a + b, Interval(0, oo)),
Contains(a - b, Interval(0, oo))),
FiniteSet(-a - b - 3, -a + b - 3, a - b - 3, a + b - 3))
eq = Abs(Abs(x + 3) - a) - b
assert invert_real(eq, 0, x)[1] == sol
reps = {a: 3, b: 1}
eqab = eq.subs(reps)
for si in sol.subs(reps):
assert not eqab.subs(x, si)
assert dumeq(solveset(Eq(sin(Abs(x)), 1), x, domain=S.Reals), Union(
Intersection(Interval(0, oo),
ImageSet(Lambda(n, (-1)**n*pi/2 + n*pi), S.Integers)),
Intersection(Interval(-oo, 0),
ImageSet(Lambda(n, n*pi - (-1)**(-n)*pi/2), S.Integers))))
def test_issue_9824():
assert dumeq(solveset(sin(x)**2 - 2*sin(x) + 1, x), ImageSet(Lambda(n, 2*n*pi + pi/2), S.Integers))
assert dumeq(solveset(cos(x)**2 - 2*cos(x) + 1, x), ImageSet(Lambda(n, 2*n*pi), S.Integers))
def test_issue_9565():
assert solveset_real(Abs((x - 1)/(x - 5)) <= Rational(1, 3), x) == Interval(-1, 2)
def test_issue_10069():
eq = abs(1/(x - 1)) - 1 > 0
assert solveset_real(eq, x) == Union(
Interval.open(0, 1), Interval.open(1, 2))
def test_real_imag_splitting():
a, b = symbols('a b', real=True)
assert solveset_real(sqrt(a**2 - b**2) - 3, a) == \
FiniteSet(-sqrt(b**2 + 9), sqrt(b**2 + 9))
assert solveset_real(sqrt(a**2 + b**2) - 3, a) != \
S.EmptySet
def test_units():
assert solveset_real(1/x - 1/(2*cm), x) == FiniteSet(2*cm)
def test_solve_only_exp_1():
y = Symbol('y', positive=True)
assert solveset_real(exp(x) - y, x) == FiniteSet(log(y))
assert solveset_real(exp(x) + exp(-x) - 4, x) == \
FiniteSet(log(-sqrt(3) + 2), log(sqrt(3) + 2))
assert solveset_real(exp(x) + exp(-x) - y, x) != S.EmptySet
def test_atan2():
# The .inverse() method on atan2 works only if x.is_real is True and the
# second argument is a real constant
assert solveset_real(atan2(x, 2) - pi/3, x) == FiniteSet(2*sqrt(3))
def test_piecewise_solveset():
eq = Piecewise((x - 2, Gt(x, 2)), (2 - x, True)) - 3
assert set(solveset_real(eq, x)) == set(FiniteSet(-1, 5))
absxm3 = Piecewise(
(x - 3, 0 <= x - 3),
(3 - x, 0 > x - 3))
y = Symbol('y', positive=True)
assert solveset_real(absxm3 - y, x) == FiniteSet(-y + 3, y + 3)
f = Piecewise(((x - 2)**2, x >= 0), (0, True))
assert solveset(f, x, domain=S.Reals) == Union(FiniteSet(2), Interval(-oo, 0, True, True))
assert solveset(
Piecewise((x + 1, x > 0), (I, True)) - I, x, S.Reals
) == Interval(-oo, 0)
assert solveset(Piecewise((x - 1, Ne(x, I)), (x, True)), x) == FiniteSet(1)
def test_solveset_complex_polynomial():
assert solveset_complex(a*x**2 + b*x + c, x) == \
FiniteSet(-b/(2*a) - sqrt(-4*a*c + b**2)/(2*a),
-b/(2*a) + sqrt(-4*a*c + b**2)/(2*a))
assert solveset_complex(x - y**3, y) == FiniteSet(
(-x**Rational(1, 3))/2 + I*sqrt(3)*x**Rational(1, 3)/2,
x**Rational(1, 3),
(-x**Rational(1, 3))/2 - I*sqrt(3)*x**Rational(1, 3)/2)
assert solveset_complex(x + 1/x - 1, x) == \
FiniteSet(S.Half + I*sqrt(3)/2, S.Half - I*sqrt(3)/2)
def test_sol_zero_complex():
assert solveset_complex(0, x) == S.Complexes
def test_solveset_complex_rational():
assert solveset_complex((x - 1)*(x - I)/(x - 3), x) == \
FiniteSet(1, I)
assert solveset_complex((x - y**3)/((y**2)*sqrt(1 - y**2)), x) == \
FiniteSet(y**3)
assert solveset_complex(-x**2 - I, x) == \
FiniteSet(-sqrt(2)/2 + sqrt(2)*I/2, sqrt(2)/2 - sqrt(2)*I/2)
def test_solve_quintics():
skip("This test is too slow")
f = x**5 - 110*x**3 - 55*x**2 + 2310*x + 979
s = solveset_complex(f, x)
for root in s:
res = f.subs(x, root.n()).n()
assert tn(res, 0)
f = x**5 + 15*x + 12
s = solveset_complex(f, x)
for root in s:
res = f.subs(x, root.n()).n()
assert tn(res, 0)
def test_solveset_complex_exp():
from sympy.abc import x, n
assert dumeq(solveset_complex(exp(x) - 1, x),
imageset(Lambda(n, I*2*n*pi), S.Integers))
assert dumeq(solveset_complex(exp(x) - I, x),
imageset(Lambda(n, I*(2*n*pi + pi/2)), S.Integers))
assert solveset_complex(1/exp(x), x) == S.EmptySet
assert dumeq(solveset_complex(sinh(x).rewrite(exp), x),
imageset(Lambda(n, n*pi*I), S.Integers))
def test_solveset_real_exp():
from sympy.abc import x, y
assert solveset(Eq((-2)**x, 4), x, S.Reals) == FiniteSet(2)
assert solveset(Eq(-2**x, 4), x, S.Reals) == S.EmptySet
assert solveset(Eq((-3)**x, 27), x, S.Reals) == S.EmptySet
assert solveset(Eq((-5)**(x+1), 625), x, S.Reals) == FiniteSet(3)
assert solveset(Eq(2**(x-3), -16), x, S.Reals) == S.EmptySet
assert solveset(Eq((-3)**(x - 3), -3**39), x, S.Reals) == FiniteSet(42)
assert solveset(Eq(2**x, y), x, S.Reals) == Intersection(S.Reals, FiniteSet(log(y)/log(2)))
assert invert_real((-2)**(2*x) - 16, 0, x) == (x, FiniteSet(2))
def test_solve_complex_log():
assert solveset_complex(log(x), x) == FiniteSet(1)
assert solveset_complex(1 - log(a + 4*x**2), x) == \
FiniteSet(-sqrt(-a + E)/2, sqrt(-a + E)/2)
def test_solve_complex_sqrt():
assert solveset_complex(sqrt(5*x + 6) - 2 - x, x) == \
FiniteSet(-S.One, S(2))
assert solveset_complex(sqrt(5*x + 6) - (2 + 2*I) - x, x) == \
FiniteSet(-S(2), 3 - 4*I)
assert solveset_complex(4*x*(1 - a * sqrt(x)), x) == \
FiniteSet(S.Zero, 1 / a ** 2)
def test_solveset_complex_tan():
s = solveset_complex(tan(x).rewrite(exp), x)
assert dumeq(s, imageset(Lambda(n, pi*n), S.Integers) - \
imageset(Lambda(n, pi*n + pi/2), S.Integers))
def test_solve_trig():
from sympy.abc import n
assert dumeq(solveset_real(sin(x), x),
Union(imageset(Lambda(n, 2*pi*n), S.Integers),
imageset(Lambda(n, 2*pi*n + pi), S.Integers)))
assert dumeq(solveset_real(sin(x) - 1, x),
imageset(Lambda(n, 2*pi*n + pi/2), S.Integers))
assert dumeq(solveset_real(cos(x), x),
Union(imageset(Lambda(n, 2*pi*n + pi/2), S.Integers),
imageset(Lambda(n, 2*pi*n + pi*Rational(3, 2)), S.Integers)))
assert dumeq(solveset_real(sin(x) + cos(x), x),
Union(imageset(Lambda(n, 2*n*pi + pi*Rational(3, 4)), S.Integers),
imageset(Lambda(n, 2*n*pi + pi*Rational(7, 4)), S.Integers)))
assert solveset_real(sin(x)**2 + cos(x)**2, x) == S.EmptySet
assert dumeq(solveset_complex(cos(x) - S.Half, x),
Union(imageset(Lambda(n, 2*n*pi + pi*Rational(5, 3)), S.Integers),
imageset(Lambda(n, 2*n*pi + pi/3), S.Integers)))
assert dumeq(solveset(sin(y + a) - sin(y), a, domain=S.Reals),
Union(ImageSet(Lambda(n, 2*n*pi), S.Integers),
Intersection(ImageSet(Lambda(n, -I*(I*(
2*n*pi + arg(-exp(-2*I*y))) +
2*im(y))), S.Integers), S.Reals)))
assert dumeq(solveset_real(sin(2*x)*cos(x) + cos(2*x)*sin(x)-1, x),
ImageSet(Lambda(n, n*pi*Rational(2, 3) + pi/6), S.Integers))
# Tests for _solve_trig2() function
assert dumeq(solveset_real(2*cos(x)*cos(2*x) - 1, x),
Union(ImageSet(Lambda(n, 2*n*pi + 2*atan(sqrt(-2*2**Rational(1, 3)*(67 +
9*sqrt(57))**Rational(2, 3) + 8*2**Rational(2, 3) + 11*(67 +
9*sqrt(57))**Rational(1, 3))/(3*(67 + 9*sqrt(57))**Rational(1, 6)))), S.Integers),
ImageSet(Lambda(n, 2*n*pi - 2*atan(sqrt(-2*2**Rational(1, 3)*(67 +
9*sqrt(57))**Rational(2, 3) + 8*2**Rational(2, 3) + 11*(67 +
9*sqrt(57))**Rational(1, 3))/(3*(67 + 9*sqrt(57))**Rational(1, 6))) +
2*pi), S.Integers)))
assert dumeq(solveset_real(2*tan(x)*sin(x) + 1, x), Union(
ImageSet(Lambda(n, 2*n*pi + atan(sqrt(2)*sqrt(-1 +sqrt(17))/
(1 - sqrt(17))) + pi), S.Integers),
ImageSet(Lambda(n, 2*n*pi - atan(sqrt(2)*sqrt(-1 + sqrt(17))/
(1 - sqrt(17))) + pi), S.Integers)))
assert dumeq(solveset_real(cos(2*x)*cos(4*x) - 1, x),
ImageSet(Lambda(n, n*pi), S.Integers))
def test_solve_hyperbolic():
# actual solver: _solve_trig1
n = Dummy('n')
assert solveset(sinh(x) + cosh(x), x) == S.EmptySet
assert solveset(sinh(x) + cos(x), x) == ConditionSet(x,
Eq(cos(x) + sinh(x), 0), S.Complexes)
assert solveset_real(sinh(x) + sech(x), x) == FiniteSet(
log(sqrt(sqrt(5) - 2)))
assert solveset_real(3*cosh(2*x) - 5, x) == FiniteSet(
log(sqrt(3)/3), log(sqrt(3)))
assert solveset_real(sinh(x - 3) - 2, x) == FiniteSet(
log((2 + sqrt(5))*exp(3)))
assert solveset_real(cosh(2*x) + 2*sinh(x) - 5, x) == FiniteSet(
log(-2 + sqrt(5)), log(1 + sqrt(2)))
assert solveset_real((coth(x) + sinh(2*x))/cosh(x) - 3, x) == FiniteSet(
log(S.Half + sqrt(5)/2), log(1 + sqrt(2)))
assert solveset_real(cosh(x)*sinh(x) - 2, x) == FiniteSet(
log(sqrt(4 + sqrt(17))))
assert solveset_real(sinh(x) + tanh(x) - 1, x) == FiniteSet(
log(sqrt(2)/2 + sqrt(-S(1)/2 + sqrt(2))))
assert dumeq(solveset_complex(sinh(x) - I/2, x), Union(
ImageSet(Lambda(n, I*(2*n*pi + 5*pi/6)), S.Integers),
ImageSet(Lambda(n, I*(2*n*pi + pi/6)), S.Integers)))
assert dumeq(solveset_complex(sinh(x) + sech(x), x), Union(
ImageSet(Lambda(n, 2*n*I*pi + log(sqrt(-2 + sqrt(5)))), S.Integers),
ImageSet(Lambda(n, I*(2*n*pi + pi/2) + log(sqrt(2 + sqrt(5)))), S.Integers),
ImageSet(Lambda(n, I*(2*n*pi + pi) + log(sqrt(-2 + sqrt(5)))), S.Integers),
ImageSet(Lambda(n, I*(2*n*pi - pi/2) + log(sqrt(2 + sqrt(5)))), S.Integers)))
# issues #9606 / #9531:
assert solveset(sinh(x), x, S.Reals) == FiniteSet(0)
assert dumeq(solveset(sinh(x), x, S.Complexes), Union(
ImageSet(Lambda(n, I*(2*n*pi + pi)), S.Integers),
ImageSet(Lambda(n, 2*n*I*pi), S.Integers)))
def test_solve_invalid_sol():
assert 0 not in solveset_real(sin(x)/x, x)
assert 0 not in solveset_complex((exp(x) - 1)/x, x)
@XFAIL
def test_solve_trig_simplified():
from sympy.abc import n
assert dumeq(solveset_real(sin(x), x),
imageset(Lambda(n, n*pi), S.Integers))
assert dumeq(solveset_real(cos(x), x),
imageset(Lambda(n, n*pi + pi/2), S.Integers))
assert dumeq(solveset_real(cos(x) + sin(x), x),
imageset(Lambda(n, n*pi - pi/4), S.Integers))
@XFAIL
def test_solve_lambert():
assert solveset_real(x*exp(x) - 1, x) == FiniteSet(LambertW(1))
assert solveset_real(exp(x) + x, x) == FiniteSet(-LambertW(1))
assert solveset_real(x + 2**x, x) == \
FiniteSet(-LambertW(log(2))/log(2))
# issue 4739
ans = solveset_real(3*x + 5 + 2**(-5*x + 3), x)
assert ans == FiniteSet(Rational(-5, 3) +
LambertW(-10240*2**Rational(1, 3)*log(2)/3)/(5*log(2)))
eq = 2*(3*x + 4)**5 - 6*7**(3*x + 9)
result = solveset_real(eq, x)
ans = FiniteSet((log(2401) +
5*LambertW(-log(7**(7*3**Rational(1, 5)/5))))/(3*log(7))/-1)
assert result == ans
assert solveset_real(eq.expand(), x) == result
assert solveset_real(5*x - 1 + 3*exp(2 - 7*x), x) == \
FiniteSet(Rational(1, 5) + LambertW(-21*exp(Rational(3, 5))/5)/7)
assert solveset_real(2*x + 5 + log(3*x - 2), x) == \
FiniteSet(Rational(2, 3) + LambertW(2*exp(Rational(-19, 3))/3)/2)
assert solveset_real(3*x + log(4*x), x) == \
FiniteSet(LambertW(Rational(3, 4))/3)
assert solveset_real(x**x - 2) == FiniteSet(exp(LambertW(log(2))))
a = Symbol('a')
assert solveset_real(-a*x + 2*x*log(x), x) == FiniteSet(exp(a/2))
a = Symbol('a', real=True)
assert solveset_real(a/x + exp(x/2), x) == \
FiniteSet(2*LambertW(-a/2))
assert solveset_real((a/x + exp(x/2)).diff(x), x) == \
FiniteSet(4*LambertW(sqrt(2)*sqrt(a)/4))
# coverage test
assert solveset_real(tanh(x + 3)*tanh(x - 3) - 1, x) == EmptySet()
assert solveset_real((x**2 - 2*x + 1).subs(x, log(x) + 3*x), x) == \
FiniteSet(LambertW(3*S.Exp1)/3)
assert solveset_real((x**2 - 2*x + 1).subs(x, (log(x) + 3*x)**2 - 1), x) == \
FiniteSet(LambertW(3*exp(-sqrt(2)))/3, LambertW(3*exp(sqrt(2)))/3)
assert solveset_real((x**2 - 2*x - 2).subs(x, log(x) + 3*x), x) == \
FiniteSet(LambertW(3*exp(1 + sqrt(3)))/3, LambertW(3*exp(-sqrt(3) + 1))/3)
assert solveset_real(x*log(x) + 3*x + 1, x) == \
FiniteSet(exp(-3 + LambertW(-exp(3))))
eq = (x*exp(x) - 3).subs(x, x*exp(x))
assert solveset_real(eq, x) == \
FiniteSet(LambertW(3*exp(-LambertW(3))))
assert solveset_real(3*log(a**(3*x + 5)) + a**(3*x + 5), x) == \
FiniteSet(-((log(a**5) + LambertW(Rational(1, 3)))/(3*log(a))))
p = symbols('p', positive=True)
assert solveset_real(3*log(p**(3*x + 5)) + p**(3*x + 5), x) == \
FiniteSet(
log((-3**Rational(1, 3) - 3**Rational(5, 6)*I)*LambertW(Rational(1, 3))**Rational(1, 3)/(2*p**Rational(5, 3)))/log(p),
log((-3**Rational(1, 3) + 3**Rational(5, 6)*I)*LambertW(Rational(1, 3))**Rational(1, 3)/(2*p**Rational(5, 3)))/log(p),
log((3*LambertW(Rational(1, 3))/p**5)**(1/(3*log(p)))),) # checked numerically
# check collection
b = Symbol('b')
eq = 3*log(a**(3*x + 5)) + b*log(a**(3*x + 5)) + a**(3*x + 5)
assert solveset_real(eq, x) == FiniteSet(
-((log(a**5) + LambertW(1/(b + 3)))/(3*log(a))))
# issue 4271
assert solveset_real((a/x + exp(x/2)).diff(x, 2), x) == FiniteSet(
6*LambertW((-1)**Rational(1, 3)*a**Rational(1, 3)/3))
assert solveset_real(x**3 - 3**x, x) == \
FiniteSet(-3/log(3)*LambertW(-log(3)/3))
assert solveset_real(3**cos(x) - cos(x)**3) == FiniteSet(
acos(-3*LambertW(-log(3)/3)/log(3)))
assert solveset_real(x**2 - 2**x, x) == \
solveset_real(-x**2 + 2**x, x)
assert solveset_real(3*log(x) - x*log(3)) == FiniteSet(
-3*LambertW(-log(3)/3)/log(3),
-3*LambertW(-log(3)/3, -1)/log(3))
assert solveset_real(LambertW(2*x) - y) == FiniteSet(
y*exp(y)/2)
@XFAIL
def test_other_lambert():
a = Rational(6, 5)
assert solveset_real(x**a - a**x, x) == FiniteSet(
a, -a*LambertW(-log(a)/a)/log(a))
def test_solveset():
f = Function('f')
raises(ValueError, lambda: solveset(x + y))
assert solveset(x, 1) == S.EmptySet
assert solveset(f(1)**2 + y + 1, f(1)
) == FiniteSet(-sqrt(-y - 1), sqrt(-y - 1))
assert solveset(f(1)**2 - 1, f(1), S.Reals) == FiniteSet(-1, 1)
assert solveset(f(1)**2 + 1, f(1)) == FiniteSet(-I, I)
assert solveset(x - 1, 1) == FiniteSet(x)
assert solveset(sin(x) - cos(x), sin(x)) == FiniteSet(cos(x))
assert solveset(0, domain=S.Reals) == S.Reals
assert solveset(1) == S.EmptySet
assert solveset(True, domain=S.Reals) == S.Reals # issue 10197
assert solveset(False, domain=S.Reals) == S.EmptySet
assert solveset(exp(x) - 1, domain=S.Reals) == FiniteSet(0)
assert solveset(exp(x) - 1, x, S.Reals) == FiniteSet(0)
assert solveset(Eq(exp(x), 1), x, S.Reals) == FiniteSet(0)
assert solveset(exp(x) - 1, exp(x), S.Reals) == FiniteSet(1)
A = Indexed('A', x)
assert solveset(A - 1, A, S.Reals) == FiniteSet(1)
assert solveset(x - 1 >= 0, x, S.Reals) == Interval(1, oo)
assert solveset(exp(x) - 1 >= 0, x, S.Reals) == Interval(0, oo)
assert dumeq(solveset(exp(x) - 1, x), imageset(Lambda(n, 2*I*pi*n), S.Integers))
assert dumeq(solveset(Eq(exp(x), 1), x), imageset(Lambda(n, 2*I*pi*n),
S.Integers))
# issue 13825
assert solveset(x**2 + f(0) + 1, x) == {-sqrt(-f(0) - 1), sqrt(-f(0) - 1)}
def test__solveset_multi():
from sympy.solvers.solveset import _solveset_multi
from sympy import Reals
# Basic univariate case:
from sympy.abc import x
assert _solveset_multi([x**2-1], [x], [S.Reals]) == FiniteSet((1,), (-1,))
# Linear systems of two equations
from sympy.abc import x, y
assert _solveset_multi([x+y, x+1], [x, y], [Reals, Reals]) == FiniteSet((-1, 1))
assert _solveset_multi([x+y, x+1], [y, x], [Reals, Reals]) == FiniteSet((1, -1))
assert _solveset_multi([x+y, x-y-1], [x, y], [Reals, Reals]) == FiniteSet((S(1)/2, -S(1)/2))
assert _solveset_multi([x-1, y-2], [x, y], [Reals, Reals]) == FiniteSet((1, 2))
# assert dumeq(_solveset_multi([x+y], [x, y], [Reals, Reals]), ImageSet(Lambda(x, (x, -x)), Reals))
assert dumeq(_solveset_multi([x+y], [x, y], [Reals, Reals]), Union(
ImageSet(Lambda(((x,),), (x, -x)), ProductSet(Reals)),
ImageSet(Lambda(((y,),), (-y, y)), ProductSet(Reals))))
assert _solveset_multi([x+y, x+y+1], [x, y], [Reals, Reals]) == S.EmptySet
assert _solveset_multi([x+y, x-y, x-1], [x, y], [Reals, Reals]) == S.EmptySet
assert _solveset_multi([x+y, x-y, x-1], [y, x], [Reals, Reals]) == S.EmptySet
# Systems of three equations:
from sympy.abc import x, y, z
assert _solveset_multi([x+y+z-1, x+y-z-2, x-y-z-3], [x, y, z], [Reals,
Reals, Reals]) == FiniteSet((2, -S.Half, -S.Half))
# Nonlinear systems:
from sympy.abc import r, theta, z, x, y
assert _solveset_multi([x**2+y**2-2, x+y], [x, y], [Reals, Reals]) == FiniteSet((-1, 1), (1, -1))
assert _solveset_multi([x**2-1, y], [x, y], [Reals, Reals]) == FiniteSet((1, 0), (-1, 0))
#assert _solveset_multi([x**2-y**2], [x, y], [Reals, Reals]) == Union(
# ImageSet(Lambda(x, (x, -x)), Reals), ImageSet(Lambda(x, (x, x)), Reals))
assert dumeq(_solveset_multi([x**2-y**2], [x, y], [Reals, Reals]), Union(
ImageSet(Lambda(((x,),), (x, -Abs(x))), ProductSet(Reals)),
ImageSet(Lambda(((x,),), (x, Abs(x))), ProductSet(Reals)),
ImageSet(Lambda(((y,),), (-Abs(y), y)), ProductSet(Reals)),
ImageSet(Lambda(((y,),), (Abs(y), y)), ProductSet(Reals))))
assert _solveset_multi([r*cos(theta)-1, r*sin(theta)], [theta, r],
[Interval(0, pi), Interval(-1, 1)]) == FiniteSet((0, 1), (pi, -1))
assert _solveset_multi([r*cos(theta)-1, r*sin(theta)], [r, theta],
[Interval(0, 1), Interval(0, pi)]) == FiniteSet((1, 0))
#assert _solveset_multi([r*cos(theta)-r, r*sin(theta)], [r, theta],
# [Interval(0, 1), Interval(0, pi)]) == ?
assert dumeq(_solveset_multi([r*cos(theta)-r, r*sin(theta)], [r, theta],
[Interval(0, 1), Interval(0, pi)]), Union(
ImageSet(Lambda(((r,),), (r, 0)), ImageSet(Lambda(r, (r,)), Interval(0, 1))),
ImageSet(Lambda(((theta,),), (0, theta)), ImageSet(Lambda(theta, (theta,)), Interval(0, pi)))))
def test_conditionset():
assert solveset(Eq(sin(x)**2 + cos(x)**2, 1), x, domain=S.Reals) == \
ConditionSet(x, True, S.Reals)
assert solveset(Eq(x**2 + x*sin(x), 1), x, domain=S.Reals
) == ConditionSet(x, Eq(x**2 + x*sin(x) - 1, 0), S.Reals)
assert dumeq(solveset(Eq(-I*(exp(I*x) - exp(-I*x))/2, 1), x
), imageset(Lambda(n, 2*n*pi + pi/2), S.Integers))
assert solveset(x + sin(x) > 1, x, domain=S.Reals
) == ConditionSet(x, x + sin(x) > 1, S.Reals)
assert solveset(Eq(sin(Abs(x)), x), x, domain=S.Reals
) == ConditionSet(x, Eq(-x + sin(Abs(x)), 0), S.Reals)
assert solveset(y**x-z, x, S.Reals) == \
ConditionSet(x, Eq(y**x - z, 0), S.Reals)
@XFAIL
def test_conditionset_equality():
''' Checking equality of different representations of ConditionSet'''
assert solveset(Eq(tan(x), y), x) == ConditionSet(x, Eq(tan(x), y), S.Complexes)
def test_solveset_domain():
assert solveset(x**2 - x - 6, x, Interval(0, oo)) == FiniteSet(3)
assert solveset(x**2 - 1, x, Interval(0, oo)) == FiniteSet(1)
assert solveset(x**4 - 16, x, Interval(0, 10)) == FiniteSet(2)
def test_improve_coverage():
from sympy.solvers.solveset import _has_rational_power
solution = solveset(exp(x) + sin(x), x, S.Reals)
unsolved_object = ConditionSet(x, Eq(exp(x) + sin(x), 0), S.Reals)
assert solution == unsolved_object
assert _has_rational_power(sin(x)*exp(x) + 1, x) == (False, S.One)
assert _has_rational_power((sin(x)**2)*(exp(x) + 1)**3, x) == (False, S.One)
def test_issue_9522():
expr1 = Eq(1/(x**2 - 4) + x, 1/(x**2 - 4) + 2)
expr2 = Eq(1/x + x, 1/x)
assert solveset(expr1, x, S.Reals) == EmptySet()
assert solveset(expr2, x, S.Reals) == EmptySet()
def test_solvify():
assert solvify(x**2 + 10, x, S.Reals) == []
assert solvify(x**3 + 1, x, S.Complexes) == [-1, S.Half - sqrt(3)*I/2,
S.Half + sqrt(3)*I/2]
assert solvify(log(x), x, S.Reals) == [1]
assert solvify(cos(x), x, S.Reals) == [pi/2, pi*Rational(3, 2)]
assert solvify(sin(x) + 1, x, S.Reals) == [pi*Rational(3, 2)]
raises(NotImplementedError, lambda: solvify(sin(exp(x)), x, S.Complexes))
def test_abs_invert_solvify():
assert solvify(sin(Abs(x)), x, S.Reals) is None
def test_linear_eq_to_matrix():
eqns1 = [2*x + y - 2*z - 3, x - y - z, x + y + 3*z - 12]
eqns2 = [Eq(3*x + 2*y - z, 1), Eq(2*x - 2*y + 4*z, -2), -2*x + y - 2*z]
A, B = linear_eq_to_matrix(eqns1, x, y, z)
assert A == Matrix([[2, 1, -2], [1, -1, -1], [1, 1, 3]])
assert B == Matrix([[3], [0], [12]])
A, B = linear_eq_to_matrix(eqns2, x, y, z)
assert A == Matrix([[3, 2, -1], [2, -2, 4], [-2, 1, -2]])
assert B == Matrix([[1], [-2], [0]])
# Pure symbolic coefficients
eqns3 = [a*b*x + b*y + c*z - d, e*x + d*x + f*y + g*z - h, i*x + j*y + k*z - l]
A, B = linear_eq_to_matrix(eqns3, x, y, z)
assert A == Matrix([[a*b, b, c], [d + e, f, g], [i, j, k]])
assert B == Matrix([[d], [h], [l]])
# raise ValueError if
# 1) no symbols are given
raises(ValueError, lambda: linear_eq_to_matrix(eqns3))
# 2) there are duplicates
raises(ValueError, lambda: linear_eq_to_matrix(eqns3, [x, x, y]))
# 3) there are non-symbols
raises(ValueError, lambda: linear_eq_to_matrix(eqns3, [x, 1/a, y]))
# 4) a nonlinear term is detected in the original expression
raises(NonlinearError, lambda: linear_eq_to_matrix(Eq(1/x + x, 1/x), [x]))
assert linear_eq_to_matrix(1, x) == (Matrix([[0]]), Matrix([[-1]]))
# issue 15195
assert linear_eq_to_matrix(x + y*(z*(3*x + 2) + 3), x) == (
Matrix([[3*y*z + 1]]), Matrix([[-y*(2*z + 3)]]))
assert linear_eq_to_matrix(Matrix(
[[a*x + b*y - 7], [5*x + 6*y - c]]), x, y) == (
Matrix([[a, b], [5, 6]]), Matrix([[7], [c]]))
# issue 15312
assert linear_eq_to_matrix(Eq(x + 2, 1), x) == (
Matrix([[1]]), Matrix([[-1]]))
def test_issue_16577():
assert linear_eq_to_matrix(Eq(a*(2*x + 3*y) + 4*y, 5), x, y) == (
Matrix([[2*a, 3*a + 4]]), Matrix([[5]]))
def test_linsolve():
x1, x2, x3, x4 = symbols('x1, x2, x3, x4')
# Test for different input forms
M = Matrix([[1, 2, 1, 1, 7], [1, 2, 2, -1, 12], [2, 4, 0, 6, 4]])
system1 = A, B = M[:, :-1], M[:, -1]
Eqns = [x1 + 2*x2 + x3 + x4 - 7, x1 + 2*x2 + 2*x3 - x4 - 12,
2*x1 + 4*x2 + 6*x4 - 4]
sol = FiniteSet((-2*x2 - 3*x4 + 2, x2, 2*x4 + 5, x4))
assert linsolve(Eqns, (x1, x2, x3, x4)) == sol
assert linsolve(Eqns, *(x1, x2, x3, x4)) == sol
assert linsolve(system1, (x1, x2, x3, x4)) == sol
assert linsolve(system1, *(x1, x2, x3, x4)) == sol
# issue 9667 - symbols can be Dummy symbols
x1, x2, x3, x4 = symbols('x:4', cls=Dummy)
assert linsolve(system1, x1, x2, x3, x4) == FiniteSet(
(-2*x2 - 3*x4 + 2, x2, 2*x4 + 5, x4))
# raise ValueError for garbage value
raises(ValueError, lambda: linsolve(Eqns))
raises(ValueError, lambda: linsolve(x1))
raises(ValueError, lambda: linsolve(x1, x2))
raises(ValueError, lambda: linsolve((A,), x1, x2))
raises(ValueError, lambda: linsolve(A, B, x1, x2))
#raise ValueError if equations are non-linear in given variables
raises(NonlinearError, lambda: linsolve([x + y - 1, x ** 2 + y - 3], [x, y]))
raises(NonlinearError, lambda: linsolve([cos(x) + y, x + y], [x, y]))
assert linsolve([x + z - 1, x ** 2 + y - 3], [z, y]) == {(-x + 1, -x**2 + 3)}
# Fully symbolic test
A = Matrix([[a, b], [c, d]])
B = Matrix([[e], [g]])
system2 = (A, B)
sol = FiniteSet(((-b*g + d*e)/(a*d - b*c), (a*g - c*e)/(a*d - b*c)))
assert linsolve(system2, [x, y]) == sol
# No solution
A = Matrix([[1, 2, 3], [2, 4, 6], [3, 6, 9]])
B = Matrix([0, 0, 1])
assert linsolve((A, B), (x, y, z)) == EmptySet()
# Issue #10056
A, B, J1, J2 = symbols('A B J1 J2')
Augmatrix = Matrix([
[2*I*J1, 2*I*J2, -2/J1],
[-2*I*J2, -2*I*J1, 2/J2],
[0, 2, 2*I/(J1*J2)],
[2, 0, 0],
])
assert linsolve(Augmatrix, A, B) == FiniteSet((0, I/(J1*J2)))
# Issue #10121 - Assignment of free variables
Augmatrix = Matrix([[0, 1, 0, 0, 0, 0], [0, 0, 0, 1, 0, 0]])
assert linsolve(Augmatrix, a, b, c, d, e) == FiniteSet((a, 0, c, 0, e))
raises(IndexError, lambda: linsolve(Augmatrix, a, b, c))
x0, x1, x2, _x0 = symbols('tau0 tau1 tau2 _tau0')
assert linsolve(Matrix([[0, 1, 0, 0, 0, 0], [0, 0, 0, 1, 0, _x0]])
) == FiniteSet((x0, 0, x1, _x0, x2))
x0, x1, x2, _x0 = symbols('_tau0 _tau1 _tau2 tau0')
assert linsolve(Matrix([[0, 1, 0, 0, 0, 0], [0, 0, 0, 1, 0, _x0]])
) == FiniteSet((x0, 0, x1, _x0, x2))
x0, x1, x2, _x0 = symbols('_tau0 _tau1 _tau2 tau1')
assert linsolve(Matrix([[0, 1, 0, 0, 0, 0], [0, 0, 0, 1, 0, _x0]])
) == FiniteSet((x0, 0, x1, _x0, x2))
# symbols can be given as generators
x0, x2, x4 = symbols('x0, x2, x4')
assert linsolve(Augmatrix, numbered_symbols('x')
) == FiniteSet((x0, 0, x2, 0, x4))
Augmatrix[-1, -1] = x0
# use Dummy to avoid clash; the names may clash but the symbols
# will not
Augmatrix[-1, -1] = symbols('_x0')
assert len(linsolve(
Augmatrix, numbered_symbols('x', cls=Dummy)).free_symbols) == 4
# Issue #12604
f = Function('f')
assert linsolve([f(x) - 5], f(x)) == FiniteSet((5,))
# Issue #14860
from sympy.physics.units import meter, newton, kilo
Eqns = [8*kilo*newton + x + y, 28*kilo*newton*meter + 3*x*meter]
assert linsolve(Eqns, x, y) == {(newton*Rational(-28000, 3), newton*Rational(4000, 3))}
# linsolve fully expands expressions, so removable singularities
# and other nonlinearity does not raise an error
assert linsolve([Eq(x, x + y)], [x, y]) == {(x, 0)}
assert linsolve([Eq(1/x, 1/x + y)], [x, y]) == {(x, 0)}
assert linsolve([Eq(y/x, y/x + y)], [x, y]) == {(x, 0)}
assert linsolve([Eq(x*(x + 1), x**2 + y)], [x, y]) == {(y, y)}
def test_linsolve_immutable():
A = ImmutableDenseMatrix([[1, 1, 2], [0, 1, 2], [0, 0, 1]])
B = ImmutableDenseMatrix([2, 1, -1])
assert linsolve([A, B], (x, y, z)) == FiniteSet((1, 3, -1))
A = ImmutableDenseMatrix([[1, 1, 7], [1, -1, 3]])
assert linsolve(A) == FiniteSet((5, 2))
def test_solve_decomposition():
n = Dummy('n')
f1 = exp(3*x) - 6*exp(2*x) + 11*exp(x) - 6
f2 = sin(x)**2 - 2*sin(x) + 1
f3 = sin(x)**2 - sin(x)
f4 = sin(x + 1)
f5 = exp(x + 2) - 1
f6 = 1/log(x)
f7 = 1/x
s1 = ImageSet(Lambda(n, 2*n*pi), S.Integers)
s2 = ImageSet(Lambda(n, 2*n*pi + pi), S.Integers)
s3 = ImageSet(Lambda(n, 2*n*pi + pi/2), S.Integers)
s4 = ImageSet(Lambda(n, 2*n*pi - 1), S.Integers)
s5 = ImageSet(Lambda(n, 2*n*pi - 1 + pi), S.Integers)
assert solve_decomposition(f1, x, S.Reals) == FiniteSet(0, log(2), log(3))
assert dumeq(solve_decomposition(f2, x, S.Reals), s3)
assert dumeq(solve_decomposition(f3, x, S.Reals), Union(s1, s2, s3))
assert dumeq(solve_decomposition(f4, x, S.Reals), Union(s4, s5))
assert solve_decomposition(f5, x, S.Reals) == FiniteSet(-2)
assert solve_decomposition(f6, x, S.Reals) == S.EmptySet
assert solve_decomposition(f7, x, S.Reals) == S.EmptySet
assert solve_decomposition(x, x, Interval(1, 2)) == S.EmptySet
# nonlinsolve testcases
def test_nonlinsolve_basic():
assert nonlinsolve([],[]) == S.EmptySet
assert nonlinsolve([],[x, y]) == S.EmptySet
system = [x, y - x - 5]
assert nonlinsolve([x],[x, y]) == FiniteSet((0, y))
assert nonlinsolve(system, [y]) == FiniteSet((x + 5,))
soln = (ImageSet(Lambda(n, 2*n*pi + pi/2), S.Integers),)
assert dumeq(nonlinsolve([sin(x) - 1], [x]), FiniteSet(tuple(soln)))
assert nonlinsolve([x**2 - 1], [x]) == FiniteSet((-1,), (1,))
soln = FiniteSet((y, y))
assert nonlinsolve([x - y, 0], x, y) == soln
assert nonlinsolve([0, x - y], x, y) == soln
assert nonlinsolve([x - y, x - y], x, y) == soln
assert nonlinsolve([x, 0], x, y) == FiniteSet((0, y))
f = Function('f')
assert nonlinsolve([f(x), 0], f(x), y) == FiniteSet((0, y))
assert nonlinsolve([f(x), 0], f(x), f(y)) == FiniteSet((0, f(y)))
A = Indexed('A', x)
assert nonlinsolve([A, 0], A, y) == FiniteSet((0, y))
assert nonlinsolve([x**2 -1], [sin(x)]) == FiniteSet((S.EmptySet,))
assert nonlinsolve([x**2 -1], sin(x)) == FiniteSet((S.EmptySet,))
assert nonlinsolve([x**2 -1], 1) == FiniteSet((x**2,))
assert nonlinsolve([x**2 -1], x + y) == FiniteSet((S.EmptySet,))
def test_nonlinsolve_abs():
soln = FiniteSet((x, Abs(x)))
assert nonlinsolve([Abs(x) - y], x, y) == soln
def test_raise_exception_nonlinsolve():
raises(IndexError, lambda: nonlinsolve([x**2 -1], []))
raises(ValueError, lambda: nonlinsolve([x**2 -1]))
raises(NotImplementedError, lambda: nonlinsolve([(x+y)**2 - 9, x**2 - y**2 - 0.75], (x, y)))
def test_trig_system():
# TODO: add more simple testcases when solveset returns
# simplified soln for Trig eq
assert nonlinsolve([sin(x) - 1, cos(x) -1 ], x) == S.EmptySet
soln1 = (ImageSet(Lambda(n, 2*n*pi + pi/2), S.Integers),)
soln = FiniteSet(soln1)
assert dumeq(nonlinsolve([sin(x) - 1, cos(x)], x), soln)
@XFAIL
def test_trig_system_fail():
# fails because solveset trig solver is not much smart.
sys = [x + y - pi/2, sin(x) + sin(y) - 1]
# solveset returns conditionset for sin(x) + sin(y) - 1
soln_1 = (ImageSet(Lambda(n, n*pi + pi/2), S.Integers),
ImageSet(Lambda(n, n*pi)), S.Integers)
soln_1 = FiniteSet(soln_1)
soln_2 = (ImageSet(Lambda(n, n*pi), S.Integers),
ImageSet(Lambda(n, n*pi+ pi/2), S.Integers))
soln_2 = FiniteSet(soln_2)
soln = soln_1 + soln_2
assert dumeq(nonlinsolve(sys, [x, y]), soln)
# Add more cases from here
# http://www.vitutor.com/geometry/trigonometry/equations_systems.html#uno
sys = [sin(x) + sin(y) - (sqrt(3)+1)/2, sin(x) - sin(y) - (sqrt(3) - 1)/2]
soln_x = Union(ImageSet(Lambda(n, 2*n*pi + pi/3), S.Integers),
ImageSet(Lambda(n, 2*n*pi + pi*Rational(2, 3)), S.Integers))
soln_y = Union(ImageSet(Lambda(n, 2*n*pi + pi/6), S.Integers),
ImageSet(Lambda(n, 2*n*pi + pi*Rational(5, 6)), S.Integers))
assert dumeq(nonlinsolve(sys, [x, y]), FiniteSet((soln_x, soln_y)))
def test_nonlinsolve_positive_dimensional():
x, y, z, a, b, c, d = symbols('x, y, z, a, b, c, d', extended_real=True)
assert nonlinsolve([x*y, x*y - x], [x, y]) == FiniteSet((0, y))
system = [a**2 + a*c, a - b]
assert nonlinsolve(system, [a, b]) == FiniteSet((0, 0), (-c, -c))
# here (a= 0, b = 0) is independent soln so both is printed.
# if symbols = [a, b, c] then only {a : -c ,b : -c}
eq1 = a + b + c + d
eq2 = a*b + b*c + c*d + d*a
eq3 = a*b*c + b*c*d + c*d*a + d*a*b
eq4 = a*b*c*d - 1
system = [eq1, eq2, eq3, eq4]
sol1 = (-1/d, -d, 1/d, FiniteSet(d) - FiniteSet(0))
sol2 = (1/d, -d, -1/d, FiniteSet(d) - FiniteSet(0))
soln = FiniteSet(sol1, sol2)
assert nonlinsolve(system, [a, b, c, d]) == soln
def test_nonlinsolve_polysys():
x, y, z = symbols('x, y, z', real=True)
assert nonlinsolve([x**2 + y - 2, x**2 + y], [x, y]) == S.EmptySet
s = (-y + 2, y)
assert nonlinsolve([(x + y)**2 - 4, x + y - 2], [x, y]) == FiniteSet(s)
system = [x**2 - y**2]
soln_real = FiniteSet((-y, y), (y, y))
soln_complex = FiniteSet((-Abs(y), y), (Abs(y), y))
soln =soln_real + soln_complex
assert nonlinsolve(system, [x, y]) == soln
system = [x**2 - y**2]
soln_real= FiniteSet((y, -y), (y, y))
soln_complex = FiniteSet((y, -Abs(y)), (y, Abs(y)))
soln = soln_real + soln_complex
assert nonlinsolve(system, [y, x]) == soln
system = [x**2 + y - 3, x - y - 4]
assert nonlinsolve(system, (x, y)) != nonlinsolve(system, (y, x))
def test_nonlinsolve_using_substitution():
x, y, z, n = symbols('x, y, z, n', real = True)
system = [(x + y)*n - y**2 + 2]
s_x = (n*y - y**2 + 2)/n
soln = (-s_x, y)
assert nonlinsolve(system, [x, y]) == FiniteSet(soln)
system = [z**2*x**2 - z**2*y**2/exp(x)]
soln_real_1 = (y, x, 0)
soln_real_2 = (-exp(x/2)*Abs(x), x, z)
soln_real_3 = (exp(x/2)*Abs(x), x, z)
soln_complex_1 = (-x*exp(x/2), x, z)
soln_complex_2 = (x*exp(x/2), x, z)
syms = [y, x, z]
soln = FiniteSet(soln_real_1, soln_complex_1, soln_complex_2,\
soln_real_2, soln_real_3)
assert nonlinsolve(system,syms) == soln
def test_nonlinsolve_complex():
n = Dummy('n')
assert dumeq(nonlinsolve([exp(x) - sin(y), 1/y - 3], [x, y]), {
(ImageSet(Lambda(n, 2*n*I*pi + log(sin(Rational(1, 3)))), S.Integers), Rational(1, 3))})
system = [exp(x) - sin(y), 1/exp(y) - 3]
assert dumeq(nonlinsolve(system, [x, y]), {
(ImageSet(Lambda(n, I*(2*n*pi + pi)
+ log(sin(log(3)))), S.Integers), -log(3)),
(ImageSet(Lambda(n, I*(2*n*pi + arg(sin(2*n*I*pi - log(3))))
+ log(Abs(sin(2*n*I*pi - log(3))))), S.Integers),
ImageSet(Lambda(n, 2*n*I*pi - log(3)), S.Integers))})
system = [exp(x) - sin(y), y**2 - 4]
assert dumeq(nonlinsolve(system, [x, y]), {
(ImageSet(Lambda(n, I*(2*n*pi + pi) + log(sin(2))), S.Integers), -2),
(ImageSet(Lambda(n, 2*n*I*pi + log(sin(2))), S.Integers), 2)})
@XFAIL
def test_solve_nonlinear_trans():
# After the transcendental equation solver these will work
x, y, z = symbols('x, y, z', real=True)
soln1 = FiniteSet((2*LambertW(y/2), y))
soln2 = FiniteSet((-x*sqrt(exp(x)), y), (x*sqrt(exp(x)), y))
soln3 = FiniteSet((x*exp(x/2), x))
soln4 = FiniteSet(2*LambertW(y/2), y)
assert nonlinsolve([x**2 - y**2/exp(x)], [x, y]) == soln1
assert nonlinsolve([x**2 - y**2/exp(x)], [y, x]) == soln2
assert nonlinsolve([x**2 - y**2/exp(x)], [y, x]) == soln3
assert nonlinsolve([x**2 - y**2/exp(x)], [x, y]) == soln4
def test_issue_5132_1():
system = [sqrt(x**2 + y**2) - sqrt(10), x + y - 4]
assert nonlinsolve(system, [x, y]) == FiniteSet((1, 3), (3, 1))
n = Dummy('n')
eqs = [exp(x)**2 - sin(y) + z**2, 1/exp(y) - 3]
s_real_y = -log(3)
s_real_z = sqrt(-exp(2*x) - sin(log(3)))
soln_real = FiniteSet((s_real_y, s_real_z), (s_real_y, -s_real_z))
lam = Lambda(n, 2*n*I*pi + -log(3))
s_complex_y = ImageSet(lam, S.Integers)
lam = Lambda(n, sqrt(-exp(2*x) + sin(2*n*I*pi + -log(3))))
s_complex_z_1 = ImageSet(lam, S.Integers)
lam = Lambda(n, -sqrt(-exp(2*x) + sin(2*n*I*pi + -log(3))))
s_complex_z_2 = ImageSet(lam, S.Integers)
soln_complex = FiniteSet(
(s_complex_y, s_complex_z_1),
(s_complex_y, s_complex_z_2)
)
soln = soln_real + soln_complex
assert dumeq(nonlinsolve(eqs, [y, z]), soln)
def test_issue_5132_2():
x, y = symbols('x, y', real=True)
eqs = [exp(x)**2 - sin(y) + z**2, 1/exp(y) - 3]
n = Dummy('n')
soln_real = (log(-z**2 + sin(y))/2, z)
lam = Lambda( n, I*(2*n*pi + arg(-z**2 + sin(y)))/2 + log(Abs(z**2 - sin(y)))/2)
img = ImageSet(lam, S.Integers)
# not sure about the complex soln. But it looks correct.
soln_complex = (img, z)
soln = FiniteSet(soln_real, soln_complex)
assert dumeq(nonlinsolve(eqs, [x, z]), soln)
system = [r - x**2 - y**2, tan(t) - y/x]
s_x = sqrt(r/(tan(t)**2 + 1))
s_y = sqrt(r/(tan(t)**2 + 1))*tan(t)
soln = FiniteSet((s_x, s_y), (-s_x, -s_y))
assert nonlinsolve(system, [x, y]) == soln
def test_issue_6752():
a,b,c,d = symbols('a, b, c, d', real=True)
assert nonlinsolve([a**2 + a, a - b], [a, b]) == {(-1, -1), (0, 0)}
@SKIP("slow")
def test_issue_5114_solveset():
# slow testcase
from sympy.abc import d, e, f, g, h, i, j, k, l, o, p, q, r
# there is no 'a' in the equation set but this is how the
# problem was originally posed
syms = [a, b, c, f, h, k, n]
eqs = [b + r/d - c/d,
c*(1/d + 1/e + 1/g) - f/g - r/d,
f*(1/g + 1/i + 1/j) - c/g - h/i,
h*(1/i + 1/l + 1/m) - f/i - k/m,
k*(1/m + 1/o + 1/p) - h/m - n/p,
n*(1/p + 1/q) - k/p]
assert len(nonlinsolve(eqs, syms)) == 1
@SKIP("Hangs")
def _test_issue_5335():
# Not able to check zero dimensional system.
# is_zero_dimensional Hangs
lam, a0, conc = symbols('lam a0 conc')
eqs = [lam + 2*y - a0*(1 - x/2)*x - 0.005*x/2*x,
a0*(1 - x/2)*x - 1*y - 0.743436700916726*y,
x + y - conc]
sym = [x, y, a0]
# there are 4 solutions but only two are valid
assert len(nonlinsolve(eqs, sym)) == 2
# float
eqs = [lam + 2*y - a0*(1 - x/2)*x - 0.005*x/2*x,
a0*(1 - x/2)*x - 1*y - 0.743436700916726*y,
x + y - conc]
sym = [x, y, a0]
assert len(nonlinsolve(eqs, sym)) == 2
def test_issue_2777():
# the equations represent two circles
x, y = symbols('x y', real=True)
e1, e2 = sqrt(x**2 + y**2) - 10, sqrt(y**2 + (-x + 10)**2) - 3
a, b = Rational(191, 20), 3*sqrt(391)/20
ans = {(a, -b), (a, b)}
assert nonlinsolve((e1, e2), (x, y)) == ans
assert nonlinsolve((e1, e2/(x - a)), (x, y)) == S.EmptySet
# make the 2nd circle's radius be -3
e2 += 6
assert nonlinsolve((e1, e2), (x, y)) == S.EmptySet
def test_issue_8828():
x1 = 0
y1 = -620
r1 = 920
x2 = 126
y2 = 276
x3 = 51
y3 = 205
r3 = 104
v = [x, y, z]
f1 = (x - x1)**2 + (y - y1)**2 - (r1 - z)**2
f2 = (x2 - x)**2 + (y2 - y)**2 - z**2
f3 = (x - x3)**2 + (y - y3)**2 - (r3 - z)**2
F = [f1, f2, f3]
g1 = sqrt((x - x1)**2 + (y - y1)**2) + z - r1
g2 = f2
g3 = sqrt((x - x3)**2 + (y - y3)**2) + z - r3
G = [g1, g2, g3]
# both soln same
A = nonlinsolve(F, v)
B = nonlinsolve(G, v)
assert A == B
def test_nonlinsolve_conditionset():
# when solveset failed to solve all the eq
# return conditionset
f = Function('f')
f1 = f(x) - pi/2
f2 = f(y) - pi*Rational(3, 2)
intermediate_system = Eq(2*f(x) - pi, 0) & Eq(2*f(y) - 3*pi, 0)
symbols = Tuple(x, y)
soln = ConditionSet(
symbols,
intermediate_system,
S.Complexes**2)
assert nonlinsolve([f1, f2], [x, y]) == soln
def test_substitution_basic():
assert substitution([], [x, y]) == S.EmptySet
assert substitution([], []) == S.EmptySet
system = [2*x**2 + 3*y**2 - 30, 3*x**2 - 2*y**2 - 19]
soln = FiniteSet((-3, -2), (-3, 2), (3, -2), (3, 2))
assert substitution(system, [x, y]) == soln
soln = FiniteSet((-1, 1))
assert substitution([x + y], [x], [{y: 1}], [y], set([]), [x, y]) == soln
assert substitution(
[x + y], [x], [{y: 1}], [y],
set([x + 1]), [y, x]) == S.EmptySet
def test_issue_5132_substitution():
x, y, z, r, t = symbols('x, y, z, r, t', real=True)
system = [r - x**2 - y**2, tan(t) - y/x]
s_x_1 = Complement(FiniteSet(-sqrt(r/(tan(t)**2 + 1))), FiniteSet(0))
s_x_2 = Complement(FiniteSet(sqrt(r/(tan(t)**2 + 1))), FiniteSet(0))
s_y = sqrt(r/(tan(t)**2 + 1))*tan(t)
soln = FiniteSet((s_x_2, s_y)) + FiniteSet((s_x_1, -s_y))
assert substitution(system, [x, y]) == soln
n = Dummy('n')
eqs = [exp(x)**2 - sin(y) + z**2, 1/exp(y) - 3]
s_real_y = -log(3)
s_real_z = sqrt(-exp(2*x) - sin(log(3)))
soln_real = FiniteSet((s_real_y, s_real_z), (s_real_y, -s_real_z))
lam = Lambda(n, 2*n*I*pi + -log(3))
s_complex_y = ImageSet(lam, S.Integers)
lam = Lambda(n, sqrt(-exp(2*x) + sin(2*n*I*pi + -log(3))))
s_complex_z_1 = ImageSet(lam, S.Integers)
lam = Lambda(n, -sqrt(-exp(2*x) + sin(2*n*I*pi + -log(3))))
s_complex_z_2 = ImageSet(lam, S.Integers)
soln_complex = FiniteSet(
(s_complex_y, s_complex_z_1),
(s_complex_y, s_complex_z_2))
soln = soln_real + soln_complex
assert dumeq(substitution(eqs, [y, z]), soln)
def test_raises_substitution():
raises(ValueError, lambda: substitution([x**2 -1], []))
raises(TypeError, lambda: substitution([x**2 -1]))
raises(ValueError, lambda: substitution([x**2 -1], [sin(x)]))
raises(TypeError, lambda: substitution([x**2 -1], x))
raises(TypeError, lambda: substitution([x**2 -1], 1))
# end of tests for nonlinsolve
def test_issue_9556():
b = Symbol('b', positive=True)
assert solveset(Abs(x) + 1, x, S.Reals) == EmptySet()
assert solveset(Abs(x) + b, x, S.Reals) == EmptySet()
assert solveset(Eq(b, -1), b, S.Reals) == EmptySet()
def test_issue_9611():
assert solveset(Eq(x - x + a, a), x, S.Reals) == S.Reals
assert solveset(Eq(y - y + a, a), y) == S.Complexes
def test_issue_9557():
assert solveset(x**2 + a, x, S.Reals) == Intersection(S.Reals,
FiniteSet(-sqrt(-a), sqrt(-a)))
def test_issue_9778():
x = Symbol('x', real=True)
y = Symbol('y', real=True)
assert solveset(x**3 + 1, x, S.Reals) == FiniteSet(-1)
assert solveset(x**Rational(3, 5) + 1, x, S.Reals) == S.EmptySet
assert solveset(x**3 + y, x, S.Reals) == \
FiniteSet(-Abs(y)**Rational(1, 3)*sign(y))
def test_issue_10214():
assert solveset(x**Rational(3, 2) + 4, x, S.Reals) == S.EmptySet
assert solveset(x**(Rational(-3, 2)) + 4, x, S.Reals) == S.EmptySet
ans = FiniteSet(-2**Rational(2, 3))
assert solveset(x**(S(3)) + 4, x, S.Reals) == ans
assert (x**(S(3)) + 4).subs(x,list(ans)[0]) == 0 # substituting ans and verifying the result.
assert (x**(S(3)) + 4).subs(x,-(-2)**Rational(2, 3)) == 0
def test_issue_9849():
assert solveset(Abs(sin(x)) + 1, x, S.Reals) == S.EmptySet
def test_issue_9953():
assert linsolve([ ], x) == S.EmptySet
def test_issue_9913():
assert solveset(2*x + 1/(x - 10)**2, x, S.Reals) == \
FiniteSet(-(3*sqrt(24081)/4 + Rational(4027, 4))**Rational(1, 3)/3 - 100/
(3*(3*sqrt(24081)/4 + Rational(4027, 4))**Rational(1, 3)) + Rational(20, 3))
def test_issue_10397():
assert solveset(sqrt(x), x, S.Complexes) == FiniteSet(0)
def test_issue_14987():
raises(ValueError, lambda: linear_eq_to_matrix(
[x**2], x))
raises(ValueError, lambda: linear_eq_to_matrix(
[x*(-3/x + 1) + 2*y - a], [x, y]))
raises(ValueError, lambda: linear_eq_to_matrix(
[(x**2 - 3*x)/(x - 3) - 3], x))
raises(ValueError, lambda: linear_eq_to_matrix(
[(x + 1)**3 - x**3 - 3*x**2 + 7], x))
raises(ValueError, lambda: linear_eq_to_matrix(
[x*(1/x + 1) + y], [x, y]))
raises(ValueError, lambda: linear_eq_to_matrix(
[(x + 1)*y], [x, y]))
raises(ValueError, lambda: linear_eq_to_matrix(
[Eq(1/x, 1/x + y)], [x, y]))
raises(ValueError, lambda: linear_eq_to_matrix(
[Eq(y/x, y/x + y)], [x, y]))
raises(ValueError, lambda: linear_eq_to_matrix(
[Eq(x*(x + 1), x**2 + y)], [x, y]))
def test_simplification():
eq = x + (a - b)/(-2*a + 2*b)
assert solveset(eq, x) == FiniteSet(S.Half)
assert solveset(eq, x, S.Reals) == Intersection({-((a - b)/(-2*a + 2*b))}, S.Reals)
# So that ap - bn is not zero:
ap = Symbol('ap', positive=True)
bn = Symbol('bn', negative=True)
eq = x + (ap - bn)/(-2*ap + 2*bn)
assert solveset(eq, x) == FiniteSet(S.Half)
assert solveset(eq, x, S.Reals) == FiniteSet(S.Half)
def test_issue_10555():
f = Function('f')
g = Function('g')
assert solveset(f(x) - pi/2, x, S.Reals) == \
ConditionSet(x, Eq(f(x) - pi/2, 0), S.Reals)
assert solveset(f(g(x)) - pi/2, g(x), S.Reals) == \
ConditionSet(g(x), Eq(f(g(x)) - pi/2, 0), S.Reals)
def test_issue_8715():
eq = x + 1/x > -2 + 1/x
assert solveset(eq, x, S.Reals) == \
(Interval.open(-2, oo) - FiniteSet(0))
assert solveset(eq.subs(x,log(x)), x, S.Reals) == \
Interval.open(exp(-2), oo) - FiniteSet(1)
def test_issue_11174():
eq = z**2 + exp(2*x) - sin(y)
soln = Intersection(S.Reals, FiniteSet(log(-z**2 + sin(y))/2))
assert solveset(eq, x, S.Reals) == soln
eq = sqrt(r)*Abs(tan(t))/sqrt(tan(t)**2 + 1) + x*tan(t)
s = -sqrt(r)*Abs(tan(t))/(sqrt(tan(t)**2 + 1)*tan(t))
soln = Intersection(S.Reals, FiniteSet(s))
assert solveset(eq, x, S.Reals) == soln
def test_issue_11534():
# eq and eq2 should give the same solution as a Complement
x = Symbol('x', real=True)
y = Symbol('y', real=True)
eq = -y + x/sqrt(-x**2 + 1)
eq2 = -y**2 + x**2/(-x**2 + 1)
soln = Complement(FiniteSet(-y/sqrt(y**2 + 1), y/sqrt(y**2 + 1)), FiniteSet(-1, 1))
assert solveset(eq, x, S.Reals) == soln
assert solveset(eq2, x, S.Reals) == soln
def test_issue_10477():
assert solveset((x**2 + 4*x - 3)/x < 2, x, S.Reals) == \
Union(Interval.open(-oo, -3), Interval.open(0, 1))
def test_issue_10671():
assert solveset(sin(y), y, Interval(0, pi)) == FiniteSet(0, pi)
i = Interval(1, 10)
assert solveset((1/x).diff(x) < 0, x, i) == i
def test_issue_11064():
eq = x + sqrt(x**2 - 5)
assert solveset(eq > 0, x, S.Reals) == \
Interval(sqrt(5), oo)
assert solveset(eq < 0, x, S.Reals) == \
Interval(-oo, -sqrt(5))
assert solveset(eq > sqrt(5), x, S.Reals) == \
Interval.Lopen(sqrt(5), oo)
def test_issue_12478():
eq = sqrt(x - 2) + 2
soln = solveset_real(eq, x)
assert soln is S.EmptySet
assert solveset(eq < 0, x, S.Reals) is S.EmptySet
assert solveset(eq > 0, x, S.Reals) == Interval(2, oo)
def test_issue_12429():
eq = solveset(log(x)/x <= 0, x, S.Reals)
sol = Interval.Lopen(0, 1)
assert eq == sol
def test_solveset_arg():
assert solveset(arg(x), x, S.Reals) == Interval.open(0, oo)
assert solveset(arg(4*x -3), x) == Interval.open(Rational(3, 4), oo)
def test__is_finite_with_finite_vars():
f = _is_finite_with_finite_vars
# issue 12482
assert all(f(1/x) is None for x in (
Dummy(), Dummy(real=True), Dummy(complex=True)))
assert f(1/Dummy(real=False)) is True # b/c it's finite but not 0
def test_issue_13550():
assert solveset(x**2 - 2*x - 15, symbol = x, domain = Interval(-oo, 0)) == FiniteSet(-3)
def test_issue_13849():
assert nonlinsolve((t*(sqrt(5) + sqrt(2)) - sqrt(2), t), t) == EmptySet()
def test_issue_14223():
assert solveset((Abs(x + Min(x, 2)) - 2).rewrite(Piecewise), x,
S.Reals) == FiniteSet(-1, 1)
assert solveset((Abs(x + Min(x, 2)) - 2).rewrite(Piecewise), x,
Interval(0, 2)) == FiniteSet(1)
def test_issue_10158():
dom = S.Reals
assert solveset(x*Max(x, 15) - 10, x, dom) == FiniteSet(Rational(2, 3))
assert solveset(x*Min(x, 15) - 10, x, dom) == FiniteSet(-sqrt(10), sqrt(10))
assert solveset(Max(Abs(x - 3) - 1, x + 2) - 3, x, dom) == FiniteSet(-1, 1)
assert solveset(Abs(x - 1) - Abs(y), x, dom) == FiniteSet(-Abs(y) + 1, Abs(y) + 1)
assert solveset(Abs(x + 4*Abs(x + 1)), x, dom) == FiniteSet(Rational(-4, 3), Rational(-4, 5))
assert solveset(2*Abs(x + Abs(x + Max(3, x))) - 2, x, S.Reals) == FiniteSet(-1, -2)
dom = S.Complexes
raises(ValueError, lambda: solveset(x*Max(x, 15) - 10, x, dom))
raises(ValueError, lambda: solveset(x*Min(x, 15) - 10, x, dom))
raises(ValueError, lambda: solveset(Max(Abs(x - 3) - 1, x + 2) - 3, x, dom))
raises(ValueError, lambda: solveset(Abs(x - 1) - Abs(y), x, dom))
raises(ValueError, lambda: solveset(Abs(x + 4*Abs(x + 1)), x, dom))
def test_issue_14300():
f = 1 - exp(-18000000*x) - y
a1 = FiniteSet(-log(-y + 1)/18000000)
assert solveset(f, x, S.Reals) == \
Intersection(S.Reals, a1)
assert dumeq(solveset(f, x),
ImageSet(Lambda(n, -I*(2*n*pi + arg(-y + 1))/18000000 -
log(Abs(y - 1))/18000000), S.Integers))
def test_issue_14454():
number = CRootOf(x**4 + x - 1, 2)
raises(ValueError, lambda: invert_real(number, 0, x, S.Reals))
assert invert_real(x**2, number, x, S.Reals) # no error
def test_issue_17882():
assert solveset(-8*x**2/(9*(x**2 - 1)**(S(4)/3)) + 4/(3*(x**2 - 1)**(S(1)/3)), x, S.Complexes) == \
FiniteSet(sqrt(3), -sqrt(3))
def test_term_factors():
assert list(_term_factors(3**x - 2)) == [-2, 3**x]
expr = 4**(x + 1) + 4**(x + 2) + 4**(x - 1) - 3**(x + 2) - 3**(x + 3)
assert set(_term_factors(expr)) == set([
3**(x + 2), 4**(x + 2), 3**(x + 3), 4**(x - 1), -1, 4**(x + 1)])
#################### tests for transolve and its helpers ###############
def test_transolve():
assert _transolve(3**x, x, S.Reals) == S.EmptySet
assert _transolve(3**x - 9**(x + 5), x, S.Reals) == FiniteSet(-10)
# exponential tests
def test_exponential_real():
from sympy.abc import x, y, z
e1 = 3**(2*x) - 2**(x + 3)
e2 = 4**(5 - 9*x) - 8**(2 - x)
e3 = 2**x + 4**x
e4 = exp(log(5)*x) - 2**x
e5 = exp(x/y)*exp(-z/y) - 2
e6 = 5**(x/2) - 2**(x/3)
e7 = 4**(x + 1) + 4**(x + 2) + 4**(x - 1) - 3**(x + 2) - 3**(x + 3)
e8 = -9*exp(-2*x + 5) + 4*exp(3*x + 1)
e9 = 2**x + 4**x + 8**x - 84
assert solveset(e1, x, S.Reals) == FiniteSet(
-3*log(2)/(-2*log(3) + log(2)))
assert solveset(e2, x, S.Reals) == FiniteSet(Rational(4, 15))
assert solveset(e3, x, S.Reals) == S.EmptySet
assert solveset(e4, x, S.Reals) == FiniteSet(0)
assert solveset(e5, x, S.Reals) == Intersection(
S.Reals, FiniteSet(y*log(2*exp(z/y))))
assert solveset(e6, x, S.Reals) == FiniteSet(0)
assert solveset(e7, x, S.Reals) == FiniteSet(2)
assert solveset(e8, x, S.Reals) == FiniteSet(-2*log(2)/5 + 2*log(3)/5 + Rational(4, 5))
assert solveset(e9, x, S.Reals) == FiniteSet(2)
assert solveset_real(-9*exp(-2*x + 5) + 2**(x + 1), x) == FiniteSet(
-((-5 - 2*log(3) + log(2))/(log(2) + 2)))
assert solveset_real(4**(x/2) - 2**(x/3), x) == FiniteSet(0)
b = sqrt(6)*sqrt(log(2))/sqrt(log(5))
assert solveset_real(5**(x/2) - 2**(3/x), x) == FiniteSet(-b, b)
# coverage test
C1, C2 = symbols('C1 C2')
f = Function('f')
assert solveset_real(C1 + C2/x**2 - exp(-f(x)), f(x)) == Intersection(
S.Reals, FiniteSet(-log(C1 + C2/x**2)))
y = symbols('y', positive=True)
assert solveset_real(x**2 - y**2/exp(x), y) == Intersection(
S.Reals, FiniteSet(-sqrt(x**2*exp(x)), sqrt(x**2*exp(x))))
p = Symbol('p', positive=True)
assert solveset_real((1/p + 1)**(p + 1), p) == EmptySet()
@XFAIL
def test_exponential_complex():
from sympy.abc import x
from sympy import Dummy
n = Dummy('n')
assert dumeq(solveset_complex(2**x + 4**x, x),imageset(
Lambda(n, I*(2*n*pi + pi)/log(2)), S.Integers))
assert solveset_complex(x**z*y**z - 2, z) == FiniteSet(
log(2)/(log(x) + log(y)))
assert dumeq(solveset_complex(4**(x/2) - 2**(x/3), x), imageset(
Lambda(n, 3*n*I*pi/log(2)), S.Integers))
assert dumeq(solveset(2**x + 32, x), imageset(
Lambda(n, (I*(2*n*pi + pi) + 5*log(2))/log(2)), S.Integers))
eq = (2**exp(y**2/x) + 2)/(x**2 + 15)
a = sqrt(x)*sqrt(-log(log(2)) + log(log(2) + 2*n*I*pi))
assert solveset_complex(eq, y) == FiniteSet(-a, a)
union1 = imageset(Lambda(n, I*(2*n*pi - pi*Rational(2, 3))/log(2)), S.Integers)
union2 = imageset(Lambda(n, I*(2*n*pi + pi*Rational(2, 3))/log(2)), S.Integers)
assert dumeq(solveset(2**x + 4**x + 8**x, x), Union(union1, union2))
eq = 4**(x + 1) + 4**(x + 2) + 4**(x - 1) - 3**(x + 2) - 3**(x + 3)
res = solveset(eq, x)
num = 2*n*I*pi - 4*log(2) + 2*log(3)
den = -2*log(2) + log(3)
ans = imageset(Lambda(n, num/den), S.Integers)
assert dumeq(res, ans)
def test_expo_conditionset():
f1 = (exp(x) + 1)**x - 2
f2 = (x + 2)**y*x - 3
f3 = 2**x - exp(x) - 3
f4 = log(x) - exp(x)
f5 = 2**x + 3**x - 5**x
assert solveset(f1, x, S.Reals) == ConditionSet(
x, Eq((exp(x) + 1)**x - 2, 0), S.Reals)
assert solveset(f2, x, S.Reals) == ConditionSet(
x, Eq(x*(x + 2)**y - 3, 0), S.Reals)
assert solveset(f3, x, S.Reals) == ConditionSet(
x, Eq(2**x - exp(x) - 3, 0), S.Reals)
assert solveset(f4, x, S.Reals) == ConditionSet(
x, Eq(-exp(x) + log(x), 0), S.Reals)
assert solveset(f5, x, S.Reals) == ConditionSet(
x, Eq(2**x + 3**x - 5**x, 0), S.Reals)
def test_exponential_symbols():
x, y, z = symbols('x y z', positive=True)
assert solveset(z**x - y, x, S.Reals) == Intersection(
S.Reals, FiniteSet(log(y)/log(z)))
f1 = 2*x**w - 4*y**w
f2 = (x/y)**w - 2
sol1 = Intersection({log(2)/(log(x) - log(y))}, S.Reals)
sol2 = Intersection({log(2)/log(x/y)}, S.Reals)
assert solveset(f1, w, S.Reals) == sol1
assert solveset(f2, w, S.Reals) == sol2
assert solveset(x**x, x, S.Reals) == S.EmptySet
assert solveset(x**y - 1, y, S.Reals) == FiniteSet(0)
assert solveset(exp(x/y)*exp(-z/y) - 2, y, S.Reals) == FiniteSet(
(x - z)/log(2)) - FiniteSet(0)
assert solveset_real(a**x - b**x, x) == ConditionSet(
x, (a > 0) & (b > 0), FiniteSet(0))
assert solveset(a**x - b**x, x) == ConditionSet(
x, Ne(a, 0) & Ne(b, 0), FiniteSet(0))
@XFAIL
def test_issue_10864():
assert solveset(x**(y*z) - x, x, S.Reals) == FiniteSet(1)
@XFAIL
def test_solve_only_exp_2():
assert solveset_real(sqrt(exp(x)) + sqrt(exp(-x)) - 4, x) == \
FiniteSet(2*log(-sqrt(3) + 2), 2*log(sqrt(3) + 2))
def test_is_exponential():
assert _is_exponential(y, x) is False
assert _is_exponential(3**x - 2, x) is True
assert _is_exponential(5**x - 7**(2 - x), x) is True
assert _is_exponential(sin(2**x) - 4*x, x) is False
assert _is_exponential(x**y - z, y) is True
assert _is_exponential(x**y - z, x) is False
assert _is_exponential(2**x + 4**x - 1, x) is True
assert _is_exponential(x**(y*z) - x, x) is False
assert _is_exponential(x**(2*x) - 3**x, x) is False
assert _is_exponential(x**y - y*z, y) is False
assert _is_exponential(x**y - x*z, y) is True
def test_solve_exponential():
assert _solve_exponential(3**(2*x) - 2**(x + 3), 0, x, S.Reals) == \
FiniteSet(-3*log(2)/(-2*log(3) + log(2)))
assert _solve_exponential(2**y + 4**y, 1, y, S.Reals) == \
FiniteSet(log(Rational(-1, 2) + sqrt(5)/2)/log(2))
assert _solve_exponential(2**y + 4**y, 0, y, S.Reals) == \
S.EmptySet
assert _solve_exponential(2**x + 3**x - 5**x, 0, x, S.Reals) == \
ConditionSet(x, Eq(2**x + 3**x - 5**x, 0), S.Reals)
# end of exponential tests
# logarithmic tests
def test_logarithmic():
assert solveset_real(log(x - 3) + log(x + 3), x) == FiniteSet(
-sqrt(10), sqrt(10))
assert solveset_real(log(x + 1) - log(2*x - 1), x) == FiniteSet(2)
assert solveset_real(log(x + 3) + log(1 + 3/x) - 3, x) == FiniteSet(
-3 + sqrt(-12 + exp(3))*exp(Rational(3, 2))/2 + exp(3)/2,
-sqrt(-12 + exp(3))*exp(Rational(3, 2))/2 - 3 + exp(3)/2)
eq = z - log(x) + log(y/(x*(-1 + y**2/x**2)))
assert solveset_real(eq, x) == \
Intersection(S.Reals, FiniteSet(-sqrt(y**2 - y*exp(z)),
sqrt(y**2 - y*exp(z)))) - \
Intersection(S.Reals, FiniteSet(-sqrt(y**2), sqrt(y**2)))
assert solveset_real(
log(3*x) - log(-x + 1) - log(4*x + 1), x) == FiniteSet(Rational(-1, 2), S.Half)
assert solveset(log(x**y) - y*log(x), x, S.Reals) == S.Reals
@XFAIL
def test_uselogcombine_2():
eq = log(exp(2*x) + 1) + log(-tanh(x) + 1) - log(2)
assert solveset_real(eq, x) == EmptySet()
eq = log(8*x) - log(sqrt(x) + 1) - 2
assert solveset_real(eq, x) == EmptySet()
def test_is_logarithmic():
assert _is_logarithmic(y, x) is False
assert _is_logarithmic(log(x), x) is True
assert _is_logarithmic(log(x) - 3, x) is True
assert _is_logarithmic(log(x)*log(y), x) is True
assert _is_logarithmic(log(x)**2, x) is False
assert _is_logarithmic(log(x - 3) + log(x + 3), x) is True
assert _is_logarithmic(log(x**y) - y*log(x), x) is True
assert _is_logarithmic(sin(log(x)), x) is False
assert _is_logarithmic(x + y, x) is False
assert _is_logarithmic(log(3*x) - log(1 - x) + 4, x) is True
assert _is_logarithmic(log(x) + log(y) + x, x) is False
assert _is_logarithmic(log(log(x - 3)) + log(x - 3), x) is True
assert _is_logarithmic(log(log(3) + x) + log(x), x) is True
assert _is_logarithmic(log(x)*(y + 3) + log(x), y) is False
def test_solve_logarithm():
y = Symbol('y')
assert _solve_logarithm(log(x**y) - y*log(x), 0, x, S.Reals) == S.Reals
y = Symbol('y', positive=True)
assert _solve_logarithm(log(x)*log(y), 0, x, S.Reals) == FiniteSet(1)
# end of logarithmic tests
def test_linear_coeffs():
from sympy.solvers.solveset import linear_coeffs
assert linear_coeffs(0, x) == [0, 0]
assert all(i is S.Zero for i in linear_coeffs(0, x))
assert linear_coeffs(x + 2*y + 3, x, y) == [1, 2, 3]
assert linear_coeffs(x + 2*y + 3, y, x) == [2, 1, 3]
assert linear_coeffs(x + 2*x**2 + 3, x, x**2) == [1, 2, 3]
raises(ValueError, lambda:
linear_coeffs(x + 2*x**2 + x**3, x, x**2))
raises(ValueError, lambda:
linear_coeffs(1/x*(x - 1) + 1/x, x))
assert linear_coeffs(a*(x + y), x, y) == [a, a, 0]
assert linear_coeffs(1.0, x, y) == [0, 0, 1.0]
# modular tests
def test_is_modular():
assert _is_modular(y, x) is False
assert _is_modular(Mod(x, 3) - 1, x) is True
assert _is_modular(Mod(x**3 - 3*x**2 - x + 1, 3) - 1, x) is True
assert _is_modular(Mod(exp(x + y), 3) - 2, x) is True
assert _is_modular(Mod(exp(x + y), 3) - log(x), x) is True
assert _is_modular(Mod(x, 3) - 1, y) is False
assert _is_modular(Mod(x, 3)**2 - 5, x) is False
assert _is_modular(Mod(x, 3)**2 - y, x) is False
assert _is_modular(exp(Mod(x, 3)) - 1, x) is False
assert _is_modular(Mod(3, y) - 1, y) is False
def test_invert_modular():
n = Dummy('n', integer=True)
from sympy.solvers.solveset import _invert_modular as invert_modular
# non invertible cases
assert invert_modular(Mod(sin(x), 7), S(5), n, x) == (Mod(sin(x), 7), 5)
assert invert_modular(Mod(exp(x), 7), S(5), n, x) == (Mod(exp(x), 7), 5)
assert invert_modular(Mod(log(x), 7), S(5), n, x) == (Mod(log(x), 7), 5)
# a is symbol
assert dumeq(invert_modular(Mod(x, 7), S(5), n, x),
(x, ImageSet(Lambda(n, 7*n + 5), S.Integers)))
# a.is_Add
assert dumeq(invert_modular(Mod(x + 8, 7), S(5), n, x),
(x, ImageSet(Lambda(n, 7*n + 4), S.Integers)))
assert invert_modular(Mod(x**2 + x, 7), S(5), n, x) == \
(Mod(x**2 + x, 7), 5)
# a.is_Mul
assert dumeq(invert_modular(Mod(3*x, 7), S(5), n, x),
(x, ImageSet(Lambda(n, 7*n + 4), S.Integers)))
assert invert_modular(Mod((x + 1)*(x + 2), 7), S(5), n, x) == \
(Mod((x + 1)*(x + 2), 7), 5)
# a.is_Pow
assert invert_modular(Mod(x**4, 7), S(5), n, x) == \
(x, EmptySet())
assert dumeq(invert_modular(Mod(3**x, 4), S(3), n, x),
(x, ImageSet(Lambda(n, 2*n + 1), S.Naturals0)))
assert dumeq(invert_modular(Mod(2**(x**2 + x + 1), 7), S(2), n, x),
(x**2 + x + 1, ImageSet(Lambda(n, 3*n + 1), S.Naturals0)))
assert invert_modular(Mod(sin(x)**4, 7), S(5), n, x) == (x, EmptySet())
def test_solve_modular():
n = Dummy('n', integer=True)
# if rhs has symbol (need to be implemented in future).
assert solveset(Mod(x, 4) - x, x, S.Integers) == \
ConditionSet(x, Eq(-x + Mod(x, 4), 0), \
S.Integers)
# when _invert_modular fails to invert
assert solveset(3 - Mod(sin(x), 7), x, S.Integers) == \
ConditionSet(x, Eq(Mod(sin(x), 7) - 3, 0), S.Integers)
assert solveset(3 - Mod(log(x), 7), x, S.Integers) == \
ConditionSet(x, Eq(Mod(log(x), 7) - 3, 0), S.Integers)
assert solveset(3 - Mod(exp(x), 7), x, S.Integers) == \
ConditionSet(x, Eq(Mod(exp(x), 7) - 3, 0), S.Integers)
# EmptySet solution definitely
assert solveset(7 - Mod(x, 5), x, S.Integers) == EmptySet()
assert solveset(5 - Mod(x, 5), x, S.Integers) == EmptySet()
# Negative m
assert dumeq(solveset(2 + Mod(x, -3), x, S.Integers),
ImageSet(Lambda(n, -3*n - 2), S.Integers))
assert solveset(4 + Mod(x, -3), x, S.Integers) == EmptySet()
# linear expression in Mod
assert dumeq(solveset(3 - Mod(x, 5), x, S.Integers),
ImageSet(Lambda(n, 5*n + 3), S.Integers))
assert dumeq(solveset(3 - Mod(5*x - 8, 7), x, S.Integers),
ImageSet(Lambda(n, 7*n + 5), S.Integers))
assert dumeq(solveset(3 - Mod(5*x, 7), x, S.Integers),
ImageSet(Lambda(n, 7*n + 2), S.Integers))
# higher degree expression in Mod
assert dumeq(solveset(Mod(x**2, 160) - 9, x, S.Integers),
Union(ImageSet(Lambda(n, 160*n + 3), S.Integers),
ImageSet(Lambda(n, 160*n + 13), S.Integers),
ImageSet(Lambda(n, 160*n + 67), S.Integers),
ImageSet(Lambda(n, 160*n + 77), S.Integers),
ImageSet(Lambda(n, 160*n + 83), S.Integers),
ImageSet(Lambda(n, 160*n + 93), S.Integers),
ImageSet(Lambda(n, 160*n + 147), S.Integers),
ImageSet(Lambda(n, 160*n + 157), S.Integers)))
assert solveset(3 - Mod(x**4, 7), x, S.Integers) == EmptySet()
assert dumeq(solveset(Mod(x**4, 17) - 13, x, S.Integers),
Union(ImageSet(Lambda(n, 17*n + 3), S.Integers),
ImageSet(Lambda(n, 17*n + 5), S.Integers),
ImageSet(Lambda(n, 17*n + 12), S.Integers),
ImageSet(Lambda(n, 17*n + 14), S.Integers)))
# a.is_Pow tests
assert dumeq(solveset(Mod(7**x, 41) - 15, x, S.Integers),
ImageSet(Lambda(n, 40*n + 3), S.Naturals0))
assert dumeq(solveset(Mod(12**x, 21) - 18, x, S.Integers),
ImageSet(Lambda(n, 6*n + 2), S.Naturals0))
assert dumeq(solveset(Mod(3**x, 4) - 3, x, S.Integers),
ImageSet(Lambda(n, 2*n + 1), S.Naturals0))
assert dumeq(solveset(Mod(2**x, 7) - 2 , x, S.Integers),
ImageSet(Lambda(n, 3*n + 1), S.Naturals0))
assert dumeq(solveset(Mod(3**(3**x), 4) - 3, x, S.Integers),
Intersection(ImageSet(Lambda(n, Intersection({log(2*n + 1)/log(3)},
S.Integers)), S.Naturals0), S.Integers))
# Implemented for m without primitive root
assert solveset(Mod(x**3, 7) - 2, x, S.Integers) == EmptySet()
assert dumeq(solveset(Mod(x**3, 8) - 1, x, S.Integers),
ImageSet(Lambda(n, 8*n + 1), S.Integers))
assert dumeq(solveset(Mod(x**4, 9) - 4, x, S.Integers),
Union(ImageSet(Lambda(n, 9*n + 4), S.Integers),
ImageSet(Lambda(n, 9*n + 5), S.Integers)))
# domain intersection
assert dumeq(solveset(3 - Mod(5*x - 8, 7), x, S.Naturals0),
Intersection(ImageSet(Lambda(n, 7*n + 5), S.Integers), S.Naturals0))
# Complex args
assert solveset(Mod(x, 3) - I, x, S.Integers) == \
EmptySet()
assert solveset(Mod(I*x, 3) - 2, x, S.Integers) == \
ConditionSet(x, Eq(Mod(I*x, 3) - 2, 0), S.Integers)
assert solveset(Mod(I + x, 3) - 2, x, S.Integers) == \
ConditionSet(x, Eq(Mod(x + I, 3) - 2, 0), S.Integers)
# issue 17373 (https://github.com/sympy/sympy/issues/17373)
assert dumeq(solveset(Mod(x**4, 14) - 11, x, S.Integers),
Union(ImageSet(Lambda(n, 14*n + 3), S.Integers),
ImageSet(Lambda(n, 14*n + 11), S.Integers)))
assert dumeq(solveset(Mod(x**31, 74) - 43, x, S.Integers),
ImageSet(Lambda(n, 74*n + 31), S.Integers))
# issue 13178
n = symbols('n', integer=True)
a = 742938285
b = 1898888478
m = 2**31 - 1
c = 20170816
assert dumeq(solveset(c - Mod(a**n*b, m), n, S.Integers),
ImageSet(Lambda(n, 2147483646*n + 100), S.Naturals0))
assert dumeq(solveset(c - Mod(a**n*b, m), n, S.Naturals0),
Intersection(ImageSet(Lambda(n, 2147483646*n + 100), S.Naturals0),
S.Naturals0))
assert dumeq(solveset(c - Mod(a**(2*n)*b, m), n, S.Integers),
Intersection(ImageSet(Lambda(n, 1073741823*n + 50), S.Naturals0),
S.Integers))
assert solveset(c - Mod(a**(2*n + 7)*b, m), n, S.Integers) == EmptySet()
assert dumeq(solveset(c - Mod(a**(n - 4)*b, m), n, S.Integers),
Intersection(ImageSet(Lambda(n, 2147483646*n + 104), S.Naturals0),
S.Integers))
# end of modular tests
def test_issue_17276():
assert nonlinsolve([Eq(x, 5**(S(1)/5)), Eq(x*y, 25*sqrt(5))], x, y) == \
FiniteSet((5**(S(1)/5), 25*5**(S(3)/10)))
|
03621a1b8862fb2f4c864ce3dd80e2682bdfc248be5bf1d698c7af054d840c7a | from sympy import (
Abs, And, Derivative, Dummy, Eq, Float, Function, Gt, I, Integral,
LambertW, Lt, Matrix, Or, Poly, Q, Rational, S, Symbol, Ne,
Wild, acos, asin, atan, atanh, binomial, cos, cosh, diff, erf, erfinv, erfc,
erfcinv, exp, im, log, pi, re, sec, sin,
sinh, solve, solve_linear, sqrt, sstr, symbols, sympify, tan, tanh,
root, atan2, arg, Mul, SparseMatrix, ask, Tuple, nsolve, oo,
E, cbrt, denom, Add, Piecewise, GoldenRatio, TribonacciConstant)
from sympy.core.function import nfloat
from sympy.solvers import solve_linear_system, solve_linear_system_LU, \
solve_undetermined_coeffs
from sympy.solvers.bivariate import _filtered_gens, _solve_lambert, _lambert
from sympy.solvers.solvers import _invert, unrad, checksol, posify, _ispow, \
det_quick, det_perm, det_minor, _simple_dens, denoms
from sympy.physics.units import cm
from sympy.polys.rootoftools import CRootOf
from sympy.testing.pytest import slow, XFAIL, SKIP, raises
from sympy.testing.randtest import verify_numerically as tn
from sympy.abc import a, b, c, d, k, h, p, x, y, z, t, q, m
def NS(e, n=15, **options):
return sstr(sympify(e).evalf(n, **options), full_prec=True)
def test_swap_back():
f, g = map(Function, 'fg')
fx, gx = f(x), g(x)
assert solve([fx + y - 2, fx - gx - 5], fx, y, gx) == \
{fx: gx + 5, y: -gx - 3}
assert solve(fx + gx*x - 2, [fx, gx], dict=True)[0] == {fx: 2, gx: 0}
assert solve(fx + gx**2*x - y, [fx, gx], dict=True) == [{fx: y - gx**2*x}]
assert solve([f(1) - 2, x + 2], dict=True) == [{x: -2, f(1): 2}]
def guess_solve_strategy(eq, symbol):
try:
solve(eq, symbol)
return True
except (TypeError, NotImplementedError):
return False
def test_guess_poly():
# polynomial equations
assert guess_solve_strategy( S(4), x ) # == GS_POLY
assert guess_solve_strategy( x, x ) # == GS_POLY
assert guess_solve_strategy( x + a, x ) # == GS_POLY
assert guess_solve_strategy( 2*x, x ) # == GS_POLY
assert guess_solve_strategy( x + sqrt(2), x) # == GS_POLY
assert guess_solve_strategy( x + 2**Rational(1, 4), x) # == GS_POLY
assert guess_solve_strategy( x**2 + 1, x ) # == GS_POLY
assert guess_solve_strategy( x**2 - 1, x ) # == GS_POLY
assert guess_solve_strategy( x*y + y, x ) # == GS_POLY
assert guess_solve_strategy( x*exp(y) + y, x) # == GS_POLY
assert guess_solve_strategy(
(x - y**3)/(y**2*sqrt(1 - y**2)), x) # == GS_POLY
def test_guess_poly_cv():
# polynomial equations via a change of variable
assert guess_solve_strategy( sqrt(x) + 1, x ) # == GS_POLY_CV_1
assert guess_solve_strategy(
x**Rational(1, 3) + sqrt(x) + 1, x ) # == GS_POLY_CV_1
assert guess_solve_strategy( 4*x*(1 - sqrt(x)), x ) # == GS_POLY_CV_1
# polynomial equation multiplying both sides by x**n
assert guess_solve_strategy( x + 1/x + y, x ) # == GS_POLY_CV_2
def test_guess_rational_cv():
# rational functions
assert guess_solve_strategy( (x + 1)/(x**2 + 2), x) # == GS_RATIONAL
assert guess_solve_strategy(
(x - y**3)/(y**2*sqrt(1 - y**2)), y) # == GS_RATIONAL_CV_1
# rational functions via the change of variable y -> x**n
assert guess_solve_strategy( (sqrt(x) + 1)/(x**Rational(1, 3) + sqrt(x) + 1), x ) \
#== GS_RATIONAL_CV_1
def test_guess_transcendental():
#transcendental functions
assert guess_solve_strategy( exp(x) + 1, x ) # == GS_TRANSCENDENTAL
assert guess_solve_strategy( 2*cos(x) - y, x ) # == GS_TRANSCENDENTAL
assert guess_solve_strategy(
exp(x) + exp(-x) - y, x ) # == GS_TRANSCENDENTAL
assert guess_solve_strategy(3**x - 10, x) # == GS_TRANSCENDENTAL
assert guess_solve_strategy(-3**x + 10, x) # == GS_TRANSCENDENTAL
assert guess_solve_strategy(a*x**b - y, x) # == GS_TRANSCENDENTAL
def test_solve_args():
# equation container, issue 5113
ans = {x: -3, y: 1}
eqs = (x + 5*y - 2, -3*x + 6*y - 15)
assert all(solve(container(eqs), x, y) == ans for container in
(tuple, list, set, frozenset))
assert solve(Tuple(*eqs), x, y) == ans
# implicit symbol to solve for
assert set(solve(x**2 - 4)) == set([S(2), -S(2)])
assert solve([x + y - 3, x - y - 5]) == {x: 4, y: -1}
assert solve(x - exp(x), x, implicit=True) == [exp(x)]
# no symbol to solve for
assert solve(42) == solve(42, x) == []
assert solve([1, 2]) == []
# duplicate symbols removed
assert solve((x - 3, y + 2), x, y, x) == {x: 3, y: -2}
# unordered symbols
# only 1
assert solve(y - 3, set([y])) == [3]
# more than 1
assert solve(y - 3, set([x, y])) == [{y: 3}]
# multiple symbols: take the first linear solution+
# - return as tuple with values for all requested symbols
assert solve(x + y - 3, [x, y]) == [(3 - y, y)]
# - unless dict is True
assert solve(x + y - 3, [x, y], dict=True) == [{x: 3 - y}]
# - or no symbols are given
assert solve(x + y - 3) == [{x: 3 - y}]
# multiple symbols might represent an undetermined coefficients system
assert solve(a + b*x - 2, [a, b]) == {a: 2, b: 0}
args = (a + b)*x - b**2 + 2, a, b
assert solve(*args) == \
[(-sqrt(2), sqrt(2)), (sqrt(2), -sqrt(2))]
assert solve(*args, set=True) == \
([a, b], set([(-sqrt(2), sqrt(2)), (sqrt(2), -sqrt(2))]))
assert solve(*args, dict=True) == \
[{b: sqrt(2), a: -sqrt(2)}, {b: -sqrt(2), a: sqrt(2)}]
eq = a*x**2 + b*x + c - ((x - h)**2 + 4*p*k)/4/p
flags = dict(dict=True)
assert solve(eq, [h, p, k], exclude=[a, b, c], **flags) == \
[{k: c - b**2/(4*a), h: -b/(2*a), p: 1/(4*a)}]
flags.update(dict(simplify=False))
assert solve(eq, [h, p, k], exclude=[a, b, c], **flags) == \
[{k: (4*a*c - b**2)/(4*a), h: -b/(2*a), p: 1/(4*a)}]
# failing undetermined system
assert solve(a*x + b**2/(x + 4) - 3*x - 4/x, a, b, dict=True) == \
[{a: (-b**2*x + 3*x**3 + 12*x**2 + 4*x + 16)/(x**2*(x + 4))}]
# failed single equation
assert solve(1/(1/x - y + exp(y))) == []
raises(
NotImplementedError, lambda: solve(exp(x) + sin(x) + exp(y) + sin(y)))
# failed system
# -- when no symbols given, 1 fails
assert solve([y, exp(x) + x]) == [{x: -LambertW(1), y: 0}]
# both fail
assert solve(
(exp(x) - x, exp(y) - y)) == [{x: -LambertW(-1), y: -LambertW(-1)}]
# -- when symbols given
solve([y, exp(x) + x], x, y) == [(-LambertW(1), 0)]
# symbol is a number
assert solve(x**2 - pi, pi) == [x**2]
# no equations
assert solve([], [x]) == []
# overdetermined system
# - nonlinear
assert solve([(x + y)**2 - 4, x + y - 2]) == [{x: -y + 2}]
# - linear
assert solve((x + y - 2, 2*x + 2*y - 4)) == {x: -y + 2}
# When one or more args are Boolean
assert solve(Eq(x**2, 0.0)) == [0] # issue 19048
assert solve([True, Eq(x, 0)], [x], dict=True) == [{x: 0}]
assert solve([Eq(x, x), Eq(x, 0), Eq(x, x+1)], [x], dict=True) == []
assert not solve([Eq(x, x+1), x < 2], x)
assert solve([Eq(x, 0), x+1<2]) == Eq(x, 0)
assert solve([Eq(x, x), Eq(x, x+1)], x) == []
assert solve(True, x) == []
assert solve([x - 1, False], [x], set=True) == ([], set())
def test_solve_polynomial1():
assert solve(3*x - 2, x) == [Rational(2, 3)]
assert solve(Eq(3*x, 2), x) == [Rational(2, 3)]
assert set(solve(x**2 - 1, x)) == set([-S.One, S.One])
assert set(solve(Eq(x**2, 1), x)) == set([-S.One, S.One])
assert solve(x - y**3, x) == [y**3]
rx = root(x, 3)
assert solve(x - y**3, y) == [
rx, -rx/2 - sqrt(3)*I*rx/2, -rx/2 + sqrt(3)*I*rx/2]
a11, a12, a21, a22, b1, b2 = symbols('a11,a12,a21,a22,b1,b2')
assert solve([a11*x + a12*y - b1, a21*x + a22*y - b2], x, y) == \
{
x: (a22*b1 - a12*b2)/(a11*a22 - a12*a21),
y: (a11*b2 - a21*b1)/(a11*a22 - a12*a21),
}
solution = {y: S.Zero, x: S.Zero}
assert solve((x - y, x + y), x, y ) == solution
assert solve((x - y, x + y), (x, y)) == solution
assert solve((x - y, x + y), [x, y]) == solution
assert set(solve(x**3 - 15*x - 4, x)) == set([
-2 + 3**S.Half,
S(4),
-2 - 3**S.Half
])
assert set(solve((x**2 - 1)**2 - a, x)) == \
set([sqrt(1 + sqrt(a)), -sqrt(1 + sqrt(a)),
sqrt(1 - sqrt(a)), -sqrt(1 - sqrt(a))])
def test_solve_polynomial2():
assert solve(4, x) == []
def test_solve_polynomial_cv_1a():
"""
Test for solving on equations that can be converted to a polynomial equation
using the change of variable y -> x**Rational(p, q)
"""
assert solve( sqrt(x) - 1, x) == [1]
assert solve( sqrt(x) - 2, x) == [4]
assert solve( x**Rational(1, 4) - 2, x) == [16]
assert solve( x**Rational(1, 3) - 3, x) == [27]
assert solve(sqrt(x) + x**Rational(1, 3) + x**Rational(1, 4), x) == [0]
def test_solve_polynomial_cv_1b():
assert set(solve(4*x*(1 - a*sqrt(x)), x)) == set([S.Zero, 1/a**2])
assert set(solve(x*(root(x, 3) - 3), x)) == set([S.Zero, S(27)])
def test_solve_polynomial_cv_2():
"""
Test for solving on equations that can be converted to a polynomial equation
multiplying both sides of the equation by x**m
"""
assert solve(x + 1/x - 1, x) in \
[[ S.Half + I*sqrt(3)/2, S.Half - I*sqrt(3)/2],
[ S.Half - I*sqrt(3)/2, S.Half + I*sqrt(3)/2]]
def test_quintics_1():
f = x**5 - 110*x**3 - 55*x**2 + 2310*x + 979
s = solve(f, check=False)
for r in s:
res = f.subs(x, r.n()).n()
assert tn(res, 0)
f = x**5 - 15*x**3 - 5*x**2 + 10*x + 20
s = solve(f)
for r in s:
assert r.func == CRootOf
# if one uses solve to get the roots of a polynomial that has a CRootOf
# solution, make sure that the use of nfloat during the solve process
# doesn't fail. Note: if you want numerical solutions to a polynomial
# it is *much* faster to use nroots to get them than to solve the
# equation only to get RootOf solutions which are then numerically
# evaluated. So for eq = x**5 + 3*x + 7 do Poly(eq).nroots() rather
# than [i.n() for i in solve(eq)] to get the numerical roots of eq.
assert nfloat(solve(x**5 + 3*x**3 + 7)[0], exponent=False) == \
CRootOf(x**5 + 3*x**3 + 7, 0).n()
def test_quintics_2():
f = x**5 + 15*x + 12
s = solve(f, check=False)
for r in s:
res = f.subs(x, r.n()).n()
assert tn(res, 0)
f = x**5 - 15*x**3 - 5*x**2 + 10*x + 20
s = solve(f)
for r in s:
assert r.func == CRootOf
assert solve(x**5 - 6*x**3 - 6*x**2 + x - 6) == [
CRootOf(x**5 - 6*x**3 - 6*x**2 + x - 6, 0),
CRootOf(x**5 - 6*x**3 - 6*x**2 + x - 6, 1),
CRootOf(x**5 - 6*x**3 - 6*x**2 + x - 6, 2),
CRootOf(x**5 - 6*x**3 - 6*x**2 + x - 6, 3),
CRootOf(x**5 - 6*x**3 - 6*x**2 + x - 6, 4)]
def test_highorder_poly():
# just testing that the uniq generator is unpacked
sol = solve(x**6 - 2*x + 2)
assert all(isinstance(i, CRootOf) for i in sol) and len(sol) == 6
def test_solve_rational():
"""Test solve for rational functions"""
assert solve( ( x - y**3 )/( (y**2)*sqrt(1 - y**2) ), x) == [y**3]
def test_solve_nonlinear():
assert solve(x**2 - y**2, x, y, dict=True) == [{x: -y}, {x: y}]
assert solve(x**2 - y**2/exp(x), y, x, dict=True) == [{y: -x*sqrt(exp(x))},
{y: x*sqrt(exp(x))}]
def test_issue_8666():
x = symbols('x')
assert solve(Eq(x**2 - 1/(x**2 - 4), 4 - 1/(x**2 - 4)), x) == []
assert solve(Eq(x + 1/x, 1/x), x) == []
def test_issue_7228():
assert solve(4**(2*(x**2) + 2*x) - 8, x) == [Rational(-3, 2), S.Half]
def test_issue_7190():
assert solve(log(x-3) + log(x+3), x) == [sqrt(10)]
def test_linear_system():
x, y, z, t, n = symbols('x, y, z, t, n')
assert solve([x - 1, x - y, x - 2*y, y - 1], [x, y]) == []
assert solve([x - 1, x - y, x - 2*y, x - 1], [x, y]) == []
assert solve([x - 1, x - 1, x - y, x - 2*y], [x, y]) == []
assert solve([x + 5*y - 2, -3*x + 6*y - 15], x, y) == {x: -3, y: 1}
M = Matrix([[0, 0, n*(n + 1), (n + 1)**2, 0],
[n + 1, n + 1, -2*n - 1, -(n + 1), 0],
[-1, 0, 1, 0, 0]])
assert solve_linear_system(M, x, y, z, t) == \
{x: -t - t/n, z: -t - t/n, y: 0}
assert solve([x + y + z + t, -z - t], x, y, z, t) == {x: -y, z: -t}
# https://github.com/sympy/sympy/issues/6420
M = Matrix([[0, 15.0, 10.0, 700.0],
[1, 1, 1, 100.0],
[0, 10.0, 5.0, 200.0],
[-5.0, 0, 0, 0 ]])
assert solve_linear_system(M, x, y, z) == {x: 0, y: -60.0, z: 160.0}
def test_linear_system_function():
a = Function('a')
assert solve([a(0, 0) + a(0, 1) + a(1, 0) + a(1, 1), -a(1, 0) - a(1, 1)],
a(0, 0), a(0, 1), a(1, 0), a(1, 1)) == {a(1, 0): -a(1, 1), a(0, 0): -a(0, 1)}
def test_linear_systemLU():
n = Symbol('n')
M = Matrix([[1, 2, 0, 1], [1, 3, 2*n, 1], [4, -1, n**2, 1]])
assert solve_linear_system_LU(M, [x, y, z]) == {z: -3/(n**2 + 18*n),
x: 1 - 12*n/(n**2 + 18*n),
y: 6*n/(n**2 + 18*n)}
# Note: multiple solutions exist for some of these equations, so the tests
# should be expected to break if the implementation of the solver changes
# in such a way that a different branch is chosen
@slow
def test_solve_transcendental():
from sympy.abc import a, b
assert solve(exp(x) - 3, x) == [log(3)]
assert set(solve((a*x + b)*(exp(x) - 3), x)) == set([-b/a, log(3)])
assert solve(cos(x) - y, x) == [-acos(y) + 2*pi, acos(y)]
assert solve(2*cos(x) - y, x) == [-acos(y/2) + 2*pi, acos(y/2)]
assert solve(Eq(cos(x), sin(x)), x) == [pi/4]
assert set(solve(exp(x) + exp(-x) - y, x)) in [set([
log(y/2 - sqrt(y**2 - 4)/2),
log(y/2 + sqrt(y**2 - 4)/2),
]), set([
log(y - sqrt(y**2 - 4)) - log(2),
log(y + sqrt(y**2 - 4)) - log(2)]),
set([
log(y/2 - sqrt((y - 2)*(y + 2))/2),
log(y/2 + sqrt((y - 2)*(y + 2))/2)])]
assert solve(exp(x) - 3, x) == [log(3)]
assert solve(Eq(exp(x), 3), x) == [log(3)]
assert solve(log(x) - 3, x) == [exp(3)]
assert solve(sqrt(3*x) - 4, x) == [Rational(16, 3)]
assert solve(3**(x + 2), x) == []
assert solve(3**(2 - x), x) == []
assert solve(x + 2**x, x) == [-LambertW(log(2))/log(2)]
assert solve(2*x + 5 + log(3*x - 2), x) == \
[Rational(2, 3) + LambertW(2*exp(Rational(-19, 3))/3)/2]
assert solve(3*x + log(4*x), x) == [LambertW(Rational(3, 4))/3]
assert set(solve((2*x + 8)*(8 + exp(x)), x)) == set([S(-4), log(8) + pi*I])
eq = 2*exp(3*x + 4) - 3
ans = solve(eq, x) # this generated a failure in flatten
assert len(ans) == 3 and all(eq.subs(x, a).n(chop=True) == 0 for a in ans)
assert solve(2*log(3*x + 4) - 3, x) == [(exp(Rational(3, 2)) - 4)/3]
assert solve(exp(x) + 1, x) == [pi*I]
eq = 2*(3*x + 4)**5 - 6*7**(3*x + 9)
result = solve(eq, x)
ans = [(log(2401) + 5*LambertW((-1 + sqrt(5) + sqrt(2)*I*sqrt(sqrt(5) + \
5))*log(7**(7*3**Rational(1, 5)/20))* -1))/(-3*log(7)), \
(log(2401) + 5*LambertW((1 + sqrt(5) - sqrt(2)*I*sqrt(5 - \
sqrt(5)))*log(7**(7*3**Rational(1, 5)/20))))/(-3*log(7)), \
(log(2401) + 5*LambertW((1 + sqrt(5) + sqrt(2)*I*sqrt(5 - \
sqrt(5)))*log(7**(7*3**Rational(1, 5)/20))))/(-3*log(7)), \
(log(2401) + 5*LambertW((-sqrt(5) + 1 + sqrt(2)*I*sqrt(sqrt(5) + \
5))*log(7**(7*3**Rational(1, 5)/20))))/(-3*log(7)), \
(log(2401) + 5*LambertW(-log(7**(7*3**Rational(1, 5)/5))))/(-3*log(7))]
assert result == ans
# it works if expanded, too
assert solve(eq.expand(), x) == result
assert solve(z*cos(x) - y, x) == [-acos(y/z) + 2*pi, acos(y/z)]
assert solve(z*cos(2*x) - y, x) == [-acos(y/z)/2 + pi, acos(y/z)/2]
assert solve(z*cos(sin(x)) - y, x) == [
pi - asin(acos(y/z)), asin(acos(y/z) - 2*pi) + pi,
-asin(acos(y/z) - 2*pi), asin(acos(y/z))]
assert solve(z*cos(x), x) == [pi/2, pi*Rational(3, 2)]
# issue 4508
assert solve(y - b*x/(a + x), x) in [[-a*y/(y - b)], [a*y/(b - y)]]
assert solve(y - b*exp(a/x), x) == [a/log(y/b)]
# issue 4507
assert solve(y - b/(1 + a*x), x) in [[(b - y)/(a*y)], [-((y - b)/(a*y))]]
# issue 4506
assert solve(y - a*x**b, x) == [(y/a)**(1/b)]
# issue 4505
assert solve(z**x - y, x) == [log(y)/log(z)]
# issue 4504
assert solve(2**x - 10, x) == [1 + log(5)/log(2)]
# issue 6744
assert solve(x*y) == [{x: 0}, {y: 0}]
assert solve([x*y]) == [{x: 0}, {y: 0}]
assert solve(x**y - 1) == [{x: 1}, {y: 0}]
assert solve([x**y - 1]) == [{x: 1}, {y: 0}]
assert solve(x*y*(x**2 - y**2)) == [{x: 0}, {x: -y}, {x: y}, {y: 0}]
assert solve([x*y*(x**2 - y**2)]) == [{x: 0}, {x: -y}, {x: y}, {y: 0}]
# issue 4739
assert solve(exp(log(5)*x) - 2**x, x) == [0]
# issue 14791
assert solve(exp(log(5)*x) - exp(log(2)*x), x) == [0]
f = Function('f')
assert solve(y*f(log(5)*x) - y*f(log(2)*x), x) == [0]
assert solve(f(x) - f(0), x) == [0]
assert solve(f(x) - f(2 - x), x) == [1]
raises(NotImplementedError, lambda: solve(f(x, y) - f(1, 2), x))
raises(NotImplementedError, lambda: solve(f(x, y) - f(2 - x, 2), x))
raises(ValueError, lambda: solve(f(x, y) - f(1 - x), x))
raises(ValueError, lambda: solve(f(x, y) - f(1), x))
# misc
# make sure that the right variables is picked up in tsolve
# shouldn't generate a GeneratorsNeeded error in _tsolve when the NaN is generated
# for eq_down. Actual answers, as determined numerically are approx. +/- 0.83
raises(NotImplementedError, lambda:
solve(sinh(x)*sinh(sinh(x)) + cosh(x)*cosh(sinh(x)) - 3))
# watch out for recursive loop in tsolve
raises(NotImplementedError, lambda: solve((x + 2)**y*x - 3, x))
# issue 7245
assert solve(sin(sqrt(x))) == [0, pi**2]
# issue 7602
a, b = symbols('a, b', real=True, negative=False)
assert str(solve(Eq(a, 0.5 - cos(pi*b)/2), b)) == \
'[2.0 - 0.318309886183791*acos(1.0 - 2.0*a), 0.318309886183791*acos(1.0 - 2.0*a)]'
# issue 15325
assert solve(y**(1/x) - z, x) == [log(y)/log(z)]
def test_solve_for_functions_derivatives():
t = Symbol('t')
x = Function('x')(t)
y = Function('y')(t)
a11, a12, a21, a22, b1, b2 = symbols('a11,a12,a21,a22,b1,b2')
soln = solve([a11*x + a12*y - b1, a21*x + a22*y - b2], x, y)
assert soln == {
x: (a22*b1 - a12*b2)/(a11*a22 - a12*a21),
y: (a11*b2 - a21*b1)/(a11*a22 - a12*a21),
}
assert solve(x - 1, x) == [1]
assert solve(3*x - 2, x) == [Rational(2, 3)]
soln = solve([a11*x.diff(t) + a12*y.diff(t) - b1, a21*x.diff(t) +
a22*y.diff(t) - b2], x.diff(t), y.diff(t))
assert soln == { y.diff(t): (a11*b2 - a21*b1)/(a11*a22 - a12*a21),
x.diff(t): (a22*b1 - a12*b2)/(a11*a22 - a12*a21) }
assert solve(x.diff(t) - 1, x.diff(t)) == [1]
assert solve(3*x.diff(t) - 2, x.diff(t)) == [Rational(2, 3)]
eqns = set((3*x - 1, 2*y - 4))
assert solve(eqns, set((x, y))) == { x: Rational(1, 3), y: 2 }
x = Symbol('x')
f = Function('f')
F = x**2 + f(x)**2 - 4*x - 1
assert solve(F.diff(x), diff(f(x), x)) == [(-x + 2)/f(x)]
# Mixed cased with a Symbol and a Function
x = Symbol('x')
y = Function('y')(t)
soln = solve([a11*x + a12*y.diff(t) - b1, a21*x +
a22*y.diff(t) - b2], x, y.diff(t))
assert soln == { y.diff(t): (a11*b2 - a21*b1)/(a11*a22 - a12*a21),
x: (a22*b1 - a12*b2)/(a11*a22 - a12*a21) }
# issue 13263
x = Symbol('x')
f = Function('f')
soln = solve([f(x).diff(x) + f(x).diff(x, 2) - 1, f(x).diff(x) - f(x).diff(x, 2)],
f(x).diff(x), f(x).diff(x, 2))
assert soln == { f(x).diff(x, 2): 1/2, f(x).diff(x): 1/2 }
soln = solve([f(x).diff(x, 2) + f(x).diff(x, 3) - 1, 1 - f(x).diff(x, 2) -
f(x).diff(x, 3), 1 - f(x).diff(x,3)], f(x).diff(x, 2), f(x).diff(x, 3))
assert soln == { f(x).diff(x, 2): 0, f(x).diff(x, 3): 1 }
def test_issue_3725():
f = Function('f')
F = x**2 + f(x)**2 - 4*x - 1
e = F.diff(x)
assert solve(e, f(x).diff(x)) in [[(2 - x)/f(x)], [-((x - 2)/f(x))]]
def test_issue_3870():
a, b, c, d = symbols('a b c d')
A = Matrix(2, 2, [a, b, c, d])
B = Matrix(2, 2, [0, 2, -3, 0])
C = Matrix(2, 2, [1, 2, 3, 4])
assert solve(A*B - C, [a, b, c, d]) == {a: 1, b: Rational(-1, 3), c: 2, d: -1}
assert solve([A*B - C], [a, b, c, d]) == {a: 1, b: Rational(-1, 3), c: 2, d: -1}
assert solve(Eq(A*B, C), [a, b, c, d]) == {a: 1, b: Rational(-1, 3), c: 2, d: -1}
assert solve([A*B - B*A], [a, b, c, d]) == {a: d, b: Rational(-2, 3)*c}
assert solve([A*C - C*A], [a, b, c, d]) == {a: d - c, b: Rational(2, 3)*c}
assert solve([A*B - B*A, A*C - C*A], [a, b, c, d]) == {a: d, b: 0, c: 0}
assert solve([Eq(A*B, B*A)], [a, b, c, d]) == {a: d, b: Rational(-2, 3)*c}
assert solve([Eq(A*C, C*A)], [a, b, c, d]) == {a: d - c, b: Rational(2, 3)*c}
assert solve([Eq(A*B, B*A), Eq(A*C, C*A)], [a, b, c, d]) == {a: d, b: 0, c: 0}
def test_solve_linear():
w = Wild('w')
assert solve_linear(x, x) == (0, 1)
assert solve_linear(x, exclude=[x]) == (0, 1)
assert solve_linear(x, symbols=[w]) == (0, 1)
assert solve_linear(x, y - 2*x) in [(x, y/3), (y, 3*x)]
assert solve_linear(x, y - 2*x, exclude=[x]) == (y, 3*x)
assert solve_linear(3*x - y, 0) in [(x, y/3), (y, 3*x)]
assert solve_linear(3*x - y, 0, [x]) == (x, y/3)
assert solve_linear(3*x - y, 0, [y]) == (y, 3*x)
assert solve_linear(x**2/y, 1) == (y, x**2)
assert solve_linear(w, x) in [(w, x), (x, w)]
assert solve_linear(cos(x)**2 + sin(x)**2 + 2 + y) == \
(y, -2 - cos(x)**2 - sin(x)**2)
assert solve_linear(cos(x)**2 + sin(x)**2 + 2 + y, symbols=[x]) == (0, 1)
assert solve_linear(Eq(x, 3)) == (x, 3)
assert solve_linear(1/(1/x - 2)) == (0, 0)
assert solve_linear((x + 1)*exp(-x), symbols=[x]) == (x, -1)
assert solve_linear((x + 1)*exp(x), symbols=[x]) == ((x + 1)*exp(x), 1)
assert solve_linear(x*exp(-x**2), symbols=[x]) == (x, 0)
assert solve_linear(0**x - 1) == (0**x - 1, 1)
assert solve_linear(1 + 1/(x - 1)) == (x, 0)
eq = y*cos(x)**2 + y*sin(x)**2 - y # = y*(1 - 1) = 0
assert solve_linear(eq) == (0, 1)
eq = cos(x)**2 + sin(x)**2 # = 1
assert solve_linear(eq) == (0, 1)
raises(ValueError, lambda: solve_linear(Eq(x, 3), 3))
def test_solve_undetermined_coeffs():
assert solve_undetermined_coeffs(a*x**2 + b*x**2 + b*x + 2*c*x + c + 1, [a, b, c], x) == \
{a: -2, b: 2, c: -1}
# Test that rational functions work
assert solve_undetermined_coeffs(a/x + b/(x + 1) - (2*x + 1)/(x**2 + x), [a, b], x) == \
{a: 1, b: 1}
# Test cancellation in rational functions
assert solve_undetermined_coeffs(((c + 1)*a*x**2 + (c + 1)*b*x**2 +
(c + 1)*b*x + (c + 1)*2*c*x + (c + 1)**2)/(c + 1), [a, b, c], x) == \
{a: -2, b: 2, c: -1}
def test_solve_inequalities():
x = Symbol('x')
sol = And(S.Zero < x, x < oo)
assert solve(x + 1 > 1) == sol
assert solve([x + 1 > 1]) == sol
assert solve([x + 1 > 1], x) == sol
assert solve([x + 1 > 1], [x]) == sol
system = [Lt(x**2 - 2, 0), Gt(x**2 - 1, 0)]
assert solve(system) == \
And(Or(And(Lt(-sqrt(2), x), Lt(x, -1)),
And(Lt(1, x), Lt(x, sqrt(2)))), Eq(0, 0))
x = Symbol('x', real=True)
system = [Lt(x**2 - 2, 0), Gt(x**2 - 1, 0)]
assert solve(system) == \
Or(And(Lt(-sqrt(2), x), Lt(x, -1)), And(Lt(1, x), Lt(x, sqrt(2))))
# issues 6627, 3448
assert solve((x - 3)/(x - 2) < 0, x) == And(Lt(2, x), Lt(x, 3))
assert solve(x/(x + 1) > 1, x) == And(Lt(-oo, x), Lt(x, -1))
assert solve(sin(x) > S.Half) == And(pi/6 < x, x < pi*Rational(5, 6))
assert solve(Eq(False, x < 1)) == (S.One <= x) & (x < oo)
assert solve(Eq(True, x < 1)) == (-oo < x) & (x < 1)
assert solve(Eq(x < 1, False)) == (S.One <= x) & (x < oo)
assert solve(Eq(x < 1, True)) == (-oo < x) & (x < 1)
assert solve(Eq(False, x)) == False
assert solve(Eq(0, x)) == [0]
assert solve(Eq(True, x)) == True
assert solve(Eq(1, x)) == [1]
assert solve(Eq(False, ~x)) == True
assert solve(Eq(True, ~x)) == False
assert solve(Ne(True, x)) == False
assert solve(Ne(1, x)) == (x > -oo) & (x < oo) & Ne(x, 1)
def test_issue_4793():
assert solve(1/x) == []
assert solve(x*(1 - 5/x)) == [5]
assert solve(x + sqrt(x) - 2) == [1]
assert solve(-(1 + x)/(2 + x)**2 + 1/(2 + x)) == []
assert solve(-x**2 - 2*x + (x + 1)**2 - 1) == []
assert solve((x/(x + 1) + 3)**(-2)) == []
assert solve(x/sqrt(x**2 + 1), x) == [0]
assert solve(exp(x) - y, x) == [log(y)]
assert solve(exp(x)) == []
assert solve(x**2 + x + sin(y)**2 + cos(y)**2 - 1, x) in [[0, -1], [-1, 0]]
eq = 4*3**(5*x + 2) - 7
ans = solve(eq, x)
assert len(ans) == 5 and all(eq.subs(x, a).n(chop=True) == 0 for a in ans)
assert solve(log(x**2) - y**2/exp(x), x, y, set=True) == (
[x, y],
{(x, sqrt(exp(x) * log(x ** 2))), (x, -sqrt(exp(x) * log(x ** 2)))})
assert solve(x**2*z**2 - z**2*y**2) == [{x: -y}, {x: y}, {z: 0}]
assert solve((x - 1)/(1 + 1/(x - 1))) == []
assert solve(x**(y*z) - x, x) == [1]
raises(NotImplementedError, lambda: solve(log(x) - exp(x), x))
raises(NotImplementedError, lambda: solve(2**x - exp(x) - 3))
def test_PR1964():
# issue 5171
assert solve(sqrt(x)) == solve(sqrt(x**3)) == [0]
assert solve(sqrt(x - 1)) == [1]
# issue 4462
a = Symbol('a')
assert solve(-3*a/sqrt(x), x) == []
# issue 4486
assert solve(2*x/(x + 2) - 1, x) == [2]
# issue 4496
assert set(solve((x**2/(7 - x)).diff(x))) == set([S.Zero, S(14)])
# issue 4695
f = Function('f')
assert solve((3 - 5*x/f(x))*f(x), f(x)) == [x*Rational(5, 3)]
# issue 4497
assert solve(1/root(5 + x, 5) - 9, x) == [Rational(-295244, 59049)]
assert solve(sqrt(x) + sqrt(sqrt(x)) - 4) == [(Rational(-1, 2) + sqrt(17)/2)**4]
assert set(solve(Poly(sqrt(exp(x)) + sqrt(exp(-x)) - 4))) in \
[
set([log((-sqrt(3) + 2)**2), log((sqrt(3) + 2)**2)]),
set([2*log(-sqrt(3) + 2), 2*log(sqrt(3) + 2)]),
set([log(-4*sqrt(3) + 7), log(4*sqrt(3) + 7)]),
]
assert set(solve(Poly(exp(x) + exp(-x) - 4))) == \
set([log(-sqrt(3) + 2), log(sqrt(3) + 2)])
assert set(solve(x**y + x**(2*y) - 1, x)) == \
set([(Rational(-1, 2) + sqrt(5)/2)**(1/y), (Rational(-1, 2) - sqrt(5)/2)**(1/y)])
assert solve(exp(x/y)*exp(-z/y) - 2, y) == [(x - z)/log(2)]
assert solve(
x**z*y**z - 2, z) in [[log(2)/(log(x) + log(y))], [log(2)/(log(x*y))]]
# if you do inversion too soon then multiple roots (as for the following)
# will be missed, e.g. if exp(3*x) = exp(3) -> 3*x = 3
E = S.Exp1
assert solve(exp(3*x) - exp(3), x) in [
[1, log(E*(Rational(-1, 2) - sqrt(3)*I/2)), log(E*(Rational(-1, 2) + sqrt(3)*I/2))],
[1, log(-E/2 - sqrt(3)*E*I/2), log(-E/2 + sqrt(3)*E*I/2)],
]
# coverage test
p = Symbol('p', positive=True)
assert solve((1/p + 1)**(p + 1)) == []
def test_issue_5197():
x = Symbol('x', real=True)
assert solve(x**2 + 1, x) == []
n = Symbol('n', integer=True, positive=True)
assert solve((n - 1)*(n + 2)*(2*n - 1), n) == [1]
x = Symbol('x', positive=True)
y = Symbol('y')
assert solve([x + 5*y - 2, -3*x + 6*y - 15], x, y) == []
# not {x: -3, y: 1} b/c x is positive
# The solution following should not contain (-sqrt(2), sqrt(2))
assert solve((x + y)*n - y**2 + 2, x, y) == [(sqrt(2), -sqrt(2))]
y = Symbol('y', positive=True)
# The solution following should not contain {y: -x*exp(x/2)}
assert solve(x**2 - y**2/exp(x), y, x, dict=True) == [{y: x*exp(x/2)}]
x, y, z = symbols('x y z', positive=True)
assert solve(z**2*x**2 - z**2*y**2/exp(x), y, x, z, dict=True) == [{y: x*exp(x/2)}]
def test_checking():
assert set(
solve(x*(x - y/x), x, check=False)) == set([sqrt(y), S.Zero, -sqrt(y)])
assert set(solve(x*(x - y/x), x, check=True)) == set([sqrt(y), -sqrt(y)])
# {x: 0, y: 4} sets denominator to 0 in the following so system should return None
assert solve((1/(1/x + 2), 1/(y - 3) - 1)) == []
# 0 sets denominator of 1/x to zero so None is returned
assert solve(1/(1/x + 2)) == []
def test_issue_4671_4463_4467():
assert solve((sqrt(x**2 - 1) - 2)) in ([sqrt(5), -sqrt(5)],
[-sqrt(5), sqrt(5)])
assert solve((2**exp(y**2/x) + 2)/(x**2 + 15), y) == [
-sqrt(x*log(1 + I*pi/log(2))), sqrt(x*log(1 + I*pi/log(2)))]
C1, C2 = symbols('C1 C2')
f = Function('f')
assert solve(C1 + C2/x**2 - exp(-f(x)), f(x)) == [log(x**2/(C1*x**2 + C2))]
a = Symbol('a')
E = S.Exp1
assert solve(1 - log(a + 4*x**2), x) in (
[-sqrt(-a + E)/2, sqrt(-a + E)/2],
[sqrt(-a + E)/2, -sqrt(-a + E)/2]
)
assert solve(log(a**(-3) - x**2)/a, x) in (
[-sqrt(-1 + a**(-3)), sqrt(-1 + a**(-3))],
[sqrt(-1 + a**(-3)), -sqrt(-1 + a**(-3))],)
assert solve(1 - log(a + 4*x**2), x) in (
[-sqrt(-a + E)/2, sqrt(-a + E)/2],
[sqrt(-a + E)/2, -sqrt(-a + E)/2],)
assert solve((a**2 + 1)*(sin(a*x) + cos(a*x)), x) == [-pi/(4*a)]
assert solve(3 - (sinh(a*x) + cosh(a*x)), x) == [log(3)/a]
assert set(solve(3 - (sinh(a*x) + cosh(a*x)**2), x)) == \
set([log(-2 + sqrt(5))/a, log(-sqrt(2) + 1)/a,
log(-sqrt(5) - 2)/a, log(1 + sqrt(2))/a])
assert solve(atan(x) - 1) == [tan(1)]
def test_issue_5132():
r, t = symbols('r,t')
assert set(solve([r - x**2 - y**2, tan(t) - y/x], [x, y])) == \
set([(
-sqrt(r*cos(t)**2), -1*sqrt(r*cos(t)**2)*tan(t)),
(sqrt(r*cos(t)**2), sqrt(r*cos(t)**2)*tan(t))])
assert solve([exp(x) - sin(y), 1/y - 3], [x, y]) == \
[(log(sin(Rational(1, 3))), Rational(1, 3))]
assert solve([exp(x) - sin(y), 1/exp(y) - 3], [x, y]) == \
[(log(-sin(log(3))), -log(3))]
assert set(solve([exp(x) - sin(y), y**2 - 4], [x, y])) == \
set([(log(-sin(2)), -S(2)), (log(sin(2)), S(2))])
eqs = [exp(x)**2 - sin(y) + z**2, 1/exp(y) - 3]
assert solve(eqs, set=True) == \
([x, y], set([
(log(-sqrt(-z**2 - sin(log(3)))), -log(3)),
(log(-z**2 - sin(log(3)))/2, -log(3))]))
assert solve(eqs, x, z, set=True) == (
[x, z],
{(log(-z**2 + sin(y))/2, z), (log(-sqrt(-z**2 + sin(y))), z)})
assert set(solve(eqs, x, y)) == \
set([
(log(-sqrt(-z**2 - sin(log(3)))), -log(3)),
(log(-z**2 - sin(log(3)))/2, -log(3))])
assert set(solve(eqs, y, z)) == \
set([
(-log(3), -sqrt(-exp(2*x) - sin(log(3)))),
(-log(3), sqrt(-exp(2*x) - sin(log(3))))])
eqs = [exp(x)**2 - sin(y) + z, 1/exp(y) - 3]
assert solve(eqs, set=True) == ([x, y], set(
[
(log(-sqrt(-z - sin(log(3)))), -log(3)),
(log(-z - sin(log(3)))/2, -log(3))]))
assert solve(eqs, x, z, set=True) == (
[x, z],
{(log(-sqrt(-z + sin(y))), z), (log(-z + sin(y))/2, z)})
assert set(solve(eqs, x, y)) == set(
[
(log(-sqrt(-z - sin(log(3)))), -log(3)),
(log(-z - sin(log(3)))/2, -log(3))])
assert solve(eqs, z, y) == \
[(-exp(2*x) - sin(log(3)), -log(3))]
assert solve((sqrt(x**2 + y**2) - sqrt(10), x + y - 4), set=True) == (
[x, y], set([(S.One, S(3)), (S(3), S.One)]))
assert set(solve((sqrt(x**2 + y**2) - sqrt(10), x + y - 4), x, y)) == \
set([(S.One, S(3)), (S(3), S.One)])
def test_issue_5335():
lam, a0, conc = symbols('lam a0 conc')
a = 0.005
b = 0.743436700916726
eqs = [lam + 2*y - a0*(1 - x/2)*x - a*x/2*x,
a0*(1 - x/2)*x - 1*y - b*y,
x + y - conc]
sym = [x, y, a0]
# there are 4 solutions obtained manually but only two are valid
assert len(solve(eqs, sym, manual=True, minimal=True)) == 2
assert len(solve(eqs, sym)) == 2 # cf below with rational=False
@SKIP("Hangs")
def _test_issue_5335_float():
# gives ZeroDivisionError: polynomial division
lam, a0, conc = symbols('lam a0 conc')
a = 0.005
b = 0.743436700916726
eqs = [lam + 2*y - a0*(1 - x/2)*x - a*x/2*x,
a0*(1 - x/2)*x - 1*y - b*y,
x + y - conc]
sym = [x, y, a0]
assert len(solve(eqs, sym, rational=False)) == 2
def test_issue_5767():
assert set(solve([x**2 + y + 4], [x])) == \
set([(-sqrt(-y - 4),), (sqrt(-y - 4),)])
def test_polysys():
assert set(solve([x**2 + 2/y - 2, x + y - 3], [x, y])) == \
set([(S.One, S(2)), (1 + sqrt(5), 2 - sqrt(5)),
(1 - sqrt(5), 2 + sqrt(5))])
assert solve([x**2 + y - 2, x**2 + y]) == []
# the ordering should be whatever the user requested
assert solve([x**2 + y - 3, x - y - 4], (x, y)) != solve([x**2 +
y - 3, x - y - 4], (y, x))
@slow
def test_unrad1():
raises(NotImplementedError, lambda:
unrad(sqrt(x) + sqrt(x + 1) + sqrt(1 - sqrt(x)) + 3))
raises(NotImplementedError, lambda:
unrad(sqrt(x) + (x + 1)**Rational(1, 3) + 2*sqrt(y)))
s = symbols('s', cls=Dummy)
# checkers to deal with possibility of answer coming
# back with a sign change (cf issue 5203)
def check(rv, ans):
assert bool(rv[1]) == bool(ans[1])
if ans[1]:
return s_check(rv, ans)
e = rv[0].expand()
a = ans[0].expand()
return e in [a, -a] and rv[1] == ans[1]
def s_check(rv, ans):
# get the dummy
rv = list(rv)
d = rv[0].atoms(Dummy)
reps = list(zip(d, [s]*len(d)))
# replace s with this dummy
rv = (rv[0].subs(reps).expand(), [rv[1][0].subs(reps), rv[1][1].subs(reps)])
ans = (ans[0].subs(reps).expand(), [ans[1][0].subs(reps), ans[1][1].subs(reps)])
return str(rv[0]) in [str(ans[0]), str(-ans[0])] and \
str(rv[1]) == str(ans[1])
assert check(unrad(sqrt(x)),
(x, []))
assert check(unrad(sqrt(x) + 1),
(x - 1, []))
assert check(unrad(sqrt(x) + root(x, 3) + 2),
(s**3 + s**2 + 2, [s, s**6 - x]))
assert check(unrad(sqrt(x)*root(x, 3) + 2),
(x**5 - 64, []))
assert check(unrad(sqrt(x) + (x + 1)**Rational(1, 3)),
(x**3 - (x + 1)**2, []))
assert check(unrad(sqrt(x) + sqrt(x + 1) + sqrt(2*x)),
(-2*sqrt(2)*x - 2*x + 1, []))
assert check(unrad(sqrt(x) + sqrt(x + 1) + 2),
(16*x - 9, []))
assert check(unrad(sqrt(x) + sqrt(x + 1) + sqrt(1 - x)),
(5*x**2 - 4*x, []))
assert check(unrad(a*sqrt(x) + b*sqrt(x) + c*sqrt(y) + d*sqrt(y)),
((a*sqrt(x) + b*sqrt(x))**2 - (c*sqrt(y) + d*sqrt(y))**2, []))
assert check(unrad(sqrt(x) + sqrt(1 - x)),
(2*x - 1, []))
assert check(unrad(sqrt(x) + sqrt(1 - x) - 3),
(x**2 - x + 16, []))
assert check(unrad(sqrt(x) + sqrt(1 - x) + sqrt(2 + x)),
(5*x**2 - 2*x + 1, []))
assert unrad(sqrt(x) + sqrt(1 - x) + sqrt(2 + x) - 3) in [
(25*x**4 + 376*x**3 + 1256*x**2 - 2272*x + 784, []),
(25*x**8 - 476*x**6 + 2534*x**4 - 1468*x**2 + 169, [])]
assert unrad(sqrt(x) + sqrt(1 - x) + sqrt(2 + x) - sqrt(1 - 2*x)) == \
(41*x**4 + 40*x**3 + 232*x**2 - 160*x + 16, []) # orig root at 0.487
assert check(unrad(sqrt(x) + sqrt(x + 1)), (S.One, []))
eq = sqrt(x) + sqrt(x + 1) + sqrt(1 - sqrt(x))
assert check(unrad(eq),
(16*x**2 - 9*x, []))
assert set(solve(eq, check=False)) == set([S.Zero, Rational(9, 16)])
assert solve(eq) == []
# but this one really does have those solutions
assert set(solve(sqrt(x) - sqrt(x + 1) + sqrt(1 - sqrt(x)))) == \
set([S.Zero, Rational(9, 16)])
assert check(unrad(sqrt(x) + root(x + 1, 3) + 2*sqrt(y), y),
(S('2*sqrt(x)*(x + 1)**(1/3) + x - 4*y + (x + 1)**(2/3)'), []))
assert check(unrad(sqrt(x/(1 - x)) + (x + 1)**Rational(1, 3)),
(x**5 - x**4 - x**3 + 2*x**2 + x - 1, []))
assert check(unrad(sqrt(x/(1 - x)) + 2*sqrt(y), y),
(4*x*y + x - 4*y, []))
assert check(unrad(sqrt(x)*sqrt(1 - x) + 2, x),
(x**2 - x + 4, []))
# http://tutorial.math.lamar.edu/
# Classes/Alg/SolveRadicalEqns.aspx#Solve_Rad_Ex2_a
assert solve(Eq(x, sqrt(x + 6))) == [3]
assert solve(Eq(x + sqrt(x - 4), 4)) == [4]
assert solve(Eq(1, x + sqrt(2*x - 3))) == []
assert set(solve(Eq(sqrt(5*x + 6) - 2, x))) == set([-S.One, S(2)])
assert set(solve(Eq(sqrt(2*x - 1) - sqrt(x - 4), 2))) == set([S(5), S(13)])
assert solve(Eq(sqrt(x + 7) + 2, sqrt(3 - x))) == [-6]
# http://www.purplemath.com/modules/solverad.htm
assert solve((2*x - 5)**Rational(1, 3) - 3) == [16]
assert set(solve(x + 1 - root(x**4 + 4*x**3 - x, 4))) == \
set([Rational(-1, 2), Rational(-1, 3)])
assert set(solve(sqrt(2*x**2 - 7) - (3 - x))) == set([-S(8), S(2)])
assert solve(sqrt(2*x + 9) - sqrt(x + 1) - sqrt(x + 4)) == [0]
assert solve(sqrt(x + 4) + sqrt(2*x - 1) - 3*sqrt(x - 1)) == [5]
assert solve(sqrt(x)*sqrt(x - 7) - 12) == [16]
assert solve(sqrt(x - 3) + sqrt(x) - 3) == [4]
assert solve(sqrt(9*x**2 + 4) - (3*x + 2)) == [0]
assert solve(sqrt(x) - 2 - 5) == [49]
assert solve(sqrt(x - 3) - sqrt(x) - 3) == []
assert solve(sqrt(x - 1) - x + 7) == [10]
assert solve(sqrt(x - 2) - 5) == [27]
assert solve(sqrt(17*x - sqrt(x**2 - 5)) - 7) == [3]
assert solve(sqrt(x) - sqrt(x - 1) + sqrt(sqrt(x))) == []
# don't posify the expression in unrad and do use _mexpand
z = sqrt(2*x + 1)/sqrt(x) - sqrt(2 + 1/x)
p = posify(z)[0]
assert solve(p) == []
assert solve(z) == []
assert solve(z + 6*I) == [Rational(-1, 11)]
assert solve(p + 6*I) == []
# issue 8622
assert unrad((root(x + 1, 5) - root(x, 3))) == (
x**5 - x**3 - 3*x**2 - 3*x - 1, [])
# issue #8679
assert check(unrad(x + root(x, 3) + root(x, 3)**2 + sqrt(y), x),
(s**3 + s**2 + s + sqrt(y), [s, s**3 - x]))
# for coverage
assert check(unrad(sqrt(x) + root(x, 3) + y),
(s**3 + s**2 + y, [s, s**6 - x]))
assert solve(sqrt(x) + root(x, 3) - 2) == [1]
raises(NotImplementedError, lambda:
solve(sqrt(x) + root(x, 3) + root(x + 1, 5) - 2))
# fails through a different code path
raises(NotImplementedError, lambda: solve(-sqrt(2) + cosh(x)/x))
# unrad some
assert solve(sqrt(x + root(x, 3))+root(x - y, 5), y) == [
x + (x**Rational(1, 3) + x)**Rational(5, 2)]
assert check(unrad(sqrt(x) - root(x + 1, 3)*sqrt(x + 2) + 2),
(s**10 + 8*s**8 + 24*s**6 - 12*s**5 - 22*s**4 - 160*s**3 - 212*s**2 -
192*s - 56, [s, s**2 - x]))
e = root(x + 1, 3) + root(x, 3)
assert unrad(e) == (2*x + 1, [])
eq = (sqrt(x) + sqrt(x + 1) + sqrt(1 - x) - 6*sqrt(5)/5)
assert check(unrad(eq),
(15625*x**4 + 173000*x**3 + 355600*x**2 - 817920*x + 331776, []))
assert check(unrad(root(x, 4) + root(x, 4)**3 - 1),
(s**3 + s - 1, [s, s**4 - x]))
assert check(unrad(root(x, 2) + root(x, 2)**3 - 1),
(x**3 + 2*x**2 + x - 1, []))
assert unrad(x**0.5) is None
assert check(unrad(t + root(x + y, 5) + root(x + y, 5)**3),
(s**3 + s + t, [s, s**5 - x - y]))
assert check(unrad(x + root(x + y, 5) + root(x + y, 5)**3, y),
(s**3 + s + x, [s, s**5 - x - y]))
assert check(unrad(x + root(x + y, 5) + root(x + y, 5)**3, x),
(s**5 + s**3 + s - y, [s, s**5 - x - y]))
assert check(unrad(root(x - 1, 3) + root(x + 1, 5) + root(2, 5)),
(s**5 + 5*2**Rational(1, 5)*s**4 + s**3 + 10*2**Rational(2, 5)*s**3 +
10*2**Rational(3, 5)*s**2 + 5*2**Rational(4, 5)*s + 4, [s, s**3 - x + 1]))
raises(NotImplementedError, lambda:
unrad((root(x, 2) + root(x, 3) + root(x, 4)).subs(x, x**5 - x + 1)))
# the simplify flag should be reset to False for unrad results;
# if it's not then this next test will take a long time
assert solve(root(x, 3) + root(x, 5) - 2) == [1]
eq = (sqrt(x) + sqrt(x + 1) + sqrt(1 - x) - 6*sqrt(5)/5)
assert check(unrad(eq),
((5*x - 4)*(3125*x**3 + 37100*x**2 + 100800*x - 82944), []))
ans = S('''
[4/5, -1484/375 + 172564/(140625*(114*sqrt(12657)/78125 +
12459439/52734375)**(1/3)) +
4*(114*sqrt(12657)/78125 + 12459439/52734375)**(1/3)]''')
assert solve(eq) == ans
# duplicate radical handling
assert check(unrad(sqrt(x + root(x + 1, 3)) - root(x + 1, 3) - 2),
(s**3 - s**2 - 3*s - 5, [s, s**3 - x - 1]))
# cov post-processing
e = root(x**2 + 1, 3) - root(x**2 - 1, 5) - 2
assert check(unrad(e),
(s**5 - 10*s**4 + 39*s**3 - 80*s**2 + 80*s - 30,
[s, s**3 - x**2 - 1]))
e = sqrt(x + root(x + 1, 2)) - root(x + 1, 3) - 2
assert check(unrad(e),
(s**6 - 2*s**5 - 7*s**4 - 3*s**3 + 26*s**2 + 40*s + 25,
[s, s**3 - x - 1]))
assert check(unrad(e, _reverse=True),
(s**6 - 14*s**5 + 73*s**4 - 187*s**3 + 276*s**2 - 228*s + 89,
[s, s**2 - x - sqrt(x + 1)]))
# this one needs r0, r1 reversal to work
assert check(unrad(sqrt(x + sqrt(root(x, 3) - 1)) - root(x, 6) - 2),
(s**12 - 2*s**8 - 8*s**7 - 8*s**6 + s**4 + 8*s**3 + 23*s**2 +
32*s + 17, [s, s**6 - x]))
# is this needed?
#assert unrad(root(cosh(x), 3)/x*root(x + 1, 5) - 1) == (
# x**15 - x**3*cosh(x)**5 - 3*x**2*cosh(x)**5 - 3*x*cosh(x)**5 - cosh(x)**5, [])
raises(NotImplementedError, lambda:
unrad(sqrt(cosh(x)/x) + root(x + 1,3)*sqrt(x) - 1))
assert unrad(S('(x+y)**(2*y/3) + (x+y)**(1/3) + 1')) is None
assert check(unrad(S('(x+y)**(2*y/3) + (x+y)**(1/3) + 1'), x),
(s**(2*y) + s + 1, [s, s**3 - x - y]))
# This tests two things: that if full unrad is attempted and fails
# the solution should still be found; also it tests that the use of
# composite
assert len(solve(sqrt(y)*x + x**3 - 1, x)) == 3
assert len(solve(-512*y**3 + 1344*(x + 2)**Rational(1, 3)*y**2 -
1176*(x + 2)**Rational(2, 3)*y - 169*x + 686, y, _unrad=False)) == 3
# watch out for when the cov doesn't involve the symbol of interest
eq = S('-x + (7*y/8 - (27*x/2 + 27*sqrt(x**2)/2)**(1/3)/3)**3 - 1')
assert solve(eq, y) == [
4*2**Rational(2, 3)*(27*x + 27*sqrt(x**2))**Rational(1, 3)/21 - (Rational(-1, 2) -
sqrt(3)*I/2)*(x*Rational(-6912, 343) + sqrt((x*Rational(-13824, 343) - Rational(13824, 343))**2)/2 -
Rational(6912, 343))**Rational(1, 3)/3, 4*2**Rational(2, 3)*(27*x + 27*sqrt(x**2))**Rational(1, 3)/21 -
(Rational(-1, 2) + sqrt(3)*I/2)*(x*Rational(-6912, 343) + sqrt((x*Rational(-13824, 343) -
Rational(13824, 343))**2)/2 - Rational(6912, 343))**Rational(1, 3)/3, 4*2**Rational(2, 3)*(27*x +
27*sqrt(x**2))**Rational(1, 3)/21 - (x*Rational(-6912, 343) + sqrt((x*Rational(-13824, 343) -
Rational(13824, 343))**2)/2 - Rational(6912, 343))**Rational(1, 3)/3]
eq = root(x + 1, 3) - (root(x, 3) + root(x, 5))
assert check(unrad(eq),
(3*s**13 + 3*s**11 + s**9 - 1, [s, s**15 - x]))
assert check(unrad(eq - 2),
(3*s**13 + 3*s**11 + 6*s**10 + s**9 + 12*s**8 + 6*s**6 + 12*s**5 +
12*s**3 + 7, [s, s**15 - x]))
assert check(unrad(root(x, 3) - root(x + 1, 4)/2 + root(x + 2, 3)),
(4096*s**13 + 960*s**12 + 48*s**11 - s**10 - 1728*s**4,
[s, s**4 - x - 1])) # orig expr has two real roots: -1, -.389
assert check(unrad(root(x, 3) + root(x + 1, 4) - root(x + 2, 3)/2),
(343*s**13 + 2904*s**12 + 1344*s**11 + 512*s**10 - 1323*s**9 -
3024*s**8 - 1728*s**7 + 1701*s**5 + 216*s**4 - 729*s, [s, s**4 - x -
1])) # orig expr has one real root: -0.048
assert check(unrad(root(x, 3)/2 - root(x + 1, 4) + root(x + 2, 3)),
(729*s**13 - 216*s**12 + 1728*s**11 - 512*s**10 + 1701*s**9 -
3024*s**8 + 1344*s**7 + 1323*s**5 - 2904*s**4 + 343*s, [s, s**4 - x -
1])) # orig expr has 2 real roots: -0.91, -0.15
assert check(unrad(root(x, 3)/2 - root(x + 1, 4) + root(x + 2, 3) - 2),
(729*s**13 + 1242*s**12 + 18496*s**10 + 129701*s**9 + 388602*s**8 +
453312*s**7 - 612864*s**6 - 3337173*s**5 - 6332418*s**4 - 7134912*s**3
- 5064768*s**2 - 2111913*s - 398034, [s, s**4 - x - 1]))
# orig expr has 1 real root: 19.53
ans = solve(sqrt(x) + sqrt(x + 1) -
sqrt(1 - x) - sqrt(2 + x))
assert len(ans) == 1 and NS(ans[0])[:4] == '0.73'
# the fence optimization problem
# https://github.com/sympy/sympy/issues/4793#issuecomment-36994519
F = Symbol('F')
eq = F - (2*x + 2*y + sqrt(x**2 + y**2))
ans = F*Rational(2, 7) - sqrt(2)*F/14
X = solve(eq, x, check=False)
for xi in reversed(X): # reverse since currently, ans is the 2nd one
Y = solve((x*y).subs(x, xi).diff(y), y, simplify=False, check=False)
if any((a - ans).expand().is_zero for a in Y):
break
else:
assert None # no answer was found
assert solve(sqrt(x + 1) + root(x, 3) - 2) == S('''
[(-11/(9*(47/54 + sqrt(93)/6)**(1/3)) + 1/3 + (47/54 +
sqrt(93)/6)**(1/3))**3]''')
assert solve(sqrt(sqrt(x + 1)) + x**Rational(1, 3) - 2) == S('''
[(-sqrt(-2*(-1/16 + sqrt(6913)/16)**(1/3) + 6/(-1/16 +
sqrt(6913)/16)**(1/3) + 17/2 + 121/(4*sqrt(-6/(-1/16 +
sqrt(6913)/16)**(1/3) + 2*(-1/16 + sqrt(6913)/16)**(1/3) + 17/4)))/2 +
sqrt(-6/(-1/16 + sqrt(6913)/16)**(1/3) + 2*(-1/16 +
sqrt(6913)/16)**(1/3) + 17/4)/2 + 9/4)**3]''')
assert solve(sqrt(x) + root(sqrt(x) + 1, 3) - 2) == S('''
[(-(81/2 + 3*sqrt(741)/2)**(1/3)/3 + (81/2 + 3*sqrt(741)/2)**(-1/3) +
2)**2]''')
eq = S('''
-x + (1/2 - sqrt(3)*I/2)*(3*x**3/2 - x*(3*x**2 - 34)/2 + sqrt((-3*x**3
+ x*(3*x**2 - 34) + 90)**2/4 - 39304/27) - 45)**(1/3) + 34/(3*(1/2 -
sqrt(3)*I/2)*(3*x**3/2 - x*(3*x**2 - 34)/2 + sqrt((-3*x**3 + x*(3*x**2
- 34) + 90)**2/4 - 39304/27) - 45)**(1/3))''')
assert check(unrad(eq),
(-s*(-s**6 + sqrt(3)*s**6*I - 153*2**Rational(2, 3)*3**Rational(1, 3)*s**4 +
51*12**Rational(1, 3)*s**4 - 102*2**Rational(2, 3)*3**Rational(5, 6)*s**4*I - 1620*s**3 +
1620*sqrt(3)*s**3*I + 13872*18**Rational(1, 3)*s**2 - 471648 +
471648*sqrt(3)*I), [s, s**3 - 306*x - sqrt(3)*sqrt(31212*x**2 -
165240*x + 61484) + 810]))
assert solve(eq) == [] # not other code errors
eq = root(x, 3) - root(y, 3) + root(x, 5)
assert check(unrad(eq),
(s**15 + 3*s**13 + 3*s**11 + s**9 - y, [s, s**15 - x]))
eq = root(x, 3) + root(y, 3) + root(x*y, 4)
assert check(unrad(eq),
(s*y*(-s**12 - 3*s**11*y - 3*s**10*y**2 - s**9*y**3 -
3*s**8*y**2 + 21*s**7*y**3 - 3*s**6*y**4 - 3*s**4*y**4 -
3*s**3*y**5 - y**6), [s, s**4 - x*y]))
raises(NotImplementedError,
lambda: unrad(root(x, 3) + root(y, 3) + root(x*y, 5)))
# Test unrad with an Equality
eq = Eq(-x**(S(1)/5) + x**(S(1)/3), -3**(S(1)/3) - (-1)**(S(3)/5)*3**(S(1)/5))
assert check(unrad(eq),
(-s**5 + s**3 - 3**(S(1)/3) - (-1)**(S(3)/5)*3**(S(1)/5), [s, s**15 - x]))
@slow
def test_unrad_slow():
# this has roots with multiplicity > 1; there should be no
# repeats in roots obtained, however
eq = (sqrt(1 + sqrt(1 - 4*x**2)) - x*((1 + sqrt(1 + 2*sqrt(1 - 4*x**2)))))
assert solve(eq) == [S.Half]
@XFAIL
def test_unrad_fail():
# this only works if we check real_root(eq.subs(x, Rational(1, 3)))
# but checksol doesn't work like that
assert solve(root(x**3 - 3*x**2, 3) + 1 - x) == [Rational(1, 3)]
assert solve(root(x + 1, 3) + root(x**2 - 2, 5) + 1) == [
-1, -1 + CRootOf(x**5 + x**4 + 5*x**3 + 8*x**2 + 10*x + 5, 0)**3]
def test_checksol():
x, y, r, t = symbols('x, y, r, t')
eq = r - x**2 - y**2
dict_var_soln = {y: - sqrt(r) / sqrt(tan(t)**2 + 1),
x: -sqrt(r)*tan(t)/sqrt(tan(t)**2 + 1)}
assert checksol(eq, dict_var_soln) == True
assert checksol(Eq(x, False), {x: False}) is True
assert checksol(Ne(x, False), {x: False}) is False
assert checksol(Eq(x < 1, True), {x: 0}) is True
assert checksol(Eq(x < 1, True), {x: 1}) is False
assert checksol(Eq(x < 1, False), {x: 1}) is True
assert checksol(Eq(x < 1, False), {x: 0}) is False
assert checksol(Eq(x + 1, x**2 + 1), {x: 1}) is True
assert checksol([x - 1, x**2 - 1], x, 1) is True
assert checksol([x - 1, x**2 - 2], x, 1) is False
assert checksol(Poly(x**2 - 1), x, 1) is True
raises(ValueError, lambda: checksol(x, 1))
raises(ValueError, lambda: checksol([], x, 1))
def test__invert():
assert _invert(x - 2) == (2, x)
assert _invert(2) == (2, 0)
assert _invert(exp(1/x) - 3, x) == (1/log(3), x)
assert _invert(exp(1/x + a/x) - 3, x) == ((a + 1)/log(3), x)
assert _invert(a, x) == (a, 0)
def test_issue_4463():
assert solve(-a*x + 2*x*log(x), x) == [exp(a/2)]
assert solve(x**x) == []
assert solve(x**x - 2) == [exp(LambertW(log(2)))]
assert solve(((x - 3)*(x - 2))**((x - 3)*(x - 4))) == [2]
@slow
def test_issue_5114_solvers():
a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q, r = symbols('a:r')
# there is no 'a' in the equation set but this is how the
# problem was originally posed
syms = a, b, c, f, h, k, n
eqs = [b + r/d - c/d,
c*(1/d + 1/e + 1/g) - f/g - r/d,
f*(1/g + 1/i + 1/j) - c/g - h/i,
h*(1/i + 1/l + 1/m) - f/i - k/m,
k*(1/m + 1/o + 1/p) - h/m - n/p,
n*(1/p + 1/q) - k/p]
assert len(solve(eqs, syms, manual=True, check=False, simplify=False)) == 1
def test_issue_5849():
I1, I2, I3, I4, I5, I6 = symbols('I1:7')
dI1, dI4, dQ2, dQ4, Q2, Q4 = symbols('dI1,dI4,dQ2,dQ4,Q2,Q4')
e = (
I1 - I2 - I3,
I3 - I4 - I5,
I4 + I5 - I6,
-I1 + I2 + I6,
-2*I1 - 2*I3 - 2*I5 - 3*I6 - dI1/2 + 12,
-I4 + dQ4,
-I2 + dQ2,
2*I3 + 2*I5 + 3*I6 - Q2,
I4 - 2*I5 + 2*Q4 + dI4
)
ans = [{
dQ4: I3 - I5,
dI1: -4*I2 - 8*I3 - 4*I5 - 6*I6 + 24,
I4: I3 - I5,
dQ2: I2,
Q2: 2*I3 + 2*I5 + 3*I6,
I1: I2 + I3,
Q4: -I3/2 + 3*I5/2 - dI4/2}]
v = I1, I4, Q2, Q4, dI1, dI4, dQ2, dQ4
assert solve(e, *v, manual=True, check=False, dict=True) == ans
assert solve(e, *v, manual=True) == []
# the matrix solver (tested below) doesn't like this because it produces
# a zero row in the matrix. Is this related to issue 4551?
assert [ei.subs(
ans[0]) for ei in e] == [0, 0, I3 - I6, -I3 + I6, 0, 0, 0, 0, 0]
# Should this work at all? Simpler examples fail e.g.:
# solve([x+y+z,x+y],[x,y]) == []
# Here a solution only exists if I3 == I6 which is not generically true.
@XFAIL
def test_issue_5849_matrix():
'''Same as test_issue_5849 but solved with the matrix solver.'''
I1, I2, I3, I4, I5, I6 = symbols('I1:7')
dI1, dI4, dQ2, dQ4, Q2, Q4 = symbols('dI1,dI4,dQ2,dQ4,Q2,Q4')
e = (
I1 - I2 - I3,
I3 - I4 - I5,
I4 + I5 - I6,
-I1 + I2 + I6,
-2*I1 - 2*I3 - 2*I5 - 3*I6 - dI1/2 + 12,
-I4 + dQ4,
-I2 + dQ2,
2*I3 + 2*I5 + 3*I6 - Q2,
I4 - 2*I5 + 2*Q4 + dI4
)
assert solve(e, I1, I4, Q2, Q4, dI1, dI4, dQ2, dQ4) == {
dI4: -I3 + 3*I5 - 2*Q4,
dI1: -4*I2 - 8*I3 - 4*I5 - 6*I6 + 24,
dQ2: I2,
I1: I2 + I3,
Q2: 2*I3 + 2*I5 + 3*I6,
dQ4: I3 - I5,
I4: I3 - I5}
def test_issue_5901():
f, g, h = map(Function, 'fgh')
a = Symbol('a')
D = Derivative(f(x), x)
G = Derivative(g(a), a)
assert solve(f(x) + f(x).diff(x), f(x)) == \
[-D]
assert solve(f(x) - 3, f(x)) == \
[3]
assert solve(f(x) - 3*f(x).diff(x), f(x)) == \
[3*D]
assert solve([f(x) - 3*f(x).diff(x)], f(x)) == \
{f(x): 3*D}
assert solve([f(x) - 3*f(x).diff(x), f(x)**2 - y + 4], f(x), y) == \
[{f(x): 3*D, y: 9*D**2 + 4}]
assert solve(-f(a)**2*g(a)**2 + f(a)**2*h(a)**2 + g(a).diff(a),
h(a), g(a), set=True) == \
([g(a)], set([
(-sqrt(h(a)**2*f(a)**2 + G)/f(a),),
(sqrt(h(a)**2*f(a)**2+ G)/f(a),)]))
args = [f(x).diff(x, 2)*(f(x) + g(x)) - g(x)**2 + 2, f(x), g(x)]
assert set(solve(*args)) == \
set([(-sqrt(2), sqrt(2)), (sqrt(2), -sqrt(2))])
eqs = [f(x)**2 + g(x) - 2*f(x).diff(x), g(x)**2 - 4]
assert solve(eqs, f(x), g(x), set=True) == \
([f(x), g(x)], set([
(-sqrt(2*D - 2), S(2)),
(sqrt(2*D - 2), S(2)),
(-sqrt(2*D + 2), -S(2)),
(sqrt(2*D + 2), -S(2))]))
# the underlying problem was in solve_linear that was not masking off
# anything but a Mul or Add; it now raises an error if it gets anything
# but a symbol and solve handles the substitutions necessary so solve_linear
# won't make this error
raises(
ValueError, lambda: solve_linear(f(x) + f(x).diff(x), symbols=[f(x)]))
assert solve_linear(f(x) + f(x).diff(x), symbols=[x]) == \
(f(x) + Derivative(f(x), x), 1)
assert solve_linear(f(x) + Integral(x, (x, y)), symbols=[x]) == \
(f(x) + Integral(x, (x, y)), 1)
assert solve_linear(f(x) + Integral(x, (x, y)) + x, symbols=[x]) == \
(x + f(x) + Integral(x, (x, y)), 1)
assert solve_linear(f(y) + Integral(x, (x, y)) + x, symbols=[x]) == \
(x, -f(y) - Integral(x, (x, y)))
assert solve_linear(x - f(x)/a + (f(x) - 1)/a, symbols=[x]) == \
(x, 1/a)
assert solve_linear(x + Derivative(2*x, x)) == \
(x, -2)
assert solve_linear(x + Integral(x, y), symbols=[x]) == \
(x, 0)
assert solve_linear(x + Integral(x, y) - 2, symbols=[x]) == \
(x, 2/(y + 1))
assert set(solve(x + exp(x)**2, exp(x))) == \
set([-sqrt(-x), sqrt(-x)])
assert solve(x + exp(x), x, implicit=True) == \
[-exp(x)]
assert solve(cos(x) - sin(x), x, implicit=True) == []
assert solve(x - sin(x), x, implicit=True) == \
[sin(x)]
assert solve(x**2 + x - 3, x, implicit=True) == \
[-x**2 + 3]
assert solve(x**2 + x - 3, x**2, implicit=True) == \
[-x + 3]
def test_issue_5912():
assert set(solve(x**2 - x - 0.1, rational=True)) == \
set([S.Half + sqrt(35)/10, -sqrt(35)/10 + S.Half])
ans = solve(x**2 - x - 0.1, rational=False)
assert len(ans) == 2 and all(a.is_Number for a in ans)
ans = solve(x**2 - x - 0.1)
assert len(ans) == 2 and all(a.is_Number for a in ans)
def test_float_handling():
def test(e1, e2):
return len(e1.atoms(Float)) == len(e2.atoms(Float))
assert solve(x - 0.5, rational=True)[0].is_Rational
assert solve(x - 0.5, rational=False)[0].is_Float
assert solve(x - S.Half, rational=False)[0].is_Rational
assert solve(x - 0.5, rational=None)[0].is_Float
assert solve(x - S.Half, rational=None)[0].is_Rational
assert test(nfloat(1 + 2*x), 1.0 + 2.0*x)
for contain in [list, tuple, set]:
ans = nfloat(contain([1 + 2*x]))
assert type(ans) is contain and test(list(ans)[0], 1.0 + 2.0*x)
k, v = list(nfloat({2*x: [1 + 2*x]}).items())[0]
assert test(k, 2*x) and test(v[0], 1.0 + 2.0*x)
assert test(nfloat(cos(2*x)), cos(2.0*x))
assert test(nfloat(3*x**2), 3.0*x**2)
assert test(nfloat(3*x**2, exponent=True), 3.0*x**2.0)
assert test(nfloat(exp(2*x)), exp(2.0*x))
assert test(nfloat(x/3), x/3.0)
assert test(nfloat(x**4 + 2*x + cos(Rational(1, 3)) + 1),
x**4 + 2.0*x + 1.94495694631474)
# don't call nfloat if there is no solution
tot = 100 + c + z + t
assert solve(((.7 + c)/tot - .6, (.2 + z)/tot - .3, t/tot - .1)) == []
def test_check_assumptions():
x = symbols('x', positive=True)
assert solve(x**2 - 1) == [1]
def test_issue_6056():
assert solve(tanh(x + 3)*tanh(x - 3) - 1) == []
assert solve(tanh(x - 1)*tanh(x + 1) + 1) == \
[I*pi*Rational(-3, 4), -I*pi/4, I*pi/4, I*pi*Rational(3, 4)]
assert solve((tanh(x + 3)*tanh(x - 3) + 1)**2) == \
[I*pi*Rational(-3, 4), -I*pi/4, I*pi/4, I*pi*Rational(3, 4)]
def test_issue_5673():
eq = -x + exp(exp(LambertW(log(x)))*LambertW(log(x)))
assert checksol(eq, x, 2) is True
assert checksol(eq, x, 2, numerical=False) is None
def test_exclude():
R, C, Ri, Vout, V1, Vminus, Vplus, s = \
symbols('R, C, Ri, Vout, V1, Vminus, Vplus, s')
Rf = symbols('Rf', positive=True) # to eliminate Rf = 0 soln
eqs = [C*V1*s + Vplus*(-2*C*s - 1/R),
Vminus*(-1/Ri - 1/Rf) + Vout/Rf,
C*Vplus*s + V1*(-C*s - 1/R) + Vout/R,
-Vminus + Vplus]
assert solve(eqs, exclude=s*C*R) == [
{
Rf: Ri*(C*R*s + 1)**2/(C*R*s),
Vminus: Vplus,
V1: 2*Vplus + Vplus/(C*R*s),
Vout: C*R*Vplus*s + 3*Vplus + Vplus/(C*R*s)},
{
Vplus: 0,
Vminus: 0,
V1: 0,
Vout: 0},
]
# TODO: Investigate why currently solution [0] is preferred over [1].
assert solve(eqs, exclude=[Vplus, s, C]) in [[{
Vminus: Vplus,
V1: Vout/2 + Vplus/2 + sqrt((Vout - 5*Vplus)*(Vout - Vplus))/2,
R: (Vout - 3*Vplus - sqrt(Vout**2 - 6*Vout*Vplus + 5*Vplus**2))/(2*C*Vplus*s),
Rf: Ri*(Vout - Vplus)/Vplus,
}, {
Vminus: Vplus,
V1: Vout/2 + Vplus/2 - sqrt((Vout - 5*Vplus)*(Vout - Vplus))/2,
R: (Vout - 3*Vplus + sqrt(Vout**2 - 6*Vout*Vplus + 5*Vplus**2))/(2*C*Vplus*s),
Rf: Ri*(Vout - Vplus)/Vplus,
}], [{
Vminus: Vplus,
Vout: (V1**2 - V1*Vplus - Vplus**2)/(V1 - 2*Vplus),
Rf: Ri*(V1 - Vplus)**2/(Vplus*(V1 - 2*Vplus)),
R: Vplus/(C*s*(V1 - 2*Vplus)),
}]]
def test_high_order_roots():
s = x**5 + 4*x**3 + 3*x**2 + Rational(7, 4)
assert set(solve(s)) == set(Poly(s*4, domain='ZZ').all_roots())
def test_minsolve_linear_system():
def count(dic):
return len([x for x in dic.values() if x == 0])
assert count(solve([x + y + z, y + z + a + t], particular=True, quick=True)) \
== 3
assert count(solve([x + y + z, y + z + a + t], particular=True, quick=False)) \
== 3
assert count(solve([x + y + z, y + z + a], particular=True, quick=True)) == 1
assert count(solve([x + y + z, y + z + a], particular=True, quick=False)) == 2
def test_real_roots():
# cf. issue 6650
x = Symbol('x', real=True)
assert len(solve(x**5 + x**3 + 1)) == 1
def test_issue_6528():
eqs = [
327600995*x**2 - 37869137*x + 1809975124*y**2 - 9998905626,
895613949*x**2 - 273830224*x*y + 530506983*y**2 - 10000000000]
# two expressions encountered are > 1400 ops long so if this hangs
# it is likely because simplification is being done
assert len(solve(eqs, y, x, check=False)) == 4
def test_overdetermined():
x = symbols('x', real=True)
eqs = [Abs(4*x - 7) - 5, Abs(3 - 8*x) - 1]
assert solve(eqs, x) == [(S.Half,)]
assert solve(eqs, x, manual=True) == [(S.Half,)]
assert solve(eqs, x, manual=True, check=False) == [(S.Half,), (S(3),)]
def test_issue_6605():
x = symbols('x')
assert solve(4**(x/2) - 2**(x/3)) == [0, 3*I*pi/log(2)]
# while the first one passed, this one failed
x = symbols('x', real=True)
assert solve(5**(x/2) - 2**(x/3)) == [0]
b = sqrt(6)*sqrt(log(2))/sqrt(log(5))
assert solve(5**(x/2) - 2**(3/x)) == [-b, b]
def test__ispow():
assert _ispow(x**2)
assert not _ispow(x)
assert not _ispow(True)
def test_issue_6644():
eq = -sqrt((m - q)**2 + (-m/(2*q) + S.Half)**2) + sqrt((-m**2/2 - sqrt(
4*m**4 - 4*m**2 + 8*m + 1)/4 - Rational(1, 4))**2 + (m**2/2 - m - sqrt(
4*m**4 - 4*m**2 + 8*m + 1)/4 - Rational(1, 4))**2)
sol = solve(eq, q, simplify=False, check=False)
assert len(sol) == 5
def test_issue_6752():
assert solve([a**2 + a, a - b], [a, b]) == [(-1, -1), (0, 0)]
assert solve([a**2 + a*c, a - b], [a, b]) == [(0, 0), (-c, -c)]
def test_issue_6792():
assert solve(x*(x - 1)**2*(x + 1)*(x**6 - x + 1)) == [
-1, 0, 1, CRootOf(x**6 - x + 1, 0), CRootOf(x**6 - x + 1, 1),
CRootOf(x**6 - x + 1, 2), CRootOf(x**6 - x + 1, 3),
CRootOf(x**6 - x + 1, 4), CRootOf(x**6 - x + 1, 5)]
def test_issues_6819_6820_6821_6248_8692():
# issue 6821
x, y = symbols('x y', real=True)
assert solve(abs(x + 3) - 2*abs(x - 3)) == [1, 9]
assert solve([abs(x) - 2, arg(x) - pi], x) == [(-2,), (2,)]
assert set(solve(abs(x - 7) - 8)) == set([-S.One, S(15)])
# issue 8692
assert solve(Eq(Abs(x + 1) + Abs(x**2 - 7), 9), x) == [
Rational(-1, 2) + sqrt(61)/2, -sqrt(69)/2 + S.Half]
# issue 7145
assert solve(2*abs(x) - abs(x - 1)) == [-1, Rational(1, 3)]
x = symbols('x')
assert solve([re(x) - 1, im(x) - 2], x) == [
{re(x): 1, x: 1 + 2*I, im(x): 2}]
# check for 'dict' handling of solution
eq = sqrt(re(x)**2 + im(x)**2) - 3
assert solve(eq) == solve(eq, x)
i = symbols('i', imaginary=True)
assert solve(abs(i) - 3) == [-3*I, 3*I]
raises(NotImplementedError, lambda: solve(abs(x) - 3))
w = symbols('w', integer=True)
assert solve(2*x**w - 4*y**w, w) == solve((x/y)**w - 2, w)
x, y = symbols('x y', real=True)
assert solve(x + y*I + 3) == {y: 0, x: -3}
# issue 2642
assert solve(x*(1 + I)) == [0]
x, y = symbols('x y', imaginary=True)
assert solve(x + y*I + 3 + 2*I) == {x: -2*I, y: 3*I}
x = symbols('x', real=True)
assert solve(x + y + 3 + 2*I) == {x: -3, y: -2*I}
# issue 6248
f = Function('f')
assert solve(f(x + 1) - f(2*x - 1)) == [2]
assert solve(log(x + 1) - log(2*x - 1)) == [2]
x = symbols('x')
assert solve(2**x + 4**x) == [I*pi/log(2)]
def test_issue_14607():
# issue 14607
s, tau_c, tau_1, tau_2, phi, K = symbols(
's, tau_c, tau_1, tau_2, phi, K')
target = (s**2*tau_1*tau_2 + s*tau_1 + s*tau_2 + 1)/(K*s*(-phi + tau_c))
K_C, tau_I, tau_D = symbols('K_C, tau_I, tau_D',
positive=True, nonzero=True)
PID = K_C*(1 + 1/(tau_I*s) + tau_D*s)
eq = (target - PID).together()
eq *= denom(eq).simplify()
eq = Poly(eq, s)
c = eq.coeffs()
vars = [K_C, tau_I, tau_D]
s = solve(c, vars, dict=True)
assert len(s) == 1
knownsolution = {K_C: -(tau_1 + tau_2)/(K*(phi - tau_c)),
tau_I: tau_1 + tau_2,
tau_D: tau_1*tau_2/(tau_1 + tau_2)}
for var in vars:
assert s[0][var].simplify() == knownsolution[var].simplify()
def test_lambert_multivariate():
from sympy.abc import x, y
assert _filtered_gens(Poly(x + 1/x + exp(x) + y), x) == set([x, exp(x)])
assert _lambert(x, x) == []
assert solve((x**2 - 2*x + 1).subs(x, log(x) + 3*x)) == [LambertW(3*S.Exp1)/3]
assert solve((x**2 - 2*x + 1).subs(x, (log(x) + 3*x)**2 - 1)) == \
[LambertW(3*exp(-sqrt(2)))/3, LambertW(3*exp(sqrt(2)))/3]
assert solve((x**2 - 2*x - 2).subs(x, log(x) + 3*x)) == \
[LambertW(3*exp(1 - sqrt(3)))/3, LambertW(3*exp(1 + sqrt(3)))/3]
eq = (x*exp(x) - 3).subs(x, x*exp(x))
assert solve(eq) == [LambertW(3*exp(-LambertW(3)))]
# coverage test
raises(NotImplementedError, lambda: solve(x - sin(x)*log(y - x), x))
ans = [3, -3*LambertW(-log(3)/3)/log(3)] # 3 and 2.478...
assert solve(x**3 - 3**x, x) == ans
assert set(solve(3*log(x) - x*log(3))) == set(ans)
assert solve(LambertW(2*x) - y, x) == [y*exp(y)/2]
@XFAIL
def test_other_lambert():
assert solve(3*sin(x) - x*sin(3), x) == [3]
assert set(solve(x**a - a**x), x) == set(
[a, -a*LambertW(-log(a)/a)/log(a)])
@slow
def test_lambert_bivariate():
# tests passing current implementation
assert solve((x**2 + x)*exp((x**2 + x)) - 1) == [
Rational(-1, 2) + sqrt(1 + 4*LambertW(1))/2,
Rational(-1, 2) - sqrt(1 + 4*LambertW(1))/2]
assert solve((x**2 + x)*exp((x**2 + x)*2) - 1) == [
Rational(-1, 2) + sqrt(1 + 2*LambertW(2))/2,
Rational(-1, 2) - sqrt(1 + 2*LambertW(2))/2]
assert solve(a/x + exp(x/2), x) == [2*LambertW(-a/2)]
assert solve((a/x + exp(x/2)).diff(x), x) == \
[4*LambertW(-sqrt(2)*sqrt(a)/4), 4*LambertW(sqrt(2)*sqrt(a)/4)]
assert solve((1/x + exp(x/2)).diff(x), x) == \
[4*LambertW(-sqrt(2)/4),
4*LambertW(sqrt(2)/4), # nsimplifies as 2*2**(141/299)*3**(206/299)*5**(205/299)*7**(37/299)/21
4*LambertW(-sqrt(2)/4, -1)]
assert solve(x*log(x) + 3*x + 1, x) == \
[exp(-3 + LambertW(-exp(3)))]
assert solve(-x**2 + 2**x, x) == [2, 4, -2*LambertW(log(2)/2)/log(2)]
assert solve(x**2 - 2**x, x) == [2, 4, -2*LambertW(log(2)/2)/log(2)]
ans = solve(3*x + 5 + 2**(-5*x + 3), x)
assert len(ans) == 1 and ans[0].expand() == \
Rational(-5, 3) + LambertW(-10240*root(2, 3)*log(2)/3)/(5*log(2))
assert solve(5*x - 1 + 3*exp(2 - 7*x), x) == \
[Rational(1, 5) + LambertW(-21*exp(Rational(3, 5))/5)/7]
assert solve((log(x) + x).subs(x, x**2 + 1)) == [
-I*sqrt(-LambertW(1) + 1), sqrt(-1 + LambertW(1))]
# check collection
ax = a**(3*x + 5)
ans = solve(3*log(ax) + b*log(ax) + ax, x)
x0 = 1/log(a)
x1 = sqrt(3)*I
x2 = b + 3
x3 = x2*LambertW(1/x2)/a**5
x4 = x3**Rational(1, 3)/2
assert ans == [
x0*log(x4*(x1 - 1)),
x0*log(-x4*(x1 + 1)),
x0*log(x3)/3]
x1 = LambertW(Rational(1, 3))
x2 = a**(-5)
x3 = 3**Rational(1, 3)
x4 = 3**Rational(5, 6)*I
x5 = x1**Rational(1, 3)*x2**Rational(1, 3)/2
ans = solve(3*log(ax) + ax, x)
assert ans == [
x0*log(3*x1*x2)/3,
x0*log(x5*(-x3 + x4)),
x0*log(-x5*(x3 + x4))]
# coverage
p = symbols('p', positive=True)
eq = 4*2**(2*p + 3) - 2*p - 3
assert _solve_lambert(eq, p, _filtered_gens(Poly(eq), p)) == [
Rational(-3, 2) - LambertW(-4*log(2))/(2*log(2))]
assert set(solve(3**cos(x) - cos(x)**3)) == set(
[acos(3), acos(-3*LambertW(-log(3)/3)/log(3))])
# should give only one solution after using `uniq`
assert solve(2*log(x) - 2*log(z) + log(z + log(x) + log(z)), x) == [
exp(-z + LambertW(2*z**4*exp(2*z))/2)/z]
# cases when p != S.One
# issue 4271
ans = solve((a/x + exp(x/2)).diff(x, 2), x)
x0 = (-a)**Rational(1, 3)
x1 = sqrt(3)*I
x2 = x0/6
assert ans == [
6*LambertW(x0/3),
6*LambertW(x2*(x1 - 1)),
6*LambertW(-x2*(x1 + 1))]
assert solve((1/x + exp(x/2)).diff(x, 2), x) == \
[6*LambertW(Rational(-1, 3)), 6*LambertW(Rational(1, 6) - sqrt(3)*I/6), \
6*LambertW(Rational(1, 6) + sqrt(3)*I/6), 6*LambertW(Rational(-1, 3), -1)]
assert solve(x**2 - y**2/exp(x), x, y, dict=True) == \
[{x: 2*LambertW(-y/2)}, {x: 2*LambertW(y/2)}]
# this is slow but not exceedingly slow
assert solve((x**3)**(x/2) + pi/2, x) == [
exp(LambertW(-2*log(2)/3 + 2*log(pi)/3 + I*pi*Rational(2, 3)))]
def test_rewrite_trig():
assert solve(sin(x) + tan(x)) == [0, -pi, pi, 2*pi]
assert solve(sin(x) + sec(x)) == [
-2*atan(Rational(-1, 2) + sqrt(2)*sqrt(1 - sqrt(3)*I)/2 + sqrt(3)*I/2),
2*atan(S.Half - sqrt(2)*sqrt(1 + sqrt(3)*I)/2 + sqrt(3)*I/2), 2*atan(S.Half
+ sqrt(2)*sqrt(1 + sqrt(3)*I)/2 + sqrt(3)*I/2), 2*atan(S.Half -
sqrt(3)*I/2 + sqrt(2)*sqrt(1 - sqrt(3)*I)/2)]
assert solve(sinh(x) + tanh(x)) == [0, I*pi]
# issue 6157
assert solve(2*sin(x) - cos(x), x) == [atan(S.Half)]
@XFAIL
def test_rewrite_trigh():
# if this import passes then the test below should also pass
from sympy import sech
assert solve(sinh(x) + sech(x)) == [
2*atanh(Rational(-1, 2) + sqrt(5)/2 - sqrt(-2*sqrt(5) + 2)/2),
2*atanh(Rational(-1, 2) + sqrt(5)/2 + sqrt(-2*sqrt(5) + 2)/2),
2*atanh(-sqrt(5)/2 - S.Half + sqrt(2 + 2*sqrt(5))/2),
2*atanh(-sqrt(2 + 2*sqrt(5))/2 - sqrt(5)/2 - S.Half)]
def test_uselogcombine():
eq = z - log(x) + log(y/(x*(-1 + y**2/x**2)))
assert solve(eq, x, force=True) == [-sqrt(y*(y - exp(z))), sqrt(y*(y - exp(z)))]
assert solve(log(x + 3) + log(1 + 3/x) - 3) in [
[-3 + sqrt(-12 + exp(3))*exp(Rational(3, 2))/2 + exp(3)/2,
-sqrt(-12 + exp(3))*exp(Rational(3, 2))/2 - 3 + exp(3)/2],
[-3 + sqrt(-36 + (-exp(3) + 6)**2)/2 + exp(3)/2,
-3 - sqrt(-36 + (-exp(3) + 6)**2)/2 + exp(3)/2],
]
assert solve(log(exp(2*x) + 1) + log(-tanh(x) + 1) - log(2)) == []
def test_atan2():
assert solve(atan2(x, 2) - pi/3, x) == [2*sqrt(3)]
def test_errorinverses():
assert solve(erf(x) - y, x) == [erfinv(y)]
assert solve(erfinv(x) - y, x) == [erf(y)]
assert solve(erfc(x) - y, x) == [erfcinv(y)]
assert solve(erfcinv(x) - y, x) == [erfc(y)]
def test_issue_2725():
R = Symbol('R')
eq = sqrt(2)*R*sqrt(1/(R + 1)) + (R + 1)*(sqrt(2)*sqrt(1/(R + 1)) - 1)
sol = solve(eq, R, set=True)[1]
assert sol == set([(Rational(5, 3) + (Rational(-1, 2) - sqrt(3)*I/2)*(Rational(251, 27) +
sqrt(111)*I/9)**Rational(1, 3) + 40/(9*((Rational(-1, 2) - sqrt(3)*I/2)*(Rational(251, 27) +
sqrt(111)*I/9)**Rational(1, 3))),), (Rational(5, 3) + 40/(9*(Rational(251, 27) +
sqrt(111)*I/9)**Rational(1, 3)) + (Rational(251, 27) + sqrt(111)*I/9)**Rational(1, 3),)])
def test_issue_5114_6611():
# See that it doesn't hang; this solves in about 2 seconds.
# Also check that the solution is relatively small.
# Note: the system in issue 6611 solves in about 5 seconds and has
# an op-count of 138336 (with simplify=False).
b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q, r = symbols('b:r')
eqs = Matrix([
[b - c/d + r/d], [c*(1/g + 1/e + 1/d) - f/g - r/d],
[-c/g + f*(1/j + 1/i + 1/g) - h/i], [-f/i + h*(1/m + 1/l + 1/i) - k/m],
[-h/m + k*(1/p + 1/o + 1/m) - n/p], [-k/p + n*(1/q + 1/p)]])
v = Matrix([f, h, k, n, b, c])
ans = solve(list(eqs), list(v), simplify=False)
# If time is taken to simplify then then 2617 below becomes
# 1168 and the time is about 50 seconds instead of 2.
assert sum([s.count_ops() for s in ans.values()]) <= 3093
def test_det_quick():
m = Matrix(3, 3, symbols('a:9'))
assert m.det() == det_quick(m) # calls det_perm
m[0, 0] = 1
assert m.det() == det_quick(m) # calls det_minor
m = Matrix(3, 3, list(range(9)))
assert m.det() == det_quick(m) # defaults to .det()
# make sure they work with Sparse
s = SparseMatrix(2, 2, (1, 2, 1, 4))
assert det_perm(s) == det_minor(s) == s.det()
def test_real_imag_splitting():
a, b = symbols('a b', real=True)
assert solve(sqrt(a**2 + b**2) - 3, a) == \
[-sqrt(-b**2 + 9), sqrt(-b**2 + 9)]
a, b = symbols('a b', imaginary=True)
assert solve(sqrt(a**2 + b**2) - 3, a) == []
def test_issue_7110():
y = -2*x**3 + 4*x**2 - 2*x + 5
assert any(ask(Q.real(i)) for i in solve(y))
def test_units():
assert solve(1/x - 1/(2*cm)) == [2*cm]
def test_issue_7547():
A, B, V = symbols('A,B,V')
eq1 = Eq(630.26*(V - 39.0)*V*(V + 39) - A + B, 0)
eq2 = Eq(B, 1.36*10**8*(V - 39))
eq3 = Eq(A, 5.75*10**5*V*(V + 39.0))
sol = Matrix(nsolve(Tuple(eq1, eq2, eq3), [A, B, V], (0, 0, 0)))
assert str(sol) == str(Matrix(
[['4442890172.68209'],
['4289299466.1432'],
['70.5389666628177']]))
def test_issue_7895():
r = symbols('r', real=True)
assert solve(sqrt(r) - 2) == [4]
def test_issue_2777():
# the equations represent two circles
x, y = symbols('x y', real=True)
e1, e2 = sqrt(x**2 + y**2) - 10, sqrt(y**2 + (-x + 10)**2) - 3
a, b = Rational(191, 20), 3*sqrt(391)/20
ans = [(a, -b), (a, b)]
assert solve((e1, e2), (x, y)) == ans
assert solve((e1, e2/(x - a)), (x, y)) == []
# make the 2nd circle's radius be -3
e2 += 6
assert solve((e1, e2), (x, y)) == []
assert solve((e1, e2), (x, y), check=False) == ans
def test_issue_7322():
number = 5.62527e-35
assert solve(x - number, x)[0] == number
def test_nsolve():
raises(ValueError, lambda: nsolve(x, (-1, 1), method='bisect'))
raises(TypeError, lambda: nsolve((x - y + 3,x + y,z - y),(x,y,z),(-50,50)))
raises(TypeError, lambda: nsolve((x + y, x - y), (0, 1)))
@slow
def test_high_order_multivariate():
assert len(solve(a*x**3 - x + 1, x)) == 3
assert len(solve(a*x**4 - x + 1, x)) == 4
assert solve(a*x**5 - x + 1, x) == [] # incomplete solution allowed
raises(NotImplementedError, lambda:
solve(a*x**5 - x + 1, x, incomplete=False))
# result checking must always consider the denominator and CRootOf
# must be checked, too
d = x**5 - x + 1
assert solve(d*(1 + 1/d)) == [CRootOf(d + 1, i) for i in range(5)]
d = x - 1
assert solve(d*(2 + 1/d)) == [S.Half]
def test_base_0_exp_0():
assert solve(0**x - 1) == [0]
assert solve(0**(x - 2) - 1) == [2]
assert solve(S('x*(1/x**0 - x)', evaluate=False)) == \
[0, 1]
def test__simple_dens():
assert _simple_dens(1/x**0, [x]) == set()
assert _simple_dens(1/x**y, [x]) == set([x**y])
assert _simple_dens(1/root(x, 3), [x]) == set([x])
def test_issue_8755():
# This tests two things: that if full unrad is attempted and fails
# the solution should still be found; also it tests the use of
# keyword `composite`.
assert len(solve(sqrt(y)*x + x**3 - 1, x)) == 3
assert len(solve(-512*y**3 + 1344*(x + 2)**Rational(1, 3)*y**2 -
1176*(x + 2)**Rational(2, 3)*y - 169*x + 686, y, _unrad=False)) == 3
@slow
def test_issue_8828():
x1 = 0
y1 = -620
r1 = 920
x2 = 126
y2 = 276
x3 = 51
y3 = 205
r3 = 104
v = x, y, z
f1 = (x - x1)**2 + (y - y1)**2 - (r1 - z)**2
f2 = (x2 - x)**2 + (y2 - y)**2 - z**2
f3 = (x - x3)**2 + (y - y3)**2 - (r3 - z)**2
F = f1,f2,f3
g1 = sqrt((x - x1)**2 + (y - y1)**2) + z - r1
g2 = f2
g3 = sqrt((x - x3)**2 + (y - y3)**2) + z - r3
G = g1,g2,g3
A = solve(F, v)
B = solve(G, v)
C = solve(G, v, manual=True)
p, q, r = [set([tuple(i.evalf(2) for i in j) for j in R]) for R in [A, B, C]]
assert p == q == r
@slow
def test_issue_2840_8155():
assert solve(sin(3*x) + sin(6*x)) == [
0, pi*Rational(-5, 3), pi*Rational(-4, 3), -pi, pi*Rational(-2, 3),
pi*Rational(-4, 9), -pi/3, pi*Rational(-2, 9), pi*Rational(2, 9),
pi/3, pi*Rational(4, 9), pi*Rational(2, 3), pi, pi*Rational(4, 3),
pi*Rational(14, 9), pi*Rational(5, 3), pi*Rational(16, 9), 2*pi,
-2*I*log(-(-1)**Rational(1, 9)), -2*I*log(-(-1)**Rational(2, 9)),
-2*I*log(-sin(pi/18) - I*cos(pi/18)),
-2*I*log(-sin(pi/18) + I*cos(pi/18)),
-2*I*log(sin(pi/18) - I*cos(pi/18)),
-2*I*log(sin(pi/18) + I*cos(pi/18))]
assert solve(2*sin(x) - 2*sin(2*x)) == [
0, pi*Rational(-5, 3), -pi, -pi/3, pi/3, pi, pi*Rational(5, 3)]
def test_issue_9567():
assert solve(1 + 1/(x - 1)) == [0]
def test_issue_11538():
assert solve(x + E) == [-E]
assert solve(x**2 + E) == [-I*sqrt(E), I*sqrt(E)]
assert solve(x**3 + 2*E) == [
-cbrt(2 * E),
cbrt(2)*cbrt(E)/2 - cbrt(2)*sqrt(3)*I*cbrt(E)/2,
cbrt(2)*cbrt(E)/2 + cbrt(2)*sqrt(3)*I*cbrt(E)/2]
assert solve([x + 4, y + E], x, y) == {x: -4, y: -E}
assert solve([x**2 + 4, y + E], x, y) == [
(-2*I, -E), (2*I, -E)]
e1 = x - y**3 + 4
e2 = x + y + 4 + 4 * E
assert len(solve([e1, e2], x, y)) == 3
@slow
def test_issue_12114():
a, b, c, d, e, f, g = symbols('a,b,c,d,e,f,g')
terms = [1 + a*b + d*e, 1 + a*c + d*f, 1 + b*c + e*f,
g - a**2 - d**2, g - b**2 - e**2, g - c**2 - f**2]
s = solve(terms, [a, b, c, d, e, f, g], dict=True)
assert s == [{a: -sqrt(-f**2 - 1), b: -sqrt(-f**2 - 1),
c: -sqrt(-f**2 - 1), d: f, e: f, g: -1},
{a: sqrt(-f**2 - 1), b: sqrt(-f**2 - 1),
c: sqrt(-f**2 - 1), d: f, e: f, g: -1},
{a: -sqrt(3)*f/2 - sqrt(-f**2 + 2)/2,
b: sqrt(3)*f/2 - sqrt(-f**2 + 2)/2, c: sqrt(-f**2 + 2),
d: -f/2 + sqrt(-3*f**2 + 6)/2,
e: -f/2 - sqrt(3)*sqrt(-f**2 + 2)/2, g: 2},
{a: -sqrt(3)*f/2 + sqrt(-f**2 + 2)/2,
b: sqrt(3)*f/2 + sqrt(-f**2 + 2)/2, c: -sqrt(-f**2 + 2),
d: -f/2 - sqrt(-3*f**2 + 6)/2,
e: -f/2 + sqrt(3)*sqrt(-f**2 + 2)/2, g: 2},
{a: sqrt(3)*f/2 - sqrt(-f**2 + 2)/2,
b: -sqrt(3)*f/2 - sqrt(-f**2 + 2)/2, c: sqrt(-f**2 + 2),
d: -f/2 - sqrt(-3*f**2 + 6)/2,
e: -f/2 + sqrt(3)*sqrt(-f**2 + 2)/2, g: 2},
{a: sqrt(3)*f/2 + sqrt(-f**2 + 2)/2,
b: -sqrt(3)*f/2 + sqrt(-f**2 + 2)/2, c: -sqrt(-f**2 + 2),
d: -f/2 + sqrt(-3*f**2 + 6)/2,
e: -f/2 - sqrt(3)*sqrt(-f**2 + 2)/2, g: 2}]
def test_inf():
assert solve(1 - oo*x) == []
assert solve(oo*x, x) == []
assert solve(oo*x - oo, x) == []
def test_issue_12448():
f = Function('f')
fun = [f(i) for i in range(15)]
sym = symbols('x:15')
reps = dict(zip(fun, sym))
(x, y, z), c = sym[:3], sym[3:]
ssym = solve([c[4*i]*x + c[4*i + 1]*y + c[4*i + 2]*z + c[4*i + 3]
for i in range(3)], (x, y, z))
(x, y, z), c = fun[:3], fun[3:]
sfun = solve([c[4*i]*x + c[4*i + 1]*y + c[4*i + 2]*z + c[4*i + 3]
for i in range(3)], (x, y, z))
assert sfun[fun[0]].xreplace(reps).count_ops() == \
ssym[sym[0]].count_ops()
def test_denoms():
assert denoms(x/2 + 1/y) == set([2, y])
assert denoms(x/2 + 1/y, y) == set([y])
assert denoms(x/2 + 1/y, [y]) == set([y])
assert denoms(1/x + 1/y + 1/z, [x, y]) == set([x, y])
assert denoms(1/x + 1/y + 1/z, x, y) == set([x, y])
assert denoms(1/x + 1/y + 1/z, set([x, y])) == set([x, y])
def test_issue_12476():
x0, x1, x2, x3, x4, x5 = symbols('x0 x1 x2 x3 x4 x5')
eqns = [x0**2 - x0, x0*x1 - x1, x0*x2 - x2, x0*x3 - x3, x0*x4 - x4, x0*x5 - x5,
x0*x1 - x1, -x0/3 + x1**2 - 2*x2/3, x1*x2 - x1/3 - x2/3 - x3/3,
x1*x3 - x2/3 - x3/3 - x4/3, x1*x4 - 2*x3/3 - x5/3, x1*x5 - x4, x0*x2 - x2,
x1*x2 - x1/3 - x2/3 - x3/3, -x0/6 - x1/6 + x2**2 - x2/6 - x3/3 - x4/6,
-x1/6 + x2*x3 - x2/3 - x3/6 - x4/6 - x5/6, x2*x4 - x2/3 - x3/3 - x4/3,
x2*x5 - x3, x0*x3 - x3, x1*x3 - x2/3 - x3/3 - x4/3,
-x1/6 + x2*x3 - x2/3 - x3/6 - x4/6 - x5/6,
-x0/6 - x1/6 - x2/6 + x3**2 - x3/3 - x4/6, -x1/3 - x2/3 + x3*x4 - x3/3,
-x2 + x3*x5, x0*x4 - x4, x1*x4 - 2*x3/3 - x5/3, x2*x4 - x2/3 - x3/3 - x4/3,
-x1/3 - x2/3 + x3*x4 - x3/3, -x0/3 - 2*x2/3 + x4**2, -x1 + x4*x5, x0*x5 - x5,
x1*x5 - x4, x2*x5 - x3, -x2 + x3*x5, -x1 + x4*x5, -x0 + x5**2, x0 - 1]
sols = [{x0: 1, x3: Rational(1, 6), x2: Rational(1, 6), x4: Rational(-2, 3), x1: Rational(-2, 3), x5: 1},
{x0: 1, x3: S.Half, x2: Rational(-1, 2), x4: 0, x1: 0, x5: -1},
{x0: 1, x3: Rational(-1, 3), x2: Rational(-1, 3), x4: Rational(1, 3), x1: Rational(1, 3), x5: 1},
{x0: 1, x3: 1, x2: 1, x4: 1, x1: 1, x5: 1},
{x0: 1, x3: Rational(-1, 3), x2: Rational(1, 3), x4: sqrt(5)/3, x1: -sqrt(5)/3, x5: -1},
{x0: 1, x3: Rational(-1, 3), x2: Rational(1, 3), x4: -sqrt(5)/3, x1: sqrt(5)/3, x5: -1}]
assert solve(eqns) == sols
def test_issue_13849():
t = symbols('t')
assert solve((t*(sqrt(5) + sqrt(2)) - sqrt(2), t), t) == []
def test_issue_14860():
from sympy.physics.units import newton, kilo
assert solve(8*kilo*newton + x + y, x) == [-8000*newton - y]
def test_issue_14721():
k, h, a, b = symbols(':4')
assert solve([
-1 + (-k + 1)**2/b**2 + (-h - 1)**2/a**2,
-1 + (-k + 1)**2/b**2 + (-h + 1)**2/a**2,
h, k + 2], h, k, a, b) == [
(0, -2, -b*sqrt(1/(b**2 - 9)), b),
(0, -2, b*sqrt(1/(b**2 - 9)), b)]
assert solve([
h, h/a + 1/b**2 - 2, -h/2 + 1/b**2 - 2], a, h, b) == [
(a, 0, -sqrt(2)/2), (a, 0, sqrt(2)/2)]
assert solve((a + b**2 - 1, a + b**2 - 2)) == []
def test_issue_14779():
x = symbols('x', real=True)
assert solve(sqrt(x**4 - 130*x**2 + 1089) + sqrt(x**4 - 130*x**2
+ 3969) - 96*Abs(x)/x,x) == [sqrt(130)]
def test_issue_15307():
assert solve((y - 2, Mul(x + 3,x - 2, evaluate=False))) == \
[{x: -3, y: 2}, {x: 2, y: 2}]
assert solve((y - 2, Mul(3, x - 2, evaluate=False))) == \
{x: 2, y: 2}
assert solve((y - 2, Add(x + 4, x - 2, evaluate=False))) == \
{x: -1, y: 2}
eq1 = Eq(12513*x + 2*y - 219093, -5726*x - y)
eq2 = Eq(-2*x + 8, 2*x - 40)
assert solve([eq1, eq2]) == {x:12, y:75}
def test_issue_15415():
assert solve(x - 3, x) == [3]
assert solve([x - 3], x) == {x:3}
assert solve(Eq(y + 3*x**2/2, y + 3*x), y) == []
assert solve([Eq(y + 3*x**2/2, y + 3*x)], y) == []
assert solve([Eq(y + 3*x**2/2, y + 3*x), Eq(x, 1)], y) == []
@slow
def test_issue_15731():
# f(x)**g(x)=c
assert solve(Eq((x**2 - 7*x + 11)**(x**2 - 13*x + 42), 1)) == [2, 3, 4, 5, 6, 7]
assert solve((x)**(x + 4) - 4) == [-2]
assert solve((-x)**(-x + 4) - 4) == [2]
assert solve((x**2 - 6)**(x**2 - 2) - 4) == [-2, 2]
assert solve((x**2 - 2*x - 1)**(x**2 - 3) - 1/(1 - 2*sqrt(2))) == [sqrt(2)]
assert solve(x**(x + S.Half) - 4*sqrt(2)) == [S(2)]
assert solve((x**2 + 1)**x - 25) == [2]
assert solve(x**(2/x) - 2) == [2, 4]
assert solve((x/2)**(2/x) - sqrt(2)) == [4, 8]
assert solve(x**(x + S.Half) - Rational(9, 4)) == [Rational(3, 2)]
# a**g(x)=c
assert solve((-sqrt(sqrt(2)))**x - 2) == [4, log(2)/(log(2**Rational(1, 4)) + I*pi)]
assert solve((sqrt(2))**x - sqrt(sqrt(2))) == [S.Half]
assert solve((-sqrt(2))**x + 2*(sqrt(2))) == [3,
(3*log(2)**2 + 4*pi**2 - 4*I*pi*log(2))/(log(2)**2 + 4*pi**2)]
assert solve((sqrt(2))**x - 2*(sqrt(2))) == [3]
assert solve(I**x + 1) == [2]
assert solve((1 + I)**x - 2*I) == [2]
assert solve((sqrt(2) + sqrt(3))**x - (2*sqrt(6) + 5)**Rational(1, 3)) == [Rational(2, 3)]
# bases of both sides are equal
b = Symbol('b')
assert solve(b**x - b**2, x) == [2]
assert solve(b**x - 1/b, x) == [-1]
assert solve(b**x - b, x) == [1]
b = Symbol('b', positive=True)
assert solve(b**x - b**2, x) == [2]
assert solve(b**x - 1/b, x) == [-1]
def test_issue_10933():
assert solve(x**4 + y*(x + 0.1), x) # doesn't fail
assert solve(I*x**4 + x**3 + x**2 + 1.) # doesn't fail
def test_Abs_handling():
x = symbols('x', real=True)
assert solve(abs(x/y), x) == [0]
def test_issue_7982():
x = Symbol('x')
# Test that no exception happens
assert solve([2*x**2 + 5*x + 20 <= 0, x >= 1.5], x) is S.false
# From #8040
assert solve([x**3 - 8.08*x**2 - 56.48*x/5 - 106 >= 0, x - 1 <= 0], [x]) is S.false
def test_issue_14645():
x, y = symbols('x y')
assert solve([x*y - x - y, x*y - x - y], [x, y]) == [(y/(y - 1), y)]
def test_issue_12024():
x, y = symbols('x y')
assert solve(Piecewise((0.0, x < 0.1), (x, x >= 0.1)) - y) == \
[{y: Piecewise((0.0, x < 0.1), (x, True))}]
def test_issue_17452():
assert solve((7**x)**x + pi, x) == [-sqrt(log(pi) + I*pi)/sqrt(log(7)),
sqrt(log(pi) + I*pi)/sqrt(log(7))]
assert solve(x**(x/11) + pi/11, x) == [exp(LambertW(-11*log(11) + 11*log(pi) + 11*I*pi))]
def test_issue_17799():
assert solve(-erf(x**(S(1)/3))**pi + I, x) == []
def test_issue_17650():
x = Symbol('x', real=True)
assert solve(abs((abs(x**2 - 1) - x)) - x) == [1, -1 + sqrt(2), 1 + sqrt(2)]
def test_issue_17882():
eq = -8*x**2/(9*(x**2 - 1)**(S(4)/3)) + 4/(3*(x**2 - 1)**(S(1)/3))
assert unrad(eq) == (4*x**2 - 12, [])
def test_issue_17949():
assert solve(exp(+x+x**2), x) == []
assert solve(exp(-x+x**2), x) == []
assert solve(exp(+x-x**2), x) == []
assert solve(exp(-x-x**2), x) == []
def test_issue_10993():
assert solve(Eq(binomial(x, 2), 3)) == [-2, 3]
assert solve(Eq(pow(x, 2) + binomial(x, 3), x)) == [-4, 0, 1]
assert solve(Eq(binomial(x, 2), 0)) == [0, 1]
assert solve(a+binomial(x, 3), a) == [-binomial(x, 3)]
assert solve(x-binomial(a, 3) + binomial(y, 2) + sin(a), x) == [-sin(a) + binomial(a, 3) - binomial(y, 2)]
assert solve((x+1)-binomial(x+1, 3), x) == [-2, -1, 3]
def test_issue_11553():
eq1 = x + y + 1
eq2 = x + GoldenRatio
assert solve([eq1, eq2], x, y) == {x: -GoldenRatio, y: -1 + GoldenRatio}
eq3 = x + 2 + TribonacciConstant
assert solve([eq1, eq3], x, y) == {x: -2 - TribonacciConstant, y: 1 + TribonacciConstant}
def test_issue_19113_19102():
t = S(1)/3
solve(cos(x)**5-sin(x)**5)
assert solve(4*cos(x)**3 - 2*sin(x)**3) == [
atan(2**(t)), -atan(2**(t)*(1 - sqrt(3)*I)/2),
-atan(2**(t)*(1 + sqrt(3)*I)/2)]
h = S.Half
assert solve(cos(x)**2 + sin(x)) == [
2*atan(-h + sqrt(5)/2 + sqrt(2)*sqrt(1 - sqrt(5))/2),
-2*atan(h + sqrt(5)/2 + sqrt(2)*sqrt(1 + sqrt(5))/2),
-2*atan(-sqrt(5)/2 + h + sqrt(2)*sqrt(1 - sqrt(5))/2),
-2*atan(-sqrt(2)*sqrt(1 + sqrt(5))/2 + h + sqrt(5)/2)]
assert solve(3*cos(x) - sin(x)) == [atan(3)]
|
194739571f687792478a01ef77be9c14b4246b92c39a4b979fbce6ad1e2b8a75 | from sympy import (acos, acosh, asinh, atan, cos, Derivative, diff,
Dummy, Eq, Ne, exp, Function, I, Integral, LambertW, log, O, pi,
Rational, rootof, S, sin, sqrt, Subs, Symbol, tan, asin, sinh,
Piecewise, symbols, Poly, sec, re, im, atan2, collect, hyper, integrate)
from sympy.solvers.ode import (classify_ode,
homogeneous_order, infinitesimals, checkinfsol,
dsolve)
from sympy.solvers.ode.subscheck import checkodesol, checksysodesol
from sympy.solvers.ode.ode import (_linear_coeff_match,
_undetermined_coefficients_match, classify_sysode,
constant_renumber, constantsimp, get_numbered_constants, solve_ics)
from sympy.functions import airyai, airybi, besselj, bessely
from sympy.solvers.deutils import ode_order
from sympy.testing.pytest import XFAIL, skip, raises, slow, ON_TRAVIS, SKIP
from sympy.utilities.misc import filldedent
C0, C1, C2, C3, C4, C5, C6, C7, C8, C9, C10 = symbols('C0:11')
u, x, y, z = symbols('u,x:z', real=True)
f = Function('f')
g = Function('g')
h = Function('h')
# Note: the tests below may fail (but still be correct) if ODE solver,
# the integral engine, solve(), or even simplify() changes. Also, in
# differently formatted solutions, the arbitrary constants might not be
# equal. Using specific hints in tests can help to avoid this.
# Tests of order higher than 1 should run the solutions through
# constant_renumber because it will normalize it (constant_renumber causes
# dsolve() to return different results on different machines)
def test_get_numbered_constants():
with raises(ValueError):
get_numbered_constants(None)
def test_dsolve_system():
eqs = [f(x).diff(x, 2), g(x).diff(x)]
with raises(ValueError):
dsolve(eqs) # NotImplementedError would be better
eqs = [f(x).diff(x) - x, f(x).diff(x) + x]
with raises(ValueError):
# Could also be NotImplementedError. f(x)=0 is a solution...
dsolve(eqs)
eqs = [f(x, y).diff(x)]
with raises(ValueError):
dsolve(eqs)
eqs = [f(x, y).diff(x)+g(x).diff(x), g(x).diff(x)]
with raises(ValueError):
dsolve(eqs)
def test_dsolve_all_hint():
eq = f(x).diff(x)
output = dsolve(eq, hint='all')
# Match the Dummy variables:
sol1 = output['separable_Integral']
_y = sol1.lhs.args[1][0]
sol1 = output['1st_homogeneous_coeff_subs_dep_div_indep_Integral']
_u1 = sol1.rhs.args[1].args[1][0]
expected = {'Bernoulli_Integral': Eq(f(x), C1 + Integral(0, x)),
'1st_homogeneous_coeff_best': Eq(f(x), C1),
'Bernoulli': Eq(f(x), C1),
'nth_algebraic': Eq(f(x), C1),
'nth_linear_euler_eq_homogeneous': Eq(f(x), C1),
'nth_linear_constant_coeff_homogeneous': Eq(f(x), C1),
'separable': Eq(f(x), C1),
'1st_homogeneous_coeff_subs_indep_div_dep': Eq(f(x), C1),
'nth_algebraic_Integral': Eq(f(x), C1),
'1st_linear': Eq(f(x), C1),
'1st_linear_Integral': Eq(f(x), C1 + Integral(0, x)),
'lie_group': Eq(f(x), C1),
'1st_homogeneous_coeff_subs_dep_div_indep': Eq(f(x), C1),
'1st_homogeneous_coeff_subs_dep_div_indep_Integral': Eq(log(x), C1 + Integral(-1/_u1, (_u1, f(x)/x))),
'1st_power_series': Eq(f(x), C1),
'separable_Integral': Eq(Integral(1, (_y, f(x))), C1 + Integral(0, x)),
'1st_homogeneous_coeff_subs_indep_div_dep_Integral': Eq(f(x), C1),
'best': Eq(f(x), C1),
'best_hint': 'nth_algebraic',
'default': 'nth_algebraic',
'order': 1}
assert output == expected
assert dsolve(eq, hint='best') == Eq(f(x), C1)
def test_dsolve_ics():
# Maybe this should just use one of the solutions instead of raising...
with raises(NotImplementedError):
dsolve(f(x).diff(x) - sqrt(f(x)), ics={f(1):1})
@slow
@XFAIL
def test_nonlinear_3eq_order1_type1():
if ON_TRAVIS:
skip("Too slow for travis.")
a, b, c = symbols('a b c')
eqs = [
a * f(x).diff(x) - (b - c) * g(x) * h(x),
b * g(x).diff(x) - (c - a) * h(x) * f(x),
c * h(x).diff(x) - (a - b) * f(x) * g(x),
]
assert dsolve(eqs) # NotImplementedError
def test_dsolve_euler_rootof():
eq = x**6 * f(x).diff(x, 6) - x*f(x).diff(x) + f(x)
sol = Eq(f(x),
C1*x
+ C2*x**rootof(x**5 - 14*x**4 + 71*x**3 - 154*x**2 + 120*x - 1, 0)
+ C3*x**rootof(x**5 - 14*x**4 + 71*x**3 - 154*x**2 + 120*x - 1, 1)
+ C4*x**rootof(x**5 - 14*x**4 + 71*x**3 - 154*x**2 + 120*x - 1, 2)
+ C5*x**rootof(x**5 - 14*x**4 + 71*x**3 - 154*x**2 + 120*x - 1, 3)
+ C6*x**rootof(x**5 - 14*x**4 + 71*x**3 - 154*x**2 + 120*x - 1, 4)
)
assert dsolve(eq) == sol
def test_linear_2eq_order1_type2_noninvertible():
# a*d - b*c == 0
eqs = [Eq(diff(f(x), x), f(x) + g(x) + 5),
Eq(diff(g(x), x), f(x) + g(x) + 7)]
sol = [Eq(f(x), C1*exp(2*x) + C2 - x - 3), Eq(g(x), C1*exp(2*x) - C2 + x - 3)]
assert dsolve(eqs) == sol
assert checksysodesol(eqs, sol) == (True, [0, 0])
def test_linear_2eq_order1_type2_fixme():
# There is a FIXME comment about this in the code that handles this case.
# The answer returned is currently incorrect as reported by checksysodesol
# below in the XFAIL test below...
# a*d - b*c == 0 and a + b*c/a = 0
eqs = [Eq(diff(f(x), x), f(x) + g(x) + 5),
Eq(diff(g(x), x), -f(x) - g(x) + 7)]
sol = [Eq(f(x), C1 + C2*(x + 1) + 12*x**2 + 5*x), Eq(g(x), -C1 - C2*x - 12*x**2 + 7*x)]
assert dsolve(eqs) == sol
# FIXME: checked in XFAIL test_linear_2eq_order1_type2_fixme_check below
@XFAIL
def test_linear_2eq_order1_type2_fixme_check():
# See test_linear_2eq_order1_type2_fixme above
eqs = [Eq(diff(f(x), x), f(x) + g(x) + 5),
Eq(diff(g(x), x), -f(x) - g(x) + 7)]
sol = [Eq(f(x), C1 + C2*(x + 1) + 12*x**2 + 5*x), Eq(g(x), -C1 - C2*x - 12*x**2 + 7*x)]
assert checksysodesol(eqs, sol) == (True, [0, 0])
def test_linear_2eq_order1_type6_path1_broken():
eqs = [Eq(diff(f(x), x), f(x) + x*g(x)),
Eq(diff(g(x), x), 2*(1 + 2/x)*f(x) + 2*(x - 1/x) * g(x))]
# FIXME: This is not the correct solution:
sol = [
Eq(f(x), (C1 + Integral(C2*x*exp(-2*Integral(1/x, x))*exp(Integral(-2*x - 1, x)), x))*exp(-Integral(-2*x - 1, x))),
Eq(g(x), C1*exp(-2*Integral(1/x, x))
+ 2*(C1 + Integral(C2*x*exp(-2*Integral(1/x, x))*exp(Integral(-2*x - 1, x)), x))*exp(-Integral(-2*x - 1, x)))
]
dsolve_sol = dsolve(eqs)
# FIXME: Comparing solutions with == doesn't work in this case...
assert [ds.lhs for ds in dsolve_sol] == [f(x), g(x)]
assert [ds.rhs.equals(ss.rhs) for ds, ss in zip(dsolve_sol, sol)]
# FIXME: checked in XFAIL test_linear_2eq_order1_type6_path1_broken_check below
@XFAIL
def test_linear_2eq_order1_type6_path1_broken_check():
# See test_linear_2eq_order1_type6_path1_broken above
eqs = [Eq(diff(f(x), x), f(x) + x*g(x)),
Eq(diff(g(x), x), 2*(1 + 2/x)*f(x) + 2*(x - 1/x) * g(x))]
# FIXME: This is not the correct solution:
sol = [
Eq(f(x), (C1 + Integral(C2*x*exp(-2*Integral(1/x, x))*exp(Integral(-2*x - 1, x)), x))*exp(-Integral(-2*x - 1, x))),
Eq(g(x), C1*exp(-2*Integral(1/x, x))
+ 2*(C1 + Integral(C2*x*exp(-2*Integral(1/x, x))*exp(Integral(-2*x - 1, x)), x))*exp(-Integral(-2*x - 1, x)))
]
assert checksysodesol(eqs, sol) == (True, [0, 0]) # XFAIL
def test_linear_2eq_order1_type6_path2_broken():
# This is the reverse of the equations above and should also be handled by
# type6.
eqs = [Eq(diff(g(x), x), 2*(1 + 2/x)*g(x) + 2*(x - 1/x) * f(x)),
Eq(diff(f(x), x), g(x) + x*f(x))]
# FIXME: This is not the correct solution:
sol = [
Eq(g(x), C1*exp(-2*Integral(1/x, x))
+ 2*(C1 + Integral(-C2*exp(-2*Integral(1/x, x))*exp(Integral(-2*x - 1, x)), x))*exp(-Integral(-2*x - 1, x))),
Eq(f(x), (C1 + Integral(-C2*exp(-2*Integral(1/x, x))*exp(Integral(-2*x - 1, x)), x))*exp(-Integral(-2*x - 1, x)))
]
dsolve_sol = dsolve(eqs)
# Comparing solutions with == doesn't work in this case...
assert [ds.lhs for ds in dsolve_sol] == [g(x), f(x)]
assert [ds.rhs.equals(ss.rhs) for ds, ss in zip(dsolve_sol, sol)]
# FIXME: checked in XFAIL test_linear_2eq_order1_type6_path2_broken_check below
@XFAIL
def test_linear_2eq_order1_type6_path2_broken_check():
# See test_linear_2eq_order1_type6_path2_broken above
eqs = [Eq(diff(g(x), x), 2*(1 + 2/x)*g(x) + 2*(x - 1/x) * f(x)),
Eq(diff(f(x), x), g(x) + x*f(x))]
sol = [
Eq(g(x), C1*exp(-2*Integral(1/x, x))
+ 2*(C1 + Integral(-C2*exp(-2*Integral(1/x, x))*exp(Integral(-2*x - 1, x)), x))*exp(-Integral(-2*x - 1, x))),
Eq(f(x), (C1 + Integral(-C2*exp(-2*Integral(1/x, x))*exp(Integral(-2*x - 1, x)), x))*exp(-Integral(-2*x - 1, x)))
]
assert checksysodesol(eqs, sol) == (True, [0, 0]) # XFAIL
def test_nth_euler_imroot():
eq = x**2 * f(x).diff(x, 2) + x * f(x).diff(x) + 4 * f(x) - 1/x
sol = Eq(f(x), C1*sin(2*log(x)) + C2*cos(2*log(x)) + 1/(5*x))
dsolve_sol = dsolve(eq, hint='nth_linear_euler_eq_nonhomogeneous_variation_of_parameters')
assert dsolve_sol == sol
assert checkodesol(eq, sol, order=2, solve_for_func=False)[0]
def test_constant_coeff_circular_atan2():
eq = f(x).diff(x, x) + y*f(x)
sol = Eq(f(x), C1*exp(-x*sqrt(-y)) + C2*exp(x*sqrt(-y)))
assert dsolve(eq) == sol
assert checkodesol(eq, sol, order=2, solve_for_func=False)[0]
def test_linear_2eq_order2_type1_broken1():
eqs = [Eq(f(x).diff(x, 2), 2*f(x) + g(x)),
Eq(g(x).diff(x, 2), -f(x))]
# FIXME: This is not the correct solution:
sol = [
Eq(f(x), 2*C1*(x + 2)*exp(x) + 2*C2*(x + 2)*exp(-x) + 2*C3*x*exp(x) + 2*C4*x*exp(-x)),
Eq(g(x), -2*C1*x*exp(x) - 2*C2*x*exp(-x) + C3*(-2*x + 4)*exp(x) + C4*(-2*x - 4)*exp(-x))
]
assert dsolve(eqs) == sol
# FIXME: checked in XFAIL test_linear_2eq_order2_type1_broken1_check below
@XFAIL
def test_linear_2eq_order2_type1_broken1_check():
# See test_linear_2eq_order2_type1_broken1 above
eqs = [Eq(f(x).diff(x, 2), 2*f(x) + g(x)),
Eq(g(x).diff(x, 2), -f(x))]
# This is the returned solution but it isn't correct:
sol = [
Eq(f(x), 2*C1*(x + 2)*exp(x) + 2*C2*(x + 2)*exp(-x) + 2*C3*x*exp(x) + 2*C4*x*exp(-x)),
Eq(g(x), -2*C1*x*exp(x) - 2*C2*x*exp(-x) + C3*(-2*x + 4)*exp(x) + C4*(-2*x - 4)*exp(-x))
]
assert checksysodesol(eqs, sol) == (True, [0, 0])
@XFAIL
def test_linear_2eq_order2_type1_broken2():
eqs = [Eq(f(x).diff(x, 2), 0),
Eq(g(x).diff(x, 2), f(x))]
sol = [
Eq(f(x), C1 + C2*x),
Eq(g(x), C4 + C3*x + C2*x**3/6 + C1*x**2/2)
]
assert dsolve(eqs) == sol # UnboundLocalError
def test_linear_2eq_order2_type1_broken2_check():
eqs = [Eq(f(x).diff(x, 2), 0),
Eq(g(x).diff(x, 2), f(x))]
sol = [
Eq(f(x), C1 + C2*x),
Eq(g(x), C4 + C3*x + C2*x**3/6 + C1*x**2/2)
]
assert checksysodesol(eqs, sol) == (True, [0, 0])
def test_linear_2eq_order2_type1():
eqs = [Eq(f(x).diff(x, 2), 2*f(x)),
Eq(g(x).diff(x, 2), -f(x) + 2*g(x))]
sol = [
Eq(f(x), 2*sqrt(2)*C1*exp(sqrt(2)*x) + 2*sqrt(2)*C2*exp(-sqrt(2)*x)),
Eq(g(x), -C1*x*exp(sqrt(2)*x) + C2*x*exp(-sqrt(2)*x) + C3*exp(sqrt(2)*x) + C4*exp(-sqrt(2)*x))
]
assert dsolve(eqs) == sol
assert checksysodesol(eqs, sol) == (True, [0, 0])
eqs = [Eq(f(x).diff(x, 2), 2*f(x) + g(x)),
Eq(g(x).diff(x, 2), + 2*g(x))]
sol = [
Eq(f(x), C1*x*exp(sqrt(2)*x) - C2*x*exp(-sqrt(2)*x) + C3*exp(sqrt(2)*x) + C4*exp(-sqrt(2)*x)),
Eq(g(x), 2*sqrt(2)*C1*exp(sqrt(2)*x) + 2*sqrt(2)*C2*exp(-sqrt(2)*x))
]
assert dsolve(eqs) == sol
assert checksysodesol(eqs, sol) == (True, [0, 0])
eqs = [Eq(f(x).diff(x, 2), f(x)),
Eq(g(x).diff(x, 2), f(x))]
sol = [Eq(f(x), C1*exp(x) + C2*exp(-x)),
Eq(g(x), C1*exp(x) + C2*exp(-x) - C3*x - C4)]
assert dsolve(eqs) == sol
assert checksysodesol(eqs, sol) == (True, [0, 0])
eqs = [Eq(f(x).diff(x, 2), f(x) + g(x)),
Eq(g(x).diff(x, 2), -f(x) - g(x))]
sol = [Eq(f(x), C1*x**3 + C2*x**2 + C3*x + C4),
Eq(g(x), -C1*x**3 + 6*C1*x - C2*x**2 + 2*C2 - C3*x - C4)]
assert dsolve(eqs) == sol
assert checksysodesol(eqs, sol) == (True, [0, 0])
def test_linear_2eq_order2_type2():
eqs = [Eq(f(x).diff(x, 2), f(x) + g(x) + 1),
Eq(g(x).diff(x, 2), f(x) + g(x) + 1)]
sol = [Eq(f(x), C1*exp(sqrt(2)*x) + C2*exp(-sqrt(2)*x) + C3*x + C4 - S.Half),
Eq(g(x), C1*exp(sqrt(2)*x) + C2*exp(-sqrt(2)*x) - C3*x - C4 - S.Half)]
assert dsolve(eqs) == sol
assert checksysodesol(eqs, sol) == (True, [0, 0])
eqs = [Eq(f(x).diff(x, 2), f(x) + g(x) + 1),
Eq(g(x).diff(x, 2), -f(x) - g(x) + 1)]
sol = [Eq(f(x), C1*x**3 + C2*x**2 + C3*x + C4 + x**4/12 + x**2/2),
Eq(g(x), -C1*x**3 + 6*C1*x - C2*x**2 + 2*C2 - C3*x - C4 - x**4/12 + x**2/2)]
assert dsolve(eqs) == sol
assert checksysodesol(eqs, sol) == (True, [0, 0])
def test_linear_2eq_order2_type4_broken():
Ca, Cb, Ra, Rb = symbols('Ca, Cb, Ra, Rb')
eq = [f(x).diff(x, 2) + 2*f(x).diff(x) + f(x) + g(x) - 2*exp(I*x),
g(x).diff(x, 2) + 2*g(x).diff(x) + f(x) + g(x) - 2*exp(I*x)]
# FIXME: This is not the correct solution:
# Solution returned with Ca, Ra etc symbols is clearly incorrect:
sol = [
Eq(f(x), C1 + C2*exp(2*x) + C3*exp(x*(1 + sqrt(3))) + C4*exp(x*(-sqrt(3) + 1)) + (I*Ca + Ra)*exp(I*x)),
Eq(g(x), -C1 - 3*C2*exp(2*x) + C3*(-3*sqrt(3) - 4 + (1 + sqrt(3))**2)*exp(x*(1 + sqrt(3)))
+ C4*(-4 + (-sqrt(3) + 1)**2 + 3*sqrt(3))*exp(x*(-sqrt(3) + 1)) + (I*Cb + Rb)*exp(I*x))
]
dsolve_sol = dsolve(eq)
assert dsolve_sol == sol
# FIXME: checked in XFAIL test_linear_2eq_order2_type4_broken_check below
@XFAIL
def test_linear_2eq_order2_type4_broken_check():
# See test_linear_2eq_order2_type4_broken above
Ca, Cb, Ra, Rb = symbols('Ca, Cb, Ra, Rb')
eq = [f(x).diff(x, 2) + 2*f(x).diff(x) + f(x) + g(x) - 2*exp(I*x),
g(x).diff(x, 2) + 2*g(x).diff(x) + f(x) + g(x) - 2*exp(I*x)]
# Solution returned with Ca, Ra etc symbols is clearly incorrect:
sol = [
Eq(f(x), C1 + C2*exp(2*x) + C3*exp(x*(1 + sqrt(3))) + C4*exp(x*(-sqrt(3) + 1)) + (I*Ca + Ra)*exp(I*x)),
Eq(g(x), -C1 - 3*C2*exp(2*x) + C3*(-3*sqrt(3) - 4 + (1 + sqrt(3))**2)*exp(x*(1 + sqrt(3)))
+ C4*(-4 + (-sqrt(3) + 1)**2 + 3*sqrt(3))*exp(x*(-sqrt(3) + 1)) + (I*Cb + Rb)*exp(I*x))
]
assert checksysodesol(eq, sol) == (True, [0, 0]) # Fails here
def test_linear_2eq_order2_type5():
eqs = [Eq(f(x).diff(x, 2), 2*(x*g(x).diff(x) - g(x))),
Eq(g(x).diff(x, 2),-2*(x*f(x).diff(x) - f(x)))]
sol = [Eq(f(x), C3*x + x*Integral((2*C1*cos(x**2) + 2*C2*sin(x**2))/x**2, x)),
Eq(g(x), C4*x + x*Integral((-2*C1*sin(x**2) + 2*C2*cos(x**2))/x**2, x))]
assert dsolve(eqs) == sol
assert checksysodesol(eqs, sol) == (True, [0, 0])
def test_linear_2eq_order2_type8():
eqs = [Eq(f(x).diff(x, 2), 2/x *(x*g(x).diff(x) - g(x))),
Eq(g(x).diff(x, 2),-2/x *(x*f(x).diff(x) - f(x)))]
# FIXME: This is what is returned but it does not seem correct:
sol = [Eq(f(x), C3*x + x*Integral((-C1*cos(Integral(-2, x)) - C2*sin(Integral(-2, x)))/x**2, x)),
Eq(g(x), C4*x + x*Integral((-C1*sin(Integral(-2, x)) + C2*cos(Integral(-2, x)))/x**2, x))]
assert dsolve(eqs) == sol
assert checksysodesol(eqs, sol) == (True, [0, 0]) # Fails here
@XFAIL
def test_nonlinear_3eq_order1_type4():
eqs = [
Eq(f(x).diff(x), (2*h(x)*g(x) - 3*g(x)*h(x))),
Eq(g(x).diff(x), (4*f(x)*h(x) - 2*h(x)*f(x))),
Eq(h(x).diff(x), (3*g(x)*f(x) - 4*f(x)*g(x))),
]
dsolve(eqs) # KeyError when matching
# sol = ?
# assert dsolve_sol == sol
# assert checksysodesol(eqs, dsolve_sol) == (True, [0, 0, 0])
@slow
@XFAIL
def test_nonlinear_3eq_order1_type3():
if ON_TRAVIS:
skip("Too slow for travis.")
eqs = [
Eq(f(x).diff(x), (2*f(x)**2 - 3 )),
Eq(g(x).diff(x), (4 - 2*h(x) )),
Eq(h(x).diff(x), (3*h(x) - 4*f(x)**2)),
]
dsolve(eqs) # Not sure if this finishes...
# sol = ?
# assert dsolve_sol == sol
# assert checksysodesol(eqs, dsolve_sol) == (True, [0, 0, 0])
@XFAIL
def test_nonlinear_3eq_order1_type5():
eqs = [
Eq(f(x).diff(x), f(x)*(2*f(x) - 3*g(x))),
Eq(g(x).diff(x), g(x)*(4*g(x) - 2*h(x))),
Eq(h(x).diff(x), h(x)*(3*h(x) - 4*f(x))),
]
dsolve(eqs) # KeyError
# sol = ?
# assert dsolve_sol == sol
# assert checksysodesol(eqs, dsolve_sol) == (True, [0, 0, 0])
def test_linear_2eq_order1():
x, y, z = symbols('x, y, z', cls=Function)
k, l, m, n = symbols('k, l, m, n', Integer=True)
t = Symbol('t')
x0, y0 = symbols('x0, y0', cls=Function)
eq1 = (Eq(diff(x(t),t), x(t) + y(t) + 9), Eq(diff(y(t),t), 2*x(t) + 5*y(t) + 23))
sol1 = [Eq(x(t), C1*exp(t*(sqrt(6) + 3)) + C2*exp(t*(-sqrt(6) + 3)) - Rational(22, 3)), \
Eq(y(t), C1*(2 + sqrt(6))*exp(t*(sqrt(6) + 3)) + C2*(-sqrt(6) + 2)*exp(t*(-sqrt(6) + 3)) - Rational(5, 3))]
assert checksysodesol(eq1, sol1) == (True, [0, 0])
eq2 = (Eq(diff(x(t),t), x(t) + y(t) + 81), Eq(diff(y(t),t), -2*x(t) + y(t) + 23))
sol2 = [Eq(x(t), (C1*cos(sqrt(2)*t) + C2*sin(sqrt(2)*t))*exp(t) - Rational(58, 3)), \
Eq(y(t), (-sqrt(2)*C1*sin(sqrt(2)*t) + sqrt(2)*C2*cos(sqrt(2)*t))*exp(t) - Rational(185, 3))]
assert checksysodesol(eq2, sol2) == (True, [0, 0])
eq3 = (Eq(diff(x(t),t), 5*t*x(t) + 2*y(t)), Eq(diff(y(t),t), 2*x(t) + 5*t*y(t)))
sol3 = [Eq(x(t), (C1*exp(2*t) + C2*exp(-2*t))*exp(Rational(5, 2)*t**2)), \
Eq(y(t), (C1*exp(2*t) - C2*exp(-2*t))*exp(Rational(5, 2)*t**2))]
assert checksysodesol(eq3, sol3) == (True, [0, 0])
eq4 = (Eq(diff(x(t),t), 5*t*x(t) + t**2*y(t)), Eq(diff(y(t),t), -t**2*x(t) + 5*t*y(t)))
sol4 = [Eq(x(t), (C1*cos((t**3)/3) + C2*sin((t**3)/3))*exp(Rational(5, 2)*t**2)), \
Eq(y(t), (-C1*sin((t**3)/3) + C2*cos((t**3)/3))*exp(Rational(5, 2)*t**2))]
assert checksysodesol(eq4, sol4) == (True, [0, 0])
eq5 = (Eq(diff(x(t),t), 5*t*x(t) + t**2*y(t)), Eq(diff(y(t),t), -t**2*x(t) + (5*t+9*t**2)*y(t)))
sol5 = [Eq(x(t), (C1*exp((sqrt(77)/2 + Rational(9, 2))*(t**3)/3) + \
C2*exp((-sqrt(77)/2 + Rational(9, 2))*(t**3)/3))*exp(Rational(5, 2)*t**2)), \
Eq(y(t), (C1*(sqrt(77)/2 + Rational(9, 2))*exp((sqrt(77)/2 + Rational(9, 2))*(t**3)/3) + \
C2*(-sqrt(77)/2 + Rational(9, 2))*exp((-sqrt(77)/2 + Rational(9, 2))*(t**3)/3))*exp(Rational(5, 2)*t**2))]
assert checksysodesol(eq5, sol5) == (True, [0, 0])
eq6 = (Eq(diff(x(t),t), 5*t*x(t) + t**2*y(t)), Eq(diff(y(t),t), (1-t**2)*x(t) + (5*t+9*t**2)*y(t)))
sol6 = [Eq(x(t), C1*x0(t) + C2*x0(t)*Integral(t**2*exp(Integral(5*t, t))*exp(Integral(9*t**2 + 5*t, t))/x0(t)**2, t)), \
Eq(y(t), C1*y0(t) + C2*(y0(t)*Integral(t**2*exp(Integral(5*t, t))*exp(Integral(9*t**2 + 5*t, t))/x0(t)**2, t) + \
exp(Integral(5*t, t))*exp(Integral(9*t**2 + 5*t, t))/x0(t)))]
s = dsolve(eq6)
assert s == sol6 # too complicated to test with subs and simplify
# assert checksysodesol(eq10, sol10) == (True, [0, 0]) # this one fails
def test_linear_2eq_order1_nonhomog_linear():
e = [Eq(diff(f(x), x), f(x) + g(x) + 5*x),
Eq(diff(g(x), x), f(x) - g(x))]
raises(NotImplementedError, lambda: dsolve(e))
def test_linear_2eq_order1_nonhomog():
# Note: once implemented, add some tests esp. with resonance
e = [Eq(diff(f(x), x), f(x) + exp(x)),
Eq(diff(g(x), x), f(x) + g(x) + x*exp(x))]
raises(NotImplementedError, lambda: dsolve(e))
def test_linear_2eq_order1_type2_degen():
e = [Eq(diff(f(x), x), f(x) + 5),
Eq(diff(g(x), x), f(x) + 7)]
s1 = [Eq(f(x), C1*exp(x) - 5), Eq(g(x), C1*exp(x) - C2 + 2*x - 5)]
assert checksysodesol(e, s1) == (True, [0, 0])
def test_dsolve_linear_2eq_order1_diag_triangular():
e = [Eq(diff(f(x), x), f(x)),
Eq(diff(g(x), x), g(x))]
s1 = [Eq(f(x), C1*exp(x)), Eq(g(x), C2*exp(x))]
assert checksysodesol(e, s1) == (True, [0, 0])
e = [Eq(diff(f(x), x), 2*f(x)),
Eq(diff(g(x), x), 3*f(x) + 7*g(x))]
s1 = [Eq(f(x), -5*C2*exp(2*x)),
Eq(g(x), 5*C1*exp(7*x) + 3*C2*exp(2*x))]
assert checksysodesol(e, s1) == (True, [0, 0])
def test_sysode_linear_2eq_order1_type1_D_lt_0():
e = [Eq(diff(f(x), x), -9*I*f(x) - 4*g(x)),
Eq(diff(g(x), x), -4*I*g(x))]
s1 = [Eq(f(x), -4*C1*exp(-4*I*x) - 4*C2*exp(-9*I*x)), \
Eq(g(x), 5*I*C1*exp(-4*I*x))]
assert checksysodesol(e, s1) == (True, [0, 0])
def test_sysode_linear_2eq_order1_type1_D_lt_0_b_eq_0():
e = [Eq(diff(f(x), x), -9*I*f(x)),
Eq(diff(g(x), x), -4*I*g(x))]
s1 = [Eq(f(x), -5*I*C2*exp(-9*I*x)), Eq(g(x), 5*I*C1*exp(-4*I*x))]
assert checksysodesol(e, s1) == (True, [0, 0])
def test_sysode_linear_2eq_order1_many_zeros():
t = Symbol('t')
corner_cases = [(0, 0, 0, 0), (1, 0, 0, 0), (0, 1, 0, 0),
(0, 0, 1, 0), (0, 0, 0, 1), (1, 0, 0, I),
(I, 0, 0, -I), (0, I, 0, 0), (0, I, I, 0)]
s1 = [[Eq(f(t), C1), Eq(g(t), C2)],
[Eq(f(t), C1*exp(t)), Eq(g(t), -C2)],
[Eq(f(t), C1 + C2*t), Eq(g(t), C2)],
[Eq(f(t), C2), Eq(g(t), C1 + C2*t)],
[Eq(f(t), -C2), Eq(g(t), C1*exp(t))],
[Eq(f(t), C1*(1 - I)*exp(t)), Eq(g(t), C2*(-1 + I)*exp(I*t))],
[Eq(f(t), 2*I*C1*exp(I*t)), Eq(g(t), -2*I*C2*exp(-I*t))],
[Eq(f(t), I*C1 + I*C2*t), Eq(g(t), C2)],
[Eq(f(t), I*C1*exp(I*t) + I*C2*exp(-I*t)), \
Eq(g(t), I*C1*exp(I*t) - I*C2*exp(-I*t))]
]
for r, sol in zip(corner_cases, s1):
eq = [Eq(diff(f(t), t), r[0]*f(t) + r[1]*g(t)),
Eq(diff(g(t), t), r[2]*f(t) + r[3]*g(t))]
assert checksysodesol(eq, sol) == (True, [0, 0])
def test_dsolve_linsystem_symbol_piecewise():
# example from https://groups.google.com/d/msg/sympy/xmzoqW6tWaE/sf0bgQrlCgAJ
i, r1, c1, r2, c2, t = symbols('i, r1, c1, r2, c2, t')
x1 = Function('x1')
x2 = Function('x2')
eq1 = r1*c1*Derivative(x1(t), t) + x1(t) - x2(t) - r1*i
eq2 = r2*c1*Derivative(x1(t), t) + r2*c2*Derivative(x2(t), t) + x2(t) - r2*i
sol = dsolve((eq1, eq2))
# FIXME: assert checksysodesol(eq, sol) == (True, [0, 0])
# Remove line below when checksysodesol works
assert all(s.has(Piecewise) for s in sol)
@slow
def test_linear_2eq_order2():
x, y, z = symbols('x, y, z', cls=Function)
k, l, m, n = symbols('k, l, m, n', Integer=True)
t, l = symbols('t, l')
x0, y0 = symbols('x0, y0', cls=Function)
eq1 = (Eq(diff(x(t),t,t), 5*x(t) + 43*y(t)), Eq(diff(y(t),t,t), x(t) + 9*y(t)))
sol1 = [Eq(x(t), 43*C1*exp(t*rootof(l**4 - 14*l**2 + 2, 0)) + 43*C2*exp(t*rootof(l**4 - 14*l**2 + 2, 1)) + \
43*C3*exp(t*rootof(l**4 - 14*l**2 + 2, 2)) + 43*C4*exp(t*rootof(l**4 - 14*l**2 + 2, 3))), \
Eq(y(t), C1*(rootof(l**4 - 14*l**2 + 2, 0)**2 - 5)*exp(t*rootof(l**4 - 14*l**2 + 2, 0)) + \
C2*(rootof(l**4 - 14*l**2 + 2, 1)**2 - 5)*exp(t*rootof(l**4 - 14*l**2 + 2, 1)) + \
C3*(rootof(l**4 - 14*l**2 + 2, 2)**2 - 5)*exp(t*rootof(l**4 - 14*l**2 + 2, 2)) + \
C4*(rootof(l**4 - 14*l**2 + 2, 3)**2 - 5)*exp(t*rootof(l**4 - 14*l**2 + 2, 3)))]
assert dsolve(eq1) == sol1
# FIXME: assert checksysodesol(eq1, sol1) == (True, [0, 0]) # this one fails
eq2 = (Eq(diff(x(t),t,t), 8*x(t)+3*y(t)+31), Eq(diff(y(t),t,t), 9*x(t)+7*y(t)+12))
sol2 = [Eq(x(t), 3*C1*exp(t*rootof(l**4 - 15*l**2 + 29, 0)) + 3*C2*exp(t*rootof(l**4 - 15*l**2 + 29, 1)) + \
3*C3*exp(t*rootof(l**4 - 15*l**2 + 29, 2)) + 3*C4*exp(t*rootof(l**4 - 15*l**2 + 29, 3)) - Rational(181, 29)), \
Eq(y(t), C1*(rootof(l**4 - 15*l**2 + 29, 0)**2 - 8)*exp(t*rootof(l**4 - 15*l**2 + 29, 0)) + \
C2*(rootof(l**4 - 15*l**2 + 29, 1)**2 - 8)*exp(t*rootof(l**4 - 15*l**2 + 29, 1)) + \
C3*(rootof(l**4 - 15*l**2 + 29, 2)**2 - 8)*exp(t*rootof(l**4 - 15*l**2 + 29, 2)) + \
C4*(rootof(l**4 - 15*l**2 + 29, 3)**2 - 8)*exp(t*rootof(l**4 - 15*l**2 + 29, 3)) + Rational(183, 29))]
assert dsolve(eq2) == sol2
# FIXME: assert checksysodesol(eq2, sol2) == (True, [0, 0]) # this one fails
eq3 = (Eq(diff(x(t),t,t) - 9*diff(y(t),t) + 7*x(t),0), Eq(diff(y(t),t,t) + 9*diff(x(t),t) + 7*y(t),0))
sol3 = [Eq(x(t), C1*cos(t*(Rational(9, 2) + sqrt(109)/2)) + C2*sin(t*(Rational(9, 2) + sqrt(109)/2)) + C3*cos(t*(-sqrt(109)/2 + Rational(9, 2))) + \
C4*sin(t*(-sqrt(109)/2 + Rational(9, 2)))), Eq(y(t), -C1*sin(t*(Rational(9, 2) + sqrt(109)/2)) + C2*cos(t*(Rational(9, 2) + sqrt(109)/2)) - \
C3*sin(t*(-sqrt(109)/2 + Rational(9, 2))) + C4*cos(t*(-sqrt(109)/2 + Rational(9, 2))))]
assert dsolve(eq3) == sol3
assert checksysodesol(eq3, sol3) == (True, [0, 0])
eq4 = (Eq(diff(x(t),t,t), 9*t*diff(y(t),t)-9*y(t)), Eq(diff(y(t),t,t),7*t*diff(x(t),t)-7*x(t)))
sol4 = [Eq(x(t), C3*t + t*Integral((9*C1*exp(3*sqrt(7)*t**2/2) + 9*C2*exp(-3*sqrt(7)*t**2/2))/t**2, t)), \
Eq(y(t), C4*t + t*Integral((3*sqrt(7)*C1*exp(3*sqrt(7)*t**2/2) - 3*sqrt(7)*C2*exp(-3*sqrt(7)*t**2/2))/t**2, t))]
assert dsolve(eq4) == sol4
assert checksysodesol(eq4, sol4) == (True, [0, 0])
eq5 = (Eq(diff(x(t),t,t), (log(t)+t**2)*diff(x(t),t)+(log(t)+t**2)*3*diff(y(t),t)), Eq(diff(y(t),t,t), \
(log(t)+t**2)*2*diff(x(t),t)+(log(t)+t**2)*9*diff(y(t),t)))
sol5 = [Eq(x(t), -sqrt(22)*(C1*Integral(exp((-sqrt(22) + 5)*Integral(t**2 + log(t), t)), t) + C2 - \
C3*Integral(exp((sqrt(22) + 5)*Integral(t**2 + log(t), t)), t) - C4 - \
(sqrt(22) + 5)*(C1*Integral(exp((-sqrt(22) + 5)*Integral(t**2 + log(t), t)), t) + C2) + \
(-sqrt(22) + 5)*(C3*Integral(exp((sqrt(22) + 5)*Integral(t**2 + log(t), t)), t) + C4))/88), \
Eq(y(t), -sqrt(22)*(C1*Integral(exp((-sqrt(22) + 5)*Integral(t**2 + log(t), t)), t) + \
C2 - C3*Integral(exp((sqrt(22) + 5)*Integral(t**2 + log(t), t)), t) - C4)/44)]
assert dsolve(eq5) == sol5
assert checksysodesol(eq5, sol5) == (True, [0, 0])
eq6 = (Eq(diff(x(t),t,t), log(t)*t*diff(y(t),t) - log(t)*y(t)), Eq(diff(y(t),t,t), log(t)*t*diff(x(t),t) - log(t)*x(t)))
sol6 = [Eq(x(t), C3*t + t*Integral((C1*exp(Integral(t*log(t), t)) + \
C2*exp(-Integral(t*log(t), t)))/t**2, t)), Eq(y(t), C4*t + t*Integral((C1*exp(Integral(t*log(t), t)) - \
C2*exp(-Integral(t*log(t), t)))/t**2, t))]
assert dsolve(eq6) == sol6
assert checksysodesol(eq6, sol6) == (True, [0, 0])
eq7 = (Eq(diff(x(t),t,t), log(t)*(t*diff(x(t),t) - x(t)) + exp(t)*(t*diff(y(t),t) - y(t))), \
Eq(diff(y(t),t,t), (t**2)*(t*diff(x(t),t) - x(t)) + (t)*(t*diff(y(t),t) - y(t))))
sol7 = [Eq(x(t), C3*t + t*Integral((C1*x0(t) + C2*x0(t)*Integral(t*exp(t)*exp(Integral(t**2, t))*\
exp(Integral(t*log(t), t))/x0(t)**2, t))/t**2, t)), Eq(y(t), C4*t + t*Integral((C1*y0(t) + \
C2*(y0(t)*Integral(t*exp(t)*exp(Integral(t**2, t))*exp(Integral(t*log(t), t))/x0(t)**2, t) + \
exp(Integral(t**2, t))*exp(Integral(t*log(t), t))/x0(t)))/t**2, t))]
assert dsolve(eq7) == sol7
# FIXME: assert checksysodesol(eq7, sol7) == (True, [0, 0])
eq8 = (Eq(diff(x(t),t,t), t*(4*x(t) + 9*y(t))), Eq(diff(y(t),t,t), t*(12*x(t) - 6*y(t))))
sol8 = [Eq(x(t), -sqrt(133)*(-4*C1*airyai(t*(-1 + sqrt(133))**(S(1)/3)) + 4*C1*airyai(-t*(1 + \
sqrt(133))**(S(1)/3)) - 4*C2*airybi(t*(-1 + sqrt(133))**(S(1)/3)) + 4*C2*airybi(-t*(1 + sqrt(133))**(S(1)/3)) +\
(-sqrt(133) - 1)*(C1*airyai(t*(-1 + sqrt(133))**(S(1)/3)) + C2*airybi(t*(-1 + sqrt(133))**(S(1)/3))) - (-1 +\
sqrt(133))*(C1*airyai(-t*(1 + sqrt(133))**(S(1)/3)) + C2*airybi(-t*(1 + sqrt(133))**(S(1)/3))))/3192), \
Eq(y(t), -sqrt(133)*(-C1*airyai(t*(-1 + sqrt(133))**(S(1)/3)) + C1*airyai(-t*(1 + sqrt(133))**(S(1)/3)) -\
C2*airybi(t*(-1 + sqrt(133))**(S(1)/3)) + C2*airybi(-t*(1 + sqrt(133))**(S(1)/3)))/266)]
assert dsolve(eq8) == sol8
assert checksysodesol(eq8, sol8) == (True, [0, 0])
assert filldedent(dsolve(eq8)) == filldedent('''
[Eq(x(t), -sqrt(133)*(-4*C1*airyai(t*(-1 + sqrt(133))**(1/3)) +
4*C1*airyai(-t*(1 + sqrt(133))**(1/3)) - 4*C2*airybi(t*(-1 +
sqrt(133))**(1/3)) + 4*C2*airybi(-t*(1 + sqrt(133))**(1/3)) +
(-sqrt(133) - 1)*(C1*airyai(t*(-1 + sqrt(133))**(1/3)) +
C2*airybi(t*(-1 + sqrt(133))**(1/3))) - (-1 +
sqrt(133))*(C1*airyai(-t*(1 + sqrt(133))**(1/3)) + C2*airybi(-t*(1 +
sqrt(133))**(1/3))))/3192), Eq(y(t), -sqrt(133)*(-C1*airyai(t*(-1 +
sqrt(133))**(1/3)) + C1*airyai(-t*(1 + sqrt(133))**(1/3)) -
C2*airybi(t*(-1 + sqrt(133))**(1/3)) + C2*airybi(-t*(1 +
sqrt(133))**(1/3)))/266)]''')
assert checksysodesol(eq8, sol8) == (True, [0, 0])
eq9 = (Eq(diff(x(t),t,t), t*(4*diff(x(t),t) + 9*diff(y(t),t))), Eq(diff(y(t),t,t), t*(12*diff(x(t),t) - 6*diff(y(t),t))))
sol9 = [Eq(x(t), -sqrt(133)*(4*C1*Integral(exp((-sqrt(133) - 1)*Integral(t, t)), t) + 4*C2 - \
4*C3*Integral(exp((-1 + sqrt(133))*Integral(t, t)), t) - 4*C4 - (-1 + sqrt(133))*(C1*Integral(exp((-sqrt(133) - \
1)*Integral(t, t)), t) + C2) + (-sqrt(133) - 1)*(C3*Integral(exp((-1 + sqrt(133))*Integral(t, t)), t) + \
C4))/3192), Eq(y(t), -sqrt(133)*(C1*Integral(exp((-sqrt(133) - 1)*Integral(t, t)), t) + C2 - \
C3*Integral(exp((-1 + sqrt(133))*Integral(t, t)), t) - C4)/266)]
assert dsolve(eq9) == sol9
assert checksysodesol(eq9, sol9) == (True, [0, 0])
eq10 = (t**2*diff(x(t),t,t) + 3*t*diff(x(t),t) + 4*t*diff(y(t),t) + 12*x(t) + 9*y(t), \
t**2*diff(y(t),t,t) + 2*t*diff(x(t),t) - 5*t*diff(y(t),t) + 15*x(t) + 8*y(t))
sol10 = [Eq(x(t), -C1*(-2*sqrt(-346/(3*(Rational(4333, 4) + 5*sqrt(70771857)/36)**Rational(1, 3)) + 4 + 2*(Rational(4333, 4) + \
5*sqrt(70771857)/36)**Rational(1, 3)) + 13 + 2*sqrt(-284/sqrt(-346/(3*(Rational(4333, 4) + 5*sqrt(70771857)/36)**Rational(1, 3)) + \
4 + 2*(Rational(4333, 4) + 5*sqrt(70771857)/36)**Rational(1, 3)) - 2*(Rational(4333, 4) + 5*sqrt(70771857)/36)**Rational(1, 3) + 8 + \
346/(3*(Rational(4333, 4) + 5*sqrt(70771857)/36)**Rational(1, 3))))*exp((-sqrt(-346/(3*(Rational(4333, 4) + 5*sqrt(70771857)/36)**Rational(1, 3)) + \
4 + 2*(Rational(4333, 4) + 5*sqrt(70771857)/36)**Rational(1, 3))/2 + 1 + sqrt(-284/sqrt(-346/(3*(Rational(4333, 4) + \
5*sqrt(70771857)/36)**Rational(1, 3)) + 4 + 2*(Rational(4333, 4) + 5*sqrt(70771857)/36)**Rational(1, 3)) - 2*(Rational(4333, 4) + \
5*sqrt(70771857)/36)**Rational(1, 3) + 8 + 346/(3*(Rational(4333, 4) + 5*sqrt(70771857)/36)**Rational(1, 3)))/2)*log(t)) - \
C2*(-2*sqrt(-346/(3*(Rational(4333, 4) + 5*sqrt(70771857)/36)**Rational(1, 3)) + 4 + 2*(Rational(4333, 4) + 5*sqrt(70771857)/36)**Rational(1, 3)) + \
13 - 2*sqrt(-284/sqrt(-346/(3*(Rational(4333, 4) + 5*sqrt(70771857)/36)**Rational(1, 3)) + 4 + 2*(Rational(4333, 4) + \
5*sqrt(70771857)/36)**Rational(1, 3)) - 2*(Rational(4333, 4) + 5*sqrt(70771857)/36)**Rational(1, 3) + 8 + 346/(3*(Rational(4333, 4) + \
5*sqrt(70771857)/36)**Rational(1, 3))))*exp((-sqrt(-346/(3*(Rational(4333, 4) + 5*sqrt(70771857)/36)**Rational(1, 3)) + 4 + \
2*(Rational(4333, 4) + 5*sqrt(70771857)/36)**Rational(1, 3))/2 + 1 - sqrt(-284/sqrt(-346/(3*(Rational(4333, 4) + 5*sqrt(70771857)/36)**Rational(1, 3)) + \
4 + 2*(Rational(4333, 4) + 5*sqrt(70771857)/36)**Rational(1, 3)) - 2*(Rational(4333, 4) + 5*sqrt(70771857)/36)**Rational(1, 3) + 8 + 346/(3*(Rational(4333, 4) + \
5*sqrt(70771857)/36)**Rational(1, 3)))/2)*log(t)) - C3*t**(1 + sqrt(-346/(3*(Rational(4333, 4) + 5*sqrt(70771857)/36)**Rational(1, 3)) + 4 + \
2*(Rational(4333, 4) + 5*sqrt(70771857)/36)**Rational(1, 3))/2 + sqrt(-2*(Rational(4333, 4) + 5*sqrt(70771857)/36)**Rational(1, 3) + 8 + 346/(3*(Rational(4333, 4) + \
5*sqrt(70771857)/36)**Rational(1, 3)) + 284/sqrt(-346/(3*(Rational(4333, 4) + 5*sqrt(70771857)/36)**Rational(1, 3)) + 4 + 2*(Rational(4333, 4) + \
5*sqrt(70771857)/36)**Rational(1, 3)))/2)*(2*sqrt(-346/(3*(Rational(4333, 4) + 5*sqrt(70771857)/36)**Rational(1, 3)) + 4 + 2*(Rational(4333, 4) + \
5*sqrt(70771857)/36)**Rational(1, 3)) + 13 + 2*sqrt(-2*(Rational(4333, 4) + 5*sqrt(70771857)/36)**Rational(1, 3) + 8 + 346/(3*(Rational(4333, 4) + \
5*sqrt(70771857)/36)**Rational(1, 3)) + 284/sqrt(-346/(3*(Rational(4333, 4) + 5*sqrt(70771857)/36)**Rational(1, 3)) + 4 + 2*(Rational(4333, 4) + \
5*sqrt(70771857)/36)**Rational(1, 3)))) - C4*t**(-sqrt(-2*(Rational(4333, 4) + 5*sqrt(70771857)/36)**Rational(1, 3) + 8 + 346/(3*(Rational(4333, 4) + \
5*sqrt(70771857)/36)**Rational(1, 3)) + 284/sqrt(-346/(3*(Rational(4333, 4) + 5*sqrt(70771857)/36)**Rational(1, 3)) + 4 + 2*(Rational(4333, 4) + \
5*sqrt(70771857)/36)**Rational(1, 3)))/2 + 1 + sqrt(-346/(3*(Rational(4333, 4) + 5*sqrt(70771857)/36)**Rational(1, 3)) + 4 + 2*(Rational(4333, 4) + \
5*sqrt(70771857)/36)**Rational(1, 3))/2)*(-2*sqrt(-2*(Rational(4333, 4) + 5*sqrt(70771857)/36)**Rational(1, 3) + 8 + 346/(3*(Rational(4333, 4) + \
5*sqrt(70771857)/36)**Rational(1, 3)) + 284/sqrt(-346/(3*(Rational(4333, 4) + 5*sqrt(70771857)/36)**Rational(1, 3)) + 4 + 2*(Rational(4333, 4) + \
5*sqrt(70771857)/36)**Rational(1, 3))) + 2*sqrt(-346/(3*(Rational(4333, 4) + 5*sqrt(70771857)/36)**Rational(1, 3)) + 4 + 2*(Rational(4333, 4) + \
5*sqrt(70771857)/36)**Rational(1, 3)) + 13)), Eq(y(t), C1*(-sqrt(-346/(3*(Rational(4333, 4) + 5*sqrt(70771857)/36)**Rational(1, 3)) + 4 + \
2*(Rational(4333, 4) + 5*sqrt(70771857)/36)**Rational(1, 3)) + 14 + (-sqrt(-346/(3*(Rational(4333, 4) + 5*sqrt(70771857)/36)**Rational(1, 3)) + 4 + \
2*(Rational(4333, 4) + 5*sqrt(70771857)/36)**Rational(1, 3))/2 + 1 + sqrt(-284/sqrt(-346/(3*(Rational(4333, 4) + 5*sqrt(70771857)/36)**Rational(1, 3)) + \
4 + 2*(Rational(4333, 4) + 5*sqrt(70771857)/36)**Rational(1, 3)) - 2*(Rational(4333, 4) + 5*sqrt(70771857)/36)**Rational(1, 3) + 8 + 346/(3*(Rational(4333, 4) + \
5*sqrt(70771857)/36)**Rational(1, 3)))/2)**2 + sqrt(-284/sqrt(-346/(3*(Rational(4333, 4) + 5*sqrt(70771857)/36)**Rational(1, 3)) + 4 + \
2*(Rational(4333, 4) + 5*sqrt(70771857)/36)**Rational(1, 3)) - 2*(Rational(4333, 4) + 5*sqrt(70771857)/36)**Rational(1, 3) + 8 + 346/(3*(Rational(4333, 4) + \
5*sqrt(70771857)/36)**Rational(1, 3))))*exp((-sqrt(-346/(3*(Rational(4333, 4) + 5*sqrt(70771857)/36)**Rational(1, 3)) + 4 + 2*(Rational(4333, 4) + \
5*sqrt(70771857)/36)**Rational(1, 3))/2 + 1 + sqrt(-284/sqrt(-346/(3*(Rational(4333, 4) + 5*sqrt(70771857)/36)**Rational(1, 3)) + 4 + \
2*(Rational(4333, 4) + 5*sqrt(70771857)/36)**Rational(1, 3)) - 2*(Rational(4333, 4) + 5*sqrt(70771857)/36)**Rational(1, 3) + 8 + 346/(3*(Rational(4333, 4) + \
5*sqrt(70771857)/36)**Rational(1, 3)))/2)*log(t)) + C2*(-sqrt(-346/(3*(Rational(4333, 4) + 5*sqrt(70771857)/36)**Rational(1, 3)) + 4 + \
2*(Rational(4333, 4) + 5*sqrt(70771857)/36)**Rational(1, 3)) + 14 - sqrt(-284/sqrt(-346/(3*(Rational(4333, 4) + 5*sqrt(70771857)/36)**Rational(1, 3)) + \
4 + 2*(Rational(4333, 4) + 5*sqrt(70771857)/36)**Rational(1, 3)) - 2*(Rational(4333, 4) + 5*sqrt(70771857)/36)**Rational(1, 3) + 8 + 346/(3*(Rational(4333, 4) + \
5*sqrt(70771857)/36)**Rational(1, 3))) + (-sqrt(-346/(3*(Rational(4333, 4) + 5*sqrt(70771857)/36)**Rational(1, 3)) + 4 + 2*(Rational(4333, 4) + \
5*sqrt(70771857)/36)**Rational(1, 3))/2 + 1 - sqrt(-284/sqrt(-346/(3*(Rational(4333, 4) + 5*sqrt(70771857)/36)**Rational(1, 3)) + 4 + \
2*(Rational(4333, 4) + 5*sqrt(70771857)/36)**Rational(1, 3)) - 2*(Rational(4333, 4) + 5*sqrt(70771857)/36)**Rational(1, 3) + 8 + 346/(3*(Rational(4333, 4) + \
5*sqrt(70771857)/36)**Rational(1, 3)))/2)**2)*exp((-sqrt(-346/(3*(Rational(4333, 4) + 5*sqrt(70771857)/36)**Rational(1, 3)) + 4 + 2*(Rational(4333, 4) + \
5*sqrt(70771857)/36)**Rational(1, 3))/2 + 1 - sqrt(-284/sqrt(-346/(3*(Rational(4333, 4) + 5*sqrt(70771857)/36)**Rational(1, 3)) + 4 + \
2*(Rational(4333, 4) + 5*sqrt(70771857)/36)**Rational(1, 3)) - 2*(Rational(4333, 4) + 5*sqrt(70771857)/36)**Rational(1, 3) + 8 + 346/(3*(Rational(4333, 4) + \
5*sqrt(70771857)/36)**Rational(1, 3)))/2)*log(t)) + C3*t**(1 + sqrt(-346/(3*(Rational(4333, 4) + 5*sqrt(70771857)/36)**Rational(1, 3)) + 4 + \
2*(Rational(4333, 4) + 5*sqrt(70771857)/36)**Rational(1, 3))/2 + sqrt(-2*(Rational(4333, 4) + 5*sqrt(70771857)/36)**Rational(1, 3) + 8 + 346/(3*(Rational(4333, 4) + \
5*sqrt(70771857)/36)**Rational(1, 3)) + 284/sqrt(-346/(3*(Rational(4333, 4) + 5*sqrt(70771857)/36)**Rational(1, 3)) + 4 + 2*(Rational(4333, 4) + \
5*sqrt(70771857)/36)**Rational(1, 3)))/2)*(sqrt(-346/(3*(Rational(4333, 4) + 5*sqrt(70771857)/36)**Rational(1, 3)) + 4 + 2*(Rational(4333, 4) + \
5*sqrt(70771857)/36)**Rational(1, 3)) + sqrt(-2*(Rational(4333, 4) + 5*sqrt(70771857)/36)**Rational(1, 3) + 8 + 346/(3*(Rational(4333, 4) + \
5*sqrt(70771857)/36)**Rational(1, 3)) + 284/sqrt(-346/(3*(Rational(4333, 4) + 5*sqrt(70771857)/36)**Rational(1, 3)) + 4 + 2*(Rational(4333, 4) + \
5*sqrt(70771857)/36)**Rational(1, 3))) + 14 + (1 + sqrt(-346/(3*(Rational(4333, 4) + 5*sqrt(70771857)/36)**Rational(1, 3)) + 4 + 2*(Rational(4333, 4) + \
5*sqrt(70771857)/36)**Rational(1, 3))/2 + sqrt(-2*(Rational(4333, 4) + 5*sqrt(70771857)/36)**Rational(1, 3) + 8 + 346/(3*(Rational(4333, 4) + \
5*sqrt(70771857)/36)**Rational(1, 3)) + 284/sqrt(-346/(3*(Rational(4333, 4) + 5*sqrt(70771857)/36)**Rational(1, 3)) + 4 + 2*(Rational(4333, 4) + \
5*sqrt(70771857)/36)**Rational(1, 3)))/2)**2) + C4*t**(-sqrt(-2*(Rational(4333, 4) + 5*sqrt(70771857)/36)**Rational(1, 3) + 8 + \
346/(3*(Rational(4333, 4) + 5*sqrt(70771857)/36)**Rational(1, 3)) + 284/sqrt(-346/(3*(Rational(4333, 4) + 5*sqrt(70771857)/36)**Rational(1, 3)) + \
4 + 2*(Rational(4333, 4) + 5*sqrt(70771857)/36)**Rational(1, 3)))/2 + 1 + sqrt(-346/(3*(Rational(4333, 4) + 5*sqrt(70771857)/36)**Rational(1, 3)) + \
4 + 2*(Rational(4333, 4) + 5*sqrt(70771857)/36)**Rational(1, 3))/2)*(-sqrt(-2*(Rational(4333, 4) + 5*sqrt(70771857)/36)**Rational(1, 3) + \
8 + 346/(3*(Rational(4333, 4) + 5*sqrt(70771857)/36)**Rational(1, 3)) + 284/sqrt(-346/(3*(Rational(4333, 4) + 5*sqrt(70771857)/36)**Rational(1, 3)) + \
4 + 2*(Rational(4333, 4) + 5*sqrt(70771857)/36)**Rational(1, 3))) + (-sqrt(-2*(Rational(4333, 4) + 5*sqrt(70771857)/36)**Rational(1, 3) + 8 + \
346/(3*(Rational(4333, 4) + 5*sqrt(70771857)/36)**Rational(1, 3)) + 284/sqrt(-346/(3*(Rational(4333, 4) + 5*sqrt(70771857)/36)**Rational(1, 3)) + \
4 + 2*(Rational(4333, 4) + 5*sqrt(70771857)/36)**Rational(1, 3)))/2 + 1 + sqrt(-346/(3*(Rational(4333, 4) + 5*sqrt(70771857)/36)**Rational(1, 3)) + \
4 + 2*(Rational(4333, 4) + 5*sqrt(70771857)/36)**Rational(1, 3))/2)**2 + sqrt(-346/(3*(Rational(4333, 4) + \
5*sqrt(70771857)/36)**Rational(1, 3)) + 4 + 2*(Rational(4333, 4) + 5*sqrt(70771857)/36)**Rational(1, 3)) + 14))]
assert dsolve(eq10) == sol10
# FIXME: assert checksysodesol(eq10, sol10) == (True, [0, 0]) # this hangs or at least takes a while...
def test_linear_3eq_order1_nonhomog():
e = [Eq(diff(f(x), x), -9*f(x) - 4*g(x)),
Eq(diff(g(x), x), -4*g(x)),
Eq(diff(h(x), x), h(x) + exp(x))]
raises(NotImplementedError, lambda: dsolve(e))
def test_nonlinear_2eq_order1():
x, y, z = symbols('x, y, z', cls=Function)
t = Symbol('t')
eq1 = (Eq(diff(x(t),t),x(t)*y(t)**3), Eq(diff(y(t),t),y(t)**5))
sol1 = [
Eq(x(t), C1*exp((-1/(4*C2 + 4*t))**(Rational(-1, 4)))),
Eq(y(t), -(-1/(4*C2 + 4*t))**Rational(1, 4)),
Eq(x(t), C1*exp(-1/(-1/(4*C2 + 4*t))**Rational(1, 4))),
Eq(y(t), (-1/(4*C2 + 4*t))**Rational(1, 4)),
Eq(x(t), C1*exp(-I/(-1/(4*C2 + 4*t))**Rational(1, 4))),
Eq(y(t), -I*(-1/(4*C2 + 4*t))**Rational(1, 4)),
Eq(x(t), C1*exp(I/(-1/(4*C2 + 4*t))**Rational(1, 4))),
Eq(y(t), I*(-1/(4*C2 + 4*t))**Rational(1, 4))]
assert dsolve(eq1) == sol1
assert checksysodesol(eq1, sol1) == (True, [0, 0])
eq2 = (Eq(diff(x(t),t), exp(3*x(t))*y(t)**3),Eq(diff(y(t),t), y(t)**5))
sol2 = [
Eq(x(t), -log(C1 - 3/(-1/(4*C2 + 4*t))**Rational(1, 4))/3),
Eq(y(t), -(-1/(4*C2 + 4*t))**Rational(1, 4)),
Eq(x(t), -log(C1 + 3/(-1/(4*C2 + 4*t))**Rational(1, 4))/3),
Eq(y(t), (-1/(4*C2 + 4*t))**Rational(1, 4)),
Eq(x(t), -log(C1 + 3*I/(-1/(4*C2 + 4*t))**Rational(1, 4))/3),
Eq(y(t), -I*(-1/(4*C2 + 4*t))**Rational(1, 4)),
Eq(x(t), -log(C1 - 3*I/(-1/(4*C2 + 4*t))**Rational(1, 4))/3),
Eq(y(t), I*(-1/(4*C2 + 4*t))**Rational(1, 4))]
assert dsolve(eq2) == sol2
assert checksysodesol(eq2, sol2) == (True, [0, 0])
eq3 = (Eq(diff(x(t),t), y(t)*x(t)), Eq(diff(y(t),t), x(t)**3))
tt = Rational(2, 3)
sol3 = [
Eq(x(t), 6**tt/(6*(-sinh(sqrt(C1)*(C2 + t)/2)/sqrt(C1))**tt)),
Eq(y(t), sqrt(C1 + C1/sinh(sqrt(C1)*(C2 + t)/2)**2)/3)]
assert dsolve(eq3) == sol3
# FIXME: assert checksysodesol(eq3, sol3) == (True, [0, 0])
eq4 = (Eq(diff(x(t),t),x(t)*y(t)*sin(t)**2), Eq(diff(y(t),t),y(t)**2*sin(t)**2))
sol4 = set([Eq(x(t), -2*exp(C1)/(C2*exp(C1) + t - sin(2*t)/2)), Eq(y(t), -2/(C1 + t - sin(2*t)/2))])
assert dsolve(eq4) == sol4
# FIXME: assert checksysodesol(eq4, sol4) == (True, [0, 0])
eq5 = (Eq(x(t),t*diff(x(t),t)+diff(x(t),t)*diff(y(t),t)), Eq(y(t),t*diff(y(t),t)+diff(y(t),t)**2))
sol5 = set([Eq(x(t), C1*C2 + C1*t), Eq(y(t), C2**2 + C2*t)])
assert dsolve(eq5) == sol5
assert checksysodesol(eq5, sol5) == (True, [0, 0])
eq6 = (Eq(diff(x(t),t),x(t)**2*y(t)**3), Eq(diff(y(t),t),y(t)**5))
sol6 = [
Eq(x(t), 1/(C1 - 1/(-1/(4*C2 + 4*t))**Rational(1, 4))),
Eq(y(t), -(-1/(4*C2 + 4*t))**Rational(1, 4)),
Eq(x(t), 1/(C1 + (-1/(4*C2 + 4*t))**(Rational(-1, 4)))),
Eq(y(t), (-1/(4*C2 + 4*t))**Rational(1, 4)),
Eq(x(t), 1/(C1 + I/(-1/(4*C2 + 4*t))**Rational(1, 4))),
Eq(y(t), -I*(-1/(4*C2 + 4*t))**Rational(1, 4)),
Eq(x(t), 1/(C1 - I/(-1/(4*C2 + 4*t))**Rational(1, 4))),
Eq(y(t), I*(-1/(4*C2 + 4*t))**Rational(1, 4))]
assert dsolve(eq6) == sol6
assert checksysodesol(eq6, sol6) == (True, [0, 0])
@slow
def test_nonlinear_3eq_order1():
x, y, z = symbols('x, y, z', cls=Function)
t, u = symbols('t u')
eq1 = (4*diff(x(t),t) + 2*y(t)*z(t), 3*diff(y(t),t) - z(t)*x(t), 5*diff(z(t),t) - x(t)*y(t))
sol1 = [Eq(4*Integral(1/(sqrt(-4*u**2 - 3*C1 + C2)*sqrt(-4*u**2 + 5*C1 - C2)), (u, x(t))),
C3 - sqrt(15)*t/15), Eq(3*Integral(1/(sqrt(-6*u**2 - C1 + 5*C2)*sqrt(3*u**2 + C1 - 4*C2)),
(u, y(t))), C3 + sqrt(5)*t/10), Eq(5*Integral(1/(sqrt(-10*u**2 - 3*C1 + C2)*
sqrt(5*u**2 + 4*C1 - C2)), (u, z(t))), C3 + sqrt(3)*t/6)]
assert [i.dummy_eq(j) for i, j in zip(dsolve(eq1), sol1)]
# FIXME: assert checksysodesol(eq1, sol1) == (True, [0, 0, 0])
eq2 = (4*diff(x(t),t) + 2*y(t)*z(t)*sin(t), 3*diff(y(t),t) - z(t)*x(t)*sin(t), 5*diff(z(t),t) - x(t)*y(t)*sin(t))
sol2 = [Eq(3*Integral(1/(sqrt(-6*u**2 - C1 + 5*C2)*sqrt(3*u**2 + C1 - 4*C2)), (u, x(t))), C3 +
sqrt(5)*cos(t)/10), Eq(4*Integral(1/(sqrt(-4*u**2 - 3*C1 + C2)*sqrt(-4*u**2 + 5*C1 - C2)),
(u, y(t))), C3 - sqrt(15)*cos(t)/15), Eq(5*Integral(1/(sqrt(-10*u**2 - 3*C1 + C2)*
sqrt(5*u**2 + 4*C1 - C2)), (u, z(t))), C3 + sqrt(3)*cos(t)/6)]
assert [i.dummy_eq(j) for i, j in zip(dsolve(eq2), sol2)]
# FIXME: assert checksysodesol(eq2, sol2) == (True, [0, 0, 0])
@slow
def test_dsolve_options():
eq = x*f(x).diff(x) + f(x)
a = dsolve(eq, hint='all')
b = dsolve(eq, hint='all', simplify=False)
c = dsolve(eq, hint='all_Integral')
keys = ['1st_exact', '1st_exact_Integral', '1st_homogeneous_coeff_best',
'1st_homogeneous_coeff_subs_dep_div_indep',
'1st_homogeneous_coeff_subs_dep_div_indep_Integral',
'1st_homogeneous_coeff_subs_indep_div_dep',
'1st_homogeneous_coeff_subs_indep_div_dep_Integral', '1st_linear',
'1st_linear_Integral', 'Bernoulli', 'Bernoulli_Integral',
'almost_linear', 'almost_linear_Integral', 'best', 'best_hint',
'default', 'lie_group',
'nth_linear_euler_eq_homogeneous', 'order',
'separable', 'separable_Integral']
Integral_keys = ['1st_exact_Integral',
'1st_homogeneous_coeff_subs_dep_div_indep_Integral',
'1st_homogeneous_coeff_subs_indep_div_dep_Integral', '1st_linear_Integral',
'Bernoulli_Integral', 'almost_linear_Integral', 'best', 'best_hint', 'default',
'nth_linear_euler_eq_homogeneous',
'order', 'separable_Integral']
assert sorted(a.keys()) == keys
assert a['order'] == ode_order(eq, f(x))
assert a['best'] == Eq(f(x), C1/x)
assert dsolve(eq, hint='best') == Eq(f(x), C1/x)
assert a['default'] == 'separable'
assert a['best_hint'] == 'separable'
assert not a['1st_exact'].has(Integral)
assert not a['separable'].has(Integral)
assert not a['1st_homogeneous_coeff_best'].has(Integral)
assert not a['1st_homogeneous_coeff_subs_dep_div_indep'].has(Integral)
assert not a['1st_homogeneous_coeff_subs_indep_div_dep'].has(Integral)
assert not a['1st_linear'].has(Integral)
assert a['1st_linear_Integral'].has(Integral)
assert a['1st_exact_Integral'].has(Integral)
assert a['1st_homogeneous_coeff_subs_dep_div_indep_Integral'].has(Integral)
assert a['1st_homogeneous_coeff_subs_indep_div_dep_Integral'].has(Integral)
assert a['separable_Integral'].has(Integral)
assert sorted(b.keys()) == keys
assert b['order'] == ode_order(eq, f(x))
assert b['best'] == Eq(f(x), C1/x)
assert dsolve(eq, hint='best', simplify=False) == Eq(f(x), C1/x)
assert b['default'] == 'separable'
assert b['best_hint'] == '1st_linear'
assert a['separable'] != b['separable']
assert a['1st_homogeneous_coeff_subs_dep_div_indep'] != \
b['1st_homogeneous_coeff_subs_dep_div_indep']
assert a['1st_homogeneous_coeff_subs_indep_div_dep'] != \
b['1st_homogeneous_coeff_subs_indep_div_dep']
assert not b['1st_exact'].has(Integral)
assert not b['separable'].has(Integral)
assert not b['1st_homogeneous_coeff_best'].has(Integral)
assert not b['1st_homogeneous_coeff_subs_dep_div_indep'].has(Integral)
assert not b['1st_homogeneous_coeff_subs_indep_div_dep'].has(Integral)
assert not b['1st_linear'].has(Integral)
assert b['1st_linear_Integral'].has(Integral)
assert b['1st_exact_Integral'].has(Integral)
assert b['1st_homogeneous_coeff_subs_dep_div_indep_Integral'].has(Integral)
assert b['1st_homogeneous_coeff_subs_indep_div_dep_Integral'].has(Integral)
assert b['separable_Integral'].has(Integral)
assert sorted(c.keys()) == Integral_keys
raises(ValueError, lambda: dsolve(eq, hint='notarealhint'))
raises(ValueError, lambda: dsolve(eq, hint='Liouville'))
assert dsolve(f(x).diff(x) - 1/f(x)**2, hint='all')['best'] == \
dsolve(f(x).diff(x) - 1/f(x)**2, hint='best')
assert dsolve(f(x) + f(x).diff(x) + sin(x).diff(x) + 1, f(x),
hint="1st_linear_Integral") == \
Eq(f(x), (C1 + Integral((-sin(x).diff(x) - 1)*
exp(Integral(1, x)), x))*exp(-Integral(1, x)))
def test_classify_ode():
assert classify_ode(f(x).diff(x, 2), f(x)) == \
(
'nth_algebraic',
'nth_linear_constant_coeff_homogeneous',
'nth_linear_euler_eq_homogeneous',
'Liouville',
'2nd_power_series_ordinary',
'nth_algebraic_Integral',
'Liouville_Integral',
)
assert classify_ode(f(x), f(x)) == ('nth_algebraic', 'nth_algebraic_Integral')
assert classify_ode(Eq(f(x).diff(x), 0), f(x)) == (
'nth_algebraic',
'separable',
'1st_linear',
'Bernoulli',
'1st_homogeneous_coeff_best',
'1st_homogeneous_coeff_subs_indep_div_dep',
'1st_homogeneous_coeff_subs_dep_div_indep',
'1st_power_series', 'lie_group',
'nth_linear_constant_coeff_homogeneous',
'nth_linear_euler_eq_homogeneous',
'nth_algebraic_Integral',
'separable_Integral',
'1st_linear_Integral',
'Bernoulli_Integral',
'1st_homogeneous_coeff_subs_indep_div_dep_Integral',
'1st_homogeneous_coeff_subs_dep_div_indep_Integral')
assert classify_ode(f(x).diff(x)**2, f(x)) == ('factorable',
'nth_algebraic',
'separable',
'1st_linear',
'Bernoulli',
'1st_homogeneous_coeff_best',
'1st_homogeneous_coeff_subs_indep_div_dep',
'1st_homogeneous_coeff_subs_dep_div_indep',
'1st_power_series',
'lie_group',
'nth_linear_constant_coeff_homogeneous',
'nth_linear_euler_eq_homogeneous',
'nth_algebraic_Integral',
'separable_Integral',
'1st_linear_Integral',
'Bernoulli_Integral',
'1st_homogeneous_coeff_subs_indep_div_dep_Integral',
'1st_homogeneous_coeff_subs_dep_div_indep_Integral')
# issue 4749: f(x) should be cleared from highest derivative before classifying
a = classify_ode(Eq(f(x).diff(x) + f(x), x), f(x))
b = classify_ode(f(x).diff(x)*f(x) + f(x)*f(x) - x*f(x), f(x))
c = classify_ode(f(x).diff(x)/f(x) + f(x)/f(x) - x/f(x), f(x))
assert a == ('1st_linear',
'Bernoulli',
'almost_linear',
'1st_power_series', "lie_group",
'nth_linear_constant_coeff_undetermined_coefficients',
'nth_linear_constant_coeff_variation_of_parameters',
'1st_linear_Integral',
'Bernoulli_Integral',
'almost_linear_Integral',
'nth_linear_constant_coeff_variation_of_parameters_Integral')
assert b == ('factorable',
'1st_linear',
'Bernoulli',
'1st_power_series',
'lie_group',
'nth_linear_constant_coeff_undetermined_coefficients',
'nth_linear_constant_coeff_variation_of_parameters',
'1st_linear_Integral',
'Bernoulli_Integral',
'nth_linear_constant_coeff_variation_of_parameters_Integral')
assert c == ('1st_linear',
'Bernoulli',
'1st_power_series',
'lie_group',
'nth_linear_constant_coeff_undetermined_coefficients',
'nth_linear_constant_coeff_variation_of_parameters',
'1st_linear_Integral',
'Bernoulli_Integral',
'nth_linear_constant_coeff_variation_of_parameters_Integral')
assert classify_ode(
2*x*f(x)*f(x).diff(x) + (1 + x)*f(x)**2 - exp(x), f(x)
) == ('Bernoulli', 'almost_linear', 'lie_group',
'Bernoulli_Integral', 'almost_linear_Integral')
assert 'Riccati_special_minus2' in \
classify_ode(2*f(x).diff(x) + f(x)**2 - f(x)/x + 3*x**(-2), f(x))
raises(ValueError, lambda: classify_ode(x + f(x, y).diff(x).diff(
y), f(x, y)))
# issue 5176
k = Symbol('k')
assert classify_ode(f(x).diff(x)/(k*f(x) + k*x*f(x)) + 2*f(x)/(k*f(x) +
k*x*f(x)) + x*f(x).diff(x)/(k*f(x) + k*x*f(x)) + z, f(x)) == \
('separable', '1st_exact', '1st_linear', 'Bernoulli',
'1st_power_series', 'lie_group', 'separable_Integral', '1st_exact_Integral',
'1st_linear_Integral', 'Bernoulli_Integral')
# preprocessing
ans = ('nth_algebraic', 'separable', '1st_exact', '1st_linear', 'Bernoulli',
'1st_homogeneous_coeff_best',
'1st_homogeneous_coeff_subs_indep_div_dep',
'1st_homogeneous_coeff_subs_dep_div_indep',
'1st_power_series', 'lie_group',
'nth_linear_constant_coeff_undetermined_coefficients',
'nth_linear_euler_eq_nonhomogeneous_undetermined_coefficients',
'nth_linear_constant_coeff_variation_of_parameters',
'nth_linear_euler_eq_nonhomogeneous_variation_of_parameters',
'nth_algebraic_Integral',
'separable_Integral', '1st_exact_Integral',
'1st_linear_Integral',
'Bernoulli_Integral',
'1st_homogeneous_coeff_subs_indep_div_dep_Integral',
'1st_homogeneous_coeff_subs_dep_div_indep_Integral',
'nth_linear_constant_coeff_variation_of_parameters_Integral',
'nth_linear_euler_eq_nonhomogeneous_variation_of_parameters_Integral')
# w/o f(x) given
assert classify_ode(diff(f(x) + x, x) + diff(f(x), x)) == ans
# w/ f(x) and prep=True
assert classify_ode(diff(f(x) + x, x) + diff(f(x), x), f(x),
prep=True) == ans
assert classify_ode(Eq(2*x**3*f(x).diff(x), 0), f(x)) == \
('factorable', 'nth_algebraic', 'separable', '1st_linear',
'Bernoulli', '1st_power_series',
'lie_group', 'nth_linear_euler_eq_homogeneous',
'nth_algebraic_Integral', 'separable_Integral',
'1st_linear_Integral', 'Bernoulli_Integral')
assert classify_ode(Eq(2*f(x)**3*f(x).diff(x), 0), f(x)) == \
('factorable', 'nth_algebraic', 'separable', '1st_linear', 'Bernoulli',
'1st_power_series', 'lie_group', 'nth_algebraic_Integral',
'separable_Integral', '1st_linear_Integral', 'Bernoulli_Integral')
# test issue 13864
assert classify_ode(Eq(diff(f(x), x) - f(x)**x, 0), f(x)) == \
('1st_power_series', 'lie_group')
assert isinstance(classify_ode(Eq(f(x), 5), f(x), dict=True), dict)
def test_classify_ode_ics():
# Dummy
eq = f(x).diff(x, x) - f(x)
# Not f(0) or f'(0)
ics = {x: 1}
raises(ValueError, lambda: classify_ode(eq, f(x), ics=ics))
############################
# f(0) type (AppliedUndef) #
############################
# Wrong function
ics = {g(0): 1}
raises(ValueError, lambda: classify_ode(eq, f(x), ics=ics))
# Contains x
ics = {f(x): 1}
raises(ValueError, lambda: classify_ode(eq, f(x), ics=ics))
# Too many args
ics = {f(0, 0): 1}
raises(ValueError, lambda: classify_ode(eq, f(x), ics=ics))
# point contains f
# XXX: Should be NotImplementedError
ics = {f(0): f(1)}
raises(ValueError, lambda: classify_ode(eq, f(x), ics=ics))
# Does not raise
ics = {f(0): 1}
classify_ode(eq, f(x), ics=ics)
#####################
# f'(0) type (Subs) #
#####################
# Wrong function
ics = {g(x).diff(x).subs(x, 0): 1}
raises(ValueError, lambda: classify_ode(eq, f(x), ics=ics))
# Contains x
ics = {f(y).diff(y).subs(y, x): 1}
raises(ValueError, lambda: classify_ode(eq, f(x), ics=ics))
# Wrong variable
ics = {f(y).diff(y).subs(y, 0): 1}
raises(ValueError, lambda: classify_ode(eq, f(x), ics=ics))
# Too many args
ics = {f(x, y).diff(x).subs(x, 0): 1}
raises(ValueError, lambda: classify_ode(eq, f(x), ics=ics))
# Derivative wrt wrong vars
ics = {Derivative(f(x), x, y).subs(x, 0): 1}
raises(ValueError, lambda: classify_ode(eq, f(x), ics=ics))
# point contains f
# XXX: Should be NotImplementedError
ics = {f(x).diff(x).subs(x, 0): f(0)}
raises(ValueError, lambda: classify_ode(eq, f(x), ics=ics))
# Does not raise
ics = {f(x).diff(x).subs(x, 0): 1}
classify_ode(eq, f(x), ics=ics)
###########################
# f'(y) type (Derivative) #
###########################
# Wrong function
ics = {g(x).diff(x).subs(x, y): 1}
raises(ValueError, lambda: classify_ode(eq, f(x), ics=ics))
# Contains x
ics = {f(y).diff(y).subs(y, x): 1}
raises(ValueError, lambda: classify_ode(eq, f(x), ics=ics))
# Too many args
ics = {f(x, y).diff(x).subs(x, y): 1}
raises(ValueError, lambda: classify_ode(eq, f(x), ics=ics))
# Derivative wrt wrong vars
ics = {Derivative(f(x), x, z).subs(x, y): 1}
raises(ValueError, lambda: classify_ode(eq, f(x), ics=ics))
# point contains f
# XXX: Should be NotImplementedError
ics = {f(x).diff(x).subs(x, y): f(0)}
raises(ValueError, lambda: classify_ode(eq, f(x), ics=ics))
# Does not raise
ics = {f(x).diff(x).subs(x, y): 1}
classify_ode(eq, f(x), ics=ics)
def test_classify_sysode():
# Here x is assumed to be x(t) and y as y(t) for simplicity.
# Similarly diff(x,t) and diff(y,y) is assumed to be x1 and y1 respectively.
k, l, m, n = symbols('k, l, m, n', Integer=True)
k1, k2, k3, l1, l2, l3, m1, m2, m3 = symbols('k1, k2, k3, l1, l2, l3, m1, m2, m3', Integer=True)
P, Q, R, p, q, r = symbols('P, Q, R, p, q, r', cls=Function)
P1, P2, P3, Q1, Q2, R1, R2 = symbols('P1, P2, P3, Q1, Q2, R1, R2', cls=Function)
x, y, z = symbols('x, y, z', cls=Function)
t = symbols('t')
x1 = diff(x(t),t) ; y1 = diff(y(t),t) ;
x2 = diff(x(t),t,t) ; y2 = diff(y(t),t,t)
eq2 = (Eq(x2, k*x(t) - l*y1), Eq(y2, l*x1 + k*y(t)))
sol2 = {'order': {y(t): 2, x(t): 2}, 'type_of_equation': 'type3', 'is_linear': True, 'eq': \
[-k*x(t) + l*Derivative(y(t), t) + Derivative(x(t), t, t), -k*y(t) - l*Derivative(x(t), t) + \
Derivative(y(t), t, t)], 'no_of_equation': 2, 'func_coeff': {(0, y(t), 0): 0, (0, x(t), 2): 1, \
(1, y(t), 1): 0, (1, y(t), 2): 1, (1, x(t), 2): 0, (0, y(t), 2): 0, (0, x(t), 0): -k, (1, x(t), 1): \
-l, (0, x(t), 1): 0, (0, y(t), 1): l, (1, x(t), 0): 0, (1, y(t), 0): -k}, 'func': [x(t), y(t)]}
assert classify_sysode(eq2) == sol2
eq3 = (Eq(x2+4*x1+3*y1+9*x(t)+7*y(t), 11*exp(I*t)), Eq(y2+5*x1+8*y1+3*x(t)+12*y(t), 2*exp(I*t)))
sol3 = {'no_of_equation': 2, 'func_coeff': {(1, x(t), 2): 0, (0, y(t), 2): 0, (0, x(t), 0): 9, \
(1, x(t), 1): 5, (0, x(t), 1): 4, (0, y(t), 1): 3, (1, x(t), 0): 3, (1, y(t), 0): 12, (0, y(t), 0): 7, \
(0, x(t), 2): 1, (1, y(t), 2): 1, (1, y(t), 1): 8}, 'type_of_equation': 'type4', 'func': [x(t), y(t)], \
'is_linear': True, 'eq': [9*x(t) + 7*y(t) - 11*exp(I*t) + 4*Derivative(x(t), t) + 3*Derivative(y(t), t) + \
Derivative(x(t), t, t), 3*x(t) + 12*y(t) - 2*exp(I*t) + 5*Derivative(x(t), t) + 8*Derivative(y(t), t) + \
Derivative(y(t), t, t)], 'order': {y(t): 2, x(t): 2}}
assert classify_sysode(eq3) == sol3
eq4 = (Eq((4*t**2 + 7*t + 1)**2*x2, 5*x(t) + 35*y(t)), Eq((4*t**2 + 7*t + 1)**2*y2, x(t) + 9*y(t)))
sol4 = {'no_of_equation': 2, 'func_coeff': {(1, x(t), 2): 0, (0, y(t), 2): 0, (0, x(t), 0): -5, \
(1, x(t), 1): 0, (0, x(t), 1): 0, (0, y(t), 1): 0, (1, x(t), 0): -1, (1, y(t), 0): -9, (0, y(t), 0): -35, \
(0, x(t), 2): 16*t**4 + 56*t**3 + 57*t**2 + 14*t + 1, (1, y(t), 2): 16*t**4 + 56*t**3 + 57*t**2 + 14*t + 1, \
(1, y(t), 1): 0}, 'type_of_equation': 'type10', 'func': [x(t), y(t)], 'is_linear': True, \
'eq': [(4*t**2 + 7*t + 1)**2*Derivative(x(t), t, t) - 5*x(t) - 35*y(t), (4*t**2 + 7*t + 1)**2*Derivative(y(t), t, t)\
- x(t) - 9*y(t)], 'order': {y(t): 2, x(t): 2}}
assert classify_sysode(eq4) == sol4
eq5 = (Eq(diff(x(t),t), x(t) + y(t) + 9), Eq(diff(y(t),t), 2*x(t) + 5*y(t) + 23))
sol5 = {'no_of_equation': 2, 'func_coeff': {(0, x(t), 0): -1, (1, x(t), 1): 0, (0, x(t), 1): 1, (1, y(t), 0): -5, \
(1, x(t), 0): -2, (0, y(t), 1): 0, (0, y(t), 0): -1, (1, y(t), 1): 1}, 'type_of_equation': 'type2', \
'func': [x(t), y(t)], 'is_linear': True, 'eq': [-x(t) - y(t) + Derivative(x(t), t) - 9, -2*x(t) - 5*y(t) + \
Derivative(y(t), t) - 23], 'order': {y(t): 1, x(t): 1}}
assert classify_sysode(eq5) == sol5
eq6 = (Eq(x1, exp(k*x(t))*P(x(t),y(t))), Eq(y1,r(y(t))*P(x(t),y(t))))
sol6 = {'no_of_equation': 2, 'func_coeff': {(0, x(t), 0): 0, (1, x(t), 1): 0, (0, x(t), 1): 1, (1, y(t), 0): 0, \
(1, x(t), 0): 0, (0, y(t), 1): 0, (0, y(t), 0): 0, (1, y(t), 1): 1}, 'type_of_equation': 'type2', 'func': \
[x(t), y(t)], 'is_linear': False, 'eq': [-P(x(t), y(t))*exp(k*x(t)) + Derivative(x(t), t), -P(x(t), \
y(t))*r(y(t)) + Derivative(y(t), t)], 'order': {y(t): 1, x(t): 1}}
assert classify_sysode(eq6) == sol6
eq7 = (Eq(x1, x(t)**2+y(t)/x(t)), Eq(y1, x(t)/y(t)))
sol7 = {'no_of_equation': 2, 'func_coeff': {(0, x(t), 0): 0, (1, x(t), 1): 0, (0, x(t), 1): 1, (1, y(t), 0): 0, \
(1, x(t), 0): -1/y(t), (0, y(t), 1): 0, (0, y(t), 0): -1/x(t), (1, y(t), 1): 1}, 'type_of_equation': 'type3', \
'func': [x(t), y(t)], 'is_linear': False, 'eq': [-x(t)**2 + Derivative(x(t), t) - y(t)/x(t), -x(t)/y(t) + \
Derivative(y(t), t)], 'order': {y(t): 1, x(t): 1}}
assert classify_sysode(eq7) == sol7
eq8 = (Eq(x1, P1(x(t))*Q1(y(t))*R(x(t),y(t),t)), Eq(y1, P1(x(t))*Q1(y(t))*R(x(t),y(t),t)))
sol8 = {'func': [x(t), y(t)], 'is_linear': False, 'type_of_equation': 'type4', 'eq': \
[-P1(x(t))*Q1(y(t))*R(x(t), y(t), t) + Derivative(x(t), t), -P1(x(t))*Q1(y(t))*R(x(t), y(t), t) + \
Derivative(y(t), t)], 'func_coeff': {(0, y(t), 1): 0, (1, y(t), 1): 1, (1, x(t), 1): 0, (0, y(t), 0): 0, \
(1, x(t), 0): 0, (0, x(t), 0): 0, (1, y(t), 0): 0, (0, x(t), 1): 1}, 'order': {y(t): 1, x(t): 1}, 'no_of_equation': 2}
assert classify_sysode(eq8) == sol8
eq10 = (x2 + log(t)*(t*x1 - x(t)) + exp(t)*(t*y1 - y(t)), y2 + (t**2)*(t*x1 - x(t)) + (t)*(t*y1 - y(t)))
sol10 = {'no_of_equation': 2, 'func_coeff': {(1, x(t), 2): 0, (0, y(t), 2): 0, (0, x(t), 0): -log(t), \
(1, x(t), 1): t**3, (0, x(t), 1): t*log(t), (0, y(t), 1): t*exp(t), (1, x(t), 0): -t**2, (1, y(t), 0): -t, \
(0, y(t), 0): -exp(t), (0, x(t), 2): 1, (1, y(t), 2): 1, (1, y(t), 1): t**2}, 'type_of_equation': 'type11', \
'func': [x(t), y(t)], 'is_linear': True, 'eq': [(t*Derivative(x(t), t) - x(t))*log(t) + (t*Derivative(y(t), t) - \
y(t))*exp(t) + Derivative(x(t), t, t), t**2*(t*Derivative(x(t), t) - x(t)) + t*(t*Derivative(y(t), t) - y(t)) \
+ Derivative(y(t), t, t)], 'order': {y(t): 2, x(t): 2}}
assert classify_sysode(eq10) == sol10
eq11 = (Eq(x1,x(t)*y(t)**3), Eq(y1,y(t)**5))
sol11 = {'no_of_equation': 2, 'func_coeff': {(0, x(t), 0): -y(t)**3, (1, x(t), 1): 0, (0, x(t), 1): 1, \
(1, y(t), 0): 0, (1, x(t), 0): 0, (0, y(t), 1): 0, (0, y(t), 0): 0, (1, y(t), 1): 1}, 'type_of_equation': \
'type1', 'func': [x(t), y(t)], 'is_linear': False, 'eq': [-x(t)*y(t)**3 + Derivative(x(t), t), \
-y(t)**5 + Derivative(y(t), t)], 'order': {y(t): 1, x(t): 1}}
assert classify_sysode(eq11) == sol11
eq13 = (Eq(x1,x(t)*y(t)*sin(t)**2), Eq(y1,y(t)**2*sin(t)**2))
sol13 = {'no_of_equation': 2, 'func_coeff': {(0, x(t), 0): -y(t)*sin(t)**2, (1, x(t), 1): 0, (0, x(t), 1): 1, \
(1, y(t), 0): 0, (1, x(t), 0): 0, (0, y(t), 1): 0, (0, y(t), 0): -x(t)*sin(t)**2, (1, y(t), 1): 1}, \
'type_of_equation': 'type4', 'func': [x(t), y(t)], 'is_linear': False, 'eq': [-x(t)*y(t)*sin(t)**2 + \
Derivative(x(t), t), -y(t)**2*sin(t)**2 + Derivative(y(t), t)], 'order': {y(t): 1, x(t): 1}}
assert classify_sysode(eq13) == sol13
def test_solve_ics():
# Basic tests that things work from dsolve.
assert dsolve(f(x).diff(x) - 1/f(x), f(x), ics={f(1): 2}) == \
Eq(f(x), sqrt(2 * x + 2))
assert dsolve(f(x).diff(x) - f(x), f(x), ics={f(0): 1}) == Eq(f(x), exp(x))
assert dsolve(f(x).diff(x) - f(x), f(x), ics={f(x).diff(x).subs(x, 0): 1}) == Eq(f(x), exp(x))
assert dsolve(f(x).diff(x, x) + f(x), f(x), ics={f(0): 1,
f(x).diff(x).subs(x, 0): 1}) == Eq(f(x), sin(x) + cos(x))
assert dsolve([f(x).diff(x) - f(x) + g(x), g(x).diff(x) - g(x) - f(x)],
[f(x), g(x)], ics={f(0): 1, g(0): 0}) == [Eq(f(x), exp(x)*cos(x)), Eq(g(x), exp(x)*sin(x))]
# Test cases where dsolve returns two solutions.
eq = (x**2*f(x)**2 - x).diff(x)
assert dsolve(eq, f(x), ics={f(1): 0}) == [Eq(f(x),
-sqrt(x - 1)/x), Eq(f(x), sqrt(x - 1)/x)]
assert dsolve(eq, f(x), ics={f(x).diff(x).subs(x, 1): 0}) == [Eq(f(x),
-sqrt(x - S.Half)/x), Eq(f(x), sqrt(x - S.Half)/x)]
eq = cos(f(x)) - (x*sin(f(x)) - f(x)**2)*f(x).diff(x)
assert dsolve(eq, f(x),
ics={f(0):1}, hint='1st_exact', simplify=False) == Eq(x*cos(f(x)) + f(x)**3/3, Rational(1, 3))
assert dsolve(eq, f(x),
ics={f(0):1}, hint='1st_exact', simplify=True) == Eq(x*cos(f(x)) + f(x)**3/3, Rational(1, 3))
assert solve_ics([Eq(f(x), C1*exp(x))], [f(x)], [C1], {f(0): 1}) == {C1: 1}
assert solve_ics([Eq(f(x), C1*sin(x) + C2*cos(x))], [f(x)], [C1, C2],
{f(0): 1, f(pi/2): 1}) == {C1: 1, C2: 1}
assert solve_ics([Eq(f(x), C1*sin(x) + C2*cos(x))], [f(x)], [C1, C2],
{f(0): 1, f(x).diff(x).subs(x, 0): 1}) == {C1: 1, C2: 1}
assert solve_ics([Eq(f(x), C1*sin(x) + C2*cos(x))], [f(x)], [C1, C2], {f(0): 1}) == \
{C2: 1}
# Some more complicated tests Refer to PR #16098
assert set(dsolve(f(x).diff(x)*(f(x).diff(x, 2)-x), ics={f(0):0, f(x).diff(x).subs(x, 1):0})) == \
{Eq(f(x), 0), Eq(f(x), x ** 3 / 6 - x / 2)}
assert set(dsolve(f(x).diff(x)*(f(x).diff(x, 2)-x), ics={f(0):0})) == \
{Eq(f(x), 0), Eq(f(x), C2*x + x**3/6)}
K, r, f0 = symbols('K r f0')
sol = Eq(f(x), K*f0*exp(r*x)/((-K + f0)*(f0*exp(r*x)/(-K + f0) - 1)))
assert (dsolve(Eq(f(x).diff(x), r * f(x) * (1 - f(x) / K)), f(x), ics={f(0): f0})) == sol
#Order dependent issues Refer to PR #16098
assert set(dsolve(f(x).diff(x)*(f(x).diff(x, 2)-x), ics={f(x).diff(x).subs(x,0):0, f(0):0})) == \
{Eq(f(x), 0), Eq(f(x), x ** 3 / 6)}
assert set(dsolve(f(x).diff(x)*(f(x).diff(x, 2)-x), ics={f(0):0, f(x).diff(x).subs(x,0):0})) == \
{Eq(f(x), 0), Eq(f(x), x ** 3 / 6)}
# XXX: Ought to be ValueError
raises(ValueError, lambda: solve_ics([Eq(f(x), C1*sin(x) + C2*cos(x))], [f(x)], [C1, C2], {f(0): 1, f(pi): 1}))
# Degenerate case. f'(0) is identically 0.
raises(ValueError, lambda: solve_ics([Eq(f(x), sqrt(C1 - x**2))], [f(x)], [C1], {f(x).diff(x).subs(x, 0): 0}))
EI, q, L = symbols('EI q L')
# eq = Eq(EI*diff(f(x), x, 4), q)
sols = [Eq(f(x), C1 + C2*x + C3*x**2 + C4*x**3 + q*x**4/(24*EI))]
funcs = [f(x)]
constants = [C1, C2, C3, C4]
# Test both cases, Derivative (the default from f(x).diff(x).subs(x, L)),
# and Subs
ics1 = {f(0): 0,
f(x).diff(x).subs(x, 0): 0,
f(L).diff(L, 2): 0,
f(L).diff(L, 3): 0}
ics2 = {f(0): 0,
f(x).diff(x).subs(x, 0): 0,
Subs(f(x).diff(x, 2), x, L): 0,
Subs(f(x).diff(x, 3), x, L): 0}
solved_constants1 = solve_ics(sols, funcs, constants, ics1)
solved_constants2 = solve_ics(sols, funcs, constants, ics2)
assert solved_constants1 == solved_constants2 == {
C1: 0,
C2: 0,
C3: L**2*q/(4*EI),
C4: -L*q/(6*EI)}
def test_ode_order():
f = Function('f')
g = Function('g')
x = Symbol('x')
assert ode_order(3*x*exp(f(x)), f(x)) == 0
assert ode_order(x*diff(f(x), x) + 3*x*f(x) - sin(x)/x, f(x)) == 1
assert ode_order(x**2*f(x).diff(x, x) + x*diff(f(x), x) - f(x), f(x)) == 2
assert ode_order(diff(x*exp(f(x)), x, x), f(x)) == 2
assert ode_order(diff(x*diff(x*exp(f(x)), x, x), x), f(x)) == 3
assert ode_order(diff(f(x), x, x), g(x)) == 0
assert ode_order(diff(f(x), x, x)*diff(g(x), x), f(x)) == 2
assert ode_order(diff(f(x), x, x)*diff(g(x), x), g(x)) == 1
assert ode_order(diff(x*diff(x*exp(f(x)), x, x), x), g(x)) == 0
# issue 5835: ode_order has to also work for unevaluated derivatives
# (ie, without using doit()).
assert ode_order(Derivative(x*f(x), x), f(x)) == 1
assert ode_order(x*sin(Derivative(x*f(x)**2, x, x)), f(x)) == 2
assert ode_order(Derivative(x*Derivative(x*exp(f(x)), x, x), x), g(x)) == 0
assert ode_order(Derivative(f(x), x, x), g(x)) == 0
assert ode_order(Derivative(x*exp(f(x)), x, x), f(x)) == 2
assert ode_order(Derivative(f(x), x, x)*Derivative(g(x), x), g(x)) == 1
assert ode_order(Derivative(x*Derivative(f(x), x, x), x), f(x)) == 3
assert ode_order(
x*sin(Derivative(x*Derivative(f(x), x)**2, x, x)), f(x)) == 3
# In all tests below, checkodesol has the order option set to prevent
# superfluous calls to ode_order(), and the solve_for_func flag set to False
# because dsolve() already tries to solve for the function, unless the
# simplify=False option is set.
def test_old_ode_tests():
# These are simple tests from the old ode module
eq1 = Eq(f(x).diff(x), 0)
eq2 = Eq(3*f(x).diff(x) - 5, 0)
eq3 = Eq(3*f(x).diff(x), 5)
eq4 = Eq(9*f(x).diff(x, x) + f(x), 0)
eq5 = Eq(9*f(x).diff(x, x), f(x))
# Type: a(x)f'(x)+b(x)*f(x)+c(x)=0
eq6 = Eq(x**2*f(x).diff(x) + 3*x*f(x) - sin(x)/x, 0)
eq7 = Eq(f(x).diff(x, x) - 3*diff(f(x), x) + 2*f(x), 0)
# Type: 2nd order, constant coefficients (two real different roots)
eq8 = Eq(f(x).diff(x, x) - 4*diff(f(x), x) + 4*f(x), 0)
# Type: 2nd order, constant coefficients (two real equal roots)
eq9 = Eq(f(x).diff(x, x) + 2*diff(f(x), x) + 3*f(x), 0)
# Type: 2nd order, constant coefficients (two complex roots)
eq10 = Eq(3*f(x).diff(x) - 1, 0)
eq11 = Eq(x*f(x).diff(x) - 1, 0)
sol1 = Eq(f(x), C1)
sol2 = Eq(f(x), C1 + x*Rational(5, 3))
sol3 = Eq(f(x), C1 + x*Rational(5, 3))
sol4 = Eq(f(x), C1*sin(x/3) + C2*cos(x/3))
sol5 = Eq(f(x), C1*exp(-x/3) + C2*exp(x/3))
sol6 = Eq(f(x), (C1 - cos(x))/x**3)
sol7 = Eq(f(x), (C1 + C2*exp(x))*exp(x))
sol8 = Eq(f(x), (C1 + C2*x)*exp(2*x))
sol9 = Eq(f(x), (C1*sin(x*sqrt(2)) + C2*cos(x*sqrt(2)))*exp(-x))
sol10 = Eq(f(x), C1 + x/3)
sol11 = Eq(f(x), C1 + log(x))
assert dsolve(eq1) == sol1
assert dsolve(eq1.lhs) == sol1
assert dsolve(eq2) == sol2
assert dsolve(eq3) == sol3
assert dsolve(eq4) == sol4
assert dsolve(eq5) == sol5
assert dsolve(eq6) == sol6
assert dsolve(eq7) == sol7
assert dsolve(eq8) == sol8
assert dsolve(eq9) == sol9
assert dsolve(eq10) == sol10
assert dsolve(eq11) == sol11
assert checkodesol(eq1, sol1, order=1, solve_for_func=False)[0]
assert checkodesol(eq2, sol2, order=1, solve_for_func=False)[0]
assert checkodesol(eq3, sol3, order=1, solve_for_func=False)[0]
assert checkodesol(eq4, sol4, order=2, solve_for_func=False)[0]
assert checkodesol(eq5, sol5, order=2, solve_for_func=False)[0]
assert checkodesol(eq6, sol6, order=1, solve_for_func=False)[0]
assert checkodesol(eq7, sol7, order=2, solve_for_func=False)[0]
assert checkodesol(eq8, sol8, order=2, solve_for_func=False)[0]
assert checkodesol(eq9, sol9, order=2, solve_for_func=False)[0]
assert checkodesol(eq10, sol10, order=1, solve_for_func=False)[0]
assert checkodesol(eq11, sol11, order=1, solve_for_func=False)[0]
@slow
def test_1st_exact1():
# Type: Exact differential equation, p(x,f) + q(x,f)*f' == 0,
# where dp/df == dq/dx
eq1 = sin(x)*cos(f(x)) + cos(x)*sin(f(x))*f(x).diff(x)
eq2 = (2*x*f(x) + 1)/f(x) + (f(x) - x)/f(x)**2*f(x).diff(x)
eq3 = 2*x + f(x)*cos(x) + (2*f(x) + sin(x) - sin(f(x)))*f(x).diff(x)
eq4 = cos(f(x)) - (x*sin(f(x)) - f(x)**2)*f(x).diff(x)
eq5 = 2*x*f(x) + (x**2 + f(x)**2)*f(x).diff(x)
sol1 = [Eq(f(x), -acos(C1/cos(x)) + 2*pi), Eq(f(x), acos(C1/cos(x)))]
sol2 = Eq(f(x), exp(C1 - x**2 + LambertW(-x*exp(-C1 + x**2))))
sol2b = Eq(log(f(x)) + x/f(x) + x**2, C1)
sol3 = Eq(f(x)*sin(x) + cos(f(x)) + x**2 + f(x)**2, C1)
sol4 = Eq(x*cos(f(x)) + f(x)**3/3, C1)
sol5 = Eq(x**2*f(x) + f(x)**3/3, C1)
assert dsolve(eq1, f(x), hint='1st_exact') == sol1
assert dsolve(eq2, f(x), hint='1st_exact') == sol2
assert dsolve(eq3, f(x), hint='1st_exact') == sol3
assert dsolve(eq4, hint='1st_exact') == sol4
assert dsolve(eq5, hint='1st_exact', simplify=False) == sol5
assert checkodesol(eq1, sol1, order=1, solve_for_func=False)[0]
# issue 5080 blocks the testing of this solution
# FIXME: assert checkodesol(eq2, sol2, order=1, solve_for_func=False)[0]
assert checkodesol(eq2, sol2b, order=1, solve_for_func=False)[0]
assert checkodesol(eq3, sol3, order=1, solve_for_func=False)[0]
assert checkodesol(eq4, sol4, order=1, solve_for_func=False)[0]
assert checkodesol(eq5, sol5, order=1, solve_for_func=False)[0]
@slow
@XFAIL
def test_1st_exact2_broken():
"""
This is an exact equation that fails under the exact engine. It is caught
by first order homogeneous albeit with a much contorted solution. The
exact engine fails because of a poorly simplified integral of q(0,y)dy,
where q is the function multiplying f'. The solutions should be
Eq(sqrt(x**2+f(x)**2)**3+y**3, C1). The equation below is
equivalent, but it is so complex that checkodesol fails, and takes a long
time to do so.
"""
if ON_TRAVIS:
skip("Too slow for travis.")
eq = (x*sqrt(x**2 + f(x)**2) - (x**2*f(x)/(f(x) -
sqrt(x**2 + f(x)**2)))*f(x).diff(x))
sol = Eq(log(x),
C1 - 9*sqrt(1 + f(x)**2/x**2)*asinh(f(x)/x)/(-27*f(x)/x +
27*sqrt(1 + f(x)**2/x**2)) - 9*sqrt(1 + f(x)**2/x**2)*
log(1 - sqrt(1 + f(x)**2/x**2)*f(x)/x + 2*f(x)**2/x**2)/
(-27*f(x)/x + 27*sqrt(1 + f(x)**2/x**2)) +
9*asinh(f(x)/x)*f(x)/(x*(-27*f(x)/x + 27*sqrt(1 + f(x)**2/x**2))) +
9*f(x)*log(1 - sqrt(1 + f(x)**2/x**2)*f(x)/x + 2*f(x)**2/x**2)/
(x*(-27*f(x)/x + 27*sqrt(1 + f(x)**2/x**2))))
assert dsolve(eq) == sol # Slow
# FIXME: Checked in test_1st_exact2_broken_check below
@slow
def test_1st_exact2_broken_check():
# See test_1st_exact2_broken above
eq = (x*sqrt(x**2 + f(x)**2) - (x**2*f(x)/(f(x) -
sqrt(x**2 + f(x)**2)))*f(x).diff(x))
sol = Eq(log(x),
C1 - 9*sqrt(1 + f(x)**2/x**2)*asinh(f(x)/x)/(-27*f(x)/x +
27*sqrt(1 + f(x)**2/x**2)) - 9*sqrt(1 + f(x)**2/x**2)*
log(1 - sqrt(1 + f(x)**2/x**2)*f(x)/x + 2*f(x)**2/x**2)/
(-27*f(x)/x + 27*sqrt(1 + f(x)**2/x**2)) +
9*asinh(f(x)/x)*f(x)/(x*(-27*f(x)/x + 27*sqrt(1 + f(x)**2/x**2))) +
9*f(x)*log(1 - sqrt(1 + f(x)**2/x**2)*f(x)/x + 2*f(x)**2/x**2)/
(x*(-27*f(x)/x + 27*sqrt(1 + f(x)**2/x**2))))
assert checkodesol(eq, sol, order=1, solve_for_func=False)[0]
def test_homogeneous_order():
assert homogeneous_order(exp(y/x) + tan(y/x), x, y) == 0
assert homogeneous_order(x**2 + sin(x)*cos(y), x, y) is None
assert homogeneous_order(x - y - x*sin(y/x), x, y) == 1
assert homogeneous_order((x*y + sqrt(x**4 + y**4) + x**2*(log(x) - log(y)))/
(pi*x**Rational(2, 3)*sqrt(y)**3), x, y) == Rational(-1, 6)
assert homogeneous_order(y/x*cos(y/x) - x/y*sin(y/x) + cos(y/x), x, y) == 0
assert homogeneous_order(f(x), x, f(x)) == 1
assert homogeneous_order(f(x)**2, x, f(x)) == 2
assert homogeneous_order(x*y*z, x, y) == 2
assert homogeneous_order(x*y*z, x, y, z) == 3
assert homogeneous_order(x**2*f(x)/sqrt(x**2 + f(x)**2), f(x)) is None
assert homogeneous_order(f(x, y)**2, x, f(x, y), y) == 2
assert homogeneous_order(f(x, y)**2, x, f(x), y) is None
assert homogeneous_order(f(x, y)**2, x, f(x, y)) is None
assert homogeneous_order(f(y, x)**2, x, y, f(x, y)) is None
assert homogeneous_order(f(y), f(x), x) is None
assert homogeneous_order(-f(x)/x + 1/sin(f(x)/ x), f(x), x) == 0
assert homogeneous_order(log(1/y) + log(x**2), x, y) is None
assert homogeneous_order(log(1/y) + log(x), x, y) == 0
assert homogeneous_order(log(x/y), x, y) == 0
assert homogeneous_order(2*log(1/y) + 2*log(x), x, y) == 0
a = Symbol('a')
assert homogeneous_order(a*log(1/y) + a*log(x), x, y) == 0
assert homogeneous_order(f(x).diff(x), x, y) is None
assert homogeneous_order(-f(x).diff(x) + x, x, y) is None
assert homogeneous_order(O(x), x, y) is None
assert homogeneous_order(x + O(x**2), x, y) is None
assert homogeneous_order(x**pi, x) == pi
assert homogeneous_order(x**x, x) is None
raises(ValueError, lambda: homogeneous_order(x*y))
@slow
def test_1st_homogeneous_coeff_ode():
# Type: First order homogeneous, y'=f(y/x)
eq1 = f(x)/x*cos(f(x)/x) - (x/f(x)*sin(f(x)/x) + cos(f(x)/x))*f(x).diff(x)
eq2 = x*f(x).diff(x) - f(x) - x*sin(f(x)/x)
eq3 = f(x) + (x*log(f(x)/x) - 2*x)*diff(f(x), x)
eq4 = 2*f(x)*exp(x/f(x)) + f(x)*f(x).diff(x) - 2*x*exp(x/f(x))*f(x).diff(x)
eq5 = 2*x**2*f(x) + f(x)**3 + (x*f(x)**2 - 2*x**3)*f(x).diff(x)
eq6 = x*exp(f(x)/x) - f(x)*sin(f(x)/x) + x*sin(f(x)/x)*f(x).diff(x)
eq7 = (x + sqrt(f(x)**2 - x*f(x)))*f(x).diff(x) - f(x)
eq8 = x + f(x) - (x - f(x))*f(x).diff(x)
sol1 = Eq(log(x), C1 - log(f(x)*sin(f(x)/x)/x))
sol2 = Eq(log(x), log(C1) + log(cos(f(x)/x) - 1)/2 - log(cos(f(x)/x) + 1)/2)
sol3 = Eq(f(x), -exp(C1)*LambertW(-x*exp(-C1 + 1)))
sol4 = Eq(log(f(x)), C1 - 2*exp(x/f(x)))
sol5 = Eq(f(x), exp(2*C1 + LambertW(-2*x**4*exp(-4*C1))/2)/x)
sol6 = Eq(log(x), C1 + exp(-f(x)/x)*sin(f(x)/x)/2 + exp(-f(x)/x)*cos(f(x)/x)/2)
sol7 = Eq(log(f(x)), C1 - 2*sqrt(-x/f(x) + 1))
sol8 = Eq(log(x), C1 - log(sqrt(1 + f(x)**2/x**2)) + atan(f(x)/x))
# indep_div_dep actually has a simpler solution for eq2,
# but it runs too slow
assert dsolve(eq1, hint='1st_homogeneous_coeff_subs_dep_div_indep') == sol1
assert dsolve(eq2, hint='1st_homogeneous_coeff_subs_dep_div_indep', simplify=False) == sol2
assert dsolve(eq3, hint='1st_homogeneous_coeff_best') == sol3
assert dsolve(eq4, hint='1st_homogeneous_coeff_best') == sol4
assert dsolve(eq5, hint='1st_homogeneous_coeff_best') == sol5
assert dsolve(eq6, hint='1st_homogeneous_coeff_subs_dep_div_indep') == sol6
assert dsolve(eq7, hint='1st_homogeneous_coeff_best') == sol7
assert dsolve(eq8, hint='1st_homogeneous_coeff_best') == sol8
# FIXME: sol3 and sol5 don't work with checkodesol (because of LambertW?)
# previous code was testing with these other solutions:
sol3b = Eq(-f(x)/(1 + log(x/f(x))), C1)
sol5b = Eq(log(C1*x*sqrt(1/x)*sqrt(f(x))) + x**2/(2*f(x)**2), 0)
assert checkodesol(eq1, sol1, order=1, solve_for_func=False)[0]
assert checkodesol(eq2, sol2, order=1, solve_for_func=False)[0]
assert checkodesol(eq3, sol3b, order=1, solve_for_func=False)[0]
assert checkodesol(eq4, sol4, order=1, solve_for_func=False)[0]
assert checkodesol(eq5, sol5b, order=1, solve_for_func=False)[0]
assert checkodesol(eq6, sol6, order=1, solve_for_func=False)[0]
assert checkodesol(eq8, sol8, order=1, solve_for_func=False)[0]
def test_1st_homogeneous_coeff_ode_check2():
eq2 = x*f(x).diff(x) - f(x) - x*sin(f(x)/x)
sol2 = Eq(x/tan(f(x)/(2*x)), C1)
assert checkodesol(eq2, sol2, order=1, solve_for_func=False)[0]
def test_1st_homogeneous_coeff_ode_check3():
eq3 = f(x) + (x*log(f(x)/x) - 2*x)*diff(f(x), x)
# This solution is correct:
sol3 = Eq(f(x), -exp(C1)*LambertW(-x*exp(1 - C1)))
assert dsolve(eq3) == sol3
# FIXME: Checked in test_1st_homogeneous_coeff_ode_check3_check below
# Alternate form:
sol3a = Eq(f(x), x*exp(1 - LambertW(C1*x)))
assert checkodesol(eq3, sol3a, solve_for_func=True)[0]
@XFAIL
def test_1st_homogeneous_coeff_ode_check3_check():
# See test_1st_homogeneous_coeff_ode_check3 above
eq3 = f(x) + (x*log(f(x)/x) - 2*x)*diff(f(x), x)
sol3 = Eq(f(x), -exp(C1)*LambertW(-x*exp(1 - C1)))
assert checkodesol(eq3, sol3) == (True, 0) # XFAIL
def test_1st_homogeneous_coeff_ode_check7():
eq7 = (x + sqrt(f(x)**2 - x*f(x)))*f(x).diff(x) - f(x)
sol7 = Eq(log(f(x)), C1 - 2*sqrt(-x/f(x) + 1))
assert dsolve(eq7) == sol7
assert checkodesol(eq7, sol7, order=1, solve_for_func=False) == (True, 0)
def test_1st_homogeneous_coeff_ode2():
eq1 = f(x).diff(x) - f(x)/x + 1/sin(f(x)/x)
eq2 = x**2 + f(x)**2 - 2*x*f(x)*f(x).diff(x)
eq3 = x*exp(f(x)/x) + f(x) - x*f(x).diff(x)
sol1 = [Eq(f(x), x*(-acos(C1 + log(x)) + 2*pi)), Eq(f(x), x*acos(C1 + log(x)))]
sol2 = Eq(log(f(x)), log(C1) + log(x/f(x)) - log(x**2/f(x)**2 - 1))
sol3 = Eq(f(x), log((1/(C1 - log(x)))**x))
# specific hints are applied for speed reasons
assert dsolve(eq1, hint='1st_homogeneous_coeff_subs_dep_div_indep') == sol1
assert dsolve(eq2, hint='1st_homogeneous_coeff_best', simplify=False) == sol2
assert dsolve(eq3, hint='1st_homogeneous_coeff_subs_dep_div_indep') == sol3
# FIXME: sol3 doesn't work with checkodesol (because of **x?)
# previous code was testing with this other solution:
sol3b = Eq(f(x), log(log(C1/x)**(-x)))
assert checkodesol(eq1, sol1, order=1, solve_for_func=False)[0]
assert checkodesol(eq2, sol2, order=1, solve_for_func=False)[0]
assert checkodesol(eq3, sol3b, order=1, solve_for_func=False)[0]
def test_1st_homogeneous_coeff_ode_check9():
_u2 = Dummy('u2')
__a = Dummy('a')
eq9 = f(x)**2 + (x*sqrt(f(x)**2 - x**2) - x*f(x))*f(x).diff(x)
sol9 = Eq(-Integral(-1/(-(1 - sqrt(1 - _u2**2))*_u2 + _u2), (_u2, __a,
x/f(x))) + log(C1*f(x)), 0)
assert checkodesol(eq9, sol9, order=1, solve_for_func=False)[0]
def test_1st_homogeneous_coeff_ode3():
# The standard integration engine cannot handle one of the integrals
# involved (see issue 4551). meijerg code comes up with an answer, but in
# unconventional form.
# checkodesol fails for this equation, so its test is in
# test_1st_homogeneous_coeff_ode_check9 above. It has to compare string
# expressions because u2 is a dummy variable.
eq = f(x)**2 + (x*sqrt(f(x)**2 - x**2) - x*f(x))*f(x).diff(x)
sol = Eq(log(f(x)), C1 + Piecewise(
(acosh(f(x)/x), abs(f(x)**2)/x**2 > 1),
(-I*asin(f(x)/x), True)))
assert dsolve(eq, hint='1st_homogeneous_coeff_subs_indep_div_dep') == sol
def test_1st_homogeneous_coeff_corner_case():
eq1 = f(x).diff(x) - f(x)/x
c1 = classify_ode(eq1, f(x))
eq2 = x*f(x).diff(x) - f(x)
c2 = classify_ode(eq2, f(x))
sdi = "1st_homogeneous_coeff_subs_dep_div_indep"
sid = "1st_homogeneous_coeff_subs_indep_div_dep"
assert sid not in c1 and sdi not in c1
assert sid not in c2 and sdi not in c2
@slow
def test_nth_linear_constant_coeff_homogeneous():
# From Exercise 20, in Ordinary Differential Equations,
# Tenenbaum and Pollard, pg. 220
a = Symbol('a', positive=True)
k = Symbol('k', real=True)
eq1 = f(x).diff(x, 2) + 2*f(x).diff(x)
eq2 = f(x).diff(x, 2) - 3*f(x).diff(x) + 2*f(x)
eq3 = f(x).diff(x, 2) - f(x)
eq4 = f(x).diff(x, 3) + f(x).diff(x, 2) - 6*f(x).diff(x)
eq5 = 6*f(x).diff(x, 2) - 11*f(x).diff(x) + 4*f(x)
eq6 = Eq(f(x).diff(x, 2) + 2*f(x).diff(x) - f(x), 0)
eq7 = diff(f(x), x, 3) + diff(f(x), x, 2) - 10*diff(f(x), x) - 6*f(x)
eq8 = f(x).diff(x, 4) - f(x).diff(x, 3) - 4*f(x).diff(x, 2) + \
4*f(x).diff(x)
eq9 = f(x).diff(x, 4) + 4*f(x).diff(x, 3) + f(x).diff(x, 2) - \
4*f(x).diff(x) - 2*f(x)
eq10 = f(x).diff(x, 4) - a**2*f(x)
eq11 = f(x).diff(x, 2) - 2*k*f(x).diff(x) - 2*f(x)
eq12 = f(x).diff(x, 2) + 4*k*f(x).diff(x) - 12*k**2*f(x)
eq13 = f(x).diff(x, 4)
eq14 = f(x).diff(x, 2) + 4*f(x).diff(x) + 4*f(x)
eq15 = 3*f(x).diff(x, 3) + 5*f(x).diff(x, 2) + f(x).diff(x) - f(x)
eq16 = f(x).diff(x, 3) - 6*f(x).diff(x, 2) + 12*f(x).diff(x) - 8*f(x)
eq17 = f(x).diff(x, 2) - 2*a*f(x).diff(x) + a**2*f(x)
eq18 = f(x).diff(x, 4) + 3*f(x).diff(x, 3)
eq19 = f(x).diff(x, 4) - 2*f(x).diff(x, 2)
eq20 = f(x).diff(x, 4) + 2*f(x).diff(x, 3) - 11*f(x).diff(x, 2) - \
12*f(x).diff(x) + 36*f(x)
eq21 = 36*f(x).diff(x, 4) - 37*f(x).diff(x, 2) + 4*f(x).diff(x) + 5*f(x)
eq22 = f(x).diff(x, 4) - 8*f(x).diff(x, 2) + 16*f(x)
eq23 = f(x).diff(x, 2) - 2*f(x).diff(x) + 5*f(x)
eq24 = f(x).diff(x, 2) - f(x).diff(x) + f(x)
eq25 = f(x).diff(x, 4) + 5*f(x).diff(x, 2) + 6*f(x)
eq26 = f(x).diff(x, 2) - 4*f(x).diff(x) + 20*f(x)
eq27 = f(x).diff(x, 4) + 4*f(x).diff(x, 2) + 4*f(x)
eq28 = f(x).diff(x, 3) + 8*f(x)
eq29 = f(x).diff(x, 4) + 4*f(x).diff(x, 2)
eq30 = f(x).diff(x, 5) + 2*f(x).diff(x, 3) + f(x).diff(x)
eq31 = f(x).diff(x, 4) + f(x).diff(x, 2) + f(x)
eq32 = f(x).diff(x, 4) + 4*f(x).diff(x, 2) + f(x)
sol1 = Eq(f(x), C1 + C2*exp(-2*x))
sol2 = Eq(f(x), (C1 + C2*exp(x))*exp(x))
sol3 = Eq(f(x), C1*exp(x) + C2*exp(-x))
sol4 = Eq(f(x), C1 + C2*exp(-3*x) + C3*exp(2*x))
sol5 = Eq(f(x), C1*exp(x/2) + C2*exp(x*Rational(4, 3)))
sol6 = Eq(f(x), C1*exp(x*(-1 + sqrt(2))) + C2*exp(x*(-sqrt(2) - 1)))
sol7 = Eq(f(x), C3*exp(3*x) + (C1*exp(-sqrt(2)*x) + C2*exp(sqrt(2)*x))*exp(-2*x))
sol8 = Eq(f(x), C1 + C2*exp(x) + C3*exp(-2*x) + C4*exp(2*x))
sol9 = Eq(f(x), C3*exp(-x) + C4*exp(x) + (C1*exp(-sqrt(2)*x) + C2*exp(sqrt(2)*x))*exp(-2*x))
sol10 = Eq(f(x),
C1*sin(x*sqrt(a)) + C2*cos(x*sqrt(a)) + C3*exp(x*sqrt(a)) +
C4*exp(-x*sqrt(a)))
sol11 = Eq(f(x),
C1*exp(x*(k - sqrt(k**2 + 2))) + C2*exp(x*(k + sqrt(k**2 + 2))))
sol12 = Eq(f(x), C1*exp(-6*k*x) + C2*exp(2*k*x))
sol13 = Eq(f(x), C1 + C2*x + C3*x**2 + C4*x**3)
sol14 = Eq(f(x), (C1 + C2*x)*exp(-2*x))
sol15 = Eq(f(x), (C1 + C2*x)*exp(-x) + C3*exp(x/3))
sol16 = Eq(f(x), (C1 + x*(C2 + C3*x))*exp(2*x))
sol17 = Eq(f(x), (C1 + C2*x)*exp(a*x))
sol18 = Eq(f(x), C1 + C2*x + C3*x**2 + C4*exp(-3*x))
sol19 = Eq(f(x), C1 + C2*x + C3*exp(x*sqrt(2)) + C4*exp(-x*sqrt(2)))
sol20 = Eq(f(x), (C1 + C2*x)*exp(-3*x) + (C3 + C4*x)*exp(2*x))
sol21 = Eq(f(x), C1*exp(x/2) + C2*exp(-x) + C3*exp(-x/3) + C4*exp(x*Rational(5, 6)))
sol22 = Eq(f(x), (C1 + C2*x)*exp(-2*x) + (C3 + C4*x)*exp(2*x))
sol23 = Eq(f(x), (C1*sin(2*x) + C2*cos(2*x))*exp(x))
sol24 = Eq(f(x), (C1*sin(x*sqrt(3)/2) + C2*cos(x*sqrt(3)/2))*exp(x/2))
sol25 = Eq(f(x),
C1*cos(x*sqrt(3)) + C2*sin(x*sqrt(3)) + C3*sin(x*sqrt(2)) +
C4*cos(x*sqrt(2)))
sol26 = Eq(f(x), (C1*sin(4*x) + C2*cos(4*x))*exp(2*x))
sol27 = Eq(f(x), (C1 + C2*x)*sin(x*sqrt(2)) + (C3 + C4*x)*cos(x*sqrt(2)))
sol28 = Eq(f(x),
(C1*sin(x*sqrt(3)) + C2*cos(x*sqrt(3)))*exp(x) + C3*exp(-2*x))
sol29 = Eq(f(x), C1 + C2*sin(2*x) + C3*cos(2*x) + C4*x)
sol30 = Eq(f(x), C1 + (C2 + C3*x)*sin(x) + (C4 + C5*x)*cos(x))
sol31 = Eq(f(x), (C1*sin(sqrt(3)*x/2) + C2*cos(sqrt(3)*x/2))/sqrt(exp(x))
+ (C3*sin(sqrt(3)*x/2) + C4*cos(sqrt(3)*x/2))*sqrt(exp(x)))
sol32 = Eq(f(x), C1*sin(x*sqrt(-sqrt(3) + 2)) + C2*sin(x*sqrt(sqrt(3) + 2))
+ C3*cos(x*sqrt(-sqrt(3) + 2)) + C4*cos(x*sqrt(sqrt(3) + 2)))
sol1s = constant_renumber(sol1)
sol2s = constant_renumber(sol2)
sol3s = constant_renumber(sol3)
sol4s = constant_renumber(sol4)
sol5s = constant_renumber(sol5)
sol6s = constant_renumber(sol6)
sol7s = constant_renumber(sol7)
sol8s = constant_renumber(sol8)
sol9s = constant_renumber(sol9)
sol10s = constant_renumber(sol10)
sol11s = constant_renumber(sol11)
sol12s = constant_renumber(sol12)
sol13s = constant_renumber(sol13)
sol14s = constant_renumber(sol14)
sol15s = constant_renumber(sol15)
sol16s = constant_renumber(sol16)
sol17s = constant_renumber(sol17)
sol18s = constant_renumber(sol18)
sol19s = constant_renumber(sol19)
sol20s = constant_renumber(sol20)
sol21s = constant_renumber(sol21)
sol22s = constant_renumber(sol22)
sol23s = constant_renumber(sol23)
sol24s = constant_renumber(sol24)
sol25s = constant_renumber(sol25)
sol26s = constant_renumber(sol26)
sol27s = constant_renumber(sol27)
sol28s = constant_renumber(sol28)
sol29s = constant_renumber(sol29)
sol30s = constant_renumber(sol30)
assert dsolve(eq1) in (sol1, sol1s)
assert dsolve(eq2) in (sol2, sol2s)
assert dsolve(eq3) in (sol3, sol3s)
assert dsolve(eq4) in (sol4, sol4s)
assert dsolve(eq5) in (sol5, sol5s)
assert dsolve(eq6) in (sol6, sol6s)
got = dsolve(eq7)
assert got in (sol7, sol7s), got
assert dsolve(eq8) in (sol8, sol8s)
got = dsolve(eq9)
assert got in (sol9, sol9s), got
assert dsolve(eq10) in (sol10, sol10s)
assert dsolve(eq11) in (sol11, sol11s)
assert dsolve(eq12) in (sol12, sol12s)
assert dsolve(eq13) in (sol13, sol13s)
assert dsolve(eq14) in (sol14, sol14s)
assert dsolve(eq15) in (sol15, sol15s)
got = dsolve(eq16)
assert got in (sol16, sol16s), got
assert dsolve(eq17) in (sol17, sol17s)
assert dsolve(eq18) in (sol18, sol18s)
assert dsolve(eq19) in (sol19, sol19s)
assert dsolve(eq20) in (sol20, sol20s)
assert dsolve(eq21) in (sol21, sol21s)
assert dsolve(eq22) in (sol22, sol22s)
assert dsolve(eq23) in (sol23, sol23s)
assert dsolve(eq24) in (sol24, sol24s)
assert dsolve(eq25) in (sol25, sol25s)
assert dsolve(eq26) in (sol26, sol26s)
assert dsolve(eq27) in (sol27, sol27s)
assert dsolve(eq28) in (sol28, sol28s)
assert dsolve(eq29) in (sol29, sol29s)
assert dsolve(eq30) in (sol30, sol30s)
assert dsolve(eq31) in (sol31,)
assert dsolve(eq32) in (sol32,)
assert checkodesol(eq1, sol1, order=2, solve_for_func=False)[0]
assert checkodesol(eq2, sol2, order=2, solve_for_func=False)[0]
assert checkodesol(eq3, sol3, order=2, solve_for_func=False)[0]
assert checkodesol(eq4, sol4, order=3, solve_for_func=False)[0]
assert checkodesol(eq5, sol5, order=2, solve_for_func=False)[0]
assert checkodesol(eq6, sol6, order=2, solve_for_func=False)[0]
assert checkodesol(eq7, sol7, order=3, solve_for_func=False)[0]
assert checkodesol(eq8, sol8, order=4, solve_for_func=False)[0]
assert checkodesol(eq9, sol9, order=4, solve_for_func=False)[0]
assert checkodesol(eq10, sol10, order=4, solve_for_func=False)[0]
assert checkodesol(eq11, sol11, order=2, solve_for_func=False)[0]
assert checkodesol(eq12, sol12, order=2, solve_for_func=False)[0]
assert checkodesol(eq13, sol13, order=4, solve_for_func=False)[0]
assert checkodesol(eq14, sol14, order=2, solve_for_func=False)[0]
assert checkodesol(eq15, sol15, order=3, solve_for_func=False)[0]
assert checkodesol(eq16, sol16, order=3, solve_for_func=False)[0]
assert checkodesol(eq17, sol17, order=2, solve_for_func=False)[0]
assert checkodesol(eq18, sol18, order=4, solve_for_func=False)[0]
assert checkodesol(eq19, sol19, order=4, solve_for_func=False)[0]
assert checkodesol(eq20, sol20, order=4, solve_for_func=False)[0]
assert checkodesol(eq21, sol21, order=4, solve_for_func=False)[0]
assert checkodesol(eq22, sol22, order=4, solve_for_func=False)[0]
assert checkodesol(eq23, sol23, order=2, solve_for_func=False)[0]
assert checkodesol(eq24, sol24, order=2, solve_for_func=False)[0]
assert checkodesol(eq25, sol25, order=4, solve_for_func=False)[0]
assert checkodesol(eq26, sol26, order=2, solve_for_func=False)[0]
assert checkodesol(eq27, sol27, order=4, solve_for_func=False)[0]
assert checkodesol(eq28, sol28, order=3, solve_for_func=False)[0]
assert checkodesol(eq29, sol29, order=4, solve_for_func=False)[0]
assert checkodesol(eq30, sol30, order=5, solve_for_func=False)[0]
assert checkodesol(eq31, sol31, order=4, solve_for_func=False)[0]
assert checkodesol(eq32, sol32, order=4, solve_for_func=False)[0]
# Issue #15237
eqn = Derivative(x*f(x), x, x, x)
hint = 'nth_linear_constant_coeff_homogeneous'
raises(ValueError, lambda: dsolve(eqn, f(x), hint, prep=True))
raises(ValueError, lambda: dsolve(eqn, f(x), hint, prep=False))
def test_nth_linear_constant_coeff_homogeneous_rootof():
# One real root, two complex conjugate pairs
eq = f(x).diff(x, 5) + 11*f(x).diff(x) - 2*f(x)
r1, r2, r3, r4, r5 = [rootof(x**5 + 11*x - 2, n) for n in range(5)]
sol = Eq(f(x),
C5*exp(r1*x)
+ exp(re(r2)*x) * (C1*sin(im(r2)*x) + C2*cos(im(r2)*x))
+ exp(re(r4)*x) * (C3*sin(im(r4)*x) + C4*cos(im(r4)*x))
)
assert dsolve(eq) == sol
# FIXME: assert checkodesol(eq, sol) == (True, [0]) # Hangs...
# Three real roots, one complex conjugate pair
eq = f(x).diff(x,5) - 3*f(x).diff(x) + f(x)
r1, r2, r3, r4, r5 = [rootof(x**5 - 3*x + 1, n) for n in range(5)]
sol = Eq(f(x),
C3*exp(r1*x) + C4*exp(r2*x) + C5*exp(r3*x)
+ exp(re(r4)*x) * (C1*sin(im(r4)*x) + C2*cos(im(r4)*x))
)
assert dsolve(eq) == sol
# FIXME: assert checkodesol(eq, sol) == (True, [0]) # Hangs...
# Five distinct real roots
eq = f(x).diff(x,5) - 100*f(x).diff(x,3) + 1000*f(x).diff(x) + f(x)
r1, r2, r3, r4, r5 = [rootof(x**5 - 100*x**3 + 1000*x + 1, n) for n in range(5)]
sol = Eq(f(x), C1*exp(r1*x) + C2*exp(r2*x) + C3*exp(r3*x) + C4*exp(r4*x) + C5*exp(r5*x))
assert dsolve(eq) == sol
# FIXME: assert checkodesol(eq, sol) == (True, [0]) # Hangs...
# Rational root and unsolvable quintic
eq = f(x).diff(x, 6) - 6*f(x).diff(x, 5) + 5*f(x).diff(x, 4) + 10*f(x).diff(x) - 50 * f(x)
r2, r3, r4, r5, r6 = [rootof(x**5 - x**4 + 10, n) for n in range(5)]
sol = Eq(f(x),
C5*exp(5*x)
+ C6*exp(x*r2)
+ exp(re(r3)*x) * (C1*sin(im(r3)*x) + C2*cos(im(r3)*x))
+ exp(re(r5)*x) * (C3*sin(im(r5)*x) + C4*cos(im(r5)*x))
)
assert dsolve(eq) == sol
# FIXME: assert checkodesol(eq, sol) == (True, [0]) # Hangs...
# Five double roots (this is (x**5 - x + 1)**2)
eq = f(x).diff(x, 10) - 2*f(x).diff(x, 6) + 2*f(x).diff(x, 5) + f(x).diff(x, 2) - 2*f(x).diff(x, 1) + f(x)
r1, r2, r3, r4, r5 = [rootof(x**5 - x + 1, n) for n in range(5)]
sol = Eq(f(x), (C1 + C2*x)*exp(x*r1) + (C10*sin(x*im(r4)) + C7*x*sin(x*im(r4)) + (
C8 + C9*x)*cos(x*im(r4)))*exp(x*re(r4)) + (C3*x*sin(x*im(r2)) + C6*sin(x*im(r2)
) + (C4 + C5*x)*cos(x*im(r2)))*exp(x*re(r2)))
got = dsolve(eq)
assert sol == got, got
# FIXME: assert checkodesol(eq, sol) == (True, [0]) # Hangs...
def test_nth_linear_constant_coeff_homogeneous_irrational():
our_hint='nth_linear_constant_coeff_homogeneous'
eq = Eq(sqrt(2) * f(x).diff(x,x,x) + f(x).diff(x), 0)
sol = Eq(f(x), C1 + C2*sin(2**Rational(3, 4)*x/2) + C3*cos(2**Rational(3, 4)*x/2))
assert our_hint in classify_ode(eq)
assert dsolve(eq, f(x), hint=our_hint) == sol
assert dsolve(eq, f(x)) == sol
assert checkodesol(eq, sol, order=3, solve_for_func=False)[0]
E = exp(1)
eq = Eq(E * f(x).diff(x,x,x) + f(x).diff(x), 0)
sol = Eq(f(x), C1 + C2*sin(x/sqrt(E)) + C3*cos(x/sqrt(E)))
assert our_hint in classify_ode(eq)
assert dsolve(eq, f(x), hint=our_hint) == sol
assert dsolve(eq, f(x)) == sol
assert checkodesol(eq, sol, order=3, solve_for_func=False)[0]
eq = Eq(pi * f(x).diff(x,x,x) + f(x).diff(x), 0)
sol = Eq(f(x), C1 + C2*sin(x/sqrt(pi)) + C3*cos(x/sqrt(pi)))
assert our_hint in classify_ode(eq)
assert dsolve(eq, f(x), hint=our_hint) == sol
assert dsolve(eq, f(x)) == sol
assert checkodesol(eq, sol, order=3, solve_for_func=False)[0]
eq = Eq(I * f(x).diff(x,x,x) + f(x).diff(x), 0)
sol = Eq(f(x), C1 + C2*exp(-sqrt(I)*x) + C3*exp(sqrt(I)*x))
assert our_hint in classify_ode(eq)
assert dsolve(eq, f(x), hint=our_hint) == sol
assert dsolve(eq, f(x)) == sol
assert checkodesol(eq, sol, order=3, solve_for_func=False)[0]
@XFAIL
@slow
def test_nth_linear_constant_coeff_homogeneous_rootof_sol():
# See https://github.com/sympy/sympy/issues/15753
if ON_TRAVIS:
skip("Too slow for travis.")
eq = f(x).diff(x, 5) + 11*f(x).diff(x) - 2*f(x)
sol = Eq(f(x),
C1*exp(x*rootof(x**5 + 11*x - 2, 0)) +
C2*exp(x*rootof(x**5 + 11*x - 2, 1)) +
C3*exp(x*rootof(x**5 + 11*x - 2, 2)) +
C4*exp(x*rootof(x**5 + 11*x - 2, 3)) +
C5*exp(x*rootof(x**5 + 11*x - 2, 4)))
assert checkodesol(eq, sol, order=5, solve_for_func=False)[0]
@XFAIL
def test_noncircularized_real_imaginary_parts():
# If this passes, lines numbered 3878-3882 (at the time of this commit)
# of sympy/solvers/ode.py for nth_linear_constant_coeff_homogeneous
# should be removed.
y = sqrt(1+x)
i, r = im(y), re(y)
assert not (i.has(atan2) and r.has(atan2))
def test_collect_respecting_exponentials():
# If this test passes, lines 1306-1311 (at the time of this commit)
# of sympy/solvers/ode.py should be removed.
sol = 1 + exp(x/2)
assert sol == collect( sol, exp(x/3))
def test_undetermined_coefficients_match():
assert _undetermined_coefficients_match(g(x), x) == {'test': False}
assert _undetermined_coefficients_match(sin(2*x + sqrt(5)), x) == \
{'test': True, 'trialset':
set([cos(2*x + sqrt(5)), sin(2*x + sqrt(5))])}
assert _undetermined_coefficients_match(sin(x)*cos(x), x) == \
{'test': False}
s = set([cos(x), x*cos(x), x**2*cos(x), x**2*sin(x), x*sin(x), sin(x)])
assert _undetermined_coefficients_match(sin(x)*(x**2 + x + 1), x) == \
{'test': True, 'trialset': s}
assert _undetermined_coefficients_match(
sin(x)*x**2 + sin(x)*x + sin(x), x) == {'test': True, 'trialset': s}
assert _undetermined_coefficients_match(
exp(2*x)*sin(x)*(x**2 + x + 1), x
) == {
'test': True, 'trialset': set([exp(2*x)*sin(x), x**2*exp(2*x)*sin(x),
cos(x)*exp(2*x), x**2*cos(x)*exp(2*x), x*cos(x)*exp(2*x),
x*exp(2*x)*sin(x)])}
assert _undetermined_coefficients_match(1/sin(x), x) == {'test': False}
assert _undetermined_coefficients_match(log(x), x) == {'test': False}
assert _undetermined_coefficients_match(2**(x)*(x**2 + x + 1), x) == \
{'test': True, 'trialset': set([2**x, x*2**x, x**2*2**x])}
assert _undetermined_coefficients_match(x**y, x) == {'test': False}
assert _undetermined_coefficients_match(exp(x)*exp(2*x + 1), x) == \
{'test': True, 'trialset': set([exp(1 + 3*x)])}
assert _undetermined_coefficients_match(sin(x)*(x**2 + x + 1), x) == \
{'test': True, 'trialset': set([x*cos(x), x*sin(x), x**2*cos(x),
x**2*sin(x), cos(x), sin(x)])}
assert _undetermined_coefficients_match(sin(x)*(x + sin(x)), x) == \
{'test': False}
assert _undetermined_coefficients_match(sin(x)*(x + sin(2*x)), x) == \
{'test': False}
assert _undetermined_coefficients_match(sin(x)*tan(x), x) == \
{'test': False}
assert _undetermined_coefficients_match(
x**2*sin(x)*exp(x) + x*sin(x) + x, x
) == {
'test': True, 'trialset': set([x**2*cos(x)*exp(x), x, cos(x), S.One,
exp(x)*sin(x), sin(x), x*exp(x)*sin(x), x*cos(x), x*cos(x)*exp(x),
x*sin(x), cos(x)*exp(x), x**2*exp(x)*sin(x)])}
assert _undetermined_coefficients_match(4*x*sin(x - 2), x) == {
'trialset': set([x*cos(x - 2), x*sin(x - 2), cos(x - 2), sin(x - 2)]),
'test': True,
}
assert _undetermined_coefficients_match(2**x*x, x) == \
{'test': True, 'trialset': set([2**x, x*2**x])}
assert _undetermined_coefficients_match(2**x*exp(2*x), x) == \
{'test': True, 'trialset': set([2**x*exp(2*x)])}
assert _undetermined_coefficients_match(exp(-x)/x, x) == \
{'test': False}
# Below are from Ordinary Differential Equations,
# Tenenbaum and Pollard, pg. 231
assert _undetermined_coefficients_match(S(4), x) == \
{'test': True, 'trialset': set([S.One])}
assert _undetermined_coefficients_match(12*exp(x), x) == \
{'test': True, 'trialset': set([exp(x)])}
assert _undetermined_coefficients_match(exp(I*x), x) == \
{'test': True, 'trialset': set([exp(I*x)])}
assert _undetermined_coefficients_match(sin(x), x) == \
{'test': True, 'trialset': set([cos(x), sin(x)])}
assert _undetermined_coefficients_match(cos(x), x) == \
{'test': True, 'trialset': set([cos(x), sin(x)])}
assert _undetermined_coefficients_match(8 + 6*exp(x) + 2*sin(x), x) == \
{'test': True, 'trialset': set([S.One, cos(x), sin(x), exp(x)])}
assert _undetermined_coefficients_match(x**2, x) == \
{'test': True, 'trialset': set([S.One, x, x**2])}
assert _undetermined_coefficients_match(9*x*exp(x) + exp(-x), x) == \
{'test': True, 'trialset': set([x*exp(x), exp(x), exp(-x)])}
assert _undetermined_coefficients_match(2*exp(2*x)*sin(x), x) == \
{'test': True, 'trialset': set([exp(2*x)*sin(x), cos(x)*exp(2*x)])}
assert _undetermined_coefficients_match(x - sin(x), x) == \
{'test': True, 'trialset': set([S.One, x, cos(x), sin(x)])}
assert _undetermined_coefficients_match(x**2 + 2*x, x) == \
{'test': True, 'trialset': set([S.One, x, x**2])}
assert _undetermined_coefficients_match(4*x*sin(x), x) == \
{'test': True, 'trialset': set([x*cos(x), x*sin(x), cos(x), sin(x)])}
assert _undetermined_coefficients_match(x*sin(2*x), x) == \
{'test': True, 'trialset':
set([x*cos(2*x), x*sin(2*x), cos(2*x), sin(2*x)])}
assert _undetermined_coefficients_match(x**2*exp(-x), x) == \
{'test': True, 'trialset': set([x*exp(-x), x**2*exp(-x), exp(-x)])}
assert _undetermined_coefficients_match(2*exp(-x) - x**2*exp(-x), x) == \
{'test': True, 'trialset': set([x*exp(-x), x**2*exp(-x), exp(-x)])}
assert _undetermined_coefficients_match(exp(-2*x) + x**2, x) == \
{'test': True, 'trialset': set([S.One, x, x**2, exp(-2*x)])}
assert _undetermined_coefficients_match(x*exp(-x), x) == \
{'test': True, 'trialset': set([x*exp(-x), exp(-x)])}
assert _undetermined_coefficients_match(x + exp(2*x), x) == \
{'test': True, 'trialset': set([S.One, x, exp(2*x)])}
assert _undetermined_coefficients_match(sin(x) + exp(-x), x) == \
{'test': True, 'trialset': set([cos(x), sin(x), exp(-x)])}
assert _undetermined_coefficients_match(exp(x), x) == \
{'test': True, 'trialset': set([exp(x)])}
# converted from sin(x)**2
assert _undetermined_coefficients_match(S.Half - cos(2*x)/2, x) == \
{'test': True, 'trialset': set([S.One, cos(2*x), sin(2*x)])}
# converted from exp(2*x)*sin(x)**2
assert _undetermined_coefficients_match(
exp(2*x)*(S.Half + cos(2*x)/2), x
) == {
'test': True, 'trialset': set([exp(2*x)*sin(2*x), cos(2*x)*exp(2*x),
exp(2*x)])}
assert _undetermined_coefficients_match(2*x + sin(x) + cos(x), x) == \
{'test': True, 'trialset': set([S.One, x, cos(x), sin(x)])}
# converted from sin(2*x)*sin(x)
assert _undetermined_coefficients_match(cos(x)/2 - cos(3*x)/2, x) == \
{'test': True, 'trialset': set([cos(x), cos(3*x), sin(x), sin(3*x)])}
assert _undetermined_coefficients_match(cos(x**2), x) == {'test': False}
assert _undetermined_coefficients_match(2**(x**2), x) == {'test': False}
def test_issue_12623():
t = symbols("t")
u = symbols("u",cls=Function)
R, L, C, E_0, alpha = symbols("R L C E_0 alpha",positive=True)
omega = Symbol('omega')
eqRLC_1 = Eq( u(t).diff(t,t) + R /L*u(t).diff(t) + 1/(L*C)*u(t), alpha)
sol_1 = Eq(u(t), C*L*alpha + C1*exp(t*(-R - sqrt(C*R**2 - 4*L)/sqrt(C))/(2*L)) + C2*exp(t*(-R + sqrt(C*R**2 - 4*L)/sqrt(C))/(2*L)))
assert dsolve(eqRLC_1) == sol_1
assert checkodesol(eqRLC_1, sol_1) == (True, 0)
eqRLC_2 = Eq( L*C*u(t).diff(t,t) + R*C*u(t).diff(t) + u(t), E_0*exp(I*omega*t) )
sol_2 = Eq(u(t), C1*exp(t*(-R - sqrt(C*R**2 - 4*L)/sqrt(C))/(2*L)) + C2*exp(t*(-R + sqrt(C*R**2 - 4*L)/sqrt(C))/(2*L)) + E_0*exp(I*omega*t)/(-C*L*omega**2 + I*C*R*omega + 1))
assert dsolve(eqRLC_2) == sol_2
assert checkodesol(eqRLC_2, sol_2) == (True, 0)
#issue-https://github.com/sympy/sympy/issues/12623
def test_issue_5787():
# This test case is to show the classification of imaginary constants under
# nth_linear_constant_coeff_undetermined_coefficients
eq = Eq(diff(f(x), x), I*f(x) + S.Half - I)
our_hint = 'nth_linear_constant_coeff_undetermined_coefficients'
assert our_hint in classify_ode(eq)
def test_nth_linear_constant_coeff_undetermined_coefficients_imaginary_exp():
# Equivalent to eq26 in
# test_nth_linear_constant_coeff_undetermined_coefficients above. This
# previously failed because the algorithm for undetermined coefficients
# didn't know to multiply exp(I*x) by sufficient x because it is linearly
# dependent on sin(x) and cos(x).
hint = 'nth_linear_constant_coeff_undetermined_coefficients'
eq26a = f(x).diff(x, 5) + 2*f(x).diff(x, 3) + f(x).diff(x) - 2*x - exp(I*x)
sol26 = Eq(f(x), C1 + x**2*(I*exp(I*x)/8 + 1) + (C2 + C3*x)*sin(x) + (C4 + C5*x)*cos(x))
assert dsolve(eq26a, hint=hint) == sol26
assert checkodesol(eq26a, sol26) == (True, 0)
@slow
def test_nth_linear_constant_coeff_variation_of_parameters():
hint = 'nth_linear_constant_coeff_variation_of_parameters'
g = exp(-x)
f2 = f(x).diff(x, 2)
c = 3*f(x).diff(x, 3) + 5*f2 + f(x).diff(x) - f(x) - x
eq1 = c - x*g
eq2 = c - g
eq3 = f(x).diff(x) - 1
eq4 = f2 + 3*f(x).diff(x) + 2*f(x) - 4
eq5 = f2 + 3*f(x).diff(x) + 2*f(x) - 12*exp(x)
eq6 = f2 - 2*f(x).diff(x) - 8*f(x) - 9*x*exp(x) - 10*exp(-x)
eq7 = f2 + 2*f(x).diff(x) + f(x) - x**2*exp(-x)
eq8 = f2 - 3*f(x).diff(x) + 2*f(x) - x*exp(-x)
eq9 = f(x).diff(x, 3) - 3*f2 + 3*f(x).diff(x) - f(x) - exp(x)
eq10 = f2 + 2*f(x).diff(x) + f(x) - exp(-x)/x
eq11 = f2 + f(x) - 1/sin(x)*1/cos(x)
eq12 = f(x).diff(x, 4) - 1/x
sol1 = Eq(f(x), C3*exp(x/3) - x + (C1 + x*(C2 - x**2/24 - 3*x/32))*exp(-x) - 1)
sol2 = Eq(f(x), C3*exp(x/3) - x + (C1 + x*(C2 - x/8))*exp(-x) - 1)
sol3 = Eq(f(x), C1 + x)
sol4 = Eq(f(x), 2 + C1*exp(-x) + C2*exp(-2*x))
sol5 = Eq(f(x), 2*exp(x) + C1*exp(-x) + C2*exp(-2*x))
sol6 = Eq(f(x), -x*exp(x) - 2*exp(-x) + C1*exp(-2*x) + C2*exp(4*x))
sol7 = Eq(f(x), (C1 + x*(C2 + x**3/12))*exp(-x))
sol8 = Eq(f(x), C1*exp(x) + C2*exp(2*x) + (6*x + 5)*exp(-x)/36)
sol9 = Eq(f(x), (C1 + x*(C2 + x*(C3 + x/6)))*exp(x))
sol10 = Eq(f(x), (C1 + x*(C2 + log(x)))*exp(-x))
sol11 = Eq(f(x), (C1 + log(sin(x) - 1)/2 - log(sin(x) + 1)/2
)*cos(x) + (C2 + log(cos(x) - 1)/2 - log(cos(x) + 1)/2)*sin(x))
sol12 = Eq(f(x), C1 + C2*x + x**3*(C3 + log(x)/6) + C4*x**2)
sol1s = constant_renumber(sol1)
sol2s = constant_renumber(sol2)
sol3s = constant_renumber(sol3)
sol4s = constant_renumber(sol4)
sol5s = constant_renumber(sol5)
sol6s = constant_renumber(sol6)
sol7s = constant_renumber(sol7)
sol8s = constant_renumber(sol8)
sol9s = constant_renumber(sol9)
sol10s = constant_renumber(sol10)
sol11s = constant_renumber(sol11)
sol12s = constant_renumber(sol12)
got = dsolve(eq1, hint=hint)
assert got in (sol1, sol1s), got
got = dsolve(eq2, hint=hint)
assert got in (sol2, sol2s), got
assert dsolve(eq3, hint=hint) in (sol3, sol3s)
assert dsolve(eq4, hint=hint) in (sol4, sol4s)
assert dsolve(eq5, hint=hint) in (sol5, sol5s)
assert dsolve(eq6, hint=hint) in (sol6, sol6s)
got = dsolve(eq7, hint=hint)
assert got in (sol7, sol7s), got
assert dsolve(eq8, hint=hint) in (sol8, sol8s)
got = dsolve(eq9, hint=hint)
assert got in (sol9, sol9s), got
assert dsolve(eq10, hint=hint) in (sol10, sol10s)
assert dsolve(eq11, hint=hint + '_Integral').doit() in (sol11, sol11s)
assert dsolve(eq12, hint=hint) in (sol12, sol12s)
assert checkodesol(eq1, sol1, order=3, solve_for_func=False)[0]
assert checkodesol(eq2, sol2, order=3, solve_for_func=False)[0]
assert checkodesol(eq3, sol3, order=1, solve_for_func=False)[0]
assert checkodesol(eq4, sol4, order=2, solve_for_func=False)[0]
assert checkodesol(eq5, sol5, order=2, solve_for_func=False)[0]
assert checkodesol(eq6, sol6, order=2, solve_for_func=False)[0]
assert checkodesol(eq7, sol7, order=2, solve_for_func=False)[0]
assert checkodesol(eq8, sol8, order=2, solve_for_func=False)[0]
assert checkodesol(eq9, sol9, order=3, solve_for_func=False)[0]
assert checkodesol(eq10, sol10, order=2, solve_for_func=False)[0]
assert checkodesol(eq12, sol12, order=4, solve_for_func=False)[0]
@slow
def test_nth_linear_constant_coeff_variation_of_parameters_simplify_False():
# solve_variation_of_parameters shouldn't attempt to simplify the
# Wronskian if simplify=False. If wronskian() ever gets good enough
# to simplify the result itself, this test might fail.
our_hint = 'nth_linear_constant_coeff_variation_of_parameters_Integral'
eq = f(x).diff(x, 5) + 2*f(x).diff(x, 3) + f(x).diff(x) - 2*x - exp(I*x)
sol_simp = dsolve(eq, f(x), hint=our_hint, simplify=True)
sol_nsimp = dsolve(eq, f(x), hint=our_hint, simplify=False)
assert sol_simp != sol_nsimp
assert checkodesol(eq, sol_simp, order=5, solve_for_func=False) == (True, 0)
assert checkodesol(eq, sol_simp, order=5, solve_for_func=False) == (True, 0)
def test_unexpanded_Liouville_ODE():
# This is the same as eq1 from test_Liouville_ODE() above.
eq1 = diff(f(x), x)/x + diff(f(x), x, x)/2 - diff(f(x), x)**2/2
eq2 = eq1*exp(-f(x))/exp(f(x))
sol2 = Eq(f(x), C1 + log(x) - log(C2 + x))
sol2s = constant_renumber(sol2)
assert dsolve(eq2) in (sol2, sol2s)
assert checkodesol(eq2, sol2, order=2, solve_for_func=False)[0]
def test_issue_4785():
from sympy.abc import A
eq = x + A*(x + diff(f(x), x) + f(x)) + diff(f(x), x) + f(x) + 2
assert classify_ode(eq, f(x)) == ('1st_linear', 'almost_linear',
'1st_power_series', 'lie_group',
'nth_linear_constant_coeff_undetermined_coefficients',
'nth_linear_constant_coeff_variation_of_parameters',
'1st_linear_Integral', 'almost_linear_Integral',
'nth_linear_constant_coeff_variation_of_parameters_Integral')
# issue 4864
eq = (x**2 + f(x)**2)*f(x).diff(x) - 2*x*f(x)
assert classify_ode(eq, f(x)) == ('1st_exact',
'1st_homogeneous_coeff_best',
'1st_homogeneous_coeff_subs_indep_div_dep',
'1st_homogeneous_coeff_subs_dep_div_indep',
'1st_power_series',
'lie_group', '1st_exact_Integral',
'1st_homogeneous_coeff_subs_indep_div_dep_Integral',
'1st_homogeneous_coeff_subs_dep_div_indep_Integral')
def test_issue_4825():
raises(ValueError, lambda: dsolve(f(x, y).diff(x) - y*f(x, y), f(x)))
assert classify_ode(f(x, y).diff(x) - y*f(x, y), f(x), dict=True) == \
{'order': 0, 'default': None, 'ordered_hints': ()}
# See also issue 3793, test Z13.
raises(ValueError, lambda: dsolve(f(x).diff(x), f(y)))
assert classify_ode(f(x).diff(x), f(y), dict=True) == \
{'order': 0, 'default': None, 'ordered_hints': ()}
def test_constant_renumber_order_issue_5308():
from sympy.utilities.iterables import variations
assert constant_renumber(C1*x + C2*y) == \
constant_renumber(C1*y + C2*x) == \
C1*x + C2*y
e = C1*(C2 + x)*(C3 + y)
for a, b, c in variations([C1, C2, C3], 3):
assert constant_renumber(a*(b + x)*(c + y)) == e
def test_issue_5770():
k = Symbol("k", real=True)
t = Symbol('t')
w = Function('w')
sol = dsolve(w(t).diff(t, 6) - k**6*w(t), w(t))
assert len([s for s in sol.free_symbols if s.name.startswith('C')]) == 6
assert constantsimp((C1*cos(x) + C2*cos(x))*exp(x), set([C1, C2])) == \
C1*cos(x)*exp(x)
assert constantsimp(C1*cos(x) + C2*cos(x) + C3*sin(x), set([C1, C2, C3])) == \
C1*cos(x) + C3*sin(x)
assert constantsimp(exp(C1 + x), set([C1])) == C1*exp(x)
assert constantsimp(x + C1 + y, set([C1, y])) == C1 + x
assert constantsimp(x + C1 + Integral(x, (x, 1, 2)), set([C1])) == C1 + x
def test_issue_5112_5430():
assert homogeneous_order(-log(x) + acosh(x), x) is None
assert homogeneous_order(y - log(x), x, y) is None
def test_issue_5095():
f = Function('f')
raises(ValueError, lambda: dsolve(f(x).diff(x)**2, f(x), 'fdsjf'))
def test_exact_enhancement():
f = Function('f')(x)
df = Derivative(f, x)
eq = f/x**2 + ((f*x - 1)/x)*df
sol = [Eq(f, (i*sqrt(C1*x**2 + 1) + 1)/x) for i in (-1, 1)]
assert set(dsolve(eq, f)) == set(sol)
assert checkodesol(eq, sol, order=1, solve_for_func=False) == [(True, 0), (True, 0)]
eq = (x*f - 1) + df*(x**2 - x*f)
sol = [Eq(f, x - sqrt(C1 + x**2 - 2*log(x))),
Eq(f, x + sqrt(C1 + x**2 - 2*log(x)))]
assert set(dsolve(eq, f)) == set(sol)
assert checkodesol(eq, sol, order=1, solve_for_func=False) == [(True, 0), (True, 0)]
eq = (x + 2)*sin(f) + df*x*cos(f)
sol = [Eq(f, -asin(C1*exp(-x)/x**2) + pi),
Eq(f, asin(C1*exp(-x)/x**2))]
assert set(dsolve(eq, f)) == set(sol)
assert checkodesol(eq, sol, order=1, solve_for_func=False) == [(True, 0), (True, 0)]
@slow
def test_separable_reduced():
f = Function('f')
x = Symbol('x')
df = f(x).diff(x)
eq = (x / f(x))*df + tan(x**2*f(x) / (x**2*f(x) - 1))
assert classify_ode(eq) == ('separable_reduced', 'lie_group',
'separable_reduced_Integral')
eq = x* df + f(x)* (1 / (x**2*f(x) - 1))
assert classify_ode(eq) == ('separable_reduced', 'lie_group',
'separable_reduced_Integral')
sol = dsolve(eq, hint = 'separable_reduced', simplify=False)
assert sol.lhs == log(x**2*f(x))/3 + log(x**2*f(x) - Rational(3, 2))/6
assert sol.rhs == C1 + log(x)
assert checkodesol(eq, sol, order=1, solve_for_func=False)[0]
eq = f(x).diff(x) + (f(x) / (x**4*f(x) - x))
assert classify_ode(eq) == ('separable_reduced', 'lie_group',
'separable_reduced_Integral')
sol = dsolve(eq, hint = 'separable_reduced')
# FIXME: This one hangs
#assert checkodesol(eq, sol, order=1, solve_for_func=False) == [(True, 0)] * 4
assert len(sol) == 4
eq = x*df + f(x)*(x**2*f(x))
sol = dsolve(eq, hint = 'separable_reduced', simplify=False)
assert sol == Eq(log(x**2*f(x))/2 - log(x**2*f(x) - 2)/2, C1 + log(x))
assert checkodesol(eq, sol, order=1, solve_for_func=False)[0]
eq = Eq(f(x).diff(x) + f(x)/x * (1 + (x**(S(2)/3)*f(x))**2), 0)
sol = dsolve(eq, hint = 'separable_reduced', simplify=False)
assert sol == Eq(-3*log(x**(S(2)/3)*f(x)) + 3*log(3*x**(S(4)/3)*f(x)**2 + 1)/2, C1 + log(x))
assert checkodesol(eq, sol, solve_for_func=False) == (True, 0)
eq = Eq(f(x).diff(x) + f(x)/x * (1 + (x*f(x))**2), 0)
sol = dsolve(eq, hint = 'separable_reduced')
assert sol == [Eq(f(x), -sqrt(2)*sqrt(1/(C1 + log(x)))/(2*x)),\
Eq(f(x), sqrt(2)*sqrt(1/(C1 + log(x)))/(2*x))]
assert checkodesol(eq, sol) == [(True, 0)]*2
eq = Eq(f(x).diff(x) + (x**4*f(x)**2 + x**2*f(x))*f(x)/(x*(x**6*f(x)**3 + x**4*f(x)**2)), 0)
sol = dsolve(eq, hint = 'separable_reduced')
assert sol == Eq(f(x), C1 + 1/(2*x**2))
assert checkodesol(eq, sol) == (True, 0)
eq = Eq(f(x).diff(x) + (f(x)**2)*f(x)/(x), 0)
sol = dsolve(eq, hint = 'separable_reduced')
assert sol == [Eq(f(x), -sqrt(2)*sqrt(1/(C1 + log(x)))/2),\
Eq(f(x), sqrt(2)*sqrt(1/(C1 + log(x)))/2)]
assert checkodesol(eq, sol) == [(True, 0), (True, 0)]
eq = Eq(f(x).diff(x) + (f(x)+3)*f(x)/(x*(f(x)+2)), 0)
sol = dsolve(eq, hint = 'separable_reduced', simplify=False)
assert sol == Eq(-log(f(x) + 3)/3 - 2*log(f(x))/3, C1 + log(x))
assert checkodesol(eq, sol, solve_for_func=False) == (True, 0)
eq = Eq(f(x).diff(x) + (f(x)+3)*f(x)/x, 0)
sol = dsolve(eq, hint = 'separable_reduced')
assert sol == Eq(f(x), 3/(C1*x**3 - 1))
assert checkodesol(eq, sol) == (True, 0)
eq = Eq(f(x).diff(x) + (f(x)**2+f(x))*f(x)/(x), 0)
sol = dsolve(eq, hint='separable_reduced', simplify=False)
assert sol == Eq(-log(f(x) + 1) + log(f(x)) + 1/f(x), C1 + log(x))
assert checkodesol(eq, sol, solve_for_func=False) == (True, 0)
def test_homogeneous_function():
f = Function('f')
eq1 = tan(x + f(x))
eq2 = sin((3*x)/(4*f(x)))
eq3 = cos(x*f(x)*Rational(3, 4))
eq4 = log((3*x + 4*f(x))/(5*f(x) + 7*x))
eq5 = exp((2*x**2)/(3*f(x)**2))
eq6 = log((3*x + 4*f(x))/(5*f(x) + 7*x) + exp((2*x**2)/(3*f(x)**2)))
eq7 = sin((3*x)/(5*f(x) + x**2))
assert homogeneous_order(eq1, x, f(x)) == None
assert homogeneous_order(eq2, x, f(x)) == 0
assert homogeneous_order(eq3, x, f(x)) == None
assert homogeneous_order(eq4, x, f(x)) == 0
assert homogeneous_order(eq5, x, f(x)) == 0
assert homogeneous_order(eq6, x, f(x)) == 0
assert homogeneous_order(eq7, x, f(x)) == None
def test_linear_coeff_match():
n, d = z*(2*x + 3*f(x) + 5), z*(7*x + 9*f(x) + 11)
rat = n/d
eq1 = sin(rat) + cos(rat.expand())
eq2 = rat
eq3 = log(sin(rat))
ans = (4, Rational(-13, 3))
assert _linear_coeff_match(eq1, f(x)) == ans
assert _linear_coeff_match(eq2, f(x)) == ans
assert _linear_coeff_match(eq3, f(x)) == ans
# no c
eq4 = (3*x)/f(x)
# not x and f(x)
eq5 = (3*x + 2)/x
# denom will be zero
eq6 = (3*x + 2*f(x) + 1)/(3*x + 2*f(x) + 5)
# not rational coefficient
eq7 = (3*x + 2*f(x) + sqrt(2))/(3*x + 2*f(x) + 5)
assert _linear_coeff_match(eq4, f(x)) is None
assert _linear_coeff_match(eq5, f(x)) is None
assert _linear_coeff_match(eq6, f(x)) is None
assert _linear_coeff_match(eq7, f(x)) is None
def test_linear_coefficients():
f = Function('f')
sol = Eq(f(x), C1/(x**2 + 6*x + 9) - Rational(3, 2))
eq = f(x).diff(x) + (3 + 2*f(x))/(x + 3)
assert dsolve(eq, hint='linear_coefficients') == sol
assert checkodesol(eq, sol, order=1, solve_for_func=False)[0]
def test_constantsimp_take_problem():
c = exp(C1) + 2
assert len(Poly(constantsimp(exp(C1) + c + c*x, [C1])).gens) == 2
def test_issue_6879():
f = Function('f')
eq = Eq(Derivative(f(x), x, 2) - 2*Derivative(f(x), x) + f(x), sin(x))
sol = (C1 + C2*x)*exp(x) + cos(x)/2
assert dsolve(eq).rhs == sol
assert checkodesol(eq, sol, order=1, solve_for_func=False)[0]
def test_issue_6989():
f = Function('f')
k = Symbol('k')
eq = f(x).diff(x) - x*exp(-k*x)
csol = Eq(f(x), C1 + Piecewise(
((-k*x - 1)*exp(-k*x)/k**2, Ne(k**2, 0)),
(x**2/2, True)
))
sol = dsolve(eq, f(x))
assert sol == csol
assert checkodesol(eq, sol, order=1, solve_for_func=False)[0]
eq = -f(x).diff(x) + x*exp(-k*x)
csol = Eq(f(x), C1 + Piecewise(
((-k*x - 1)*exp(-k*x)/k**2, Ne(k**2, 0)),
(x**2/2, True)
))
sol = dsolve(eq, f(x))
assert sol == csol
assert checkodesol(eq, sol, order=1, solve_for_func=False)[0]
def test_heuristic1():
y, a, b, c, a4, a3, a2, a1, a0 = symbols("y a b c a4 a3 a2 a1 a0")
f = Function('f')
xi = Function('xi')
eta = Function('eta')
df = f(x).diff(x)
eq = Eq(df, x**2*f(x))
eq1 = f(x).diff(x) + a*f(x) - c*exp(b*x)
eq2 = f(x).diff(x) + 2*x*f(x) - x*exp(-x**2)
eq3 = (1 + 2*x)*df + 2 - 4*exp(-f(x))
eq4 = f(x).diff(x) - (a4*x**4 + a3*x**3 + a2*x**2 + a1*x + a0)**Rational(-1, 2)
eq5 = x**2*df - f(x) + x**2*exp(x - (1/x))
eqlist = [eq, eq1, eq2, eq3, eq4, eq5]
i = infinitesimals(eq, hint='abaco1_simple')
assert i == [{eta(x, f(x)): exp(x**3/3), xi(x, f(x)): 0},
{eta(x, f(x)): f(x), xi(x, f(x)): 0},
{eta(x, f(x)): 0, xi(x, f(x)): x**(-2)}]
i1 = infinitesimals(eq1, hint='abaco1_simple')
assert i1 == [{eta(x, f(x)): exp(-a*x), xi(x, f(x)): 0}]
i2 = infinitesimals(eq2, hint='abaco1_simple')
assert i2 == [{eta(x, f(x)): exp(-x**2), xi(x, f(x)): 0}]
i3 = infinitesimals(eq3, hint='abaco1_simple')
assert i3 == [{eta(x, f(x)): 0, xi(x, f(x)): 2*x + 1},
{eta(x, f(x)): 0, xi(x, f(x)): 1/(exp(f(x)) - 2)}]
i4 = infinitesimals(eq4, hint='abaco1_simple')
assert i4 == [{eta(x, f(x)): 1, xi(x, f(x)): 0},
{eta(x, f(x)): 0,
xi(x, f(x)): sqrt(a0 + a1*x + a2*x**2 + a3*x**3 + a4*x**4)}]
i5 = infinitesimals(eq5, hint='abaco1_simple')
assert i5 == [{xi(x, f(x)): 0, eta(x, f(x)): exp(-1/x)}]
ilist = [i, i1, i2, i3, i4, i5]
for eq, i in (zip(eqlist, ilist)):
check = checkinfsol(eq, i)
assert check[0]
def test_issue_6247():
eq = x**2*f(x)**2 + x*Derivative(f(x), x)
sol = Eq(f(x), 2*C1/(C1*x**2 - 1))
assert dsolve(eq, hint = 'separable_reduced') == sol
assert checkodesol(eq, sol, order=1)[0]
eq = f(x).diff(x, x) + 4*f(x)
sol = Eq(f(x), C1*sin(2*x) + C2*cos(2*x))
assert dsolve(eq) == sol
assert checkodesol(eq, sol, order=1)[0]
def test_heuristic2():
xi = Function('xi')
eta = Function('eta')
df = f(x).diff(x)
# This ODE can be solved by the Lie Group method, when there are
# better assumptions
eq = df - (f(x)/x)*(x*log(x**2/f(x)) + 2)
i = infinitesimals(eq, hint='abaco1_product')
assert i == [{eta(x, f(x)): f(x)*exp(-x), xi(x, f(x)): 0}]
assert checkinfsol(eq, i)[0]
@slow
def test_heuristic3():
xi = Function('xi')
eta = Function('eta')
a, b = symbols("a b")
df = f(x).diff(x)
eq = x**2*df + x*f(x) + f(x)**2 + x**2
i = infinitesimals(eq, hint='bivariate')
assert i == [{eta(x, f(x)): f(x), xi(x, f(x)): x}]
assert checkinfsol(eq, i)[0]
eq = x**2*(-f(x)**2 + df)- a*x**2*f(x) + 2 - a*x
i = infinitesimals(eq, hint='bivariate')
assert checkinfsol(eq, i)[0]
def test_heuristic_4():
y, a = symbols("y a")
eq = x*(f(x).diff(x)) + 1 - f(x)**2
i = infinitesimals(eq, hint='chi')
assert checkinfsol(eq, i)[0]
def test_heuristic_function_sum():
xi = Function('xi')
eta = Function('eta')
eq = f(x).diff(x) - (3*(1 + x**2/f(x)**2)*atan(f(x)/x) + (1 - 2*f(x))/x +
(1 - 3*f(x))*(x/f(x)**2))
i = infinitesimals(eq, hint='function_sum')
assert i == [{eta(x, f(x)): f(x)**(-2) + x**(-2), xi(x, f(x)): 0}]
assert checkinfsol(eq, i)[0]
def test_heuristic_abaco2_similar():
xi = Function('xi')
eta = Function('eta')
F = Function('F')
a, b = symbols("a b")
eq = f(x).diff(x) - F(a*x + b*f(x))
i = infinitesimals(eq, hint='abaco2_similar')
assert i == [{eta(x, f(x)): -a/b, xi(x, f(x)): 1}]
assert checkinfsol(eq, i)[0]
eq = f(x).diff(x) - (f(x)**2 / (sin(f(x) - x) - x**2 + 2*x*f(x)))
i = infinitesimals(eq, hint='abaco2_similar')
assert i == [{eta(x, f(x)): f(x)**2, xi(x, f(x)): f(x)**2}]
assert checkinfsol(eq, i)[0]
def test_heuristic_abaco2_unique_unknown():
xi = Function('xi')
eta = Function('eta')
F = Function('F')
a, b = symbols("a b")
x = Symbol("x", positive=True)
eq = f(x).diff(x) - x**(a - 1)*(f(x)**(1 - b))*F(x**a/a + f(x)**b/b)
i = infinitesimals(eq, hint='abaco2_unique_unknown')
assert i == [{eta(x, f(x)): -f(x)*f(x)**(-b), xi(x, f(x)): x*x**(-a)}]
assert checkinfsol(eq, i)[0]
eq = f(x).diff(x) + tan(F(x**2 + f(x)**2) + atan(x/f(x)))
i = infinitesimals(eq, hint='abaco2_unique_unknown')
assert i == [{eta(x, f(x)): x, xi(x, f(x)): -f(x)}]
assert checkinfsol(eq, i)[0]
eq = (x*f(x).diff(x) + f(x) + 2*x)**2 -4*x*f(x) -4*x**2 -4*a
i = infinitesimals(eq, hint='abaco2_unique_unknown')
assert checkinfsol(eq, i)[0]
def test_heuristic_linear():
a, b, m, n = symbols("a b m n")
eq = x**(n*(m + 1) - m)*(f(x).diff(x)) - a*f(x)**n -b*x**(n*(m + 1))
i = infinitesimals(eq, hint='linear')
assert checkinfsol(eq, i)[0]
@XFAIL
def test_kamke():
a, b, alpha, c = symbols("a b alpha c")
eq = x**2*(a*f(x)**2+(f(x).diff(x))) + b*x**alpha + c
i = infinitesimals(eq, hint='sum_function') # XFAIL
assert checkinfsol(eq, i)[0]
def test_series():
C1 = Symbol("C1")
eq = f(x).diff(x) - f(x)
sol = Eq(f(x), C1 + C1*x + C1*x**2/2 + C1*x**3/6 + C1*x**4/24 +
C1*x**5/120 + O(x**6))
assert dsolve(eq, hint='1st_power_series') == sol
assert checkodesol(eq, sol, order=1)[0]
eq = f(x).diff(x) - x*f(x)
sol = Eq(f(x), C1*x**4/8 + C1*x**2/2 + C1 + O(x**6))
assert dsolve(eq, hint='1st_power_series') == sol
assert checkodesol(eq, sol, order=1)[0]
eq = f(x).diff(x) - sin(x*f(x))
sol = Eq(f(x), (x - 2)**2*(1+ sin(4))*cos(4) + (x - 2)*sin(4) + 2 + O(x**3))
assert dsolve(eq, hint='1st_power_series', ics={f(2): 2}, n=3) == sol
# FIXME: The solution here should be O((x-2)**3) so is incorrect
#assert checkodesol(eq, sol, order=1)[0]
@XFAIL
@SKIP
def test_lie_group_issue17322_1():
eq=x*f(x).diff(x)*(f(x)+4) + (f(x)**2) -2*f(x)-2*x
sol = dsolve(eq, f(x)) # Hangs
assert checkodesol(eq, sol) == (True, 0)
@XFAIL
@SKIP
def test_lie_group_issue17322_2():
eq=x*f(x).diff(x)*(f(x)+4) + (f(x)**2) -2*f(x)-2*x
sol = dsolve(eq) # Hangs
assert checkodesol(eq, sol) == (True, 0)
@XFAIL
@SKIP
def test_lie_group_issue17322_3():
eq=Eq(x**7*Derivative(f(x), x) + 5*x**3*f(x)**2 - (2*x**2 + 2)*f(x)**3, 0)
sol = dsolve(eq) # Hangs
assert checkodesol(eq, sol) == (True, 0)
@XFAIL
def test_lie_group_issue17322_4():
eq=f(x).diff(x) - (f(x) - x*log(x))**2/x**2 + log(x)
sol = dsolve(eq) # NotImplementedError
assert checkodesol(eq, sol) == (True, 0)
@slow
def test_lie_group():
C1 = Symbol("C1")
x = Symbol("x") # assuming x is real generates an error!
a, b, c = symbols("a b c")
eq = f(x).diff(x)**2
sol = dsolve(eq, f(x), hint='lie_group')
assert checkodesol(eq, sol) == (True, 0)
eq = Eq(f(x).diff(x), x**2*f(x))
sol = dsolve(eq, f(x), hint='lie_group')
assert sol == Eq(f(x), C1*exp(x**3)**Rational(1, 3))
assert checkodesol(eq, sol) == (True, 0)
eq = f(x).diff(x) + a*f(x) - c*exp(b*x)
sol = dsolve(eq, f(x), hint='lie_group')
assert checkodesol(eq, sol) == (True, 0)
eq = f(x).diff(x) + 2*x*f(x) - x*exp(-x**2)
sol = dsolve(eq, f(x), hint='lie_group')
actual_sol = Eq(f(x), (C1 + x**2/2)*exp(-x**2))
errstr = str(eq)+' : '+str(sol)+' == '+str(actual_sol)
assert sol == actual_sol, errstr
assert checkodesol(eq, sol) == (True, 0)
eq = (1 + 2*x)*(f(x).diff(x)) + 2 - 4*exp(-f(x))
sol = dsolve(eq, f(x), hint='lie_group')
assert sol == Eq(f(x), log(C1/(2*x + 1) + 2))
assert checkodesol(eq, sol) == (True, 0)
eq = x**2*(f(x).diff(x)) - f(x) + x**2*exp(x - (1/x))
sol = dsolve(eq, f(x), hint='lie_group')
assert checkodesol(eq, sol)[0]
eq = x**2*f(x)**2 + x*Derivative(f(x), x)
sol = dsolve(eq, f(x), hint='lie_group')
assert sol == Eq(f(x), 2/(C1 + x**2))
assert checkodesol(eq, sol) == (True, 0)
eq=diff(f(x),x) + 2*x*f(x) - x*exp(-x**2)
sol = Eq(f(x), exp(-x**2)*(C1 + x**2/2))
assert sol == dsolve(eq, hint='lie_group')
assert checkodesol(eq, sol) == (True, 0)
eq = diff(f(x),x) + f(x)*cos(x) - exp(2*x)
sol = Eq(f(x), exp(-sin(x))*(C1 + Integral(exp(2*x)*exp(sin(x)), x)))
assert sol == dsolve(eq, hint='lie_group')
assert checkodesol(eq, sol) == (True, 0)
eq = diff(f(x),x) + f(x)*cos(x) - sin(2*x)/2
sol = Eq(f(x), C1*exp(-sin(x)) + sin(x) - 1)
assert sol == dsolve(eq, hint='lie_group')
assert checkodesol(eq, sol) == (True, 0)
eq = x*diff(f(x),x) + f(x) - x*sin(x)
sol = Eq(f(x), (C1 - x*cos(x) + sin(x))/x)
assert sol == dsolve(eq, hint='lie_group')
assert checkodesol(eq, sol) == (True, 0)
eq = x*diff(f(x),x) - f(x) - x/log(x)
sol = Eq(f(x), x*(C1 + log(log(x))))
assert sol == dsolve(eq, hint='lie_group')
assert checkodesol(eq, sol) == (True, 0)
eq = (f(x).diff(x)-f(x)) * (f(x).diff(x)+f(x))
sol = [Eq(f(x), C1*exp(x)), Eq(f(x), C1*exp(-x))]
assert set(sol) == set(dsolve(eq, hint='lie_group'))
assert checkodesol(eq, sol[0]) == (True, 0)
assert checkodesol(eq, sol[1]) == (True, 0)
eq = f(x).diff(x) * (f(x).diff(x) - f(x))
sol = [Eq(f(x), C1*exp(x)), Eq(f(x), C1)]
assert set(sol) == set(dsolve(eq, hint='lie_group'))
assert checkodesol(eq, sol[0]) == (True, 0)
assert checkodesol(eq, sol[1]) == (True, 0)
@XFAIL
def test_lie_group_issue15219():
eqn = exp(f(x).diff(x)-f(x))
assert 'lie_group' not in classify_ode(eqn, f(x))
def test_user_infinitesimals():
x = Symbol("x") # assuming x is real generates an error
eq = x*(f(x).diff(x)) + 1 - f(x)**2
sol = Eq(f(x), (C1 + x**2)/(C1 - x**2))
infinitesimals = {'xi':sqrt(f(x) - 1)/sqrt(f(x) + 1), 'eta':0}
assert dsolve(eq, hint='lie_group', **infinitesimals) == sol
assert checkodesol(eq, sol) == (True, 0)
def test_issue_7081():
eq = x*(f(x).diff(x)) + 1 - f(x)**2
s = Eq(f(x), -1/(-C1 + x**2)*(C1 + x**2))
assert dsolve(eq) == s
assert checkodesol(eq, s) == (True, 0)
@slow
def test_2nd_power_series_ordinary():
C1, C2 = symbols("C1 C2")
eq = f(x).diff(x, 2) - x*f(x)
assert classify_ode(eq) == ('2nd_linear_airy', '2nd_power_series_ordinary')
sol = Eq(f(x), C2*(x**3/6 + 1) + C1*x*(x**3/12 + 1) + O(x**6))
assert dsolve(eq, hint='2nd_power_series_ordinary') == sol
assert checkodesol(eq, sol) == (True, 0)
sol = Eq(f(x), C2*((x + 2)**4/6 + (x + 2)**3/6 - (x + 2)**2 + 1)
+ C1*(x + (x + 2)**4/12 - (x + 2)**3/3 + S(2))
+ O(x**6))
assert dsolve(eq, hint='2nd_power_series_ordinary', x0=-2) == sol
# FIXME: Solution should be O((x+2)**6)
# assert checkodesol(eq, sol) == (True, 0)
sol = Eq(f(x), C2*x + C1 + O(x**2))
assert dsolve(eq, hint='2nd_power_series_ordinary', n=2) == sol
assert checkodesol(eq, sol) == (True, 0)
eq = (1 + x**2)*(f(x).diff(x, 2)) + 2*x*(f(x).diff(x)) -2*f(x)
assert classify_ode(eq) == ('2nd_power_series_ordinary',)
sol = Eq(f(x), C2*(-x**4/3 + x**2 + 1) + C1*x + O(x**6))
assert dsolve(eq) == sol
assert checkodesol(eq, sol) == (True, 0)
eq = f(x).diff(x, 2) + x*(f(x).diff(x)) + f(x)
assert classify_ode(eq) == ('2nd_power_series_ordinary',)
sol = Eq(f(x), C2*(x**4/8 - x**2/2 + 1) + C1*x*(-x**2/3 + 1) + O(x**6))
assert dsolve(eq) == sol
# FIXME: checkodesol fails for this solution...
# assert checkodesol(eq, sol) == (True, 0)
eq = f(x).diff(x, 2) + f(x).diff(x) - x*f(x)
assert classify_ode(eq) == ('2nd_power_series_ordinary',)
sol = Eq(f(x), C2*(-x**4/24 + x**3/6 + 1)
+ C1*x*(x**3/24 + x**2/6 - x/2 + 1) + O(x**6))
assert dsolve(eq) == sol
# FIXME: checkodesol fails for this solution...
# assert checkodesol(eq, sol) == (True, 0)
eq = f(x).diff(x, 2) + x*f(x)
assert classify_ode(eq) == ('2nd_linear_airy', '2nd_power_series_ordinary')
sol = Eq(f(x), C2*(x**6/180 - x**3/6 + 1) + C1*x*(-x**3/12 + 1) + O(x**7))
assert dsolve(eq, hint='2nd_power_series_ordinary', n=7) == sol
assert checkodesol(eq, sol) == (True, 0)
def test_Airy_equation():
eq = f(x).diff(x, 2) - x*f(x)
sol = Eq(f(x), C1*airyai(x) + C2*airybi(x))
sols = constant_renumber(sol)
assert classify_ode(eq) == ("2nd_linear_airy",'2nd_power_series_ordinary')
assert checkodesol(eq, sol) == (True, 0)
assert dsolve(eq, f(x)) in (sol, sols)
assert dsolve(eq, f(x), hint='2nd_linear_airy') in (sol, sols)
eq = f(x).diff(x, 2) + 2*x*f(x)
sol = Eq(f(x), C1*airyai(-2**(S(1)/3)*x) + C2*airybi(-2**(S(1)/3)*x))
sols = constant_renumber(sol)
assert classify_ode(eq) == ("2nd_linear_airy",'2nd_power_series_ordinary')
assert checkodesol(eq, sol) == (True, 0)
assert dsolve(eq, f(x)) in (sol, sols)
assert dsolve(eq, f(x), hint='2nd_linear_airy') in (sol, sols)
def test_2nd_power_series_regular():
C1, C2 = symbols("C1 C2")
eq = x**2*(f(x).diff(x, 2)) - 3*x*(f(x).diff(x)) + (4*x + 4)*f(x)
sol = Eq(f(x), C1*x**2*(-16*x**3/9 + 4*x**2 - 4*x + 1) + O(x**6))
assert dsolve(eq, hint='2nd_power_series_regular') == sol
assert checkodesol(eq, sol) == (True, 0)
eq = 4*x**2*(f(x).diff(x, 2)) -8*x**2*(f(x).diff(x)) + (4*x**2 +
1)*f(x)
sol = Eq(f(x), C1*sqrt(x)*(x**4/24 + x**3/6 + x**2/2 + x + 1) + O(x**6))
assert dsolve(eq, hint='2nd_power_series_regular') == sol
assert checkodesol(eq, sol) == (True, 0)
eq = x**2*(f(x).diff(x, 2)) - x**2*(f(x).diff(x)) + (
x**2 - 2)*f(x)
sol = Eq(f(x), C1*(-x**6/720 - 3*x**5/80 - x**4/8 + x**2/2 + x/2 + 1)/x +
C2*x**2*(-x**3/60 + x**2/20 + x/2 + 1) + O(x**6))
assert dsolve(eq) == sol
assert checkodesol(eq, sol) == (True, 0)
eq = x**2*(f(x).diff(x, 2)) + x*(f(x).diff(x)) + (x**2 - Rational(1, 4))*f(x)
sol = Eq(f(x), C1*(x**4/24 - x**2/2 + 1)/sqrt(x) +
C2*sqrt(x)*(x**4/120 - x**2/6 + 1) + O(x**6))
assert dsolve(eq, hint='2nd_power_series_regular') == sol
assert checkodesol(eq, sol) == (True, 0)
def test_2nd_linear_bessel_equation():
eq = x**2*(f(x).diff(x, 2)) + x*(f(x).diff(x)) + (x**2 - 4)*f(x)
sol = Eq(f(x), C1*besselj(2, x) + C2*bessely(2, x))
sols = constant_renumber(sol)
assert dsolve(eq, f(x)) in (sol, sols)
assert dsolve(eq, f(x), hint='2nd_linear_bessel') in (sol, sols)
assert checkodesol(eq, sol, order=2, solve_for_func=False) == (True, 0)
eq = x**2*(f(x).diff(x, 2)) + x*(f(x).diff(x)) + (x**2 +25)*f(x)
sol = Eq(f(x), C1*besselj(5*I, x) + C2*bessely(5*I, x))
sols = constant_renumber(sol)
assert dsolve(eq, f(x)) in (sol, sols)
assert dsolve(eq, f(x), hint='2nd_linear_bessel') in (sol, sols)
checkodesol(eq, sol, order=2, solve_for_func=False) == (True, 0)
eq = x**2*(f(x).diff(x, 2)) + x*(f(x).diff(x)) + (x**2)*f(x)
sol = Eq(f(x), C1*besselj(0, x) + C2*bessely(0, x))
sols = constant_renumber(sol)
assert dsolve(eq, f(x)) in (sol, sols)
assert dsolve(eq, f(x), hint='2nd_linear_bessel') in (sol, sols)
assert checkodesol(eq, sol, order=2, solve_for_func=False) == (True, 0)
eq = x**2*(f(x).diff(x, 2)) + x*(f(x).diff(x)) + (81*x**2 -S(1)/9)*f(x)
sol = Eq(f(x), C1*besselj(S(1)/3, 9*x) + C2*bessely(S(1)/3, 9*x))
sols = constant_renumber(sol)
assert dsolve(eq, f(x)) in (sol, sols)
assert dsolve(eq, f(x), hint='2nd_linear_bessel') in (sol, sols)
checkodesol(eq, sol, order=2, solve_for_func=False) == (True, 0)
eq = x**2*(f(x).diff(x, 2)) + x*(f(x).diff(x)) + (x**4 - 4)*f(x)
sol = Eq(f(x), C1*besselj(1, x**2/2) + C2*bessely(1, x**2/2))
sols = constant_renumber(sol)
assert dsolve(eq, f(x)) in (sol, sols)
assert dsolve(eq, f(x), hint='2nd_linear_bessel') in (sol, sols)
assert checkodesol(eq, sol, order=2, solve_for_func=False) == (True, 0)
eq = x**2*(f(x).diff(x, 2)) + 2*x*(f(x).diff(x)) + (x**4 - 4)*f(x)
sol = Eq(f(x), (C1*besselj(sqrt(17)/4, x**2/2) + C2*bessely(sqrt(17)/4, x**2/2))/sqrt(x))
sols = constant_renumber(sol)
assert dsolve(eq, f(x)) in (sol, sols)
assert dsolve(eq, f(x), hint='2nd_linear_bessel') in (sol, sols)
assert checkodesol(eq, sol, order=2, solve_for_func=False) == (True, 0)
eq = x**2*(f(x).diff(x, 2)) + x*(f(x).diff(x)) + (x**2 - S(1)/4)*f(x)
sol = Eq(f(x), C1*besselj(S(1)/2, x) + C2*bessely(S(1)/2, x))
sols = constant_renumber(sol)
assert dsolve(eq, f(x)) in (sol, sols)
assert dsolve(eq, f(x), hint='2nd_linear_bessel') in (sol, sols)
assert checkodesol(eq, sol, order=2, solve_for_func=False) == (True, 0)
eq = x**2*(f(x).diff(x, 2)) - 3*x*(f(x).diff(x)) + (4*x + 4)*f(x)
sol = Eq(f(x), x**2*(C1*besselj(0, 4*sqrt(x)) + C2*bessely(0, 4*sqrt(x))))
sols = constant_renumber(sol)
assert dsolve(eq, f(x)) in (sol, sols)
assert dsolve(eq, f(x), hint='2nd_linear_bessel') in (sol, sols)
assert checkodesol(eq, sol, order=2, solve_for_func=False) == (True, 0)
eq = x*(f(x).diff(x, 2)) - f(x).diff(x) + 4*x**3*f(x)
sol = Eq(f(x), x*(C1*besselj(S(1)/2, x**2) + C2*bessely(S(1)/2, x**2)))
sols = constant_renumber(sol)
assert dsolve(eq, f(x)) in (sol, sols)
assert dsolve(eq, f(x), hint='2nd_linear_bessel') in (sol, sols)
assert checkodesol(eq, sol, order=2, solve_for_func=False) == (True, 0)
eq = (x-2)**2*(f(x).diff(x, 2)) - (x-2)*f(x).diff(x) + 4*(x-2)**2*f(x)
sol = Eq(f(x), (x - 2)*(C1*besselj(1, 2*x - 4) + C2*bessely(1, 2*x - 4)))
sols = constant_renumber(sol)
assert dsolve(eq, f(x)) in (sol, sols)
assert dsolve(eq, f(x), hint='2nd_linear_bessel') in (sol, sols)
assert checkodesol(eq, sol, order=2, solve_for_func=False) == (True, 0)
def test_issue_7093():
x = Symbol("x") # assuming x is real leads to an error
sol = [Eq(f(x), C1 - 2*x*sqrt(x**3)/5),
Eq(f(x), C1 + 2*x*sqrt(x**3)/5)]
eq = Derivative(f(x), x)**2 - x**3
assert set(dsolve(eq)) == set(sol)
assert checkodesol(eq, sol) == [(True, 0)] * 2
def test_dsolve_linsystem_symbol():
eps = Symbol('epsilon', positive=True)
eq1 = (Eq(diff(f(x), x), -eps*g(x)), Eq(diff(g(x), x), eps*f(x)))
sol1 = [Eq(f(x), -C1*eps*cos(eps*x) - C2*eps*sin(eps*x)),
Eq(g(x), -C1*eps*sin(eps*x) + C2*eps*cos(eps*x))]
assert checksysodesol(eq1, sol1) == (True, [0, 0])
def test_C1_function_9239():
t = Symbol('t')
C1 = Function('C1')
C2 = Function('C2')
C3 = Symbol('C3')
C4 = Symbol('C4')
eq = (Eq(diff(C1(t), t), 9*C2(t)), Eq(diff(C2(t), t), 12*C1(t)))
sol = [Eq(C1(t), 9*C3*exp(6*sqrt(3)*t) + 9*C4*exp(-6*sqrt(3)*t)),
Eq(C2(t), 6*sqrt(3)*C3*exp(6*sqrt(3)*t) - 6*sqrt(3)*C4*exp(-6*sqrt(3)*t))]
assert checksysodesol(eq, sol) == (True, [0, 0])
def test_issue_15056():
t = Symbol('t')
C3 = Symbol('C3')
assert get_numbered_constants(Symbol('C1') * Function('C2')(t)) == C3
def test_issue_10379():
t,y = symbols('t,y')
eq = f(t).diff(t)-(1-51.05*y*f(t))
sol = Eq(f(t), (0.019588638589618*exp(y*(C1 - 51.05*t)) + 0.019588638589618)/y)
dsolve_sol = dsolve(eq, rational=False)
assert str(dsolve_sol) == str(sol)
assert checkodesol(eq, dsolve_sol)[0]
def test_issue_10867():
x = Symbol('x')
eq = Eq(g(x).diff(x).diff(x), (x-2)**2 + (x-3)**3)
sol = Eq(g(x), C1 + C2*x + x**5/20 - 2*x**4/3 + 23*x**3/6 - 23*x**2/2)
assert dsolve(eq, g(x)) == sol
assert checkodesol(eq, sol, order=2, solve_for_func=False) == (True, 0)
def test_issue_11290():
eq = cos(f(x)) - (x*sin(f(x)) - f(x)**2)*f(x).diff(x)
sol_1 = dsolve(eq, f(x), simplify=False, hint='1st_exact_Integral')
sol_0 = dsolve(eq, f(x), simplify=False, hint='1st_exact')
assert sol_1.dummy_eq(Eq(Subs(
Integral(u**2 - x*sin(u) - Integral(-sin(u), x), u) +
Integral(cos(u), x), u, f(x)), C1))
assert sol_1.doit() == sol_0
assert checkodesol(eq, sol_0, order=1, solve_for_func=False)
assert checkodesol(eq, sol_1, order=1, solve_for_func=False)
def test_issue_4838():
# Issue #15999
eq = f(x).diff(x) - C1*f(x)
sol = Eq(f(x), C2*exp(C1*x))
assert dsolve(eq, f(x)) == sol
assert checkodesol(eq, sol, order=1, solve_for_func=False) == (True, 0)
# Issue #13691
eq = f(x).diff(x) - C1*g(x).diff(x)
sol = Eq(f(x), C2 + C1*g(x))
assert dsolve(eq, f(x)) == sol
assert checkodesol(eq, sol, f(x), order=1, solve_for_func=False) == (True, 0)
# Issue #4838
eq = f(x).diff(x) - 3*C1 - 3*x**2
sol = Eq(f(x), C2 + 3*C1*x + x**3)
assert dsolve(eq, f(x)) == sol
assert checkodesol(eq, sol, order=1, solve_for_func=False) == (True, 0)
@slow
def test_issue_14395():
eq = Derivative(f(x), x, x) + 9*f(x) - sec(x)
sol = Eq(f(x), (C1 - x/3 + sin(2*x)/3)*sin(3*x) + (C2 + log(cos(x))
- 2*log(cos(x)**2)/3 + 2*cos(x)**2/3)*cos(3*x))
assert dsolve(eq, f(x)) == sol
# FIXME: assert checkodesol(eq, sol, order=2, solve_for_func=False) == (True, 0)
# Needs to be a way to know how to combine derivatives in the expression
def test_factoring_ode():
from sympy import Mul
eqn = Derivative(x*f(x), x, x, x) + Derivative(f(x), x, x, x)
# 2-arg Mul!
soln = Eq(f(x), C1 + C2*x + C3/Mul(2, (x + 1), evaluate=False))
assert checkodesol(eqn, soln, order=2, solve_for_func=False)[0]
assert soln == dsolve(eqn, f(x))
def test_issue_11542():
m = 96
g = 9.8
k = .2
f1 = g * m
t = Symbol('t')
v = Function('v')
v_equation = dsolve(f1 - k * (v(t) ** 2) - m * Derivative(v(t)), 0)
assert str(v_equation) == \
'Eq(v(t), -68.585712797929/tanh(C1 - 0.142886901662352*t))'
def test_issue_15913():
eq = -C1/x - 2*x*f(x) - f(x) + Derivative(f(x), x)
sol = C2*exp(x**2 + x) + exp(x**2 + x)*Integral(C1*exp(-x**2 - x)/x, x)
assert checkodesol(eq, sol) == (True, 0)
sol = C1 + C2*exp(-x*y)
eq = Derivative(y*f(x), x) + f(x).diff(x, 2)
assert checkodesol(eq, sol, f(x)) == (True, 0)
def test_issue_16146():
raises(ValueError, lambda: dsolve([f(x).diff(x), g(x).diff(x)], [f(x), g(x), h(x)]))
raises(ValueError, lambda: dsolve([f(x).diff(x), g(x).diff(x)], [f(x)]))
def test_dsolve_remove_redundant_solutions():
eq = (f(x)-2)*f(x).diff(x)
sol = Eq(f(x), C1)
assert dsolve(eq) == sol
eq = (f(x)-sin(x))*(f(x).diff(x, 2))
sol = {Eq(f(x), C1 + C2*x), Eq(f(x), sin(x))}
assert set(dsolve(eq)) == sol
eq = (f(x)**2-2*f(x)+1)*f(x).diff(x, 3)
sol = Eq(f(x), C1 + C2*x + C3*x**2)
assert dsolve(eq) == sol
def test_issue_17322():
eq = (f(x).diff(x)-f(x)) * (f(x).diff(x)+f(x))
sol = [Eq(f(x), C1*exp(-x)), Eq(f(x), C1*exp(x))]
assert set(sol) == set(dsolve(eq, hint='lie_group'))
assert checkodesol(eq, sol) == 2*[(True, 0)]
eq = f(x).diff(x)*(f(x).diff(x)+f(x))
sol = [Eq(f(x), C1), Eq(f(x), C1*exp(-x))]
assert set(sol) == set(dsolve(eq, hint='lie_group'))
assert checkodesol(eq, sol) == 2*[(True, 0)]
def test_2nd_2F1_hypergeometric():
eq = x*(x-1)*f(x).diff(x, 2) + (S(3)/2 -2*x)*f(x).diff(x) + 2*f(x)
sol = Eq(f(x), C1*x**(S(5)/2)*hyper((S(3)/2, S(1)/2), (S(7)/2,), x) + C2*hyper((-1, -2), (-S(3)/2,), x))
assert sol == dsolve(eq, hint='2nd_hypergeometric')
assert checkodesol(eq, sol) == (True, 0)
eq = x*(x-1)*f(x).diff(x, 2) + (S(7)/2*x)*f(x).diff(x) + f(x)
sol = Eq(f(x), (C1*(1 - x)**(S(5)/2)*hyper((S(1)/2, 2), (S(7)/2,), 1 - x) +
C2*hyper((-S(1)/2, -2), (-S(3)/2,), 1 - x))/(x - 1)**(S(5)/2))
assert sol == dsolve(eq, hint='2nd_hypergeometric')
assert checkodesol(eq, sol) == (True, 0)
eq = x*(x-1)*f(x).diff(x, 2) + (S(3)+ S(7)/2*x)*f(x).diff(x) + f(x)
sol = Eq(f(x), (C1*(1 - x)**(S(11)/2)*hyper((S(1)/2, 2), (S(13)/2,), 1 - x) +
C2*hyper((-S(7)/2, -5), (-S(9)/2,), 1 - x))/(x - 1)**(S(11)/2))
assert sol == dsolve(eq, hint='2nd_hypergeometric')
assert checkodesol(eq, sol) == (True, 0)
eq = x*(x-1)*f(x).diff(x, 2) + (-1+ S(7)/2*x)*f(x).diff(x) + f(x)
sol = Eq(f(x), (C1 + C2*Integral(exp(Integral((1 - x/2)/(x*(x - 1)), x))/(1 -
x/2)**2, x))*exp(Integral(1/(x - 1), x)/4)*exp(-Integral(7/(x -
1), x)/4)*hyper((S(1)/2, -1), (1,), x))
assert sol == dsolve(eq, hint='2nd_hypergeometric_Integral')
assert checkodesol(eq, sol) == (True, 0)
eq = -x**(S(5)/7)*(-416*x**(S(9)/7)/9 - 2385*x**(S(5)/7)/49 + S(298)*x/3)*f(x)/(196*(-x**(S(6)/7) +
x)**2*(x**(S(6)/7) + x)**2) + Derivative(f(x), (x, 2))
sol = Eq(f(x), x**(S(45)/98)*(C1*x**(S(4)/49)*hyper((S(1)/3, -S(1)/2), (S(9)/7,), x**(S(2)/7)) +
C2*hyper((S(1)/21, -S(11)/14), (S(5)/7,), x**(S(2)/7)))/(x**(S(2)/7) - 1)**(S(19)/84))
assert sol == dsolve(eq, hint='2nd_hypergeometric')
# assert checkodesol(eq, sol) == (True, 0) #issue-https://github.com/sympy/sympy/issues/17702
def test_issue_5096():
eq = f(x).diff(x, x) + f(x) - x*sin(x - 2)
sol = Eq(f(x), C1*sin(x) + C2*cos(x) - x**2*cos(x - 2)/4 + x*sin(x - 2)/4)
assert sol == dsolve(eq, hint='nth_linear_constant_coeff_undetermined_coefficients')
assert checkodesol(eq, sol) == (True, 0)
eq = f(x).diff(x, 2) + f(x) - x**4*sin(x-1)
sol = Eq(f(x), C1*sin(x) + C2*cos(x) - x**5*cos(x - 1)/10 + x**4*sin(x - 1)/4 + x**3*cos(x - 1)/2 - 3*x**2*sin(x - 1)/4 - 3*x*cos(x - 1)/4)
assert sol == dsolve(eq, hint='nth_linear_constant_coeff_undetermined_coefficients')
assert checkodesol(eq, sol) == (True, 0)
eq = f(x).diff(x, 2) - f(x) - exp(x - 1)
sol = Eq(f(x), C2*exp(-x) + (C1 + x*exp(-1)/2)*exp(x))
got = dsolve(eq, hint='nth_linear_constant_coeff_undetermined_coefficients')
assert sol == got, got
assert checkodesol(eq, sol) == (True, 0)
eq = f(x).diff(x, 2)+f(x)-(sin(x-2)+1)
sol = Eq(f(x), C1*sin(x) + C2*cos(x) - x*cos(x - 2)/2 + 1)
assert sol == dsolve(eq, hint='nth_linear_constant_coeff_undetermined_coefficients')
assert checkodesol(eq, sol) == (True, 0)
eq = 2*x**2*f(x).diff(x, 2) + f(x) + sqrt(2*x)*sin(log(2*x)/2)
sol = Eq(f(x), sqrt(x)*(C1*sin(log(x)/2) + C2*cos(log(x)/2) + sqrt(2)*log(x)*cos(log(2*x)/2)/2))
assert sol == dsolve(eq, hint='nth_linear_euler_eq_nonhomogeneous_undetermined_coefficients')
assert checkodesol(eq, sol) == (True, 0)
eq = 2*x**2*f(x).diff(x, 2) + f(x) + sin(log(2*x)/2)
sol = Eq(f(x), C1*sqrt(x)*sin(log(x)/2) + C2*sqrt(x)*cos(log(x)/2) - 2*sin(log(2*x)/2)/5 - 4*cos(log(2*x)/2)/5)
assert sol == dsolve(eq, hint='nth_linear_euler_eq_nonhomogeneous_undetermined_coefficients')
assert checkodesol(eq, sol) == (True, 0)
def test_issue_15996():
eq = f(x).diff(x, 5) + 2*f(x).diff(x, 3) + f(x).diff(x) - 2*x - exp(I*x)
sol = Eq(f(x), C1 + x**2 + (C2 + x*(C3 - x/8 + 3*exp(I*x)/2 + 3*exp(-I*x)/2) + 5*exp(2*I*x)/16 + 2*I*exp(I*x) - 2*I*exp(-I*x))*sin(x) + (C4 + x*(C5 + I*x/8 + 3*I*exp(I*x)/2 - 3*I*exp(-I*x)/2) + 5*I*exp(2*I*x)/16 - 2*exp(I*x) - 2*exp(-I*x))*cos(x) - I*exp(I*x))
got = dsolve(eq, hint='nth_linear_constant_coeff_variation_of_parameters')
assert sol == got, got
assert checkodesol(eq, sol) == (True, 0)
eq = f(x).diff(x, 5) + 2*f(x).diff(x, 3) + f(x).diff(x) - exp(I*x)
sol = Eq(f(x), C1 + (C2 + x*(C3 - x/8) + 5*exp(2*I*x)/16)*sin(x) + (C4 + x*(C5 + I*x/8) + 5*I*exp(2*I*x)/16)*cos(x) - I*exp(I*x))
got = dsolve(eq, hint='nth_linear_constant_coeff_variation_of_parameters')
assert sol == got, got
assert checkodesol(eq, sol) == (True, 0)
def test_issue_18408():
eq = f(x).diff(x, 3) - f(x).diff(x) - sinh(x)
sol = Eq(f(x), C1 + C2*exp(-x) + C3*exp(x) + x*sinh(x)/2)
assert sol == dsolve(eq, hint='nth_linear_constant_coeff_undetermined_coefficients')
assert checkodesol(eq, sol) == (True, 0)
eq = f(x).diff(x, 2) - 49*f(x) - sinh(3*x)
sol = Eq(f(x), C1*exp(-7*x) + C2*exp(7*x) - sinh(3*x)/40)
assert sol == dsolve(eq, hint='nth_linear_constant_coeff_undetermined_coefficients')
assert checkodesol(eq, sol) == (True, 0)
eq = f(x).diff(x, 3) - f(x).diff(x) - sinh(x) - exp(x)
sol = Eq(f(x), C1 + C3*exp(-x) + x*sinh(x)/2 + (C2 + x/2)*exp(x))
assert sol == dsolve(eq, hint='nth_linear_constant_coeff_undetermined_coefficients')
assert checkodesol(eq, sol) == (True, 0)
def test_issue_9446():
f = Function('f')
assert dsolve(Eq(f(2 * x), sin(Derivative(f(x)))), f(x)) == \
[Eq(f(x), C1 + pi*x - Integral(asin(f(2*x)), x)), Eq(f(x), C1 + Integral(asin(f(2*x)), x))]
assert integrate(-asin(f(2*x)+pi), x) == -Integral(asin(pi + f(2*x)), x)
|
922676b22dc10de788021c4684a6619fdf1b66c36808b5ceba8d388f9faa7303 | #
# The main tests for the code in single.py are currently located in
# sympy/solvers/tests/test_ode.py
#
r"""
This File contains test functions for the individual hints used for solving ODEs.
Examples of each solver will be returned by _get_examples_ode_sol_name_of_solver.
Examples should have a key 'XFAIL' which stores the list of hints if they are
expected to fail for that hint.
Functions that are for internal use:
1) _ode_solver_test(ode_examples) - It takes dictionary of examples returned by
_get_examples method and tests them with their respective hints.
2) _test_particular_example(our_hint, example_name) - It tests the ODE example corresponding
to the hint provided.
3) _test_all_hints(runxfail=False) - It is used to test all the examples with all the hints
currently implemented. It calls _test_all_examples_for_one_hint() which outputs whether the
given hint functions properly if it classifies the ODE example.
If runxfail flag is set to True then it will only test the examples which are expected to fail.
Everytime the ODE of partiular solver are added then _test_all_hints() is to execuetd to find
the possible failures of different solver hints.
4) _test_all_examples_for_one_hint(our_hint, all_examples) - It takes hint as argument and checks
this hint against all the ODE examples and gives output as the number of ODEs matched, number
of ODEs which were solved correctly, list of ODEs which gives incorrect solution and list of
ODEs which raises exception.
"""
from sympy import (acos, asin, atan, cos, Derivative, Dummy, diff,
E, Eq, exp, I, log, pi, Piecewise, Rational, S, sin, sinh, tan,
sqrt, symbols, Ei, erfi)
from sympy.core import Function, Symbol
from sympy.functions import airyai, airybi, besselj, bessely
from sympy.integrals.risch import NonElementaryIntegral
from sympy.solvers.ode import classify_ode, dsolve
from sympy.solvers.ode.ode import allhints, _remove_redundant_solutions
from sympy.solvers.ode.single import (FirstLinear, ODEMatchError,
SingleODEProblem, SingleODESolver)
from sympy.solvers.ode.subscheck import checkodesol
from sympy.testing.pytest import raises, slow
import traceback
x = Symbol('x')
u = Symbol('u')
y = Symbol('y')
f = Function('f')
g = Function('g')
C1, C2, C3, C4, C5 = symbols('C1:6')
hint_message = """\
Hint did not match the example {example}.
The ODE is:
{eq}.
The expected hint was
{our_hint}\
"""
expected_sol_message = """\
Different solution found from dsolve for example {example}.
The ODE is:
{eq}
The expected solution was
{sol}
What dsolve returned is:
{dsolve_sol}\
"""
checkodesol_msg = """\
solution found is not correct for example {example}.
The ODE is:
{eq}\
"""
dsol_incorrect_msg = """\
solution returned by dsolve is incorrect when using {hint}.
The ODE is:
{eq}
The expected solution was
{sol}
what dsolve returned is:
{dsolve_sol}
You can test this with:
eq = {eq}
sol = dsolve(eq, hint='{hint}')
print(sol)
print(checkodesol(eq, sol))
"""
exception_msg = """\
dsolve raised exception : {e}
when using {hint} for the example {example}
You can test this with:
from sympy.solvers.ode.tests.test_single import _test_an_example
_test_an_example('{hint}', example_name = '{example}')
The ODE is:
{eq}
\
"""
check_hint_msg = """\
Tested hint was : {hint}
Total of {matched} examples matched with this hint.
Out of which {solve} gave correct results.
Examples which gave incorrect results are {unsolve}.
Examples which raised exceptions are {exceptions}
\
"""
def _ode_solver_test(ode_examples, run_slow_test=False):
our_hint = ode_examples['hint']
for example in ode_examples['examples']:
temp = {
'eq': ode_examples['examples'][example]['eq'],
'sol': ode_examples['examples'][example]['sol'],
'XFAIL': ode_examples['examples'][example].get('XFAIL', []),
'func': ode_examples['examples'][example].get('func',ode_examples['func']),
'example_name': example,
'slow': ode_examples['examples'][example].get('slow', False),
'checkodesol_XFAIL': ode_examples['examples'][example].get('checkodesol_XFAIL', False)
}
if (not run_slow_test) and temp['slow']:
continue
result = _test_particular_example(our_hint, temp, solver_flag=True)
if result['xpass_msg'] != "":
print(result['xpass_msg'])
def _test_all_hints(runxfail=False):
all_hints = list(allhints)+["default"]
all_examples = _get_all_examples()
for our_hint in all_hints:
if our_hint.endswith('_Integral') or 'series' in our_hint:
continue
_test_all_examples_for_one_hint(our_hint, all_examples, runxfail)
def _test_dummy_sol(expected_sol,dsolve_sol):
if type(dsolve_sol)==list:
return any(expected_sol.dummy_eq(sub_dsol) for sub_dsol in dsolve_sol)
else:
return expected_sol.dummy_eq(dsolve_sol)
def _test_an_example(our_hint, example_name):
all_examples = _get_all_examples()
for example in all_examples:
if example['example_name'] == example_name:
_test_particular_example(our_hint, example)
def _test_particular_example(our_hint, ode_example, solver_flag=False):
eq = ode_example['eq']
expected_sol = ode_example['sol']
example = ode_example['example_name']
xfail = our_hint in ode_example['XFAIL']
func = ode_example['func']
result = {'msg': '', 'xpass_msg': ''}
checkodesol_XFAIL = ode_example['checkodesol_XFAIL']
xpass = True
if solver_flag:
if our_hint not in classify_ode(eq, func):
message = hint_message.format(example=example, eq=eq, our_hint=our_hint)
raise AssertionError(message)
if our_hint in classify_ode(eq, func):
result['match_list'] = example
try:
dsolve_sol = dsolve(eq, func, hint=our_hint)
except Exception as e:
dsolve_sol = []
result['exception_list'] = example
if not solver_flag:
traceback.print_exc()
result['msg'] = exception_msg.format(e=str(e), hint=our_hint, example=example, eq=eq)
xpass = False
if solver_flag and dsolve_sol!=[]:
expect_sol_check = False
if type(dsolve_sol)==list:
for sub_sol in expected_sol:
if sub_sol.has(Dummy):
expect_sol_check = not _test_dummy_sol(sub_sol, dsolve_sol)
else:
expect_sol_check = sub_sol not in dsolve_sol
if expect_sol_check:
break
else:
expect_sol_check = dsolve_sol not in expected_sol
for sub_sol in expected_sol:
if sub_sol.has(Dummy):
expect_sol_check = not _test_dummy_sol(sub_sol, dsolve_sol)
if expect_sol_check:
message = expected_sol_message.format(example=example, eq=eq, sol=expected_sol, dsolve_sol=dsolve_sol)
raise AssertionError(message)
expected_checkodesol = [(True, 0) for i in range(len(expected_sol))]
if len(expected_sol) == 1:
expected_checkodesol = (True, 0)
if not checkodesol_XFAIL:
if checkodesol(eq, dsolve_sol, solve_for_func=False) != expected_checkodesol:
result['unsolve_list'] = example
xpass = False
message = dsol_incorrect_msg.format(hint=our_hint, eq=eq, sol=expected_sol,dsolve_sol=dsolve_sol)
if solver_flag:
message = checkodesol_msg.format(example=example, eq=eq)
raise AssertionError(message)
else:
result['msg'] = 'AssertionError: ' + message
if xpass and xfail:
result['xpass_msg'] = example + "is now passing for the hint" + our_hint
return result
def _test_all_examples_for_one_hint(our_hint, all_examples=[], runxfail=None):
if all_examples == []:
all_examples = _get_all_examples()
match_list, unsolve_list, exception_list = [], [], []
for ode_example in all_examples:
xfail = our_hint in ode_example['XFAIL']
if runxfail and not xfail:
continue
if xfail:
continue
result = _test_particular_example(our_hint, ode_example)
match_list += result.get('match_list',[])
unsolve_list += result.get('unsolve_list',[])
exception_list += result.get('exception_list',[])
if runxfail is not None:
msg = result['msg']
if msg!='':
print(result['msg'])
# print(result.get('xpass_msg',''))
if runxfail is None:
match_count = len(match_list)
solved = len(match_list)-len(unsolve_list)-len(exception_list)
msg = check_hint_msg.format(hint=our_hint, matched=match_count, solve=solved, unsolve=unsolve_list, exceptions=exception_list)
print(msg)
def test_SingleODESolver():
# Test that not implemented methods give NotImplementedError
# Subclasses should override these methods.
problem = SingleODEProblem(f(x).diff(x), f(x), x)
solver = SingleODESolver(problem)
raises(NotImplementedError, lambda: solver.matches())
raises(NotImplementedError, lambda: solver.get_general_solution())
raises(NotImplementedError, lambda: solver._matches())
raises(NotImplementedError, lambda: solver._get_general_solution())
# This ODE can not be solved by the FirstLinear solver. Here we test that
# it does not match and the asking for a general solution gives
# ODEMatchError
problem = SingleODEProblem(f(x).diff(x) + f(x)*f(x), f(x), x)
solver = FirstLinear(problem)
raises(ODEMatchError, lambda: solver.get_general_solution())
solver = FirstLinear(problem)
assert solver.matches() is False
#These are just test for order of ODE
problem = SingleODEProblem(f(x).diff(x) + f(x), f(x), x)
assert problem.order == 1
problem = SingleODEProblem(f(x).diff(x,4) + f(x).diff(x,2) - f(x).diff(x,3), f(x), x)
assert problem.order == 4
def test_nth_algebraic():
eqn = f(x) + f(x)*f(x).diff(x)
solns = [Eq(f(x), exp(x)),
Eq(f(x), C1*exp(C2*x))]
solns_final = _remove_redundant_solutions(eqn, solns, 2, x)
assert solns_final == [Eq(f(x), C1*exp(C2*x))]
_ode_solver_test(_get_examples_ode_sol_nth_algebraic())
@slow
def test_slow_examples_nth_order_reducible():
_ode_solver_test(_get_examples_ode_sol_nth_order_reducible(), run_slow_test=True)
@slow
def test_slow_examples_nth_linear_constant_coeff_undetermined_coefficients():
_ode_solver_test(_get_examples_ode_sol_nth_linear_undetermined_coefficients(), run_slow_test=True)
@slow
def test_slow_examples_separable():
_ode_solver_test(_get_examples_ode_sol_separable(), run_slow_test=True)
def test_nth_linear_constant_coeff_undetermined_coefficients():
_ode_solver_test(_get_examples_ode_sol_nth_linear_undetermined_coefficients())
def test_nth_order_reducible():
from sympy.solvers.ode.ode import _nth_order_reducible_match
F = lambda eq: _nth_order_reducible_match(eq, f(x))
D = Derivative
assert F(D(y*f(x), x, y) + D(f(x), x)) is None
assert F(D(y*f(y), y, y) + D(f(y), y)) is None
assert F(f(x)*D(f(x), x) + D(f(x), x, 2)) is None
assert F(D(x*f(y), y, 2) + D(u*y*f(x), x, 3)) is None # no simplification by design
assert F(D(f(y), y, 2) + D(f(y), y, 3) + D(f(x), x, 4)) is None
assert F(D(f(x), x, 2) + D(f(x), x, 3)) == dict(n=2)
_ode_solver_test(_get_examples_ode_sol_nth_order_reducible())
def test_separable():
_ode_solver_test(_get_examples_ode_sol_separable())
def test_factorable():
_ode_solver_test(_get_examples_ode_sol_factorable())
def test_Riccati_special_minus2():
_ode_solver_test(_get_examples_ode_sol_riccati())
def test_Bernoulli():
_ode_solver_test(_get_examples_ode_sol_bernoulli())
def test_1st_linear():
_ode_solver_test(_get_examples_ode_sol_1st_linear())
def test_almost_linear():
_ode_solver_test(_get_examples_ode_sol_almost_linear())
def test_Liouville_ODE():
hint = 'Liouville'
not_Liouville1 = classify_ode(diff(f(x), x)/x + f(x)*diff(f(x), x, x)/2 -
diff(f(x), x)**2/2, f(x))
not_Liouville2 = classify_ode(diff(f(x), x)/x + diff(f(x), x, x)/2 -
x*diff(f(x), x)**2/2, f(x))
assert hint not in not_Liouville1
assert hint not in not_Liouville2
assert hint + '_Integral' not in not_Liouville1
assert hint + '_Integral' not in not_Liouville2
_ode_solver_test(_get_examples_ode_sol_liouville())
def test_nth_order_linear_euler_eq_homogeneous():
x, t, a, b, c = symbols('x t a b c')
y = Function('y')
our_hint = "nth_linear_euler_eq_homogeneous"
eq = diff(f(t), t, 4)*t**4 - 13*diff(f(t), t, 2)*t**2 + 36*f(t)
assert our_hint in classify_ode(eq)
eq = a*y(t) + b*t*diff(y(t), t) + c*t**2*diff(y(t), t, 2)
assert our_hint in classify_ode(eq)
_ode_solver_test(_get_examples_ode_sol_euler_homogeneous())
def test_nth_order_linear_euler_eq_nonhomogeneous_undetermined_coefficients():
x, t = symbols('x t')
a, b, c, d = symbols('a b c d', integer=True)
our_hint = "nth_linear_euler_eq_nonhomogeneous_undetermined_coefficients"
eq = x**4*diff(f(x), x, 4) - 13*x**2*diff(f(x), x, 2) + 36*f(x) + x
assert our_hint in classify_ode(eq, f(x))
eq = a*x**2*diff(f(x), x, 2) + b*x*diff(f(x), x) + c*f(x) + d*log(x)
assert our_hint in classify_ode(eq, f(x))
_ode_solver_test(_get_examples_ode_sol_euler_undetermined_coeff())
def test_nth_order_linear_euler_eq_nonhomogeneous_variation_of_parameters():
x, t = symbols('x, t')
a, b, c, d = symbols('a, b, c, d', integer=True)
our_hint = "nth_linear_euler_eq_nonhomogeneous_variation_of_parameters"
eq = Eq(x**2*diff(f(x),x,2) - 8*x*diff(f(x),x) + 12*f(x), x**2)
assert our_hint in classify_ode(eq, f(x))
eq = Eq(a*x**3*diff(f(x),x,3) + b*x**2*diff(f(x),x,2) + c*x*diff(f(x),x) + d*f(x), x*log(x))
assert our_hint in classify_ode(eq, f(x))
_ode_solver_test(_get_examples_ode_sol_euler_var_para())
def _get_examples_ode_sol_euler_homogeneous():
return {
'hint': "nth_linear_euler_eq_homogeneous",
'func': f(x),
'examples':{
'euler_hom_01': {
'eq': Eq(-3*diff(f(x), x)*x + 2*x**2*diff(f(x), x, x), 0),
'sol': [Eq(f(x), C1 + C2*x**Rational(5, 2))],
},
'euler_hom_02': {
'eq': Eq(3*f(x) - 5*diff(f(x), x)*x + 2*x**2*diff(f(x), x, x), 0),
'sol': [Eq(f(x), C1*sqrt(x) + C2*x**3)]
},
'euler_hom_03': {
'eq': Eq(4*f(x) + 5*diff(f(x), x)*x + x**2*diff(f(x), x, x), 0),
'sol': [Eq(f(x), (C1 + C2*log(x))/x**2)]
},
'euler_hom_04': {
'eq': Eq(6*f(x) - 6*diff(f(x), x)*x + 1*x**2*diff(f(x), x, x) + x**3*diff(f(x), x, x, x), 0),
'sol': [Eq(f(x), C1/x**2 + C2*x + C3*x**3)]
},
'euler_hom_05': {
'eq': Eq(-125*f(x) + 61*diff(f(x), x)*x - 12*x**2*diff(f(x), x, x) + x**3*diff(f(x), x, x, x), 0),
'sol': [Eq(f(x), x**5*(C1 + C2*log(x) + C3*log(x)**2))]
},
'euler_hom_06': {
'eq': x**2*diff(f(x), x, 2) + x*diff(f(x), x) - 9*f(x),
'sol': [Eq(f(x), C1*x**-3 + C2*x**3)]
},
'euler_hom_07': {
'eq': sin(x)*x**2*f(x).diff(x, 2) + sin(x)*x*f(x).diff(x) + sin(x)*f(x),
'sol': [Eq(f(x), C1*sin(log(x)) + C2*cos(log(x)))],
'XFAIL': ['2nd_power_series_regular','nth_linear_euler_eq_nonhomogeneous_undetermined_coefficients']
},
}
}
def _get_examples_ode_sol_euler_undetermined_coeff():
return {
'hint': "nth_linear_euler_eq_nonhomogeneous_undetermined_coefficients",
'func': f(x),
'examples':{
'euler_undet_01': {
'eq': Eq(x**2*diff(f(x), x, x) + x*diff(f(x), x), 1),
'sol': [Eq(f(x), C1 + C2*log(x) + log(x)**2/2)]
},
'euler_undet_02': {
'eq': Eq(x**2*diff(f(x), x, x) - 2*x*diff(f(x), x) + 2*f(x), x**3),
'sol': [Eq(f(x), x*(C1 + C2*x + Rational(1, 2)*x**2))]
},
'euler_undet_03': {
'eq': Eq(x**2*diff(f(x), x, x) - x*diff(f(x), x) - 3*f(x), log(x)/x),
'sol': [Eq(f(x), (C1 + C2*x**4 - log(x)**2/8 - log(x)/16)/x)]
},
'euler_undet_04': {
'eq': Eq(x**2*diff(f(x), x, x) + 3*x*diff(f(x), x) - 8*f(x), log(x)**3 - log(x)),
'sol': [Eq(f(x), C1/x**4 + C2*x**2 - Rational(1,8)*log(x)**3 - Rational(3,32)*log(x)**2 - Rational(1,64)*log(x) - Rational(7, 256))]
},
'euler_undet_05': {
'eq': Eq(x**3*diff(f(x), x, x, x) - 3*x**2*diff(f(x), x, x) + 6*x*diff(f(x), x) - 6*f(x), log(x)),
'sol': [Eq(f(x), C1*x + C2*x**2 + C3*x**3 - Rational(1, 6)*log(x) - Rational(11, 36))]
},
}
}
def _get_examples_ode_sol_euler_var_para():
return {
'hint': "nth_linear_euler_eq_nonhomogeneous_variation_of_parameters",
'func': f(x),
'examples':{
'euler_var_01': {
'eq': Eq(x**2*Derivative(f(x), x, x) - 2*x*Derivative(f(x), x) + 2*f(x), x**4),
'sol': [Eq(f(x), x*(C1 + C2*x + x**3/6))]
},
'euler_var_02': {
'eq': Eq(3*x**2*diff(f(x), x, x) + 6*x*diff(f(x), x) - 6*f(x), x**3*exp(x)),
'sol': [Eq(f(x), C1/x**2 + C2*x + x*exp(x)/3 - 4*exp(x)/3 + 8*exp(x)/(3*x) - 8*exp(x)/(3*x**2))]
},
'euler_var_03': {
'eq': Eq(x**2*Derivative(f(x), x, x) - 2*x*Derivative(f(x), x) + 2*f(x), x**4*exp(x)),
'sol': [Eq(f(x), x*(C1 + C2*x + x*exp(x) - 2*exp(x)))]
},
'euler_var_04': {
'eq': x**2*Derivative(f(x), x, x) - 2*x*Derivative(f(x), x) + 2*f(x) - log(x),
'sol': [Eq(f(x), C1*x + C2*x**2 + log(x)/2 + Rational(3, 4))]
},
'euler_var_05': {
'eq': -exp(x) + (x*Derivative(f(x), (x, 2)) + Derivative(f(x), x))/x,
'sol': [Eq(f(x), C1 + C2*log(x) + exp(x) - Ei(x))]
},
}
}
def _get_examples_ode_sol_bernoulli():
# Type: Bernoulli, f'(x) + p(x)*f(x) == q(x)*f(x)**n
return {
'hint': "Bernoulli",
'func': f(x),
'examples':{
'bernoulli_01': {
'eq': Eq(x*f(x).diff(x) + f(x) - f(x)**2, 0),
'sol': [Eq(f(x), 1/(C1*x + 1))],
'XFAIL': ['separable_reduced']
},
'bernoulli_02': {
'eq': f(x).diff(x) - y*f(x),
'sol': [Eq(f(x), C1*exp(x*y))]
},
'bernoulli_03': {
'eq': f(x)*f(x).diff(x) - 1,
'sol': [Eq(f(x), -sqrt(C1 + 2*x)), Eq(f(x), sqrt(C1 + 2*x))]
},
}
}
def _get_examples_ode_sol_riccati():
# Type: Riccati special alpha = -2, a*dy/dx + b*y**2 + c*y/x +d/x**2
return {
'hint': "Riccati_special_minus2",
'func': f(x),
'examples':{
'riccati_01': {
'eq': 2*f(x).diff(x) + f(x)**2 - f(x)/x + 3*x**(-2),
'sol': [Eq(f(x), (-sqrt(3)*tan(C1 + sqrt(3)*log(x)/4) + 3)/(2*x))],
},
},
}
def _get_examples_ode_sol_1st_linear():
# Type: first order linear form f'(x)+p(x)f(x)=q(x)
return {
'hint': "1st_linear",
'func': f(x),
'examples':{
'linear_01': {
'eq': Eq(f(x).diff(x) + x*f(x), x**2),
'sol': [Eq(f(x), (C1 + x*exp(x**2/2)- sqrt(2)*sqrt(pi)*erfi(sqrt(2)*x/2)/2)*exp(-x**2/2))],
},
},
}
def _get_examples_ode_sol_factorable():
""" some hints are marked as xfail for examples because they missed additional algebraic solution
which could be found by Factorable hint. Fact_01 raise exception for
nth_linear_constant_coeff_undetermined_coefficients"""
y = Dummy('y')
return {
'hint': "factorable",
'func': f(x),
'examples':{
'fact_01': {
'eq': f(x) + f(x)*f(x).diff(x),
'sol': [Eq(f(x), 0), Eq(f(x), C1 - x)],
'XFAIL': ['separable', '1st_exact', '1st_linear', 'Bernoulli', '1st_homogeneous_coeff_best',
'1st_homogeneous_coeff_subs_indep_div_dep', '1st_homogeneous_coeff_subs_dep_div_indep',
'lie_group', 'nth_linear_euler_eq_nonhomogeneous_undetermined_coefficients',
'nth_linear_constant_coeff_variation_of_parameters',
'nth_linear_euler_eq_nonhomogeneous_variation_of_parameters',
'nth_linear_constant_coeff_undetermined_coefficients']
},
'fact_02': {
'eq': f(x)*(f(x).diff(x)+f(x)*x+2),
'sol': [Eq(f(x), (C1 - sqrt(2)*sqrt(pi)*erfi(sqrt(2)*x/2))*exp(-x**2/2)), Eq(f(x), 0)],
'XFAIL': ['Bernoulli', '1st_linear', 'lie_group']
},
'fact_03': {
'eq': (f(x).diff(x)+f(x)*x**2)*(f(x).diff(x, 2) + x*f(x)),
'sol': [Eq(f(x), C1*airyai(-x) + C2*airybi(-x)),Eq(f(x), C1*exp(-x**3/3))]
},
'fact_04': {
'eq': (f(x).diff(x)+f(x)*x**2)*(f(x).diff(x, 2) + f(x)),
'sol': [Eq(f(x), C1*exp(-x**3/3)), Eq(f(x), C1*sin(x) + C2*cos(x))]
},
'fact_05': {
'eq': (f(x).diff(x)**2-1)*(f(x).diff(x)**2-4),
'sol': [Eq(f(x), C1 - x), Eq(f(x), C1 + x), Eq(f(x), C1 + 2*x), Eq(f(x), C1 - 2*x)]
},
'fact_06': {
'eq': (f(x).diff(x, 2)-exp(f(x)))*f(x).diff(x),
'sol': [Eq(f(x), C1)]
},
'fact_07': {
'eq': (f(x).diff(x)**2-1)*(f(x)*f(x).diff(x)-1),
'sol': [Eq(f(x), C1 - x), Eq(f(x), -sqrt(C1 + 2*x)),Eq(f(x), sqrt(C1 + 2*x)), Eq(f(x), C1 + x)]
},
'fact_08': {
'eq': Derivative(f(x), x)**4 - 2*Derivative(f(x), x)**2 + 1,
'sol': [Eq(f(x), C1 - x), Eq(f(x), C1 + x)]
},
'fact_09': {
'eq': f(x)**2*Derivative(f(x), x)**6 - 2*f(x)**2*Derivative(f(x),
x)**4 + f(x)**2*Derivative(f(x), x)**2 - 2*f(x)*Derivative(f(x),
x)**5 + 4*f(x)*Derivative(f(x), x)**3 - 2*f(x)*Derivative(f(x),
x) + Derivative(f(x), x)**4 - 2*Derivative(f(x), x)**2 + 1,
'sol': [Eq(f(x), C1 - x), Eq(f(x), -sqrt(C1 + 2*x)),
Eq(f(x), sqrt(C1 + 2*x)), Eq(f(x), C1 + x)]
},
'fact_10': {
'eq': x**4*f(x)**2 + 2*x**4*f(x)*Derivative(f(x), (x, 2)) + x**4*Derivative(f(x),
(x, 2))**2 + 2*x**3*f(x)*Derivative(f(x), x) + 2*x**3*Derivative(f(x),
x)*Derivative(f(x), (x, 2)) - 7*x**2*f(x)**2 - 7*x**2*f(x)*Derivative(f(x),
(x, 2)) + x**2*Derivative(f(x), x)**2 - 7*x*f(x)*Derivative(f(x), x) + 12*f(x)**2,
'sol': [Eq(f(x), C1*besselj(2, x) + C2*bessely(2, x)), Eq(f(x), C1*besselj(sqrt(3),
x) + C2*bessely(sqrt(3), x))]
},
'fact_11': {
'eq': (f(x).diff(x, 2)-exp(f(x)))*(f(x).diff(x, 2)+exp(f(x))),
'sol': [], #currently dsolve doesn't return any solution for this example
'XFAIL': ['factorable']
},
#Below examples were added for the issue: https://github.com/sympy/sympy/issues/15889
'fact_12': {
'eq': exp(f(x).diff(x))-f(x)**2,
'sol': [Eq(NonElementaryIntegral(1/log(y**2), (y, f(x))), C1 + x)],
'XFAIL': ['lie_group'] #It shows not implemented error for lie_group.
},
'fact_13': {
'eq': f(x).diff(x)**2 - f(x)**3,
'sol': [Eq(f(x), 4/(C1**2 - 2*C1*x + x**2))],
'XFAIL': ['lie_group'] #It shows not implemented error for lie_group.
},
'fact_14': {
'eq': f(x).diff(x)**2 - f(x),
'sol': [Eq(f(x), C1**2/4 - C1*x/2 + x**2/4)]
},
'fact_15': {
'eq': f(x).diff(x)**2 - f(x)**2,
'sol': [Eq(f(x), C1*exp(x)), Eq(f(x), C1*exp(-x))]
},
'fact_16': {
'eq': f(x).diff(x)**2 - f(x)**3,
'sol': [Eq(f(x), 4/(C1**2 - 2*C1*x + x**2))]
},
}
}
def _get_examples_ode_sol_almost_linear():
from sympy import Ei
A = Symbol('A', positive=True)
f = Function('f')
d = f(x).diff(x)
return {
'hint': "almost_linear",
'func': f(x),
'examples':{
'almost_lin_01': {
'eq': x**2*f(x)**2*d + f(x)**3 + 1,
'sol': [Eq(f(x), (C1*exp(3/x) - 1)**Rational(1, 3)),
Eq(f(x), (-1 - sqrt(3)*I)*(C1*exp(3/x) - 1)**Rational(1, 3)/2),
Eq(f(x), (-1 + sqrt(3)*I)*(C1*exp(3/x) - 1)**Rational(1, 3)/2)],
},
'almost_lin_02': {
'eq': x*f(x)*d + 2*x*f(x)**2 + 1,
'sol': [Eq(f(x), -sqrt((C1 - 2*Ei(4*x))*exp(-4*x))), Eq(f(x), sqrt((C1 - 2*Ei(4*x))*exp(-4*x)))]
},
'almost_lin_03': {
'eq': x*d + x*f(x) + 1,
'sol': [Eq(f(x), (C1 - Ei(x))*exp(-x))]
},
'almost_lin_04': {
'eq': x*exp(f(x))*d + exp(f(x)) + 3*x,
'sol': [Eq(f(x), log(C1/x - x*Rational(3, 2)))],
},
'almost_lin_05': {
'eq': x + A*(x + diff(f(x), x) + f(x)) + diff(f(x), x) + f(x) + 2,
'sol': [Eq(f(x), (C1 + Piecewise(
(x, Eq(A + 1, 0)), ((-A*x + A - x - 1)*exp(x)/(A + 1), True)))*exp(-x))],
},
}
}
def _get_examples_ode_sol_liouville():
return {
'hint': "Liouville",
'func': f(x),
'examples':{
'liouville_01': {
'eq': diff(f(x), x)/x + diff(f(x), x, x)/2 - diff(f(x), x)**2/2,
'sol': [Eq(f(x), log(x/(C1 + C2*x)))],
},
'liouville_02': {
'eq': diff(x*exp(-f(x)), x, x),
'sol': [Eq(f(x), log(x/(C1 + C2*x)))]
},
'liouville_03': {
'eq': ((diff(f(x), x)/x + diff(f(x), x, x)/2 - diff(f(x), x)**2/2)*exp(-f(x))/exp(f(x))).expand(),
'sol': [Eq(f(x), log(x/(C1 + C2*x)))]
},
'liouville_04': {
'eq': diff(f(x), x, x) + 1/f(x)*(diff(f(x), x))**2 + 1/x*diff(f(x), x),
'sol': [Eq(f(x), -sqrt(C1 + C2*log(x))), Eq(f(x), sqrt(C1 + C2*log(x)))],
},
'liouville_05': {
'eq': x*diff(f(x), x, x) + x/f(x)*diff(f(x), x)**2 + x*diff(f(x), x),
'sol': [Eq(f(x), -sqrt(C1 + C2*exp(-x))), Eq(f(x), sqrt(C1 + C2*exp(-x)))],
},
'liouville_06': {
'eq': Eq((x*exp(f(x))).diff(x, x), 0),
'sol': [Eq(f(x), log(C1 + C2/x))],
},
}
}
def _get_examples_ode_sol_nth_algebraic():
M, m, r, t = symbols('M m r t')
phi = Function('phi')
# This one needs a substitution f' = g.
# 'algeb_12': {
# 'eq': -exp(x) + (x*Derivative(f(x), (x, 2)) + Derivative(f(x), x))/x,
# 'sol': [Eq(f(x), C1 + C2*log(x) + exp(x) - Ei(x))],
# },
return {
'hint': "nth_algebraic",
'func': f(x),
'examples':{
'algeb_01': {
'eq': f(x) * f(x).diff(x) * f(x).diff(x, x) * (f(x) - 1) * (f(x).diff(x) - x),
'sol': [Eq(f(x), C1 + x**2/2), Eq(f(x), C1 + C2*x)]
},
'algeb_02': {
'eq': f(x) * f(x).diff(x) * f(x).diff(x, x) * (f(x) - 1),
'sol': [Eq(f(x), C1 + C2*x)]
},
'algeb_03': {
'eq': f(x) * f(x).diff(x) * f(x).diff(x, x),
'sol': [Eq(f(x), C1 + C2*x)]
},
'algeb_04': {
'eq': Eq(-M * phi(t).diff(t),
Rational(3, 2) * m * r**2 * phi(t).diff(t) * phi(t).diff(t,t)),
'sol': [Eq(phi(t), C1), Eq(phi(t), C1 + C2*t - M*t**2/(3*m*r**2))],
'func': phi(t)
},
'algeb_05': {
'eq': (1 - sin(f(x))) * f(x).diff(x),
'sol': [Eq(f(x), C1)],
'XFAIL': ['separable'] #It raised exception.
},
'algeb_06': {
'eq': (diff(f(x)) - x)*(diff(f(x)) + x),
'sol': [Eq(f(x), C1 - x**2/2), Eq(f(x), C1 + x**2/2)]
},
'algeb_07': {
'eq': Eq(Derivative(f(x), x), Derivative(g(x), x)),
'sol': [Eq(f(x), C1 + g(x))],
},
'algeb_08': {
'eq': f(x).diff(x) - C1, #this example is from issue 15999
'sol': [Eq(f(x), C1*x + C2)],
},
'algeb_09': {
'eq': f(x)*f(x).diff(x),
'sol': [Eq(f(x), C1)],
},
'algeb_10': {
'eq': (diff(f(x)) - x)*(diff(f(x)) + x),
'sol': [Eq(f(x), C1 - x**2/2), Eq(f(x), C1 + x**2/2)],
},
'algeb_11': {
'eq': f(x) + f(x)*f(x).diff(x),
'sol': [Eq(f(x), 0), Eq(f(x), C1 - x)],
'XFAIL': ['separable', '1st_exact', '1st_linear', 'Bernoulli', '1st_homogeneous_coeff_best',
'1st_homogeneous_coeff_subs_indep_div_dep', '1st_homogeneous_coeff_subs_dep_div_indep',
'lie_group', 'nth_linear_constant_coeff_undetermined_coefficients',
'nth_linear_euler_eq_nonhomogeneous_undetermined_coefficients',
'nth_linear_constant_coeff_variation_of_parameters',
'nth_linear_euler_eq_nonhomogeneous_variation_of_parameters']
#nth_linear_constant_coeff_undetermined_coefficients raises exception rest all of them misses a solution.
},
'algeb_12': {
'eq': Derivative(x*f(x), x, x, x),
'sol': [Eq(f(x), (C1 + C2*x + C3*x**2) / x)],
'XFAIL': ['nth_algebraic'] # It passes only when prep=False is set in dsolve.
},
'algeb_13': {
'eq': Eq(Derivative(x*Derivative(f(x), x), x)/x, exp(x)),
'sol': [Eq(f(x), C1 + C2*log(x) + exp(x) - Ei(x))],
'XFAIL': ['nth_algebraic'] # It passes only when prep=False is set in dsolve.
},
}
}
def _get_examples_ode_sol_nth_order_reducible():
return {
'hint': "nth_order_reducible",
'func': f(x),
'examples':{
'reducible_01': {
'eq': Eq(x*Derivative(f(x), x)**2 + Derivative(f(x), x, 2), 0),
'sol': [Eq(f(x),C1 - sqrt(-1/C2)*log(-C2*sqrt(-1/C2) + x) +
sqrt(-1/C2)*log(C2*sqrt(-1/C2) + x))],
'slow': True,
},
'reducible_02': {
'eq': -exp(x) + (x*Derivative(f(x), (x, 2)) + Derivative(f(x), x))/x,
'sol': [Eq(f(x), C1 + C2*log(x) + exp(x) - Ei(x))],
'slow': True,
},
'reducible_03': {
'eq': Eq(sqrt(2) * f(x).diff(x,x,x) + f(x).diff(x), 0),
'sol': [Eq(f(x), C1 + C2*sin(2**Rational(3, 4)*x/2) + C3*cos(2**Rational(3, 4)*x/2))],
'slow': True,
},
'reducible_04': {
'eq': f(x).diff(x, 2) + 2*f(x).diff(x),
'sol': [Eq(f(x), C1 + C2*exp(-2*x))],
},
'reducible_05': {
'eq': f(x).diff(x, 3) + f(x).diff(x, 2) - 6*f(x).diff(x),
'sol': [Eq(f(x), C1 + C2*exp(-3*x) + C3*exp(2*x))],
'slow': True,
},
'reducible_06': {
'eq': f(x).diff(x, 4) - f(x).diff(x, 3) - 4*f(x).diff(x, 2) + \
4*f(x).diff(x),
'sol': [Eq(f(x), C1 + C2*exp(-2*x) + C3*exp(x) + C4*exp(2*x))],
'slow': True,
},
'reducible_07': {
'eq': f(x).diff(x, 4) + 3*f(x).diff(x, 3),
'sol': [Eq(f(x), C1 + C2*x + C3*x**2 + C4*exp(-3*x))],
'slow': True,
},
'reducible_08': {
'eq': f(x).diff(x, 4) - 2*f(x).diff(x, 2),
'sol': [Eq(f(x), C1 + C2*x + C3*exp(-sqrt(2)*x) + C4*exp(sqrt(2)*x))],
'slow': True,
},
'reducible_09': {
'eq': f(x).diff(x, 4) + 4*f(x).diff(x, 2),
'sol': [Eq(f(x), C1 + C2*x + C3*sin(2*x) + C4*cos(2*x))],
'slow': True,
},
'reducible_10': {
'eq': f(x).diff(x, 5) + 2*f(x).diff(x, 3) + f(x).diff(x),
'sol': [Eq(f(x), C1 + C2*(x*sin(x) + cos(x)) + C3*(-x*cos(x) + sin(x)) + C4*sin(x) + C5*cos(x))],
'slow': True,
},
'reducible_11': {
'eq': f(x).diff(x, 2) - f(x).diff(x)**3,
'sol': [Eq(f(x), C1 - sqrt(2)*I*(C2 + x)*sqrt(1/(C2 + x))),
Eq(f(x), C1 + sqrt(2)*I*(C2 + x)*sqrt(1/(C2 + x)))],
'slow': True,
},
}
}
def _get_examples_ode_sol_nth_linear_undetermined_coefficients():
# examples 3-27 below are from Ordinary Differential Equations,
# Tenenbaum and Pollard, pg. 231
g = exp(-x)
f2 = f(x).diff(x, 2)
c = 3*f(x).diff(x, 3) + 5*f2 + f(x).diff(x) - f(x) - x
return {
'hint': "nth_linear_constant_coeff_undetermined_coefficients",
'func': f(x),
'examples':{
'undet_01': {
'eq': c - x*g,
'sol': [Eq(f(x), C3*exp(x/3) - x + (C1 + x*(C2 - x**2/24 - 3*x/32))*exp(-x) - 1)],
'slow': True,
},
'undet_02': {
'eq': c - g,
'sol': [Eq(f(x), C3*exp(x/3) - x + (C1 + x*(C2 - x/8))*exp(-x) - 1)],
'slow': True,
},
'undet_03': {
'eq': f2 + 3*f(x).diff(x) + 2*f(x) - 4,
'sol': [Eq(f(x), C1*exp(-2*x) + C2*exp(-x) + 2)],
'slow': True,
},
'undet_04': {
'eq': f2 + 3*f(x).diff(x) + 2*f(x) - 12*exp(x),
'sol': [Eq(f(x), C1*exp(-2*x) + C2*exp(-x) + 2*exp(x))],
'slow': True,
},
'undet_05': {
'eq': f2 + 3*f(x).diff(x) + 2*f(x) - exp(I*x),
'sol': [Eq(f(x), C1*exp(-2*x) + C2*exp(-x) + exp(I*x)/10 - 3*I*exp(I*x)/10)],
'slow': True,
},
'undet_06': {
'eq': f2 + 3*f(x).diff(x) + 2*f(x) - sin(x),
'sol': [Eq(f(x), C1*exp(-2*x) + C2*exp(-x) + sin(x)/10 - 3*cos(x)/10)],
'slow': True,
},
'undet_07': {
'eq': f2 + 3*f(x).diff(x) + 2*f(x) - cos(x),
'sol': [Eq(f(x), C1*exp(-2*x) + C2*exp(-x) + 3*sin(x)/10 + cos(x)/10)],
'slow': True,
},
'undet_08': {
'eq': f2 + 3*f(x).diff(x) + 2*f(x) - (8 + 6*exp(x) + 2*sin(x)),
'sol': [Eq(f(x), C1*exp(-2*x) + C2*exp(-x) + exp(x) + sin(x)/5 - 3*cos(x)/5 + 4)],
'slow': True,
},
'undet_09': {
'eq': f2 + f(x).diff(x) + f(x) - x**2,
'sol': [Eq(f(x), -2*x + x**2 + (C1*sin(x*sqrt(3)/2) + C2*cos(x*sqrt(3)/2))*exp(-x/2))],
'slow': True,
},
'undet_10': {
'eq': f2 - 2*f(x).diff(x) - 8*f(x) - 9*x*exp(x) - 10*exp(-x),
'sol': [Eq(f(x), -x*exp(x) - 2*exp(-x) + C1*exp(-2*x) + C2*exp(4*x))],
'slow': True,
},
'undet_11': {
'eq': f2 - 3*f(x).diff(x) - 2*exp(2*x)*sin(x),
'sol': [Eq(f(x), C1 + C2*exp(3*x) - 3*exp(2*x)*sin(x)/5 - exp(2*x)*cos(x)/5)],
'slow': True,
},
'undet_12': {
'eq': f(x).diff(x, 4) - 2*f2 + f(x) - x + sin(x),
'sol': [Eq(f(x), x - sin(x)/4 + (C1 + C2*x)*exp(-x) + (C3 + C4*x)*exp(x))],
'slow': True,
},
'undet_13': {
'eq': f2 + f(x).diff(x) - x**2 - 2*x,
'sol': [Eq(f(x), C1 + x**3/3 + C2*exp(-x))],
'slow': True,
},
'undet_14': {
'eq': f2 + f(x).diff(x) - x - sin(2*x),
'sol': [Eq(f(x), C1 - x - sin(2*x)/5 - cos(2*x)/10 + x**2/2 + C2*exp(-x))],
'slow': True,
},
'undet_15': {
'eq': f2 + f(x) - 4*x*sin(x),
'sol': [Eq(f(x), (C1 - x**2)*cos(x) + (C2 + x)*sin(x))],
'slow': True,
},
'undet_16': {
'eq': f2 + 4*f(x) - x*sin(2*x),
'sol': [Eq(f(x), (C1 - x**2/8)*cos(2*x) + (C2 + x/16)*sin(2*x))],
'slow': True,
},
'undet_17': {
'eq': f2 + 2*f(x).diff(x) + f(x) - x**2*exp(-x),
'sol': [Eq(f(x), (C1 + x*(C2 + x**3/12))*exp(-x))],
'slow': True,
},
'undet_18': {
'eq': f(x).diff(x, 3) + 3*f2 + 3*f(x).diff(x) + f(x) - 2*exp(-x) + \
x**2*exp(-x),
'sol': [Eq(f(x), (C1 + x*(C2 + x*(C3 - x**3/60 + x/3)))*exp(-x))],
'slow': True,
},
'undet_19': {
'eq': f2 + 3*f(x).diff(x) + 2*f(x) - exp(-2*x) - x**2,
'sol': [Eq(f(x), C2*exp(-x) + x**2/2 - x*Rational(3,2) + (C1 - x)*exp(-2*x) + Rational(7,4))],
'slow': True,
},
'undet_20': {
'eq': f2 - 3*f(x).diff(x) + 2*f(x) - x*exp(-x),
'sol': [Eq(f(x), C1*exp(x) + C2*exp(2*x) + (6*x + 5)*exp(-x)/36)],
'slow': True,
},
'undet_21': {
'eq': f2 + f(x).diff(x) - 6*f(x) - x - exp(2*x),
'sol': [Eq(f(x), Rational(-1, 36) - x/6 + C2*exp(-3*x) + (C1 + x/5)*exp(2*x))],
'slow': True,
},
'undet_22': {
'eq': f2 + f(x) - sin(x) - exp(-x),
'sol': [Eq(f(x), C2*sin(x) + (C1 - x/2)*cos(x) + exp(-x)/2)],
'slow': True,
},
'undet_23': {
'eq': f(x).diff(x, 3) - 3*f2 + 3*f(x).diff(x) - f(x) - exp(x),
'sol': [Eq(f(x), (C1 + x*(C2 + x*(C3 + x/6)))*exp(x))],
'slow': True,
},
'undet_24': {
'eq': f2 + f(x) - S.Half - cos(2*x)/2,
'sol': [Eq(f(x), S.Half - cos(2*x)/6 + C1*sin(x) + C2*cos(x))],
'slow': True,
},
'undet_25': {
'eq': f(x).diff(x, 3) - f(x).diff(x) - exp(2*x)*(S.Half - cos(2*x)/2),
'sol': [Eq(f(x), C1 + C2*exp(-x) + C3*exp(x) + (-21*sin(2*x) + 27*cos(2*x) + 130)*exp(2*x)/1560)],
'slow': True,
},
'undet_26': {
'eq': (f(x).diff(x, 5) + 2*f(x).diff(x, 3) + f(x).diff(x) - 2*x -
sin(x) - cos(x)),
'sol': [Eq(f(x), C1 + x**2 + (C2 + x*(C3 - x/8))*sin(x) + (C4 + x*(C5 + x/8))*cos(x))],
'slow': True,
},
'undet_27': {
'eq': f2 + f(x) - cos(x)/2 + cos(3*x)/2,
'sol': [Eq(f(x), cos(3*x)/16 + C2*cos(x) + (C1 + x/4)*sin(x))],
'slow': True,
},
'undet_28': {
'eq': f(x).diff(x) - 1,
'sol': [Eq(f(x), C1 + x)],
'slow': True,
},
# https://github.com/sympy/sympy/issues/19358
'undet_29': {
'eq': f2 + f(x).diff(x) + exp(x-C1),
'sol': [Eq(f(x), C2 + C3*exp(-x) - exp(-C1 + x)/2)],
'slow': True,
},
}
}
def _get_examples_ode_sol_separable():
# test_separable1-5 are from Ordinary Differential Equations, Tenenbaum and
# Pollard, pg. 55
a = Symbol('a')
return {
'hint': "separable",
'func': f(x),
'examples':{
'separable_01': {
'eq': f(x).diff(x) - f(x),
'sol': [Eq(f(x), C1*exp(x))],
},
'separable_02': {
'eq': x*f(x).diff(x) - f(x),
'sol': [Eq(f(x), C1*x)],
},
'separable_03': {
'eq': f(x).diff(x) + sin(x),
'sol': [Eq(f(x), C1 + cos(x))],
},
'separable_04': {
'eq': f(x)**2 + 1 - (x**2 + 1)*f(x).diff(x),
'sol': [Eq(f(x), tan(C1 + atan(x)))],
},
'separable_05': {
'eq': f(x).diff(x)/tan(x) - f(x) - 2,
'sol': [Eq(f(x), C1/cos(x) - 2)],
},
'separable_06': {
'eq': f(x).diff(x) * (1 - sin(f(x))) - 1,
'sol': [Eq(-x + f(x) + cos(f(x)), C1)],
},
'separable_07': {
'eq': f(x)*x**2*f(x).diff(x) - f(x)**3 - 2*x**2*f(x).diff(x),
'sol': [Eq(f(x), (-x + sqrt(x*(4*C1*x + x - 4)))/(C1*x - 1)/2),
Eq(f(x), -((x + sqrt(x*(4*C1*x + x - 4)))/(C1*x - 1))/2)],
'slow': True,
},
'separable_08': {
'eq': f(x)**2 - 1 - (2*f(x) + x*f(x))*f(x).diff(x),
'sol': [Eq(f(x), -sqrt(C1*x**2 + 4*C1*x + 4*C1 + 1)),
Eq(f(x), sqrt(C1*x**2 + 4*C1*x + 4*C1 + 1))],
'slow': True,
},
'separable_09': {
'eq': x*log(x)*f(x).diff(x) + sqrt(1 + f(x)**2),
'sol': [Eq(f(x), sinh(C1 - log(log(x))))], #One more solution is f(x)=I
'slow': True,
'checkodesol_XFAIL': True,
},
'separable_10': {
'eq': exp(x + 1)*tan(f(x)) + cos(f(x))*f(x).diff(x),
'sol': [Eq(E*exp(x) + log(cos(f(x)) - 1)/2 - log(cos(f(x)) + 1)/2 + cos(f(x)), C1)],
'slow': True,
},
'separable_11': {
'eq': (x*cos(f(x)) + x**2*sin(f(x))*f(x).diff(x) - a**2*sin(f(x))*f(x).diff(x)),
'sol': [Eq(f(x), -acos(C1*sqrt(-a**2 + x**2)) + 2*pi),
Eq(f(x), acos(C1*sqrt(-a**2 + x**2)))],
'slow': True,
},
'separable_12': {
'eq': f(x).diff(x) - f(x)*tan(x),
'sol': [Eq(f(x), C1/cos(x))],
},
'separable_13': {
'eq': (x - 1)*cos(f(x))*f(x).diff(x) - 2*x*sin(f(x)),
'sol': [Eq(f(x), pi - asin(C1*(x**2 - 2*x + 1)*exp(2*x))),
Eq(f(x), asin(C1*(x**2 - 2*x + 1)*exp(2*x)))],
},
'separable_14': {
'eq': f(x).diff(x) - f(x)*log(f(x))/tan(x),
'sol': [Eq(f(x), exp(C1*sin(x)))],
},
'separable_15': {
'eq': x*f(x).diff(x) + (1 + f(x)**2)*atan(f(x)),
'sol': [Eq(f(x), tan(C1/x))], #Two more solutions are f(x)=0 and f(x)=I
'slow': True,
'checkodesol_XFAIL': True,
},
'separable_16': {
'eq': f(x).diff(x) + x*(f(x) + 1),
'sol': [Eq(f(x), -1 + C1*exp(-x**2/2))],
},
'separable_17': {
'eq': exp(f(x)**2)*(x**2 + 2*x + 1) + (x*f(x) + f(x))*f(x).diff(x),
'sol': [Eq(f(x), -sqrt(log(1/(C1 + x**2 + 2*x)))),
Eq(f(x), sqrt(log(1/(C1 + x**2 + 2*x))))],
},
'separable_18': {
'eq': f(x).diff(x) + f(x),
'sol': [Eq(f(x), C1*exp(-x))],
},
'separable_19': {
'eq': sin(x)*cos(2*f(x)) + cos(x)*sin(2*f(x))*f(x).diff(x),
'sol': [Eq(f(x), pi - acos(C1/cos(x)**2)/2), Eq(f(x), acos(C1/cos(x)**2)/2)],
},
'separable_20': {
'eq': (1 - x)*f(x).diff(x) - x*(f(x) + 1),
'sol': [Eq(f(x), (C1*exp(-x) - x + 1)/(x - 1))],
},
'separable_21': {
'eq': f(x)*diff(f(x), x) + x - 3*x*f(x)**2,
'sol': [Eq(f(x), -sqrt(3)*sqrt(C1*exp(3*x**2) + 1)/3),
Eq(f(x), sqrt(3)*sqrt(C1*exp(3*x**2) + 1)/3)],
},
'separable_22': {
'eq': f(x).diff(x) - exp(x + f(x)),
'sol': [Eq(f(x), log(-1/(C1 + exp(x))))],
'XFAIL': ['lie_group'] #It shows 'NoneType' object is not subscriptable for lie_group.
},
}
}
def _get_all_examples():
all_solvers = [_get_examples_ode_sol_euler_homogeneous(),
_get_examples_ode_sol_euler_undetermined_coeff(),
_get_examples_ode_sol_euler_var_para(),
_get_examples_ode_sol_factorable(),
_get_examples_ode_sol_bernoulli(),
_get_examples_ode_sol_nth_algebraic(),
_get_examples_ode_sol_riccati(),
_get_examples_ode_sol_1st_linear(),
_get_examples_ode_sol_almost_linear(),
_get_examples_ode_sol_nth_order_reducible(),
_get_examples_ode_sol_nth_linear_undetermined_coefficients(),
_get_examples_ode_sol_liouville(),
_get_examples_ode_sol_separable(),
]
all_examples = []
for solver in all_solvers:
for example in solver['examples']:
temp = {
'hint': solver['hint'],
'func': solver['examples'][example].get('func',solver['func']),
'eq': solver['examples'][example]['eq'],
'sol': solver['examples'][example]['sol'],
'XFAIL': solver['examples'][example].get('XFAIL',[]),
'checkodesol_XFAIL': solver['examples'][example].get('checkodesol_XFAIL', False),
'example_name': example,
}
all_examples.append(temp)
return all_examples
|
f2107eceb32dfd97d11efadeeed0029418cd38e369eaadf592ba32d0d7e9282f | import glob
import os
import shutil
import subprocess
import sys
import tempfile
import warnings
from distutils.errors import CompileError
from distutils.sysconfig import get_config_var
from .runners import (
CCompilerRunner,
CppCompilerRunner,
FortranCompilerRunner
)
from .util import (
get_abspath, make_dirs, copy, Glob, ArbitraryDepthGlob,
glob_at_depth, import_module_from_file, pyx_is_cplus,
sha256_of_string, sha256_of_file
)
sharedext = get_config_var('EXT_SUFFIX' if sys.version_info >= (3, 3) else 'SO')
if os.name == 'posix':
objext = '.o'
elif os.name == 'nt':
objext = '.obj'
else:
warnings.warn("Unknown os.name: {}".format(os.name))
objext = '.o'
def compile_sources(files, Runner=None, destdir=None, cwd=None, keep_dir_struct=False,
per_file_kwargs=None, **kwargs):
""" Compile source code files to object files.
Parameters
==========
files : iterable of str
Paths to source files, if ``cwd`` is given, the paths are taken as relative.
Runner: CompilerRunner subclass (optional)
Could be e.g. ``FortranCompilerRunner``. Will be inferred from filename
extensions if missing.
destdir: str
Output directory, if cwd is given, the path is taken as relative.
cwd: str
Working directory. Specify to have compiler run in other directory.
also used as root of relative paths.
keep_dir_struct: bool
Reproduce directory structure in `destdir`. default: ``False``
per_file_kwargs: dict
Dict mapping instances in ``files`` to keyword arguments.
\\*\\*kwargs: dict
Default keyword arguments to pass to ``Runner``.
"""
_per_file_kwargs = {}
if per_file_kwargs is not None:
for k, v in per_file_kwargs.items():
if isinstance(k, Glob):
for path in glob.glob(k.pathname):
_per_file_kwargs[path] = v
elif isinstance(k, ArbitraryDepthGlob):
for path in glob_at_depth(k.filename, cwd):
_per_file_kwargs[path] = v
else:
_per_file_kwargs[k] = v
# Set up destination directory
destdir = destdir or '.'
if not os.path.isdir(destdir):
if os.path.exists(destdir):
raise OSError("{} is not a directory".format(destdir))
else:
make_dirs(destdir)
if cwd is None:
cwd = '.'
for f in files:
copy(f, destdir, only_update=True, dest_is_dir=True)
# Compile files and return list of paths to the objects
dstpaths = []
for f in files:
if keep_dir_struct:
name, ext = os.path.splitext(f)
else:
name, ext = os.path.splitext(os.path.basename(f))
file_kwargs = kwargs.copy()
file_kwargs.update(_per_file_kwargs.get(f, {}))
dstpaths.append(src2obj(f, Runner, cwd=cwd, **file_kwargs))
return dstpaths
def get_mixed_fort_c_linker(vendor=None, cplus=False, cwd=None):
vendor = vendor or os.environ.get('SYMPY_COMPILER_VENDOR', 'gnu')
if vendor.lower() == 'intel':
if cplus:
return (FortranCompilerRunner,
{'flags': ['-nofor_main', '-cxxlib']}, vendor)
else:
return (FortranCompilerRunner,
{'flags': ['-nofor_main']}, vendor)
elif vendor.lower() == 'gnu' or 'llvm':
if cplus:
return (CppCompilerRunner,
{'lib_options': ['fortran']}, vendor)
else:
return (FortranCompilerRunner,
{}, vendor)
else:
raise ValueError("No vendor found.")
def link(obj_files, out_file=None, shared=False, Runner=None,
cwd=None, cplus=False, fort=False, **kwargs):
""" Link object files.
Parameters
==========
obj_files: iterable of str
Paths to object files.
out_file: str (optional)
Path to executable/shared library, if ``None`` it will be
deduced from the last item in obj_files.
shared: bool
Generate a shared library?
Runner: CompilerRunner subclass (optional)
If not given the ``cplus`` and ``fort`` flags will be inspected
(fallback is the C compiler).
cwd: str
Path to the root of relative paths and working directory for compiler.
cplus: bool
C++ objects? default: ``False``.
fort: bool
Fortran objects? default: ``False``.
\\*\\*kwargs: dict
Keyword arguments passed to ``Runner``.
Returns
=======
The absolute path to the generated shared object / executable.
"""
if out_file is None:
out_file, ext = os.path.splitext(os.path.basename(obj_files[-1]))
if shared:
out_file += sharedext
if not Runner:
if fort:
Runner, extra_kwargs, vendor = \
get_mixed_fort_c_linker(
vendor=kwargs.get('vendor', None),
cplus=cplus,
cwd=cwd,
)
for k, v in extra_kwargs.items():
if k in kwargs:
kwargs[k].expand(v)
else:
kwargs[k] = v
else:
if cplus:
Runner = CppCompilerRunner
else:
Runner = CCompilerRunner
flags = kwargs.pop('flags', [])
if shared:
if '-shared' not in flags:
flags.append('-shared')
run_linker = kwargs.pop('run_linker', True)
if not run_linker:
raise ValueError("run_linker was set to False (nonsensical).")
out_file = get_abspath(out_file, cwd=cwd)
runner = Runner(obj_files, out_file, flags, cwd=cwd, **kwargs)
runner.run()
return out_file
def link_py_so(obj_files, so_file=None, cwd=None, libraries=None,
cplus=False, fort=False, **kwargs):
""" Link python extension module (shared object) for importing
Parameters
==========
obj_files: iterable of str
Paths to object files to be linked.
so_file: str
Name (path) of shared object file to create. If not specified it will
have the basname of the last object file in `obj_files` but with the
extension '.so' (Unix).
cwd: path string
Root of relative paths and working directory of linker.
libraries: iterable of strings
Libraries to link against, e.g. ['m'].
cplus: bool
Any C++ objects? default: ``False``.
fort: bool
Any Fortran objects? default: ``False``.
kwargs**: dict
Keyword arguments passed to ``link(...)``.
Returns
=======
Absolute path to the generate shared object.
"""
libraries = libraries or []
include_dirs = kwargs.pop('include_dirs', [])
library_dirs = kwargs.pop('library_dirs', [])
# from distutils/command/build_ext.py:
if sys.platform == "win32":
warnings.warn("Windows not yet supported.")
elif sys.platform == 'darwin':
# Don't use the default code below
pass
elif sys.platform[:3] == 'aix':
# Don't use the default code below
pass
else:
from distutils import sysconfig
if sysconfig.get_config_var('Py_ENABLE_SHARED'):
ABIFLAGS = sysconfig.get_config_var('ABIFLAGS')
pythonlib = 'python{}.{}{}'.format(
sys.hexversion >> 24, (sys.hexversion >> 16) & 0xff,
ABIFLAGS or '')
libraries += [pythonlib]
else:
pass
flags = kwargs.pop('flags', [])
needed_flags = ('-pthread',)
for flag in needed_flags:
if flag not in flags:
flags.append(flag)
return link(obj_files, shared=True, flags=flags, cwd=cwd,
cplus=cplus, fort=fort, include_dirs=include_dirs,
libraries=libraries, library_dirs=library_dirs, **kwargs)
def simple_cythonize(src, destdir=None, cwd=None, **cy_kwargs):
""" Generates a C file from a Cython source file.
Parameters
==========
src: str
Path to Cython source.
destdir: str (optional)
Path to output directory (default: '.').
cwd: path string (optional)
Root of relative paths (default: '.').
**cy_kwargs:
Second argument passed to cy_compile. Generates a .cpp file if ``cplus=True`` in ``cy_kwargs``,
else a .c file.
"""
from Cython.Compiler.Main import (
default_options, CompilationOptions
)
from Cython.Compiler.Main import compile as cy_compile
assert src.lower().endswith('.pyx') or src.lower().endswith('.py')
cwd = cwd or '.'
destdir = destdir or '.'
ext = '.cpp' if cy_kwargs.get('cplus', False) else '.c'
c_name = os.path.splitext(os.path.basename(src))[0] + ext
dstfile = os.path.join(destdir, c_name)
if cwd:
ori_dir = os.getcwd()
else:
ori_dir = '.'
os.chdir(cwd)
try:
cy_options = CompilationOptions(default_options)
cy_options.__dict__.update(cy_kwargs)
cy_result = cy_compile([src], cy_options)
if cy_result.num_errors > 0:
raise ValueError("Cython compilation failed.")
if os.path.abspath(os.path.dirname(src)) != os.path.abspath(destdir):
if os.path.exists(dstfile):
os.unlink(dstfile)
shutil.move(os.path.join(os.path.dirname(src), c_name), destdir)
finally:
os.chdir(ori_dir)
return dstfile
extension_mapping = {
'.c': (CCompilerRunner, None),
'.cpp': (CppCompilerRunner, None),
'.cxx': (CppCompilerRunner, None),
'.f': (FortranCompilerRunner, None),
'.for': (FortranCompilerRunner, None),
'.ftn': (FortranCompilerRunner, None),
'.f90': (FortranCompilerRunner, None), # ifort only knows about .f90
'.f95': (FortranCompilerRunner, 'f95'),
'.f03': (FortranCompilerRunner, 'f2003'),
'.f08': (FortranCompilerRunner, 'f2008'),
}
def src2obj(srcpath, Runner=None, objpath=None, cwd=None, inc_py=False, **kwargs):
""" Compiles a source code file to an object file.
Files ending with '.pyx' assumed to be cython files and
are dispatched to pyx2obj.
Parameters
==========
srcpath: str
Path to source file.
Runner: CompilerRunner subclass (optional)
If ``None``: deduced from extension of srcpath.
objpath : str (optional)
Path to generated object. If ``None``: deduced from ``srcpath``.
cwd: str (optional)
Working directory and root of relative paths. If ``None``: current dir.
inc_py: bool
Add Python include path to kwarg "include_dirs". Default: False
\\*\\*kwargs: dict
keyword arguments passed to Runner or pyx2obj
"""
name, ext = os.path.splitext(os.path.basename(srcpath))
if objpath is None:
if os.path.isabs(srcpath):
objpath = '.'
else:
objpath = os.path.dirname(srcpath)
objpath = objpath or '.' # avoid objpath == ''
if os.path.isdir(objpath):
objpath = os.path.join(objpath, name + objext)
include_dirs = kwargs.pop('include_dirs', [])
if inc_py:
from distutils.sysconfig import get_python_inc
py_inc_dir = get_python_inc()
if py_inc_dir not in include_dirs:
include_dirs.append(py_inc_dir)
if ext.lower() == '.pyx':
return pyx2obj(srcpath, objpath=objpath, include_dirs=include_dirs, cwd=cwd,
**kwargs)
if Runner is None:
Runner, std = extension_mapping[ext.lower()]
if 'std' not in kwargs:
kwargs['std'] = std
flags = kwargs.pop('flags', [])
needed_flags = ('-fPIC',)
for flag in needed_flags:
if flag not in flags:
flags.append(flag)
# src2obj implies not running the linker...
run_linker = kwargs.pop('run_linker', False)
if run_linker:
raise CompileError("src2obj called with run_linker=True")
runner = Runner([srcpath], objpath, include_dirs=include_dirs,
run_linker=run_linker, cwd=cwd, flags=flags, **kwargs)
runner.run()
return objpath
def pyx2obj(pyxpath, objpath=None, destdir=None, cwd=None,
include_dirs=None, cy_kwargs=None, cplus=None, **kwargs):
"""
Convenience function
If cwd is specified, pyxpath and dst are taken to be relative
If only_update is set to `True` the modification time is checked
and compilation is only run if the source is newer than the
destination
Parameters
==========
pyxpath: str
Path to Cython source file.
objpath: str (optional)
Path to object file to generate.
destdir: str (optional)
Directory to put generated C file. When ``None``: directory of ``objpath``.
cwd: str (optional)
Working directory and root of relative paths.
include_dirs: iterable of path strings (optional)
Passed onto src2obj and via cy_kwargs['include_path']
to simple_cythonize.
cy_kwargs: dict (optional)
Keyword arguments passed onto `simple_cythonize`
cplus: bool (optional)
Indicate whether C++ is used. default: auto-detect using ``.util.pyx_is_cplus``.
compile_kwargs: dict
keyword arguments passed onto src2obj
Returns
=======
Absolute path of generated object file.
"""
assert pyxpath.endswith('.pyx')
cwd = cwd or '.'
objpath = objpath or '.'
destdir = destdir or os.path.dirname(objpath)
abs_objpath = get_abspath(objpath, cwd=cwd)
if os.path.isdir(abs_objpath):
pyx_fname = os.path.basename(pyxpath)
name, ext = os.path.splitext(pyx_fname)
objpath = os.path.join(objpath, name + objext)
cy_kwargs = cy_kwargs or {}
cy_kwargs['output_dir'] = cwd
if cplus is None:
cplus = pyx_is_cplus(pyxpath)
cy_kwargs['cplus'] = cplus
interm_c_file = simple_cythonize(pyxpath, destdir=destdir, cwd=cwd, **cy_kwargs)
include_dirs = include_dirs or []
flags = kwargs.pop('flags', [])
needed_flags = ('-fwrapv', '-pthread', '-fPIC')
for flag in needed_flags:
if flag not in flags:
flags.append(flag)
options = kwargs.pop('options', [])
if kwargs.pop('strict_aliasing', False):
raise CompileError("Cython requires strict aliasing to be disabled.")
# Let's be explicit about standard
if cplus:
std = kwargs.pop('std', 'c++98')
else:
std = kwargs.pop('std', 'c99')
return src2obj(interm_c_file, objpath=objpath, cwd=cwd,
include_dirs=include_dirs, flags=flags, std=std,
options=options, inc_py=True, strict_aliasing=False,
**kwargs)
def _any_X(srcs, cls):
for src in srcs:
name, ext = os.path.splitext(src)
key = ext.lower()
if key in extension_mapping:
if extension_mapping[key][0] == cls:
return True
return False
def any_fortran_src(srcs):
return _any_X(srcs, FortranCompilerRunner)
def any_cplus_src(srcs):
return _any_X(srcs, CppCompilerRunner)
def compile_link_import_py_ext(sources, extname=None, build_dir='.', compile_kwargs=None,
link_kwargs=None):
""" Compiles sources to a shared object (python extension) and imports it
Sources in ``sources`` which is imported. If shared object is newer than the sources, they
are not recompiled but instead it is imported.
Parameters
==========
sources : string
List of paths to sources.
extname : string
Name of extension (default: ``None``).
If ``None``: taken from the last file in ``sources`` without extension.
build_dir: str
Path to directory in which objects files etc. are generated.
compile_kwargs: dict
keyword arguments passed to ``compile_sources``
link_kwargs: dict
keyword arguments passed to ``link_py_so``
Returns
=======
The imported module from of the python extension.
"""
if extname is None:
extname = os.path.splitext(os.path.basename(sources[-1]))[0]
compile_kwargs = compile_kwargs or {}
link_kwargs = link_kwargs or {}
try:
mod = import_module_from_file(os.path.join(build_dir, extname), sources)
except ImportError:
objs = compile_sources(list(map(get_abspath, sources)), destdir=build_dir,
cwd=build_dir, **compile_kwargs)
so = link_py_so(objs, cwd=build_dir, fort=any_fortran_src(sources),
cplus=any_cplus_src(sources), **link_kwargs)
mod = import_module_from_file(so)
return mod
def _write_sources_to_build_dir(sources, build_dir):
build_dir = build_dir or tempfile.mkdtemp()
if not os.path.isdir(build_dir):
raise OSError("Non-existent directory: ", build_dir)
source_files = []
for name, src in sources:
dest = os.path.join(build_dir, name)
differs = True
sha256_in_mem = sha256_of_string(src.encode('utf-8')).hexdigest()
if os.path.exists(dest):
if os.path.exists(dest + '.sha256'):
sha256_on_disk = open(dest + '.sha256').read()
else:
sha256_on_disk = sha256_of_file(dest).hexdigest()
differs = sha256_on_disk != sha256_in_mem
if differs:
with open(dest, 'wt') as fh:
fh.write(src)
open(dest + '.sha256', 'wt').write(sha256_in_mem)
source_files.append(dest)
return source_files, build_dir
def compile_link_import_strings(sources, build_dir=None, **kwargs):
""" Compiles, links and imports extension module from source.
Parameters
==========
sources : iterable of name/source pair tuples
build_dir : string (default: None)
Path. ``None`` implies use a temporary directory.
**kwargs:
Keyword arguments passed onto `compile_link_import_py_ext`.
Returns
=======
mod : module
The compiled and imported extension module.
info : dict
Containing ``build_dir`` as 'build_dir'.
"""
source_files, build_dir = _write_sources_to_build_dir(sources, build_dir)
mod = compile_link_import_py_ext(source_files, build_dir=build_dir, **kwargs)
info = dict(build_dir=build_dir)
return mod, info
def compile_run_strings(sources, build_dir=None, clean=False, compile_kwargs=None, link_kwargs=None):
""" Compiles, links and runs a program built from sources.
Parameters
==========
sources : iterable of name/source pair tuples
build_dir : string (default: None)
Path. ``None`` implies use a temporary directory.
clean : bool
Whether to remove build_dir after use. This will only have an
effect if ``build_dir`` is ``None`` (which creates a temporary directory).
Passing ``clean == True`` and ``build_dir != None`` raises a ``ValueError``.
This will also set ``build_dir`` in returned info dictionary to ``None``.
compile_kwargs: dict
Keyword arguments passed onto ``compile_sources``
link_kwargs: dict
Keyword arguments passed onto ``link``
Returns
=======
(stdout, stderr): pair of strings
info: dict
Containing exit status as 'exit_status' and ``build_dir`` as 'build_dir'
"""
if clean and build_dir is not None:
raise ValueError("Automatic removal of build_dir is only available for temporary directory.")
try:
source_files, build_dir = _write_sources_to_build_dir(sources, build_dir)
objs = compile_sources(list(map(get_abspath, source_files)), destdir=build_dir,
cwd=build_dir, **(compile_kwargs or {}))
prog = link(objs, cwd=build_dir,
fort=any_fortran_src(source_files),
cplus=any_cplus_src(source_files), **(link_kwargs or {}))
p = subprocess.Popen([prog], stdout=subprocess.PIPE, stderr=subprocess.PIPE)
exit_status = p.wait()
stdout, stderr = [txt.decode('utf-8') for txt in p.communicate()]
finally:
if clean and os.path.isdir(build_dir):
shutil.rmtree(build_dir)
build_dir = None
info = dict(exit_status=exit_status, build_dir=build_dir)
return (stdout, stderr), info
|
c0da37cac18c1c06a6aba393ed82e686f426fe5de4d45a1a757988c0947afc2c | """ Tests from Michael Wester's 1999 paper "Review of CAS mathematical
capabilities".
http://www.math.unm.edu/~wester/cas/book/Wester.pdf
See also http://math.unm.edu/~wester/cas_review.html for detailed output of
each tested system.
"""
from sympy import (Rational, symbols, Dummy, factorial, sqrt, log, exp, oo, zoo,
product, binomial, rf, pi, gamma, igcd, factorint, radsimp, combsimp,
npartitions, totient, primerange, factor, simplify, gcd, resultant, expand,
I, trigsimp, tan, sin, cos, cot, diff, nan, limit, EulerGamma, polygamma,
bernoulli, hyper, hyperexpand, besselj, asin, assoc_legendre, Function, re,
im, DiracDelta, chebyshevt, legendre_poly, polylog, series, O,
atan, sinh, cosh, tanh, floor, ceiling, solve, asinh, acot, csc, sec,
LambertW, N, apart, sqrtdenest, factorial2, powdenest, Mul, S, ZZ,
Poly, expand_func, E, Q, And, Lt, Min, ask, refine, AlgebraicNumber,
continued_fraction_iterator as cf_i, continued_fraction_periodic as cf_p,
continued_fraction_convergents as cf_c, continued_fraction_reduce as cf_r,
FiniteSet, elliptic_e, elliptic_f, powsimp, hessian, wronskian, fibonacci,
sign, Lambda, Piecewise, Subs, residue, Derivative, logcombine, Symbol,
Intersection, Union, EmptySet, Interval, idiff, ImageSet, acos, Max,
MatMul, conjugate)
import mpmath
from sympy.functions.combinatorial.numbers import stirling
from sympy.functions.special.delta_functions import Heaviside
from sympy.functions.special.error_functions import Ci, Si, erf
from sympy.functions.special.zeta_functions import zeta
from sympy.testing.pytest import (XFAIL, slow, SKIP, skip, ON_TRAVIS,
raises)
from sympy.utilities.iterables import partitions
from mpmath import mpi, mpc
from sympy.matrices import Matrix, GramSchmidt, eye
from sympy.matrices.expressions.blockmatrix import BlockMatrix, block_collapse
from sympy.matrices.expressions import MatrixSymbol, ZeroMatrix
from sympy.physics.quantum import Commutator
from sympy.assumptions import assuming
from sympy.polys.rings import PolyRing
from sympy.polys.fields import FracField
from sympy.polys.solvers import solve_lin_sys
from sympy.concrete import Sum
from sympy.concrete.products import Product
from sympy.integrals import integrate
from sympy.integrals.transforms import laplace_transform,\
inverse_laplace_transform, LaplaceTransform, fourier_transform,\
mellin_transform
from sympy.solvers.recurr import rsolve
from sympy.solvers.solveset import solveset, solveset_real, linsolve
from sympy.solvers.ode import dsolve
from sympy.core.relational import Equality
from itertools import islice, takewhile
from sympy.series.formal import fps
from sympy.series.fourier import fourier_series
from sympy.calculus.util import minimum
R = Rational
x, y, z = symbols('x y z')
i, j, k, l, m, n = symbols('i j k l m n', integer=True)
f = Function('f')
g = Function('g')
# A. Boolean Logic and Quantifier Elimination
# Not implemented.
# B. Set Theory
def test_B1():
assert (FiniteSet(i, j, j, k, k, k) | FiniteSet(l, k, j) |
FiniteSet(j, m, j)) == FiniteSet(i, j, k, l, m)
def test_B2():
assert (FiniteSet(i, j, j, k, k, k) & FiniteSet(l, k, j) &
FiniteSet(j, m, j)) == Intersection({j, m}, {i, j, k}, {j, k, l})
# Previous output below. Not sure why that should be the expected output.
# There should probably be a way to rewrite Intersections that way but I
# don't see why an Intersection should evaluate like that:
#
# == Union({j}, Intersection({m}, Union({j, k}, Intersection({i}, {l}))))
def test_B3():
assert (FiniteSet(i, j, k, l, m) - FiniteSet(j) ==
FiniteSet(i, k, l, m))
def test_B4():
assert (FiniteSet(*(FiniteSet(i, j)*FiniteSet(k, l))) ==
FiniteSet((i, k), (i, l), (j, k), (j, l)))
# C. Numbers
def test_C1():
assert (factorial(50) ==
30414093201713378043612608166064768844377641568960512000000000000)
def test_C2():
assert (factorint(factorial(50)) == {2: 47, 3: 22, 5: 12, 7: 8,
11: 4, 13: 3, 17: 2, 19: 2, 23: 2, 29: 1, 31: 1, 37: 1,
41: 1, 43: 1, 47: 1})
def test_C3():
assert (factorial2(10), factorial2(9)) == (3840, 945)
# Base conversions; not really implemented by sympy
# Whatever. Take credit!
def test_C4():
assert 0xABC == 2748
def test_C5():
assert 123 == int('234', 7)
def test_C6():
assert int('677', 8) == int('1BF', 16) == 447
def test_C7():
assert log(32768, 8) == 5
def test_C8():
# Modular multiplicative inverse. Would be nice if divmod could do this.
assert ZZ.invert(5, 7) == 3
assert ZZ.invert(5, 6) == 5
def test_C9():
assert igcd(igcd(1776, 1554), 5698) == 74
def test_C10():
x = 0
for n in range(2, 11):
x += R(1, n)
assert x == R(4861, 2520)
def test_C11():
assert R(1, 7) == S('0.[142857]')
def test_C12():
assert R(7, 11) * R(22, 7) == 2
def test_C13():
test = R(10, 7) * (1 + R(29, 1000)) ** R(1, 3)
good = 3 ** R(1, 3)
assert test == good
def test_C14():
assert sqrtdenest(sqrt(2*sqrt(3) + 4)) == 1 + sqrt(3)
def test_C15():
test = sqrtdenest(sqrt(14 + 3*sqrt(3 + 2*sqrt(5 - 12*sqrt(3 - 2*sqrt(2))))))
good = sqrt(2) + 3
assert test == good
def test_C16():
test = sqrtdenest(sqrt(10 + 2*sqrt(6) + 2*sqrt(10) + 2*sqrt(15)))
good = sqrt(2) + sqrt(3) + sqrt(5)
assert test == good
def test_C17():
test = radsimp((sqrt(3) + sqrt(2)) / (sqrt(3) - sqrt(2)))
good = 5 + 2*sqrt(6)
assert test == good
def test_C18():
assert simplify((sqrt(-2 + sqrt(-5)) * sqrt(-2 - sqrt(-5))).expand(complex=True)) == 3
@XFAIL
def test_C19():
assert radsimp(simplify((90 + 34*sqrt(7)) ** R(1, 3))) == 3 + sqrt(7)
def test_C20():
inside = (135 + 78*sqrt(3))
test = AlgebraicNumber((inside**R(2, 3) + 3) * sqrt(3) / inside**R(1, 3))
assert simplify(test) == AlgebraicNumber(12)
def test_C21():
assert simplify(AlgebraicNumber((41 + 29*sqrt(2)) ** R(1, 5))) == \
AlgebraicNumber(1 + sqrt(2))
@XFAIL
def test_C22():
test = simplify(((6 - 4*sqrt(2))*log(3 - 2*sqrt(2)) + (3 - 2*sqrt(2))*log(17
- 12*sqrt(2)) + 32 - 24*sqrt(2)) / (48*sqrt(2) - 72))
good = sqrt(2)/3 - log(sqrt(2) - 1)/3
assert test == good
def test_C23():
assert 2 * oo - 3 is oo
@XFAIL
def test_C24():
raise NotImplementedError("2**aleph_null == aleph_1")
# D. Numerical Analysis
def test_D1():
assert 0.0 / sqrt(2) == 0.0
def test_D2():
assert str(exp(-1000000).evalf()) == '3.29683147808856e-434295'
def test_D3():
assert exp(pi*sqrt(163)).evalf(50).num.ae(262537412640768744)
def test_D4():
assert floor(R(-5, 3)) == -2
assert ceiling(R(-5, 3)) == -1
@XFAIL
def test_D5():
raise NotImplementedError("cubic_spline([1, 2, 4, 5], [1, 4, 2, 3], x)(3) == 27/8")
@XFAIL
def test_D6():
raise NotImplementedError("translate sum(a[i]*x**i, (i,1,n)) to FORTRAN")
@XFAIL
def test_D7():
raise NotImplementedError("translate sum(a[i]*x**i, (i,1,n)) to C")
@XFAIL
def test_D8():
# One way is to cheat by converting the sum to a string,
# and replacing the '[' and ']' with ''.
# E.g., horner(S(str(_).replace('[','').replace(']','')))
raise NotImplementedError("apply Horner's rule to sum(a[i]*x**i, (i,1,5))")
@XFAIL
def test_D9():
raise NotImplementedError("translate D8 to FORTRAN")
@XFAIL
def test_D10():
raise NotImplementedError("translate D8 to C")
@XFAIL
def test_D11():
#Is there a way to use count_ops?
raise NotImplementedError("flops(sum(product(f[i][k], (i,1,k)), (k,1,n)))")
@XFAIL
def test_D12():
assert (mpi(-4, 2) * x + mpi(1, 3)) ** 2 == mpi(-8, 16)*x**2 + mpi(-24, 12)*x + mpi(1, 9)
@XFAIL
def test_D13():
raise NotImplementedError("discretize a PDE: diff(f(x,t),t) == diff(diff(f(x,t),x),x)")
# E. Statistics
# See scipy; all of this is numerical.
# F. Combinatorial Theory.
def test_F1():
assert rf(x, 3) == x*(1 + x)*(2 + x)
def test_F2():
assert expand_func(binomial(n, 3)) == n*(n - 1)*(n - 2)/6
@XFAIL
def test_F3():
assert combsimp(2**n * factorial(n) * factorial2(2*n - 1)) == factorial(2*n)
@XFAIL
def test_F4():
assert combsimp(2**n * factorial(n) * product(2*k - 1, (k, 1, n))) == factorial(2*n)
@XFAIL
def test_F5():
assert gamma(n + R(1, 2)) / sqrt(pi) / factorial(n) == factorial(2*n)/2**(2*n)/factorial(n)**2
def test_F6():
partTest = [p.copy() for p in partitions(4)]
partDesired = [{4: 1}, {1: 1, 3: 1}, {2: 2}, {1: 2, 2:1}, {1: 4}]
assert partTest == partDesired
def test_F7():
assert npartitions(4) == 5
def test_F8():
assert stirling(5, 2, signed=True) == -50 # if signed, then kind=1
def test_F9():
assert totient(1776) == 576
# G. Number Theory
def test_G1():
assert list(primerange(999983, 1000004)) == [999983, 1000003]
@XFAIL
def test_G2():
raise NotImplementedError("find the primitive root of 191 == 19")
@XFAIL
def test_G3():
raise NotImplementedError("(a+b)**p mod p == a**p + b**p mod p; p prime")
# ... G14 Modular equations are not implemented.
def test_G15():
assert Rational(sqrt(3).evalf()).limit_denominator(15) == R(26, 15)
assert list(takewhile(lambda x: x.q <= 15, cf_c(cf_i(sqrt(3)))))[-1] == \
R(26, 15)
def test_G16():
assert list(islice(cf_i(pi),10)) == [3, 7, 15, 1, 292, 1, 1, 1, 2, 1]
def test_G17():
assert cf_p(0, 1, 23) == [4, [1, 3, 1, 8]]
def test_G18():
assert cf_p(1, 2, 5) == [[1]]
assert cf_r([[1]]).expand() == S.Half + sqrt(5)/2
@XFAIL
def test_G19():
s = symbols('s', integer=True, positive=True)
it = cf_i((exp(1/s) - 1)/(exp(1/s) + 1))
assert list(islice(it, 5)) == [0, 2*s, 6*s, 10*s, 14*s]
def test_G20():
s = symbols('s', integer=True, positive=True)
# Wester erroneously has this as -s + sqrt(s**2 + 1)
assert cf_r([[2*s]]) == s + sqrt(s**2 + 1)
@XFAIL
def test_G20b():
s = symbols('s', integer=True, positive=True)
assert cf_p(s, 1, s**2 + 1) == [[2*s]]
# H. Algebra
def test_H1():
assert simplify(2*2**n) == simplify(2**(n + 1))
assert powdenest(2*2**n) == simplify(2**(n + 1))
def test_H2():
assert powsimp(4 * 2**n) == 2**(n + 2)
def test_H3():
assert (-1)**(n*(n + 1)) == 1
def test_H4():
expr = factor(6*x - 10)
assert type(expr) is Mul
assert expr.args[0] == 2
assert expr.args[1] == 3*x - 5
p1 = 64*x**34 - 21*x**47 - 126*x**8 - 46*x**5 - 16*x**60 - 81
p2 = 72*x**60 - 25*x**25 - 19*x**23 - 22*x**39 - 83*x**52 + 54*x**10 + 81
q = 34*x**19 - 25*x**16 + 70*x**7 + 20*x**3 - 91*x - 86
def test_H5():
assert gcd(p1, p2, x) == 1
def test_H6():
assert gcd(expand(p1 * q), expand(p2 * q)) == q
def test_H7():
p1 = 24*x*y**19*z**8 - 47*x**17*y**5*z**8 + 6*x**15*y**9*z**2 - 3*x**22 + 5
p2 = 34*x**5*y**8*z**13 + 20*x**7*y**7*z**7 + 12*x**9*y**16*z**4 + 80*y**14*z
assert gcd(p1, p2, x, y, z) == 1
def test_H8():
p1 = 24*x*y**19*z**8 - 47*x**17*y**5*z**8 + 6*x**15*y**9*z**2 - 3*x**22 + 5
p2 = 34*x**5*y**8*z**13 + 20*x**7*y**7*z**7 + 12*x**9*y**16*z**4 + 80*y**14*z
q = 11*x**12*y**7*z**13 - 23*x**2*y**8*z**10 + 47*x**17*y**5*z**8
assert gcd(p1 * q, p2 * q, x, y, z) == q
def test_H9():
p1 = 2*x**(n + 4) - x**(n + 2)
p2 = 4*x**(n + 1) + 3*x**n
assert gcd(p1, p2) == x**n
def test_H10():
p1 = 3*x**4 + 3*x**3 + x**2 - x - 2
p2 = x**3 - 3*x**2 + x + 5
assert resultant(p1, p2, x) == 0
def test_H11():
assert resultant(p1 * q, p2 * q, x) == 0
def test_H12():
num = x**2 - 4
den = x**2 + 4*x + 4
assert simplify(num/den) == (x - 2)/(x + 2)
@XFAIL
def test_H13():
assert simplify((exp(x) - 1) / (exp(x/2) + 1)) == exp(x/2) - 1
def test_H14():
p = (x + 1) ** 20
ep = expand(p)
assert ep == (1 + 20*x + 190*x**2 + 1140*x**3 + 4845*x**4 + 15504*x**5
+ 38760*x**6 + 77520*x**7 + 125970*x**8 + 167960*x**9 + 184756*x**10
+ 167960*x**11 + 125970*x**12 + 77520*x**13 + 38760*x**14 + 15504*x**15
+ 4845*x**16 + 1140*x**17 + 190*x**18 + 20*x**19 + x**20)
dep = diff(ep, x)
assert dep == (20 + 380*x + 3420*x**2 + 19380*x**3 + 77520*x**4
+ 232560*x**5 + 542640*x**6 + 1007760*x**7 + 1511640*x**8 + 1847560*x**9
+ 1847560*x**10 + 1511640*x**11 + 1007760*x**12 + 542640*x**13
+ 232560*x**14 + 77520*x**15 + 19380*x**16 + 3420*x**17 + 380*x**18
+ 20*x**19)
assert factor(dep) == 20*(1 + x)**19
def test_H15():
assert simplify(Mul(*[x - r for r in solveset(x**3 + x**2 - 7)])) == x**3 + x**2 - 7
def test_H16():
assert factor(x**100 - 1) == ((x - 1)*(x + 1)*(x**2 + 1)*(x**4 - x**3
+ x**2 - x + 1)*(x**4 + x**3 + x**2 + x + 1)*(x**8 - x**6 + x**4
- x**2 + 1)*(x**20 - x**15 + x**10 - x**5 + 1)*(x**20 + x**15 + x**10
+ x**5 + 1)*(x**40 - x**30 + x**20 - x**10 + 1))
def test_H17():
assert simplify(factor(expand(p1 * p2)) - p1*p2) == 0
@XFAIL
def test_H18():
# Factor over complex rationals.
test = factor(4*x**4 + 8*x**3 + 77*x**2 + 18*x + 153)
good = (2*x + 3*I)*(2*x - 3*I)*(x + 1 - 4*I)*(x + 1 + 4*I)
assert test == good
def test_H19():
a = symbols('a')
# The idea is to let a**2 == 2, then solve 1/(a-1). Answer is a+1")
assert Poly(a - 1).invert(Poly(a**2 - 2)) == a + 1
@XFAIL
def test_H20():
raise NotImplementedError("let a**2==2; (x**3 + (a-2)*x**2 - "
+ "(2*a+3)*x - 3*a) / (x**2-2) = (x**2 - 2*x - 3) / (x-a)")
@XFAIL
def test_H21():
raise NotImplementedError("evaluate (b+c)**4 assuming b**3==2, c**2==3. \
Answer is 2*b + 8*c + 18*b**2 + 12*b*c + 9")
def test_H22():
assert factor(x**4 - 3*x**2 + 1, modulus=5) == (x - 2)**2 * (x + 2)**2
def test_H23():
f = x**11 + x + 1
g = (x**2 + x + 1) * (x**9 - x**8 + x**6 - x**5 + x**3 - x**2 + 1)
assert factor(f, modulus=65537) == g
def test_H24():
phi = AlgebraicNumber(S.GoldenRatio.expand(func=True), alias='phi')
assert factor(x**4 - 3*x**2 + 1, extension=phi) == \
(x - phi)*(x + 1 - phi)*(x - 1 + phi)*(x + phi)
def test_H25():
e = (x - 2*y**2 + 3*z**3) ** 20
assert factor(expand(e)) == e
def test_H26():
g = expand((sin(x) - 2*cos(y)**2 + 3*tan(z)**3)**20)
assert factor(g, expand=False) == (-sin(x) + 2*cos(y)**2 - 3*tan(z)**3)**20
def test_H27():
f = 24*x*y**19*z**8 - 47*x**17*y**5*z**8 + 6*x**15*y**9*z**2 - 3*x**22 + 5
g = 34*x**5*y**8*z**13 + 20*x**7*y**7*z**7 + 12*x**9*y**16*z**4 + 80*y**14*z
h = -2*z*y**7 \
*(6*x**9*y**9*z**3 + 10*x**7*z**6 + 17*y*x**5*z**12 + 40*y**7) \
*(3*x**22 + 47*x**17*y**5*z**8 - 6*x**15*y**9*z**2 - 24*x*y**19*z**8 - 5)
assert factor(expand(f*g)) == h
@XFAIL
def test_H28():
raise NotImplementedError("expand ((1 - c**2)**5 * (1 - s**2)**5 * "
+ "(c**2 + s**2)**10) with c**2 + s**2 = 1. Answer is c**10*s**10.")
@XFAIL
def test_H29():
assert factor(4*x**2 - 21*x*y + 20*y**2, modulus=3) == (x + y)*(x - y)
def test_H30():
test = factor(x**3 + y**3, extension=sqrt(-3))
answer = (x + y)*(x + y*(-R(1, 2) - sqrt(3)/2*I))*(x + y*(-R(1, 2) + sqrt(3)/2*I))
assert answer == test
def test_H31():
f = (x**2 + 2*x + 3)/(x**3 + 4*x**2 + 5*x + 2)
g = 2 / (x + 1)**2 - 2 / (x + 1) + 3 / (x + 2)
assert apart(f) == g
@XFAIL
def test_H32(): # issue 6558
raise NotImplementedError("[A*B*C - (A*B*C)**(-1)]*A*C*B (product \
of a non-commuting product and its inverse)")
def test_H33():
A, B, C = symbols('A, B, C', commutative=False)
assert (Commutator(A, Commutator(B, C))
+ Commutator(B, Commutator(C, A))
+ Commutator(C, Commutator(A, B))).doit().expand() == 0
# I. Trigonometry
def test_I1():
assert tan(pi*R(7, 10)) == -sqrt(1 + 2/sqrt(5))
@XFAIL
def test_I2():
assert sqrt((1 + cos(6))/2) == -cos(3)
def test_I3():
assert cos(n*pi) + sin((4*n - 1)*pi/2) == (-1)**n - 1
def test_I4():
assert refine(cos(pi*cos(n*pi)) + sin(pi/2*cos(n*pi)), Q.integer(n)) == (-1)**n - 1
@XFAIL
def test_I5():
assert sin((n**5/5 + n**4/2 + n**3/3 - n/30) * pi) == 0
@XFAIL
def test_I6():
raise NotImplementedError("assuming -3*pi<x<-5*pi/2, abs(cos(x)) == -cos(x), abs(sin(x)) == -sin(x)")
@XFAIL
def test_I7():
assert cos(3*x)/cos(x) == cos(x)**2 - 3*sin(x)**2
@XFAIL
def test_I8():
assert cos(3*x)/cos(x) == 2*cos(2*x) - 1
@XFAIL
def test_I9():
# Supposed to do this with rewrite rules.
assert cos(3*x)/cos(x) == cos(x)**2 - 3*sin(x)**2
def test_I10():
assert trigsimp((tan(x)**2 + 1 - cos(x)**-2) / (sin(x)**2 + cos(x)**2 - 1)) is nan
@SKIP("hangs")
@XFAIL
def test_I11():
assert limit((tan(x)**2 + 1 - cos(x)**-2) / (sin(x)**2 + cos(x)**2 - 1), x, 0) != 0
@XFAIL
def test_I12():
# This should fail or return nan or something.
res = diff((tan(x)**2 + 1 - cos(x)**-2) / (sin(x)**2 + cos(x)**2 - 1), x)
assert res is nan # trigsimp(res) gives nan
# J. Special functions.
def test_J1():
assert bernoulli(16) == R(-3617, 510)
def test_J2():
assert diff(elliptic_e(x, y**2), y) == (elliptic_e(x, y**2) - elliptic_f(x, y**2))/y
@XFAIL
def test_J3():
raise NotImplementedError("Jacobi elliptic functions: diff(dn(u,k), u) == -k**2*sn(u,k)*cn(u,k)")
def test_J4():
assert gamma(R(-1, 2)) == -2*sqrt(pi)
def test_J5():
assert polygamma(0, R(1, 3)) == -log(3) - sqrt(3)*pi/6 - EulerGamma - log(sqrt(3))
def test_J6():
assert mpmath.besselj(2, 1 + 1j).ae(mpc('0.04157988694396212', '0.24739764151330632'))
def test_J7():
assert simplify(besselj(R(-5,2), pi/2)) == 12/(pi**2)
def test_J8():
p = besselj(R(3,2), z)
q = (sin(z)/z - cos(z))/sqrt(pi*z/2)
assert simplify(expand_func(p) -q) == 0
def test_J9():
assert besselj(0, z).diff(z) == - besselj(1, z)
def test_J10():
mu, nu = symbols('mu, nu', integer=True)
assert assoc_legendre(nu, mu, 0) == 2**mu*sqrt(pi)/gamma((nu - mu)/2 + 1)/gamma((-nu - mu + 1)/2)
def test_J11():
assert simplify(assoc_legendre(3, 1, x)) == simplify(-R(3, 2)*sqrt(1 - x**2)*(5*x**2 - 1))
@slow
def test_J12():
assert simplify(chebyshevt(1008, x) - 2*x*chebyshevt(1007, x) + chebyshevt(1006, x)) == 0
def test_J13():
a = symbols('a', integer=True, negative=False)
assert chebyshevt(a, -1) == (-1)**a
def test_J14():
p = hyper([S.Half, S.Half], [R(3, 2)], z**2)
assert hyperexpand(p) == asin(z)/z
@XFAIL
def test_J15():
raise NotImplementedError("F((n+2)/2,-(n-2)/2,R(3,2),sin(z)**2) == sin(n*z)/(n*sin(z)*cos(z)); F(.) is hypergeometric function")
@XFAIL
def test_J16():
raise NotImplementedError("diff(zeta(x), x) @ x=0 == -log(2*pi)/2")
def test_J17():
assert integrate(f((x + 2)/5)*DiracDelta((x - 2)/3) - g(x)*diff(DiracDelta(x - 1), x), (x, 0, 3)) == 3*f(R(4, 5)) + Subs(Derivative(g(x), x), x, 1)
@XFAIL
def test_J18():
raise NotImplementedError("define an antisymmetric function")
# K. The Complex Domain
def test_K1():
z1, z2 = symbols('z1, z2', complex=True)
assert re(z1 + I*z2) == -im(z2) + re(z1)
assert im(z1 + I*z2) == im(z1) + re(z2)
def test_K2():
assert abs(3 - sqrt(7) + I*sqrt(6*sqrt(7) - 15)) == 1
@XFAIL
def test_K3():
a, b = symbols('a, b', real=True)
assert simplify(abs(1/(a + I/a + I*b))) == 1/sqrt(a**2 + (I/a + b)**2)
def test_K4():
assert log(3 + 4*I).expand(complex=True) == log(5) + I*atan(R(4, 3))
def test_K5():
x, y = symbols('x, y', real=True)
assert tan(x + I*y).expand(complex=True) == (sin(2*x)/(cos(2*x) +
cosh(2*y)) + I*sinh(2*y)/(cos(2*x) + cosh(2*y)))
def test_K6():
assert sqrt(x*y*abs(z)**2)/(sqrt(x)*abs(z)) == sqrt(x*y)/sqrt(x)
assert sqrt(x*y*abs(z)**2)/(sqrt(x)*abs(z)) != sqrt(y)
def test_K7():
y = symbols('y', real=True, negative=False)
expr = sqrt(x*y*abs(z)**2)/(sqrt(x)*abs(z))
sexpr = simplify(expr)
assert sexpr == sqrt(y)
def test_K8():
z = symbols('z', complex=True)
assert simplify(sqrt(1/z) - 1/sqrt(z)) != 0 # Passes
z = symbols('z', complex=True, negative=False)
assert simplify(sqrt(1/z) - 1/sqrt(z)) == 0 # Fails
def test_K9():
z = symbols('z', real=True, positive=True)
assert simplify(sqrt(1/z) - 1/sqrt(z)) == 0
def test_K10():
z = symbols('z', real=True, negative=True)
assert simplify(sqrt(1/z) + 1/sqrt(z)) == 0
# This goes up to K25
# L. Determining Zero Equivalence
def test_L1():
assert sqrt(997) - (997**3)**R(1, 6) == 0
def test_L2():
assert sqrt(999983) - (999983**3)**R(1, 6) == 0
def test_L3():
assert simplify((2**R(1, 3) + 4**R(1, 3))**3 - 6*(2**R(1, 3) + 4**R(1, 3)) - 6) == 0
def test_L4():
assert trigsimp(cos(x)**3 + cos(x)*sin(x)**2 - cos(x)) == 0
@XFAIL
def test_L5():
assert log(tan(R(1, 2)*x + pi/4)) - asinh(tan(x)) == 0
def test_L6():
assert (log(tan(x/2 + pi/4)) - asinh(tan(x))).diff(x).subs({x: 0}) == 0
@XFAIL
def test_L7():
assert simplify(log((2*sqrt(x) + 1)/(sqrt(4*x + 4*sqrt(x) + 1)))) == 0
@XFAIL
def test_L8():
assert simplify((4*x + 4*sqrt(x) + 1)**(sqrt(x)/(2*sqrt(x) + 1)) \
*(2*sqrt(x) + 1)**(1/(2*sqrt(x) + 1)) - 2*sqrt(x) - 1) == 0
@XFAIL
def test_L9():
z = symbols('z', complex=True)
assert simplify(2**(1 - z)*gamma(z)*zeta(z)*cos(z*pi/2) - pi**2*zeta(1 - z)) == 0
# M. Equations
@XFAIL
def test_M1():
assert Equality(x, 2)/2 + Equality(1, 1) == Equality(x/2 + 1, 2)
def test_M2():
# The roots of this equation should all be real. Note that this
# doesn't test that they are correct.
sol = solveset(3*x**3 - 18*x**2 + 33*x - 19, x)
assert all(s.expand(complex=True).is_real for s in sol)
@XFAIL
def test_M5():
assert solveset(x**6 - 9*x**4 - 4*x**3 + 27*x**2 - 36*x - 23, x) == FiniteSet(2**(1/3) + sqrt(3), 2**(1/3) - sqrt(3), +sqrt(3) - 1/2**(2/3) + I*sqrt(3)/2**(2/3), +sqrt(3) - 1/2**(2/3) - I*sqrt(3)/2**(2/3), -sqrt(3) - 1/2**(2/3) + I*sqrt(3)/2**(2/3), -sqrt(3) - 1/2**(2/3) - I*sqrt(3)/2**(2/3))
def test_M6():
assert set(solveset(x**7 - 1, x)) == \
{cos(n*pi*R(2, 7)) + I*sin(n*pi*R(2, 7)) for n in range(0, 7)}
# The paper asks for exp terms, but sin's and cos's may be acceptable;
# if the results are simplified, exp terms appear for all but
# -sin(pi/14) - I*cos(pi/14) and -sin(pi/14) + I*cos(pi/14) which
# will simplify if you apply the transformation foo.rewrite(exp).expand()
def test_M7():
# TODO: Replace solve with solveset, as of now test fails for solveset
sol = solve(x**8 - 8*x**7 + 34*x**6 - 92*x**5 + 175*x**4 - 236*x**3 +
226*x**2 - 140*x + 46, x)
assert [s.simplify() for s in sol] == [
1 - sqrt(-6 - 2*I*sqrt(3 + 4*sqrt(3)))/2,
1 + sqrt(-6 - 2*I*sqrt(3 + 4*sqrt(3)))/2,
1 - sqrt(-6 + 2*I*sqrt(3 + 4*sqrt(3)))/2,
1 + sqrt(-6 + 2*I*sqrt(3 + 4*sqrt (3)))/2,
1 - sqrt(-6 + 2*sqrt(-3 + 4*sqrt(3)))/2,
1 + sqrt(-6 + 2*sqrt(-3 + 4*sqrt(3)))/2,
1 - sqrt(-6 - 2*sqrt(-3 + 4*sqrt(3)))/2,
1 + sqrt(-6 - 2*sqrt(-3 + 4*sqrt(3)))/2]
@XFAIL # There are an infinite number of solutions.
def test_M8():
x = Symbol('x')
z = symbols('z', complex=True)
assert solveset(exp(2*x) + 2*exp(x) + 1 - z, x, S.Reals) == \
FiniteSet(log(1 + z - 2*sqrt(z))/2, log(1 + z + 2*sqrt(z))/2)
# This one could be simplified better (the 1/2 could be pulled into the log
# as a sqrt, and the function inside the log can be factored as a square,
# giving [log(sqrt(z) - 1), log(sqrt(z) + 1)]). Also, there should be an
# infinite number of solutions.
# x = {log(sqrt(z) - 1), log(sqrt(z) + 1) + i pi} [+ n 2 pi i, + n 2 pi i]
# where n is an arbitrary integer. See url of detailed output above.
@XFAIL
def test_M9():
# x = symbols('x')
raise NotImplementedError("solveset(exp(2-x**2)-exp(-x),x) has complex solutions.")
def test_M10():
# TODO: Replace solve with solveset, as of now test fails for solveset
assert solve(exp(x) - x, x) == [-LambertW(-1)]
@XFAIL
def test_M11():
assert solveset(x**x - x, x) == FiniteSet(-1, 1)
def test_M12():
# TODO: x = [-1, 2*(+/-asinh(1)*I + n*pi}, 3*(pi/6 + n*pi/3)]
# TODO: Replace solve with solveset, as of now test fails for solveset
assert solve((x + 1)*(sin(x)**2 + 1)**2*cos(3*x)**3, x) == [
-1, pi/6, pi/2,
- I*log(1 + sqrt(2)), I*log(1 + sqrt(2)),
pi - I*log(1 + sqrt(2)), pi + I*log(1 + sqrt(2)),
]
@XFAIL
def test_M13():
n = Dummy('n')
assert solveset_real(sin(x) - cos(x), x) == ImageSet(Lambda(n, n*pi - pi*R(7, 4)), S.Integers)
@XFAIL
def test_M14():
n = Dummy('n')
assert solveset_real(tan(x) - 1, x) == ImageSet(Lambda(n, n*pi + pi/4), S.Integers)
def test_M15():
n = Dummy('n')
got = solveset(sin(x) - S.Half)
assert any(got.dummy_eq(i) for i in (
Union(ImageSet(Lambda(n, 2*n*pi + pi/6), S.Integers),
ImageSet(Lambda(n, 2*n*pi + pi*R(5, 6)), S.Integers)),
Union(ImageSet(Lambda(n, 2*n*pi + pi*R(5, 6)), S.Integers),
ImageSet(Lambda(n, 2*n*pi + pi/6), S.Integers))))
@XFAIL
def test_M16():
n = Dummy('n')
assert solveset(sin(x) - tan(x), x) == ImageSet(Lambda(n, n*pi), S.Integers)
@XFAIL
def test_M17():
assert solveset_real(asin(x) - atan(x), x) == FiniteSet(0)
@XFAIL
def test_M18():
assert solveset_real(acos(x) - atan(x), x) == FiniteSet(sqrt((sqrt(5) - 1)/2))
def test_M19():
# TODO: Replace solve with solveset, as of now test fails for solveset
assert solve((x - 2)/x**R(1, 3), x) == [2]
def test_M20():
assert solveset(sqrt(x**2 + 1) - x + 2, x) == EmptySet
def test_M21():
assert solveset(x + sqrt(x) - 2) == FiniteSet(1)
def test_M22():
assert solveset(2*sqrt(x) + 3*x**R(1, 4) - 2) == FiniteSet(R(1, 16))
def test_M23():
x = symbols('x', complex=True)
# TODO: Replace solve with solveset, as of now test fails for solveset
assert solve(x - 1/sqrt(1 + x**2)) == [
-I*sqrt(S.Half + sqrt(5)/2), sqrt(Rational(-1, 2) + sqrt(5)/2)]
def test_M24():
# TODO: Replace solve with solveset, as of now test fails for solveset
solution = solve(1 - binomial(m, 2)*2**k, k)
answer = log(2/(m*(m - 1)), 2)
assert solution[0].expand() == answer.expand()
def test_M25():
a, b, c, d = symbols(':d', positive=True)
x = symbols('x')
# TODO: Replace solve with solveset, as of now test fails for solveset
assert solve(a*b**x - c*d**x, x)[0].expand() == (log(c/a)/log(b/d)).expand()
def test_M26():
# TODO: Replace solve with solveset, as of now test fails for solveset
assert solve(sqrt(log(x)) - log(sqrt(x))) == [1, exp(4)]
def test_M27():
x = symbols('x', real=True)
b = symbols('b', real=True)
with assuming(Q.is_true(sin(cos(1/E**2) + 1) + b > 0)):
# TODO: Replace solve with solveset
solve(log(acos(asin(x**R(2, 3) - b) - 1)) + 2, x) == [-b - sin(1 + cos(1/E**2))**R(3/2), b + sin(1 + cos(1/E**2))**R(3/2)]
@XFAIL
def test_M28():
assert solveset_real(5*x + exp((x - 5)/2) - 8*x**3, x, assume=Q.real(x)) == [-0.784966, -0.016291, 0.802557]
def test_M29():
x = symbols('x')
assert solveset(abs(x - 1) - 2, domain=S.Reals) == FiniteSet(-1, 3)
def test_M30():
# TODO: Replace solve with solveset, as of now
# solveset doesn't supports assumptions
# assert solve(abs(2*x + 5) - abs(x - 2),x, assume=Q.real(x)) == [-1, -7]
assert solveset_real(abs(2*x + 5) - abs(x - 2), x) == FiniteSet(-1, -7)
def test_M31():
# TODO: Replace solve with solveset, as of now
# solveset doesn't supports assumptions
# assert solve(1 - abs(x) - max(-x - 2, x - 2),x, assume=Q.real(x)) == [-3/2, 3/2]
assert solveset_real(1 - abs(x) - Max(-x - 2, x - 2), x) == FiniteSet(R(-3, 2), R(3, 2))
@XFAIL
def test_M32():
# TODO: Replace solve with solveset, as of now
# solveset doesn't supports assumptions
assert solveset_real(Max(2 - x**2, x)- Max(-x, (x**3)/9), x) == FiniteSet(-1, 3)
@XFAIL
def test_M33():
# TODO: Replace solve with solveset, as of now
# solveset doesn't supports assumptions
# Second answer can be written in another form. The second answer is the root of x**3 + 9*x**2 - 18 = 0 in the interval (-2, -1).
assert solveset_real(Max(2 - x**2, x) - x**3/9, x) == FiniteSet(-3, -1.554894, 3)
@XFAIL
def test_M34():
z = symbols('z', complex=True)
assert solveset((1 + I) * z + (2 - I) * conjugate(z) + 3*I, z) == FiniteSet(2 + 3*I)
def test_M35():
x, y = symbols('x y', real=True)
assert linsolve((3*x - 2*y - I*y + 3*I).as_real_imag(), y, x) == FiniteSet((3, 2))
def test_M36():
# TODO: Replace solve with solveset, as of now
# solveset doesn't supports solving for function
# assert solve(f**2 + f - 2, x) == [Eq(f(x), 1), Eq(f(x), -2)]
assert solveset(f(x)**2 + f(x) - 2, f(x)) == FiniteSet(-2, 1)
def test_M37():
assert linsolve([x + y + z - 6, 2*x + y + 2*z - 10, x + 3*y + z - 10 ], x, y, z) == \
FiniteSet((-z + 4, 2, z))
def test_M38():
a, b, c = symbols('a, b, c')
domain = FracField([a, b, c], ZZ).to_domain()
ring = PolyRing('k1:50', domain)
(k1, k2, k3, k4, k5, k6, k7, k8, k9, k10,
k11, k12, k13, k14, k15, k16, k17, k18, k19, k20,
k21, k22, k23, k24, k25, k26, k27, k28, k29, k30,
k31, k32, k33, k34, k35, k36, k37, k38, k39, k40,
k41, k42, k43, k44, k45, k46, k47, k48, k49) = ring.gens
system = [
-b*k8/a + c*k8/a, -b*k11/a + c*k11/a, -b*k10/a + c*k10/a + k2, -k3 - b*k9/a + c*k9/a,
-b*k14/a + c*k14/a, -b*k15/a + c*k15/a, -b*k18/a + c*k18/a - k2, -b*k17/a + c*k17/a,
-b*k16/a + c*k16/a + k4, -b*k13/a + c*k13/a - b*k21/a + c*k21/a + b*k5/a - c*k5/a,
b*k44/a - c*k44/a, -b*k45/a + c*k45/a, -b*k20/a + c*k20/a, -b*k44/a + c*k44/a,
b*k46/a - c*k46/a, b**2*k47/a**2 - 2*b*c*k47/a**2 + c**2*k47/a**2, k3, -k4,
-b*k12/a + c*k12/a - a*k6/b + c*k6/b, -b*k19/a + c*k19/a + a*k7/c - b*k7/c,
b*k45/a - c*k45/a, -b*k46/a + c*k46/a, -k48 + c*k48/a + c*k48/b - c**2*k48/(a*b),
-k49 + b*k49/a + b*k49/c - b**2*k49/(a*c), a*k1/b - c*k1/b, a*k4/b - c*k4/b,
a*k3/b - c*k3/b + k9, -k10 + a*k2/b - c*k2/b, a*k7/b - c*k7/b, -k9, k11,
b*k12/a - c*k12/a + a*k6/b - c*k6/b, a*k15/b - c*k15/b, k10 + a*k18/b - c*k18/b,
-k11 + a*k17/b - c*k17/b, a*k16/b - c*k16/b, -a*k13/b + c*k13/b + a*k21/b - c*k21/b + a*k5/b - c*k5/b,
-a*k44/b + c*k44/b, a*k45/b - c*k45/b, a*k14/c - b*k14/c + a*k20/b - c*k20/b,
a*k44/b - c*k44/b, -a*k46/b + c*k46/b, -k47 + c*k47/a + c*k47/b - c**2*k47/(a*b),
a*k19/b - c*k19/b, -a*k45/b + c*k45/b, a*k46/b - c*k46/b, a**2*k48/b**2 - 2*a*c*k48/b**2 + c**2*k48/b**2,
-k49 + a*k49/b + a*k49/c - a**2*k49/(b*c), k16, -k17, -a*k1/c + b*k1/c,
-k16 - a*k4/c + b*k4/c, -a*k3/c + b*k3/c, k18 - a*k2/c + b*k2/c, b*k19/a - c*k19/a - a*k7/c + b*k7/c,
-a*k6/c + b*k6/c, -a*k8/c + b*k8/c, -a*k11/c + b*k11/c + k17, -a*k10/c + b*k10/c - k18,
-a*k9/c + b*k9/c, -a*k14/c + b*k14/c - a*k20/b + c*k20/b, -a*k13/c + b*k13/c + a*k21/c - b*k21/c - a*k5/c + b*k5/c,
a*k44/c - b*k44/c, -a*k45/c + b*k45/c, -a*k44/c + b*k44/c, a*k46/c - b*k46/c,
-k47 + b*k47/a + b*k47/c - b**2*k47/(a*c), -a*k12/c + b*k12/c, a*k45/c - b*k45/c,
-a*k46/c + b*k46/c, -k48 + a*k48/b + a*k48/c - a**2*k48/(b*c),
a**2*k49/c**2 - 2*a*b*k49/c**2 + b**2*k49/c**2, k8, k11, -k15, k10 - k18,
-k17, k9, -k16, -k29, k14 - k32, -k21 + k23 - k31, -k24 - k30, -k35, k44,
-k45, k36, k13 - k23 + k39, -k20 + k38, k25 + k37, b*k26/a - c*k26/a - k34 + k42,
-2*k44, k45, k46, b*k47/a - c*k47/a, k41, k44, -k46, -b*k47/a + c*k47/a,
k12 + k24, -k19 - k25, -a*k27/b + c*k27/b - k33, k45, -k46, -a*k48/b + c*k48/b,
a*k28/c - b*k28/c + k40, -k45, k46, a*k48/b - c*k48/b, a*k49/c - b*k49/c,
-a*k49/c + b*k49/c, -k1, -k4, -k3, k15, k18 - k2, k17, k16, k22, k25 - k7,
k24 + k30, k21 + k23 - k31, k28, -k44, k45, -k30 - k6, k20 + k32, k27 + b*k33/a - c*k33/a,
k44, -k46, -b*k47/a + c*k47/a, -k36, k31 - k39 - k5, -k32 - k38, k19 - k37,
k26 - a*k34/b + c*k34/b - k42, k44, -2*k45, k46, a*k48/b - c*k48/b,
a*k35/c - b*k35/c - k41, -k44, k46, b*k47/a - c*k47/a, -a*k49/c + b*k49/c,
-k40, k45, -k46, -a*k48/b + c*k48/b, a*k49/c - b*k49/c, k1, k4, k3, -k8,
-k11, -k10 + k2, -k9, k37 + k7, -k14 - k38, -k22, -k25 - k37, -k24 + k6,
-k13 - k23 + k39, -k28 + b*k40/a - c*k40/a, k44, -k45, -k27, -k44, k46,
b*k47/a - c*k47/a, k29, k32 + k38, k31 - k39 + k5, -k12 + k30, k35 - a*k41/b + c*k41/b,
-k44, k45, -k26 + k34 + a*k42/c - b*k42/c, k44, k45, -2*k46, -b*k47/a + c*k47/a,
-a*k48/b + c*k48/b, a*k49/c - b*k49/c, k33, -k45, k46, a*k48/b - c*k48/b,
-a*k49/c + b*k49/c
]
solution = {
k49: 0, k48: 0, k47: 0, k46: 0, k45: 0, k44: 0, k41: 0, k40: 0,
k38: 0, k37: 0, k36: 0, k35: 0, k33: 0, k32: 0, k30: 0, k29: 0,
k28: 0, k27: 0, k25: 0, k24: 0, k22: 0, k21: 0, k20: 0, k19: 0,
k18: 0, k17: 0, k16: 0, k15: 0, k14: 0, k13: 0, k12: 0, k11: 0,
k10: 0, k9: 0, k8: 0, k7: 0, k6: 0, k5: 0, k4: 0, k3: 0,
k2: 0, k1: 0,
k34: b/c*k42, k31: k39, k26: a/c*k42, k23: k39
}
assert solve_lin_sys(system, ring) == solution
def test_M39():
x, y, z = symbols('x y z', complex=True)
# TODO: Replace solve with solveset, as of now
# solveset doesn't supports non-linear multivariate
assert solve([x**2*y + 3*y*z - 4, -3*x**2*z + 2*y**2 + 1, 2*y*z**2 - z**2 - 1 ]) ==\
[{y: 1, z: 1, x: -1}, {y: 1, z: 1, x: 1},\
{y: sqrt(2)*I, z: R(1,3) - sqrt(2)*I/3, x: -sqrt(-1 - sqrt(2)*I)},\
{y: sqrt(2)*I, z: R(1,3) - sqrt(2)*I/3, x: sqrt(-1 - sqrt(2)*I)},\
{y: -sqrt(2)*I, z: R(1,3) + sqrt(2)*I/3, x: -sqrt(-1 + sqrt(2)*I)},\
{y: -sqrt(2)*I, z: R(1,3) + sqrt(2)*I/3, x: sqrt(-1 + sqrt(2)*I)}]
# N. Inequalities
def test_N1():
assert ask(Q.is_true(E**pi > pi**E))
@XFAIL
def test_N2():
x = symbols('x', real=True)
assert ask(Q.is_true(x**4 - x + 1 > 0)) is True
assert ask(Q.is_true(x**4 - x + 1 > 1)) is False
@XFAIL
def test_N3():
x = symbols('x', real=True)
assert ask(Q.is_true(And(Lt(-1, x), Lt(x, 1))), Q.is_true(abs(x) < 1 ))
@XFAIL
def test_N4():
x, y = symbols('x y', real=True)
assert ask(Q.is_true(2*x**2 > 2*y**2), Q.is_true((x > y) & (y > 0))) is True
@XFAIL
def test_N5():
x, y, k = symbols('x y k', real=True)
assert ask(Q.is_true(k*x**2 > k*y**2), Q.is_true((x > y) & (y > 0) & (k > 0))) is True
@XFAIL
def test_N6():
x, y, k, n = symbols('x y k n', real=True)
assert ask(Q.is_true(k*x**n > k*y**n), Q.is_true((x > y) & (y > 0) & (k > 0) & (n > 0))) is True
@XFAIL
def test_N7():
x, y = symbols('x y', real=True)
assert ask(Q.is_true(y > 0), Q.is_true((x > 1) & (y >= x - 1))) is True
@XFAIL
def test_N8():
x, y, z = symbols('x y z', real=True)
assert ask(Q.is_true((x == y) & (y == z)),
Q.is_true((x >= y) & (y >= z) & (z >= x)))
def test_N9():
x = Symbol('x')
assert solveset(abs(x - 1) > 2, domain=S.Reals) == Union(Interval(-oo, -1, False, True),
Interval(3, oo, True))
def test_N10():
x = Symbol('x')
p = (x - 1)*(x - 2)*(x - 3)*(x - 4)*(x - 5)
assert solveset(expand(p) < 0, domain=S.Reals) == Union(Interval(-oo, 1, True, True),
Interval(2, 3, True, True),
Interval(4, 5, True, True))
def test_N11():
x = Symbol('x')
assert solveset(6/(x - 3) <= 3, domain=S.Reals) == Union(Interval(-oo, 3, True, True), Interval(5, oo))
def test_N12():
x = Symbol('x')
assert solveset(sqrt(x) < 2, domain=S.Reals) == Interval(0, 4, False, True)
def test_N13():
x = Symbol('x')
assert solveset(sin(x) < 2, domain=S.Reals) == S.Reals
@XFAIL
def test_N14():
x = Symbol('x')
# Gives 'Union(Interval(Integer(0), Mul(Rational(1, 2), pi), false, true),
# Interval(Mul(Rational(1, 2), pi), Mul(Integer(2), pi), true, false))'
# which is not the correct answer, but the provided also seems wrong.
assert solveset(sin(x) < 1, x, domain=S.Reals) == Union(Interval(-oo, pi/2, True, True),
Interval(pi/2, oo, True, True))
def test_N15():
r, t = symbols('r t')
# raises NotImplementedError: only univariate inequalities are supported
solveset(abs(2*r*(cos(t) - 1) + 1) <= 1, r, S.Reals)
def test_N16():
r, t = symbols('r t')
solveset((r**2)*((cos(t) - 4)**2)*sin(t)**2 < 9, r, S.Reals)
@XFAIL
def test_N17():
# currently only univariate inequalities are supported
assert solveset((x + y > 0, x - y < 0), (x, y)) == (abs(x) < y)
def test_O1():
M = Matrix((1 + I, -2, 3*I))
assert sqrt(expand(M.dot(M.H))) == sqrt(15)
def test_O2():
assert Matrix((2, 2, -3)).cross(Matrix((1, 3, 1))) == Matrix([[11],
[-5],
[4]])
# The vector module has no way of representing vectors symbolically (without
# respect to a basis)
@XFAIL
def test_O3():
# assert (va ^ vb) | (vc ^ vd) == -(va | vc)*(vb | vd) + (va | vd)*(vb | vc)
raise NotImplementedError("""The vector module has no way of representing
vectors symbolically (without respect to a basis)""")
def test_O4():
from sympy.vector import CoordSys3D, Del
N = CoordSys3D("N")
delop = Del()
i, j, k = N.base_vectors()
x, y, z = N.base_scalars()
F = i*(x*y*z) + j*((x*y*z)**2) + k*((y**2)*(z**3))
assert delop.cross(F).doit() == (-2*x**2*y**2*z + 2*y*z**3)*i + x*y*j + (2*x*y**2*z**2 - x*z)*k
@XFAIL
def test_O5():
#assert grad|(f^g)-g|(grad^f)+f|(grad^g) == 0
raise NotImplementedError("""The vector module has no way of representing
vectors symbolically (without respect to a basis)""")
#testO8-O9 MISSING!!
def test_O10():
L = [Matrix([2, 3, 5]), Matrix([3, 6, 2]), Matrix([8, 3, 6])]
assert GramSchmidt(L) == [Matrix([
[2],
[3],
[5]]),
Matrix([
[R(23, 19)],
[R(63, 19)],
[R(-47, 19)]]),
Matrix([
[R(1692, 353)],
[R(-1551, 706)],
[R(-423, 706)]])]
def test_P1():
assert Matrix(3, 3, lambda i, j: j - i).diagonal(-1) == Matrix(
1, 2, [-1, -1])
def test_P2():
M = Matrix([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
M.row_del(1)
M.col_del(2)
assert M == Matrix([[1, 2],
[7, 8]])
def test_P3():
A = Matrix([
[11, 12, 13, 14],
[21, 22, 23, 24],
[31, 32, 33, 34],
[41, 42, 43, 44]])
A11 = A[0:3, 1:4]
A12 = A[(0, 1, 3), (2, 0, 3)]
A21 = A
A221 = -A[0:2, 2:4]
A222 = -A[(3, 0), (2, 1)]
A22 = BlockMatrix([[A221, A222]]).T
rows = [[-A11, A12], [A21, A22]]
raises(ValueError, lambda: BlockMatrix(rows))
B = Matrix(rows)
assert B == Matrix([
[-12, -13, -14, 13, 11, 14],
[-22, -23, -24, 23, 21, 24],
[-32, -33, -34, 43, 41, 44],
[11, 12, 13, 14, -13, -23],
[21, 22, 23, 24, -14, -24],
[31, 32, 33, 34, -43, -13],
[41, 42, 43, 44, -42, -12]])
@XFAIL
def test_P4():
raise NotImplementedError("Block matrix diagonalization not supported")
def test_P5():
M = Matrix([[7, 11],
[3, 8]])
assert M % 2 == Matrix([[1, 1],
[1, 0]])
def test_P6():
M = Matrix([[cos(x), sin(x)],
[-sin(x), cos(x)]])
assert M.diff(x, 2) == Matrix([[-cos(x), -sin(x)],
[sin(x), -cos(x)]])
def test_P7():
M = Matrix([[x, y]])*(
z*Matrix([[1, 3, 5],
[2, 4, 6]]) + Matrix([[7, -9, 11],
[-8, 10, -12]]))
assert M == Matrix([[x*(z + 7) + y*(2*z - 8), x*(3*z - 9) + y*(4*z + 10),
x*(5*z + 11) + y*(6*z - 12)]])
def test_P8():
M = Matrix([[1, -2*I],
[-3*I, 4]])
assert M.norm(ord=S.Infinity) == 7
def test_P9():
a, b, c = symbols('a b c', nonzero=True)
M = Matrix([[a/(b*c), 1/c, 1/b],
[1/c, b/(a*c), 1/a],
[1/b, 1/a, c/(a*b)]])
assert factor(M.norm('fro')) == (a**2 + b**2 + c**2)/(abs(a)*abs(b)*abs(c))
@XFAIL
def test_P10():
M = Matrix([[1, 2 + 3*I],
[f(4 - 5*I), 6]])
# conjugate(f(4 - 5*i)) is not simplified to f(4+5*I)
assert M.H == Matrix([[1, f(4 + 5*I)],
[2 + 3*I, 6]])
@XFAIL
def test_P11():
# raises NotImplementedError("Matrix([[x,y],[1,x*y]]).inv()
# not simplifying to extract common factor")
assert Matrix([[x, y],
[1, x*y]]).inv() == (1/(x**2 - 1))*Matrix([[x, -1],
[-1/y, x/y]])
def test_P11_workaround():
# This test was changed to inverse method ADJ because it depended on the
# specific form of inverse returned from the 'GE' method which has changed.
M = Matrix([[x, y], [1, x*y]]).inv('ADJ')
c = gcd(tuple(M))
assert MatMul(c, M/c, evaluate=False) == MatMul(c, Matrix([
[x*y, -y],
[ -1, x]]), evaluate=False)
def test_P12():
A11 = MatrixSymbol('A11', n, n)
A12 = MatrixSymbol('A12', n, n)
A22 = MatrixSymbol('A22', n, n)
B = BlockMatrix([[A11, A12],
[ZeroMatrix(n, n), A22]])
assert block_collapse(B.I) == BlockMatrix([[A11.I, (-1)*A11.I*A12*A22.I],
[ZeroMatrix(n, n), A22.I]])
def test_P13():
M = Matrix([[1, x - 2, x - 3],
[x - 1, x**2 - 3*x + 6, x**2 - 3*x - 2],
[x - 2, x**2 - 8, 2*(x**2) - 12*x + 14]])
L, U, _ = M.LUdecomposition()
assert simplify(L) == Matrix([[1, 0, 0],
[x - 1, 1, 0],
[x - 2, x - 3, 1]])
assert simplify(U) == Matrix([[1, x - 2, x - 3],
[0, 4, x - 5],
[0, 0, x - 7]])
def test_P14():
M = Matrix([[1, 2, 3, 1, 3],
[3, 2, 1, 1, 7],
[0, 2, 4, 1, 1],
[1, 1, 1, 1, 4]])
R, _ = M.rref()
assert R == Matrix([[1, 0, -1, 0, 2],
[0, 1, 2, 0, -1],
[0, 0, 0, 1, 3],
[0, 0, 0, 0, 0]])
def test_P15():
M = Matrix([[-1, 3, 7, -5],
[4, -2, 1, 3],
[2, 4, 15, -7]])
assert M.rank() == 2
def test_P16():
M = Matrix([[2*sqrt(2), 8],
[6*sqrt(6), 24*sqrt(3)]])
assert M.rank() == 1
def test_P17():
t = symbols('t', real=True)
M=Matrix([
[sin(2*t), cos(2*t)],
[2*(1 - (cos(t)**2))*cos(t), (1 - 2*(sin(t)**2))*sin(t)]])
assert M.rank() == 1
def test_P18():
M = Matrix([[1, 0, -2, 0],
[-2, 1, 0, 3],
[-1, 2, -6, 6]])
assert M.nullspace() == [Matrix([[2],
[4],
[1],
[0]]),
Matrix([[0],
[-3],
[0],
[1]])]
def test_P19():
w = symbols('w')
M = Matrix([[1, 1, 1, 1],
[w, x, y, z],
[w**2, x**2, y**2, z**2],
[w**3, x**3, y**3, z**3]])
assert M.det() == (w**3*x**2*y - w**3*x**2*z - w**3*x*y**2 + w**3*x*z**2
+ w**3*y**2*z - w**3*y*z**2 - w**2*x**3*y + w**2*x**3*z
+ w**2*x*y**3 - w**2*x*z**3 - w**2*y**3*z + w**2*y*z**3
+ w*x**3*y**2 - w*x**3*z**2 - w*x**2*y**3 + w*x**2*z**3
+ w*y**3*z**2 - w*y**2*z**3 - x**3*y**2*z + x**3*y*z**2
+ x**2*y**3*z - x**2*y*z**3 - x*y**3*z**2 + x*y**2*z**3
)
@XFAIL
def test_P20():
raise NotImplementedError("Matrix minimal polynomial not supported")
def test_P21():
M = Matrix([[5, -3, -7],
[-2, 1, 2],
[2, -3, -4]])
assert M.charpoly(x).as_expr() == x**3 - 2*x**2 - 5*x + 6
def test_P22():
d = 100
M = (2 - x)*eye(d)
assert M.eigenvals() == {-x + 2: d}
def test_P23():
M = Matrix([
[2, 1, 0, 0, 0],
[1, 2, 1, 0, 0],
[0, 1, 2, 1, 0],
[0, 0, 1, 2, 1],
[0, 0, 0, 1, 2]])
assert M.eigenvals() == {
S('1'): 1,
S('2'): 1,
S('3'): 1,
S('sqrt(3) + 2'): 1,
S('-sqrt(3) + 2'): 1}
def test_P24():
M = Matrix([[611, 196, -192, 407, -8, -52, -49, 29],
[196, 899, 113, -192, -71, -43, -8, -44],
[-192, 113, 899, 196, 61, 49, 8, 52],
[ 407, -192, 196, 611, 8, 44, 59, -23],
[ -8, -71, 61, 8, 411, -599, 208, 208],
[ -52, -43, 49, 44, -599, 411, 208, 208],
[ -49, -8, 8, 59, 208, 208, 99, -911],
[ 29, -44, 52, -23, 208, 208, -911, 99]])
assert M.eigenvals() == {
S('0'): 1,
S('10*sqrt(10405)'): 1,
S('100*sqrt(26) + 510'): 1,
S('1000'): 2,
S('-100*sqrt(26) + 510'): 1,
S('-10*sqrt(10405)'): 1,
S('1020'): 1}
def test_P25():
MF = N(Matrix([[ 611, 196, -192, 407, -8, -52, -49, 29],
[ 196, 899, 113, -192, -71, -43, -8, -44],
[-192, 113, 899, 196, 61, 49, 8, 52],
[ 407, -192, 196, 611, 8, 44, 59, -23],
[ -8, -71, 61, 8, 411, -599, 208, 208],
[ -52, -43, 49, 44, -599, 411, 208, 208],
[ -49, -8, 8, 59, 208, 208, 99, -911],
[ 29, -44, 52, -23, 208, 208, -911, 99]]))
ev_1 = sorted(MF.eigenvals(multiple=True))
ev_2 = sorted(
[-1020.0490184299969, 0.0, 0.09804864072151699, 1000.0, 1000.0,
1019.9019513592784, 1020.0, 1020.0490184299969])
for x, y in zip(ev_1, ev_2):
assert abs(x - y) < 1e-12
def test_P26():
a0, a1, a2, a3, a4 = symbols('a0 a1 a2 a3 a4')
M = Matrix([[-a4, -a3, -a2, -a1, -a0, 0, 0, 0, 0],
[ 1, 0, 0, 0, 0, 0, 0, 0, 0],
[ 0, 1, 0, 0, 0, 0, 0, 0, 0],
[ 0, 0, 1, 0, 0, 0, 0, 0, 0],
[ 0, 0, 0, 1, 0, 0, 0, 0, 0],
[ 0, 0, 0, 0, 0, -1, -1, 0, 0],
[ 0, 0, 0, 0, 0, 1, 0, 0, 0],
[ 0, 0, 0, 0, 0, 0, 1, -1, -1],
[ 0, 0, 0, 0, 0, 0, 0, 1, 0]])
assert M.eigenvals(error_when_incomplete=False) == {
S('-1/2 - sqrt(3)*I/2'): 2,
S('-1/2 + sqrt(3)*I/2'): 2}
def test_P27():
a = symbols('a')
M = Matrix([[a, 0, 0, 0, 0],
[0, 0, 0, 0, 1],
[0, 0, a, 0, 0],
[0, 0, 0, a, 0],
[0, -2, 0, 0, 2]])
assert M.eigenvects() == [(a, 3, [Matrix([[1],
[0],
[0],
[0],
[0]]),
Matrix([[0],
[0],
[1],
[0],
[0]]),
Matrix([[0],
[0],
[0],
[1],
[0]])]),
(1 - I, 1, [Matrix([[ 0],
[S(1)/2 + I/2],
[ 0],
[ 0],
[ 1]])]),
(1 + I, 1, [Matrix([[ 0],
[S(1)/2 - I/2],
[ 0],
[ 0],
[ 1]])])]
@XFAIL
def test_P28():
raise NotImplementedError("Generalized eigenvectors not supported \
https://github.com/sympy/sympy/issues/5293")
@XFAIL
def test_P29():
raise NotImplementedError("Generalized eigenvectors not supported \
https://github.com/sympy/sympy/issues/5293")
def test_P30():
M = Matrix([[1, 0, 0, 1, -1],
[0, 1, -2, 3, -3],
[0, 0, -1, 2, -2],
[1, -1, 1, 0, 1],
[1, -1, 1, -1, 2]])
_, J = M.jordan_form()
assert J == Matrix([[-1, 0, 0, 0, 0],
[0, 1, 1, 0, 0],
[0, 0, 1, 0, 0],
[0, 0, 0, 1, 1],
[0, 0, 0, 0, 1]])
@XFAIL
def test_P31():
raise NotImplementedError("Smith normal form not implemented")
def test_P32():
M = Matrix([[1, -2],
[2, 1]])
assert exp(M).rewrite(cos).simplify() == Matrix([[E*cos(2), -E*sin(2)],
[E*sin(2), E*cos(2)]])
def test_P33():
w, t = symbols('w t')
M = Matrix([[0, 1, 0, 0],
[0, 0, 0, 2*w],
[0, 0, 0, 1],
[0, -2*w, 3*w**2, 0]])
assert exp(M*t).rewrite(cos).expand() == Matrix([
[1, -3*t + 4*sin(t*w)/w, 6*t*w - 6*sin(t*w), -2*cos(t*w)/w + 2/w],
[0, 4*cos(t*w) - 3, -6*w*cos(t*w) + 6*w, 2*sin(t*w)],
[0, 2*cos(t*w)/w - 2/w, -3*cos(t*w) + 4, sin(t*w)/w],
[0, -2*sin(t*w), 3*w*sin(t*w), cos(t*w)]])
@XFAIL
def test_P34():
a, b, c = symbols('a b c', real=True)
M = Matrix([[a, 1, 0, 0, 0, 0],
[0, a, 0, 0, 0, 0],
[0, 0, b, 0, 0, 0],
[0, 0, 0, c, 1, 0],
[0, 0, 0, 0, c, 1],
[0, 0, 0, 0, 0, c]])
# raises exception, sin(M) not supported. exp(M*I) also not supported
# https://github.com/sympy/sympy/issues/6218
assert sin(M) == Matrix([[sin(a), cos(a), 0, 0, 0, 0],
[0, sin(a), 0, 0, 0, 0],
[0, 0, sin(b), 0, 0, 0],
[0, 0, 0, sin(c), cos(c), -sin(c)/2],
[0, 0, 0, 0, sin(c), cos(c)],
[0, 0, 0, 0, 0, sin(c)]])
@XFAIL
def test_P35():
M = pi/2*Matrix([[2, 1, 1],
[2, 3, 2],
[1, 1, 2]])
# raises exception, sin(M) not supported. exp(M*I) also not supported
# https://github.com/sympy/sympy/issues/6218
assert sin(M) == eye(3)
@XFAIL
def test_P36():
M = Matrix([[10, 7],
[7, 17]])
assert sqrt(M) == Matrix([[3, 1],
[1, 4]])
def test_P37():
M = Matrix([[1, 1, 0],
[0, 1, 0],
[0, 0, 1]])
assert M**S.Half == Matrix([[1, R(1, 2), 0],
[0, 1, 0],
[0, 0, 1]])
@XFAIL
def test_P38():
M=Matrix([[0, 1, 0],
[0, 0, 0],
[0, 0, 0]])
#raises ValueError: Matrix det == 0; not invertible
M**S.Half
@XFAIL
def test_P39():
"""
M=Matrix([
[1, 1],
[2, 2],
[3, 3]])
M.SVD()
"""
raise NotImplementedError("Singular value decomposition not implemented")
def test_P40():
r, t = symbols('r t', real=True)
M = Matrix([r*cos(t), r*sin(t)])
assert M.jacobian(Matrix([r, t])) == Matrix([[cos(t), -r*sin(t)],
[sin(t), r*cos(t)]])
def test_P41():
r, t = symbols('r t', real=True)
assert hessian(r**2*sin(t),(r,t)) == Matrix([[ 2*sin(t), 2*r*cos(t)],
[2*r*cos(t), -r**2*sin(t)]])
def test_P42():
assert wronskian([cos(x), sin(x)], x).simplify() == 1
def test_P43():
def __my_jacobian(M, Y):
return Matrix([M.diff(v).T for v in Y]).T
r, t = symbols('r t', real=True)
M = Matrix([r*cos(t), r*sin(t)])
assert __my_jacobian(M,[r,t]) == Matrix([[cos(t), -r*sin(t)],
[sin(t), r*cos(t)]])
def test_P44():
def __my_hessian(f, Y):
V = Matrix([diff(f, v) for v in Y])
return Matrix([V.T.diff(v) for v in Y])
r, t = symbols('r t', real=True)
assert __my_hessian(r**2*sin(t), (r, t)) == Matrix([
[ 2*sin(t), 2*r*cos(t)],
[2*r*cos(t), -r**2*sin(t)]])
def test_P45():
def __my_wronskian(Y, v):
M = Matrix([Matrix(Y).T.diff(x, n) for n in range(0, len(Y))])
return M.det()
assert __my_wronskian([cos(x), sin(x)], x).simplify() == 1
# Q1-Q6 Tensor tests missing
@XFAIL
def test_R1():
i, j, n = symbols('i j n', integer=True, positive=True)
xn = MatrixSymbol('xn', n, 1)
Sm = Sum((xn[i, 0] - Sum(xn[j, 0], (j, 0, n - 1))/n)**2, (i, 0, n - 1))
# sum does not calculate
# Unknown result
Sm.doit()
raise NotImplementedError('Unknown result')
@XFAIL
def test_R2():
m, b = symbols('m b')
i, n = symbols('i n', integer=True, positive=True)
xn = MatrixSymbol('xn', n, 1)
yn = MatrixSymbol('yn', n, 1)
f = Sum((yn[i, 0] - m*xn[i, 0] - b)**2, (i, 0, n - 1))
f1 = diff(f, m)
f2 = diff(f, b)
# raises TypeError: solveset() takes at most 2 arguments (3 given)
solveset((f1, f2), (m, b), domain=S.Reals)
@XFAIL
def test_R3():
n, k = symbols('n k', integer=True, positive=True)
sk = ((-1)**k) * (binomial(2*n, k))**2
Sm = Sum(sk, (k, 1, oo))
T = Sm.doit()
T2 = T.combsimp()
# returns -((-1)**n*factorial(2*n)
# - (factorial(n))**2)*exp_polar(-I*pi)/(factorial(n))**2
assert T2 == (-1)**n*binomial(2*n, n)
@XFAIL
def test_R4():
# Macsyma indefinite sum test case:
#(c15) /* Check whether the full Gosper algorithm is implemented
# => 1/2^(n + 1) binomial(n, k - 1) */
#closedform(indefsum(binomial(n, k)/2^n - binomial(n + 1, k)/2^(n + 1), k));
#Time= 2690 msecs
# (- n + k - 1) binomial(n + 1, k)
#(d15) - --------------------------------
# n
# 2 2 (n + 1)
#
#(c16) factcomb(makefact(%));
#Time= 220 msecs
# n!
#(d16) ----------------
# n
# 2 k! 2 (n - k)!
# Might be possible after fixing https://github.com/sympy/sympy/pull/1879
raise NotImplementedError("Indefinite sum not supported")
@XFAIL
def test_R5():
a, b, c, n, k = symbols('a b c n k', integer=True, positive=True)
sk = ((-1)**k)*(binomial(a + b, a + k)
*binomial(b + c, b + k)*binomial(c + a, c + k))
Sm = Sum(sk, (k, 1, oo))
T = Sm.doit() # hypergeometric series not calculated
assert T == factorial(a+b+c)/(factorial(a)*factorial(b)*factorial(c))
def test_R6():
n, k = symbols('n k', integer=True, positive=True)
gn = MatrixSymbol('gn', n + 2, 1)
Sm = Sum(gn[k, 0] - gn[k - 1, 0], (k, 1, n + 1))
assert Sm.doit() == -gn[0, 0] + gn[n + 1, 0]
def test_R7():
n, k = symbols('n k', integer=True, positive=True)
T = Sum(k**3,(k,1,n)).doit()
assert T.factor() == n**2*(n + 1)**2/4
@XFAIL
def test_R8():
n, k = symbols('n k', integer=True, positive=True)
Sm = Sum(k**2*binomial(n, k), (k, 1, n))
T = Sm.doit() #returns Piecewise function
assert T.combsimp() == n*(n + 1)*2**(n - 2)
def test_R9():
n, k = symbols('n k', integer=True, positive=True)
Sm = Sum(binomial(n, k - 1)/k, (k, 1, n + 1))
assert Sm.doit().simplify() == (2**(n + 1) - 1)/(n + 1)
@XFAIL
def test_R10():
n, m, r, k = symbols('n m r k', integer=True, positive=True)
Sm = Sum(binomial(n, k)*binomial(m, r - k), (k, 0, r))
T = Sm.doit()
T2 = T.combsimp().rewrite(factorial)
assert T2 == factorial(m + n)/(factorial(r)*factorial(m + n - r))
assert T2 == binomial(m + n, r).rewrite(factorial)
# rewrite(binomial) is not working.
# https://github.com/sympy/sympy/issues/7135
T3 = T2.rewrite(binomial)
assert T3 == binomial(m + n, r)
@XFAIL
def test_R11():
n, k = symbols('n k', integer=True, positive=True)
sk = binomial(n, k)*fibonacci(k)
Sm = Sum(sk, (k, 0, n))
T = Sm.doit()
# Fibonacci simplification not implemented
# https://github.com/sympy/sympy/issues/7134
assert T == fibonacci(2*n)
@XFAIL
def test_R12():
n, k = symbols('n k', integer=True, positive=True)
Sm = Sum(fibonacci(k)**2, (k, 0, n))
T = Sm.doit()
assert T == fibonacci(n)*fibonacci(n + 1)
@XFAIL
def test_R13():
n, k = symbols('n k', integer=True, positive=True)
Sm = Sum(sin(k*x), (k, 1, n))
T = Sm.doit() # Sum is not calculated
assert T.simplify() == cot(x/2)/2 - cos(x*(2*n + 1)/2)/(2*sin(x/2))
@XFAIL
def test_R14():
n, k = symbols('n k', integer=True, positive=True)
Sm = Sum(sin((2*k - 1)*x), (k, 1, n))
T = Sm.doit() # Sum is not calculated
assert T.simplify() == sin(n*x)**2/sin(x)
@XFAIL
def test_R15():
n, k = symbols('n k', integer=True, positive=True)
Sm = Sum(binomial(n - k, k), (k, 0, floor(n/2)))
T = Sm.doit() # Sum is not calculated
assert T.simplify() == fibonacci(n + 1)
def test_R16():
k = symbols('k', integer=True, positive=True)
Sm = Sum(1/k**2 + 1/k**3, (k, 1, oo))
assert Sm.doit() == zeta(3) + pi**2/6
def test_R17():
k = symbols('k', integer=True, positive=True)
assert abs(float(Sum(1/k**2 + 1/k**3, (k, 1, oo)))
- 2.8469909700078206) < 1e-15
def test_R18():
k = symbols('k', integer=True, positive=True)
Sm = Sum(1/(2**k*k**2), (k, 1, oo))
T = Sm.doit()
assert T.simplify() == -log(2)**2/2 + pi**2/12
@slow
@XFAIL
def test_R19():
k = symbols('k', integer=True, positive=True)
Sm = Sum(1/((3*k + 1)*(3*k + 2)*(3*k + 3)), (k, 0, oo))
T = Sm.doit()
# assert fails, T not simplified
assert T.simplify() == -log(3)/4 + sqrt(3)*pi/12
@XFAIL
def test_R20():
n, k = symbols('n k', integer=True, positive=True)
Sm = Sum(binomial(n, 4*k), (k, 0, oo))
T = Sm.doit()
# assert fails, T not simplified
assert T.simplify() == 2**(n/2)*cos(pi*n/4)/2 + 2**(n - 1)/2
@XFAIL
def test_R21():
k = symbols('k', integer=True, positive=True)
Sm = Sum(1/(sqrt(k*(k + 1)) * (sqrt(k) + sqrt(k + 1))), (k, 1, oo))
T = Sm.doit() # Sum not calculated
assert T.simplify() == 1
# test_R22 answer not available in Wester samples
# Sum(Sum(binomial(n, k)*binomial(n - k, n - 2*k)*x**n*y**(n - 2*k),
# (k, 0, floor(n/2))), (n, 0, oo)) with abs(x*y)<1?
@XFAIL
def test_R23():
n, k = symbols('n k', integer=True, positive=True)
Sm = Sum(Sum((factorial(n)/(factorial(k)**2*factorial(n - 2*k)))*
(x/y)**k*(x*y)**(n - k), (n, 2*k, oo)), (k, 0, oo))
# Missing how to express constraint abs(x*y)<1?
T = Sm.doit() # Sum not calculated
assert T == -1/sqrt(x**2*y**2 - 4*x**2 - 2*x*y + 1)
def test_R24():
m, k = symbols('m k', integer=True, positive=True)
Sm = Sum(Product(k/(2*k - 1), (k, 1, m)), (m, 2, oo))
assert Sm.doit() == pi/2
def test_S1():
k = symbols('k', integer=True, positive=True)
Pr = Product(gamma(k/3), (k, 1, 8))
assert Pr.doit().simplify() == 640*sqrt(3)*pi**3/6561
def test_S2():
n, k = symbols('n k', integer=True, positive=True)
assert Product(k, (k, 1, n)).doit() == factorial(n)
def test_S3():
n, k = symbols('n k', integer=True, positive=True)
assert Product(x**k, (k, 1, n)).doit().simplify() == x**(n*(n + 1)/2)
def test_S4():
n, k = symbols('n k', integer=True, positive=True)
assert Product(1 + 1/k, (k, 1, n -1)).doit().simplify() == n
def test_S5():
n, k = symbols('n k', integer=True, positive=True)
assert (Product((2*k - 1)/(2*k), (k, 1, n)).doit().gammasimp() ==
gamma(n + S.Half)/(sqrt(pi)*gamma(n + 1)))
@XFAIL
def test_S6():
n, k = symbols('n k', integer=True, positive=True)
# Product does not evaluate
assert (Product(x**2 -2*x*cos(k*pi/n) + 1, (k, 1, n - 1)).doit().simplify()
== (x**(2*n) - 1)/(x**2 - 1))
@XFAIL
def test_S7():
k = symbols('k', integer=True, positive=True)
Pr = Product((k**3 - 1)/(k**3 + 1), (k, 2, oo))
T = Pr.doit() # Product does not evaluate
assert T.simplify() == R(2, 3)
@XFAIL
def test_S8():
k = symbols('k', integer=True, positive=True)
Pr = Product(1 - 1/(2*k)**2, (k, 1, oo))
T = Pr.doit()
# Product does not evaluate
assert T.simplify() == 2/pi
@XFAIL
def test_S9():
k = symbols('k', integer=True, positive=True)
Pr = Product(1 + (-1)**(k + 1)/(2*k - 1), (k, 1, oo))
T = Pr.doit()
# Product produces 0
# https://github.com/sympy/sympy/issues/7133
assert T.simplify() == sqrt(2)
@XFAIL
def test_S10():
k = symbols('k', integer=True, positive=True)
Pr = Product((k*(k + 1) + 1 + I)/(k*(k + 1) + 1 - I), (k, 0, oo))
T = Pr.doit()
# Product does not evaluate
assert T.simplify() == -1
def test_T1():
assert limit((1 + 1/n)**n, n, oo) == E
assert limit((1 - cos(x))/x**2, x, 0) == S.Half
def test_T2():
assert limit((3**x + 5**x)**(1/x), x, oo) == 5
def test_T3():
assert limit(log(x)/(log(x) + sin(x)), x, oo) == 1
def test_T4():
assert limit((exp(x*exp(-x)/(exp(-x) + exp(-2*x**2/(x + 1))))
- exp(x))/x, x, oo) == -exp(2)
def test_T5():
assert limit(x*log(x)*log(x*exp(x) - x**2)**2/log(log(x**2
+ 2*exp(exp(3*x**3*log(x))))), x, oo) == R(1, 3)
def test_T6():
assert limit(1/n * factorial(n)**(1/n), n, oo) == exp(-1)
def test_T7():
limit(1/n * gamma(n + 1)**(1/n), n, oo)
def test_T8():
a, z = symbols('a z', real=True, positive=True)
assert limit(gamma(z + a)/gamma(z)*exp(-a*log(z)), z, oo) == 1
@XFAIL
def test_T9():
z, k = symbols('z k', real=True, positive=True)
# raises NotImplementedError:
# Don't know how to calculate the mrv of '(1, k)'
assert limit(hyper((1, k), (1,), z/k), k, oo) == exp(z)
@XFAIL
def test_T10():
# No longer raises PoleError, but should return euler-mascheroni constant
assert limit(zeta(x) - 1/(x - 1), x, 1) == integrate(-1/x + 1/floor(x), (x, 1, oo))
@XFAIL
def test_T11():
n, k = symbols('n k', integer=True, positive=True)
# evaluates to 0
assert limit(n**x/(x*product((1 + x/k), (k, 1, n))), n, oo) == gamma(x)
@XFAIL
def test_T12():
x, t = symbols('x t', real=True)
# Does not evaluate the limit but returns an expression with erf
assert limit(x * integrate(exp(-t**2), (t, 0, x))/(1 - exp(-x**2)),
x, 0) == 1
def test_T13():
x = symbols('x', real=True)
assert [limit(x/abs(x), x, 0, dir='-'),
limit(x/abs(x), x, 0, dir='+')] == [-1, 1]
def test_T14():
x = symbols('x', real=True)
assert limit(atan(-log(x)), x, 0, dir='+') == pi/2
def test_U1():
x = symbols('x', real=True)
assert diff(abs(x), x) == sign(x)
def test_U2():
f = Lambda(x, Piecewise((-x, x < 0), (x, x >= 0)))
assert diff(f(x), x) == Piecewise((-1, x < 0), (1, x >= 0))
def test_U3():
f = Lambda(x, Piecewise((x**2 - 1, x == 1), (x**3, x != 1)))
f1 = Lambda(x, diff(f(x), x))
assert f1(x) == 3*x**2
assert f1(1) == 3
@XFAIL
def test_U4():
n = symbols('n', integer=True, positive=True)
x = symbols('x', real=True)
d = diff(x**n, x, n)
assert d.rewrite(factorial) == factorial(n)
def test_U5():
# issue 6681
t = symbols('t')
ans = (
Derivative(f(g(t)), g(t))*Derivative(g(t), (t, 2)) +
Derivative(f(g(t)), (g(t), 2))*Derivative(g(t), t)**2)
assert f(g(t)).diff(t, 2) == ans
assert ans.doit() == ans
def test_U6():
h = Function('h')
T = integrate(f(y), (y, h(x), g(x)))
assert T.diff(x) == (
f(g(x))*Derivative(g(x), x) - f(h(x))*Derivative(h(x), x))
@XFAIL
def test_U7():
p, t = symbols('p t', real=True)
# Exact differential => d(V(P, T)) => dV/dP DP + dV/dT DT
# raises ValueError: Since there is more than one variable in the
# expression, the variable(s) of differentiation must be supplied to
# differentiate f(p,t)
diff(f(p, t))
def test_U8():
x, y = symbols('x y', real=True)
eq = cos(x*y) + x
# If SymPy had implicit_diff() function this hack could be avoided
# TODO: Replace solve with solveset, current test fails for solveset
assert idiff(y - eq, y, x) == (-y*sin(x*y) + 1)/(x*sin(x*y) + 1)
def test_U9():
# Wester sample case for Maple:
# O29 := diff(f(x, y), x) + diff(f(x, y), y);
# /d \ /d \
# |-- f(x, y)| + |-- f(x, y)|
# \dx / \dy /
#
# O30 := factor(subs(f(x, y) = g(x^2 + y^2), %));
# 2 2
# 2 D(g)(x + y ) (x + y)
x, y = symbols('x y', real=True)
su = diff(f(x, y), x) + diff(f(x, y), y)
s2 = su.subs(f(x, y), g(x**2 + y**2))
s3 = s2.doit().factor()
# Subs not performed, s3 = 2*(x + y)*Subs(Derivative(
# g(_xi_1), _xi_1), _xi_1, x**2 + y**2)
# Derivative(g(x*2 + y**2), x**2 + y**2) is not valid in SymPy,
# and probably will remain that way. You can take derivatives with respect
# to other expressions only if they are atomic, like a symbol or a
# function.
# D operator should be added to SymPy
# See https://github.com/sympy/sympy/issues/4719.
assert s3 == (x + y)*Subs(Derivative(g(x), x), x, x**2 + y**2)*2
def test_U10():
# see issue 2519:
assert residue((z**3 + 5)/((z**4 - 1)*(z + 1)), z, -1) == R(-9, 4)
@XFAIL
def test_U11():
# assert (2*dx + dz) ^ (3*dx + dy + dz) ^ (dx + dy + 4*dz) == 8*dx ^ dy ^dz
raise NotImplementedError
@XFAIL
def test_U12():
# Wester sample case:
# (c41) /* d(3 x^5 dy /\ dz + 5 x y^2 dz /\ dx + 8 z dx /\ dy)
# => (15 x^4 + 10 x y + 8) dx /\ dy /\ dz */
# factor(ext_diff(3*x^5 * dy ~ dz + 5*x*y^2 * dz ~ dx + 8*z * dx ~ dy));
# 4
# (d41) (10 x y + 15 x + 8) dx dy dz
raise NotImplementedError(
"External diff of differential form not supported")
def test_U13():
assert minimum(x**4 - x + 1, x) == -3*2**R(1,3)/8 + 1
@XFAIL
def test_U14():
#f = 1/(x**2 + y**2 + 1)
#assert [minimize(f), maximize(f)] == [0,1]
raise NotImplementedError("minimize(), maximize() not supported")
@XFAIL
def test_U15():
raise NotImplementedError("minimize() not supported and also solve does \
not support multivariate inequalities")
@XFAIL
def test_U16():
raise NotImplementedError("minimize() not supported in SymPy and also \
solve does not support multivariate inequalities")
@XFAIL
def test_U17():
raise NotImplementedError("Linear programming, symbolic simplex not \
supported in SymPy")
def test_V1():
x = symbols('x', real=True)
assert integrate(abs(x), x) == Piecewise((-x**2/2, x <= 0), (x**2/2, True))
def test_V2():
assert integrate(Piecewise((-x, x < 0), (x, x >= 0)), x
) == Piecewise((-x**2/2, x < 0), (x**2/2, True))
def test_V3():
assert integrate(1/(x**3 + 2),x).diff().simplify() == 1/(x**3 + 2)
def test_V4():
assert integrate(2**x/sqrt(1 + 4**x), x) == asinh(2**x)/log(2)
@XFAIL
def test_V5():
# Returns (-45*x**2 + 80*x - 41)/(5*sqrt(2*x - 1)*(4*x**2 - 4*x + 1))
assert (integrate((3*x - 5)**2/(2*x - 1)**R(7, 2), x).simplify() ==
(-41 + 80*x - 45*x**2)/(5*(2*x - 1)**R(5, 2)))
@XFAIL
def test_V6():
# returns RootSum(40*_z**2 - 1, Lambda(_i, _i*log(-4*_i + exp(-m*x))))/m
assert (integrate(1/(2*exp(m*x) - 5*exp(-m*x)), x) == sqrt(10)*(
log(2*exp(m*x) - sqrt(10)) - log(2*exp(m*x) + sqrt(10)))/(20*m))
def test_V7():
r1 = integrate(sinh(x)**4/cosh(x)**2)
assert r1.simplify() == x*R(-3, 2) + sinh(x)**3/(2*cosh(x)) + 3*tanh(x)/2
@XFAIL
def test_V8_V9():
#Macsyma test case:
#(c27) /* This example involves several symbolic parameters
# => 1/sqrt(b^2 - a^2) log([sqrt(b^2 - a^2) tan(x/2) + a + b]/
# [sqrt(b^2 - a^2) tan(x/2) - a - b]) (a^2 < b^2)
# [Gradshteyn and Ryzhik 2.553(3)] */
#assume(b^2 > a^2)$
#(c28) integrate(1/(a + b*cos(x)), x);
#(c29) trigsimp(ratsimp(diff(%, x)));
# 1
#(d29) ------------
# b cos(x) + a
raise NotImplementedError(
"Integrate with assumption not supported")
def test_V10():
assert integrate(1/(3 + 3*cos(x) + 4*sin(x)), x) == log(tan(x/2) + R(3, 4))/4
def test_V11():
r1 = integrate(1/(4 + 3*cos(x) + 4*sin(x)), x)
r2 = factor(r1)
assert (logcombine(r2, force=True) ==
log(((tan(x/2) + 1)/(tan(x/2) + 7))**R(1, 3)))
def test_V12():
r1 = integrate(1/(5 + 3*cos(x) + 4*sin(x)), x)
assert r1 == -1/(tan(x/2) + 2)
@XFAIL
def test_V13():
r1 = integrate(1/(6 + 3*cos(x) + 4*sin(x)), x)
# expression not simplified, returns: -sqrt(11)*I*log(tan(x/2) + 4/3
# - sqrt(11)*I/3)/11 + sqrt(11)*I*log(tan(x/2) + 4/3 + sqrt(11)*I/3)/11
assert r1.simplify() == 2*sqrt(11)*atan(sqrt(11)*(3*tan(x/2) + 4)/11)/11
@slow
@XFAIL
def test_V14():
r1 = integrate(log(abs(x**2 - y**2)), x)
# Piecewise result does not simplify to the desired result.
assert (r1.simplify() == x*log(abs(x**2 - y**2))
+ y*log(x + y) - y*log(x - y) - 2*x)
def test_V15():
r1 = integrate(x*acot(x/y), x)
assert simplify(r1 - (x*y + (x**2 + y**2)*acot(x/y))/2) == 0
@XFAIL
def test_V16():
# Integral not calculated
assert integrate(cos(5*x)*Ci(2*x), x) == Ci(2*x)*sin(5*x)/5 - (Si(3*x) + Si(7*x))/10
@XFAIL
def test_V17():
r1 = integrate((diff(f(x), x)*g(x)
- f(x)*diff(g(x), x))/(f(x)**2 - g(x)**2), x)
# integral not calculated
assert simplify(r1 - (f(x) - g(x))/(f(x) + g(x))/2) == 0
@XFAIL
def test_W1():
# The function has a pole at y.
# The integral has a Cauchy principal value of zero but SymPy returns -I*pi
# https://github.com/sympy/sympy/issues/7159
assert integrate(1/(x - y), (x, y - 1, y + 1)) == 0
@XFAIL
def test_W2():
# The function has a pole at y.
# The integral is divergent but SymPy returns -2
# https://github.com/sympy/sympy/issues/7160
# Test case in Macsyma:
# (c6) errcatch(integrate(1/(x - a)^2, x, a - 1, a + 1));
# Integral is divergent
assert integrate(1/(x - y)**2, (x, y - 1, y + 1)) is zoo
@XFAIL
@slow
def test_W3():
# integral is not calculated
# https://github.com/sympy/sympy/issues/7161
assert integrate(sqrt(x + 1/x - 2), (x, 0, 1)) == R(4, 3)
@XFAIL
@slow
def test_W4():
# integral is not calculated
assert integrate(sqrt(x + 1/x - 2), (x, 1, 2)) == -2*sqrt(2)/3 + R(4, 3)
@XFAIL
@slow
def test_W5():
# integral is not calculated
assert integrate(sqrt(x + 1/x - 2), (x, 0, 2)) == -2*sqrt(2)/3 + R(8, 3)
@XFAIL
@slow
def test_W6():
# integral is not calculated
assert integrate(sqrt(2 - 2*cos(2*x))/2, (x, pi*R(-3, 4), -pi/4)) == sqrt(2)
def test_W7():
a = symbols('a', real=True, positive=True)
r1 = integrate(cos(x)/(x**2 + a**2), (x, -oo, oo))
assert r1.simplify() == pi*exp(-a)/a
@XFAIL
def test_W8():
# Test case in Mathematica:
# In[19]:= Integrate[t^(a - 1)/(1 + t), {t, 0, Infinity},
# Assumptions -> 0 < a < 1]
# Out[19]= Pi Csc[a Pi]
raise NotImplementedError(
"Integrate with assumption 0 < a < 1 not supported")
@XFAIL
def test_W9():
# Integrand with a residue at infinity => -2 pi [sin(pi/5) + sin(2pi/5)]
# (principal value) [Levinson and Redheffer, p. 234] *)
r1 = integrate(5*x**3/(1 + x + x**2 + x**3 + x**4), (x, -oo, oo))
r2 = r1.doit()
assert r2 == -2*pi*(sqrt(-sqrt(5)/8 + 5/8) + sqrt(sqrt(5)/8 + 5/8))
@XFAIL
def test_W10():
# integrate(1/[1 + x + x^2 + ... + x^(2 n)], x = -infinity..infinity) =
# 2 pi/(2 n + 1) [1 + cos(pi/[2 n + 1])] csc(2 pi/[2 n + 1])
# [Levinson and Redheffer, p. 255] => 2 pi/5 [1 + cos(pi/5)] csc(2 pi/5) */
r1 = integrate(x/(1 + x + x**2 + x**4), (x, -oo, oo))
r2 = r1.doit()
assert r2 == 2*pi*(sqrt(5)/4 + 5/4)*csc(pi*R(2, 5))/5
@XFAIL
def test_W11():
# integral not calculated
assert (integrate(sqrt(1 - x**2)/(1 + x**2), (x, -1, 1)) ==
pi*(-1 + sqrt(2)))
def test_W12():
p = symbols('p', real=True, positive=True)
q = symbols('q', real=True)
r1 = integrate(x*exp(-p*x**2 + 2*q*x), (x, -oo, oo))
assert r1.simplify() == sqrt(pi)*q*exp(q**2/p)/p**R(3, 2)
@XFAIL
def test_W13():
# Integral not calculated. Expected result is 2*(Euler_mascheroni_constant)
r1 = integrate(1/log(x) + 1/(1 - x) - log(log(1/x)), (x, 0, 1))
assert r1 == 2*EulerGamma
def test_W14():
assert integrate(sin(x)/x*exp(2*I*x), (x, -oo, oo)) == 0
@XFAIL
def test_W15():
# integral not calculated
assert integrate(log(gamma(x))*cos(6*pi*x), (x, 0, 1)) == R(1, 12)
def test_W16():
assert integrate((1 + x)**3*legendre_poly(1, x)*legendre_poly(2, x),
(x, -1, 1)) == R(36, 35)
def test_W17():
a, b = symbols('a b', real=True, positive=True)
assert integrate(exp(-a*x)*besselj(0, b*x),
(x, 0, oo)) == 1/(b*sqrt(a**2/b**2 + 1))
def test_W18():
assert integrate((besselj(1, x)/x)**2, (x, 0, oo)) == 4/(3*pi)
@XFAIL
def test_W19():
# Integral not calculated
# Expected result is (cos 7 - 1)/7 [Gradshteyn and Ryzhik 6.782(3)]
assert integrate(Ci(x)*besselj(0, 2*sqrt(7*x)), (x, 0, oo)) == (cos(7) - 1)/7
@XFAIL
def test_W20():
# integral not calculated
assert (integrate(x**2*polylog(3, 1/(x + 1)), (x, 0, 1)) ==
-pi**2/36 - R(17, 108) + zeta(3)/4 +
(-pi**2/2 - 4*log(2) + log(2)**2 + 35/3)*log(2)/9)
def test_W21():
assert abs(N(integrate(x**2*polylog(3, 1/(x + 1)), (x, 0, 1)))
- 0.210882859565594) < 1e-15
def test_W22():
t, u = symbols('t u', real=True)
s = Lambda(x, Piecewise((1, And(x >= 1, x <= 2)), (0, True)))
assert integrate(s(t)*cos(t), (t, 0, u)) == Piecewise(
(0, u < 0),
(-sin(Min(1, u)) + sin(Min(2, u)), True))
@slow
def test_W23():
a, b = symbols('a b', real=True, positive=True)
r1 = integrate(integrate(x/(x**2 + y**2), (x, a, b)), (y, -oo, oo))
assert r1.collect(pi) == pi*(-a + b)
def test_W23b():
# like W23 but limits are reversed
a, b = symbols('a b', real=True, positive=True)
r2 = integrate(integrate(x/(x**2 + y**2), (y, -oo, oo)), (x, a, b))
assert r2.collect(pi) == pi*(-a + b)
@XFAIL
@slow
def test_W24():
if ON_TRAVIS:
skip("Too slow for travis.")
# Not that slow, but does not fully evaluate so simplify is slow.
# Maybe also require doit()
x, y = symbols('x y', real=True)
r1 = integrate(integrate(sqrt(x**2 + y**2), (x, 0, 1)), (y, 0, 1))
assert (r1 - (sqrt(2) + asinh(1))/3).simplify() == 0
@XFAIL
@slow
def test_W25():
if ON_TRAVIS:
skip("Too slow for travis.")
a, x, y = symbols('a x y', real=True)
i1 = integrate(
sin(a)*sin(y)/sqrt(1 - sin(a)**2*sin(x)**2*sin(y)**2),
(x, 0, pi/2))
i2 = integrate(i1, (y, 0, pi/2))
assert (i2 - pi*a/2).simplify() == 0
def test_W26():
x, y = symbols('x y', real=True)
assert integrate(integrate(abs(y - x**2), (y, 0, 2)),
(x, -1, 1)) == R(46, 15)
def test_W27():
a, b, c = symbols('a b c')
assert integrate(integrate(integrate(1, (z, 0, c*(1 - x/a - y/b))),
(y, 0, b*(1 - x/a))),
(x, 0, a)) == a*b*c/6
def test_X1():
v, c = symbols('v c', real=True)
assert (series(1/sqrt(1 - (v/c)**2), v, x0=0, n=8) ==
5*v**6/(16*c**6) + 3*v**4/(8*c**4) + v**2/(2*c**2) + 1 + O(v**8))
def test_X2():
v, c = symbols('v c', real=True)
s1 = series(1/sqrt(1 - (v/c)**2), v, x0=0, n=8)
assert (1/s1**2).series(v, x0=0, n=8) == -v**2/c**2 + 1 + O(v**8)
def test_X3():
s1 = (sin(x).series()/cos(x).series()).series()
s2 = tan(x).series()
assert s2 == x + x**3/3 + 2*x**5/15 + O(x**6)
assert s1 == s2
def test_X4():
s1 = log(sin(x)/x).series()
assert s1 == -x**2/6 - x**4/180 + O(x**6)
assert log(series(sin(x)/x)).series() == s1
@XFAIL
def test_X5():
# test case in Mathematica syntax:
# In[21]:= (* => [a f'(a d) + g(b d) + integrate(h(c y), y = 0..d)]
# + [a^2 f''(a d) + b g'(b d) + h(c d)] (x - d) *)
# In[22]:= D[f[a*x], x] + g[b*x] + Integrate[h[c*y], {y, 0, x}]
# Out[22]= g[b x] + Integrate[h[c y], {y, 0, x}] + a f'[a x]
# In[23]:= Series[%, {x, d, 1}]
# Out[23]= (g[b d] + Integrate[h[c y], {y, 0, d}] + a f'[a d]) +
# 2 2
# (h[c d] + b g'[b d] + a f''[a d]) (-d + x) + O[-d + x]
h = Function('h')
a, b, c, d = symbols('a b c d', real=True)
# series() raises NotImplementedError:
# The _eval_nseries method should be added to <class
# 'sympy.core.function.Subs'> to give terms up to O(x**n) at x=0
series(diff(f(a*x), x) + g(b*x) + integrate(h(c*y), (y, 0, x)),
x, x0=d, n=2)
# assert missing, until exception is removed
def test_X6():
# Taylor series of nonscalar objects (noncommutative multiplication)
# expected result => (B A - A B) t^2/2 + O(t^3) [Stanly Steinberg]
a, b = symbols('a b', commutative=False, scalar=False)
assert (series(exp((a + b)*x) - exp(a*x) * exp(b*x), x, x0=0, n=3) ==
x**2*(-a*b/2 + b*a/2) + O(x**3))
def test_X7():
# => sum( Bernoulli[k]/k! x^(k - 2), k = 1..infinity )
# = 1/x^2 - 1/(2 x) + 1/12 - x^2/720 + x^4/30240 + O(x^6)
# [Levinson and Redheffer, p. 173]
assert (series(1/(x*(exp(x) - 1)), x, 0, 7) == x**(-2) - 1/(2*x) +
R(1, 12) - x**2/720 + x**4/30240 - x**6/1209600 + O(x**7))
def test_X8():
# Puiseux series (terms with fractional degree):
# => 1/sqrt(x - 3/2 pi) + (x - 3/2 pi)^(3/2) / 12 + O([x - 3/2 pi]^(7/2))
# see issue 7167:
x = symbols('x', real=True)
assert (series(sqrt(sec(x)), x, x0=pi*3/2, n=4) ==
1/sqrt(x - pi*R(3, 2)) + (x - pi*R(3, 2))**R(3, 2)/12 +
(x - pi*R(3, 2))**R(7, 2)/160 + O((x - pi*R(3, 2))**4, (x, pi*R(3, 2))))
def test_X9():
assert (series(x**x, x, x0=0, n=4) == 1 + x*log(x) + x**2*log(x)**2/2 +
x**3*log(x)**3/6 + O(x**4*log(x)**4))
def test_X10():
z, w = symbols('z w')
assert (series(log(sinh(z)) + log(cosh(z + w)), z, x0=0, n=2) ==
log(cosh(w)) + log(z) + z*sinh(w)/cosh(w) + O(z**2))
def test_X11():
z, w = symbols('z w')
assert (series(log(sinh(z) * cosh(z + w)), z, x0=0, n=2) ==
log(cosh(w)) + log(z) + z*sinh(w)/cosh(w) + O(z**2))
@XFAIL
def test_X12():
# Look at the generalized Taylor series around x = 1
# Result => (x - 1)^a/e^b [1 - (a + 2 b) (x - 1) / 2 + O((x - 1)^2)]
a, b, x = symbols('a b x', real=True)
# series returns O(log(x-1)**2)
# https://github.com/sympy/sympy/issues/7168
assert (series(log(x)**a*exp(-b*x), x, x0=1, n=2) ==
(x - 1)**a/exp(b)*(1 - (a + 2*b)*(x - 1)/2 + O((x - 1)**2)))
def test_X13():
assert series(sqrt(2*x**2 + 1), x, x0=oo, n=1) == sqrt(2)*x + O(1/x, (x, oo))
@XFAIL
def test_X14():
# Wallis' product => 1/sqrt(pi n) + ... [Knopp, p. 385]
assert series(1/2**(2*n)*binomial(2*n, n),
n, x==oo, n=1) == 1/(sqrt(pi)*sqrt(n)) + O(1/x, (x, oo))
@SKIP("https://github.com/sympy/sympy/issues/7164")
def test_X15():
# => 0!/x - 1!/x^2 + 2!/x^3 - 3!/x^4 + O(1/x^5) [Knopp, p. 544]
x, t = symbols('x t', real=True)
# raises RuntimeError: maximum recursion depth exceeded
# https://github.com/sympy/sympy/issues/7164
# 2019-02-17: Raises
# PoleError:
# Asymptotic expansion of Ei around [-oo] is not implemented.
e1 = integrate(exp(-t)/t, (t, x, oo))
assert (series(e1, x, x0=oo, n=5) ==
6/x**4 + 2/x**3 - 1/x**2 + 1/x + O(x**(-5), (x, oo)))
def test_X16():
# Multivariate Taylor series expansion => 1 - (x^2 + 2 x y + y^2)/2 + O(x^4)
assert (series(cos(x + y), x + y, x0=0, n=4) == 1 - (x + y)**2/2 +
O(x**4 + x**3*y + x**2*y**2 + x*y**3 + y**4, x, y))
@XFAIL
def test_X17():
# Power series (compute the general formula)
# (c41) powerseries(log(sin(x)/x), x, 0);
# /aquarius/data2/opt/local/macsyma_422/library1/trgred.so being loaded.
# inf
# ==== i1 2 i1 2 i1
# \ (- 1) 2 bern(2 i1) x
# (d41) > ------------------------------
# / 2 i1 (2 i1)!
# ====
# i1 = 1
# fps does not calculate
assert fps(log(sin(x)/x)) == \
Sum((-1)**k*2**(2*k - 1)*bernoulli(2*k)*x**(2*k)/(k*factorial(2*k)), (k, 1, oo))
@XFAIL
def test_X18():
# Power series (compute the general formula). Maple FPS:
# > FormalPowerSeries(exp(-x)*sin(x), x = 0);
# infinity
# ----- (1/2 k) k
# \ 2 sin(3/4 k Pi) x
# ) -------------------------
# / k!
# -----
#
# Now, sympy returns
# oo
# _____
# \ `
# \ / k k\
# \ k |I*(-1 - I) I*(-1 + I) |
# \ x *|----------- - -----------|
# / \ 2 2 /
# / ------------------------------
# / k!
# /____,
# k = 0
k = Dummy('k')
assert fps(exp(-x)*sin(x)) == \
Sum(2**(S.Half*k)*sin(R(3, 4)*k*pi)*x**k/factorial(k), (k, 0, oo))
@XFAIL
def test_X19():
# (c45) /* Derive an explicit Taylor series solution of y as a function of
# x from the following implicit relation:
# y = x - 1 + (x - 1)^2/2 + 2/3 (x - 1)^3 + (x - 1)^4 +
# 17/10 (x - 1)^5 + ...
# */
# x = sin(y) + cos(y);
# Time= 0 msecs
# (d45) x = sin(y) + cos(y)
#
# (c46) taylor_revert(%, y, 7);
raise NotImplementedError("Solve using series not supported. \
Inverse Taylor series expansion also not supported")
@XFAIL
def test_X20():
# Pade (rational function) approximation => (2 - x)/(2 + x)
# > numapprox[pade](exp(-x), x = 0, [1, 1]);
# bytes used=9019816, alloc=3669344, time=13.12
# 1 - 1/2 x
# ---------
# 1 + 1/2 x
# mpmath support numeric Pade approximant but there is
# no symbolic implementation in SymPy
# https://en.wikipedia.org/wiki/Pad%C3%A9_approximant
raise NotImplementedError("Symbolic Pade approximant not supported")
def test_X21():
"""
Test whether `fourier_series` of x periodical on the [-p, p] interval equals
`- (2 p / pi) sum( (-1)^n / n sin(n pi x / p), n = 1..infinity )`.
"""
p = symbols('p', positive=True)
n = symbols('n', positive=True, integer=True)
s = fourier_series(x, (x, -p, p))
# All cosine coefficients are equal to 0
assert s.an.formula == 0
# Check for sine coefficients
assert s.bn.formula.subs(s.bn.variables[0], 0) == 0
assert s.bn.formula.subs(s.bn.variables[0], n) == \
-2*p/pi * (-1)**n / n * sin(n*pi*x/p)
@XFAIL
def test_X22():
# (c52) /* => p / 2
# - (2 p / pi^2) sum( [1 - (-1)^n] cos(n pi x / p) / n^2,
# n = 1..infinity ) */
# fourier_series(abs(x), x, p);
# p
# (e52) a = -
# 0 2
#
# %nn
# (2 (- 1) - 2) p
# (e53) a = ------------------
# %nn 2 2
# %pi %nn
#
# (e54) b = 0
# %nn
#
# Time= 5290 msecs
# inf %nn %pi %nn x
# ==== (2 (- 1) - 2) cos(---------)
# \ p
# p > -------------------------------
# / 2
# ==== %nn
# %nn = 1 p
# (d54) ----------------------------------------- + -
# 2 2
# %pi
raise NotImplementedError("Fourier series not supported")
def test_Y1():
t = symbols('t', real=True, positive=True)
w = symbols('w', real=True)
s = symbols('s')
F, _, _ = laplace_transform(cos((w - 1)*t), t, s)
assert F == s/(s**2 + (w - 1)**2)
def test_Y2():
t = symbols('t', real=True, positive=True)
w = symbols('w', real=True)
s = symbols('s')
f = inverse_laplace_transform(s/(s**2 + (w - 1)**2), s, t)
assert f == cos(t*w - t)
def test_Y3():
t = symbols('t', real=True, positive=True)
w = symbols('w', real=True)
s = symbols('s')
F, _, _ = laplace_transform(sinh(w*t)*cosh(w*t), t, s)
assert F == w/(s**2 - 4*w**2)
def test_Y4():
t = symbols('t', real=True, positive=True)
s = symbols('s')
F, _, _ = laplace_transform(erf(3/sqrt(t)), t, s)
assert F == (1 - exp(-6*sqrt(s)))/s
@XFAIL
def test_Y5_Y6():
# Solve y'' + y = 4 [H(t - 1) - H(t - 2)], y(0) = 1, y'(0) = 0 where H is the
# Heaviside (unit step) function (the RHS describes a pulse of magnitude 4 and
# duration 1). See David A. Sanchez, Richard C. Allen, Jr. and Walter T.
# Kyner, _Differential Equations: An Introduction_, Addison-Wesley Publishing
# Company, 1983, p. 211. First, take the Laplace transform of the ODE
# => s^2 Y(s) - s + Y(s) = 4/s [e^(-s) - e^(-2 s)]
# where Y(s) is the Laplace transform of y(t)
t = symbols('t', real=True, positive=True)
s = symbols('s')
y = Function('y')
F, _, _ = laplace_transform(diff(y(t), t, 2)
+ y(t)
- 4*(Heaviside(t - 1)
- Heaviside(t - 2)), t, s)
# Laplace transform for diff() not calculated
# https://github.com/sympy/sympy/issues/7176
assert (F == s**2*LaplaceTransform(y(t), t, s) - s
+ LaplaceTransform(y(t), t, s) - 4*exp(-s)/s + 4*exp(-2*s)/s)
# TODO implement second part of test case
# Now, solve for Y(s) and then take the inverse Laplace transform
# => Y(s) = s/(s^2 + 1) + 4 [1/s - s/(s^2 + 1)] [e^(-s) - e^(-2 s)]
# => y(t) = cos t + 4 {[1 - cos(t - 1)] H(t - 1) - [1 - cos(t - 2)] H(t - 2)}
@XFAIL
def test_Y7():
# What is the Laplace transform of an infinite square wave?
# => 1/s + 2 sum( (-1)^n e^(- s n a)/s, n = 1..infinity )
# [Sanchez, Allen and Kyner, p. 213]
t = symbols('t', real=True, positive=True)
a = symbols('a', real=True)
s = symbols('s')
F, _, _ = laplace_transform(1 + 2*Sum((-1)**n*Heaviside(t - n*a),
(n, 1, oo)), t, s)
# returns 2*LaplaceTransform(Sum((-1)**n*Heaviside(-a*n + t),
# (n, 1, oo)), t, s) + 1/s
# https://github.com/sympy/sympy/issues/7177
assert F == 2*Sum((-1)**n*exp(-a*n*s)/s, (n, 1, oo)) + 1/s
@XFAIL
def test_Y8():
assert fourier_transform(1, x, z) == DiracDelta(z)
def test_Y9():
assert (fourier_transform(exp(-9*x**2), x, z) ==
sqrt(pi)*exp(-pi**2*z**2/9)/3)
def test_Y10():
assert (fourier_transform(abs(x)*exp(-3*abs(x)), x, z) ==
(-8*pi**2*z**2 + 18)/(16*pi**4*z**4 + 72*pi**2*z**2 + 81))
@SKIP("https://github.com/sympy/sympy/issues/7181")
@slow
def test_Y11():
# => pi cot(pi s) (0 < Re s < 1) [Gradshteyn and Ryzhik 17.43(5)]
x, s = symbols('x s')
# raises RuntimeError: maximum recursion depth exceeded
# https://github.com/sympy/sympy/issues/7181
# Update 2019-02-17 raises:
# TypeError: cannot unpack non-iterable MellinTransform object
F, _, _ = mellin_transform(1/(1 - x), x, s)
assert F == pi*cot(pi*s)
@XFAIL
def test_Y12():
# => 2^(s - 4) gamma(s/2)/gamma(4 - s/2) (0 < Re s < 1)
# [Gradshteyn and Ryzhik 17.43(16)]
x, s = symbols('x s')
# returns Wrong value -2**(s - 4)*gamma(s/2 - 3)/gamma(-s/2 + 1)
# https://github.com/sympy/sympy/issues/7182
F, _, _ = mellin_transform(besselj(3, x)/x**3, x, s)
assert F == -2**(s - 4)*gamma(s/2)/gamma(-s/2 + 4)
@XFAIL
def test_Y13():
# Z[H(t - m T)] => z/[z^m (z - 1)] (H is the Heaviside (unit step) function) z
raise NotImplementedError("z-transform not supported")
@XFAIL
def test_Y14():
# Z[H(t - m T)] => z/[z^m (z - 1)] (H is the Heaviside (unit step) function)
raise NotImplementedError("z-transform not supported")
def test_Z1():
r = Function('r')
assert (rsolve(r(n + 2) - 2*r(n + 1) + r(n) - 2, r(n),
{r(0): 1, r(1): m}).simplify() == n**2 + n*(m - 2) + 1)
def test_Z2():
r = Function('r')
assert (rsolve(r(n) - (5*r(n - 1) - 6*r(n - 2)), r(n), {r(0): 0, r(1): 1})
== -2**n + 3**n)
def test_Z3():
# => r(n) = Fibonacci[n + 1] [Cohen, p. 83]
r = Function('r')
# recurrence solution is correct, Wester expects it to be simplified to
# fibonacci(n+1), but that is quite hard
assert (rsolve(r(n) - (r(n - 1) + r(n - 2)), r(n),
{r(1): 1, r(2): 2}).simplify()
== 2**(-n)*((1 + sqrt(5))**n*(sqrt(5) + 5) +
(-sqrt(5) + 1)**n*(-sqrt(5) + 5))/10)
@XFAIL
def test_Z4():
# => [c^(n+1) [c^(n+1) - 2 c - 2] + (n+1) c^2 + 2 c - n] / [(c-1)^3 (c+1)]
# [Joan Z. Yu and Robert Israel in sci.math.symbolic]
r = Function('r')
c = symbols('c')
# raises ValueError: Polynomial or rational function expected,
# got '(c**2 - c**n)/(c - c**n)
s = rsolve(r(n) - ((1 + c - c**(n-1) - c**(n+1))/(1 - c**n)*r(n - 1)
- c*(1 - c**(n-2))/(1 - c**(n-1))*r(n - 2) + 1),
r(n), {r(1): 1, r(2): (2 + 2*c + c**2)/(1 + c)})
assert (s - (c*(n + 1)*(c*(n + 1) - 2*c - 2) +
(n + 1)*c**2 + 2*c - n)/((c-1)**3*(c+1)) == 0)
@XFAIL
def test_Z5():
# Second order ODE with initial conditions---solve directly
# transform: f(t) = sin(2 t)/8 - t cos(2 t)/4
C1, C2 = symbols('C1 C2')
# initial conditions not supported, this is a manual workaround
# https://github.com/sympy/sympy/issues/4720
eq = Derivative(f(x), x, 2) + 4*f(x) - sin(2*x)
sol = dsolve(eq, f(x))
f0 = Lambda(x, sol.rhs)
assert f0(x) == C2*sin(2*x) + (C1 - x/4)*cos(2*x)
f1 = Lambda(x, diff(f0(x), x))
# TODO: Replace solve with solveset, when it works for solveset
const_dict = solve((f0(0), f1(0)))
result = f0(x).subs(C1, const_dict[C1]).subs(C2, const_dict[C2])
assert result == -x*cos(2*x)/4 + sin(2*x)/8
# Result is OK, but ODE solving with initial conditions should be
# supported without all this manual work
raise NotImplementedError('ODE solving with initial conditions \
not supported')
@XFAIL
def test_Z6():
# Second order ODE with initial conditions---solve using Laplace
# transform: f(t) = sin(2 t)/8 - t cos(2 t)/4
t = symbols('t', real=True, positive=True)
s = symbols('s')
eq = Derivative(f(t), t, 2) + 4*f(t) - sin(2*t)
F, _, _ = laplace_transform(eq, t, s)
# Laplace transform for diff() not calculated
# https://github.com/sympy/sympy/issues/7176
assert (F == s**2*LaplaceTransform(f(t), t, s) +
4*LaplaceTransform(f(t), t, s) - 2/(s**2 + 4))
# rest of test case not implemented
|
eda64c75ea3c5d3beb2536a5bb5d0b96195e234651884a1930305bb6d4940a1b | from __future__ import print_function, division
import itertools
from sympy.core import S
from sympy.core.containers import Tuple
from sympy.core.function import _coeff_isneg
from sympy.core.mul import Mul
from sympy.core.numbers import Number, Rational
from sympy.core.power import Pow
from sympy.core.symbol import Symbol
from sympy.core.sympify import SympifyError
from sympy.printing.conventions import requires_partial
from sympy.printing.precedence import PRECEDENCE, precedence, precedence_traditional
from sympy.printing.printer import Printer
from sympy.printing.str import sstr
from sympy.utilities import default_sort_key
from sympy.utilities.iterables import has_variety
from sympy.utilities.exceptions import SymPyDeprecationWarning
from sympy.printing.pretty.stringpict import prettyForm, stringPict
from sympy.printing.pretty.pretty_symbology import xstr, hobj, vobj, xobj, \
xsym, pretty_symbol, pretty_atom, pretty_use_unicode, greek_unicode, U, \
pretty_try_use_unicode, annotated
# rename for usage from outside
pprint_use_unicode = pretty_use_unicode
pprint_try_use_unicode = pretty_try_use_unicode
class PrettyPrinter(Printer):
"""Printer, which converts an expression into 2D ASCII-art figure."""
printmethod = "_pretty"
_default_settings = {
"order": None,
"full_prec": "auto",
"use_unicode": None,
"wrap_line": True,
"num_columns": None,
"use_unicode_sqrt_char": True,
"root_notation": True,
"mat_symbol_style": "plain",
"imaginary_unit": "i",
"perm_cyclic": True
}
def __init__(self, settings=None):
Printer.__init__(self, settings)
if not isinstance(self._settings['imaginary_unit'], str):
raise TypeError("'imaginary_unit' must a string, not {}".format(self._settings['imaginary_unit']))
elif self._settings['imaginary_unit'] not in ["i", "j"]:
raise ValueError("'imaginary_unit' must be either 'i' or 'j', not '{}'".format(self._settings['imaginary_unit']))
def emptyPrinter(self, expr):
return prettyForm(xstr(expr))
@property
def _use_unicode(self):
if self._settings['use_unicode']:
return True
else:
return pretty_use_unicode()
def doprint(self, expr):
return self._print(expr).render(**self._settings)
# empty op so _print(stringPict) returns the same
def _print_stringPict(self, e):
return e
def _print_basestring(self, e):
return prettyForm(e)
def _print_atan2(self, e):
pform = prettyForm(*self._print_seq(e.args).parens())
pform = prettyForm(*pform.left('atan2'))
return pform
def _print_Symbol(self, e, bold_name=False):
symb = pretty_symbol(e.name, bold_name)
return prettyForm(symb)
_print_RandomSymbol = _print_Symbol
def _print_MatrixSymbol(self, e):
return self._print_Symbol(e, self._settings['mat_symbol_style'] == "bold")
def _print_Float(self, e):
# we will use StrPrinter's Float printer, but we need to handle the
# full_prec ourselves, according to the self._print_level
full_prec = self._settings["full_prec"]
if full_prec == "auto":
full_prec = self._print_level == 1
return prettyForm(sstr(e, full_prec=full_prec))
def _print_Cross(self, e):
vec1 = e._expr1
vec2 = e._expr2
pform = self._print(vec2)
pform = prettyForm(*pform.left('('))
pform = prettyForm(*pform.right(')'))
pform = prettyForm(*pform.left(self._print(U('MULTIPLICATION SIGN'))))
pform = prettyForm(*pform.left(')'))
pform = prettyForm(*pform.left(self._print(vec1)))
pform = prettyForm(*pform.left('('))
return pform
def _print_Curl(self, e):
vec = e._expr
pform = self._print(vec)
pform = prettyForm(*pform.left('('))
pform = prettyForm(*pform.right(')'))
pform = prettyForm(*pform.left(self._print(U('MULTIPLICATION SIGN'))))
pform = prettyForm(*pform.left(self._print(U('NABLA'))))
return pform
def _print_Divergence(self, e):
vec = e._expr
pform = self._print(vec)
pform = prettyForm(*pform.left('('))
pform = prettyForm(*pform.right(')'))
pform = prettyForm(*pform.left(self._print(U('DOT OPERATOR'))))
pform = prettyForm(*pform.left(self._print(U('NABLA'))))
return pform
def _print_Dot(self, e):
vec1 = e._expr1
vec2 = e._expr2
pform = self._print(vec2)
pform = prettyForm(*pform.left('('))
pform = prettyForm(*pform.right(')'))
pform = prettyForm(*pform.left(self._print(U('DOT OPERATOR'))))
pform = prettyForm(*pform.left(')'))
pform = prettyForm(*pform.left(self._print(vec1)))
pform = prettyForm(*pform.left('('))
return pform
def _print_Gradient(self, e):
func = e._expr
pform = self._print(func)
pform = prettyForm(*pform.left('('))
pform = prettyForm(*pform.right(')'))
pform = prettyForm(*pform.left(self._print(U('NABLA'))))
return pform
def _print_Laplacian(self, e):
func = e._expr
pform = self._print(func)
pform = prettyForm(*pform.left('('))
pform = prettyForm(*pform.right(')'))
pform = prettyForm(*pform.left(self._print(U('INCREMENT'))))
return pform
def _print_Atom(self, e):
try:
# print atoms like Exp1 or Pi
return prettyForm(pretty_atom(e.__class__.__name__, printer=self))
except KeyError:
return self.emptyPrinter(e)
# Infinity inherits from Number, so we have to override _print_XXX order
_print_Infinity = _print_Atom
_print_NegativeInfinity = _print_Atom
_print_EmptySet = _print_Atom
_print_Naturals = _print_Atom
_print_Naturals0 = _print_Atom
_print_Integers = _print_Atom
_print_Rationals = _print_Atom
_print_Complexes = _print_Atom
_print_EmptySequence = _print_Atom
def _print_Reals(self, e):
if self._use_unicode:
return self._print_Atom(e)
else:
inf_list = ['-oo', 'oo']
return self._print_seq(inf_list, '(', ')')
def _print_subfactorial(self, e):
x = e.args[0]
pform = self._print(x)
# Add parentheses if needed
if not ((x.is_Integer and x.is_nonnegative) or x.is_Symbol):
pform = prettyForm(*pform.parens())
pform = prettyForm(*pform.left('!'))
return pform
def _print_factorial(self, e):
x = e.args[0]
pform = self._print(x)
# Add parentheses if needed
if not ((x.is_Integer and x.is_nonnegative) or x.is_Symbol):
pform = prettyForm(*pform.parens())
pform = prettyForm(*pform.right('!'))
return pform
def _print_factorial2(self, e):
x = e.args[0]
pform = self._print(x)
# Add parentheses if needed
if not ((x.is_Integer and x.is_nonnegative) or x.is_Symbol):
pform = prettyForm(*pform.parens())
pform = prettyForm(*pform.right('!!'))
return pform
def _print_binomial(self, e):
n, k = e.args
n_pform = self._print(n)
k_pform = self._print(k)
bar = ' '*max(n_pform.width(), k_pform.width())
pform = prettyForm(*k_pform.above(bar))
pform = prettyForm(*pform.above(n_pform))
pform = prettyForm(*pform.parens('(', ')'))
pform.baseline = (pform.baseline + 1)//2
return pform
def _print_Relational(self, e):
op = prettyForm(' ' + xsym(e.rel_op) + ' ')
l = self._print(e.lhs)
r = self._print(e.rhs)
pform = prettyForm(*stringPict.next(l, op, r))
return pform
def _print_Not(self, e):
from sympy import Equivalent, Implies
if self._use_unicode:
arg = e.args[0]
pform = self._print(arg)
if isinstance(arg, Equivalent):
return self._print_Equivalent(arg, altchar=u"\N{LEFT RIGHT DOUBLE ARROW WITH STROKE}")
if isinstance(arg, Implies):
return self._print_Implies(arg, altchar=u"\N{RIGHTWARDS ARROW WITH STROKE}")
if arg.is_Boolean and not arg.is_Not:
pform = prettyForm(*pform.parens())
return prettyForm(*pform.left(u"\N{NOT SIGN}"))
else:
return self._print_Function(e)
def __print_Boolean(self, e, char, sort=True):
args = e.args
if sort:
args = sorted(e.args, key=default_sort_key)
arg = args[0]
pform = self._print(arg)
if arg.is_Boolean and not arg.is_Not:
pform = prettyForm(*pform.parens())
for arg in args[1:]:
pform_arg = self._print(arg)
if arg.is_Boolean and not arg.is_Not:
pform_arg = prettyForm(*pform_arg.parens())
pform = prettyForm(*pform.right(u' %s ' % char))
pform = prettyForm(*pform.right(pform_arg))
return pform
def _print_And(self, e):
if self._use_unicode:
return self.__print_Boolean(e, u"\N{LOGICAL AND}")
else:
return self._print_Function(e, sort=True)
def _print_Or(self, e):
if self._use_unicode:
return self.__print_Boolean(e, u"\N{LOGICAL OR}")
else:
return self._print_Function(e, sort=True)
def _print_Xor(self, e):
if self._use_unicode:
return self.__print_Boolean(e, u"\N{XOR}")
else:
return self._print_Function(e, sort=True)
def _print_Nand(self, e):
if self._use_unicode:
return self.__print_Boolean(e, u"\N{NAND}")
else:
return self._print_Function(e, sort=True)
def _print_Nor(self, e):
if self._use_unicode:
return self.__print_Boolean(e, u"\N{NOR}")
else:
return self._print_Function(e, sort=True)
def _print_Implies(self, e, altchar=None):
if self._use_unicode:
return self.__print_Boolean(e, altchar or u"\N{RIGHTWARDS ARROW}", sort=False)
else:
return self._print_Function(e)
def _print_Equivalent(self, e, altchar=None):
if self._use_unicode:
return self.__print_Boolean(e, altchar or u"\N{LEFT RIGHT DOUBLE ARROW}")
else:
return self._print_Function(e, sort=True)
def _print_conjugate(self, e):
pform = self._print(e.args[0])
return prettyForm( *pform.above( hobj('_', pform.width())) )
def _print_Abs(self, e):
pform = self._print(e.args[0])
pform = prettyForm(*pform.parens('|', '|'))
return pform
_print_Determinant = _print_Abs
def _print_floor(self, e):
if self._use_unicode:
pform = self._print(e.args[0])
pform = prettyForm(*pform.parens('lfloor', 'rfloor'))
return pform
else:
return self._print_Function(e)
def _print_ceiling(self, e):
if self._use_unicode:
pform = self._print(e.args[0])
pform = prettyForm(*pform.parens('lceil', 'rceil'))
return pform
else:
return self._print_Function(e)
def _print_Derivative(self, deriv):
if requires_partial(deriv.expr) and self._use_unicode:
deriv_symbol = U('PARTIAL DIFFERENTIAL')
else:
deriv_symbol = r'd'
x = None
count_total_deriv = 0
for sym, num in reversed(deriv.variable_count):
s = self._print(sym)
ds = prettyForm(*s.left(deriv_symbol))
count_total_deriv += num
if (not num.is_Integer) or (num > 1):
ds = ds**prettyForm(str(num))
if x is None:
x = ds
else:
x = prettyForm(*x.right(' '))
x = prettyForm(*x.right(ds))
f = prettyForm(
binding=prettyForm.FUNC, *self._print(deriv.expr).parens())
pform = prettyForm(deriv_symbol)
if (count_total_deriv > 1) != False:
pform = pform**prettyForm(str(count_total_deriv))
pform = prettyForm(*pform.below(stringPict.LINE, x))
pform.baseline = pform.baseline + 1
pform = prettyForm(*stringPict.next(pform, f))
pform.binding = prettyForm.MUL
return pform
def _print_Cycle(self, dc):
from sympy.combinatorics.permutations import Permutation, Cycle
# for Empty Cycle
if dc == Cycle():
cyc = stringPict('')
return prettyForm(*cyc.parens())
dc_list = Permutation(dc.list()).cyclic_form
# for Identity Cycle
if dc_list == []:
cyc = self._print(dc.size - 1)
return prettyForm(*cyc.parens())
cyc = stringPict('')
for i in dc_list:
l = self._print(str(tuple(i)).replace(',', ''))
cyc = prettyForm(*cyc.right(l))
return cyc
def _print_Permutation(self, expr):
from sympy.combinatorics.permutations import Permutation, Cycle
perm_cyclic = Permutation.print_cyclic
if perm_cyclic is not None:
SymPyDeprecationWarning(
feature="Permutation.print_cyclic = {}".format(perm_cyclic),
useinstead="init_printing(perm_cyclic={})"
.format(perm_cyclic),
issue=15201,
deprecated_since_version="1.6").warn()
else:
perm_cyclic = self._settings.get("perm_cyclic", True)
if perm_cyclic:
return self._print_Cycle(Cycle(expr))
lower = expr.array_form
upper = list(range(len(lower)))
result = stringPict('')
first = True
for u, l in zip(upper, lower):
s1 = self._print(u)
s2 = self._print(l)
col = prettyForm(*s1.below(s2))
if first:
first = False
else:
col = prettyForm(*col.left(" "))
result = prettyForm(*result.right(col))
return prettyForm(*result.parens())
def _print_Integral(self, integral):
f = integral.function
# Add parentheses if arg involves addition of terms and
# create a pretty form for the argument
prettyF = self._print(f)
# XXX generalize parens
if f.is_Add:
prettyF = prettyForm(*prettyF.parens())
# dx dy dz ...
arg = prettyF
for x in integral.limits:
prettyArg = self._print(x[0])
# XXX qparens (parens if needs-parens)
if prettyArg.width() > 1:
prettyArg = prettyForm(*prettyArg.parens())
arg = prettyForm(*arg.right(' d', prettyArg))
# \int \int \int ...
firstterm = True
s = None
for lim in integral.limits:
x = lim[0]
# Create bar based on the height of the argument
h = arg.height()
H = h + 2
# XXX hack!
ascii_mode = not self._use_unicode
if ascii_mode:
H += 2
vint = vobj('int', H)
# Construct the pretty form with the integral sign and the argument
pform = prettyForm(vint)
pform.baseline = arg.baseline + (
H - h)//2 # covering the whole argument
if len(lim) > 1:
# Create pretty forms for endpoints, if definite integral.
# Do not print empty endpoints.
if len(lim) == 2:
prettyA = prettyForm("")
prettyB = self._print(lim[1])
if len(lim) == 3:
prettyA = self._print(lim[1])
prettyB = self._print(lim[2])
if ascii_mode: # XXX hack
# Add spacing so that endpoint can more easily be
# identified with the correct integral sign
spc = max(1, 3 - prettyB.width())
prettyB = prettyForm(*prettyB.left(' ' * spc))
spc = max(1, 4 - prettyA.width())
prettyA = prettyForm(*prettyA.right(' ' * spc))
pform = prettyForm(*pform.above(prettyB))
pform = prettyForm(*pform.below(prettyA))
if not ascii_mode: # XXX hack
pform = prettyForm(*pform.right(' '))
if firstterm:
s = pform # first term
firstterm = False
else:
s = prettyForm(*s.left(pform))
pform = prettyForm(*arg.left(s))
pform.binding = prettyForm.MUL
return pform
def _print_Product(self, expr):
func = expr.term
pretty_func = self._print(func)
horizontal_chr = xobj('_', 1)
corner_chr = xobj('_', 1)
vertical_chr = xobj('|', 1)
if self._use_unicode:
# use unicode corners
horizontal_chr = xobj('-', 1)
corner_chr = u'\N{BOX DRAWINGS LIGHT DOWN AND HORIZONTAL}'
func_height = pretty_func.height()
first = True
max_upper = 0
sign_height = 0
for lim in expr.limits:
pretty_lower, pretty_upper = self.__print_SumProduct_Limits(lim)
width = (func_height + 2) * 5 // 3 - 2
sign_lines = [horizontal_chr + corner_chr + (horizontal_chr * (width-2)) + corner_chr + horizontal_chr]
for _ in range(func_height + 1):
sign_lines.append(' ' + vertical_chr + (' ' * (width-2)) + vertical_chr + ' ')
pretty_sign = stringPict('')
pretty_sign = prettyForm(*pretty_sign.stack(*sign_lines))
max_upper = max(max_upper, pretty_upper.height())
if first:
sign_height = pretty_sign.height()
pretty_sign = prettyForm(*pretty_sign.above(pretty_upper))
pretty_sign = prettyForm(*pretty_sign.below(pretty_lower))
if first:
pretty_func.baseline = 0
first = False
height = pretty_sign.height()
padding = stringPict('')
padding = prettyForm(*padding.stack(*[' ']*(height - 1)))
pretty_sign = prettyForm(*pretty_sign.right(padding))
pretty_func = prettyForm(*pretty_sign.right(pretty_func))
pretty_func.baseline = max_upper + sign_height//2
pretty_func.binding = prettyForm.MUL
return pretty_func
def __print_SumProduct_Limits(self, lim):
def print_start(lhs, rhs):
op = prettyForm(' ' + xsym("==") + ' ')
l = self._print(lhs)
r = self._print(rhs)
pform = prettyForm(*stringPict.next(l, op, r))
return pform
prettyUpper = self._print(lim[2])
prettyLower = print_start(lim[0], lim[1])
return prettyLower, prettyUpper
def _print_Sum(self, expr):
ascii_mode = not self._use_unicode
def asum(hrequired, lower, upper, use_ascii):
def adjust(s, wid=None, how='<^>'):
if not wid or len(s) > wid:
return s
need = wid - len(s)
if how == '<^>' or how == "<" or how not in list('<^>'):
return s + ' '*need
half = need//2
lead = ' '*half
if how == ">":
return " "*need + s
return lead + s + ' '*(need - len(lead))
h = max(hrequired, 2)
d = h//2
w = d + 1
more = hrequired % 2
lines = []
if use_ascii:
lines.append("_"*(w) + ' ')
lines.append(r"\%s`" % (' '*(w - 1)))
for i in range(1, d):
lines.append('%s\\%s' % (' '*i, ' '*(w - i)))
if more:
lines.append('%s)%s' % (' '*(d), ' '*(w - d)))
for i in reversed(range(1, d)):
lines.append('%s/%s' % (' '*i, ' '*(w - i)))
lines.append("/" + "_"*(w - 1) + ',')
return d, h + more, lines, more
else:
w = w + more
d = d + more
vsum = vobj('sum', 4)
lines.append("_"*(w))
for i in range(0, d):
lines.append('%s%s%s' % (' '*i, vsum[2], ' '*(w - i - 1)))
for i in reversed(range(0, d)):
lines.append('%s%s%s' % (' '*i, vsum[4], ' '*(w - i - 1)))
lines.append(vsum[8]*(w))
return d, h + 2*more, lines, more
f = expr.function
prettyF = self._print(f)
if f.is_Add: # add parens
prettyF = prettyForm(*prettyF.parens())
H = prettyF.height() + 2
# \sum \sum \sum ...
first = True
max_upper = 0
sign_height = 0
for lim in expr.limits:
prettyLower, prettyUpper = self.__print_SumProduct_Limits(lim)
max_upper = max(max_upper, prettyUpper.height())
# Create sum sign based on the height of the argument
d, h, slines, adjustment = asum(
H, prettyLower.width(), prettyUpper.width(), ascii_mode)
prettySign = stringPict('')
prettySign = prettyForm(*prettySign.stack(*slines))
if first:
sign_height = prettySign.height()
prettySign = prettyForm(*prettySign.above(prettyUpper))
prettySign = prettyForm(*prettySign.below(prettyLower))
if first:
# change F baseline so it centers on the sign
prettyF.baseline -= d - (prettyF.height()//2 -
prettyF.baseline)
first = False
# put padding to the right
pad = stringPict('')
pad = prettyForm(*pad.stack(*[' ']*h))
prettySign = prettyForm(*prettySign.right(pad))
# put the present prettyF to the right
prettyF = prettyForm(*prettySign.right(prettyF))
# adjust baseline of ascii mode sigma with an odd height so that it is
# exactly through the center
ascii_adjustment = ascii_mode if not adjustment else 0
prettyF.baseline = max_upper + sign_height//2 + ascii_adjustment
prettyF.binding = prettyForm.MUL
return prettyF
def _print_Limit(self, l):
e, z, z0, dir = l.args
E = self._print(e)
if precedence(e) <= PRECEDENCE["Mul"]:
E = prettyForm(*E.parens('(', ')'))
Lim = prettyForm('lim')
LimArg = self._print(z)
if self._use_unicode:
LimArg = prettyForm(*LimArg.right(u'\N{BOX DRAWINGS LIGHT HORIZONTAL}\N{RIGHTWARDS ARROW}'))
else:
LimArg = prettyForm(*LimArg.right('->'))
LimArg = prettyForm(*LimArg.right(self._print(z0)))
if str(dir) == '+-' or z0 in (S.Infinity, S.NegativeInfinity):
dir = ""
else:
if self._use_unicode:
dir = u'\N{SUPERSCRIPT PLUS SIGN}' if str(dir) == "+" else u'\N{SUPERSCRIPT MINUS}'
LimArg = prettyForm(*LimArg.right(self._print(dir)))
Lim = prettyForm(*Lim.below(LimArg))
Lim = prettyForm(*Lim.right(E), binding=prettyForm.MUL)
return Lim
def _print_matrix_contents(self, e):
"""
This method factors out what is essentially grid printing.
"""
M = e # matrix
Ms = {} # i,j -> pretty(M[i,j])
for i in range(M.rows):
for j in range(M.cols):
Ms[i, j] = self._print(M[i, j])
# h- and v- spacers
hsep = 2
vsep = 1
# max width for columns
maxw = [-1] * M.cols
for j in range(M.cols):
maxw[j] = max([Ms[i, j].width() for i in range(M.rows)] or [0])
# drawing result
D = None
for i in range(M.rows):
D_row = None
for j in range(M.cols):
s = Ms[i, j]
# reshape s to maxw
# XXX this should be generalized, and go to stringPict.reshape ?
assert s.width() <= maxw[j]
# hcenter it, +0.5 to the right 2
# ( it's better to align formula starts for say 0 and r )
# XXX this is not good in all cases -- maybe introduce vbaseline?
wdelta = maxw[j] - s.width()
wleft = wdelta // 2
wright = wdelta - wleft
s = prettyForm(*s.right(' '*wright))
s = prettyForm(*s.left(' '*wleft))
# we don't need vcenter cells -- this is automatically done in
# a pretty way because when their baselines are taking into
# account in .right()
if D_row is None:
D_row = s # first box in a row
continue
D_row = prettyForm(*D_row.right(' '*hsep)) # h-spacer
D_row = prettyForm(*D_row.right(s))
if D is None:
D = D_row # first row in a picture
continue
# v-spacer
for _ in range(vsep):
D = prettyForm(*D.below(' '))
D = prettyForm(*D.below(D_row))
if D is None:
D = prettyForm('') # Empty Matrix
return D
def _print_MatrixBase(self, e):
D = self._print_matrix_contents(e)
D.baseline = D.height()//2
D = prettyForm(*D.parens('[', ']'))
return D
_print_ImmutableMatrix = _print_MatrixBase
_print_Matrix = _print_MatrixBase
def _print_TensorProduct(self, expr):
# This should somehow share the code with _print_WedgeProduct:
circled_times = "\u2297"
return self._print_seq(expr.args, None, None, circled_times,
parenthesize=lambda x: precedence_traditional(x) <= PRECEDENCE["Mul"])
def _print_WedgeProduct(self, expr):
# This should somehow share the code with _print_TensorProduct:
wedge_symbol = u"\u2227"
return self._print_seq(expr.args, None, None, wedge_symbol,
parenthesize=lambda x: precedence_traditional(x) <= PRECEDENCE["Mul"])
def _print_Trace(self, e):
D = self._print(e.arg)
D = prettyForm(*D.parens('(',')'))
D.baseline = D.height()//2
D = prettyForm(*D.left('\n'*(0) + 'tr'))
return D
def _print_MatrixElement(self, expr):
from sympy.matrices import MatrixSymbol
from sympy import Symbol
if (isinstance(expr.parent, MatrixSymbol)
and expr.i.is_number and expr.j.is_number):
return self._print(
Symbol(expr.parent.name + '_%d%d' % (expr.i, expr.j)))
else:
prettyFunc = self._print(expr.parent)
prettyFunc = prettyForm(*prettyFunc.parens())
prettyIndices = self._print_seq((expr.i, expr.j), delimiter=', '
).parens(left='[', right=']')[0]
pform = prettyForm(binding=prettyForm.FUNC,
*stringPict.next(prettyFunc, prettyIndices))
# store pform parts so it can be reassembled e.g. when powered
pform.prettyFunc = prettyFunc
pform.prettyArgs = prettyIndices
return pform
def _print_MatrixSlice(self, m):
# XXX works only for applied functions
from sympy.matrices import MatrixSymbol
prettyFunc = self._print(m.parent)
if not isinstance(m.parent, MatrixSymbol):
prettyFunc = prettyForm(*prettyFunc.parens())
def ppslice(x, dim):
x = list(x)
if x[2] == 1:
del x[2]
if x[0] == 0:
x[0] = ''
if x[1] == dim:
x[1] = ''
return prettyForm(*self._print_seq(x, delimiter=':'))
prettyArgs = self._print_seq((ppslice(m.rowslice, m.parent.rows),
ppslice(m.colslice, m.parent.cols)), delimiter=', ').parens(left='[', right=']')[0]
pform = prettyForm(
binding=prettyForm.FUNC, *stringPict.next(prettyFunc, prettyArgs))
# store pform parts so it can be reassembled e.g. when powered
pform.prettyFunc = prettyFunc
pform.prettyArgs = prettyArgs
return pform
def _print_Transpose(self, expr):
pform = self._print(expr.arg)
from sympy.matrices import MatrixSymbol
if not isinstance(expr.arg, MatrixSymbol):
pform = prettyForm(*pform.parens())
pform = pform**(prettyForm('T'))
return pform
def _print_Adjoint(self, expr):
pform = self._print(expr.arg)
if self._use_unicode:
dag = prettyForm(u'\N{DAGGER}')
else:
dag = prettyForm('+')
from sympy.matrices import MatrixSymbol
if not isinstance(expr.arg, MatrixSymbol):
pform = prettyForm(*pform.parens())
pform = pform**dag
return pform
def _print_BlockMatrix(self, B):
if B.blocks.shape == (1, 1):
return self._print(B.blocks[0, 0])
return self._print(B.blocks)
def _print_MatAdd(self, expr):
s = None
for item in expr.args:
pform = self._print(item)
if s is None:
s = pform # First element
else:
coeff = item.as_coeff_mmul()[0]
if _coeff_isneg(S(coeff)):
s = prettyForm(*stringPict.next(s, ' '))
pform = self._print(item)
else:
s = prettyForm(*stringPict.next(s, ' + '))
s = prettyForm(*stringPict.next(s, pform))
return s
def _print_MatMul(self, expr):
args = list(expr.args)
from sympy import Add, MatAdd, HadamardProduct, KroneckerProduct
for i, a in enumerate(args):
if (isinstance(a, (Add, MatAdd, HadamardProduct, KroneckerProduct))
and len(expr.args) > 1):
args[i] = prettyForm(*self._print(a).parens())
else:
args[i] = self._print(a)
return prettyForm.__mul__(*args)
def _print_Identity(self, expr):
if self._use_unicode:
return prettyForm(u'\N{MATHEMATICAL DOUBLE-STRUCK CAPITAL I}')
else:
return prettyForm('I')
def _print_ZeroMatrix(self, expr):
if self._use_unicode:
return prettyForm(u'\N{MATHEMATICAL DOUBLE-STRUCK DIGIT ZERO}')
else:
return prettyForm('0')
def _print_OneMatrix(self, expr):
if self._use_unicode:
return prettyForm(u'\N{MATHEMATICAL DOUBLE-STRUCK DIGIT ONE}')
else:
return prettyForm('1')
def _print_DotProduct(self, expr):
args = list(expr.args)
for i, a in enumerate(args):
args[i] = self._print(a)
return prettyForm.__mul__(*args)
def _print_MatPow(self, expr):
pform = self._print(expr.base)
from sympy.matrices import MatrixSymbol
if not isinstance(expr.base, MatrixSymbol):
pform = prettyForm(*pform.parens())
pform = pform**(self._print(expr.exp))
return pform
def _print_HadamardProduct(self, expr):
from sympy import MatAdd, MatMul, HadamardProduct
if self._use_unicode:
delim = pretty_atom('Ring')
else:
delim = '.*'
return self._print_seq(expr.args, None, None, delim,
parenthesize=lambda x: isinstance(x, (MatAdd, MatMul, HadamardProduct)))
def _print_HadamardPower(self, expr):
# from sympy import MatAdd, MatMul
if self._use_unicode:
circ = pretty_atom('Ring')
else:
circ = self._print('.')
pretty_base = self._print(expr.base)
pretty_exp = self._print(expr.exp)
if precedence(expr.exp) < PRECEDENCE["Mul"]:
pretty_exp = prettyForm(*pretty_exp.parens())
pretty_circ_exp = prettyForm(
binding=prettyForm.LINE,
*stringPict.next(circ, pretty_exp)
)
return pretty_base**pretty_circ_exp
def _print_KroneckerProduct(self, expr):
from sympy import MatAdd, MatMul
if self._use_unicode:
delim = u' \N{N-ARY CIRCLED TIMES OPERATOR} '
else:
delim = ' x '
return self._print_seq(expr.args, None, None, delim,
parenthesize=lambda x: isinstance(x, (MatAdd, MatMul)))
def _print_FunctionMatrix(self, X):
D = self._print(X.lamda.expr)
D = prettyForm(*D.parens('[', ']'))
return D
def _print_BasisDependent(self, expr):
from sympy.vector import Vector
if not self._use_unicode:
raise NotImplementedError("ASCII pretty printing of BasisDependent is not implemented")
if expr == expr.zero:
return prettyForm(expr.zero._pretty_form)
o1 = []
vectstrs = []
if isinstance(expr, Vector):
items = expr.separate().items()
else:
items = [(0, expr)]
for system, vect in items:
inneritems = list(vect.components.items())
inneritems.sort(key = lambda x: x[0].__str__())
for k, v in inneritems:
#if the coef of the basis vector is 1
#we skip the 1
if v == 1:
o1.append(u"" +
k._pretty_form)
#Same for -1
elif v == -1:
o1.append(u"(-1) " +
k._pretty_form)
#For a general expr
else:
#We always wrap the measure numbers in
#parentheses
arg_str = self._print(
v).parens()[0]
o1.append(arg_str + ' ' + k._pretty_form)
vectstrs.append(k._pretty_form)
#outstr = u("").join(o1)
if o1[0].startswith(u" + "):
o1[0] = o1[0][3:]
elif o1[0].startswith(" "):
o1[0] = o1[0][1:]
#Fixing the newlines
lengths = []
strs = ['']
flag = []
for i, partstr in enumerate(o1):
flag.append(0)
# XXX: What is this hack?
if '\n' in partstr:
tempstr = partstr
tempstr = tempstr.replace(vectstrs[i], '')
if u'\N{right parenthesis extension}' in tempstr: # If scalar is a fraction
for paren in range(len(tempstr)):
flag[i] = 1
if tempstr[paren] == u'\N{right parenthesis extension}':
tempstr = tempstr[:paren] + u'\N{right parenthesis extension}'\
+ ' ' + vectstrs[i] + tempstr[paren + 1:]
break
elif u'\N{RIGHT PARENTHESIS LOWER HOOK}' in tempstr:
flag[i] = 1
tempstr = tempstr.replace(u'\N{RIGHT PARENTHESIS LOWER HOOK}',
u'\N{RIGHT PARENTHESIS LOWER HOOK}'
+ ' ' + vectstrs[i])
else:
tempstr = tempstr.replace(u'\N{RIGHT PARENTHESIS UPPER HOOK}',
u'\N{RIGHT PARENTHESIS UPPER HOOK}'
+ ' ' + vectstrs[i])
o1[i] = tempstr
o1 = [x.split('\n') for x in o1]
n_newlines = max([len(x) for x in o1]) # Width of part in its pretty form
if 1 in flag: # If there was a fractional scalar
for i, parts in enumerate(o1):
if len(parts) == 1: # If part has no newline
parts.insert(0, ' ' * (len(parts[0])))
flag[i] = 1
for i, parts in enumerate(o1):
lengths.append(len(parts[flag[i]]))
for j in range(n_newlines):
if j+1 <= len(parts):
if j >= len(strs):
strs.append(' ' * (sum(lengths[:-1]) +
3*(len(lengths)-1)))
if j == flag[i]:
strs[flag[i]] += parts[flag[i]] + ' + '
else:
strs[j] += parts[j] + ' '*(lengths[-1] -
len(parts[j])+
3)
else:
if j >= len(strs):
strs.append(' ' * (sum(lengths[:-1]) +
3*(len(lengths)-1)))
strs[j] += ' '*(lengths[-1]+3)
return prettyForm(u'\n'.join([s[:-3] for s in strs]))
def _print_NDimArray(self, expr):
from sympy import ImmutableMatrix
if expr.rank() == 0:
return self._print(expr[()])
level_str = [[]] + [[] for i in range(expr.rank())]
shape_ranges = [list(range(i)) for i in expr.shape]
# leave eventual matrix elements unflattened
mat = lambda x: ImmutableMatrix(x, evaluate=False)
for outer_i in itertools.product(*shape_ranges):
level_str[-1].append(expr[outer_i])
even = True
for back_outer_i in range(expr.rank()-1, -1, -1):
if len(level_str[back_outer_i+1]) < expr.shape[back_outer_i]:
break
if even:
level_str[back_outer_i].append(level_str[back_outer_i+1])
else:
level_str[back_outer_i].append(mat(
level_str[back_outer_i+1]))
if len(level_str[back_outer_i + 1]) == 1:
level_str[back_outer_i][-1] = mat(
[[level_str[back_outer_i][-1]]])
even = not even
level_str[back_outer_i+1] = []
out_expr = level_str[0][0]
if expr.rank() % 2 == 1:
out_expr = mat([out_expr])
return self._print(out_expr)
_print_ImmutableDenseNDimArray = _print_NDimArray
_print_ImmutableSparseNDimArray = _print_NDimArray
_print_MutableDenseNDimArray = _print_NDimArray
_print_MutableSparseNDimArray = _print_NDimArray
def _printer_tensor_indices(self, name, indices, index_map={}):
center = stringPict(name)
top = stringPict(" "*center.width())
bot = stringPict(" "*center.width())
last_valence = None
prev_map = None
for i, index in enumerate(indices):
indpic = self._print(index.args[0])
if ((index in index_map) or prev_map) and last_valence == index.is_up:
if index.is_up:
top = prettyForm(*stringPict.next(top, ","))
else:
bot = prettyForm(*stringPict.next(bot, ","))
if index in index_map:
indpic = prettyForm(*stringPict.next(indpic, "="))
indpic = prettyForm(*stringPict.next(indpic, self._print(index_map[index])))
prev_map = True
else:
prev_map = False
if index.is_up:
top = stringPict(*top.right(indpic))
center = stringPict(*center.right(" "*indpic.width()))
bot = stringPict(*bot.right(" "*indpic.width()))
else:
bot = stringPict(*bot.right(indpic))
center = stringPict(*center.right(" "*indpic.width()))
top = stringPict(*top.right(" "*indpic.width()))
last_valence = index.is_up
pict = prettyForm(*center.above(top))
pict = prettyForm(*pict.below(bot))
return pict
def _print_Tensor(self, expr):
name = expr.args[0].name
indices = expr.get_indices()
return self._printer_tensor_indices(name, indices)
def _print_TensorElement(self, expr):
name = expr.expr.args[0].name
indices = expr.expr.get_indices()
index_map = expr.index_map
return self._printer_tensor_indices(name, indices, index_map)
def _print_TensMul(self, expr):
sign, args = expr._get_args_for_traditional_printer()
args = [
prettyForm(*self._print(i).parens()) if
precedence_traditional(i) < PRECEDENCE["Mul"] else self._print(i)
for i in args
]
pform = prettyForm.__mul__(*args)
if sign:
return prettyForm(*pform.left(sign))
else:
return pform
def _print_TensAdd(self, expr):
args = [
prettyForm(*self._print(i).parens()) if
precedence_traditional(i) < PRECEDENCE["Mul"] else self._print(i)
for i in expr.args
]
return prettyForm.__add__(*args)
def _print_TensorIndex(self, expr):
sym = expr.args[0]
if not expr.is_up:
sym = -sym
return self._print(sym)
def _print_PartialDerivative(self, deriv):
if self._use_unicode:
deriv_symbol = U('PARTIAL DIFFERENTIAL')
else:
deriv_symbol = r'd'
x = None
for variable in reversed(deriv.variables):
s = self._print(variable)
ds = prettyForm(*s.left(deriv_symbol))
if x is None:
x = ds
else:
x = prettyForm(*x.right(' '))
x = prettyForm(*x.right(ds))
f = prettyForm(
binding=prettyForm.FUNC, *self._print(deriv.expr).parens())
pform = prettyForm(deriv_symbol)
if len(deriv.variables) > 1:
pform = pform**self._print(len(deriv.variables))
pform = prettyForm(*pform.below(stringPict.LINE, x))
pform.baseline = pform.baseline + 1
pform = prettyForm(*stringPict.next(pform, f))
pform.binding = prettyForm.MUL
return pform
def _print_Piecewise(self, pexpr):
P = {}
for n, ec in enumerate(pexpr.args):
P[n, 0] = self._print(ec.expr)
if ec.cond == True:
P[n, 1] = prettyForm('otherwise')
else:
P[n, 1] = prettyForm(
*prettyForm('for ').right(self._print(ec.cond)))
hsep = 2
vsep = 1
len_args = len(pexpr.args)
# max widths
maxw = [max([P[i, j].width() for i in range(len_args)])
for j in range(2)]
# FIXME: Refactor this code and matrix into some tabular environment.
# drawing result
D = None
for i in range(len_args):
D_row = None
for j in range(2):
p = P[i, j]
assert p.width() <= maxw[j]
wdelta = maxw[j] - p.width()
wleft = wdelta // 2
wright = wdelta - wleft
p = prettyForm(*p.right(' '*wright))
p = prettyForm(*p.left(' '*wleft))
if D_row is None:
D_row = p
continue
D_row = prettyForm(*D_row.right(' '*hsep)) # h-spacer
D_row = prettyForm(*D_row.right(p))
if D is None:
D = D_row # first row in a picture
continue
# v-spacer
for _ in range(vsep):
D = prettyForm(*D.below(' '))
D = prettyForm(*D.below(D_row))
D = prettyForm(*D.parens('{', ''))
D.baseline = D.height()//2
D.binding = prettyForm.OPEN
return D
def _print_ITE(self, ite):
from sympy.functions.elementary.piecewise import Piecewise
return self._print(ite.rewrite(Piecewise))
def _hprint_vec(self, v):
D = None
for a in v:
p = a
if D is None:
D = p
else:
D = prettyForm(*D.right(', '))
D = prettyForm(*D.right(p))
if D is None:
D = stringPict(' ')
return D
def _hprint_vseparator(self, p1, p2):
tmp = prettyForm(*p1.right(p2))
sep = stringPict(vobj('|', tmp.height()), baseline=tmp.baseline)
return prettyForm(*p1.right(sep, p2))
def _print_hyper(self, e):
# FIXME refactor Matrix, Piecewise, and this into a tabular environment
ap = [self._print(a) for a in e.ap]
bq = [self._print(b) for b in e.bq]
P = self._print(e.argument)
P.baseline = P.height()//2
# Drawing result - first create the ap, bq vectors
D = None
for v in [ap, bq]:
D_row = self._hprint_vec(v)
if D is None:
D = D_row # first row in a picture
else:
D = prettyForm(*D.below(' '))
D = prettyForm(*D.below(D_row))
# make sure that the argument `z' is centred vertically
D.baseline = D.height()//2
# insert horizontal separator
P = prettyForm(*P.left(' '))
D = prettyForm(*D.right(' '))
# insert separating `|`
D = self._hprint_vseparator(D, P)
# add parens
D = prettyForm(*D.parens('(', ')'))
# create the F symbol
above = D.height()//2 - 1
below = D.height() - above - 1
sz, t, b, add, img = annotated('F')
F = prettyForm('\n' * (above - t) + img + '\n' * (below - b),
baseline=above + sz)
add = (sz + 1)//2
F = prettyForm(*F.left(self._print(len(e.ap))))
F = prettyForm(*F.right(self._print(len(e.bq))))
F.baseline = above + add
D = prettyForm(*F.right(' ', D))
return D
def _print_meijerg(self, e):
# FIXME refactor Matrix, Piecewise, and this into a tabular environment
v = {}
v[(0, 0)] = [self._print(a) for a in e.an]
v[(0, 1)] = [self._print(a) for a in e.aother]
v[(1, 0)] = [self._print(b) for b in e.bm]
v[(1, 1)] = [self._print(b) for b in e.bother]
P = self._print(e.argument)
P.baseline = P.height()//2
vp = {}
for idx in v:
vp[idx] = self._hprint_vec(v[idx])
for i in range(2):
maxw = max(vp[(0, i)].width(), vp[(1, i)].width())
for j in range(2):
s = vp[(j, i)]
left = (maxw - s.width()) // 2
right = maxw - left - s.width()
s = prettyForm(*s.left(' ' * left))
s = prettyForm(*s.right(' ' * right))
vp[(j, i)] = s
D1 = prettyForm(*vp[(0, 0)].right(' ', vp[(0, 1)]))
D1 = prettyForm(*D1.below(' '))
D2 = prettyForm(*vp[(1, 0)].right(' ', vp[(1, 1)]))
D = prettyForm(*D1.below(D2))
# make sure that the argument `z' is centred vertically
D.baseline = D.height()//2
# insert horizontal separator
P = prettyForm(*P.left(' '))
D = prettyForm(*D.right(' '))
# insert separating `|`
D = self._hprint_vseparator(D, P)
# add parens
D = prettyForm(*D.parens('(', ')'))
# create the G symbol
above = D.height()//2 - 1
below = D.height() - above - 1
sz, t, b, add, img = annotated('G')
F = prettyForm('\n' * (above - t) + img + '\n' * (below - b),
baseline=above + sz)
pp = self._print(len(e.ap))
pq = self._print(len(e.bq))
pm = self._print(len(e.bm))
pn = self._print(len(e.an))
def adjust(p1, p2):
diff = p1.width() - p2.width()
if diff == 0:
return p1, p2
elif diff > 0:
return p1, prettyForm(*p2.left(' '*diff))
else:
return prettyForm(*p1.left(' '*-diff)), p2
pp, pm = adjust(pp, pm)
pq, pn = adjust(pq, pn)
pu = prettyForm(*pm.right(', ', pn))
pl = prettyForm(*pp.right(', ', pq))
ht = F.baseline - above - 2
if ht > 0:
pu = prettyForm(*pu.below('\n'*ht))
p = prettyForm(*pu.below(pl))
F.baseline = above
F = prettyForm(*F.right(p))
F.baseline = above + add
D = prettyForm(*F.right(' ', D))
return D
def _print_ExpBase(self, e):
# TODO should exp_polar be printed differently?
# what about exp_polar(0), exp_polar(1)?
base = prettyForm(pretty_atom('Exp1', 'e'))
return base ** self._print(e.args[0])
def _print_Function(self, e, sort=False, func_name=None):
# optional argument func_name for supplying custom names
# XXX works only for applied functions
return self._helper_print_function(e.func, e.args, sort=sort, func_name=func_name)
def _print_mathieuc(self, e):
return self._print_Function(e, func_name='C')
def _print_mathieus(self, e):
return self._print_Function(e, func_name='S')
def _print_mathieucprime(self, e):
return self._print_Function(e, func_name="C'")
def _print_mathieusprime(self, e):
return self._print_Function(e, func_name="S'")
def _helper_print_function(self, func, args, sort=False, func_name=None, delimiter=', ', elementwise=False):
if sort:
args = sorted(args, key=default_sort_key)
if not func_name and hasattr(func, "__name__"):
func_name = func.__name__
if func_name:
prettyFunc = self._print(Symbol(func_name))
else:
prettyFunc = prettyForm(*self._print(func).parens())
if elementwise:
if self._use_unicode:
circ = pretty_atom('Modifier Letter Low Ring')
else:
circ = '.'
circ = self._print(circ)
prettyFunc = prettyForm(
binding=prettyForm.LINE,
*stringPict.next(prettyFunc, circ)
)
prettyArgs = prettyForm(*self._print_seq(args, delimiter=delimiter).parens())
pform = prettyForm(
binding=prettyForm.FUNC, *stringPict.next(prettyFunc, prettyArgs))
# store pform parts so it can be reassembled e.g. when powered
pform.prettyFunc = prettyFunc
pform.prettyArgs = prettyArgs
return pform
def _print_ElementwiseApplyFunction(self, e):
func = e.function
arg = e.expr
args = [arg]
return self._helper_print_function(func, args, delimiter="", elementwise=True)
@property
def _special_function_classes(self):
from sympy.functions.special.tensor_functions import KroneckerDelta
from sympy.functions.special.gamma_functions import gamma, lowergamma
from sympy.functions.special.zeta_functions import lerchphi
from sympy.functions.special.beta_functions import beta
from sympy.functions.special.delta_functions import DiracDelta
from sympy.functions.special.error_functions import Chi
return {KroneckerDelta: [greek_unicode['delta'], 'delta'],
gamma: [greek_unicode['Gamma'], 'Gamma'],
lerchphi: [greek_unicode['Phi'], 'lerchphi'],
lowergamma: [greek_unicode['gamma'], 'gamma'],
beta: [greek_unicode['Beta'], 'B'],
DiracDelta: [greek_unicode['delta'], 'delta'],
Chi: ['Chi', 'Chi']}
def _print_FunctionClass(self, expr):
for cls in self._special_function_classes:
if issubclass(expr, cls) and expr.__name__ == cls.__name__:
if self._use_unicode:
return prettyForm(self._special_function_classes[cls][0])
else:
return prettyForm(self._special_function_classes[cls][1])
func_name = expr.__name__
return prettyForm(pretty_symbol(func_name))
def _print_GeometryEntity(self, expr):
# GeometryEntity is based on Tuple but should not print like a Tuple
return self.emptyPrinter(expr)
def _print_lerchphi(self, e):
func_name = greek_unicode['Phi'] if self._use_unicode else 'lerchphi'
return self._print_Function(e, func_name=func_name)
def _print_dirichlet_eta(self, e):
func_name = greek_unicode['eta'] if self._use_unicode else 'dirichlet_eta'
return self._print_Function(e, func_name=func_name)
def _print_Heaviside(self, e):
func_name = greek_unicode['theta'] if self._use_unicode else 'Heaviside'
return self._print_Function(e, func_name=func_name)
def _print_fresnels(self, e):
return self._print_Function(e, func_name="S")
def _print_fresnelc(self, e):
return self._print_Function(e, func_name="C")
def _print_airyai(self, e):
return self._print_Function(e, func_name="Ai")
def _print_airybi(self, e):
return self._print_Function(e, func_name="Bi")
def _print_airyaiprime(self, e):
return self._print_Function(e, func_name="Ai'")
def _print_airybiprime(self, e):
return self._print_Function(e, func_name="Bi'")
def _print_LambertW(self, e):
return self._print_Function(e, func_name="W")
def _print_Lambda(self, e):
expr = e.expr
sig = e.signature
if self._use_unicode:
arrow = u" \N{RIGHTWARDS ARROW FROM BAR} "
else:
arrow = " -> "
if len(sig) == 1 and sig[0].is_symbol:
sig = sig[0]
var_form = self._print(sig)
return prettyForm(*stringPict.next(var_form, arrow, self._print(expr)), binding=8)
def _print_Order(self, expr):
pform = self._print(expr.expr)
if (expr.point and any(p != S.Zero for p in expr.point)) or \
len(expr.variables) > 1:
pform = prettyForm(*pform.right("; "))
if len(expr.variables) > 1:
pform = prettyForm(*pform.right(self._print(expr.variables)))
elif len(expr.variables):
pform = prettyForm(*pform.right(self._print(expr.variables[0])))
if self._use_unicode:
pform = prettyForm(*pform.right(u" \N{RIGHTWARDS ARROW} "))
else:
pform = prettyForm(*pform.right(" -> "))
if len(expr.point) > 1:
pform = prettyForm(*pform.right(self._print(expr.point)))
else:
pform = prettyForm(*pform.right(self._print(expr.point[0])))
pform = prettyForm(*pform.parens())
pform = prettyForm(*pform.left("O"))
return pform
def _print_SingularityFunction(self, e):
if self._use_unicode:
shift = self._print(e.args[0]-e.args[1])
n = self._print(e.args[2])
base = prettyForm("<")
base = prettyForm(*base.right(shift))
base = prettyForm(*base.right(">"))
pform = base**n
return pform
else:
n = self._print(e.args[2])
shift = self._print(e.args[0]-e.args[1])
base = self._print_seq(shift, "<", ">", ' ')
return base**n
def _print_beta(self, e):
func_name = greek_unicode['Beta'] if self._use_unicode else 'B'
return self._print_Function(e, func_name=func_name)
def _print_gamma(self, e):
func_name = greek_unicode['Gamma'] if self._use_unicode else 'Gamma'
return self._print_Function(e, func_name=func_name)
def _print_uppergamma(self, e):
func_name = greek_unicode['Gamma'] if self._use_unicode else 'Gamma'
return self._print_Function(e, func_name=func_name)
def _print_lowergamma(self, e):
func_name = greek_unicode['gamma'] if self._use_unicode else 'lowergamma'
return self._print_Function(e, func_name=func_name)
def _print_DiracDelta(self, e):
if self._use_unicode:
if len(e.args) == 2:
a = prettyForm(greek_unicode['delta'])
b = self._print(e.args[1])
b = prettyForm(*b.parens())
c = self._print(e.args[0])
c = prettyForm(*c.parens())
pform = a**b
pform = prettyForm(*pform.right(' '))
pform = prettyForm(*pform.right(c))
return pform
pform = self._print(e.args[0])
pform = prettyForm(*pform.parens())
pform = prettyForm(*pform.left(greek_unicode['delta']))
return pform
else:
return self._print_Function(e)
def _print_expint(self, e):
from sympy import Function
if e.args[0].is_Integer and self._use_unicode:
return self._print_Function(Function('E_%s' % e.args[0])(e.args[1]))
return self._print_Function(e)
def _print_Chi(self, e):
# This needs a special case since otherwise it comes out as greek
# letter chi...
prettyFunc = prettyForm("Chi")
prettyArgs = prettyForm(*self._print_seq(e.args).parens())
pform = prettyForm(
binding=prettyForm.FUNC, *stringPict.next(prettyFunc, prettyArgs))
# store pform parts so it can be reassembled e.g. when powered
pform.prettyFunc = prettyFunc
pform.prettyArgs = prettyArgs
return pform
def _print_elliptic_e(self, e):
pforma0 = self._print(e.args[0])
if len(e.args) == 1:
pform = pforma0
else:
pforma1 = self._print(e.args[1])
pform = self._hprint_vseparator(pforma0, pforma1)
pform = prettyForm(*pform.parens())
pform = prettyForm(*pform.left('E'))
return pform
def _print_elliptic_k(self, e):
pform = self._print(e.args[0])
pform = prettyForm(*pform.parens())
pform = prettyForm(*pform.left('K'))
return pform
def _print_elliptic_f(self, e):
pforma0 = self._print(e.args[0])
pforma1 = self._print(e.args[1])
pform = self._hprint_vseparator(pforma0, pforma1)
pform = prettyForm(*pform.parens())
pform = prettyForm(*pform.left('F'))
return pform
def _print_elliptic_pi(self, e):
name = greek_unicode['Pi'] if self._use_unicode else 'Pi'
pforma0 = self._print(e.args[0])
pforma1 = self._print(e.args[1])
if len(e.args) == 2:
pform = self._hprint_vseparator(pforma0, pforma1)
else:
pforma2 = self._print(e.args[2])
pforma = self._hprint_vseparator(pforma1, pforma2)
pforma = prettyForm(*pforma.left('; '))
pform = prettyForm(*pforma.left(pforma0))
pform = prettyForm(*pform.parens())
pform = prettyForm(*pform.left(name))
return pform
def _print_GoldenRatio(self, expr):
if self._use_unicode:
return prettyForm(pretty_symbol('phi'))
return self._print(Symbol("GoldenRatio"))
def _print_EulerGamma(self, expr):
if self._use_unicode:
return prettyForm(pretty_symbol('gamma'))
return self._print(Symbol("EulerGamma"))
def _print_Mod(self, expr):
pform = self._print(expr.args[0])
if pform.binding > prettyForm.MUL:
pform = prettyForm(*pform.parens())
pform = prettyForm(*pform.right(' mod '))
pform = prettyForm(*pform.right(self._print(expr.args[1])))
pform.binding = prettyForm.OPEN
return pform
def _print_Add(self, expr, order=None):
terms = self._as_ordered_terms(expr, order=order)
pforms, indices = [], []
def pretty_negative(pform, index):
"""Prepend a minus sign to a pretty form. """
#TODO: Move this code to prettyForm
if index == 0:
if pform.height() > 1:
pform_neg = '- '
else:
pform_neg = '-'
else:
pform_neg = ' - '
if (pform.binding > prettyForm.NEG
or pform.binding == prettyForm.ADD):
p = stringPict(*pform.parens())
else:
p = pform
p = stringPict.next(pform_neg, p)
# Lower the binding to NEG, even if it was higher. Otherwise, it
# will print as a + ( - (b)), instead of a - (b).
return prettyForm(binding=prettyForm.NEG, *p)
for i, term in enumerate(terms):
if term.is_Mul and _coeff_isneg(term):
coeff, other = term.as_coeff_mul(rational=False)
if coeff == -1:
negterm = Mul(*other, evaluate=False)
else:
negterm = Mul(-coeff, *other, evaluate=False)
pform = self._print(negterm)
pforms.append(pretty_negative(pform, i))
elif term.is_Rational and term.q > 1:
pforms.append(None)
indices.append(i)
elif term.is_Number and term < 0:
pform = self._print(-term)
pforms.append(pretty_negative(pform, i))
elif term.is_Relational:
pforms.append(prettyForm(*self._print(term).parens()))
else:
pforms.append(self._print(term))
if indices:
large = True
for pform in pforms:
if pform is not None and pform.height() > 1:
break
else:
large = False
for i in indices:
term, negative = terms[i], False
if term < 0:
term, negative = -term, True
if large:
pform = prettyForm(str(term.p))/prettyForm(str(term.q))
else:
pform = self._print(term)
if negative:
pform = pretty_negative(pform, i)
pforms[i] = pform
return prettyForm.__add__(*pforms)
def _print_Mul(self, product):
from sympy.physics.units import Quantity
# Check for unevaluated Mul. In this case we need to make sure the
# identities are visible, multiple Rational factors are not combined
# etc so we display in a straight-forward form that fully preserves all
# args and their order.
args = product.args
if args[0] is S.One or any(isinstance(arg, Number) for arg in args[1:]):
strargs = list(map(self._print, args))
# XXX: This is a hack to work around the fact that
# prettyForm.__mul__ absorbs a leading -1 in the args. Probably it
# would be better to fix this in prettyForm.__mul__ instead.
negone = strargs[0] == '-1'
if negone:
strargs[0] = prettyForm('1', 0, 0)
obj = prettyForm.__mul__(*strargs)
if negone:
obj = prettyForm('-' + obj.s, obj.baseline, obj.binding)
return obj
a = [] # items in the numerator
b = [] # items that are in the denominator (if any)
if self.order not in ('old', 'none'):
args = product.as_ordered_factors()
else:
args = list(product.args)
# If quantities are present append them at the back
args = sorted(args, key=lambda x: isinstance(x, Quantity) or
(isinstance(x, Pow) and isinstance(x.base, Quantity)))
# Gather terms for numerator/denominator
for item in args:
if item.is_commutative and item.is_Pow and item.exp.is_Rational and item.exp.is_negative:
if item.exp != -1:
b.append(Pow(item.base, -item.exp, evaluate=False))
else:
b.append(Pow(item.base, -item.exp))
elif item.is_Rational and item is not S.Infinity:
if item.p != 1:
a.append( Rational(item.p) )
if item.q != 1:
b.append( Rational(item.q) )
else:
a.append(item)
from sympy import Integral, Piecewise, Product, Sum
# Convert to pretty forms. Add parens to Add instances if there
# is more than one term in the numer/denom
for i in range(0, len(a)):
if (a[i].is_Add and len(a) > 1) or (i != len(a) - 1 and
isinstance(a[i], (Integral, Piecewise, Product, Sum))):
a[i] = prettyForm(*self._print(a[i]).parens())
elif a[i].is_Relational:
a[i] = prettyForm(*self._print(a[i]).parens())
else:
a[i] = self._print(a[i])
for i in range(0, len(b)):
if (b[i].is_Add and len(b) > 1) or (i != len(b) - 1 and
isinstance(b[i], (Integral, Piecewise, Product, Sum))):
b[i] = prettyForm(*self._print(b[i]).parens())
else:
b[i] = self._print(b[i])
# Construct a pretty form
if len(b) == 0:
return prettyForm.__mul__(*a)
else:
if len(a) == 0:
a.append( self._print(S.One) )
return prettyForm.__mul__(*a)/prettyForm.__mul__(*b)
# A helper function for _print_Pow to print x**(1/n)
def _print_nth_root(self, base, expt):
bpretty = self._print(base)
# In very simple cases, use a single-char root sign
if (self._settings['use_unicode_sqrt_char'] and self._use_unicode
and expt is S.Half and bpretty.height() == 1
and (bpretty.width() == 1
or (base.is_Integer and base.is_nonnegative))):
return prettyForm(*bpretty.left(u'\N{SQUARE ROOT}'))
# Construct root sign, start with the \/ shape
_zZ = xobj('/', 1)
rootsign = xobj('\\', 1) + _zZ
# Make exponent number to put above it
if isinstance(expt, Rational):
exp = str(expt.q)
if exp == '2':
exp = ''
else:
exp = str(expt.args[0])
exp = exp.ljust(2)
if len(exp) > 2:
rootsign = ' '*(len(exp) - 2) + rootsign
# Stack the exponent
rootsign = stringPict(exp + '\n' + rootsign)
rootsign.baseline = 0
# Diagonal: length is one less than height of base
linelength = bpretty.height() - 1
diagonal = stringPict('\n'.join(
' '*(linelength - i - 1) + _zZ + ' '*i
for i in range(linelength)
))
# Put baseline just below lowest line: next to exp
diagonal.baseline = linelength - 1
# Make the root symbol
rootsign = prettyForm(*rootsign.right(diagonal))
# Det the baseline to match contents to fix the height
# but if the height of bpretty is one, the rootsign must be one higher
rootsign.baseline = max(1, bpretty.baseline)
#build result
s = prettyForm(hobj('_', 2 + bpretty.width()))
s = prettyForm(*bpretty.above(s))
s = prettyForm(*s.left(rootsign))
return s
def _print_Pow(self, power):
from sympy.simplify.simplify import fraction
b, e = power.as_base_exp()
if power.is_commutative:
if e is S.NegativeOne:
return prettyForm("1")/self._print(b)
n, d = fraction(e)
if n is S.One and d.is_Atom and not e.is_Integer and self._settings['root_notation']:
return self._print_nth_root(b, e)
if e.is_Rational and e < 0:
return prettyForm("1")/self._print(Pow(b, -e, evaluate=False))
if b.is_Relational:
return prettyForm(*self._print(b).parens()).__pow__(self._print(e))
return self._print(b)**self._print(e)
def _print_UnevaluatedExpr(self, expr):
return self._print(expr.args[0])
def __print_numer_denom(self, p, q):
if q == 1:
if p < 0:
return prettyForm(str(p), binding=prettyForm.NEG)
else:
return prettyForm(str(p))
elif abs(p) >= 10 and abs(q) >= 10:
# If more than one digit in numer and denom, print larger fraction
if p < 0:
return prettyForm(str(p), binding=prettyForm.NEG)/prettyForm(str(q))
# Old printing method:
#pform = prettyForm(str(-p))/prettyForm(str(q))
#return prettyForm(binding=prettyForm.NEG, *pform.left('- '))
else:
return prettyForm(str(p))/prettyForm(str(q))
else:
return None
def _print_Rational(self, expr):
result = self.__print_numer_denom(expr.p, expr.q)
if result is not None:
return result
else:
return self.emptyPrinter(expr)
def _print_Fraction(self, expr):
result = self.__print_numer_denom(expr.numerator, expr.denominator)
if result is not None:
return result
else:
return self.emptyPrinter(expr)
def _print_ProductSet(self, p):
if len(p.sets) >= 1 and not has_variety(p.sets):
return self._print(p.sets[0]) ** self._print(len(p.sets))
else:
prod_char = u"\N{MULTIPLICATION SIGN}" if self._use_unicode else 'x'
return self._print_seq(p.sets, None, None, ' %s ' % prod_char,
parenthesize=lambda set: set.is_Union or
set.is_Intersection or set.is_ProductSet)
def _print_FiniteSet(self, s):
items = sorted(s.args, key=default_sort_key)
return self._print_seq(items, '{', '}', ', ' )
def _print_Range(self, s):
if self._use_unicode:
dots = u"\N{HORIZONTAL ELLIPSIS}"
else:
dots = '...'
if s.start.is_infinite and s.stop.is_infinite:
if s.step.is_positive:
printset = dots, -1, 0, 1, dots
else:
printset = dots, 1, 0, -1, dots
elif s.start.is_infinite:
printset = dots, s[-1] - s.step, s[-1]
elif s.stop.is_infinite:
it = iter(s)
printset = next(it), next(it), dots
elif len(s) > 4:
it = iter(s)
printset = next(it), next(it), dots, s[-1]
else:
printset = tuple(s)
return self._print_seq(printset, '{', '}', ', ' )
def _print_Interval(self, i):
if i.start == i.end:
return self._print_seq(i.args[:1], '{', '}')
else:
if i.left_open:
left = '('
else:
left = '['
if i.right_open:
right = ')'
else:
right = ']'
return self._print_seq(i.args[:2], left, right)
def _print_AccumulationBounds(self, i):
left = '<'
right = '>'
return self._print_seq(i.args[:2], left, right)
def _print_Intersection(self, u):
delimiter = ' %s ' % pretty_atom('Intersection', 'n')
return self._print_seq(u.args, None, None, delimiter,
parenthesize=lambda set: set.is_ProductSet or
set.is_Union or set.is_Complement)
def _print_Union(self, u):
union_delimiter = ' %s ' % pretty_atom('Union', 'U')
return self._print_seq(u.args, None, None, union_delimiter,
parenthesize=lambda set: set.is_ProductSet or
set.is_Intersection or set.is_Complement)
def _print_SymmetricDifference(self, u):
if not self._use_unicode:
raise NotImplementedError("ASCII pretty printing of SymmetricDifference is not implemented")
sym_delimeter = ' %s ' % pretty_atom('SymmetricDifference')
return self._print_seq(u.args, None, None, sym_delimeter)
def _print_Complement(self, u):
delimiter = r' \ '
return self._print_seq(u.args, None, None, delimiter,
parenthesize=lambda set: set.is_ProductSet or set.is_Intersection
or set.is_Union)
def _print_ImageSet(self, ts):
if self._use_unicode:
inn = u"\N{SMALL ELEMENT OF}"
else:
inn = 'in'
fun = ts.lamda
sets = ts.base_sets
signature = fun.signature
expr = self._print(fun.expr)
bar = self._print("|")
if len(signature) == 1:
return self._print_seq((expr, bar, signature[0], inn, sets[0]), "{", "}", ' ')
else:
pargs = tuple(j for var, setv in zip(signature, sets) for j in (var, inn, setv, ","))
return self._print_seq((expr, bar) + pargs[:-1], "{", "}", ' ')
def _print_ConditionSet(self, ts):
if self._use_unicode:
inn = u"\N{SMALL ELEMENT OF}"
# using _and because and is a keyword and it is bad practice to
# overwrite them
_and = u"\N{LOGICAL AND}"
else:
inn = 'in'
_and = 'and'
variables = self._print_seq(Tuple(ts.sym))
as_expr = getattr(ts.condition, 'as_expr', None)
if as_expr is not None:
cond = self._print(ts.condition.as_expr())
else:
cond = self._print(ts.condition)
if self._use_unicode:
cond = self._print(cond)
cond = prettyForm(*cond.parens())
bar = self._print("|")
if ts.base_set is S.UniversalSet:
return self._print_seq((variables, bar, cond), "{", "}", ' ')
base = self._print(ts.base_set)
return self._print_seq((variables, bar, variables, inn,
base, _and, cond), "{", "}", ' ')
def _print_ComplexRegion(self, ts):
if self._use_unicode:
inn = u"\N{SMALL ELEMENT OF}"
else:
inn = 'in'
variables = self._print_seq(ts.variables)
expr = self._print(ts.expr)
bar = self._print("|")
prodsets = self._print(ts.sets)
return self._print_seq((expr, bar, variables, inn, prodsets), "{", "}", ' ')
def _print_Contains(self, e):
var, set = e.args
if self._use_unicode:
el = u" \N{ELEMENT OF} "
return prettyForm(*stringPict.next(self._print(var),
el, self._print(set)), binding=8)
else:
return prettyForm(sstr(e))
def _print_FourierSeries(self, s):
if self._use_unicode:
dots = u"\N{HORIZONTAL ELLIPSIS}"
else:
dots = '...'
return self._print_Add(s.truncate()) + self._print(dots)
def _print_FormalPowerSeries(self, s):
return self._print_Add(s.infinite)
def _print_SetExpr(self, se):
pretty_set = prettyForm(*self._print(se.set).parens())
pretty_name = self._print(Symbol("SetExpr"))
return prettyForm(*pretty_name.right(pretty_set))
def _print_SeqFormula(self, s):
if self._use_unicode:
dots = u"\N{HORIZONTAL ELLIPSIS}"
else:
dots = '...'
if len(s.start.free_symbols) > 0 or len(s.stop.free_symbols) > 0:
raise NotImplementedError("Pretty printing of sequences with symbolic bound not implemented")
if s.start is S.NegativeInfinity:
stop = s.stop
printset = (dots, s.coeff(stop - 3), s.coeff(stop - 2),
s.coeff(stop - 1), s.coeff(stop))
elif s.stop is S.Infinity or s.length > 4:
printset = s[:4]
printset.append(dots)
printset = tuple(printset)
else:
printset = tuple(s)
return self._print_list(printset)
_print_SeqPer = _print_SeqFormula
_print_SeqAdd = _print_SeqFormula
_print_SeqMul = _print_SeqFormula
def _print_seq(self, seq, left=None, right=None, delimiter=', ',
parenthesize=lambda x: False):
s = None
try:
for item in seq:
pform = self._print(item)
if parenthesize(item):
pform = prettyForm(*pform.parens())
if s is None:
# first element
s = pform
else:
# XXX: Under the tests from #15686 this raises:
# AttributeError: 'Fake' object has no attribute 'baseline'
# This is caught below but that is not the right way to
# fix it.
s = prettyForm(*stringPict.next(s, delimiter))
s = prettyForm(*stringPict.next(s, pform))
if s is None:
s = stringPict('')
except AttributeError:
s = None
for item in seq:
pform = self.doprint(item)
if parenthesize(item):
pform = prettyForm(*pform.parens())
if s is None:
# first element
s = pform
else :
s = prettyForm(*stringPict.next(s, delimiter))
s = prettyForm(*stringPict.next(s, pform))
if s is None:
s = stringPict('')
s = prettyForm(*s.parens(left, right, ifascii_nougly=True))
return s
def join(self, delimiter, args):
pform = None
for arg in args:
if pform is None:
pform = arg
else:
pform = prettyForm(*pform.right(delimiter))
pform = prettyForm(*pform.right(arg))
if pform is None:
return prettyForm("")
else:
return pform
def _print_list(self, l):
return self._print_seq(l, '[', ']')
def _print_tuple(self, t):
if len(t) == 1:
ptuple = prettyForm(*stringPict.next(self._print(t[0]), ','))
return prettyForm(*ptuple.parens('(', ')', ifascii_nougly=True))
else:
return self._print_seq(t, '(', ')')
def _print_Tuple(self, expr):
return self._print_tuple(expr)
def _print_dict(self, d):
keys = sorted(d.keys(), key=default_sort_key)
items = []
for k in keys:
K = self._print(k)
V = self._print(d[k])
s = prettyForm(*stringPict.next(K, ': ', V))
items.append(s)
return self._print_seq(items, '{', '}')
def _print_Dict(self, d):
return self._print_dict(d)
def _print_set(self, s):
if not s:
return prettyForm('set()')
items = sorted(s, key=default_sort_key)
pretty = self._print_seq(items)
pretty = prettyForm(*pretty.parens('{', '}', ifascii_nougly=True))
return pretty
def _print_frozenset(self, s):
if not s:
return prettyForm('frozenset()')
items = sorted(s, key=default_sort_key)
pretty = self._print_seq(items)
pretty = prettyForm(*pretty.parens('{', '}', ifascii_nougly=True))
pretty = prettyForm(*pretty.parens('(', ')', ifascii_nougly=True))
pretty = prettyForm(*stringPict.next(type(s).__name__, pretty))
return pretty
def _print_UniversalSet(self, s):
if self._use_unicode:
return prettyForm(u"\N{MATHEMATICAL DOUBLE-STRUCK CAPITAL U}")
else:
return prettyForm('UniversalSet')
def _print_PolyRing(self, ring):
return prettyForm(sstr(ring))
def _print_FracField(self, field):
return prettyForm(sstr(field))
def _print_FreeGroupElement(self, elm):
return prettyForm(str(elm))
def _print_PolyElement(self, poly):
return prettyForm(sstr(poly))
def _print_FracElement(self, frac):
return prettyForm(sstr(frac))
def _print_AlgebraicNumber(self, expr):
if expr.is_aliased:
return self._print(expr.as_poly().as_expr())
else:
return self._print(expr.as_expr())
def _print_ComplexRootOf(self, expr):
args = [self._print_Add(expr.expr, order='lex'), expr.index]
pform = prettyForm(*self._print_seq(args).parens())
pform = prettyForm(*pform.left('CRootOf'))
return pform
def _print_RootSum(self, expr):
args = [self._print_Add(expr.expr, order='lex')]
if expr.fun is not S.IdentityFunction:
args.append(self._print(expr.fun))
pform = prettyForm(*self._print_seq(args).parens())
pform = prettyForm(*pform.left('RootSum'))
return pform
def _print_FiniteField(self, expr):
if self._use_unicode:
form = u'\N{DOUBLE-STRUCK CAPITAL Z}_%d'
else:
form = 'GF(%d)'
return prettyForm(pretty_symbol(form % expr.mod))
def _print_IntegerRing(self, expr):
if self._use_unicode:
return prettyForm(u'\N{DOUBLE-STRUCK CAPITAL Z}')
else:
return prettyForm('ZZ')
def _print_RationalField(self, expr):
if self._use_unicode:
return prettyForm(u'\N{DOUBLE-STRUCK CAPITAL Q}')
else:
return prettyForm('QQ')
def _print_RealField(self, domain):
if self._use_unicode:
prefix = u'\N{DOUBLE-STRUCK CAPITAL R}'
else:
prefix = 'RR'
if domain.has_default_precision:
return prettyForm(prefix)
else:
return self._print(pretty_symbol(prefix + "_" + str(domain.precision)))
def _print_ComplexField(self, domain):
if self._use_unicode:
prefix = u'\N{DOUBLE-STRUCK CAPITAL C}'
else:
prefix = 'CC'
if domain.has_default_precision:
return prettyForm(prefix)
else:
return self._print(pretty_symbol(prefix + "_" + str(domain.precision)))
def _print_PolynomialRing(self, expr):
args = list(expr.symbols)
if not expr.order.is_default:
order = prettyForm(*prettyForm("order=").right(self._print(expr.order)))
args.append(order)
pform = self._print_seq(args, '[', ']')
pform = prettyForm(*pform.left(self._print(expr.domain)))
return pform
def _print_FractionField(self, expr):
args = list(expr.symbols)
if not expr.order.is_default:
order = prettyForm(*prettyForm("order=").right(self._print(expr.order)))
args.append(order)
pform = self._print_seq(args, '(', ')')
pform = prettyForm(*pform.left(self._print(expr.domain)))
return pform
def _print_PolynomialRingBase(self, expr):
g = expr.symbols
if str(expr.order) != str(expr.default_order):
g = g + ("order=" + str(expr.order),)
pform = self._print_seq(g, '[', ']')
pform = prettyForm(*pform.left(self._print(expr.domain)))
return pform
def _print_GroebnerBasis(self, basis):
exprs = [ self._print_Add(arg, order=basis.order)
for arg in basis.exprs ]
exprs = prettyForm(*self.join(", ", exprs).parens(left="[", right="]"))
gens = [ self._print(gen) for gen in basis.gens ]
domain = prettyForm(
*prettyForm("domain=").right(self._print(basis.domain)))
order = prettyForm(
*prettyForm("order=").right(self._print(basis.order)))
pform = self.join(", ", [exprs] + gens + [domain, order])
pform = prettyForm(*pform.parens())
pform = prettyForm(*pform.left(basis.__class__.__name__))
return pform
def _print_Subs(self, e):
pform = self._print(e.expr)
pform = prettyForm(*pform.parens())
h = pform.height() if pform.height() > 1 else 2
rvert = stringPict(vobj('|', h), baseline=pform.baseline)
pform = prettyForm(*pform.right(rvert))
b = pform.baseline
pform.baseline = pform.height() - 1
pform = prettyForm(*pform.right(self._print_seq([
self._print_seq((self._print(v[0]), xsym('=='), self._print(v[1])),
delimiter='') for v in zip(e.variables, e.point) ])))
pform.baseline = b
return pform
def _print_number_function(self, e, name):
# Print name_arg[0] for one argument or name_arg[0](arg[1])
# for more than one argument
pform = prettyForm(name)
arg = self._print(e.args[0])
pform_arg = prettyForm(" "*arg.width())
pform_arg = prettyForm(*pform_arg.below(arg))
pform = prettyForm(*pform.right(pform_arg))
if len(e.args) == 1:
return pform
m, x = e.args
# TODO: copy-pasted from _print_Function: can we do better?
prettyFunc = pform
prettyArgs = prettyForm(*self._print_seq([x]).parens())
pform = prettyForm(
binding=prettyForm.FUNC, *stringPict.next(prettyFunc, prettyArgs))
pform.prettyFunc = prettyFunc
pform.prettyArgs = prettyArgs
return pform
def _print_euler(self, e):
return self._print_number_function(e, "E")
def _print_catalan(self, e):
return self._print_number_function(e, "C")
def _print_bernoulli(self, e):
return self._print_number_function(e, "B")
_print_bell = _print_bernoulli
def _print_lucas(self, e):
return self._print_number_function(e, "L")
def _print_fibonacci(self, e):
return self._print_number_function(e, "F")
def _print_tribonacci(self, e):
return self._print_number_function(e, "T")
def _print_stieltjes(self, e):
if self._use_unicode:
return self._print_number_function(e, u'\N{GREEK SMALL LETTER GAMMA}')
else:
return self._print_number_function(e, "stieltjes")
def _print_KroneckerDelta(self, e):
pform = self._print(e.args[0])
pform = prettyForm(*pform.right((prettyForm(','))))
pform = prettyForm(*pform.right((self._print(e.args[1]))))
if self._use_unicode:
a = stringPict(pretty_symbol('delta'))
else:
a = stringPict('d')
b = pform
top = stringPict(*b.left(' '*a.width()))
bot = stringPict(*a.right(' '*b.width()))
return prettyForm(binding=prettyForm.POW, *bot.below(top))
def _print_RandomDomain(self, d):
if hasattr(d, 'as_boolean'):
pform = self._print('Domain: ')
pform = prettyForm(*pform.right(self._print(d.as_boolean())))
return pform
elif hasattr(d, 'set'):
pform = self._print('Domain: ')
pform = prettyForm(*pform.right(self._print(d.symbols)))
pform = prettyForm(*pform.right(self._print(' in ')))
pform = prettyForm(*pform.right(self._print(d.set)))
return pform
elif hasattr(d, 'symbols'):
pform = self._print('Domain on ')
pform = prettyForm(*pform.right(self._print(d.symbols)))
return pform
else:
return self._print(None)
def _print_DMP(self, p):
try:
if p.ring is not None:
# TODO incorporate order
return self._print(p.ring.to_sympy(p))
except SympifyError:
pass
return self._print(repr(p))
def _print_DMF(self, p):
return self._print_DMP(p)
def _print_Object(self, object):
return self._print(pretty_symbol(object.name))
def _print_Morphism(self, morphism):
arrow = xsym("-->")
domain = self._print(morphism.domain)
codomain = self._print(morphism.codomain)
tail = domain.right(arrow, codomain)[0]
return prettyForm(tail)
def _print_NamedMorphism(self, morphism):
pretty_name = self._print(pretty_symbol(morphism.name))
pretty_morphism = self._print_Morphism(morphism)
return prettyForm(pretty_name.right(":", pretty_morphism)[0])
def _print_IdentityMorphism(self, morphism):
from sympy.categories import NamedMorphism
return self._print_NamedMorphism(
NamedMorphism(morphism.domain, morphism.codomain, "id"))
def _print_CompositeMorphism(self, morphism):
circle = xsym(".")
# All components of the morphism have names and it is thus
# possible to build the name of the composite.
component_names_list = [pretty_symbol(component.name) for
component in morphism.components]
component_names_list.reverse()
component_names = circle.join(component_names_list) + ":"
pretty_name = self._print(component_names)
pretty_morphism = self._print_Morphism(morphism)
return prettyForm(pretty_name.right(pretty_morphism)[0])
def _print_Category(self, category):
return self._print(pretty_symbol(category.name))
def _print_Diagram(self, diagram):
if not diagram.premises:
# This is an empty diagram.
return self._print(S.EmptySet)
pretty_result = self._print(diagram.premises)
if diagram.conclusions:
results_arrow = " %s " % xsym("==>")
pretty_conclusions = self._print(diagram.conclusions)[0]
pretty_result = pretty_result.right(
results_arrow, pretty_conclusions)
return prettyForm(pretty_result[0])
def _print_DiagramGrid(self, grid):
from sympy.matrices import Matrix
from sympy import Symbol
matrix = Matrix([[grid[i, j] if grid[i, j] else Symbol(" ")
for j in range(grid.width)]
for i in range(grid.height)])
return self._print_matrix_contents(matrix)
def _print_FreeModuleElement(self, m):
# Print as row vector for convenience, for now.
return self._print_seq(m, '[', ']')
def _print_SubModule(self, M):
return self._print_seq(M.gens, '<', '>')
def _print_FreeModule(self, M):
return self._print(M.ring)**self._print(M.rank)
def _print_ModuleImplementedIdeal(self, M):
return self._print_seq([x for [x] in M._module.gens], '<', '>')
def _print_QuotientRing(self, R):
return self._print(R.ring) / self._print(R.base_ideal)
def _print_QuotientRingElement(self, R):
return self._print(R.data) + self._print(R.ring.base_ideal)
def _print_QuotientModuleElement(self, m):
return self._print(m.data) + self._print(m.module.killed_module)
def _print_QuotientModule(self, M):
return self._print(M.base) / self._print(M.killed_module)
def _print_MatrixHomomorphism(self, h):
matrix = self._print(h._sympy_matrix())
matrix.baseline = matrix.height() // 2
pform = prettyForm(*matrix.right(' : ', self._print(h.domain),
' %s> ' % hobj('-', 2), self._print(h.codomain)))
return pform
def _print_Manifold(self, manifold):
return self._print(manifold.name)
def _print_Patch(self, patch):
return self._print(patch.name)
def _print_CoordSystem(self, coords):
return self._print(coords.name)
def _print_BaseScalarField(self, field):
string = field._coord_sys._names[field._index]
return self._print(pretty_symbol(string))
def _print_BaseVectorField(self, field):
s = U('PARTIAL DIFFERENTIAL') + '_' + field._coord_sys._names[field._index]
return self._print(pretty_symbol(s))
def _print_Differential(self, diff):
field = diff._form_field
if hasattr(field, '_coord_sys'):
string = field._coord_sys._names[field._index]
return self._print(u'\N{DOUBLE-STRUCK ITALIC SMALL D} ' + pretty_symbol(string))
else:
pform = self._print(field)
pform = prettyForm(*pform.parens())
return prettyForm(*pform.left(u"\N{DOUBLE-STRUCK ITALIC SMALL D}"))
def _print_Tr(self, p):
#TODO: Handle indices
pform = self._print(p.args[0])
pform = prettyForm(*pform.left('%s(' % (p.__class__.__name__)))
pform = prettyForm(*pform.right(')'))
return pform
def _print_primenu(self, e):
pform = self._print(e.args[0])
pform = prettyForm(*pform.parens())
if self._use_unicode:
pform = prettyForm(*pform.left(greek_unicode['nu']))
else:
pform = prettyForm(*pform.left('nu'))
return pform
def _print_primeomega(self, e):
pform = self._print(e.args[0])
pform = prettyForm(*pform.parens())
if self._use_unicode:
pform = prettyForm(*pform.left(greek_unicode['Omega']))
else:
pform = prettyForm(*pform.left('Omega'))
return pform
def _print_Quantity(self, e):
if e.name.name == 'degree':
pform = self._print(u"\N{DEGREE SIGN}")
return pform
else:
return self.emptyPrinter(e)
def _print_AssignmentBase(self, e):
op = prettyForm(' ' + xsym(e.op) + ' ')
l = self._print(e.lhs)
r = self._print(e.rhs)
pform = prettyForm(*stringPict.next(l, op, r))
return pform
def pretty(expr, **settings):
"""Returns a string containing the prettified form of expr.
For information on keyword arguments see pretty_print function.
"""
pp = PrettyPrinter(settings)
# XXX: this is an ugly hack, but at least it works
use_unicode = pp._settings['use_unicode']
uflag = pretty_use_unicode(use_unicode)
try:
return pp.doprint(expr)
finally:
pretty_use_unicode(uflag)
def pretty_print(expr, **kwargs):
"""Prints expr in pretty form.
pprint is just a shortcut for this function.
Parameters
==========
expr : expression
The expression to print.
wrap_line : bool, optional (default=True)
Line wrapping enabled/disabled.
num_columns : int or None, optional (default=None)
Number of columns before line breaking (default to None which reads
the terminal width), useful when using SymPy without terminal.
use_unicode : bool or None, optional (default=None)
Use unicode characters, such as the Greek letter pi instead of
the string pi.
full_prec : bool or string, optional (default="auto")
Use full precision.
order : bool or string, optional (default=None)
Set to 'none' for long expressions if slow; default is None.
use_unicode_sqrt_char : bool, optional (default=True)
Use compact single-character square root symbol (when unambiguous).
root_notation : bool, optional (default=True)
Set to 'False' for printing exponents of the form 1/n in fractional form.
By default exponent is printed in root form.
mat_symbol_style : string, optional (default="plain")
Set to "bold" for printing MatrixSymbols using a bold mathematical symbol face.
By default the standard face is used.
imaginary_unit : string, optional (default="i")
Letter to use for imaginary unit when use_unicode is True.
Can be "i" (default) or "j".
"""
print(pretty(expr, **kwargs))
pprint = pretty_print
def pager_print(expr, **settings):
"""Prints expr using the pager, in pretty form.
This invokes a pager command using pydoc. Lines are not wrapped
automatically. This routine is meant to be used with a pager that allows
sideways scrolling, like ``less -S``.
Parameters are the same as for ``pretty_print``. If you wish to wrap lines,
pass ``num_columns=None`` to auto-detect the width of the terminal.
"""
from pydoc import pager
from locale import getpreferredencoding
if 'num_columns' not in settings:
settings['num_columns'] = 500000 # disable line wrap
pager(pretty(expr, **settings).encode(getpreferredencoding()))
|
1b5b2cadb177f25a0eb3a2e6f9807dc364e24c170292bec771afa0a72c71ed09 | from typing import Any, Dict
from sympy.testing.pytest import raises
from sympy import (symbols, sympify, Function, Integer, Matrix, Abs,
Rational, Float, S, WildFunction, ImmutableDenseMatrix, sin, true, false, ones,
sqrt, root, AlgebraicNumber, Symbol, Dummy, Wild, MatrixSymbol)
from sympy.combinatorics import Cycle, Permutation
from sympy.core.compatibility import exec_
from sympy.geometry import Point, Ellipse
from sympy.printing import srepr
from sympy.polys import ring, field, ZZ, QQ, lex, grlex, Poly
from sympy.polys.polyclasses import DMP
from sympy.polys.agca.extensions import FiniteExtension
x, y = symbols('x,y')
# eval(srepr(expr)) == expr has to succeed in the right environment. The right
# environment is the scope of "from sympy import *" for most cases.
ENV = {} # type: Dict[str, Any]
exec_("from sympy import *", ENV)
def sT(expr, string, import_stmt=None):
"""
sT := sreprTest
Tests that srepr delivers the expected string and that
the condition eval(srepr(expr))==expr holds.
"""
if import_stmt is None:
ENV2 = ENV
else:
ENV2 = ENV.copy()
exec_(import_stmt, ENV2)
assert srepr(expr) == string
assert eval(string, ENV2) == expr
def test_printmethod():
class R(Abs):
def _sympyrepr(self, printer):
return "foo(%s)" % printer._print(self.args[0])
assert srepr(R(x)) == "foo(Symbol('x'))"
def test_Add():
sT(x + y, "Add(Symbol('x'), Symbol('y'))")
assert srepr(x**2 + 1, order='lex') == "Add(Pow(Symbol('x'), Integer(2)), Integer(1))"
assert srepr(x**2 + 1, order='old') == "Add(Integer(1), Pow(Symbol('x'), Integer(2)))"
assert srepr(sympify('x + 3 - 2', evaluate=False), order='none') == "Add(Symbol('x'), Integer(3), Mul(Integer(-1), Integer(2)))"
def test_more_than_255_args_issue_10259():
from sympy import Add, Mul
for op in (Add, Mul):
expr = op(*symbols('x:256'))
assert eval(srepr(expr)) == expr
def test_Function():
sT(Function("f")(x), "Function('f')(Symbol('x'))")
# test unapplied Function
sT(Function('f'), "Function('f')")
sT(sin(x), "sin(Symbol('x'))")
sT(sin, "sin")
def test_Geometry():
sT(Point(0, 0), "Point2D(Integer(0), Integer(0))")
sT(Ellipse(Point(0, 0), 5, 1),
"Ellipse(Point2D(Integer(0), Integer(0)), Integer(5), Integer(1))")
# TODO more tests
def test_Singletons():
sT(S.Catalan, 'Catalan')
sT(S.ComplexInfinity, 'zoo')
sT(S.EulerGamma, 'EulerGamma')
sT(S.Exp1, 'E')
sT(S.GoldenRatio, 'GoldenRatio')
sT(S.TribonacciConstant, 'TribonacciConstant')
sT(S.Half, 'Rational(1, 2)')
sT(S.ImaginaryUnit, 'I')
sT(S.Infinity, 'oo')
sT(S.NaN, 'nan')
sT(S.NegativeInfinity, '-oo')
sT(S.NegativeOne, 'Integer(-1)')
sT(S.One, 'Integer(1)')
sT(S.Pi, 'pi')
sT(S.Zero, 'Integer(0)')
def test_Integer():
sT(Integer(4), "Integer(4)")
def test_list():
sT([x, Integer(4)], "[Symbol('x'), Integer(4)]")
def test_Matrix():
for cls, name in [(Matrix, "MutableDenseMatrix"), (ImmutableDenseMatrix, "ImmutableDenseMatrix")]:
sT(cls([[x**+1, 1], [y, x + y]]),
"%s([[Symbol('x'), Integer(1)], [Symbol('y'), Add(Symbol('x'), Symbol('y'))]])" % name)
sT(cls(), "%s([])" % name)
sT(cls([[x**+1, 1], [y, x + y]]), "%s([[Symbol('x'), Integer(1)], [Symbol('y'), Add(Symbol('x'), Symbol('y'))]])" % name)
def test_empty_Matrix():
sT(ones(0, 3), "MutableDenseMatrix(0, 3, [])")
sT(ones(4, 0), "MutableDenseMatrix(4, 0, [])")
sT(ones(0, 0), "MutableDenseMatrix([])")
def test_Rational():
sT(Rational(1, 3), "Rational(1, 3)")
sT(Rational(-1, 3), "Rational(-1, 3)")
def test_Float():
sT(Float('1.23', dps=3), "Float('1.22998', precision=13)")
sT(Float('1.23456789', dps=9), "Float('1.23456788994', precision=33)")
sT(Float('1.234567890123456789', dps=19),
"Float('1.234567890123456789013', precision=66)")
sT(Float('0.60038617995049726', dps=15),
"Float('0.60038617995049726', precision=53)")
sT(Float('1.23', precision=13), "Float('1.22998', precision=13)")
sT(Float('1.23456789', precision=33),
"Float('1.23456788994', precision=33)")
sT(Float('1.234567890123456789', precision=66),
"Float('1.234567890123456789013', precision=66)")
sT(Float('0.60038617995049726', precision=53),
"Float('0.60038617995049726', precision=53)")
sT(Float('0.60038617995049726', 15),
"Float('0.60038617995049726', precision=53)")
def test_Symbol():
sT(x, "Symbol('x')")
sT(y, "Symbol('y')")
sT(Symbol('x', negative=True), "Symbol('x', negative=True)")
def test_Symbol_two_assumptions():
x = Symbol('x', negative=0, integer=1)
# order could vary
s1 = "Symbol('x', integer=True, negative=False)"
s2 = "Symbol('x', negative=False, integer=True)"
assert srepr(x) in (s1, s2)
assert eval(srepr(x), ENV) == x
def test_Symbol_no_special_commutative_treatment():
sT(Symbol('x'), "Symbol('x')")
sT(Symbol('x', commutative=False), "Symbol('x', commutative=False)")
sT(Symbol('x', commutative=0), "Symbol('x', commutative=False)")
sT(Symbol('x', commutative=True), "Symbol('x', commutative=True)")
sT(Symbol('x', commutative=1), "Symbol('x', commutative=True)")
def test_Wild():
sT(Wild('x', even=True), "Wild('x', even=True)")
def test_Dummy():
d = Dummy('d')
sT(d, "Dummy('d', dummy_index=%s)" % str(d.dummy_index))
def test_Dummy_assumption():
d = Dummy('d', nonzero=True)
assert d == eval(srepr(d))
s1 = "Dummy('d', dummy_index=%s, nonzero=True)" % str(d.dummy_index)
s2 = "Dummy('d', nonzero=True, dummy_index=%s)" % str(d.dummy_index)
assert srepr(d) in (s1, s2)
def test_Dummy_from_Symbol():
# should not get the full dictionary of assumptions
n = Symbol('n', integer=True)
d = n.as_dummy()
assert srepr(d
) == "Dummy('n', dummy_index=%s)" % str(d.dummy_index)
def test_tuple():
sT((x,), "(Symbol('x'),)")
sT((x, y), "(Symbol('x'), Symbol('y'))")
def test_WildFunction():
sT(WildFunction('w'), "WildFunction('w')")
def test_settins():
raises(TypeError, lambda: srepr(x, method="garbage"))
def test_Mul():
sT(3*x**3*y, "Mul(Integer(3), Pow(Symbol('x'), Integer(3)), Symbol('y'))")
assert srepr(3*x**3*y, order='old') == "Mul(Integer(3), Symbol('y'), Pow(Symbol('x'), Integer(3)))"
assert srepr(sympify('(x+4)*2*x*7', evaluate=False), order='none') == "Mul(Add(Symbol('x'), Integer(4)), Integer(2), Symbol('x'), Integer(7))"
def test_AlgebraicNumber():
a = AlgebraicNumber(sqrt(2))
sT(a, "AlgebraicNumber(Pow(Integer(2), Rational(1, 2)), [Integer(1), Integer(0)])")
a = AlgebraicNumber(root(-2, 3))
sT(a, "AlgebraicNumber(Pow(Integer(-2), Rational(1, 3)), [Integer(1), Integer(0)])")
def test_PolyRing():
assert srepr(ring("x", ZZ, lex)[0]) == "PolyRing((Symbol('x'),), ZZ, lex)"
assert srepr(ring("x,y", QQ, grlex)[0]) == "PolyRing((Symbol('x'), Symbol('y')), QQ, grlex)"
assert srepr(ring("x,y,z", ZZ["t"], lex)[0]) == "PolyRing((Symbol('x'), Symbol('y'), Symbol('z')), ZZ[t], lex)"
def test_FracField():
assert srepr(field("x", ZZ, lex)[0]) == "FracField((Symbol('x'),), ZZ, lex)"
assert srepr(field("x,y", QQ, grlex)[0]) == "FracField((Symbol('x'), Symbol('y')), QQ, grlex)"
assert srepr(field("x,y,z", ZZ["t"], lex)[0]) == "FracField((Symbol('x'), Symbol('y'), Symbol('z')), ZZ[t], lex)"
def test_PolyElement():
R, x, y = ring("x,y", ZZ)
assert srepr(3*x**2*y + 1) == "PolyElement(PolyRing((Symbol('x'), Symbol('y')), ZZ, lex), [((2, 1), 3), ((0, 0), 1)])"
def test_FracElement():
F, x, y = field("x,y", ZZ)
assert srepr((3*x**2*y + 1)/(x - y**2)) == "FracElement(FracField((Symbol('x'), Symbol('y')), ZZ, lex), [((2, 1), 3), ((0, 0), 1)], [((1, 0), 1), ((0, 2), -1)])"
def test_FractionField():
assert srepr(QQ.frac_field(x)) == \
"FractionField(FracField((Symbol('x'),), QQ, lex))"
assert srepr(QQ.frac_field(x, y, order=grlex)) == \
"FractionField(FracField((Symbol('x'), Symbol('y')), QQ, grlex))"
def test_PolynomialRingBase():
assert srepr(ZZ.old_poly_ring(x)) == \
"GlobalPolynomialRing(ZZ, Symbol('x'))"
assert srepr(ZZ[x].old_poly_ring(y)) == \
"GlobalPolynomialRing(ZZ[x], Symbol('y'))"
assert srepr(QQ.frac_field(x).old_poly_ring(y)) == \
"GlobalPolynomialRing(FractionField(FracField((Symbol('x'),), QQ, lex)), Symbol('y'))"
def test_DMP():
assert srepr(DMP([1, 2], ZZ)) == 'DMP([1, 2], ZZ)'
assert srepr(ZZ.old_poly_ring(x)([1, 2])) == \
"DMP([1, 2], ZZ, ring=GlobalPolynomialRing(ZZ, Symbol('x')))"
def test_FiniteExtension():
assert srepr(FiniteExtension(Poly(x**2 + 1, x))) == \
"FiniteExtension(Poly(x**2 + 1, x, domain='ZZ'))"
def test_ExtensionElement():
A = FiniteExtension(Poly(x**2 + 1, x))
assert srepr(A.generator) == \
"ExtElem(DMP([1, 0], ZZ, ring=GlobalPolynomialRing(ZZ, Symbol('x'))), FiniteExtension(Poly(x**2 + 1, x, domain='ZZ')))"
def test_BooleanAtom():
assert srepr(true) == "true"
assert srepr(false) == "false"
def test_Integers():
sT(S.Integers, "Integers")
def test_Naturals():
sT(S.Naturals, "Naturals")
def test_Naturals0():
sT(S.Naturals0, "Naturals0")
def test_Reals():
sT(S.Reals, "Reals")
def test_matrix_expressions():
n = symbols('n', integer=True)
A = MatrixSymbol("A", n, n)
B = MatrixSymbol("B", n, n)
sT(A, "MatrixSymbol(Symbol('A'), Symbol('n', integer=True), Symbol('n', integer=True))")
sT(A*B, "MatMul(MatrixSymbol(Symbol('A'), Symbol('n', integer=True), Symbol('n', integer=True)), MatrixSymbol(Symbol('B'), Symbol('n', integer=True), Symbol('n', integer=True)))")
sT(A + B, "MatAdd(MatrixSymbol(Symbol('A'), Symbol('n', integer=True), Symbol('n', integer=True)), MatrixSymbol(Symbol('B'), Symbol('n', integer=True), Symbol('n', integer=True)))")
def test_Cycle():
# FIXME: sT fails because Cycle is not immutable and calling srepr(Cycle(1, 2))
# adds keys to the Cycle dict (GH-17661)
#import_stmt = "from sympy.combinatorics import Cycle"
#sT(Cycle(1, 2), "Cycle(1, 2)", import_stmt)
assert srepr(Cycle(1, 2)) == "Cycle(1, 2)"
def test_Permutation():
import_stmt = "from sympy.combinatorics import Permutation"
sT(Permutation(1, 2), "Permutation(1, 2)", import_stmt)
def test_diffgeom():
from sympy.diffgeom import Manifold, Patch, CoordSystem, BaseScalarField
m = Manifold('M', 2)
assert srepr(m) == "Manifold('M', 2)"
p = Patch('P', m)
assert srepr(p) == "Patch('P', Manifold('M', 2))"
rect = CoordSystem('rect', p)
assert srepr(rect) == "CoordSystem('rect', Patch('P', Manifold('M', 2)), ('rect_0', 'rect_1'))"
b = BaseScalarField(rect, 0)
assert srepr(b) == "BaseScalarField(CoordSystem('rect', Patch('P', Manifold('M', 2)), ('rect_0', 'rect_1')), Integer(0))"
def test_dict():
from sympy import srepr
from sympy.abc import x, y, z
d = {}
assert srepr(d) == "{}"
d = {x: y}
assert srepr(d) == "{Symbol('x'): Symbol('y')}"
d = {x: y, y: z}
assert srepr(d) in (
"{Symbol('x'): Symbol('y'), Symbol('y'): Symbol('z')}",
"{Symbol('y'): Symbol('z'), Symbol('x'): Symbol('y')}",
)
d = {x: {y: z}}
assert srepr(d) == "{Symbol('x'): {Symbol('y'): Symbol('z')}}"
def test_set():
from sympy import srepr
from sympy.abc import x, y
s = set()
assert srepr(s) == "set()"
s = {x, y}
assert srepr(s) in ("{Symbol('x'), Symbol('y')}", "{Symbol('y'), Symbol('x')}")
|
053d913c52fcdc5afc129f693d6429d333dea4b380aa0a1e1f4e8118b0b24a3e | from sympy import (Add, Abs, Catalan, cos, Derivative, E, EulerGamma, exp,
factorial, factorial2, Function, GoldenRatio, TribonacciConstant, I,
Integer, Integral, Interval, Lambda, Limit, Matrix, nan, O, oo, pi, Pow,
Rational, Float, Rel, S, sin, SparseMatrix, sqrt, summation, Sum, Symbol,
symbols, Wild, WildFunction, zeta, zoo, Dummy, Dict, Tuple, FiniteSet, factor,
subfactorial, true, false, Equivalent, Xor, Complement, SymmetricDifference,
AccumBounds, UnevaluatedExpr, Eq, Ne, Quaternion, Subs, MatrixSymbol, MatrixSlice)
from sympy.core import Expr, Mul
from sympy.physics.units import second, joule
from sympy.polys import Poly, rootof, RootSum, groebner, ring, field, ZZ, QQ, lex, grlex
from sympy.geometry import Point, Circle, Polygon, Ellipse, Triangle
from sympy.testing.pytest import raises
from sympy.printing import sstr, sstrrepr, StrPrinter
from sympy.core.trace import Tr
x, y, z, w, t = symbols('x,y,z,w,t')
d = Dummy('d')
def test_printmethod():
class R(Abs):
def _sympystr(self, printer):
return "foo(%s)" % printer._print(self.args[0])
assert sstr(R(x)) == "foo(x)"
class R(Abs):
def _sympystr(self, printer):
return "foo"
assert sstr(R(x)) == "foo"
def test_Abs():
assert str(Abs(x)) == "Abs(x)"
assert str(Abs(Rational(1, 6))) == "1/6"
assert str(Abs(Rational(-1, 6))) == "1/6"
def test_Add():
assert str(x + y) == "x + y"
assert str(x + 1) == "x + 1"
assert str(x + x**2) == "x**2 + x"
assert str(Add(0, 1, evaluate=False)) == "0 + 1"
assert str(Add(0, 0, 1, evaluate=False)) == "0 + 0 + 1"
assert str(1.0*x) == "1.0*x"
assert str(5 + x + y + x*y + x**2 + y**2) == "x**2 + x*y + x + y**2 + y + 5"
assert str(1 + x + x**2/2 + x**3/3) == "x**3/3 + x**2/2 + x + 1"
assert str(2*x - 7*x**2 + 2 + 3*y) == "-7*x**2 + 2*x + 3*y + 2"
assert str(x - y) == "x - y"
assert str(2 - x) == "2 - x"
assert str(x - 2) == "x - 2"
assert str(x - y - z - w) == "-w + x - y - z"
assert str(x - z*y**2*z*w) == "-w*y**2*z**2 + x"
assert str(x - 1*y*x*y) == "-x*y**2 + x"
assert str(sin(x).series(x, 0, 15)) == "x - x**3/6 + x**5/120 - x**7/5040 + x**9/362880 - x**11/39916800 + x**13/6227020800 + O(x**15)"
def test_Catalan():
assert str(Catalan) == "Catalan"
def test_ComplexInfinity():
assert str(zoo) == "zoo"
def test_Derivative():
assert str(Derivative(x, y)) == "Derivative(x, y)"
assert str(Derivative(x**2, x, evaluate=False)) == "Derivative(x**2, x)"
assert str(Derivative(
x**2/y, x, y, evaluate=False)) == "Derivative(x**2/y, x, y)"
def test_dict():
assert str({1: 1 + x}) == sstr({1: 1 + x}) == "{1: x + 1}"
assert str({1: x**2, 2: y*x}) in ("{1: x**2, 2: x*y}", "{2: x*y, 1: x**2}")
assert sstr({1: x**2, 2: y*x}) == "{1: x**2, 2: x*y}"
def test_Dict():
assert str(Dict({1: 1 + x})) == sstr({1: 1 + x}) == "{1: x + 1}"
assert str(Dict({1: x**2, 2: y*x})) in (
"{1: x**2, 2: x*y}", "{2: x*y, 1: x**2}")
assert sstr(Dict({1: x**2, 2: y*x})) == "{1: x**2, 2: x*y}"
def test_Dummy():
assert str(d) == "_d"
assert str(d + x) == "_d + x"
def test_EulerGamma():
assert str(EulerGamma) == "EulerGamma"
def test_Exp():
assert str(E) == "E"
def test_factorial():
n = Symbol('n', integer=True)
assert str(factorial(-2)) == "zoo"
assert str(factorial(0)) == "1"
assert str(factorial(7)) == "5040"
assert str(factorial(n)) == "factorial(n)"
assert str(factorial(2*n)) == "factorial(2*n)"
assert str(factorial(factorial(n))) == 'factorial(factorial(n))'
assert str(factorial(factorial2(n))) == 'factorial(factorial2(n))'
assert str(factorial2(factorial(n))) == 'factorial2(factorial(n))'
assert str(factorial2(factorial2(n))) == 'factorial2(factorial2(n))'
assert str(subfactorial(3)) == "2"
assert str(subfactorial(n)) == "subfactorial(n)"
assert str(subfactorial(2*n)) == "subfactorial(2*n)"
def test_Function():
f = Function('f')
fx = f(x)
w = WildFunction('w')
assert str(f) == "f"
assert str(fx) == "f(x)"
assert str(w) == "w_"
def test_Geometry():
assert sstr(Point(0, 0)) == 'Point2D(0, 0)'
assert sstr(Circle(Point(0, 0), 3)) == 'Circle(Point2D(0, 0), 3)'
assert sstr(Ellipse(Point(1, 2), 3, 4)) == 'Ellipse(Point2D(1, 2), 3, 4)'
assert sstr(Triangle(Point(1, 1), Point(7, 8), Point(0, -1))) == \
'Triangle(Point2D(1, 1), Point2D(7, 8), Point2D(0, -1))'
assert sstr(Polygon(Point(5, 6), Point(-2, -3), Point(0, 0), Point(4, 7))) == \
'Polygon(Point2D(5, 6), Point2D(-2, -3), Point2D(0, 0), Point2D(4, 7))'
assert sstr(Triangle(Point(0, 0), Point(1, 0), Point(0, 1)), sympy_integers=True) == \
'Triangle(Point2D(S(0), S(0)), Point2D(S(1), S(0)), Point2D(S(0), S(1)))'
assert sstr(Ellipse(Point(1, 2), 3, 4), sympy_integers=True) == \
'Ellipse(Point2D(S(1), S(2)), S(3), S(4))'
def test_GoldenRatio():
assert str(GoldenRatio) == "GoldenRatio"
def test_TribonacciConstant():
assert str(TribonacciConstant) == "TribonacciConstant"
def test_ImaginaryUnit():
assert str(I) == "I"
def test_Infinity():
assert str(oo) == "oo"
assert str(oo*I) == "oo*I"
def test_Integer():
assert str(Integer(-1)) == "-1"
assert str(Integer(1)) == "1"
assert str(Integer(-3)) == "-3"
assert str(Integer(0)) == "0"
assert str(Integer(25)) == "25"
def test_Integral():
assert str(Integral(sin(x), y)) == "Integral(sin(x), y)"
assert str(Integral(sin(x), (y, 0, 1))) == "Integral(sin(x), (y, 0, 1))"
def test_Interval():
n = (S.NegativeInfinity, 1, 2, S.Infinity)
for i in range(len(n)):
for j in range(i + 1, len(n)):
for l in (True, False):
for r in (True, False):
ival = Interval(n[i], n[j], l, r)
assert S(str(ival)) == ival
def test_AccumBounds():
a = Symbol('a', real=True)
assert str(AccumBounds(0, a)) == "AccumBounds(0, a)"
assert str(AccumBounds(0, 1)) == "AccumBounds(0, 1)"
def test_Lambda():
assert str(Lambda(d, d**2)) == "Lambda(_d, _d**2)"
# issue 2908
assert str(Lambda((), 1)) == "Lambda((), 1)"
assert str(Lambda((), x)) == "Lambda((), x)"
assert str(Lambda((x, y), x+y)) == "Lambda((x, y), x + y)"
assert str(Lambda(((x, y),), x+y)) == "Lambda(((x, y),), x + y)"
def test_Limit():
assert str(Limit(sin(x)/x, x, y)) == "Limit(sin(x)/x, x, y)"
assert str(Limit(1/x, x, 0)) == "Limit(1/x, x, 0)"
assert str(
Limit(sin(x)/x, x, y, dir="-")) == "Limit(sin(x)/x, x, y, dir='-')"
def test_list():
assert str([x]) == sstr([x]) == "[x]"
assert str([x**2, x*y + 1]) == sstr([x**2, x*y + 1]) == "[x**2, x*y + 1]"
assert str([x**2, [y + x]]) == sstr([x**2, [y + x]]) == "[x**2, [x + y]]"
def test_Matrix_str():
M = Matrix([[x**+1, 1], [y, x + y]])
assert str(M) == "Matrix([[x, 1], [y, x + y]])"
assert sstr(M) == "Matrix([\n[x, 1],\n[y, x + y]])"
M = Matrix([[1]])
assert str(M) == sstr(M) == "Matrix([[1]])"
M = Matrix([[1, 2]])
assert str(M) == sstr(M) == "Matrix([[1, 2]])"
M = Matrix()
assert str(M) == sstr(M) == "Matrix(0, 0, [])"
M = Matrix(0, 1, lambda i, j: 0)
assert str(M) == sstr(M) == "Matrix(0, 1, [])"
def test_Mul():
assert str(x/y) == "x/y"
assert str(y/x) == "y/x"
assert str(x/y/z) == "x/(y*z)"
assert str((x + 1)/(y + 2)) == "(x + 1)/(y + 2)"
assert str(2*x/3) == '2*x/3'
assert str(-2*x/3) == '-2*x/3'
assert str(-1.0*x) == '-1.0*x'
assert str(1.0*x) == '1.0*x'
assert str(Mul(0, 1, evaluate=False)) == '0*1'
assert str(Mul(1, 0, evaluate=False)) == '1*0'
assert str(Mul(1, 1, evaluate=False)) == '1*1'
assert str(Mul(1, 1, 1, evaluate=False)) == '1*1*1'
assert str(Mul(1, 2, evaluate=False)) == '1*2'
assert str(Mul(1, S.Half, evaluate=False)) == '1*(1/2)'
assert str(Mul(1, 1, S.Half, evaluate=False)) == '1*1*(1/2)'
assert str(Mul(1, 1, 2, 3, x, evaluate=False)) == '1*1*2*3*x'
assert str(Mul(1, -1, evaluate=False)) == '1*(-1)'
assert str(Mul(-1, 1, evaluate=False)) == '(-1)*1'
assert str(Mul(4, 3, 2, 1, 0, y, x, evaluate=False)) == '4*3*2*1*0*y*x'
assert str(Mul(4, 3, 2, 1+z, 0, y, x, evaluate=False)) == '4*3*2*(z + 1)*0*y*x'
assert str(Mul(Rational(2, 3), Rational(5, 7), evaluate=False)) == '(2/3)*(5/7)'
# For issue 14160
assert str(Mul(-2, x, Pow(Mul(y,y,evaluate=False), -1, evaluate=False),
evaluate=False)) == '-2*x/(y*y)'
class CustomClass1(Expr):
is_commutative = True
class CustomClass2(Expr):
is_commutative = True
cc1 = CustomClass1()
cc2 = CustomClass2()
assert str(Rational(2)*cc1) == '2*CustomClass1()'
assert str(cc1*Rational(2)) == '2*CustomClass1()'
assert str(cc1*Float("1.5")) == '1.5*CustomClass1()'
assert str(cc2*Rational(2)) == '2*CustomClass2()'
assert str(cc2*Rational(2)*cc1) == '2*CustomClass1()*CustomClass2()'
assert str(cc1*Rational(2)*cc2) == '2*CustomClass1()*CustomClass2()'
def test_NaN():
assert str(nan) == "nan"
def test_NegativeInfinity():
assert str(-oo) == "-oo"
def test_Order():
assert str(O(x)) == "O(x)"
assert str(O(x**2)) == "O(x**2)"
assert str(O(x*y)) == "O(x*y, x, y)"
assert str(O(x, x)) == "O(x)"
assert str(O(x, (x, 0))) == "O(x)"
assert str(O(x, (x, oo))) == "O(x, (x, oo))"
assert str(O(x, x, y)) == "O(x, x, y)"
assert str(O(x, x, y)) == "O(x, x, y)"
assert str(O(x, (x, oo), (y, oo))) == "O(x, (x, oo), (y, oo))"
def test_Permutation_Cycle():
from sympy.combinatorics import Permutation, Cycle
# general principle: economically, canonically show all moved elements
# and the size of the permutation.
for p, s in [
(Cycle(),
'()'),
(Cycle(2),
'(2)'),
(Cycle(2, 1),
'(1 2)'),
(Cycle(1, 2)(5)(6, 7)(10),
'(1 2)(6 7)(10)'),
(Cycle(3, 4)(1, 2)(3, 4),
'(1 2)(4)'),
]:
assert sstr(p) == s
for p, s in [
(Permutation([]),
'Permutation([])'),
(Permutation([], size=1),
'Permutation([0])'),
(Permutation([], size=2),
'Permutation([0, 1])'),
(Permutation([], size=10),
'Permutation([], size=10)'),
(Permutation([1, 0, 2]),
'Permutation([1, 0, 2])'),
(Permutation([1, 0, 2, 3, 4, 5]),
'Permutation([1, 0], size=6)'),
(Permutation([1, 0, 2, 3, 4, 5], size=10),
'Permutation([1, 0], size=10)'),
]:
assert sstr(p, perm_cyclic=False) == s
for p, s in [
(Permutation([]),
'()'),
(Permutation([], size=1),
'(0)'),
(Permutation([], size=2),
'(1)'),
(Permutation([], size=10),
'(9)'),
(Permutation([1, 0, 2]),
'(2)(0 1)'),
(Permutation([1, 0, 2, 3, 4, 5]),
'(5)(0 1)'),
(Permutation([1, 0, 2, 3, 4, 5], size=10),
'(9)(0 1)'),
(Permutation([0, 1, 3, 2, 4, 5], size=10),
'(9)(2 3)'),
]:
assert sstr(p) == s
def test_Pi():
assert str(pi) == "pi"
def test_Poly():
assert str(Poly(0, x)) == "Poly(0, x, domain='ZZ')"
assert str(Poly(1, x)) == "Poly(1, x, domain='ZZ')"
assert str(Poly(x, x)) == "Poly(x, x, domain='ZZ')"
assert str(Poly(2*x + 1, x)) == "Poly(2*x + 1, x, domain='ZZ')"
assert str(Poly(2*x - 1, x)) == "Poly(2*x - 1, x, domain='ZZ')"
assert str(Poly(-1, x)) == "Poly(-1, x, domain='ZZ')"
assert str(Poly(-x, x)) == "Poly(-x, x, domain='ZZ')"
assert str(Poly(-2*x + 1, x)) == "Poly(-2*x + 1, x, domain='ZZ')"
assert str(Poly(-2*x - 1, x)) == "Poly(-2*x - 1, x, domain='ZZ')"
assert str(Poly(x - 1, x)) == "Poly(x - 1, x, domain='ZZ')"
assert str(Poly(2*x + x**5, x)) == "Poly(x**5 + 2*x, x, domain='ZZ')"
assert str(Poly(3**(2*x), 3**x)) == "Poly((3**x)**2, 3**x, domain='ZZ')"
assert str(Poly((x**2)**x)) == "Poly(((x**2)**x), (x**2)**x, domain='ZZ')"
assert str(Poly((x + y)**3, (x + y), expand=False)
) == "Poly((x + y)**3, x + y, domain='ZZ')"
assert str(Poly((x - 1)**2, (x - 1), expand=False)
) == "Poly((x - 1)**2, x - 1, domain='ZZ')"
assert str(
Poly(x**2 + 1 + y, x)) == "Poly(x**2 + y + 1, x, domain='ZZ[y]')"
assert str(
Poly(x**2 - 1 + y, x)) == "Poly(x**2 + y - 1, x, domain='ZZ[y]')"
assert str(Poly(x**2 + I*x, x)) == "Poly(x**2 + I*x, x, domain='EX')"
assert str(Poly(x**2 - I*x, x)) == "Poly(x**2 - I*x, x, domain='EX')"
assert str(Poly(-x*y*z + x*y - 1, x, y, z)
) == "Poly(-x*y*z + x*y - 1, x, y, z, domain='ZZ')"
assert str(Poly(-w*x**21*y**7*z + (1 + w)*z**3 - 2*x*z + 1, x, y, z)) == \
"Poly(-w*x**21*y**7*z - 2*x*z + (w + 1)*z**3 + 1, x, y, z, domain='ZZ[w]')"
assert str(Poly(x**2 + 1, x, modulus=2)) == "Poly(x**2 + 1, x, modulus=2)"
assert str(Poly(2*x**2 + 3*x + 4, x, modulus=17)) == "Poly(2*x**2 + 3*x + 4, x, modulus=17)"
def test_PolyRing():
assert str(ring("x", ZZ, lex)[0]) == "Polynomial ring in x over ZZ with lex order"
assert str(ring("x,y", QQ, grlex)[0]) == "Polynomial ring in x, y over QQ with grlex order"
assert str(ring("x,y,z", ZZ["t"], lex)[0]) == "Polynomial ring in x, y, z over ZZ[t] with lex order"
def test_FracField():
assert str(field("x", ZZ, lex)[0]) == "Rational function field in x over ZZ with lex order"
assert str(field("x,y", QQ, grlex)[0]) == "Rational function field in x, y over QQ with grlex order"
assert str(field("x,y,z", ZZ["t"], lex)[0]) == "Rational function field in x, y, z over ZZ[t] with lex order"
def test_PolyElement():
Ruv, u,v = ring("u,v", ZZ)
Rxyz, x,y,z = ring("x,y,z", Ruv)
assert str(x - x) == "0"
assert str(x - 1) == "x - 1"
assert str(x + 1) == "x + 1"
assert str(x**2) == "x**2"
assert str(x**(-2)) == "x**(-2)"
assert str(x**QQ(1, 2)) == "x**(1/2)"
assert str((u**2 + 3*u*v + 1)*x**2*y + u + 1) == "(u**2 + 3*u*v + 1)*x**2*y + u + 1"
assert str((u**2 + 3*u*v + 1)*x**2*y + (u + 1)*x) == "(u**2 + 3*u*v + 1)*x**2*y + (u + 1)*x"
assert str((u**2 + 3*u*v + 1)*x**2*y + (u + 1)*x + 1) == "(u**2 + 3*u*v + 1)*x**2*y + (u + 1)*x + 1"
assert str((-u**2 + 3*u*v - 1)*x**2*y - (u + 1)*x - 1) == "-(u**2 - 3*u*v + 1)*x**2*y - (u + 1)*x - 1"
assert str(-(v**2 + v + 1)*x + 3*u*v + 1) == "-(v**2 + v + 1)*x + 3*u*v + 1"
assert str(-(v**2 + v + 1)*x - 3*u*v + 1) == "-(v**2 + v + 1)*x - 3*u*v + 1"
def test_FracElement():
Fuv, u,v = field("u,v", ZZ)
Fxyzt, x,y,z,t = field("x,y,z,t", Fuv)
assert str(x - x) == "0"
assert str(x - 1) == "x - 1"
assert str(x + 1) == "x + 1"
assert str(x/3) == "x/3"
assert str(x/z) == "x/z"
assert str(x*y/z) == "x*y/z"
assert str(x/(z*t)) == "x/(z*t)"
assert str(x*y/(z*t)) == "x*y/(z*t)"
assert str((x - 1)/y) == "(x - 1)/y"
assert str((x + 1)/y) == "(x + 1)/y"
assert str((-x - 1)/y) == "(-x - 1)/y"
assert str((x + 1)/(y*z)) == "(x + 1)/(y*z)"
assert str(-y/(x + 1)) == "-y/(x + 1)"
assert str(y*z/(x + 1)) == "y*z/(x + 1)"
assert str(((u + 1)*x*y + 1)/((v - 1)*z - 1)) == "((u + 1)*x*y + 1)/((v - 1)*z - 1)"
assert str(((u + 1)*x*y + 1)/((v - 1)*z - t*u*v - 1)) == "((u + 1)*x*y + 1)/((v - 1)*z - u*v*t - 1)"
def test_Pow():
assert str(x**-1) == "1/x"
assert str(x**-2) == "x**(-2)"
assert str(x**2) == "x**2"
assert str((x + y)**-1) == "1/(x + y)"
assert str((x + y)**-2) == "(x + y)**(-2)"
assert str((x + y)**2) == "(x + y)**2"
assert str((x + y)**(1 + x)) == "(x + y)**(x + 1)"
assert str(x**Rational(1, 3)) == "x**(1/3)"
assert str(1/x**Rational(1, 3)) == "x**(-1/3)"
assert str(sqrt(sqrt(x))) == "x**(1/4)"
# not the same as x**-1
assert str(x**-1.0) == 'x**(-1.0)'
# see issue #2860
assert str(Pow(S(2), -1.0, evaluate=False)) == '2**(-1.0)'
def test_sqrt():
assert str(sqrt(x)) == "sqrt(x)"
assert str(sqrt(x**2)) == "sqrt(x**2)"
assert str(1/sqrt(x)) == "1/sqrt(x)"
assert str(1/sqrt(x**2)) == "1/sqrt(x**2)"
assert str(y/sqrt(x)) == "y/sqrt(x)"
assert str(x**0.5) == "x**0.5"
assert str(1/x**0.5) == "x**(-0.5)"
def test_Rational():
n1 = Rational(1, 4)
n2 = Rational(1, 3)
n3 = Rational(2, 4)
n4 = Rational(2, -4)
n5 = Rational(0)
n7 = Rational(3)
n8 = Rational(-3)
assert str(n1*n2) == "1/12"
assert str(n1*n2) == "1/12"
assert str(n3) == "1/2"
assert str(n1*n3) == "1/8"
assert str(n1 + n3) == "3/4"
assert str(n1 + n2) == "7/12"
assert str(n1 + n4) == "-1/4"
assert str(n4*n4) == "1/4"
assert str(n4 + n2) == "-1/6"
assert str(n4 + n5) == "-1/2"
assert str(n4*n5) == "0"
assert str(n3 + n4) == "0"
assert str(n1**n7) == "1/64"
assert str(n2**n7) == "1/27"
assert str(n2**n8) == "27"
assert str(n7**n8) == "1/27"
assert str(Rational("-25")) == "-25"
assert str(Rational("1.25")) == "5/4"
assert str(Rational("-2.6e-2")) == "-13/500"
assert str(S("25/7")) == "25/7"
assert str(S("-123/569")) == "-123/569"
assert str(S("0.1[23]", rational=1)) == "61/495"
assert str(S("5.1[666]", rational=1)) == "31/6"
assert str(S("-5.1[666]", rational=1)) == "-31/6"
assert str(S("0.[9]", rational=1)) == "1"
assert str(S("-0.[9]", rational=1)) == "-1"
assert str(sqrt(Rational(1, 4))) == "1/2"
assert str(sqrt(Rational(1, 36))) == "1/6"
assert str((123**25) ** Rational(1, 25)) == "123"
assert str((123**25 + 1)**Rational(1, 25)) != "123"
assert str((123**25 - 1)**Rational(1, 25)) != "123"
assert str((123**25 - 1)**Rational(1, 25)) != "122"
assert str(sqrt(Rational(81, 36))**3) == "27/8"
assert str(1/sqrt(Rational(81, 36))**3) == "8/27"
assert str(sqrt(-4)) == str(2*I)
assert str(2**Rational(1, 10**10)) == "2**(1/10000000000)"
assert sstr(Rational(2, 3), sympy_integers=True) == "S(2)/3"
x = Symbol("x")
assert sstr(x**Rational(2, 3), sympy_integers=True) == "x**(S(2)/3)"
assert sstr(Eq(x, Rational(2, 3)), sympy_integers=True) == "Eq(x, S(2)/3)"
assert sstr(Limit(x, x, Rational(7, 2)), sympy_integers=True) == \
"Limit(x, x, S(7)/2)"
def test_Float():
# NOTE dps is the whole number of decimal digits
assert str(Float('1.23', dps=1 + 2)) == '1.23'
assert str(Float('1.23456789', dps=1 + 8)) == '1.23456789'
assert str(
Float('1.234567890123456789', dps=1 + 18)) == '1.234567890123456789'
assert str(pi.evalf(1 + 2)) == '3.14'
assert str(pi.evalf(1 + 14)) == '3.14159265358979'
assert str(pi.evalf(1 + 64)) == ('3.141592653589793238462643383279'
'5028841971693993751058209749445923')
assert str(pi.round(-1)) == '0.0'
assert str((pi**400 - (pi**400).round(1)).n(2)) == '-0.e+88'
assert sstr(Float("100"), full_prec=False, min=-2, max=2) == '1.0e+2'
assert sstr(Float("100"), full_prec=False, min=-2, max=3) == '100.0'
assert sstr(Float("0.1"), full_prec=False, min=-2, max=3) == '0.1'
assert sstr(Float("0.099"), min=-2, max=3) == '9.90000000000000e-2'
def test_Relational():
assert str(Rel(x, y, "<")) == "x < y"
assert str(Rel(x + y, y, "==")) == "Eq(x + y, y)"
assert str(Rel(x, y, "!=")) == "Ne(x, y)"
assert str(Eq(x, 1) | Eq(x, 2)) == "Eq(x, 1) | Eq(x, 2)"
assert str(Ne(x, 1) & Ne(x, 2)) == "Ne(x, 1) & Ne(x, 2)"
def test_CRootOf():
assert str(rootof(x**5 + 2*x - 1, 0)) == "CRootOf(x**5 + 2*x - 1, 0)"
def test_RootSum():
f = x**5 + 2*x - 1
assert str(
RootSum(f, Lambda(z, z), auto=False)) == "RootSum(x**5 + 2*x - 1)"
assert str(RootSum(f, Lambda(
z, z**2), auto=False)) == "RootSum(x**5 + 2*x - 1, Lambda(z, z**2))"
def test_GroebnerBasis():
assert str(groebner(
[], x, y)) == "GroebnerBasis([], x, y, domain='ZZ', order='lex')"
F = [x**2 - 3*y - x + 1, y**2 - 2*x + y - 1]
assert str(groebner(F, order='grlex')) == \
"GroebnerBasis([x**2 - x - 3*y + 1, y**2 - 2*x + y - 1], x, y, domain='ZZ', order='grlex')"
assert str(groebner(F, order='lex')) == \
"GroebnerBasis([2*x - y**2 - y + 1, y**4 + 2*y**3 - 3*y**2 - 16*y + 7], x, y, domain='ZZ', order='lex')"
def test_set():
assert sstr(set()) == 'set()'
assert sstr(frozenset()) == 'frozenset()'
assert sstr(set([1])) == '{1}'
assert sstr(frozenset([1])) == 'frozenset({1})'
assert sstr(set([1, 2, 3])) == '{1, 2, 3}'
assert sstr(frozenset([1, 2, 3])) == 'frozenset({1, 2, 3})'
assert sstr(
set([1, x, x**2, x**3, x**4])) == '{1, x, x**2, x**3, x**4}'
assert sstr(
frozenset([1, x, x**2, x**3, x**4])) == 'frozenset({1, x, x**2, x**3, x**4})'
def test_SparseMatrix():
M = SparseMatrix([[x**+1, 1], [y, x + y]])
assert str(M) == "Matrix([[x, 1], [y, x + y]])"
assert sstr(M) == "Matrix([\n[x, 1],\n[y, x + y]])"
def test_Sum():
assert str(summation(cos(3*z), (z, x, y))) == "Sum(cos(3*z), (z, x, y))"
assert str(Sum(x*y**2, (x, -2, 2), (y, -5, 5))) == \
"Sum(x*y**2, (x, -2, 2), (y, -5, 5))"
def test_Symbol():
assert str(y) == "y"
assert str(x) == "x"
e = x
assert str(e) == "x"
def test_tuple():
assert str((x,)) == sstr((x,)) == "(x,)"
assert str((x + y, 1 + x)) == sstr((x + y, 1 + x)) == "(x + y, x + 1)"
assert str((x + y, (
1 + x, x**2))) == sstr((x + y, (1 + x, x**2))) == "(x + y, (x + 1, x**2))"
def test_Quaternion_str_printer():
q = Quaternion(x, y, z, t)
assert str(q) == "x + y*i + z*j + t*k"
q = Quaternion(x,y,z,x*t)
assert str(q) == "x + y*i + z*j + t*x*k"
q = Quaternion(x,y,z,x+t)
assert str(q) == "x + y*i + z*j + (t + x)*k"
def test_Quantity_str():
assert sstr(second, abbrev=True) == "s"
assert sstr(joule, abbrev=True) == "J"
assert str(second) == "second"
assert str(joule) == "joule"
def test_wild_str():
# Check expressions containing Wild not causing infinite recursion
w = Wild('x')
assert str(w + 1) == 'x_ + 1'
assert str(exp(2**w) + 5) == 'exp(2**x_) + 5'
assert str(3*w + 1) == '3*x_ + 1'
assert str(1/w + 1) == '1 + 1/x_'
assert str(w**2 + 1) == 'x_**2 + 1'
assert str(1/(1 - w)) == '1/(1 - x_)'
def test_zeta():
assert str(zeta(3)) == "zeta(3)"
def test_issue_3101():
e = x - y
a = str(e)
b = str(e)
assert a == b
def test_issue_3103():
e = -2*sqrt(x) - y/sqrt(x)/2
assert str(e) not in ["(-2)*x**1/2(-1/2)*x**(-1/2)*y",
"-2*x**1/2(-1/2)*x**(-1/2)*y", "-2*x**1/2-1/2*x**-1/2*w"]
assert str(e) == "-2*sqrt(x) - y/(2*sqrt(x))"
def test_issue_4021():
e = Integral(x, x) + 1
assert str(e) == 'Integral(x, x) + 1'
def test_sstrrepr():
assert sstr('abc') == 'abc'
assert sstrrepr('abc') == "'abc'"
e = ['a', 'b', 'c', x]
assert sstr(e) == "[a, b, c, x]"
assert sstrrepr(e) == "['a', 'b', 'c', x]"
def test_infinity():
assert sstr(oo*I) == "oo*I"
def test_full_prec():
assert sstr(S("0.3"), full_prec=True) == "0.300000000000000"
assert sstr(S("0.3"), full_prec="auto") == "0.300000000000000"
assert sstr(S("0.3"), full_prec=False) == "0.3"
assert sstr(S("0.3")*x, full_prec=True) in [
"0.300000000000000*x",
"x*0.300000000000000"
]
assert sstr(S("0.3")*x, full_prec="auto") in [
"0.3*x",
"x*0.3"
]
assert sstr(S("0.3")*x, full_prec=False) in [
"0.3*x",
"x*0.3"
]
def test_noncommutative():
A, B, C = symbols('A,B,C', commutative=False)
assert sstr(A*B*C**-1) == "A*B*C**(-1)"
assert sstr(C**-1*A*B) == "C**(-1)*A*B"
assert sstr(A*C**-1*B) == "A*C**(-1)*B"
assert sstr(sqrt(A)) == "sqrt(A)"
assert sstr(1/sqrt(A)) == "A**(-1/2)"
def test_empty_printer():
str_printer = StrPrinter()
assert str_printer.emptyPrinter("foo") == "foo"
assert str_printer.emptyPrinter(x*y) == "x*y"
assert str_printer.emptyPrinter(32) == "32"
def test_settings():
raises(TypeError, lambda: sstr(S(4), method="garbage"))
def test_RandomDomain():
from sympy.stats import Normal, Die, Exponential, pspace, where
X = Normal('x1', 0, 1)
assert str(where(X > 0)) == "Domain: (0 < x1) & (x1 < oo)"
D = Die('d1', 6)
assert str(where(D > 4)) == "Domain: Eq(d1, 5) | Eq(d1, 6)"
A = Exponential('a', 1)
B = Exponential('b', 1)
assert str(pspace(Tuple(A, B)).domain) == "Domain: (0 <= a) & (0 <= b) & (a < oo) & (b < oo)"
def test_FiniteSet():
assert str(FiniteSet(*range(1, 51))) == (
'FiniteSet(1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17,'
' 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34,'
' 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50)'
)
assert str(FiniteSet(*range(1, 6))) == 'FiniteSet(1, 2, 3, 4, 5)'
def test_UniversalSet():
assert str(S.UniversalSet) == 'UniversalSet'
def test_PrettyPoly():
from sympy.polys.domains import QQ
F = QQ.frac_field(x, y)
R = QQ[x, y]
assert sstr(F.convert(x/(x + y))) == sstr(x/(x + y))
assert sstr(R.convert(x + y)) == sstr(x + y)
def test_categories():
from sympy.categories import (Object, NamedMorphism,
IdentityMorphism, Category)
A = Object("A")
B = Object("B")
f = NamedMorphism(A, B, "f")
id_A = IdentityMorphism(A)
K = Category("K")
assert str(A) == 'Object("A")'
assert str(f) == 'NamedMorphism(Object("A"), Object("B"), "f")'
assert str(id_A) == 'IdentityMorphism(Object("A"))'
assert str(K) == 'Category("K")'
def test_Tr():
A, B = symbols('A B', commutative=False)
t = Tr(A*B)
assert str(t) == 'Tr(A*B)'
def test_issue_6387():
assert str(factor(-3.0*z + 3)) == '-3.0*(1.0*z - 1.0)'
def test_MatMul_MatAdd():
from sympy import MatrixSymbol
X, Y = MatrixSymbol("X", 2, 2), MatrixSymbol("Y", 2, 2)
assert str(2*(X + Y)) == "2*(X + Y)"
assert str(I*X) == "I*X"
assert str(-I*X) == "-I*X"
assert str((1 + I)*X) == '(1 + I)*X'
assert str(-(1 + I)*X) == '(-1 - I)*X'
def test_MatrixSlice():
n = Symbol('n', integer=True)
X = MatrixSymbol('X', n, n)
Y = MatrixSymbol('Y', 10, 10)
Z = MatrixSymbol('Z', 10, 10)
assert str(MatrixSlice(X, (None, None, None), (None, None, None))) == 'X[:, :]'
assert str(X[x:x + 1, y:y + 1]) == 'X[x:x + 1, y:y + 1]'
assert str(X[x:x + 1:2, y:y + 1:2]) == 'X[x:x + 1:2, y:y + 1:2]'
assert str(X[:x, y:]) == 'X[:x, y:]'
assert str(X[:x, y:]) == 'X[:x, y:]'
assert str(X[x:, :y]) == 'X[x:, :y]'
assert str(X[x:y, z:w]) == 'X[x:y, z:w]'
assert str(X[x:y:t, w:t:x]) == 'X[x:y:t, w:t:x]'
assert str(X[x::y, t::w]) == 'X[x::y, t::w]'
assert str(X[:x:y, :t:w]) == 'X[:x:y, :t:w]'
assert str(X[::x, ::y]) == 'X[::x, ::y]'
assert str(MatrixSlice(X, (0, None, None), (0, None, None))) == 'X[:, :]'
assert str(MatrixSlice(X, (None, n, None), (None, n, None))) == 'X[:, :]'
assert str(MatrixSlice(X, (0, n, None), (0, n, None))) == 'X[:, :]'
assert str(MatrixSlice(X, (0, n, 2), (0, n, 2))) == 'X[::2, ::2]'
assert str(X[1:2:3, 4:5:6]) == 'X[1:2:3, 4:5:6]'
assert str(X[1:3:5, 4:6:8]) == 'X[1:3:5, 4:6:8]'
assert str(X[1:10:2]) == 'X[1:10:2, :]'
assert str(Y[:5, 1:9:2]) == 'Y[:5, 1:9:2]'
assert str(Y[:5, 1:10:2]) == 'Y[:5, 1::2]'
assert str(Y[5, :5:2]) == 'Y[5:6, :5:2]'
assert str(X[0:1, 0:1]) == 'X[:1, :1]'
assert str(X[0:1:2, 0:1:2]) == 'X[:1:2, :1:2]'
assert str((Y + Z)[2:, 2:]) == '(Y + Z)[2:, 2:]'
def test_true_false():
assert str(true) == repr(true) == sstr(true) == "True"
assert str(false) == repr(false) == sstr(false) == "False"
def test_Equivalent():
assert str(Equivalent(y, x)) == "Equivalent(x, y)"
def test_Xor():
assert str(Xor(y, x, evaluate=False)) == "x ^ y"
def test_Complement():
assert str(Complement(S.Reals, S.Naturals)) == 'Complement(Reals, Naturals)'
def test_SymmetricDifference():
assert str(SymmetricDifference(Interval(2, 3), Interval(3, 4),evaluate=False)) == \
'SymmetricDifference(Interval(2, 3), Interval(3, 4))'
def test_UnevaluatedExpr():
a, b = symbols("a b")
expr1 = 2*UnevaluatedExpr(a+b)
assert str(expr1) == "2*(a + b)"
def test_MatrixElement_printing():
# test cases for issue #11821
A = MatrixSymbol("A", 1, 3)
B = MatrixSymbol("B", 1, 3)
C = MatrixSymbol("C", 1, 3)
assert(str(A[0, 0]) == "A[0, 0]")
assert(str(3 * A[0, 0]) == "3*A[0, 0]")
F = C[0, 0].subs(C, A - B)
assert str(F) == "(A - B)[0, 0]"
def test_MatrixSymbol_printing():
A = MatrixSymbol("A", 3, 3)
B = MatrixSymbol("B", 3, 3)
assert str(A - A*B - B) == "A - A*B - B"
assert str(A*B - (A+B)) == "-(A + B) + A*B"
assert str(A**(-1)) == "A**(-1)"
assert str(A**3) == "A**3"
def test_MatrixExpressions():
n = Symbol('n', integer=True)
X = MatrixSymbol('X', n, n)
assert str(X) == "X"
# Apply function elementwise (`ElementwiseApplyFunc`):
expr = (X.T*X).applyfunc(sin)
assert str(expr) == 'Lambda(_d, sin(_d)).(X.T*X)'
lamda = Lambda(x, 1/x)
expr = (n*X).applyfunc(lamda)
assert str(expr) == 'Lambda(x, 1/x).(n*X)'
def test_Subs_printing():
assert str(Subs(x, (x,), (1,))) == 'Subs(x, x, 1)'
assert str(Subs(x + y, (x, y), (1, 2))) == 'Subs(x + y, (x, y), (1, 2))'
def test_issue_15716():
e = Integral(factorial(x), (x, -oo, oo))
assert e.as_terms() == ([(e, ((1.0, 0.0), (1,), ()))], [e])
def test_str_special_matrices():
from sympy.matrices import Identity, ZeroMatrix, OneMatrix
assert str(Identity(4)) == 'I'
assert str(ZeroMatrix(2, 2)) == '0'
assert str(OneMatrix(2, 2)) == '1'
def test_issue_14567():
assert factorial(Sum(-1, (x, 0, 0))) + y # doesn't raise an error
def test_diffgeom():
from sympy.diffgeom import Manifold, Patch, CoordSystem, BaseScalarField
m = Manifold('M', 2)
assert str(m) == "M"
p = Patch('P', m)
assert str(p) == "P"
rect = CoordSystem('rect', p)
assert str(rect) == "rect"
b = BaseScalarField(rect, 0)
assert str(b) == "rect_0"
|
0572e4462d6f87689fa965ab44b16b5258d6f64e0bda3b4ae61ef508d40c6962 | from sympy.tensor.toperators import PartialDerivative
from sympy import (
Abs, Chi, Ci, CosineTransform, Dict, Ei, Eq, FallingFactorial,
FiniteSet, Float, FourierTransform, Function, Indexed, IndexedBase, Integral,
Interval, InverseCosineTransform, InverseFourierTransform, Derivative,
InverseLaplaceTransform, InverseMellinTransform, InverseSineTransform,
Lambda, LaplaceTransform, Limit, Matrix, Max, MellinTransform, Min, Mul,
Order, Piecewise, Poly, ring, field, ZZ, Pow, Product, Range, Rational,
RisingFactorial, rootof, RootSum, S, Shi, Si, SineTransform, Subs,
Sum, Symbol, ImageSet, Tuple, Ynm, Znm, arg, asin, acsc, asinh, Mod,
assoc_laguerre, assoc_legendre, beta, binomial, catalan, ceiling,
chebyshevt, chebyshevu, conjugate, cot, coth, diff, dirichlet_eta, euler,
exp, expint, factorial, factorial2, floor, gamma, gegenbauer, hermite,
hyper, im, jacobi, laguerre, legendre, lerchphi, log, frac,
meijerg, oo, polar_lift, polylog, re, root, sin, sqrt, symbols,
uppergamma, zeta, subfactorial, totient, elliptic_k, elliptic_f,
elliptic_e, elliptic_pi, cos, tan, Wild, true, false, Equivalent, Not,
Contains, divisor_sigma, SeqPer, SeqFormula, MatrixSlice,
SeqAdd, SeqMul, fourier_series, pi, ConditionSet, ComplexRegion, fps,
AccumBounds, reduced_totient, primenu, primeomega, SingularityFunction,
stieltjes, mathieuc, mathieus, mathieucprime, mathieusprime,
UnevaluatedExpr, Quaternion, I, KroneckerProduct, LambertW)
from sympy.ntheory.factor_ import udivisor_sigma
from sympy.abc import mu, tau
from sympy.printing.latex import (latex, translate, greek_letters_set,
tex_greek_dictionary, multiline_latex)
from sympy.tensor.array import (ImmutableDenseNDimArray,
ImmutableSparseNDimArray,
MutableSparseNDimArray,
MutableDenseNDimArray,
tensorproduct)
from sympy.testing.pytest import XFAIL, raises
from sympy.functions import DiracDelta, Heaviside, KroneckerDelta, LeviCivita
from sympy.functions.combinatorial.numbers import bernoulli, bell, lucas, \
fibonacci, tribonacci
from sympy.logic import Implies
from sympy.logic.boolalg import And, Or, Xor
from sympy.physics.quantum import Commutator, Operator
from sympy.physics.units import meter, gibibyte, microgram, second
from sympy.core.trace import Tr
from sympy.combinatorics.permutations import \
Cycle, Permutation, AppliedPermutation
from sympy.matrices.expressions.permutation import PermutationMatrix
from sympy import MatrixSymbol, ln
from sympy.vector import CoordSys3D, Cross, Curl, Dot, Divergence, Gradient, Laplacian
from sympy.sets.setexpr import SetExpr
from sympy.sets.sets import \
Union, Intersection, Complement, SymmetricDifference, ProductSet
import sympy as sym
class lowergamma(sym.lowergamma):
pass # testing notation inheritance by a subclass with same name
x, y, z, t, a, b, c = symbols('x y z t a b c')
k, m, n = symbols('k m n', integer=True)
def test_printmethod():
class R(Abs):
def _latex(self, printer):
return "foo(%s)" % printer._print(self.args[0])
assert latex(R(x)) == "foo(x)"
class R(Abs):
def _latex(self, printer):
return "foo"
assert latex(R(x)) == "foo"
def test_latex_basic():
assert latex(1 + x) == "x + 1"
assert latex(x**2) == "x^{2}"
assert latex(x**(1 + x)) == "x^{x + 1}"
assert latex(x**3 + x + 1 + x**2) == "x^{3} + x^{2} + x + 1"
assert latex(2*x*y) == "2 x y"
assert latex(2*x*y, mul_symbol='dot') == r"2 \cdot x \cdot y"
assert latex(3*x**2*y, mul_symbol='\\,') == r"3\,x^{2}\,y"
assert latex(1.5*3**x, mul_symbol='\\,') == r"1.5 \cdot 3^{x}"
assert latex(Mul(0, 1, evaluate=False)) == r'0 \cdot 1'
assert latex(Mul(1, 0, evaluate=False)) == r'1 \cdot 0'
assert latex(Mul(1, 1, evaluate=False)) == r'1 \cdot 1'
assert latex(Mul(-1, 1, evaluate=False)) == r'\left(-1\right) 1'
assert latex(Mul(1, 1, 1, evaluate=False)) == r'1 \cdot 1 \cdot 1'
assert latex(Mul(1, 2, evaluate=False)) == r'1 \cdot 2'
assert latex(Mul(1, S.Half, evaluate=False)) == r'1 \frac{1}{2}'
assert latex(Mul(1, 1, S.Half, evaluate=False)) == \
r'1 \cdot 1 \frac{1}{2}'
assert latex(Mul(1, 1, 2, 3, x, evaluate=False)) == \
r'1 \cdot 1 \cdot 2 \cdot 3 x'
assert latex(Mul(1, -1, evaluate=False)) == r'1 \left(-1\right)'
assert latex(Mul(4, 3, 2, 1, 0, y, x, evaluate=False)) == \
r'4 \cdot 3 \cdot 2 \cdot 1 \cdot 0 y x'
assert latex(Mul(4, 3, 2, 1+z, 0, y, x, evaluate=False)) == \
r'4 \cdot 3 \cdot 2 \left(z + 1\right) 0 y x'
assert latex(Mul(Rational(2, 3), Rational(5, 7), evaluate=False)) == \
r'\frac{2}{3} \frac{5}{7}'
assert latex(1/x) == r"\frac{1}{x}"
assert latex(1/x, fold_short_frac=True) == "1 / x"
assert latex(-S(3)/2) == r"- \frac{3}{2}"
assert latex(-S(3)/2, fold_short_frac=True) == r"- 3 / 2"
assert latex(1/x**2) == r"\frac{1}{x^{2}}"
assert latex(1/(x + y)/2) == r"\frac{1}{2 \left(x + y\right)}"
assert latex(x/2) == r"\frac{x}{2}"
assert latex(x/2, fold_short_frac=True) == "x / 2"
assert latex((x + y)/(2*x)) == r"\frac{x + y}{2 x}"
assert latex((x + y)/(2*x), fold_short_frac=True) == \
r"\left(x + y\right) / 2 x"
assert latex((x + y)/(2*x), long_frac_ratio=0) == \
r"\frac{1}{2 x} \left(x + y\right)"
assert latex((x + y)/x) == r"\frac{x + y}{x}"
assert latex((x + y)/x, long_frac_ratio=3) == r"\frac{x + y}{x}"
assert latex((2*sqrt(2)*x)/3) == r"\frac{2 \sqrt{2} x}{3}"
assert latex((2*sqrt(2)*x)/3, long_frac_ratio=2) == \
r"\frac{2 x}{3} \sqrt{2}"
assert latex(binomial(x, y)) == r"{\binom{x}{y}}"
x_star = Symbol('x^*')
f = Function('f')
assert latex(x_star**2) == r"\left(x^{*}\right)^{2}"
assert latex(x_star**2, parenthesize_super=False) == r"{x^{*}}^{2}"
assert latex(Derivative(f(x_star), x_star,2)) == r"\frac{d^{2}}{d \left(x^{*}\right)^{2}} f{\left(x^{*} \right)}"
assert latex(Derivative(f(x_star), x_star,2), parenthesize_super=False) == r"\frac{d^{2}}{d {x^{*}}^{2}} f{\left(x^{*} \right)}"
assert latex(2*Integral(x, x)/3) == r"\frac{2 \int x\, dx}{3}"
assert latex(2*Integral(x, x)/3, fold_short_frac=True) == \
r"\left(2 \int x\, dx\right) / 3"
assert latex(sqrt(x)) == r"\sqrt{x}"
assert latex(x**Rational(1, 3)) == r"\sqrt[3]{x}"
assert latex(x**Rational(1, 3), root_notation=False) == r"x^{\frac{1}{3}}"
assert latex(sqrt(x)**3) == r"x^{\frac{3}{2}}"
assert latex(sqrt(x), itex=True) == r"\sqrt{x}"
assert latex(x**Rational(1, 3), itex=True) == r"\root{3}{x}"
assert latex(sqrt(x)**3, itex=True) == r"x^{\frac{3}{2}}"
assert latex(x**Rational(3, 4)) == r"x^{\frac{3}{4}}"
assert latex(x**Rational(3, 4), fold_frac_powers=True) == "x^{3/4}"
assert latex((x + 1)**Rational(3, 4)) == \
r"\left(x + 1\right)^{\frac{3}{4}}"
assert latex((x + 1)**Rational(3, 4), fold_frac_powers=True) == \
r"\left(x + 1\right)^{3/4}"
assert latex(1.5e20*x) == r"1.5 \cdot 10^{20} x"
assert latex(1.5e20*x, mul_symbol='dot') == r"1.5 \cdot 10^{20} \cdot x"
assert latex(1.5e20*x, mul_symbol='times') == \
r"1.5 \times 10^{20} \times x"
assert latex(1/sin(x)) == r"\frac{1}{\sin{\left(x \right)}}"
assert latex(sin(x)**-1) == r"\frac{1}{\sin{\left(x \right)}}"
assert latex(sin(x)**Rational(3, 2)) == \
r"\sin^{\frac{3}{2}}{\left(x \right)}"
assert latex(sin(x)**Rational(3, 2), fold_frac_powers=True) == \
r"\sin^{3/2}{\left(x \right)}"
assert latex(~x) == r"\neg x"
assert latex(x & y) == r"x \wedge y"
assert latex(x & y & z) == r"x \wedge y \wedge z"
assert latex(x | y) == r"x \vee y"
assert latex(x | y | z) == r"x \vee y \vee z"
assert latex((x & y) | z) == r"z \vee \left(x \wedge y\right)"
assert latex(Implies(x, y)) == r"x \Rightarrow y"
assert latex(~(x >> ~y)) == r"x \not\Rightarrow \neg y"
assert latex(Implies(Or(x,y), z)) == r"\left(x \vee y\right) \Rightarrow z"
assert latex(Implies(z, Or(x,y))) == r"z \Rightarrow \left(x \vee y\right)"
assert latex(~(x & y)) == r"\neg \left(x \wedge y\right)"
assert latex(~x, symbol_names={x: "x_i"}) == r"\neg x_i"
assert latex(x & y, symbol_names={x: "x_i", y: "y_i"}) == \
r"x_i \wedge y_i"
assert latex(x & y & z, symbol_names={x: "x_i", y: "y_i", z: "z_i"}) == \
r"x_i \wedge y_i \wedge z_i"
assert latex(x | y, symbol_names={x: "x_i", y: "y_i"}) == r"x_i \vee y_i"
assert latex(x | y | z, symbol_names={x: "x_i", y: "y_i", z: "z_i"}) == \
r"x_i \vee y_i \vee z_i"
assert latex((x & y) | z, symbol_names={x: "x_i", y: "y_i", z: "z_i"}) == \
r"z_i \vee \left(x_i \wedge y_i\right)"
assert latex(Implies(x, y), symbol_names={x: "x_i", y: "y_i"}) == \
r"x_i \Rightarrow y_i"
p = Symbol('p', positive=True)
assert latex(exp(-p)*log(p)) == r"e^{- p} \log{\left(p \right)}"
def test_latex_builtins():
assert latex(True) == r"\text{True}"
assert latex(False) == r"\text{False}"
assert latex(None) == r"\text{None}"
assert latex(true) == r"\text{True}"
assert latex(false) == r'\text{False}'
def test_latex_SingularityFunction():
assert latex(SingularityFunction(x, 4, 5)) == \
r"{\left\langle x - 4 \right\rangle}^{5}"
assert latex(SingularityFunction(x, -3, 4)) == \
r"{\left\langle x + 3 \right\rangle}^{4}"
assert latex(SingularityFunction(x, 0, 4)) == \
r"{\left\langle x \right\rangle}^{4}"
assert latex(SingularityFunction(x, a, n)) == \
r"{\left\langle - a + x \right\rangle}^{n}"
assert latex(SingularityFunction(x, 4, -2)) == \
r"{\left\langle x - 4 \right\rangle}^{-2}"
assert latex(SingularityFunction(x, 4, -1)) == \
r"{\left\langle x - 4 \right\rangle}^{-1}"
def test_latex_cycle():
assert latex(Cycle(1, 2, 4)) == r"\left( 1\; 2\; 4\right)"
assert latex(Cycle(1, 2)(4, 5, 6)) == \
r"\left( 1\; 2\right)\left( 4\; 5\; 6\right)"
assert latex(Cycle()) == r"\left( \right)"
def test_latex_permutation():
assert latex(Permutation(1, 2, 4)) == r"\left( 1\; 2\; 4\right)"
assert latex(Permutation(1, 2)(4, 5, 6)) == \
r"\left( 1\; 2\right)\left( 4\; 5\; 6\right)"
assert latex(Permutation()) == r"\left( \right)"
assert latex(Permutation(2, 4)*Permutation(5)) == \
r"\left( 2\; 4\right)\left( 5\right)"
assert latex(Permutation(5)) == r"\left( 5\right)"
assert latex(Permutation(0, 1), perm_cyclic=False) == \
r"\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}"
assert latex(Permutation(0, 1)(2, 3), perm_cyclic=False) == \
r"\begin{pmatrix} 0 & 1 & 2 & 3 \\ 1 & 0 & 3 & 2 \end{pmatrix}"
assert latex(Permutation(), perm_cyclic=False) == \
r"\left( \right)"
def test_latex_Float():
assert latex(Float(1.0e100)) == r"1.0 \cdot 10^{100}"
assert latex(Float(1.0e-100)) == r"1.0 \cdot 10^{-100}"
assert latex(Float(1.0e-100), mul_symbol="times") == \
r"1.0 \times 10^{-100}"
assert latex(Float('10000.0'), full_prec=False, min=-2, max=2) == \
r"1.0 \cdot 10^{4}"
assert latex(Float('10000.0'), full_prec=False, min=-2, max=4) == \
r"1.0 \cdot 10^{4}"
assert latex(Float('10000.0'), full_prec=False, min=-2, max=5) == \
r"10000.0"
assert latex(Float('0.099999'), full_prec=True, min=-2, max=5) == \
r"9.99990000000000 \cdot 10^{-2}"
def test_latex_vector_expressions():
A = CoordSys3D('A')
assert latex(Cross(A.i, A.j*A.x*3+A.k)) == \
r"\mathbf{\hat{i}_{A}} \times \left((3 \mathbf{{x}_{A}})\mathbf{\hat{j}_{A}} + \mathbf{\hat{k}_{A}}\right)"
assert latex(Cross(A.i, A.j)) == \
r"\mathbf{\hat{i}_{A}} \times \mathbf{\hat{j}_{A}}"
assert latex(x*Cross(A.i, A.j)) == \
r"x \left(\mathbf{\hat{i}_{A}} \times \mathbf{\hat{j}_{A}}\right)"
assert latex(Cross(x*A.i, A.j)) == \
r'- \mathbf{\hat{j}_{A}} \times \left((x)\mathbf{\hat{i}_{A}}\right)'
assert latex(Curl(3*A.x*A.j)) == \
r"\nabla\times \left((3 \mathbf{{x}_{A}})\mathbf{\hat{j}_{A}}\right)"
assert latex(Curl(3*A.x*A.j+A.i)) == \
r"\nabla\times \left(\mathbf{\hat{i}_{A}} + (3 \mathbf{{x}_{A}})\mathbf{\hat{j}_{A}}\right)"
assert latex(Curl(3*x*A.x*A.j)) == \
r"\nabla\times \left((3 \mathbf{{x}_{A}} x)\mathbf{\hat{j}_{A}}\right)"
assert latex(x*Curl(3*A.x*A.j)) == \
r"x \left(\nabla\times \left((3 \mathbf{{x}_{A}})\mathbf{\hat{j}_{A}}\right)\right)"
assert latex(Divergence(3*A.x*A.j+A.i)) == \
r"\nabla\cdot \left(\mathbf{\hat{i}_{A}} + (3 \mathbf{{x}_{A}})\mathbf{\hat{j}_{A}}\right)"
assert latex(Divergence(3*A.x*A.j)) == \
r"\nabla\cdot \left((3 \mathbf{{x}_{A}})\mathbf{\hat{j}_{A}}\right)"
assert latex(x*Divergence(3*A.x*A.j)) == \
r"x \left(\nabla\cdot \left((3 \mathbf{{x}_{A}})\mathbf{\hat{j}_{A}}\right)\right)"
assert latex(Dot(A.i, A.j*A.x*3+A.k)) == \
r"\mathbf{\hat{i}_{A}} \cdot \left((3 \mathbf{{x}_{A}})\mathbf{\hat{j}_{A}} + \mathbf{\hat{k}_{A}}\right)"
assert latex(Dot(A.i, A.j)) == \
r"\mathbf{\hat{i}_{A}} \cdot \mathbf{\hat{j}_{A}}"
assert latex(Dot(x*A.i, A.j)) == \
r"\mathbf{\hat{j}_{A}} \cdot \left((x)\mathbf{\hat{i}_{A}}\right)"
assert latex(x*Dot(A.i, A.j)) == \
r"x \left(\mathbf{\hat{i}_{A}} \cdot \mathbf{\hat{j}_{A}}\right)"
assert latex(Gradient(A.x)) == r"\nabla \mathbf{{x}_{A}}"
assert latex(Gradient(A.x + 3*A.y)) == \
r"\nabla \left(\mathbf{{x}_{A}} + 3 \mathbf{{y}_{A}}\right)"
assert latex(x*Gradient(A.x)) == r"x \left(\nabla \mathbf{{x}_{A}}\right)"
assert latex(Gradient(x*A.x)) == r"\nabla \left(\mathbf{{x}_{A}} x\right)"
assert latex(Laplacian(A.x)) == r"\triangle \mathbf{{x}_{A}}"
assert latex(Laplacian(A.x + 3*A.y)) == \
r"\triangle \left(\mathbf{{x}_{A}} + 3 \mathbf{{y}_{A}}\right)"
assert latex(x*Laplacian(A.x)) == r"x \left(\triangle \mathbf{{x}_{A}}\right)"
assert latex(Laplacian(x*A.x)) == r"\triangle \left(\mathbf{{x}_{A}} x\right)"
def test_latex_symbols():
Gamma, lmbda, rho = symbols('Gamma, lambda, rho')
tau, Tau, TAU, taU = symbols('tau, Tau, TAU, taU')
assert latex(tau) == r"\tau"
assert latex(Tau) == "T"
assert latex(TAU) == r"\tau"
assert latex(taU) == r"\tau"
# Check that all capitalized greek letters are handled explicitly
capitalized_letters = set(l.capitalize() for l in greek_letters_set)
assert len(capitalized_letters - set(tex_greek_dictionary.keys())) == 0
assert latex(Gamma + lmbda) == r"\Gamma + \lambda"
assert latex(Gamma * lmbda) == r"\Gamma \lambda"
assert latex(Symbol('q1')) == r"q_{1}"
assert latex(Symbol('q21')) == r"q_{21}"
assert latex(Symbol('epsilon0')) == r"\epsilon_{0}"
assert latex(Symbol('omega1')) == r"\omega_{1}"
assert latex(Symbol('91')) == r"91"
assert latex(Symbol('alpha_new')) == r"\alpha_{new}"
assert latex(Symbol('C^orig')) == r"C^{orig}"
assert latex(Symbol('x^alpha')) == r"x^{\alpha}"
assert latex(Symbol('beta^alpha')) == r"\beta^{\alpha}"
assert latex(Symbol('e^Alpha')) == r"e^{A}"
assert latex(Symbol('omega_alpha^beta')) == r"\omega^{\beta}_{\alpha}"
assert latex(Symbol('omega') ** Symbol('beta')) == r"\omega^{\beta}"
@XFAIL
def test_latex_symbols_failing():
rho, mass, volume = symbols('rho, mass, volume')
assert latex(
volume * rho == mass) == r"\rho \mathrm{volume} = \mathrm{mass}"
assert latex(volume / mass * rho == 1) == \
r"\rho \mathrm{volume} {\mathrm{mass}}^{(-1)} = 1"
assert latex(mass**3 * volume**3) == \
r"{\mathrm{mass}}^{3} \cdot {\mathrm{volume}}^{3}"
def test_latex_functions():
assert latex(exp(x)) == "e^{x}"
assert latex(exp(1) + exp(2)) == "e + e^{2}"
f = Function('f')
assert latex(f(x)) == r'f{\left(x \right)}'
assert latex(f) == r'f'
g = Function('g')
assert latex(g(x, y)) == r'g{\left(x,y \right)}'
assert latex(g) == r'g'
h = Function('h')
assert latex(h(x, y, z)) == r'h{\left(x,y,z \right)}'
assert latex(h) == r'h'
Li = Function('Li')
assert latex(Li) == r'\operatorname{Li}'
assert latex(Li(x)) == r'\operatorname{Li}{\left(x \right)}'
mybeta = Function('beta')
# not to be confused with the beta function
assert latex(mybeta(x, y, z)) == r"\beta{\left(x,y,z \right)}"
assert latex(beta(x, y)) == r'\operatorname{B}\left(x, y\right)'
assert latex(beta(x, y)**2) == r'\operatorname{B}^{2}\left(x, y\right)'
assert latex(mybeta(x)) == r"\beta{\left(x \right)}"
assert latex(mybeta) == r"\beta"
g = Function('gamma')
# not to be confused with the gamma function
assert latex(g(x, y, z)) == r"\gamma{\left(x,y,z \right)}"
assert latex(g(x)) == r"\gamma{\left(x \right)}"
assert latex(g) == r"\gamma"
a1 = Function('a_1')
assert latex(a1) == r"\operatorname{a_{1}}"
assert latex(a1(x)) == r"\operatorname{a_{1}}{\left(x \right)}"
# issue 5868
omega1 = Function('omega1')
assert latex(omega1) == r"\omega_{1}"
assert latex(omega1(x)) == r"\omega_{1}{\left(x \right)}"
assert latex(sin(x)) == r"\sin{\left(x \right)}"
assert latex(sin(x), fold_func_brackets=True) == r"\sin {x}"
assert latex(sin(2*x**2), fold_func_brackets=True) == \
r"\sin {2 x^{2}}"
assert latex(sin(x**2), fold_func_brackets=True) == \
r"\sin {x^{2}}"
assert latex(asin(x)**2) == r"\operatorname{asin}^{2}{\left(x \right)}"
assert latex(asin(x)**2, inv_trig_style="full") == \
r"\arcsin^{2}{\left(x \right)}"
assert latex(asin(x)**2, inv_trig_style="power") == \
r"\sin^{-1}{\left(x \right)}^{2}"
assert latex(asin(x**2), inv_trig_style="power",
fold_func_brackets=True) == \
r"\sin^{-1} {x^{2}}"
assert latex(acsc(x), inv_trig_style="full") == \
r"\operatorname{arccsc}{\left(x \right)}"
assert latex(asinh(x), inv_trig_style="full") == \
r"\operatorname{arcsinh}{\left(x \right)}"
assert latex(factorial(k)) == r"k!"
assert latex(factorial(-k)) == r"\left(- k\right)!"
assert latex(factorial(k)**2) == r"k!^{2}"
assert latex(subfactorial(k)) == r"!k"
assert latex(subfactorial(-k)) == r"!\left(- k\right)"
assert latex(subfactorial(k)**2) == r"\left(!k\right)^{2}"
assert latex(factorial2(k)) == r"k!!"
assert latex(factorial2(-k)) == r"\left(- k\right)!!"
assert latex(factorial2(k)**2) == r"k!!^{2}"
assert latex(binomial(2, k)) == r"{\binom{2}{k}}"
assert latex(binomial(2, k)**2) == r"{\binom{2}{k}}^{2}"
assert latex(FallingFactorial(3, k)) == r"{\left(3\right)}_{k}"
assert latex(RisingFactorial(3, k)) == r"{3}^{\left(k\right)}"
assert latex(floor(x)) == r"\left\lfloor{x}\right\rfloor"
assert latex(ceiling(x)) == r"\left\lceil{x}\right\rceil"
assert latex(frac(x)) == r"\operatorname{frac}{\left(x\right)}"
assert latex(floor(x)**2) == r"\left\lfloor{x}\right\rfloor^{2}"
assert latex(ceiling(x)**2) == r"\left\lceil{x}\right\rceil^{2}"
assert latex(frac(x)**2) == r"\operatorname{frac}{\left(x\right)}^{2}"
assert latex(Min(x, 2, x**3)) == r"\min\left(2, x, x^{3}\right)"
assert latex(Min(x, y)**2) == r"\min\left(x, y\right)^{2}"
assert latex(Max(x, 2, x**3)) == r"\max\left(2, x, x^{3}\right)"
assert latex(Max(x, y)**2) == r"\max\left(x, y\right)^{2}"
assert latex(Abs(x)) == r"\left|{x}\right|"
assert latex(Abs(x)**2) == r"\left|{x}\right|^{2}"
assert latex(re(x)) == r"\operatorname{re}{\left(x\right)}"
assert latex(re(x + y)) == \
r"\operatorname{re}{\left(x\right)} + \operatorname{re}{\left(y\right)}"
assert latex(im(x)) == r"\operatorname{im}{\left(x\right)}"
assert latex(conjugate(x)) == r"\overline{x}"
assert latex(conjugate(x)**2) == r"\overline{x}^{2}"
assert latex(conjugate(x**2)) == r"\overline{x}^{2}"
assert latex(gamma(x)) == r"\Gamma\left(x\right)"
w = Wild('w')
assert latex(gamma(w)) == r"\Gamma\left(w\right)"
assert latex(Order(x)) == r"O\left(x\right)"
assert latex(Order(x, x)) == r"O\left(x\right)"
assert latex(Order(x, (x, 0))) == r"O\left(x\right)"
assert latex(Order(x, (x, oo))) == r"O\left(x; x\rightarrow \infty\right)"
assert latex(Order(x - y, (x, y))) == \
r"O\left(x - y; x\rightarrow y\right)"
assert latex(Order(x, x, y)) == \
r"O\left(x; \left( x, \ y\right)\rightarrow \left( 0, \ 0\right)\right)"
assert latex(Order(x, x, y)) == \
r"O\left(x; \left( x, \ y\right)\rightarrow \left( 0, \ 0\right)\right)"
assert latex(Order(x, (x, oo), (y, oo))) == \
r"O\left(x; \left( x, \ y\right)\rightarrow \left( \infty, \ \infty\right)\right)"
assert latex(lowergamma(x, y)) == r'\gamma\left(x, y\right)'
assert latex(lowergamma(x, y)**2) == r'\gamma^{2}\left(x, y\right)'
assert latex(uppergamma(x, y)) == r'\Gamma\left(x, y\right)'
assert latex(uppergamma(x, y)**2) == r'\Gamma^{2}\left(x, y\right)'
assert latex(cot(x)) == r'\cot{\left(x \right)}'
assert latex(coth(x)) == r'\coth{\left(x \right)}'
assert latex(re(x)) == r'\operatorname{re}{\left(x\right)}'
assert latex(im(x)) == r'\operatorname{im}{\left(x\right)}'
assert latex(root(x, y)) == r'x^{\frac{1}{y}}'
assert latex(arg(x)) == r'\arg{\left(x \right)}'
assert latex(zeta(x)) == r"\zeta\left(x\right)"
assert latex(zeta(x)**2) == r"\zeta^{2}\left(x\right)"
assert latex(zeta(x, y)) == r"\zeta\left(x, y\right)"
assert latex(zeta(x, y)**2) == r"\zeta^{2}\left(x, y\right)"
assert latex(dirichlet_eta(x)) == r"\eta\left(x\right)"
assert latex(dirichlet_eta(x)**2) == r"\eta^{2}\left(x\right)"
assert latex(polylog(x, y)) == r"\operatorname{Li}_{x}\left(y\right)"
assert latex(
polylog(x, y)**2) == r"\operatorname{Li}_{x}^{2}\left(y\right)"
assert latex(lerchphi(x, y, n)) == r"\Phi\left(x, y, n\right)"
assert latex(lerchphi(x, y, n)**2) == r"\Phi^{2}\left(x, y, n\right)"
assert latex(stieltjes(x)) == r"\gamma_{x}"
assert latex(stieltjes(x)**2) == r"\gamma_{x}^{2}"
assert latex(stieltjes(x, y)) == r"\gamma_{x}\left(y\right)"
assert latex(stieltjes(x, y)**2) == r"\gamma_{x}\left(y\right)^{2}"
assert latex(elliptic_k(z)) == r"K\left(z\right)"
assert latex(elliptic_k(z)**2) == r"K^{2}\left(z\right)"
assert latex(elliptic_f(x, y)) == r"F\left(x\middle| y\right)"
assert latex(elliptic_f(x, y)**2) == r"F^{2}\left(x\middle| y\right)"
assert latex(elliptic_e(x, y)) == r"E\left(x\middle| y\right)"
assert latex(elliptic_e(x, y)**2) == r"E^{2}\left(x\middle| y\right)"
assert latex(elliptic_e(z)) == r"E\left(z\right)"
assert latex(elliptic_e(z)**2) == r"E^{2}\left(z\right)"
assert latex(elliptic_pi(x, y, z)) == r"\Pi\left(x; y\middle| z\right)"
assert latex(elliptic_pi(x, y, z)**2) == \
r"\Pi^{2}\left(x; y\middle| z\right)"
assert latex(elliptic_pi(x, y)) == r"\Pi\left(x\middle| y\right)"
assert latex(elliptic_pi(x, y)**2) == r"\Pi^{2}\left(x\middle| y\right)"
assert latex(Ei(x)) == r'\operatorname{Ei}{\left(x \right)}'
assert latex(Ei(x)**2) == r'\operatorname{Ei}^{2}{\left(x \right)}'
assert latex(expint(x, y)) == r'\operatorname{E}_{x}\left(y\right)'
assert latex(expint(x, y)**2) == r'\operatorname{E}_{x}^{2}\left(y\right)'
assert latex(Shi(x)**2) == r'\operatorname{Shi}^{2}{\left(x \right)}'
assert latex(Si(x)**2) == r'\operatorname{Si}^{2}{\left(x \right)}'
assert latex(Ci(x)**2) == r'\operatorname{Ci}^{2}{\left(x \right)}'
assert latex(Chi(x)**2) == r'\operatorname{Chi}^{2}\left(x\right)'
assert latex(Chi(x)) == r'\operatorname{Chi}\left(x\right)'
assert latex(jacobi(n, a, b, x)) == \
r'P_{n}^{\left(a,b\right)}\left(x\right)'
assert latex(jacobi(n, a, b, x)**2) == \
r'\left(P_{n}^{\left(a,b\right)}\left(x\right)\right)^{2}'
assert latex(gegenbauer(n, a, x)) == \
r'C_{n}^{\left(a\right)}\left(x\right)'
assert latex(gegenbauer(n, a, x)**2) == \
r'\left(C_{n}^{\left(a\right)}\left(x\right)\right)^{2}'
assert latex(chebyshevt(n, x)) == r'T_{n}\left(x\right)'
assert latex(chebyshevt(n, x)**2) == \
r'\left(T_{n}\left(x\right)\right)^{2}'
assert latex(chebyshevu(n, x)) == r'U_{n}\left(x\right)'
assert latex(chebyshevu(n, x)**2) == \
r'\left(U_{n}\left(x\right)\right)^{2}'
assert latex(legendre(n, x)) == r'P_{n}\left(x\right)'
assert latex(legendre(n, x)**2) == r'\left(P_{n}\left(x\right)\right)^{2}'
assert latex(assoc_legendre(n, a, x)) == \
r'P_{n}^{\left(a\right)}\left(x\right)'
assert latex(assoc_legendre(n, a, x)**2) == \
r'\left(P_{n}^{\left(a\right)}\left(x\right)\right)^{2}'
assert latex(laguerre(n, x)) == r'L_{n}\left(x\right)'
assert latex(laguerre(n, x)**2) == r'\left(L_{n}\left(x\right)\right)^{2}'
assert latex(assoc_laguerre(n, a, x)) == \
r'L_{n}^{\left(a\right)}\left(x\right)'
assert latex(assoc_laguerre(n, a, x)**2) == \
r'\left(L_{n}^{\left(a\right)}\left(x\right)\right)^{2}'
assert latex(hermite(n, x)) == r'H_{n}\left(x\right)'
assert latex(hermite(n, x)**2) == r'\left(H_{n}\left(x\right)\right)^{2}'
theta = Symbol("theta", real=True)
phi = Symbol("phi", real=True)
assert latex(Ynm(n, m, theta, phi)) == r'Y_{n}^{m}\left(\theta,\phi\right)'
assert latex(Ynm(n, m, theta, phi)**3) == \
r'\left(Y_{n}^{m}\left(\theta,\phi\right)\right)^{3}'
assert latex(Znm(n, m, theta, phi)) == r'Z_{n}^{m}\left(\theta,\phi\right)'
assert latex(Znm(n, m, theta, phi)**3) == \
r'\left(Z_{n}^{m}\left(\theta,\phi\right)\right)^{3}'
# Test latex printing of function names with "_"
assert latex(polar_lift(0)) == \
r"\operatorname{polar\_lift}{\left(0 \right)}"
assert latex(polar_lift(0)**3) == \
r"\operatorname{polar\_lift}^{3}{\left(0 \right)}"
assert latex(totient(n)) == r'\phi\left(n\right)'
assert latex(totient(n) ** 2) == r'\left(\phi\left(n\right)\right)^{2}'
assert latex(reduced_totient(n)) == r'\lambda\left(n\right)'
assert latex(reduced_totient(n) ** 2) == \
r'\left(\lambda\left(n\right)\right)^{2}'
assert latex(divisor_sigma(x)) == r"\sigma\left(x\right)"
assert latex(divisor_sigma(x)**2) == r"\sigma^{2}\left(x\right)"
assert latex(divisor_sigma(x, y)) == r"\sigma_y\left(x\right)"
assert latex(divisor_sigma(x, y)**2) == r"\sigma^{2}_y\left(x\right)"
assert latex(udivisor_sigma(x)) == r"\sigma^*\left(x\right)"
assert latex(udivisor_sigma(x)**2) == r"\sigma^*^{2}\left(x\right)"
assert latex(udivisor_sigma(x, y)) == r"\sigma^*_y\left(x\right)"
assert latex(udivisor_sigma(x, y)**2) == r"\sigma^*^{2}_y\left(x\right)"
assert latex(primenu(n)) == r'\nu\left(n\right)'
assert latex(primenu(n) ** 2) == r'\left(\nu\left(n\right)\right)^{2}'
assert latex(primeomega(n)) == r'\Omega\left(n\right)'
assert latex(primeomega(n) ** 2) == \
r'\left(\Omega\left(n\right)\right)^{2}'
assert latex(LambertW(n)) == r'W\left(n\right)'
assert latex(LambertW(n, -1)) == r'W_{-1}\left(n\right)'
assert latex(LambertW(n, k)) == r'W_{k}\left(n\right)'
assert latex(Mod(x, 7)) == r'x\bmod{7}'
assert latex(Mod(x + 1, 7)) == r'\left(x + 1\right)\bmod{7}'
assert latex(Mod(2 * x, 7)) == r'2 x\bmod{7}'
assert latex(Mod(x, 7) + 1) == r'\left(x\bmod{7}\right) + 1'
assert latex(2 * Mod(x, 7)) == r'2 \left(x\bmod{7}\right)'
# some unknown function name should get rendered with \operatorname
fjlkd = Function('fjlkd')
assert latex(fjlkd(x)) == r'\operatorname{fjlkd}{\left(x \right)}'
# even when it is referred to without an argument
assert latex(fjlkd) == r'\operatorname{fjlkd}'
# test that notation passes to subclasses of the same name only
def test_function_subclass_different_name():
class mygamma(gamma):
pass
assert latex(mygamma) == r"\operatorname{mygamma}"
assert latex(mygamma(x)) == r"\operatorname{mygamma}{\left(x \right)}"
def test_hyper_printing():
from sympy import pi
from sympy.abc import x, z
assert latex(meijerg(Tuple(pi, pi, x), Tuple(1),
(0, 1), Tuple(1, 2, 3/pi), z)) == \
r'{G_{4, 5}^{2, 3}\left(\begin{matrix} \pi, \pi, x & 1 \\0, 1 & 1, 2, '\
r'\frac{3}{\pi} \end{matrix} \middle| {z} \right)}'
assert latex(meijerg(Tuple(), Tuple(1), (0,), Tuple(), z)) == \
r'{G_{1, 1}^{1, 0}\left(\begin{matrix} & 1 \\0 & \end{matrix} \middle| {z} \right)}'
assert latex(hyper((x, 2), (3,), z)) == \
r'{{}_{2}F_{1}\left(\begin{matrix} x, 2 ' \
r'\\ 3 \end{matrix}\middle| {z} \right)}'
assert latex(hyper(Tuple(), Tuple(1), z)) == \
r'{{}_{0}F_{1}\left(\begin{matrix} ' \
r'\\ 1 \end{matrix}\middle| {z} \right)}'
def test_latex_bessel():
from sympy.functions.special.bessel import (besselj, bessely, besseli,
besselk, hankel1, hankel2,
jn, yn, hn1, hn2)
from sympy.abc import z
assert latex(besselj(n, z**2)**k) == r'J^{k}_{n}\left(z^{2}\right)'
assert latex(bessely(n, z)) == r'Y_{n}\left(z\right)'
assert latex(besseli(n, z)) == r'I_{n}\left(z\right)'
assert latex(besselk(n, z)) == r'K_{n}\left(z\right)'
assert latex(hankel1(n, z**2)**2) == \
r'\left(H^{(1)}_{n}\left(z^{2}\right)\right)^{2}'
assert latex(hankel2(n, z)) == r'H^{(2)}_{n}\left(z\right)'
assert latex(jn(n, z)) == r'j_{n}\left(z\right)'
assert latex(yn(n, z)) == r'y_{n}\left(z\right)'
assert latex(hn1(n, z)) == r'h^{(1)}_{n}\left(z\right)'
assert latex(hn2(n, z)) == r'h^{(2)}_{n}\left(z\right)'
def test_latex_fresnel():
from sympy.functions.special.error_functions import (fresnels, fresnelc)
from sympy.abc import z
assert latex(fresnels(z)) == r'S\left(z\right)'
assert latex(fresnelc(z)) == r'C\left(z\right)'
assert latex(fresnels(z)**2) == r'S^{2}\left(z\right)'
assert latex(fresnelc(z)**2) == r'C^{2}\left(z\right)'
def test_latex_brackets():
assert latex((-1)**x) == r"\left(-1\right)^{x}"
def test_latex_indexed():
Psi_symbol = Symbol('Psi_0', complex=True, real=False)
Psi_indexed = IndexedBase(Symbol('Psi', complex=True, real=False))
symbol_latex = latex(Psi_symbol * conjugate(Psi_symbol))
indexed_latex = latex(Psi_indexed[0] * conjugate(Psi_indexed[0]))
# \\overline{{\\Psi}_{0}} {\\Psi}_{0} vs. \\Psi_{0} \\overline{\\Psi_{0}}
assert symbol_latex == '\\Psi_{0} \\overline{\\Psi_{0}}'
assert indexed_latex == '\\overline{{\\Psi}_{0}} {\\Psi}_{0}'
# Symbol('gamma') gives r'\gamma'
assert latex(Indexed('x1', Symbol('i'))) == '{x_{1}}_{i}'
assert latex(IndexedBase('gamma')) == r'\gamma'
assert latex(IndexedBase('a b')) == 'a b'
assert latex(IndexedBase('a_b')) == 'a_{b}'
def test_latex_derivatives():
# regular "d" for ordinary derivatives
assert latex(diff(x**3, x, evaluate=False)) == \
r"\frac{d}{d x} x^{3}"
assert latex(diff(sin(x) + x**2, x, evaluate=False)) == \
r"\frac{d}{d x} \left(x^{2} + \sin{\left(x \right)}\right)"
assert latex(diff(diff(sin(x) + x**2, x, evaluate=False), evaluate=False))\
== \
r"\frac{d^{2}}{d x^{2}} \left(x^{2} + \sin{\left(x \right)}\right)"
assert latex(diff(diff(diff(sin(x) + x**2, x, evaluate=False), evaluate=False), evaluate=False)) == \
r"\frac{d^{3}}{d x^{3}} \left(x^{2} + \sin{\left(x \right)}\right)"
# \partial for partial derivatives
assert latex(diff(sin(x * y), x, evaluate=False)) == \
r"\frac{\partial}{\partial x} \sin{\left(x y \right)}"
assert latex(diff(sin(x * y) + x**2, x, evaluate=False)) == \
r"\frac{\partial}{\partial x} \left(x^{2} + \sin{\left(x y \right)}\right)"
assert latex(diff(diff(sin(x*y) + x**2, x, evaluate=False), x, evaluate=False)) == \
r"\frac{\partial^{2}}{\partial x^{2}} \left(x^{2} + \sin{\left(x y \right)}\right)"
assert latex(diff(diff(diff(sin(x*y) + x**2, x, evaluate=False), x, evaluate=False), x, evaluate=False)) == \
r"\frac{\partial^{3}}{\partial x^{3}} \left(x^{2} + \sin{\left(x y \right)}\right)"
# mixed partial derivatives
f = Function("f")
assert latex(diff(diff(f(x, y), x, evaluate=False), y, evaluate=False)) == \
r"\frac{\partial^{2}}{\partial y\partial x} " + latex(f(x, y))
assert latex(diff(diff(diff(f(x, y), x, evaluate=False), x, evaluate=False), y, evaluate=False)) == \
r"\frac{\partial^{3}}{\partial y\partial x^{2}} " + latex(f(x, y))
# for negative nested Derivative
assert latex(diff(-diff(y**2,x,evaluate=False),x,evaluate=False)) == r'\frac{d}{d x} \left(- \frac{d}{d x} y^{2}\right)'
assert latex(diff(diff(-diff(diff(y,x,evaluate=False),x,evaluate=False),x,evaluate=False),x,evaluate=False)) == \
r'\frac{d^{2}}{d x^{2}} \left(- \frac{d^{2}}{d x^{2}} y\right)'
# use ordinary d when one of the variables has been integrated out
assert latex(diff(Integral(exp(-x*y), (x, 0, oo)), y, evaluate=False)) == \
r"\frac{d}{d y} \int\limits_{0}^{\infty} e^{- x y}\, dx"
# Derivative wrapped in power:
assert latex(diff(x, x, evaluate=False)**2) == \
r"\left(\frac{d}{d x} x\right)^{2}"
assert latex(diff(f(x), x)**2) == \
r"\left(\frac{d}{d x} f{\left(x \right)}\right)^{2}"
assert latex(diff(f(x), (x, n))) == \
r"\frac{d^{n}}{d x^{n}} f{\left(x \right)}"
x1 = Symbol('x1')
x2 = Symbol('x2')
assert latex(diff(f(x1, x2), x1)) == r'\frac{\partial}{\partial x_{1}} f{\left(x_{1},x_{2} \right)}'
n1 = Symbol('n1')
assert latex(diff(f(x), (x, n1))) == r'\frac{d^{n_{1}}}{d x^{n_{1}}} f{\left(x \right)}'
n2 = Symbol('n2')
assert latex(diff(f(x), (x, Max(n1, n2)))) == \
r'\frac{d^{\max\left(n_{1}, n_{2}\right)}}{d x^{\max\left(n_{1}, n_{2}\right)}} f{\left(x \right)}'
def test_latex_subs():
assert latex(Subs(x*y, (
x, y), (1, 2))) == r'\left. x y \right|_{\substack{ x=1\\ y=2 }}'
def test_latex_integrals():
assert latex(Integral(log(x), x)) == r"\int \log{\left(x \right)}\, dx"
assert latex(Integral(x**2, (x, 0, 1))) == \
r"\int\limits_{0}^{1} x^{2}\, dx"
assert latex(Integral(x**2, (x, 10, 20))) == \
r"\int\limits_{10}^{20} x^{2}\, dx"
assert latex(Integral(y*x**2, (x, 0, 1), y)) == \
r"\int\int\limits_{0}^{1} x^{2} y\, dx\, dy"
assert latex(Integral(y*x**2, (x, 0, 1), y), mode='equation*') == \
r"\begin{equation*}\int\int\limits_{0}^{1} x^{2} y\, dx\, dy\end{equation*}"
assert latex(Integral(y*x**2, (x, 0, 1), y), mode='equation*', itex=True) \
== r"$$\int\int_{0}^{1} x^{2} y\, dx\, dy$$"
assert latex(Integral(x, (x, 0))) == r"\int\limits^{0} x\, dx"
assert latex(Integral(x*y, x, y)) == r"\iint x y\, dx\, dy"
assert latex(Integral(x*y*z, x, y, z)) == r"\iiint x y z\, dx\, dy\, dz"
assert latex(Integral(x*y*z*t, x, y, z, t)) == \
r"\iiiint t x y z\, dx\, dy\, dz\, dt"
assert latex(Integral(x, x, x, x, x, x, x)) == \
r"\int\int\int\int\int\int x\, dx\, dx\, dx\, dx\, dx\, dx"
assert latex(Integral(x, x, y, (z, 0, 1))) == \
r"\int\limits_{0}^{1}\int\int x\, dx\, dy\, dz"
# for negative nested Integral
assert latex(Integral(-Integral(y**2,x),x)) == \
r'\int \left(- \int y^{2}\, dx\right)\, dx'
assert latex(Integral(-Integral(-Integral(y,x),x),x)) == \
r'\int \left(- \int \left(- \int y\, dx\right)\, dx\right)\, dx'
# fix issue #10806
assert latex(Integral(z, z)**2) == r"\left(\int z\, dz\right)^{2}"
assert latex(Integral(x + z, z)) == r"\int \left(x + z\right)\, dz"
assert latex(Integral(x+z/2, z)) == \
r"\int \left(x + \frac{z}{2}\right)\, dz"
assert latex(Integral(x**y, z)) == r"\int x^{y}\, dz"
def test_latex_sets():
for s in (frozenset, set):
assert latex(s([x*y, x**2])) == r"\left\{x^{2}, x y\right\}"
assert latex(s(range(1, 6))) == r"\left\{1, 2, 3, 4, 5\right\}"
assert latex(s(range(1, 13))) == \
r"\left\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12\right\}"
s = FiniteSet
assert latex(s(*[x*y, x**2])) == r"\left\{x^{2}, x y\right\}"
assert latex(s(*range(1, 6))) == r"\left\{1, 2, 3, 4, 5\right\}"
assert latex(s(*range(1, 13))) == \
r"\left\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12\right\}"
def test_latex_SetExpr():
iv = Interval(1, 3)
se = SetExpr(iv)
assert latex(se) == r"SetExpr\left(\left[1, 3\right]\right)"
def test_latex_Range():
assert latex(Range(1, 51)) == r'\left\{1, 2, \ldots, 50\right\}'
assert latex(Range(1, 4)) == r'\left\{1, 2, 3\right\}'
assert latex(Range(0, 3, 1)) == r'\left\{0, 1, 2\right\}'
assert latex(Range(0, 30, 1)) == r'\left\{0, 1, \ldots, 29\right\}'
assert latex(Range(30, 1, -1)) == r'\left\{30, 29, \ldots, 2\right\}'
assert latex(Range(0, oo, 2)) == r'\left\{0, 2, \ldots\right\}'
assert latex(Range(oo, -2, -2)) == r'\left\{\ldots, 2, 0\right\}'
assert latex(Range(-2, -oo, -1)) == r'\left\{-2, -3, \ldots\right\}'
assert latex(Range(-oo, oo)) == r'\left\{\ldots, -1, 0, 1, \ldots\right\}'
assert latex(Range(oo, -oo, -1)) == \
r'\left\{\ldots, 1, 0, -1, \ldots\right\}'
a, b, c = symbols('a:c')
assert latex(Range(a, b, c)) == r'Range\left(a, b, c\right)'
assert latex(Range(a, 10, 1)) == r'Range\left(a, 10, 1\right)'
assert latex(Range(0, b, 1)) == r'Range\left(0, b, 1\right)'
assert latex(Range(0, 10, c)) == r'Range\left(0, 10, c\right)'
def test_latex_sequences():
s1 = SeqFormula(a**2, (0, oo))
s2 = SeqPer((1, 2))
latex_str = r'\left[0, 1, 4, 9, \ldots\right]'
assert latex(s1) == latex_str
latex_str = r'\left[1, 2, 1, 2, \ldots\right]'
assert latex(s2) == latex_str
s3 = SeqFormula(a**2, (0, 2))
s4 = SeqPer((1, 2), (0, 2))
latex_str = r'\left[0, 1, 4\right]'
assert latex(s3) == latex_str
latex_str = r'\left[1, 2, 1\right]'
assert latex(s4) == latex_str
s5 = SeqFormula(a**2, (-oo, 0))
s6 = SeqPer((1, 2), (-oo, 0))
latex_str = r'\left[\ldots, 9, 4, 1, 0\right]'
assert latex(s5) == latex_str
latex_str = r'\left[\ldots, 2, 1, 2, 1\right]'
assert latex(s6) == latex_str
latex_str = r'\left[1, 3, 5, 11, \ldots\right]'
assert latex(SeqAdd(s1, s2)) == latex_str
latex_str = r'\left[1, 3, 5\right]'
assert latex(SeqAdd(s3, s4)) == latex_str
latex_str = r'\left[\ldots, 11, 5, 3, 1\right]'
assert latex(SeqAdd(s5, s6)) == latex_str
latex_str = r'\left[0, 2, 4, 18, \ldots\right]'
assert latex(SeqMul(s1, s2)) == latex_str
latex_str = r'\left[0, 2, 4\right]'
assert latex(SeqMul(s3, s4)) == latex_str
latex_str = r'\left[\ldots, 18, 4, 2, 0\right]'
assert latex(SeqMul(s5, s6)) == latex_str
# Sequences with symbolic limits, issue 12629
s7 = SeqFormula(a**2, (a, 0, x))
latex_str = r'\left\{a^{2}\right\}_{a=0}^{x}'
assert latex(s7) == latex_str
b = Symbol('b')
s8 = SeqFormula(b*a**2, (a, 0, 2))
latex_str = r'\left[0, b, 4 b\right]'
assert latex(s8) == latex_str
def test_latex_FourierSeries():
latex_str = \
r'2 \sin{\left(x \right)} - \sin{\left(2 x \right)} + \frac{2 \sin{\left(3 x \right)}}{3} + \ldots'
assert latex(fourier_series(x, (x, -pi, pi))) == latex_str
def test_latex_FormalPowerSeries():
latex_str = r'\sum_{k=1}^{\infty} - \frac{\left(-1\right)^{- k} x^{k}}{k}'
assert latex(fps(log(1 + x))) == latex_str
def test_latex_intervals():
a = Symbol('a', real=True)
assert latex(Interval(0, 0)) == r"\left\{0\right\}"
assert latex(Interval(0, a)) == r"\left[0, a\right]"
assert latex(Interval(0, a, False, False)) == r"\left[0, a\right]"
assert latex(Interval(0, a, True, False)) == r"\left(0, a\right]"
assert latex(Interval(0, a, False, True)) == r"\left[0, a\right)"
assert latex(Interval(0, a, True, True)) == r"\left(0, a\right)"
def test_latex_AccumuBounds():
a = Symbol('a', real=True)
assert latex(AccumBounds(0, 1)) == r"\left\langle 0, 1\right\rangle"
assert latex(AccumBounds(0, a)) == r"\left\langle 0, a\right\rangle"
assert latex(AccumBounds(a + 1, a + 2)) == \
r"\left\langle a + 1, a + 2\right\rangle"
def test_latex_emptyset():
assert latex(S.EmptySet) == r"\emptyset"
def test_latex_universalset():
assert latex(S.UniversalSet) == r"\mathbb{U}"
def test_latex_commutator():
A = Operator('A')
B = Operator('B')
comm = Commutator(B, A)
assert latex(comm.doit()) == r"- (A B - B A)"
def test_latex_union():
assert latex(Union(Interval(0, 1), Interval(2, 3))) == \
r"\left[0, 1\right] \cup \left[2, 3\right]"
assert latex(Union(Interval(1, 1), Interval(2, 2), Interval(3, 4))) == \
r"\left\{1, 2\right\} \cup \left[3, 4\right]"
def test_latex_intersection():
assert latex(Intersection(Interval(0, 1), Interval(x, y))) == \
r"\left[0, 1\right] \cap \left[x, y\right]"
def test_latex_symmetric_difference():
assert latex(SymmetricDifference(Interval(2, 5), Interval(4, 7),
evaluate=False)) == \
r'\left[2, 5\right] \triangle \left[4, 7\right]'
def test_latex_Complement():
assert latex(Complement(S.Reals, S.Naturals)) == \
r"\mathbb{R} \setminus \mathbb{N}"
def test_latex_productset():
line = Interval(0, 1)
bigline = Interval(0, 10)
fset = FiniteSet(1, 2, 3)
assert latex(line**2) == r"%s^{2}" % latex(line)
assert latex(line**10) == r"%s^{10}" % latex(line)
assert latex((line * bigline * fset).flatten()) == r"%s \times %s \times %s" % (
latex(line), latex(bigline), latex(fset))
def test_set_operators_parenthesis():
a, b, c, d = symbols('a:d')
A = FiniteSet(a)
B = FiniteSet(b)
C = FiniteSet(c)
D = FiniteSet(d)
U1 = Union(A, B, evaluate=False)
U2 = Union(C, D, evaluate=False)
I1 = Intersection(A, B, evaluate=False)
I2 = Intersection(C, D, evaluate=False)
C1 = Complement(A, B, evaluate=False)
C2 = Complement(C, D, evaluate=False)
D1 = SymmetricDifference(A, B, evaluate=False)
D2 = SymmetricDifference(C, D, evaluate=False)
# XXX ProductSet does not support evaluate keyword
P1 = ProductSet(A, B)
P2 = ProductSet(C, D)
assert latex(Intersection(A, U2, evaluate=False)) == \
'\\left\\{a\\right\\} \\cap ' \
'\\left(\\left\\{c\\right\\} \\cup \\left\\{d\\right\\}\\right)'
assert latex(Intersection(U1, U2, evaluate=False)) == \
'\\left(\\left\\{a\\right\\} \\cup \\left\\{b\\right\\}\\right) ' \
'\\cap \\left(\\left\\{c\\right\\} \\cup \\left\\{d\\right\\}\\right)'
assert latex(Intersection(C1, C2, evaluate=False)) == \
'\\left(\\left\\{a\\right\\} \\setminus ' \
'\\left\\{b\\right\\}\\right) \\cap \\left(\\left\\{c\\right\\} ' \
'\\setminus \\left\\{d\\right\\}\\right)'
assert latex(Intersection(D1, D2, evaluate=False)) == \
'\\left(\\left\\{a\\right\\} \\triangle ' \
'\\left\\{b\\right\\}\\right) \\cap \\left(\\left\\{c\\right\\} ' \
'\\triangle \\left\\{d\\right\\}\\right)'
assert latex(Intersection(P1, P2, evaluate=False)) == \
'\\left(\\left\\{a\\right\\} \\times \\left\\{b\\right\\}\\right) ' \
'\\cap \\left(\\left\\{c\\right\\} \\times ' \
'\\left\\{d\\right\\}\\right)'
assert latex(Union(A, I2, evaluate=False)) == \
'\\left\\{a\\right\\} \\cup ' \
'\\left(\\left\\{c\\right\\} \\cap \\left\\{d\\right\\}\\right)'
assert latex(Union(I1, I2, evaluate=False)) == \
'\\left(\\left\\{a\\right\\} \\cap ''\\left\\{b\\right\\}\\right) ' \
'\\cup \\left(\\left\\{c\\right\\} \\cap \\left\\{d\\right\\}\\right)'
assert latex(Union(C1, C2, evaluate=False)) == \
'\\left(\\left\\{a\\right\\} \\setminus ' \
'\\left\\{b\\right\\}\\right) \\cup \\left(\\left\\{c\\right\\} ' \
'\\setminus \\left\\{d\\right\\}\\right)'
assert latex(Union(D1, D2, evaluate=False)) == \
'\\left(\\left\\{a\\right\\} \\triangle ' \
'\\left\\{b\\right\\}\\right) \\cup \\left(\\left\\{c\\right\\} ' \
'\\triangle \\left\\{d\\right\\}\\right)'
assert latex(Union(P1, P2, evaluate=False)) == \
'\\left(\\left\\{a\\right\\} \\times \\left\\{b\\right\\}\\right) ' \
'\\cup \\left(\\left\\{c\\right\\} \\times ' \
'\\left\\{d\\right\\}\\right)'
assert latex(Complement(A, C2, evaluate=False)) == \
'\\left\\{a\\right\\} \\setminus \\left(\\left\\{c\\right\\} ' \
'\\setminus \\left\\{d\\right\\}\\right)'
assert latex(Complement(U1, U2, evaluate=False)) == \
'\\left(\\left\\{a\\right\\} \\cup \\left\\{b\\right\\}\\right) ' \
'\\setminus \\left(\\left\\{c\\right\\} \\cup ' \
'\\left\\{d\\right\\}\\right)'
assert latex(Complement(I1, I2, evaluate=False)) == \
'\\left(\\left\\{a\\right\\} \\cap \\left\\{b\\right\\}\\right) ' \
'\\setminus \\left(\\left\\{c\\right\\} \\cap ' \
'\\left\\{d\\right\\}\\right)'
assert latex(Complement(D1, D2, evaluate=False)) == \
'\\left(\\left\\{a\\right\\} \\triangle ' \
'\\left\\{b\\right\\}\\right) \\setminus ' \
'\\left(\\left\\{c\\right\\} \\triangle \\left\\{d\\right\\}\\right)'
assert latex(Complement(P1, P2, evaluate=False)) == \
'\\left(\\left\\{a\\right\\} \\times \\left\\{b\\right\\}\\right) '\
'\\setminus \\left(\\left\\{c\\right\\} \\times '\
'\\left\\{d\\right\\}\\right)'
assert latex(SymmetricDifference(A, D2, evaluate=False)) == \
'\\left\\{a\\right\\} \\triangle \\left(\\left\\{c\\right\\} ' \
'\\triangle \\left\\{d\\right\\}\\right)'
assert latex(SymmetricDifference(U1, U2, evaluate=False)) == \
'\\left(\\left\\{a\\right\\} \\cup \\left\\{b\\right\\}\\right) ' \
'\\triangle \\left(\\left\\{c\\right\\} \\cup ' \
'\\left\\{d\\right\\}\\right)'
assert latex(SymmetricDifference(I1, I2, evaluate=False)) == \
'\\left(\\left\\{a\\right\\} \\cap \\left\\{b\\right\\}\\right) ' \
'\\triangle \\left(\\left\\{c\\right\\} \\cap ' \
'\\left\\{d\\right\\}\\right)'
assert latex(SymmetricDifference(C1, C2, evaluate=False)) == \
'\\left(\\left\\{a\\right\\} \\setminus ' \
'\\left\\{b\\right\\}\\right) \\triangle ' \
'\\left(\\left\\{c\\right\\} \\setminus \\left\\{d\\right\\}\\right)'
assert latex(SymmetricDifference(P1, P2, evaluate=False)) == \
'\\left(\\left\\{a\\right\\} \\times \\left\\{b\\right\\}\\right) ' \
'\\triangle \\left(\\left\\{c\\right\\} \\times ' \
'\\left\\{d\\right\\}\\right)'
# XXX This can be incorrect since cartesian product is not associative
assert latex(ProductSet(A, P2).flatten()) == \
'\\left\\{a\\right\\} \\times \\left\\{c\\right\\} \\times ' \
'\\left\\{d\\right\\}'
assert latex(ProductSet(U1, U2)) == \
'\\left(\\left\\{a\\right\\} \\cup \\left\\{b\\right\\}\\right) ' \
'\\times \\left(\\left\\{c\\right\\} \\cup ' \
'\\left\\{d\\right\\}\\right)'
assert latex(ProductSet(I1, I2)) == \
'\\left(\\left\\{a\\right\\} \\cap \\left\\{b\\right\\}\\right) ' \
'\\times \\left(\\left\\{c\\right\\} \\cap ' \
'\\left\\{d\\right\\}\\right)'
assert latex(ProductSet(C1, C2)) == \
'\\left(\\left\\{a\\right\\} \\setminus ' \
'\\left\\{b\\right\\}\\right) \\times \\left(\\left\\{c\\right\\} ' \
'\\setminus \\left\\{d\\right\\}\\right)'
assert latex(ProductSet(D1, D2)) == \
'\\left(\\left\\{a\\right\\} \\triangle ' \
'\\left\\{b\\right\\}\\right) \\times \\left(\\left\\{c\\right\\} ' \
'\\triangle \\left\\{d\\right\\}\\right)'
def test_latex_Complexes():
assert latex(S.Complexes) == r"\mathbb{C}"
def test_latex_Naturals():
assert latex(S.Naturals) == r"\mathbb{N}"
def test_latex_Naturals0():
assert latex(S.Naturals0) == r"\mathbb{N}_0"
def test_latex_Integers():
assert latex(S.Integers) == r"\mathbb{Z}"
def test_latex_ImageSet():
x = Symbol('x')
assert latex(ImageSet(Lambda(x, x**2), S.Naturals)) == \
r"\left\{x^{2}\; |\; x \in \mathbb{N}\right\}"
y = Symbol('y')
imgset = ImageSet(Lambda((x, y), x + y), {1, 2, 3}, {3, 4})
assert latex(imgset) == \
r"\left\{x + y\; |\; x \in \left\{1, 2, 3\right\} , y \in \left\{3, 4\right\}\right\}"
imgset = ImageSet(Lambda(((x, y),), x + y), ProductSet({1, 2, 3}, {3, 4}))
assert latex(imgset) == \
r"\left\{x + y\; |\; \left( x, \ y\right) \in \left\{1, 2, 3\right\} \times \left\{3, 4\right\}\right\}"
def test_latex_ConditionSet():
x = Symbol('x')
assert latex(ConditionSet(x, Eq(x**2, 1), S.Reals)) == \
r"\left\{x \mid x \in \mathbb{R} \wedge x^{2} = 1 \right\}"
assert latex(ConditionSet(x, Eq(x**2, 1), S.UniversalSet)) == \
r"\left\{x \mid x^{2} = 1 \right\}"
def test_latex_ComplexRegion():
assert latex(ComplexRegion(Interval(3, 5)*Interval(4, 6))) == \
r"\left\{x + y i\; |\; x, y \in \left[3, 5\right] \times \left[4, 6\right] \right\}"
assert latex(ComplexRegion(Interval(0, 1)*Interval(0, 2*pi), polar=True)) == \
r"\left\{r \left(i \sin{\left(\theta \right)} + \cos{\left(\theta "\
r"\right)}\right)\; |\; r, \theta \in \left[0, 1\right] \times \left[0, 2 \pi\right) \right\}"
def test_latex_Contains():
x = Symbol('x')
assert latex(Contains(x, S.Naturals)) == r"x \in \mathbb{N}"
def test_latex_sum():
assert latex(Sum(x*y**2, (x, -2, 2), (y, -5, 5))) == \
r"\sum_{\substack{-2 \leq x \leq 2\\-5 \leq y \leq 5}} x y^{2}"
assert latex(Sum(x**2, (x, -2, 2))) == \
r"\sum_{x=-2}^{2} x^{2}"
assert latex(Sum(x**2 + y, (x, -2, 2))) == \
r"\sum_{x=-2}^{2} \left(x^{2} + y\right)"
assert latex(Sum(x**2 + y, (x, -2, 2))**2) == \
r"\left(\sum_{x=-2}^{2} \left(x^{2} + y\right)\right)^{2}"
def test_latex_product():
assert latex(Product(x*y**2, (x, -2, 2), (y, -5, 5))) == \
r"\prod_{\substack{-2 \leq x \leq 2\\-5 \leq y \leq 5}} x y^{2}"
assert latex(Product(x**2, (x, -2, 2))) == \
r"\prod_{x=-2}^{2} x^{2}"
assert latex(Product(x**2 + y, (x, -2, 2))) == \
r"\prod_{x=-2}^{2} \left(x^{2} + y\right)"
assert latex(Product(x, (x, -2, 2))**2) == \
r"\left(\prod_{x=-2}^{2} x\right)^{2}"
def test_latex_limits():
assert latex(Limit(x, x, oo)) == r"\lim_{x \to \infty} x"
# issue 8175
f = Function('f')
assert latex(Limit(f(x), x, 0)) == r"\lim_{x \to 0^+} f{\left(x \right)}"
assert latex(Limit(f(x), x, 0, "-")) == \
r"\lim_{x \to 0^-} f{\left(x \right)}"
# issue #10806
assert latex(Limit(f(x), x, 0)**2) == \
r"\left(\lim_{x \to 0^+} f{\left(x \right)}\right)^{2}"
# bi-directional limit
assert latex(Limit(f(x), x, 0, dir='+-')) == \
r"\lim_{x \to 0} f{\left(x \right)}"
def test_latex_log():
assert latex(log(x)) == r"\log{\left(x \right)}"
assert latex(ln(x)) == r"\log{\left(x \right)}"
assert latex(log(x), ln_notation=True) == r"\ln{\left(x \right)}"
assert latex(log(x)+log(y)) == \
r"\log{\left(x \right)} + \log{\left(y \right)}"
assert latex(log(x)+log(y), ln_notation=True) == \
r"\ln{\left(x \right)} + \ln{\left(y \right)}"
assert latex(pow(log(x), x)) == r"\log{\left(x \right)}^{x}"
assert latex(pow(log(x), x), ln_notation=True) == \
r"\ln{\left(x \right)}^{x}"
def test_issue_3568():
beta = Symbol(r'\beta')
y = beta + x
assert latex(y) in [r'\beta + x', r'x + \beta']
beta = Symbol(r'beta')
y = beta + x
assert latex(y) in [r'\beta + x', r'x + \beta']
def test_latex():
assert latex((2*tau)**Rational(7, 2)) == "8 \\sqrt{2} \\tau^{\\frac{7}{2}}"
assert latex((2*mu)**Rational(7, 2), mode='equation*') == \
"\\begin{equation*}8 \\sqrt{2} \\mu^{\\frac{7}{2}}\\end{equation*}"
assert latex((2*mu)**Rational(7, 2), mode='equation', itex=True) == \
"$$8 \\sqrt{2} \\mu^{\\frac{7}{2}}$$"
assert latex([2/x, y]) == r"\left[ \frac{2}{x}, \ y\right]"
def test_latex_dict():
d = {Rational(1): 1, x**2: 2, x: 3, x**3: 4}
assert latex(d) == \
r'\left\{ 1 : 1, \ x : 3, \ x^{2} : 2, \ x^{3} : 4\right\}'
D = Dict(d)
assert latex(D) == \
r'\left\{ 1 : 1, \ x : 3, \ x^{2} : 2, \ x^{3} : 4\right\}'
def test_latex_list():
ll = [Symbol('omega1'), Symbol('a'), Symbol('alpha')]
assert latex(ll) == r'\left[ \omega_{1}, \ a, \ \alpha\right]'
def test_latex_rational():
# tests issue 3973
assert latex(-Rational(1, 2)) == "- \\frac{1}{2}"
assert latex(Rational(-1, 2)) == "- \\frac{1}{2}"
assert latex(Rational(1, -2)) == "- \\frac{1}{2}"
assert latex(-Rational(-1, 2)) == "\\frac{1}{2}"
assert latex(-Rational(1, 2)*x) == "- \\frac{x}{2}"
assert latex(-Rational(1, 2)*x + Rational(-2, 3)*y) == \
"- \\frac{x}{2} - \\frac{2 y}{3}"
def test_latex_inverse():
# tests issue 4129
assert latex(1/x) == "\\frac{1}{x}"
assert latex(1/(x + y)) == "\\frac{1}{x + y}"
def test_latex_DiracDelta():
assert latex(DiracDelta(x)) == r"\delta\left(x\right)"
assert latex(DiracDelta(x)**2) == r"\left(\delta\left(x\right)\right)^{2}"
assert latex(DiracDelta(x, 0)) == r"\delta\left(x\right)"
assert latex(DiracDelta(x, 5)) == \
r"\delta^{\left( 5 \right)}\left( x \right)"
assert latex(DiracDelta(x, 5)**2) == \
r"\left(\delta^{\left( 5 \right)}\left( x \right)\right)^{2}"
def test_latex_Heaviside():
assert latex(Heaviside(x)) == r"\theta\left(x\right)"
assert latex(Heaviside(x)**2) == r"\left(\theta\left(x\right)\right)^{2}"
def test_latex_KroneckerDelta():
assert latex(KroneckerDelta(x, y)) == r"\delta_{x y}"
assert latex(KroneckerDelta(x, y + 1)) == r"\delta_{x, y + 1}"
# issue 6578
assert latex(KroneckerDelta(x + 1, y)) == r"\delta_{y, x + 1}"
assert latex(Pow(KroneckerDelta(x, y), 2, evaluate=False)) == \
r"\left(\delta_{x y}\right)^{2}"
def test_latex_LeviCivita():
assert latex(LeviCivita(x, y, z)) == r"\varepsilon_{x y z}"
assert latex(LeviCivita(x, y, z)**2) == \
r"\left(\varepsilon_{x y z}\right)^{2}"
assert latex(LeviCivita(x, y, z + 1)) == r"\varepsilon_{x, y, z + 1}"
assert latex(LeviCivita(x, y + 1, z)) == r"\varepsilon_{x, y + 1, z}"
assert latex(LeviCivita(x + 1, y, z)) == r"\varepsilon_{x + 1, y, z}"
def test_mode():
expr = x + y
assert latex(expr) == 'x + y'
assert latex(expr, mode='plain') == 'x + y'
assert latex(expr, mode='inline') == '$x + y$'
assert latex(
expr, mode='equation*') == '\\begin{equation*}x + y\\end{equation*}'
assert latex(
expr, mode='equation') == '\\begin{equation}x + y\\end{equation}'
raises(ValueError, lambda: latex(expr, mode='foo'))
def test_latex_mathieu():
assert latex(mathieuc(x, y, z)) == r"C\left(x, y, z\right)"
assert latex(mathieus(x, y, z)) == r"S\left(x, y, z\right)"
assert latex(mathieuc(x, y, z)**2) == r"C\left(x, y, z\right)^{2}"
assert latex(mathieus(x, y, z)**2) == r"S\left(x, y, z\right)^{2}"
assert latex(mathieucprime(x, y, z)) == r"C^{\prime}\left(x, y, z\right)"
assert latex(mathieusprime(x, y, z)) == r"S^{\prime}\left(x, y, z\right)"
assert latex(mathieucprime(x, y, z)**2) == r"C^{\prime}\left(x, y, z\right)^{2}"
assert latex(mathieusprime(x, y, z)**2) == r"S^{\prime}\left(x, y, z\right)^{2}"
def test_latex_Piecewise():
p = Piecewise((x, x < 1), (x**2, True))
assert latex(p) == "\\begin{cases} x & \\text{for}\\: x < 1 \\\\x^{2} &" \
" \\text{otherwise} \\end{cases}"
assert latex(p, itex=True) == \
"\\begin{cases} x & \\text{for}\\: x \\lt 1 \\\\x^{2} &" \
" \\text{otherwise} \\end{cases}"
p = Piecewise((x, x < 0), (0, x >= 0))
assert latex(p) == '\\begin{cases} x & \\text{for}\\: x < 0 \\\\0 &' \
' \\text{otherwise} \\end{cases}'
A, B = symbols("A B", commutative=False)
p = Piecewise((A**2, Eq(A, B)), (A*B, True))
s = r"\begin{cases} A^{2} & \text{for}\: A = B \\A B & \text{otherwise} \end{cases}"
assert latex(p) == s
assert latex(A*p) == r"A \left(%s\right)" % s
assert latex(p*A) == r"\left(%s\right) A" % s
assert latex(Piecewise((x, x < 1), (x**2, x < 2))) == \
'\\begin{cases} x & ' \
'\\text{for}\\: x < 1 \\\\x^{2} & \\text{for}\\: x < 2 \\end{cases}'
def test_latex_Matrix():
M = Matrix([[1 + x, y], [y, x - 1]])
assert latex(M) == \
r'\left[\begin{matrix}x + 1 & y\\y & x - 1\end{matrix}\right]'
assert latex(M, mode='inline') == \
r'$\left[\begin{smallmatrix}x + 1 & y\\' \
r'y & x - 1\end{smallmatrix}\right]$'
assert latex(M, mat_str='array') == \
r'\left[\begin{array}{cc}x + 1 & y\\y & x - 1\end{array}\right]'
assert latex(M, mat_str='bmatrix') == \
r'\left[\begin{bmatrix}x + 1 & y\\y & x - 1\end{bmatrix}\right]'
assert latex(M, mat_delim=None, mat_str='bmatrix') == \
r'\begin{bmatrix}x + 1 & y\\y & x - 1\end{bmatrix}'
M2 = Matrix(1, 11, range(11))
assert latex(M2) == \
r'\left[\begin{array}{ccccccccccc}' \
r'0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10\end{array}\right]'
def test_latex_matrix_with_functions():
t = symbols('t')
theta1 = symbols('theta1', cls=Function)
M = Matrix([[sin(theta1(t)), cos(theta1(t))],
[cos(theta1(t).diff(t)), sin(theta1(t).diff(t))]])
expected = (r'\left[\begin{matrix}\sin{\left('
r'\theta_{1}{\left(t \right)} \right)} & '
r'\cos{\left(\theta_{1}{\left(t \right)} \right)'
r'}\\\cos{\left(\frac{d}{d t} \theta_{1}{\left(t '
r'\right)} \right)} & \sin{\left(\frac{d}{d t} '
r'\theta_{1}{\left(t \right)} \right'
r')}\end{matrix}\right]')
assert latex(M) == expected
def test_latex_NDimArray():
x, y, z, w = symbols("x y z w")
for ArrayType in (ImmutableDenseNDimArray, ImmutableSparseNDimArray,
MutableDenseNDimArray, MutableSparseNDimArray):
# Basic: scalar array
M = ArrayType(x)
assert latex(M) == "x"
M = ArrayType([[1 / x, y], [z, w]])
M1 = ArrayType([1 / x, y, z])
M2 = tensorproduct(M1, M)
M3 = tensorproduct(M, M)
assert latex(M) == \
'\\left[\\begin{matrix}\\frac{1}{x} & y\\\\z & w\\end{matrix}\\right]'
assert latex(M1) == \
"\\left[\\begin{matrix}\\frac{1}{x} & y & z\\end{matrix}\\right]"
assert latex(M2) == \
r"\left[\begin{matrix}" \
r"\left[\begin{matrix}\frac{1}{x^{2}} & \frac{y}{x}\\\frac{z}{x} & \frac{w}{x}\end{matrix}\right] & " \
r"\left[\begin{matrix}\frac{y}{x} & y^{2}\\y z & w y\end{matrix}\right] & " \
r"\left[\begin{matrix}\frac{z}{x} & y z\\z^{2} & w z\end{matrix}\right]" \
r"\end{matrix}\right]"
assert latex(M3) == \
r"""\left[\begin{matrix}"""\
r"""\left[\begin{matrix}\frac{1}{x^{2}} & \frac{y}{x}\\\frac{z}{x} & \frac{w}{x}\end{matrix}\right] & """\
r"""\left[\begin{matrix}\frac{y}{x} & y^{2}\\y z & w y\end{matrix}\right]\\"""\
r"""\left[\begin{matrix}\frac{z}{x} & y z\\z^{2} & w z\end{matrix}\right] & """\
r"""\left[\begin{matrix}\frac{w}{x} & w y\\w z & w^{2}\end{matrix}\right]"""\
r"""\end{matrix}\right]"""
Mrow = ArrayType([[x, y, 1/z]])
Mcolumn = ArrayType([[x], [y], [1/z]])
Mcol2 = ArrayType([Mcolumn.tolist()])
assert latex(Mrow) == \
r"\left[\left[\begin{matrix}x & y & \frac{1}{z}\end{matrix}\right]\right]"
assert latex(Mcolumn) == \
r"\left[\begin{matrix}x\\y\\\frac{1}{z}\end{matrix}\right]"
assert latex(Mcol2) == \
r'\left[\begin{matrix}\left[\begin{matrix}x\\y\\\frac{1}{z}\end{matrix}\right]\end{matrix}\right]'
def test_latex_mul_symbol():
assert latex(4*4**x, mul_symbol='times') == "4 \\times 4^{x}"
assert latex(4*4**x, mul_symbol='dot') == "4 \\cdot 4^{x}"
assert latex(4*4**x, mul_symbol='ldot') == r"4 \,.\, 4^{x}"
assert latex(4*x, mul_symbol='times') == "4 \\times x"
assert latex(4*x, mul_symbol='dot') == "4 \\cdot x"
assert latex(4*x, mul_symbol='ldot') == r"4 \,.\, x"
def test_latex_issue_4381():
y = 4*4**log(2)
assert latex(y) == r'4 \cdot 4^{\log{\left(2 \right)}}'
assert latex(1/y) == r'\frac{1}{4 \cdot 4^{\log{\left(2 \right)}}}'
def test_latex_issue_4576():
assert latex(Symbol("beta_13_2")) == r"\beta_{13 2}"
assert latex(Symbol("beta_132_20")) == r"\beta_{132 20}"
assert latex(Symbol("beta_13")) == r"\beta_{13}"
assert latex(Symbol("x_a_b")) == r"x_{a b}"
assert latex(Symbol("x_1_2_3")) == r"x_{1 2 3}"
assert latex(Symbol("x_a_b1")) == r"x_{a b1}"
assert latex(Symbol("x_a_1")) == r"x_{a 1}"
assert latex(Symbol("x_1_a")) == r"x_{1 a}"
assert latex(Symbol("x_1^aa")) == r"x^{aa}_{1}"
assert latex(Symbol("x_1__aa")) == r"x^{aa}_{1}"
assert latex(Symbol("x_11^a")) == r"x^{a}_{11}"
assert latex(Symbol("x_11__a")) == r"x^{a}_{11}"
assert latex(Symbol("x_a_a_a_a")) == r"x_{a a a a}"
assert latex(Symbol("x_a_a^a^a")) == r"x^{a a}_{a a}"
assert latex(Symbol("x_a_a__a__a")) == r"x^{a a}_{a a}"
assert latex(Symbol("alpha_11")) == r"\alpha_{11}"
assert latex(Symbol("alpha_11_11")) == r"\alpha_{11 11}"
assert latex(Symbol("alpha_alpha")) == r"\alpha_{\alpha}"
assert latex(Symbol("alpha^aleph")) == r"\alpha^{\aleph}"
assert latex(Symbol("alpha__aleph")) == r"\alpha^{\aleph}"
def test_latex_pow_fraction():
x = Symbol('x')
# Testing exp
assert 'e^{-x}' in latex(exp(-x)/2).replace(' ', '') # Remove Whitespace
# Testing e^{-x} in case future changes alter behavior of muls or fracs
# In particular current output is \frac{1}{2}e^{- x} but perhaps this will
# change to \frac{e^{-x}}{2}
# Testing general, non-exp, power
assert '3^{-x}' in latex(3**-x/2).replace(' ', '')
def test_noncommutative():
A, B, C = symbols('A,B,C', commutative=False)
assert latex(A*B*C**-1) == "A B C^{-1}"
assert latex(C**-1*A*B) == "C^{-1} A B"
assert latex(A*C**-1*B) == "A C^{-1} B"
def test_latex_order():
expr = x**3 + x**2*y + y**4 + 3*x*y**3
assert latex(expr, order='lex') == "x^{3} + x^{2} y + 3 x y^{3} + y^{4}"
assert latex(
expr, order='rev-lex') == "y^{4} + 3 x y^{3} + x^{2} y + x^{3}"
assert latex(expr, order='none') == "x^{3} + y^{4} + y x^{2} + 3 x y^{3}"
def test_latex_Lambda():
assert latex(Lambda(x, x + 1)) == \
r"\left( x \mapsto x + 1 \right)"
assert latex(Lambda((x, y), x + 1)) == \
r"\left( \left( x, \ y\right) \mapsto x + 1 \right)"
assert latex(Lambda(x, x)) == \
r"\left( x \mapsto x \right)"
def test_latex_PolyElement():
Ruv, u, v = ring("u,v", ZZ)
Rxyz, x, y, z = ring("x,y,z", Ruv)
assert latex(x - x) == r"0"
assert latex(x - 1) == r"x - 1"
assert latex(x + 1) == r"x + 1"
assert latex((u**2 + 3*u*v + 1)*x**2*y + u + 1) == \
r"\left({u}^{2} + 3 u v + 1\right) {x}^{2} y + u + 1"
assert latex((u**2 + 3*u*v + 1)*x**2*y + (u + 1)*x) == \
r"\left({u}^{2} + 3 u v + 1\right) {x}^{2} y + \left(u + 1\right) x"
assert latex((u**2 + 3*u*v + 1)*x**2*y + (u + 1)*x + 1) == \
r"\left({u}^{2} + 3 u v + 1\right) {x}^{2} y + \left(u + 1\right) x + 1"
assert latex((-u**2 + 3*u*v - 1)*x**2*y - (u + 1)*x - 1) == \
r"-\left({u}^{2} - 3 u v + 1\right) {x}^{2} y - \left(u + 1\right) x - 1"
assert latex(-(v**2 + v + 1)*x + 3*u*v + 1) == \
r"-\left({v}^{2} + v + 1\right) x + 3 u v + 1"
assert latex(-(v**2 + v + 1)*x - 3*u*v + 1) == \
r"-\left({v}^{2} + v + 1\right) x - 3 u v + 1"
def test_latex_FracElement():
Fuv, u, v = field("u,v", ZZ)
Fxyzt, x, y, z, t = field("x,y,z,t", Fuv)
assert latex(x - x) == r"0"
assert latex(x - 1) == r"x - 1"
assert latex(x + 1) == r"x + 1"
assert latex(x/3) == r"\frac{x}{3}"
assert latex(x/z) == r"\frac{x}{z}"
assert latex(x*y/z) == r"\frac{x y}{z}"
assert latex(x/(z*t)) == r"\frac{x}{z t}"
assert latex(x*y/(z*t)) == r"\frac{x y}{z t}"
assert latex((x - 1)/y) == r"\frac{x - 1}{y}"
assert latex((x + 1)/y) == r"\frac{x + 1}{y}"
assert latex((-x - 1)/y) == r"\frac{-x - 1}{y}"
assert latex((x + 1)/(y*z)) == r"\frac{x + 1}{y z}"
assert latex(-y/(x + 1)) == r"\frac{-y}{x + 1}"
assert latex(y*z/(x + 1)) == r"\frac{y z}{x + 1}"
assert latex(((u + 1)*x*y + 1)/((v - 1)*z - 1)) == \
r"\frac{\left(u + 1\right) x y + 1}{\left(v - 1\right) z - 1}"
assert latex(((u + 1)*x*y + 1)/((v - 1)*z - t*u*v - 1)) == \
r"\frac{\left(u + 1\right) x y + 1}{\left(v - 1\right) z - u v t - 1}"
def test_latex_Poly():
assert latex(Poly(x**2 + 2 * x, x)) == \
r"\operatorname{Poly}{\left( x^{2} + 2 x, x, domain=\mathbb{Z} \right)}"
assert latex(Poly(x/y, x)) == \
r"\operatorname{Poly}{\left( \frac{1}{y} x, x, domain=\mathbb{Z}\left(y\right) \right)}"
assert latex(Poly(2.0*x + y)) == \
r"\operatorname{Poly}{\left( 2.0 x + 1.0 y, x, y, domain=\mathbb{R} \right)}"
def test_latex_Poly_order():
assert latex(Poly([a, 1, b, 2, c, 3], x)) == \
'\\operatorname{Poly}{\\left( a x^{5} + x^{4} + b x^{3} + 2 x^{2} + c'\
' x + 3, x, domain=\\mathbb{Z}\\left[a, b, c\\right] \\right)}'
assert latex(Poly([a, 1, b+c, 2, 3], x)) == \
'\\operatorname{Poly}{\\left( a x^{4} + x^{3} + \\left(b + c\\right) '\
'x^{2} + 2 x + 3, x, domain=\\mathbb{Z}\\left[a, b, c\\right] \\right)}'
assert latex(Poly(a*x**3 + x**2*y - x*y - c*y**3 - b*x*y**2 + y - a*x + b,
(x, y))) == \
'\\operatorname{Poly}{\\left( a x^{3} + x^{2}y - b xy^{2} - xy - '\
'a x - c y^{3} + y + b, x, y, domain=\\mathbb{Z}\\left[a, b, c\\right] \\right)}'
def test_latex_ComplexRootOf():
assert latex(rootof(x**5 + x + 3, 0)) == \
r"\operatorname{CRootOf} {\left(x^{5} + x + 3, 0\right)}"
def test_latex_RootSum():
assert latex(RootSum(x**5 + x + 3, sin)) == \
r"\operatorname{RootSum} {\left(x^{5} + x + 3, \left( x \mapsto \sin{\left(x \right)} \right)\right)}"
def test_settings():
raises(TypeError, lambda: latex(x*y, method="garbage"))
def test_latex_numbers():
assert latex(catalan(n)) == r"C_{n}"
assert latex(catalan(n)**2) == r"C_{n}^{2}"
assert latex(bernoulli(n)) == r"B_{n}"
assert latex(bernoulli(n, x)) == r"B_{n}\left(x\right)"
assert latex(bernoulli(n)**2) == r"B_{n}^{2}"
assert latex(bernoulli(n, x)**2) == r"B_{n}^{2}\left(x\right)"
assert latex(bell(n)) == r"B_{n}"
assert latex(bell(n, x)) == r"B_{n}\left(x\right)"
assert latex(bell(n, m, (x, y))) == r"B_{n, m}\left(x, y\right)"
assert latex(bell(n)**2) == r"B_{n}^{2}"
assert latex(bell(n, x)**2) == r"B_{n}^{2}\left(x\right)"
assert latex(bell(n, m, (x, y))**2) == r"B_{n, m}^{2}\left(x, y\right)"
assert latex(fibonacci(n)) == r"F_{n}"
assert latex(fibonacci(n, x)) == r"F_{n}\left(x\right)"
assert latex(fibonacci(n)**2) == r"F_{n}^{2}"
assert latex(fibonacci(n, x)**2) == r"F_{n}^{2}\left(x\right)"
assert latex(lucas(n)) == r"L_{n}"
assert latex(lucas(n)**2) == r"L_{n}^{2}"
assert latex(tribonacci(n)) == r"T_{n}"
assert latex(tribonacci(n, x)) == r"T_{n}\left(x\right)"
assert latex(tribonacci(n)**2) == r"T_{n}^{2}"
assert latex(tribonacci(n, x)**2) == r"T_{n}^{2}\left(x\right)"
def test_latex_euler():
assert latex(euler(n)) == r"E_{n}"
assert latex(euler(n, x)) == r"E_{n}\left(x\right)"
assert latex(euler(n, x)**2) == r"E_{n}^{2}\left(x\right)"
def test_lamda():
assert latex(Symbol('lamda')) == r"\lambda"
assert latex(Symbol('Lamda')) == r"\Lambda"
def test_custom_symbol_names():
x = Symbol('x')
y = Symbol('y')
assert latex(x) == "x"
assert latex(x, symbol_names={x: "x_i"}) == "x_i"
assert latex(x + y, symbol_names={x: "x_i"}) == "x_i + y"
assert latex(x**2, symbol_names={x: "x_i"}) == "x_i^{2}"
assert latex(x + y, symbol_names={x: "x_i", y: "y_j"}) == "x_i + y_j"
def test_matAdd():
from sympy import MatrixSymbol
from sympy.printing.latex import LatexPrinter
C = MatrixSymbol('C', 5, 5)
B = MatrixSymbol('B', 5, 5)
l = LatexPrinter()
assert l._print(C - 2*B) in ['- 2 B + C', 'C -2 B']
assert l._print(C + 2*B) in ['2 B + C', 'C + 2 B']
assert l._print(B - 2*C) in ['B - 2 C', '- 2 C + B']
assert l._print(B + 2*C) in ['B + 2 C', '2 C + B']
def test_matMul():
from sympy import MatrixSymbol
from sympy.printing.latex import LatexPrinter
A = MatrixSymbol('A', 5, 5)
B = MatrixSymbol('B', 5, 5)
x = Symbol('x')
lp = LatexPrinter()
assert lp._print_MatMul(2*A) == '2 A'
assert lp._print_MatMul(2*x*A) == '2 x A'
assert lp._print_MatMul(-2*A) == '- 2 A'
assert lp._print_MatMul(1.5*A) == '1.5 A'
assert lp._print_MatMul(sqrt(2)*A) == r'\sqrt{2} A'
assert lp._print_MatMul(-sqrt(2)*A) == r'- \sqrt{2} A'
assert lp._print_MatMul(2*sqrt(2)*x*A) == r'2 \sqrt{2} x A'
assert lp._print_MatMul(-2*A*(A + 2*B)) in [r'- 2 A \left(A + 2 B\right)',
r'- 2 A \left(2 B + A\right)']
def test_latex_MatrixSlice():
n = Symbol('n', integer=True)
x, y, z, w, t, = symbols('x y z w t')
X = MatrixSymbol('X', n, n)
Y = MatrixSymbol('Y', 10, 10)
Z = MatrixSymbol('Z', 10, 10)
assert latex(MatrixSlice(X, (None, None, None), (None, None, None))) == r'X\left[:, :\right]'
assert latex(X[x:x + 1, y:y + 1]) == r'X\left[x:x + 1, y:y + 1\right]'
assert latex(X[x:x + 1:2, y:y + 1:2]) == r'X\left[x:x + 1:2, y:y + 1:2\right]'
assert latex(X[:x, y:]) == r'X\left[:x, y:\right]'
assert latex(X[:x, y:]) == r'X\left[:x, y:\right]'
assert latex(X[x:, :y]) == r'X\left[x:, :y\right]'
assert latex(X[x:y, z:w]) == r'X\left[x:y, z:w\right]'
assert latex(X[x:y:t, w:t:x]) == r'X\left[x:y:t, w:t:x\right]'
assert latex(X[x::y, t::w]) == r'X\left[x::y, t::w\right]'
assert latex(X[:x:y, :t:w]) == r'X\left[:x:y, :t:w\right]'
assert latex(X[::x, ::y]) == r'X\left[::x, ::y\right]'
assert latex(MatrixSlice(X, (0, None, None), (0, None, None))) == r'X\left[:, :\right]'
assert latex(MatrixSlice(X, (None, n, None), (None, n, None))) == r'X\left[:, :\right]'
assert latex(MatrixSlice(X, (0, n, None), (0, n, None))) == r'X\left[:, :\right]'
assert latex(MatrixSlice(X, (0, n, 2), (0, n, 2))) == r'X\left[::2, ::2\right]'
assert latex(X[1:2:3, 4:5:6]) == r'X\left[1:2:3, 4:5:6\right]'
assert latex(X[1:3:5, 4:6:8]) == r'X\left[1:3:5, 4:6:8\right]'
assert latex(X[1:10:2]) == r'X\left[1:10:2, :\right]'
assert latex(Y[:5, 1:9:2]) == r'Y\left[:5, 1:9:2\right]'
assert latex(Y[:5, 1:10:2]) == r'Y\left[:5, 1::2\right]'
assert latex(Y[5, :5:2]) == r'Y\left[5:6, :5:2\right]'
assert latex(X[0:1, 0:1]) == r'X\left[:1, :1\right]'
assert latex(X[0:1:2, 0:1:2]) == r'X\left[:1:2, :1:2\right]'
assert latex((Y + Z)[2:, 2:]) == r'\left(Y + Z\right)\left[2:, 2:\right]'
def test_latex_RandomDomain():
from sympy.stats import Normal, Die, Exponential, pspace, where
from sympy.stats.rv import RandomDomain
X = Normal('x1', 0, 1)
assert latex(where(X > 0)) == r"\text{Domain: }0 < x_{1} \wedge x_{1} < \infty"
D = Die('d1', 6)
assert latex(where(D > 4)) == r"\text{Domain: }d_{1} = 5 \vee d_{1} = 6"
A = Exponential('a', 1)
B = Exponential('b', 1)
assert latex(
pspace(Tuple(A, B)).domain) == \
r"\text{Domain: }0 \leq a \wedge 0 \leq b \wedge a < \infty \wedge b < \infty"
assert latex(RandomDomain(FiniteSet(x), FiniteSet(1, 2))) == \
r'\text{Domain: }\left\{x\right\}\text{ in }\left\{1, 2\right\}'
def test_PrettyPoly():
from sympy.polys.domains import QQ
F = QQ.frac_field(x, y)
R = QQ[x, y]
assert latex(F.convert(x/(x + y))) == latex(x/(x + y))
assert latex(R.convert(x + y)) == latex(x + y)
def test_integral_transforms():
x = Symbol("x")
k = Symbol("k")
f = Function("f")
a = Symbol("a")
b = Symbol("b")
assert latex(MellinTransform(f(x), x, k)) == \
r"\mathcal{M}_{x}\left[f{\left(x \right)}\right]\left(k\right)"
assert latex(InverseMellinTransform(f(k), k, x, a, b)) == \
r"\mathcal{M}^{-1}_{k}\left[f{\left(k \right)}\right]\left(x\right)"
assert latex(LaplaceTransform(f(x), x, k)) == \
r"\mathcal{L}_{x}\left[f{\left(x \right)}\right]\left(k\right)"
assert latex(InverseLaplaceTransform(f(k), k, x, (a, b))) == \
r"\mathcal{L}^{-1}_{k}\left[f{\left(k \right)}\right]\left(x\right)"
assert latex(FourierTransform(f(x), x, k)) == \
r"\mathcal{F}_{x}\left[f{\left(x \right)}\right]\left(k\right)"
assert latex(InverseFourierTransform(f(k), k, x)) == \
r"\mathcal{F}^{-1}_{k}\left[f{\left(k \right)}\right]\left(x\right)"
assert latex(CosineTransform(f(x), x, k)) == \
r"\mathcal{COS}_{x}\left[f{\left(x \right)}\right]\left(k\right)"
assert latex(InverseCosineTransform(f(k), k, x)) == \
r"\mathcal{COS}^{-1}_{k}\left[f{\left(k \right)}\right]\left(x\right)"
assert latex(SineTransform(f(x), x, k)) == \
r"\mathcal{SIN}_{x}\left[f{\left(x \right)}\right]\left(k\right)"
assert latex(InverseSineTransform(f(k), k, x)) == \
r"\mathcal{SIN}^{-1}_{k}\left[f{\left(k \right)}\right]\left(x\right)"
def test_PolynomialRingBase():
from sympy.polys.domains import QQ
assert latex(QQ.old_poly_ring(x, y)) == r"\mathbb{Q}\left[x, y\right]"
assert latex(QQ.old_poly_ring(x, y, order="ilex")) == \
r"S_<^{-1}\mathbb{Q}\left[x, y\right]"
def test_categories():
from sympy.categories import (Object, IdentityMorphism,
NamedMorphism, Category, Diagram,
DiagramGrid)
A1 = Object("A1")
A2 = Object("A2")
A3 = Object("A3")
f1 = NamedMorphism(A1, A2, "f1")
f2 = NamedMorphism(A2, A3, "f2")
id_A1 = IdentityMorphism(A1)
K1 = Category("K1")
assert latex(A1) == "A_{1}"
assert latex(f1) == "f_{1}:A_{1}\\rightarrow A_{2}"
assert latex(id_A1) == "id:A_{1}\\rightarrow A_{1}"
assert latex(f2*f1) == "f_{2}\\circ f_{1}:A_{1}\\rightarrow A_{3}"
assert latex(K1) == r"\mathbf{K_{1}}"
d = Diagram()
assert latex(d) == r"\emptyset"
d = Diagram({f1: "unique", f2: S.EmptySet})
assert latex(d) == r"\left\{ f_{2}\circ f_{1}:A_{1}" \
r"\rightarrow A_{3} : \emptyset, \ id:A_{1}\rightarrow " \
r"A_{1} : \emptyset, \ id:A_{2}\rightarrow A_{2} : " \
r"\emptyset, \ id:A_{3}\rightarrow A_{3} : \emptyset, " \
r"\ f_{1}:A_{1}\rightarrow A_{2} : \left\{unique\right\}, " \
r"\ f_{2}:A_{2}\rightarrow A_{3} : \emptyset\right\}"
d = Diagram({f1: "unique", f2: S.EmptySet}, {f2 * f1: "unique"})
assert latex(d) == r"\left\{ f_{2}\circ f_{1}:A_{1}" \
r"\rightarrow A_{3} : \emptyset, \ id:A_{1}\rightarrow " \
r"A_{1} : \emptyset, \ id:A_{2}\rightarrow A_{2} : " \
r"\emptyset, \ id:A_{3}\rightarrow A_{3} : \emptyset, " \
r"\ f_{1}:A_{1}\rightarrow A_{2} : \left\{unique\right\}," \
r" \ f_{2}:A_{2}\rightarrow A_{3} : \emptyset\right\}" \
r"\Longrightarrow \left\{ f_{2}\circ f_{1}:A_{1}" \
r"\rightarrow A_{3} : \left\{unique\right\}\right\}"
# A linear diagram.
A = Object("A")
B = Object("B")
C = Object("C")
f = NamedMorphism(A, B, "f")
g = NamedMorphism(B, C, "g")
d = Diagram([f, g])
grid = DiagramGrid(d)
assert latex(grid) == "\\begin{array}{cc}\n" \
"A & B \\\\\n" \
" & C \n" \
"\\end{array}\n"
def test_Modules():
from sympy.polys.domains import QQ
from sympy.polys.agca import homomorphism
R = QQ.old_poly_ring(x, y)
F = R.free_module(2)
M = F.submodule([x, y], [1, x**2])
assert latex(F) == r"{\mathbb{Q}\left[x, y\right]}^{2}"
assert latex(M) == \
r"\left\langle {\left[ {x},{y} \right]},{\left[ {1},{x^{2}} \right]} \right\rangle"
I = R.ideal(x**2, y)
assert latex(I) == r"\left\langle {x^{2}},{y} \right\rangle"
Q = F / M
assert latex(Q) == \
r"\frac{{\mathbb{Q}\left[x, y\right]}^{2}}{\left\langle {\left[ {x},"\
r"{y} \right]},{\left[ {1},{x^{2}} \right]} \right\rangle}"
assert latex(Q.submodule([1, x**3/2], [2, y])) == \
r"\left\langle {{\left[ {1},{\frac{x^{3}}{2}} \right]} + {\left"\
r"\langle {\left[ {x},{y} \right]},{\left[ {1},{x^{2}} \right]} "\
r"\right\rangle}},{{\left[ {2},{y} \right]} + {\left\langle {\left[ "\
r"{x},{y} \right]},{\left[ {1},{x^{2}} \right]} \right\rangle}} \right\rangle"
h = homomorphism(QQ.old_poly_ring(x).free_module(2),
QQ.old_poly_ring(x).free_module(2), [0, 0])
assert latex(h) == \
r"{\left[\begin{matrix}0 & 0\\0 & 0\end{matrix}\right]} : "\
r"{{\mathbb{Q}\left[x\right]}^{2}} \to {{\mathbb{Q}\left[x\right]}^{2}}"
def test_QuotientRing():
from sympy.polys.domains import QQ
R = QQ.old_poly_ring(x)/[x**2 + 1]
assert latex(R) == \
r"\frac{\mathbb{Q}\left[x\right]}{\left\langle {x^{2} + 1} \right\rangle}"
assert latex(R.one) == r"{1} + {\left\langle {x^{2} + 1} \right\rangle}"
def test_Tr():
#TODO: Handle indices
A, B = symbols('A B', commutative=False)
t = Tr(A*B)
assert latex(t) == r'\operatorname{tr}\left(A B\right)'
def test_Adjoint():
from sympy.matrices import MatrixSymbol, Adjoint, Inverse, Transpose
X = MatrixSymbol('X', 2, 2)
Y = MatrixSymbol('Y', 2, 2)
assert latex(Adjoint(X)) == r'X^{\dagger}'
assert latex(Adjoint(X + Y)) == r'\left(X + Y\right)^{\dagger}'
assert latex(Adjoint(X) + Adjoint(Y)) == r'X^{\dagger} + Y^{\dagger}'
assert latex(Adjoint(X*Y)) == r'\left(X Y\right)^{\dagger}'
assert latex(Adjoint(Y)*Adjoint(X)) == r'Y^{\dagger} X^{\dagger}'
assert latex(Adjoint(X**2)) == r'\left(X^{2}\right)^{\dagger}'
assert latex(Adjoint(X)**2) == r'\left(X^{\dagger}\right)^{2}'
assert latex(Adjoint(Inverse(X))) == r'\left(X^{-1}\right)^{\dagger}'
assert latex(Inverse(Adjoint(X))) == r'\left(X^{\dagger}\right)^{-1}'
assert latex(Adjoint(Transpose(X))) == r'\left(X^{T}\right)^{\dagger}'
assert latex(Transpose(Adjoint(X))) == r'\left(X^{\dagger}\right)^{T}'
assert latex(Transpose(Adjoint(X) + Y)) == r'\left(X^{\dagger} + Y\right)^{T}'
def test_Transpose():
from sympy.matrices import Transpose, MatPow, HadamardPower
X = MatrixSymbol('X', 2, 2)
Y = MatrixSymbol('Y', 2, 2)
assert latex(Transpose(X)) == r'X^{T}'
assert latex(Transpose(X + Y)) == r'\left(X + Y\right)^{T}'
assert latex(Transpose(HadamardPower(X, 2))) == \
r'\left(X^{\circ {2}}\right)^{T}'
assert latex(HadamardPower(Transpose(X), 2)) == \
r'\left(X^{T}\right)^{\circ {2}}'
assert latex(Transpose(MatPow(X, 2))) == \
r'\left(X^{2}\right)^{T}'
assert latex(MatPow(Transpose(X), 2)) == \
r'\left(X^{T}\right)^{2}'
def test_Hadamard():
from sympy.matrices import MatrixSymbol, HadamardProduct, HadamardPower
from sympy.matrices.expressions import MatAdd, MatMul, MatPow
X = MatrixSymbol('X', 2, 2)
Y = MatrixSymbol('Y', 2, 2)
assert latex(HadamardProduct(X, Y*Y)) == r'X \circ Y^{2}'
assert latex(HadamardProduct(X, Y)*Y) == r'\left(X \circ Y\right) Y'
assert latex(HadamardPower(X, 2)) == r'X^{\circ {2}}'
assert latex(HadamardPower(X, -1)) == r'X^{\circ \left({-1}\right)}'
assert latex(HadamardPower(MatAdd(X, Y), 2)) == \
r'\left(X + Y\right)^{\circ {2}}'
assert latex(HadamardPower(MatMul(X, Y), 2)) == \
r'\left(X Y\right)^{\circ {2}}'
assert latex(HadamardPower(MatPow(X, -1), -1)) == \
r'\left(X^{-1}\right)^{\circ \left({-1}\right)}'
assert latex(MatPow(HadamardPower(X, -1), -1)) == \
r'\left(X^{\circ \left({-1}\right)}\right)^{-1}'
assert latex(HadamardPower(X, n+1)) == \
r'X^{\circ \left({n + 1}\right)}'
def test_ElementwiseApplyFunction():
from sympy.matrices import MatrixSymbol
X = MatrixSymbol('X', 2, 2)
expr = (X.T*X).applyfunc(sin)
assert latex(expr) == r"{\left( d \mapsto \sin{\left(d \right)} \right)}_{\circ}\left({X^{T} X}\right)"
expr = X.applyfunc(Lambda(x, 1/x))
assert latex(expr) == r'{\left( x \mapsto \frac{1}{x} \right)}_{\circ}\left({X}\right)'
def test_ZeroMatrix():
from sympy import ZeroMatrix
assert latex(ZeroMatrix(1, 1), mat_symbol_style='plain') == r"\mathbb{0}"
assert latex(ZeroMatrix(1, 1), mat_symbol_style='bold') == r"\mathbf{0}"
def test_OneMatrix():
from sympy import OneMatrix
assert latex(OneMatrix(3, 4), mat_symbol_style='plain') == r"\mathbb{1}"
assert latex(OneMatrix(3, 4), mat_symbol_style='bold') == r"\mathbf{1}"
def test_Identity():
from sympy import Identity
assert latex(Identity(1), mat_symbol_style='plain') == r"\mathbb{I}"
assert latex(Identity(1), mat_symbol_style='bold') == r"\mathbf{I}"
def test_boolean_args_order():
syms = symbols('a:f')
expr = And(*syms)
assert latex(expr) == 'a \\wedge b \\wedge c \\wedge d \\wedge e \\wedge f'
expr = Or(*syms)
assert latex(expr) == 'a \\vee b \\vee c \\vee d \\vee e \\vee f'
expr = Equivalent(*syms)
assert latex(expr) == \
'a \\Leftrightarrow b \\Leftrightarrow c \\Leftrightarrow d \\Leftrightarrow e \\Leftrightarrow f'
expr = Xor(*syms)
assert latex(expr) == \
'a \\veebar b \\veebar c \\veebar d \\veebar e \\veebar f'
def test_imaginary():
i = sqrt(-1)
assert latex(i) == r'i'
def test_builtins_without_args():
assert latex(sin) == r'\sin'
assert latex(cos) == r'\cos'
assert latex(tan) == r'\tan'
assert latex(log) == r'\log'
assert latex(Ei) == r'\operatorname{Ei}'
assert latex(zeta) == r'\zeta'
def test_latex_greek_functions():
# bug because capital greeks that have roman equivalents should not use
# \Alpha, \Beta, \Eta, etc.
s = Function('Alpha')
assert latex(s) == r'A'
assert latex(s(x)) == r'A{\left(x \right)}'
s = Function('Beta')
assert latex(s) == r'B'
s = Function('Eta')
assert latex(s) == r'H'
assert latex(s(x)) == r'H{\left(x \right)}'
# bug because sympy.core.numbers.Pi is special
p = Function('Pi')
# assert latex(p(x)) == r'\Pi{\left(x \right)}'
assert latex(p) == r'\Pi'
# bug because not all greeks are included
c = Function('chi')
assert latex(c(x)) == r'\chi{\left(x \right)}'
assert latex(c) == r'\chi'
def test_translate():
s = 'Alpha'
assert translate(s) == 'A'
s = 'Beta'
assert translate(s) == 'B'
s = 'Eta'
assert translate(s) == 'H'
s = 'omicron'
assert translate(s) == 'o'
s = 'Pi'
assert translate(s) == r'\Pi'
s = 'pi'
assert translate(s) == r'\pi'
s = 'LamdaHatDOT'
assert translate(s) == r'\dot{\hat{\Lambda}}'
def test_other_symbols():
from sympy.printing.latex import other_symbols
for s in other_symbols:
assert latex(symbols(s)) == "\\"+s
def test_modifiers():
# Test each modifier individually in the simplest case
# (with funny capitalizations)
assert latex(symbols("xMathring")) == r"\mathring{x}"
assert latex(symbols("xCheck")) == r"\check{x}"
assert latex(symbols("xBreve")) == r"\breve{x}"
assert latex(symbols("xAcute")) == r"\acute{x}"
assert latex(symbols("xGrave")) == r"\grave{x}"
assert latex(symbols("xTilde")) == r"\tilde{x}"
assert latex(symbols("xPrime")) == r"{x}'"
assert latex(symbols("xddDDot")) == r"\ddddot{x}"
assert latex(symbols("xDdDot")) == r"\dddot{x}"
assert latex(symbols("xDDot")) == r"\ddot{x}"
assert latex(symbols("xBold")) == r"\boldsymbol{x}"
assert latex(symbols("xnOrM")) == r"\left\|{x}\right\|"
assert latex(symbols("xAVG")) == r"\left\langle{x}\right\rangle"
assert latex(symbols("xHat")) == r"\hat{x}"
assert latex(symbols("xDot")) == r"\dot{x}"
assert latex(symbols("xBar")) == r"\bar{x}"
assert latex(symbols("xVec")) == r"\vec{x}"
assert latex(symbols("xAbs")) == r"\left|{x}\right|"
assert latex(symbols("xMag")) == r"\left|{x}\right|"
assert latex(symbols("xPrM")) == r"{x}'"
assert latex(symbols("xBM")) == r"\boldsymbol{x}"
# Test strings that are *only* the names of modifiers
assert latex(symbols("Mathring")) == r"Mathring"
assert latex(symbols("Check")) == r"Check"
assert latex(symbols("Breve")) == r"Breve"
assert latex(symbols("Acute")) == r"Acute"
assert latex(symbols("Grave")) == r"Grave"
assert latex(symbols("Tilde")) == r"Tilde"
assert latex(symbols("Prime")) == r"Prime"
assert latex(symbols("DDot")) == r"\dot{D}"
assert latex(symbols("Bold")) == r"Bold"
assert latex(symbols("NORm")) == r"NORm"
assert latex(symbols("AVG")) == r"AVG"
assert latex(symbols("Hat")) == r"Hat"
assert latex(symbols("Dot")) == r"Dot"
assert latex(symbols("Bar")) == r"Bar"
assert latex(symbols("Vec")) == r"Vec"
assert latex(symbols("Abs")) == r"Abs"
assert latex(symbols("Mag")) == r"Mag"
assert latex(symbols("PrM")) == r"PrM"
assert latex(symbols("BM")) == r"BM"
assert latex(symbols("hbar")) == r"\hbar"
# Check a few combinations
assert latex(symbols("xvecdot")) == r"\dot{\vec{x}}"
assert latex(symbols("xDotVec")) == r"\vec{\dot{x}}"
assert latex(symbols("xHATNorm")) == r"\left\|{\hat{x}}\right\|"
# Check a couple big, ugly combinations
assert latex(symbols('xMathringBm_yCheckPRM__zbreveAbs')) == \
r"\boldsymbol{\mathring{x}}^{\left|{\breve{z}}\right|}_{{\check{y}}'}"
assert latex(symbols('alphadothat_nVECDOT__tTildePrime')) == \
r"\hat{\dot{\alpha}}^{{\tilde{t}}'}_{\dot{\vec{n}}}"
def test_greek_symbols():
assert latex(Symbol('alpha')) == r'\alpha'
assert latex(Symbol('beta')) == r'\beta'
assert latex(Symbol('gamma')) == r'\gamma'
assert latex(Symbol('delta')) == r'\delta'
assert latex(Symbol('epsilon')) == r'\epsilon'
assert latex(Symbol('zeta')) == r'\zeta'
assert latex(Symbol('eta')) == r'\eta'
assert latex(Symbol('theta')) == r'\theta'
assert latex(Symbol('iota')) == r'\iota'
assert latex(Symbol('kappa')) == r'\kappa'
assert latex(Symbol('lambda')) == r'\lambda'
assert latex(Symbol('mu')) == r'\mu'
assert latex(Symbol('nu')) == r'\nu'
assert latex(Symbol('xi')) == r'\xi'
assert latex(Symbol('omicron')) == r'o'
assert latex(Symbol('pi')) == r'\pi'
assert latex(Symbol('rho')) == r'\rho'
assert latex(Symbol('sigma')) == r'\sigma'
assert latex(Symbol('tau')) == r'\tau'
assert latex(Symbol('upsilon')) == r'\upsilon'
assert latex(Symbol('phi')) == r'\phi'
assert latex(Symbol('chi')) == r'\chi'
assert latex(Symbol('psi')) == r'\psi'
assert latex(Symbol('omega')) == r'\omega'
assert latex(Symbol('Alpha')) == r'A'
assert latex(Symbol('Beta')) == r'B'
assert latex(Symbol('Gamma')) == r'\Gamma'
assert latex(Symbol('Delta')) == r'\Delta'
assert latex(Symbol('Epsilon')) == r'E'
assert latex(Symbol('Zeta')) == r'Z'
assert latex(Symbol('Eta')) == r'H'
assert latex(Symbol('Theta')) == r'\Theta'
assert latex(Symbol('Iota')) == r'I'
assert latex(Symbol('Kappa')) == r'K'
assert latex(Symbol('Lambda')) == r'\Lambda'
assert latex(Symbol('Mu')) == r'M'
assert latex(Symbol('Nu')) == r'N'
assert latex(Symbol('Xi')) == r'\Xi'
assert latex(Symbol('Omicron')) == r'O'
assert latex(Symbol('Pi')) == r'\Pi'
assert latex(Symbol('Rho')) == r'P'
assert latex(Symbol('Sigma')) == r'\Sigma'
assert latex(Symbol('Tau')) == r'T'
assert latex(Symbol('Upsilon')) == r'\Upsilon'
assert latex(Symbol('Phi')) == r'\Phi'
assert latex(Symbol('Chi')) == r'X'
assert latex(Symbol('Psi')) == r'\Psi'
assert latex(Symbol('Omega')) == r'\Omega'
assert latex(Symbol('varepsilon')) == r'\varepsilon'
assert latex(Symbol('varkappa')) == r'\varkappa'
assert latex(Symbol('varphi')) == r'\varphi'
assert latex(Symbol('varpi')) == r'\varpi'
assert latex(Symbol('varrho')) == r'\varrho'
assert latex(Symbol('varsigma')) == r'\varsigma'
assert latex(Symbol('vartheta')) == r'\vartheta'
def test_fancyset_symbols():
assert latex(S.Rationals) == '\\mathbb{Q}'
assert latex(S.Naturals) == '\\mathbb{N}'
assert latex(S.Naturals0) == '\\mathbb{N}_0'
assert latex(S.Integers) == '\\mathbb{Z}'
assert latex(S.Reals) == '\\mathbb{R}'
assert latex(S.Complexes) == '\\mathbb{C}'
@XFAIL
def test_builtin_without_args_mismatched_names():
assert latex(CosineTransform) == r'\mathcal{COS}'
def test_builtin_no_args():
assert latex(Chi) == r'\operatorname{Chi}'
assert latex(beta) == r'\operatorname{B}'
assert latex(gamma) == r'\Gamma'
assert latex(KroneckerDelta) == r'\delta'
assert latex(DiracDelta) == r'\delta'
assert latex(lowergamma) == r'\gamma'
def test_issue_6853():
p = Function('Pi')
assert latex(p(x)) == r"\Pi{\left(x \right)}"
def test_Mul():
e = Mul(-2, x + 1, evaluate=False)
assert latex(e) == r'- 2 \left(x + 1\right)'
e = Mul(2, x + 1, evaluate=False)
assert latex(e) == r'2 \left(x + 1\right)'
e = Mul(S.Half, x + 1, evaluate=False)
assert latex(e) == r'\frac{x + 1}{2}'
e = Mul(y, x + 1, evaluate=False)
assert latex(e) == r'y \left(x + 1\right)'
e = Mul(-y, x + 1, evaluate=False)
assert latex(e) == r'- y \left(x + 1\right)'
e = Mul(-2, x + 1)
assert latex(e) == r'- 2 x - 2'
e = Mul(2, x + 1)
assert latex(e) == r'2 x + 2'
def test_Pow():
e = Pow(2, 2, evaluate=False)
assert latex(e) == r'2^{2}'
assert latex(x**(Rational(-1, 3))) == r'\frac{1}{\sqrt[3]{x}}'
x2 = Symbol(r'x^2')
assert latex(x2**2) == r'\left(x^{2}\right)^{2}'
def test_issue_7180():
assert latex(Equivalent(x, y)) == r"x \Leftrightarrow y"
assert latex(Not(Equivalent(x, y))) == r"x \not\Leftrightarrow y"
def test_issue_8409():
assert latex(S.Half**n) == r"\left(\frac{1}{2}\right)^{n}"
def test_issue_8470():
from sympy.parsing.sympy_parser import parse_expr
e = parse_expr("-B*A", evaluate=False)
assert latex(e) == r"A \left(- B\right)"
def test_issue_15439():
x = MatrixSymbol('x', 2, 2)
y = MatrixSymbol('y', 2, 2)
assert latex((x * y).subs(y, -y)) == r"x \left(- y\right)"
assert latex((x * y).subs(y, -2*y)) == r"x \left(- 2 y\right)"
assert latex((x * y).subs(x, -x)) == r"- x y"
def test_issue_2934():
assert latex(Symbol(r'\frac{a_1}{b_1}')) == '\\frac{a_1}{b_1}'
def test_issue_10489():
latexSymbolWithBrace = 'C_{x_{0}}'
s = Symbol(latexSymbolWithBrace)
assert latex(s) == latexSymbolWithBrace
assert latex(cos(s)) == r'\cos{\left(C_{x_{0}} \right)}'
def test_issue_12886():
m__1, l__1 = symbols('m__1, l__1')
assert latex(m__1**2 + l__1**2) == \
r'\left(l^{1}\right)^{2} + \left(m^{1}\right)^{2}'
def test_issue_13559():
from sympy.parsing.sympy_parser import parse_expr
expr = parse_expr('5/1', evaluate=False)
assert latex(expr) == r"\frac{5}{1}"
def test_issue_13651():
expr = c + Mul(-1, a + b, evaluate=False)
assert latex(expr) == r"c - \left(a + b\right)"
def test_latex_UnevaluatedExpr():
x = symbols("x")
he = UnevaluatedExpr(1/x)
assert latex(he) == latex(1/x) == r"\frac{1}{x}"
assert latex(he**2) == r"\left(\frac{1}{x}\right)^{2}"
assert latex(he + 1) == r"1 + \frac{1}{x}"
assert latex(x*he) == r"x \frac{1}{x}"
def test_MatrixElement_printing():
# test cases for issue #11821
A = MatrixSymbol("A", 1, 3)
B = MatrixSymbol("B", 1, 3)
C = MatrixSymbol("C", 1, 3)
assert latex(A[0, 0]) == r"A_{0, 0}"
assert latex(3 * A[0, 0]) == r"3 A_{0, 0}"
F = C[0, 0].subs(C, A - B)
assert latex(F) == r"\left(A - B\right)_{0, 0}"
i, j, k = symbols("i j k")
M = MatrixSymbol("M", k, k)
N = MatrixSymbol("N", k, k)
assert latex((M*N)[i, j]) == \
r'\sum_{i_{1}=0}^{k - 1} M_{i, i_{1}} N_{i_{1}, j}'
def test_MatrixSymbol_printing():
# test cases for issue #14237
A = MatrixSymbol("A", 3, 3)
B = MatrixSymbol("B", 3, 3)
C = MatrixSymbol("C", 3, 3)
assert latex(-A) == r"- A"
assert latex(A - A*B - B) == r"A - A B - B"
assert latex(-A*B - A*B*C - B) == r"- A B - A B C - B"
def test_KroneckerProduct_printing():
A = MatrixSymbol('A', 3, 3)
B = MatrixSymbol('B', 2, 2)
assert latex(KroneckerProduct(A, B)) == r'A \otimes B'
def test_Quaternion_latex_printing():
q = Quaternion(x, y, z, t)
assert latex(q) == "x + y i + z j + t k"
q = Quaternion(x, y, z, x*t)
assert latex(q) == "x + y i + z j + t x k"
q = Quaternion(x, y, z, x + t)
assert latex(q) == r"x + y i + z j + \left(t + x\right) k"
def test_TensorProduct_printing():
from sympy.tensor.functions import TensorProduct
A = MatrixSymbol("A", 3, 3)
B = MatrixSymbol("B", 3, 3)
assert latex(TensorProduct(A, B)) == r"A \otimes B"
def test_WedgeProduct_printing():
from sympy.diffgeom.rn import R2
from sympy.diffgeom import WedgeProduct
wp = WedgeProduct(R2.dx, R2.dy)
assert latex(wp) == r"\operatorname{d}x \wedge \operatorname{d}y"
def test_issue_14041():
import sympy.physics.mechanics as me
A_frame = me.ReferenceFrame('A')
thetad, phid = me.dynamicsymbols('theta, phi', 1)
L = Symbol('L')
assert latex(L*(phid + thetad)**2*A_frame.x) == \
r"L \left(\dot{\phi} + \dot{\theta}\right)^{2}\mathbf{\hat{a}_x}"
assert latex((phid + thetad)**2*A_frame.x) == \
r"\left(\dot{\phi} + \dot{\theta}\right)^{2}\mathbf{\hat{a}_x}"
assert latex((phid*thetad)**a*A_frame.x) == \
r"\left(\dot{\phi} \dot{\theta}\right)^{a}\mathbf{\hat{a}_x}"
def test_issue_9216():
expr_1 = Pow(1, -1, evaluate=False)
assert latex(expr_1) == r"1^{-1}"
expr_2 = Pow(1, Pow(1, -1, evaluate=False), evaluate=False)
assert latex(expr_2) == r"1^{1^{-1}}"
expr_3 = Pow(3, -2, evaluate=False)
assert latex(expr_3) == r"\frac{1}{9}"
expr_4 = Pow(1, -2, evaluate=False)
assert latex(expr_4) == r"1^{-2}"
def test_latex_printer_tensor():
from sympy.tensor.tensor import TensorIndexType, tensor_indices, TensorHead, tensor_heads
L = TensorIndexType("L")
i, j, k, l = tensor_indices("i j k l", L)
i0 = tensor_indices("i_0", L)
A, B, C, D = tensor_heads("A B C D", [L])
H = TensorHead("H", [L, L])
K = TensorHead("K", [L, L, L, L])
assert latex(i) == "{}^{i}"
assert latex(-i) == "{}_{i}"
expr = A(i)
assert latex(expr) == "A{}^{i}"
expr = A(i0)
assert latex(expr) == "A{}^{i_{0}}"
expr = A(-i)
assert latex(expr) == "A{}_{i}"
expr = -3*A(i)
assert latex(expr) == r"-3A{}^{i}"
expr = K(i, j, -k, -i0)
assert latex(expr) == "K{}^{ij}{}_{ki_{0}}"
expr = K(i, -j, -k, i0)
assert latex(expr) == "K{}^{i}{}_{jk}{}^{i_{0}}"
expr = K(i, -j, k, -i0)
assert latex(expr) == "K{}^{i}{}_{j}{}^{k}{}_{i_{0}}"
expr = H(i, -j)
assert latex(expr) == "H{}^{i}{}_{j}"
expr = H(i, j)
assert latex(expr) == "H{}^{ij}"
expr = H(-i, -j)
assert latex(expr) == "H{}_{ij}"
expr = (1+x)*A(i)
assert latex(expr) == r"\left(x + 1\right)A{}^{i}"
expr = H(i, -i)
assert latex(expr) == "H{}^{L_{0}}{}_{L_{0}}"
expr = H(i, -j)*A(j)*B(k)
assert latex(expr) == "H{}^{i}{}_{L_{0}}A{}^{L_{0}}B{}^{k}"
expr = A(i) + 3*B(i)
assert latex(expr) == "3B{}^{i} + A{}^{i}"
# Test ``TensorElement``:
from sympy.tensor.tensor import TensorElement
expr = TensorElement(K(i, j, k, l), {i: 3, k: 2})
assert latex(expr) == 'K{}^{i=3,j,k=2,l}'
expr = TensorElement(K(i, j, k, l), {i: 3})
assert latex(expr) == 'K{}^{i=3,jkl}'
expr = TensorElement(K(i, -j, k, l), {i: 3, k: 2})
assert latex(expr) == 'K{}^{i=3}{}_{j}{}^{k=2,l}'
expr = TensorElement(K(i, -j, k, -l), {i: 3, k: 2})
assert latex(expr) == 'K{}^{i=3}{}_{j}{}^{k=2}{}_{l}'
expr = TensorElement(K(i, j, -k, -l), {i: 3, -k: 2})
assert latex(expr) == 'K{}^{i=3,j}{}_{k=2,l}'
expr = TensorElement(K(i, j, -k, -l), {i: 3})
assert latex(expr) == 'K{}^{i=3,j}{}_{kl}'
expr = PartialDerivative(A(i), A(i))
assert latex(expr) == r"\frac{\partial}{\partial {A{}^{L_{0}}}}{A{}^{L_{0}}}"
expr = PartialDerivative(A(-i), A(-j))
assert latex(expr) == r"\frac{\partial}{\partial {A{}_{j}}}{A{}_{i}}"
expr = PartialDerivative(K(i, j, -k, -l), A(m), A(-n))
assert latex(expr) == r"\frac{\partial^{2}}{\partial {A{}^{m}} \partial {A{}_{n}}}{K{}^{ij}{}_{kl}}"
expr = PartialDerivative(B(-i) + A(-i), A(-j), A(-n))
assert latex(expr) == r"\frac{\partial^{2}}{\partial {A{}_{j}} \partial {A{}_{n}}}{\left(A{}_{i} + B{}_{i}\right)}"
expr = PartialDerivative(3*A(-i), A(-j), A(-n))
assert latex(expr) == r"\frac{\partial^{2}}{\partial {A{}_{j}} \partial {A{}_{n}}}{\left(3A{}_{i}\right)}"
def test_multiline_latex():
a, b, c, d, e, f = symbols('a b c d e f')
expr = -a + 2*b -3*c +4*d -5*e
expected = r"\begin{eqnarray}" + "\n"\
r"f & = &- a \nonumber\\" + "\n"\
r"& & + 2 b \nonumber\\" + "\n"\
r"& & - 3 c \nonumber\\" + "\n"\
r"& & + 4 d \nonumber\\" + "\n"\
r"& & - 5 e " + "\n"\
r"\end{eqnarray}"
assert multiline_latex(f, expr, environment="eqnarray") == expected
expected2 = r'\begin{eqnarray}' + '\n'\
r'f & = &- a + 2 b \nonumber\\' + '\n'\
r'& & - 3 c + 4 d \nonumber\\' + '\n'\
r'& & - 5 e ' + '\n'\
r'\end{eqnarray}'
assert multiline_latex(f, expr, 2, environment="eqnarray") == expected2
expected3 = r'\begin{eqnarray}' + '\n'\
r'f & = &- a + 2 b - 3 c \nonumber\\'+ '\n'\
r'& & + 4 d - 5 e ' + '\n'\
r'\end{eqnarray}'
assert multiline_latex(f, expr, 3, environment="eqnarray") == expected3
expected3dots = r'\begin{eqnarray}' + '\n'\
r'f & = &- a + 2 b - 3 c \dots\nonumber\\'+ '\n'\
r'& & + 4 d - 5 e ' + '\n'\
r'\end{eqnarray}'
assert multiline_latex(f, expr, 3, environment="eqnarray", use_dots=True) == expected3dots
expected3align = r'\begin{align*}' + '\n'\
r'f = &- a + 2 b - 3 c \\'+ '\n'\
r'& + 4 d - 5 e ' + '\n'\
r'\end{align*}'
assert multiline_latex(f, expr, 3) == expected3align
assert multiline_latex(f, expr, 3, environment='align*') == expected3align
expected2ieee = r'\begin{IEEEeqnarray}{rCl}' + '\n'\
r'f & = &- a + 2 b \nonumber\\' + '\n'\
r'& & - 3 c + 4 d \nonumber\\' + '\n'\
r'& & - 5 e ' + '\n'\
r'\end{IEEEeqnarray}'
assert multiline_latex(f, expr, 2, environment="IEEEeqnarray") == expected2ieee
raises(ValueError, lambda: multiline_latex(f, expr, environment="foo"))
def test_issue_15353():
from sympy import ConditionSet, Tuple, S, sin, cos
a, x = symbols('a x')
# Obtained from nonlinsolve([(sin(a*x)),cos(a*x)],[x,a])
sol = ConditionSet(
Tuple(x, a), Eq(sin(a*x), 0) & Eq(cos(a*x), 0), S.Complexes**2)
assert latex(sol) == \
r'\left\{\left( x, \ a\right) \mid \left( x, \ a\right) \in ' \
r'\mathbb{C}^{2} \wedge \sin{\left(a x \right)} = 0 \wedge ' \
r'\cos{\left(a x \right)} = 0 \right\}'
def test_trace():
# Issue 15303
from sympy import trace
A = MatrixSymbol("A", 2, 2)
assert latex(trace(A)) == r"\operatorname{tr}\left(A \right)"
assert latex(trace(A**2)) == r"\operatorname{tr}\left(A^{2} \right)"
def test_print_basic():
# Issue 15303
from sympy import Basic, Expr
# dummy class for testing printing where the function is not
# implemented in latex.py
class UnimplementedExpr(Expr):
def __new__(cls, e):
return Basic.__new__(cls, e)
# dummy function for testing
def unimplemented_expr(expr):
return UnimplementedExpr(expr).doit()
# override class name to use superscript / subscript
def unimplemented_expr_sup_sub(expr):
result = UnimplementedExpr(expr)
result.__class__.__name__ = 'UnimplementedExpr_x^1'
return result
assert latex(unimplemented_expr(x)) == r'UnimplementedExpr\left(x\right)'
assert latex(unimplemented_expr(x**2)) == \
r'UnimplementedExpr\left(x^{2}\right)'
assert latex(unimplemented_expr_sup_sub(x)) == \
r'UnimplementedExpr^{1}_{x}\left(x\right)'
def test_MatrixSymbol_bold():
# Issue #15871
from sympy import trace
A = MatrixSymbol("A", 2, 2)
assert latex(trace(A), mat_symbol_style='bold') == \
r"\operatorname{tr}\left(\mathbf{A} \right)"
assert latex(trace(A), mat_symbol_style='plain') == \
r"\operatorname{tr}\left(A \right)"
A = MatrixSymbol("A", 3, 3)
B = MatrixSymbol("B", 3, 3)
C = MatrixSymbol("C", 3, 3)
assert latex(-A, mat_symbol_style='bold') == r"- \mathbf{A}"
assert latex(A - A*B - B, mat_symbol_style='bold') == \
r"\mathbf{A} - \mathbf{A} \mathbf{B} - \mathbf{B}"
assert latex(-A*B - A*B*C - B, mat_symbol_style='bold') == \
r"- \mathbf{A} \mathbf{B} - \mathbf{A} \mathbf{B} \mathbf{C} - \mathbf{B}"
A_k = MatrixSymbol("A_k", 3, 3)
assert latex(A_k, mat_symbol_style='bold') == r"\mathbf{A}_{k}"
A = MatrixSymbol(r"\nabla_k", 3, 3)
assert latex(A, mat_symbol_style='bold') == r"\mathbf{\nabla}_{k}"
def test_AppliedPermutation():
p = Permutation(0, 1, 2)
x = Symbol('x')
assert latex(AppliedPermutation(p, x)) == \
r'\sigma_{\left( 0\; 1\; 2\right)}(x)'
def test_PermutationMatrix():
p = Permutation(0, 1, 2)
assert latex(PermutationMatrix(p)) == r'P_{\left( 0\; 1\; 2\right)}'
p = Permutation(0, 3)(1, 2)
assert latex(PermutationMatrix(p)) == \
r'P_{\left( 0\; 3\right)\left( 1\; 2\right)}'
def test_imaginary_unit():
assert latex(1 + I) == '1 + i'
assert latex(1 + I, imaginary_unit='i') == '1 + i'
assert latex(1 + I, imaginary_unit='j') == '1 + j'
assert latex(1 + I, imaginary_unit='foo') == '1 + foo'
assert latex(I, imaginary_unit="ti") == '\\text{i}'
assert latex(I, imaginary_unit="tj") == '\\text{j}'
def test_text_re_im():
assert latex(im(x), gothic_re_im=True) == r'\Im{\left(x\right)}'
assert latex(im(x), gothic_re_im=False) == r'\operatorname{im}{\left(x\right)}'
assert latex(re(x), gothic_re_im=True) == r'\Re{\left(x\right)}'
assert latex(re(x), gothic_re_im=False) == r'\operatorname{re}{\left(x\right)}'
def test_latex_diffgeom():
from sympy.diffgeom import Manifold, Patch, CoordSystem, BaseScalarField, Differential
from sympy.diffgeom.rn import R2
m = Manifold('M', 2)
assert latex(m) == r'\text{M}'
p = Patch('P', m)
assert latex(p) == r'\text{P}_{\text{M}}'
rect = CoordSystem('rect', p)
assert latex(rect) == r'\text{rect}^{\text{P}}_{\text{M}}'
b = BaseScalarField(rect, 0)
assert latex(b) == r'\mathbf{rect_{0}}'
g = Function('g')
s_field = g(R2.x, R2.y)
assert latex(Differential(s_field)) == \
r'\operatorname{d}\left(g{\left(\mathbf{x},\mathbf{y} \right)}\right)'
def test_unit_printing():
assert latex(5*meter) == r'5 \text{m}'
assert latex(3*gibibyte) == r'3 \text{gibibyte}'
assert latex(4*microgram/second) == r'\frac{4 \mu\text{g}}{\text{s}}'
def test_issue_17092():
x_star = Symbol('x^*')
assert latex(Derivative(x_star, x_star,2)) == r'\frac{d^{2}}{d \left(x^{*}\right)^{2}} x^{*}'
def test_latex_decimal_separator():
x, y, z, t = symbols('x y z t')
k, m, n = symbols('k m n', integer=True)
f, g, h = symbols('f g h', cls=Function)
# comma decimal_separator
assert(latex([1, 2.3, 4.5], decimal_separator='comma') == r'\left[ 1; \ 2{,}3; \ 4{,}5\right]')
assert(latex(FiniteSet(1, 2.3, 4.5), decimal_separator='comma') == r'\left\{1; 2{,}3; 4{,}5\right\}')
assert(latex((1, 2.3, 4.6), decimal_separator = 'comma') == r'\left( 1; \ 2{,}3; \ 4{,}6\right)')
assert(latex((1,), decimal_separator='comma') == r'\left( 1;\right)')
# period decimal_separator
assert(latex([1, 2.3, 4.5], decimal_separator='period') == r'\left[ 1, \ 2.3, \ 4.5\right]' )
assert(latex(FiniteSet(1, 2.3, 4.5), decimal_separator='period') == r'\left\{1, 2.3, 4.5\right\}')
assert(latex((1, 2.3, 4.6), decimal_separator = 'period') == r'\left( 1, \ 2.3, \ 4.6\right)')
assert(latex((1,), decimal_separator='period') == r'\left( 1,\right)')
# default decimal_separator
assert(latex([1, 2.3, 4.5]) == r'\left[ 1, \ 2.3, \ 4.5\right]')
assert(latex(FiniteSet(1, 2.3, 4.5)) == r'\left\{1, 2.3, 4.5\right\}')
assert(latex((1, 2.3, 4.6)) == r'\left( 1, \ 2.3, \ 4.6\right)')
assert(latex((1,)) == r'\left( 1,\right)')
assert(latex(Mul(3.4,5.3), decimal_separator = 'comma') ==r'18{,}02')
assert(latex(3.4*5.3, decimal_separator = 'comma')==r'18{,}02')
x = symbols('x')
y = symbols('y')
z = symbols('z')
assert(latex(x*5.3 + 2**y**3.4 + 4.5 + z, decimal_separator = 'comma')== r'2^{y^{3{,}4}} + 5{,}3 x + z + 4{,}5')
assert(latex(0.987, decimal_separator='comma') == r'0{,}987')
assert(latex(S(0.987), decimal_separator='comma')== r'0{,}987')
assert(latex(.3, decimal_separator='comma')== r'0{,}3')
assert(latex(S(.3), decimal_separator='comma')== r'0{,}3')
assert(latex(5.8*10**(-7), decimal_separator='comma') ==r'5{,}8e-07')
assert(latex(S(5.7)*10**(-7), decimal_separator='comma')==r'5{,}7 \cdot 10^{-7}')
assert(latex(S(5.7*10**(-7)), decimal_separator='comma')==r'5{,}7 \cdot 10^{-7}')
x = symbols('x')
assert(latex(1.2*x+3.4, decimal_separator='comma')==r'1{,}2 x + 3{,}4')
assert(latex(FiniteSet(1, 2.3, 4.5), decimal_separator='period')==r'\left\{1, 2.3, 4.5\right\}')
# Error Handling tests
raises(ValueError, lambda: latex([1,2.3,4.5], decimal_separator='non_existing_decimal_separator_in_list'))
raises(ValueError, lambda: latex(FiniteSet(1,2.3,4.5), decimal_separator='non_existing_decimal_separator_in_set'))
raises(ValueError, lambda: latex((1,2.3,4.5), decimal_separator='non_existing_decimal_separator_in_tuple'))
|
276a74ef13cc7e7cac2a43b483769dd055d271f4264c8d93894c95e9edf4a0b3 | import random
from sympy import symbols, Derivative
from sympy.codegen.array_utils import (CodegenArrayTensorProduct, CodegenArrayElementwiseAdd,
CodegenArrayPermuteDims, CodegenArrayDiagonal, parse_matrix_expression)
from sympy.core.relational import Eq, Ne, Ge, Gt, Le, Lt
from sympy.external import import_module
from sympy.functions import \
Abs, ceiling, exp, floor, sign, sin, asin, sqrt, cos, \
acos, tan, atan, atan2, cosh, acosh, sinh, asinh, tanh, atanh, \
re, im, arg, erf, loggamma, log
from sympy.matrices import Matrix, MatrixBase, eye, randMatrix
from sympy.matrices.expressions import \
Determinant, HadamardProduct, Inverse, MatrixSymbol, Trace
from sympy.printing.tensorflow import tensorflow_code
from sympy.utilities.lambdify import lambdify
from sympy.testing.pytest import skip
from sympy.testing.pytest import XFAIL
tf = tensorflow = import_module("tensorflow")
if tensorflow:
# Hide Tensorflow warnings
import os
os.environ['TF_CPP_MIN_LOG_LEVEL'] = '2'
M = MatrixSymbol("M", 3, 3)
N = MatrixSymbol("N", 3, 3)
P = MatrixSymbol("P", 3, 3)
Q = MatrixSymbol("Q", 3, 3)
x, y, z, t = symbols("x y z t")
if tf is not None:
llo = [[j for j in range(i, i+3)] for i in range(0, 9, 3)]
m3x3 = tf.constant(llo)
m3x3sympy = Matrix(llo)
def _compare_tensorflow_matrix(variables, expr, use_float=False):
f = lambdify(variables, expr, 'tensorflow')
if not use_float:
random_matrices = [randMatrix(v.rows, v.cols) for v in variables]
else:
random_matrices = [randMatrix(v.rows, v.cols)/100. for v in variables]
graph = tf.Graph()
r = None
with graph.as_default():
random_variables = [eval(tensorflow_code(i)) for i in random_matrices]
session = tf.compat.v1.Session(graph=graph)
r = session.run(f(*random_variables))
e = expr.subs({k: v for k, v in zip(variables, random_matrices)})
e = e.doit()
if e.is_Matrix:
if not isinstance(e, MatrixBase):
e = e.as_explicit()
e = e.tolist()
if not use_float:
assert (r == e).all()
else:
r = [i for row in r for i in row]
e = [i for row in e for i in row]
assert all(
abs(a-b) < 10**-(4-int(log(abs(a), 10))) for a, b in zip(r, e))
# Creating a custom inverse test.
# See https://github.com/sympy/sympy/issues/18469
def _compare_tensorflow_matrix_inverse(variables, expr, use_float=False):
f = lambdify(variables, expr, 'tensorflow')
if not use_float:
random_matrices = [eye(v.rows, v.cols)*4 for v in variables]
else:
random_matrices = [eye(v.rows, v.cols)*3.14 for v in variables]
graph = tf.Graph()
r = None
with graph.as_default():
random_variables = [eval(tensorflow_code(i)) for i in random_matrices]
session = tf.compat.v1.Session(graph=graph)
r = session.run(f(*random_variables))
e = expr.subs({k: v for k, v in zip(variables, random_matrices)})
e = e.doit()
if e.is_Matrix:
if not isinstance(e, MatrixBase):
e = e.as_explicit()
e = e.tolist()
if not use_float:
assert (r == e).all()
else:
r = [i for row in r for i in row]
e = [i for row in e for i in row]
assert all(
abs(a-b) < 10**-(4-int(log(abs(a), 10))) for a, b in zip(r, e))
def _compare_tensorflow_matrix_scalar(variables, expr):
f = lambdify(variables, expr, 'tensorflow')
random_matrices = [
randMatrix(v.rows, v.cols).evalf() / 100 for v in variables]
graph = tf.Graph()
r = None
with graph.as_default():
random_variables = [eval(tensorflow_code(i)) for i in random_matrices]
session = tf.compat.v1.Session(graph=graph)
r = session.run(f(*random_variables))
e = expr.subs({k: v for k, v in zip(variables, random_matrices)})
e = e.doit()
assert abs(r-e) < 10**-6
def _compare_tensorflow_scalar(
variables, expr, rng=lambda: random.randint(0, 10)):
f = lambdify(variables, expr, 'tensorflow')
rvs = [rng() for v in variables]
graph = tf.Graph()
r = None
with graph.as_default():
tf_rvs = [eval(tensorflow_code(i)) for i in rvs]
session = tf.compat.v1.Session(graph=graph)
r = session.run(f(*tf_rvs))
e = expr.subs({k: v for k, v in zip(variables, rvs)}).evalf().doit()
assert abs(r-e) < 10**-6
def _compare_tensorflow_relational(
variables, expr, rng=lambda: random.randint(0, 10)):
f = lambdify(variables, expr, 'tensorflow')
rvs = [rng() for v in variables]
graph = tf.Graph()
r = None
with graph.as_default():
tf_rvs = [eval(tensorflow_code(i)) for i in rvs]
session = tf.compat.v1.Session(graph=graph)
r = session.run(f(*tf_rvs))
e = expr.subs({k: v for k, v in zip(variables, rvs)}).doit()
assert r == e
def test_tensorflow_printing():
assert tensorflow_code(eye(3)) == \
"tensorflow.constant([[1, 0, 0], [0, 1, 0], [0, 0, 1]])"
expr = Matrix([[x, sin(y)], [exp(z), -t]])
assert tensorflow_code(expr) == \
"tensorflow.Variable(" \
"[[x, tensorflow.math.sin(y)]," \
" [tensorflow.math.exp(z), -t]])"
# This (random) test is XFAIL because it fails occasionally
# See https://github.com/sympy/sympy/issues/18469
@XFAIL
def test_tensorflow_math():
if not tf:
skip("TensorFlow not installed")
expr = Abs(x)
assert tensorflow_code(expr) == "tensorflow.math.abs(x)"
_compare_tensorflow_scalar((x,), expr)
expr = sign(x)
assert tensorflow_code(expr) == "tensorflow.math.sign(x)"
_compare_tensorflow_scalar((x,), expr)
expr = ceiling(x)
assert tensorflow_code(expr) == "tensorflow.math.ceil(x)"
_compare_tensorflow_scalar((x,), expr, rng=lambda: random.random())
expr = floor(x)
assert tensorflow_code(expr) == "tensorflow.math.floor(x)"
_compare_tensorflow_scalar((x,), expr, rng=lambda: random.random())
expr = exp(x)
assert tensorflow_code(expr) == "tensorflow.math.exp(x)"
_compare_tensorflow_scalar((x,), expr, rng=lambda: random.random())
expr = sqrt(x)
assert tensorflow_code(expr) == "tensorflow.math.sqrt(x)"
_compare_tensorflow_scalar((x,), expr, rng=lambda: random.random())
expr = x ** 4
assert tensorflow_code(expr) == "tensorflow.math.pow(x, 4)"
_compare_tensorflow_scalar((x,), expr, rng=lambda: random.random())
expr = cos(x)
assert tensorflow_code(expr) == "tensorflow.math.cos(x)"
_compare_tensorflow_scalar((x,), expr, rng=lambda: random.random())
expr = acos(x)
assert tensorflow_code(expr) == "tensorflow.math.acos(x)"
_compare_tensorflow_scalar((x,), expr, rng=lambda: random.uniform(0, 0.95))
expr = sin(x)
assert tensorflow_code(expr) == "tensorflow.math.sin(x)"
_compare_tensorflow_scalar((x,), expr, rng=lambda: random.random())
expr = asin(x)
assert tensorflow_code(expr) == "tensorflow.math.asin(x)"
_compare_tensorflow_scalar((x,), expr, rng=lambda: random.random())
expr = tan(x)
assert tensorflow_code(expr) == "tensorflow.math.tan(x)"
_compare_tensorflow_scalar((x,), expr, rng=lambda: random.random())
expr = atan(x)
assert tensorflow_code(expr) == "tensorflow.math.atan(x)"
_compare_tensorflow_scalar((x,), expr, rng=lambda: random.random())
expr = atan2(y, x)
assert tensorflow_code(expr) == "tensorflow.math.atan2(y, x)"
_compare_tensorflow_scalar((y, x), expr, rng=lambda: random.random())
expr = cosh(x)
assert tensorflow_code(expr) == "tensorflow.math.cosh(x)"
_compare_tensorflow_scalar((x,), expr, rng=lambda: random.random())
expr = acosh(x)
assert tensorflow_code(expr) == "tensorflow.math.acosh(x)"
_compare_tensorflow_scalar((x,), expr, rng=lambda: random.uniform(1, 2))
expr = sinh(x)
assert tensorflow_code(expr) == "tensorflow.math.sinh(x)"
_compare_tensorflow_scalar((x,), expr, rng=lambda: random.uniform(1, 2))
expr = asinh(x)
assert tensorflow_code(expr) == "tensorflow.math.asinh(x)"
_compare_tensorflow_scalar((x,), expr, rng=lambda: random.uniform(1, 2))
expr = tanh(x)
assert tensorflow_code(expr) == "tensorflow.math.tanh(x)"
_compare_tensorflow_scalar((x,), expr, rng=lambda: random.uniform(1, 2))
expr = atanh(x)
assert tensorflow_code(expr) == "tensorflow.math.atanh(x)"
_compare_tensorflow_scalar(
(x,), expr, rng=lambda: random.uniform(-.5, .5))
expr = erf(x)
assert tensorflow_code(expr) == "tensorflow.math.erf(x)"
_compare_tensorflow_scalar(
(x,), expr, rng=lambda: random.random())
expr = loggamma(x)
assert tensorflow_code(expr) == "tensorflow.math.lgamma(x)"
_compare_tensorflow_scalar(
(x,), expr, rng=lambda: random.random())
def test_tensorflow_complexes():
assert tensorflow_code(re(x)) == "tensorflow.math.real(x)"
assert tensorflow_code(im(x)) == "tensorflow.math.imag(x)"
assert tensorflow_code(arg(x)) == "tensorflow.math.angle(x)"
def test_tensorflow_relational():
if not tf:
skip("TensorFlow not installed")
expr = Eq(x, y)
assert tensorflow_code(expr) == "tensorflow.math.equal(x, y)"
_compare_tensorflow_relational((x, y), expr)
expr = Ne(x, y)
assert tensorflow_code(expr) == "tensorflow.math.not_equal(x, y)"
_compare_tensorflow_relational((x, y), expr)
expr = Ge(x, y)
assert tensorflow_code(expr) == "tensorflow.math.greater_equal(x, y)"
_compare_tensorflow_relational((x, y), expr)
expr = Gt(x, y)
assert tensorflow_code(expr) == "tensorflow.math.greater(x, y)"
_compare_tensorflow_relational((x, y), expr)
expr = Le(x, y)
assert tensorflow_code(expr) == "tensorflow.math.less_equal(x, y)"
_compare_tensorflow_relational((x, y), expr)
expr = Lt(x, y)
assert tensorflow_code(expr) == "tensorflow.math.less(x, y)"
_compare_tensorflow_relational((x, y), expr)
# This (random) test is XFAIL because it fails occasionally
# See https://github.com/sympy/sympy/issues/18469
@XFAIL
def test_tensorflow_matrices():
if not tf:
skip("TensorFlow not installed")
expr = M
assert tensorflow_code(expr) == "M"
_compare_tensorflow_matrix((M,), expr)
expr = M + N
assert tensorflow_code(expr) == "tensorflow.math.add(M, N)"
_compare_tensorflow_matrix((M, N), expr)
expr = M * N
assert tensorflow_code(expr) == "tensorflow.linalg.matmul(M, N)"
_compare_tensorflow_matrix((M, N), expr)
expr = HadamardProduct(M, N)
assert tensorflow_code(expr) == "tensorflow.math.multiply(M, N)"
_compare_tensorflow_matrix((M, N), expr)
expr = M*N*P*Q
assert tensorflow_code(expr) == \
"tensorflow.linalg.matmul(" \
"tensorflow.linalg.matmul(" \
"tensorflow.linalg.matmul(M, N), P), Q)"
_compare_tensorflow_matrix((M, N, P, Q), expr)
expr = M**3
assert tensorflow_code(expr) == \
"tensorflow.linalg.matmul(tensorflow.linalg.matmul(M, M), M)"
_compare_tensorflow_matrix((M,), expr)
expr = Trace(M)
assert tensorflow_code(expr) == "tensorflow.linalg.trace(M)"
_compare_tensorflow_matrix((M,), expr)
expr = Determinant(M)
assert tensorflow_code(expr) == "tensorflow.linalg.det(M)"
_compare_tensorflow_matrix_scalar((M,), expr)
expr = Inverse(M)
assert tensorflow_code(expr) == "tensorflow.linalg.inv(M)"
_compare_tensorflow_matrix_inverse((M,), expr, use_float=True)
expr = M.T
assert tensorflow_code(expr, tensorflow_version='1.14') == \
"tensorflow.linalg.matrix_transpose(M)"
assert tensorflow_code(expr, tensorflow_version='1.13') == \
"tensorflow.matrix_transpose(M)"
_compare_tensorflow_matrix((M,), expr)
def test_codegen_einsum():
if not tf:
skip("TensorFlow not installed")
graph = tf.Graph()
with graph.as_default():
session = tf.compat.v1.Session(graph=graph)
M = MatrixSymbol("M", 2, 2)
N = MatrixSymbol("N", 2, 2)
cg = parse_matrix_expression(M*N)
f = lambdify((M, N), cg, 'tensorflow')
ma = tf.constant([[1, 2], [3, 4]])
mb = tf.constant([[1,-2], [-1, 3]])
y = session.run(f(ma, mb))
c = session.run(tf.matmul(ma, mb))
assert (y == c).all()
def test_codegen_extra():
if not tf:
skip("TensorFlow not installed")
graph = tf.Graph()
with graph.as_default():
session = tf.compat.v1.Session()
M = MatrixSymbol("M", 2, 2)
N = MatrixSymbol("N", 2, 2)
P = MatrixSymbol("P", 2, 2)
Q = MatrixSymbol("Q", 2, 2)
ma = tf.constant([[1, 2], [3, 4]])
mb = tf.constant([[1,-2], [-1, 3]])
mc = tf.constant([[2, 0], [1, 2]])
md = tf.constant([[1,-1], [4, 7]])
cg = CodegenArrayTensorProduct(M, N)
assert tensorflow_code(cg) == \
'tensorflow.linalg.einsum("ab,cd", M, N)'
f = lambdify((M, N), cg, 'tensorflow')
y = session.run(f(ma, mb))
c = session.run(tf.einsum("ij,kl", ma, mb))
assert (y == c).all()
cg = CodegenArrayElementwiseAdd(M, N)
assert tensorflow_code(cg) == 'tensorflow.math.add(M, N)'
f = lambdify((M, N), cg, 'tensorflow')
y = session.run(f(ma, mb))
c = session.run(ma + mb)
assert (y == c).all()
cg = CodegenArrayElementwiseAdd(M, N, P)
assert tensorflow_code(cg) == \
'tensorflow.math.add(tensorflow.math.add(M, N), P)'
f = lambdify((M, N, P), cg, 'tensorflow')
y = session.run(f(ma, mb, mc))
c = session.run(ma + mb + mc)
assert (y == c).all()
cg = CodegenArrayElementwiseAdd(M, N, P, Q)
assert tensorflow_code(cg) == \
'tensorflow.math.add(' \
'tensorflow.math.add(tensorflow.math.add(M, N), P), Q)'
f = lambdify((M, N, P, Q), cg, 'tensorflow')
y = session.run(f(ma, mb, mc, md))
c = session.run(ma + mb + mc + md)
assert (y == c).all()
cg = CodegenArrayPermuteDims(M, [1, 0])
assert tensorflow_code(cg) == 'tensorflow.transpose(M, [1, 0])'
f = lambdify((M,), cg, 'tensorflow')
y = session.run(f(ma))
c = session.run(tf.transpose(ma))
assert (y == c).all()
cg = CodegenArrayPermuteDims(CodegenArrayTensorProduct(M, N), [1, 2, 3, 0])
assert tensorflow_code(cg) == \
'tensorflow.transpose(' \
'tensorflow.linalg.einsum("ab,cd", M, N), [1, 2, 3, 0])'
f = lambdify((M, N), cg, 'tensorflow')
y = session.run(f(ma, mb))
c = session.run(tf.transpose(tf.einsum("ab,cd", ma, mb), [1, 2, 3, 0]))
assert (y == c).all()
cg = CodegenArrayDiagonal(CodegenArrayTensorProduct(M, N), (1, 2))
assert tensorflow_code(cg) == \
'tensorflow.linalg.einsum("ab,bc->acb", M, N)'
f = lambdify((M, N), cg, 'tensorflow')
y = session.run(f(ma, mb))
c = session.run(tf.einsum("ab,bc->acb", ma, mb))
assert (y == c).all()
def test_MatrixElement_printing():
A = MatrixSymbol("A", 1, 3)
B = MatrixSymbol("B", 1, 3)
C = MatrixSymbol("C", 1, 3)
assert tensorflow_code(A[0, 0]) == "A[0, 0]"
assert tensorflow_code(3 * A[0, 0]) == "3*A[0, 0]"
F = C[0, 0].subs(C, A - B)
assert tensorflow_code(F) == "(tensorflow.math.add((-1)*B, A))[0, 0]"
def test_tensorflow_Derivative():
expr = Derivative(sin(x), x)
assert tensorflow_code(expr) == \
"tensorflow.gradients(tensorflow.math.sin(x), x)[0]"
|
142488a0b2065f89d4c18c76aa1078941672e71eaa5d09cb8340ed6c201951a1 | from sympy import (
Piecewise, lambdify, Equality, Unequality, Sum, Mod, sqrt,
MatrixSymbol, BlockMatrix, Identity
)
from sympy import eye
from sympy.abc import x, i, j, a, b, c, d
from sympy.core import Pow
from sympy.codegen.matrix_nodes import MatrixSolve
from sympy.codegen.cfunctions import log1p, expm1, hypot, log10, exp2, log2, Sqrt
from sympy.codegen.array_utils import (CodegenArrayTensorProduct, CodegenArrayDiagonal,
CodegenArrayPermuteDims, CodegenArrayElementwiseAdd, parse_matrix_expression)
from sympy.printing.lambdarepr import NumPyPrinter
from sympy.testing.pytest import warns_deprecated_sympy
from sympy.testing.pytest import skip, raises
from sympy.external import import_module
np = import_module('numpy')
def test_numpy_piecewise_regression():
"""
NumPyPrinter needs to print Piecewise()'s choicelist as a list to avoid
breaking compatibility with numpy 1.8. This is not necessary in numpy 1.9+.
See gh-9747 and gh-9749 for details.
"""
printer = NumPyPrinter()
p = Piecewise((1, x < 0), (0, True))
assert printer.doprint(p) == \
'numpy.select([numpy.less(x, 0),True], [1,0], default=numpy.nan)'
assert printer.module_imports == {'numpy': {'select', 'less', 'nan'}}
def test_sum():
if not np:
skip("NumPy not installed")
s = Sum(x ** i, (i, a, b))
f = lambdify((a, b, x), s, 'numpy')
a_, b_ = 0, 10
x_ = np.linspace(-1, +1, 10)
assert np.allclose(f(a_, b_, x_), sum(x_ ** i_ for i_ in range(a_, b_ + 1)))
s = Sum(i * x, (i, a, b))
f = lambdify((a, b, x), s, 'numpy')
a_, b_ = 0, 10
x_ = np.linspace(-1, +1, 10)
assert np.allclose(f(a_, b_, x_), sum(i_ * x_ for i_ in range(a_, b_ + 1)))
def test_multiple_sums():
if not np:
skip("NumPy not installed")
s = Sum((x + j) * i, (i, a, b), (j, c, d))
f = lambdify((a, b, c, d, x), s, 'numpy')
a_, b_ = 0, 10
c_, d_ = 11, 21
x_ = np.linspace(-1, +1, 10)
assert np.allclose(f(a_, b_, c_, d_, x_),
sum((x_ + j_) * i_ for i_ in range(a_, b_ + 1) for j_ in range(c_, d_ + 1)))
def test_codegen_einsum():
if not np:
skip("NumPy not installed")
M = MatrixSymbol("M", 2, 2)
N = MatrixSymbol("N", 2, 2)
cg = parse_matrix_expression(M*N)
f = lambdify((M, N), cg, 'numpy')
ma = np.matrix([[1, 2], [3, 4]])
mb = np.matrix([[1,-2], [-1, 3]])
assert (f(ma, mb) == ma*mb).all()
def test_codegen_extra():
if not np:
skip("NumPy not installed")
M = MatrixSymbol("M", 2, 2)
N = MatrixSymbol("N", 2, 2)
P = MatrixSymbol("P", 2, 2)
Q = MatrixSymbol("Q", 2, 2)
ma = np.matrix([[1, 2], [3, 4]])
mb = np.matrix([[1,-2], [-1, 3]])
mc = np.matrix([[2, 0], [1, 2]])
md = np.matrix([[1,-1], [4, 7]])
cg = CodegenArrayTensorProduct(M, N)
f = lambdify((M, N), cg, 'numpy')
assert (f(ma, mb) == np.einsum(ma, [0, 1], mb, [2, 3])).all()
cg = CodegenArrayElementwiseAdd(M, N)
f = lambdify((M, N), cg, 'numpy')
assert (f(ma, mb) == ma+mb).all()
cg = CodegenArrayElementwiseAdd(M, N, P)
f = lambdify((M, N, P), cg, 'numpy')
assert (f(ma, mb, mc) == ma+mb+mc).all()
cg = CodegenArrayElementwiseAdd(M, N, P, Q)
f = lambdify((M, N, P, Q), cg, 'numpy')
assert (f(ma, mb, mc, md) == ma+mb+mc+md).all()
cg = CodegenArrayPermuteDims(M, [1, 0])
f = lambdify((M,), cg, 'numpy')
assert (f(ma) == ma.T).all()
cg = CodegenArrayPermuteDims(CodegenArrayTensorProduct(M, N), [1, 2, 3, 0])
f = lambdify((M, N), cg, 'numpy')
assert (f(ma, mb) == np.transpose(np.einsum(ma, [0, 1], mb, [2, 3]), (1, 2, 3, 0))).all()
cg = CodegenArrayDiagonal(CodegenArrayTensorProduct(M, N), (1, 2))
f = lambdify((M, N), cg, 'numpy')
assert (f(ma, mb) == np.diagonal(np.einsum(ma, [0, 1], mb, [2, 3]), axis1=1, axis2=2)).all()
def test_relational():
if not np:
skip("NumPy not installed")
e = Equality(x, 1)
f = lambdify((x,), e)
x_ = np.array([0, 1, 2])
assert np.array_equal(f(x_), [False, True, False])
e = Unequality(x, 1)
f = lambdify((x,), e)
x_ = np.array([0, 1, 2])
assert np.array_equal(f(x_), [True, False, True])
e = (x < 1)
f = lambdify((x,), e)
x_ = np.array([0, 1, 2])
assert np.array_equal(f(x_), [True, False, False])
e = (x <= 1)
f = lambdify((x,), e)
x_ = np.array([0, 1, 2])
assert np.array_equal(f(x_), [True, True, False])
e = (x > 1)
f = lambdify((x,), e)
x_ = np.array([0, 1, 2])
assert np.array_equal(f(x_), [False, False, True])
e = (x >= 1)
f = lambdify((x,), e)
x_ = np.array([0, 1, 2])
assert np.array_equal(f(x_), [False, True, True])
def test_mod():
if not np:
skip("NumPy not installed")
e = Mod(a, b)
f = lambdify((a, b), e)
a_ = np.array([0, 1, 2, 3])
b_ = 2
assert np.array_equal(f(a_, b_), [0, 1, 0, 1])
a_ = np.array([0, 1, 2, 3])
b_ = np.array([2, 2, 2, 2])
assert np.array_equal(f(a_, b_), [0, 1, 0, 1])
a_ = np.array([2, 3, 4, 5])
b_ = np.array([2, 3, 4, 5])
assert np.array_equal(f(a_, b_), [0, 0, 0, 0])
def test_pow():
if not np:
skip('NumPy not installed')
expr = Pow(2, -1, evaluate=False)
f = lambdify([], expr, 'numpy')
assert f() == 0.5
def test_expm1():
if not np:
skip("NumPy not installed")
f = lambdify((a,), expm1(a), 'numpy')
assert abs(f(1e-10) - 1e-10 - 5e-21) < 1e-22
def test_log1p():
if not np:
skip("NumPy not installed")
f = lambdify((a,), log1p(a), 'numpy')
assert abs(f(1e-99) - 1e-99) < 1e-100
def test_hypot():
if not np:
skip("NumPy not installed")
assert abs(lambdify((a, b), hypot(a, b), 'numpy')(3, 4) - 5) < 1e-16
def test_log10():
if not np:
skip("NumPy not installed")
assert abs(lambdify((a,), log10(a), 'numpy')(100) - 2) < 1e-16
def test_exp2():
if not np:
skip("NumPy not installed")
assert abs(lambdify((a,), exp2(a), 'numpy')(5) - 32) < 1e-16
def test_log2():
if not np:
skip("NumPy not installed")
assert abs(lambdify((a,), log2(a), 'numpy')(256) - 8) < 1e-16
def test_Sqrt():
if not np:
skip("NumPy not installed")
assert abs(lambdify((a,), Sqrt(a), 'numpy')(4) - 2) < 1e-16
def test_sqrt():
if not np:
skip("NumPy not installed")
assert abs(lambdify((a,), sqrt(a), 'numpy')(4) - 2) < 1e-16
def test_matsolve():
if not np:
skip("NumPy not installed")
M = MatrixSymbol("M", 3, 3)
x = MatrixSymbol("x", 3, 1)
expr = M**(-1) * x + x
matsolve_expr = MatrixSolve(M, x) + x
f = lambdify((M, x), expr)
f_matsolve = lambdify((M, x), matsolve_expr)
m0 = np.array([[1, 2, 3], [3, 2, 5], [5, 6, 7]])
assert np.linalg.matrix_rank(m0) == 3
x0 = np.array([3, 4, 5])
assert np.allclose(f_matsolve(m0, x0), f(m0, x0))
def test_issue_15601():
if not np:
skip("Numpy not installed")
M = MatrixSymbol("M", 3, 3)
N = MatrixSymbol("N", 3, 3)
expr = M*N
f = lambdify((M, N), expr, "numpy")
with warns_deprecated_sympy():
ans = f(eye(3), eye(3))
assert np.array_equal(ans, np.array([1, 0, 0, 0, 1, 0, 0, 0, 1]))
def test_16857():
if not np:
skip("NumPy not installed")
a_1 = MatrixSymbol('a_1', 10, 3)
a_2 = MatrixSymbol('a_2', 10, 3)
a_3 = MatrixSymbol('a_3', 10, 3)
a_4 = MatrixSymbol('a_4', 10, 3)
A = BlockMatrix([[a_1, a_2], [a_3, a_4]])
assert A.shape == (20, 6)
printer = NumPyPrinter()
assert printer.doprint(A) == 'numpy.block([[a_1, a_2], [a_3, a_4]])'
def test_issue_17006():
if not np:
skip("NumPy not installed")
M = MatrixSymbol("M", 2, 2)
f = lambdify(M, M + Identity(2))
ma = np.array([[1, 2], [3, 4]])
mr = np.array([[2, 2], [3, 5]])
assert (f(ma) == mr).all()
from sympy import symbols
n = symbols('n', integer=True)
N = MatrixSymbol("M", n, n)
raises(NotImplementedError, lambda: lambdify(N, N + Identity(n)))
|
07bef2bc9d6da22b96e9c054160e498fcc7c6c490ab49e64870dd7aaa591677e | # -*- coding: utf-8 -*-
from sympy import (
Add, And, Basic, Derivative, Dict, Eq, Equivalent, FF,
FiniteSet, Function, Ge, Gt, I, Implies, Integral, SingularityFunction,
Lambda, Le, Limit, Lt, Matrix, Mul, Nand, Ne, Nor, Not, O, Or,
Pow, Product, QQ, RR, Rational, Ray, rootof, RootSum, S,
Segment, Subs, Sum, Symbol, Tuple, Trace, Xor, ZZ, conjugate,
groebner, oo, pi, symbols, ilex, grlex, Range, Contains,
SeqPer, SeqFormula, SeqAdd, SeqMul, fourier_series, fps, ITE,
Complement, Interval, Intersection, Union, EulerGamma, GoldenRatio,
LambertW, airyai, airybi, airyaiprime, airybiprime, fresnelc, fresnels,
Heaviside, dirichlet_eta, diag, MatrixSlice)
from sympy.codegen.ast import (Assignment, AddAugmentedAssignment,
SubAugmentedAssignment, MulAugmentedAssignment, DivAugmentedAssignment, ModAugmentedAssignment)
from sympy.core.compatibility import u_decode as u
from sympy.core.expr import UnevaluatedExpr
from sympy.core.trace import Tr
from sympy.functions import (Abs, Chi, Ci, Ei, KroneckerDelta,
Piecewise, Shi, Si, atan2, beta, binomial, catalan, ceiling, cos,
euler, exp, expint, factorial, factorial2, floor, gamma, hyper, log,
meijerg, sin, sqrt, subfactorial, tan, uppergamma, lerchphi,
elliptic_k, elliptic_f, elliptic_e, elliptic_pi, DiracDelta, bell,
bernoulli, fibonacci, tribonacci, lucas, stieltjes, mathieuc, mathieus,
mathieusprime, mathieucprime)
from sympy.matrices import Adjoint, Inverse, MatrixSymbol, Transpose, KroneckerProduct
from sympy.matrices.expressions import hadamard_power
from sympy.physics import mechanics
from sympy.physics.units import joule, degree
from sympy.printing.pretty import pprint, pretty as xpretty
from sympy.printing.pretty.pretty_symbology import center_accent, is_combining
from sympy import ConditionSet
from sympy.sets import ImageSet, ProductSet
from sympy.sets.setexpr import SetExpr
from sympy.tensor.array import (ImmutableDenseNDimArray, ImmutableSparseNDimArray,
MutableDenseNDimArray, MutableSparseNDimArray, tensorproduct)
from sympy.tensor.functions import TensorProduct
from sympy.tensor.tensor import (TensorIndexType, tensor_indices, TensorHead,
TensorElement, tensor_heads)
from sympy.testing.pytest import raises
from sympy.vector import CoordSys3D, Gradient, Curl, Divergence, Dot, Cross, Laplacian
import sympy as sym
class lowergamma(sym.lowergamma):
pass # testing notation inheritance by a subclass with same name
a, b, c, d, x, y, z, k, n = symbols('a,b,c,d,x,y,z,k,n')
f = Function("f")
th = Symbol('theta')
ph = Symbol('phi')
"""
Expressions whose pretty-printing is tested here:
(A '#' to the right of an expression indicates that its various acceptable
orderings are accounted for by the tests.)
BASIC EXPRESSIONS:
oo
(x**2)
1/x
y*x**-2
x**Rational(-5,2)
(-2)**x
Pow(3, 1, evaluate=False)
(x**2 + x + 1) #
1-x #
1-2*x #
x/y
-x/y
(x+2)/y #
(1+x)*y #3
-5*x/(x+10) # correct placement of negative sign
1 - Rational(3,2)*(x+1)
-(-x + 5)*(-x - 2*sqrt(2) + 5) - (-y + 5)*(-y + 5) # issue 5524
ORDERING:
x**2 + x + 1
1 - x
1 - 2*x
2*x**4 + y**2 - x**2 + y**3
RELATIONAL:
Eq(x, y)
Lt(x, y)
Gt(x, y)
Le(x, y)
Ge(x, y)
Ne(x/(y+1), y**2) #
RATIONAL NUMBERS:
y*x**-2
y**Rational(3,2) * x**Rational(-5,2)
sin(x)**3/tan(x)**2
FUNCTIONS (ABS, CONJ, EXP, FUNCTION BRACES, FACTORIAL, FLOOR, CEILING):
(2*x + exp(x)) #
Abs(x)
Abs(x/(x**2+1)) #
Abs(1 / (y - Abs(x)))
factorial(n)
factorial(2*n)
subfactorial(n)
subfactorial(2*n)
factorial(factorial(factorial(n)))
factorial(n+1) #
conjugate(x)
conjugate(f(x+1)) #
f(x)
f(x, y)
f(x/(y+1), y) #
f(x**x**x**x**x**x)
sin(x)**2
conjugate(a+b*I)
conjugate(exp(a+b*I))
conjugate( f(1 + conjugate(f(x))) ) #
f(x/(y+1), y) # denom of first arg
floor(1 / (y - floor(x)))
ceiling(1 / (y - ceiling(x)))
SQRT:
sqrt(2)
2**Rational(1,3)
2**Rational(1,1000)
sqrt(x**2 + 1)
(1 + sqrt(5))**Rational(1,3)
2**(1/x)
sqrt(2+pi)
(2+(1+x**2)/(2+x))**Rational(1,4)+(1+x**Rational(1,1000))/sqrt(3+x**2)
DERIVATIVES:
Derivative(log(x), x, evaluate=False)
Derivative(log(x), x, evaluate=False) + x #
Derivative(log(x) + x**2, x, y, evaluate=False)
Derivative(2*x*y, y, x, evaluate=False) + x**2 #
beta(alpha).diff(alpha)
INTEGRALS:
Integral(log(x), x)
Integral(x**2, x)
Integral((sin(x))**2 / (tan(x))**2)
Integral(x**(2**x), x)
Integral(x**2, (x,1,2))
Integral(x**2, (x,Rational(1,2),10))
Integral(x**2*y**2, x,y)
Integral(x**2, (x, None, 1))
Integral(x**2, (x, 1, None))
Integral(sin(th)/cos(ph), (th,0,pi), (ph, 0, 2*pi))
MATRICES:
Matrix([[x**2+1, 1], [y, x+y]]) #
Matrix([[x/y, y, th], [0, exp(I*k*ph), 1]])
PIECEWISE:
Piecewise((x,x<1),(x**2,True))
ITE:
ITE(x, y, z)
SEQUENCES (TUPLES, LISTS, DICTIONARIES):
()
[]
{}
(1/x,)
[x**2, 1/x, x, y, sin(th)**2/cos(ph)**2]
(x**2, 1/x, x, y, sin(th)**2/cos(ph)**2)
{x: sin(x)}
{1/x: 1/y, x: sin(x)**2} #
[x**2]
(x**2,)
{x**2: 1}
LIMITS:
Limit(x, x, oo)
Limit(x**2, x, 0)
Limit(1/x, x, 0)
Limit(sin(x)/x, x, 0)
UNITS:
joule => kg*m**2/s
SUBS:
Subs(f(x), x, ph**2)
Subs(f(x).diff(x), x, 0)
Subs(f(x).diff(x)/y, (x, y), (0, Rational(1, 2)))
ORDER:
O(1)
O(1/x)
O(x**2 + y**2)
"""
def pretty(expr, order=None):
"""ASCII pretty-printing"""
return xpretty(expr, order=order, use_unicode=False, wrap_line=False)
def upretty(expr, order=None):
"""Unicode pretty-printing"""
return xpretty(expr, order=order, use_unicode=True, wrap_line=False)
def test_pretty_ascii_str():
assert pretty( 'xxx' ) == 'xxx'
assert pretty( "xxx" ) == 'xxx'
assert pretty( 'xxx\'xxx' ) == 'xxx\'xxx'
assert pretty( 'xxx"xxx' ) == 'xxx\"xxx'
assert pretty( 'xxx\"xxx' ) == 'xxx\"xxx'
assert pretty( "xxx'xxx" ) == 'xxx\'xxx'
assert pretty( "xxx\'xxx" ) == 'xxx\'xxx'
assert pretty( "xxx\"xxx" ) == 'xxx\"xxx'
assert pretty( "xxx\"xxx\'xxx" ) == 'xxx"xxx\'xxx'
assert pretty( "xxx\nxxx" ) == 'xxx\nxxx'
def test_pretty_unicode_str():
assert pretty( u'xxx' ) == u'xxx'
assert pretty( u'xxx' ) == u'xxx'
assert pretty( u'xxx\'xxx' ) == u'xxx\'xxx'
assert pretty( u'xxx"xxx' ) == u'xxx\"xxx'
assert pretty( u'xxx\"xxx' ) == u'xxx\"xxx'
assert pretty( u"xxx'xxx" ) == u'xxx\'xxx'
assert pretty( u"xxx\'xxx" ) == u'xxx\'xxx'
assert pretty( u"xxx\"xxx" ) == u'xxx\"xxx'
assert pretty( u"xxx\"xxx\'xxx" ) == u'xxx"xxx\'xxx'
assert pretty( u"xxx\nxxx" ) == u'xxx\nxxx'
def test_upretty_greek():
assert upretty( oo ) == u'∞'
assert upretty( Symbol('alpha^+_1') ) == u'α⁺₁'
assert upretty( Symbol('beta') ) == u'β'
assert upretty(Symbol('lambda')) == u'λ'
def test_upretty_multiindex():
assert upretty( Symbol('beta12') ) == u'β₁₂'
assert upretty( Symbol('Y00') ) == u'Y₀₀'
assert upretty( Symbol('Y_00') ) == u'Y₀₀'
assert upretty( Symbol('F^+-') ) == u'F⁺⁻'
def test_upretty_sub_super():
assert upretty( Symbol('beta_1_2') ) == u'β₁ ₂'
assert upretty( Symbol('beta^1^2') ) == u'β¹ ²'
assert upretty( Symbol('beta_1^2') ) == u'β²₁'
assert upretty( Symbol('beta_10_20') ) == u'β₁₀ ₂₀'
assert upretty( Symbol('beta_ax_gamma^i') ) == u'βⁱₐₓ ᵧ'
assert upretty( Symbol("F^1^2_3_4") ) == u'F¹ ²₃ ₄'
assert upretty( Symbol("F_1_2^3^4") ) == u'F³ ⁴₁ ₂'
assert upretty( Symbol("F_1_2_3_4") ) == u'F₁ ₂ ₃ ₄'
assert upretty( Symbol("F^1^2^3^4") ) == u'F¹ ² ³ ⁴'
def test_upretty_subs_missing_in_24():
assert upretty( Symbol('F_beta') ) == u'Fᵦ'
assert upretty( Symbol('F_gamma') ) == u'Fᵧ'
assert upretty( Symbol('F_rho') ) == u'Fᵨ'
assert upretty( Symbol('F_phi') ) == u'Fᵩ'
assert upretty( Symbol('F_chi') ) == u'Fᵪ'
assert upretty( Symbol('F_a') ) == u'Fₐ'
assert upretty( Symbol('F_e') ) == u'Fₑ'
assert upretty( Symbol('F_i') ) == u'Fᵢ'
assert upretty( Symbol('F_o') ) == u'Fₒ'
assert upretty( Symbol('F_u') ) == u'Fᵤ'
assert upretty( Symbol('F_r') ) == u'Fᵣ'
assert upretty( Symbol('F_v') ) == u'Fᵥ'
assert upretty( Symbol('F_x') ) == u'Fₓ'
def test_missing_in_2X_issue_9047():
assert upretty( Symbol('F_h') ) == u'Fₕ'
assert upretty( Symbol('F_k') ) == u'Fₖ'
assert upretty( Symbol('F_l') ) == u'Fₗ'
assert upretty( Symbol('F_m') ) == u'Fₘ'
assert upretty( Symbol('F_n') ) == u'Fₙ'
assert upretty( Symbol('F_p') ) == u'Fₚ'
assert upretty( Symbol('F_s') ) == u'Fₛ'
assert upretty( Symbol('F_t') ) == u'Fₜ'
def test_upretty_modifiers():
# Accents
assert upretty( Symbol('Fmathring') ) == u'F̊'
assert upretty( Symbol('Fddddot') ) == u'F⃜'
assert upretty( Symbol('Fdddot') ) == u'F⃛'
assert upretty( Symbol('Fddot') ) == u'F̈'
assert upretty( Symbol('Fdot') ) == u'Ḟ'
assert upretty( Symbol('Fcheck') ) == u'F̌'
assert upretty( Symbol('Fbreve') ) == u'F̆'
assert upretty( Symbol('Facute') ) == u'F́'
assert upretty( Symbol('Fgrave') ) == u'F̀'
assert upretty( Symbol('Ftilde') ) == u'F̃'
assert upretty( Symbol('Fhat') ) == u'F̂'
assert upretty( Symbol('Fbar') ) == u'F̅'
assert upretty( Symbol('Fvec') ) == u'F⃗'
assert upretty( Symbol('Fprime') ) == u'F′'
assert upretty( Symbol('Fprm') ) == u'F′'
# No faces are actually implemented, but test to make sure the modifiers are stripped
assert upretty( Symbol('Fbold') ) == u'Fbold'
assert upretty( Symbol('Fbm') ) == u'Fbm'
assert upretty( Symbol('Fcal') ) == u'Fcal'
assert upretty( Symbol('Fscr') ) == u'Fscr'
assert upretty( Symbol('Ffrak') ) == u'Ffrak'
# Brackets
assert upretty( Symbol('Fnorm') ) == u'‖F‖'
assert upretty( Symbol('Favg') ) == u'⟨F⟩'
assert upretty( Symbol('Fabs') ) == u'|F|'
assert upretty( Symbol('Fmag') ) == u'|F|'
# Combinations
assert upretty( Symbol('xvecdot') ) == u'x⃗̇'
assert upretty( Symbol('xDotVec') ) == u'ẋ⃗'
assert upretty( Symbol('xHATNorm') ) == u'‖x̂‖'
assert upretty( Symbol('xMathring_yCheckPRM__zbreveAbs') ) == u'x̊_y̌′__|z̆|'
assert upretty( Symbol('alphadothat_nVECDOT__tTildePrime') ) == u'α̇̂_n⃗̇__t̃′'
assert upretty( Symbol('x_dot') ) == u'x_dot'
assert upretty( Symbol('x__dot') ) == u'x__dot'
def test_pretty_Cycle():
from sympy.combinatorics.permutations import Cycle
assert pretty(Cycle(1, 2)) == '(1 2)'
assert pretty(Cycle(2)) == '(2)'
assert pretty(Cycle(1, 3)(4, 5)) == '(1 3)(4 5)'
assert pretty(Cycle()) == '()'
def test_pretty_Permutation():
from sympy.combinatorics.permutations import Permutation
p1 = Permutation(1, 2)(3, 4)
assert xpretty(p1, perm_cyclic=True, use_unicode=True) == "(1 2)(3 4)"
assert xpretty(p1, perm_cyclic=True, use_unicode=False) == "(1 2)(3 4)"
assert xpretty(p1, perm_cyclic=False, use_unicode=True) == \
u'⎛0 1 2 3 4⎞\n'\
u'⎝0 2 1 4 3⎠'
assert xpretty(p1, perm_cyclic=False, use_unicode=False) == \
"/0 1 2 3 4\\\n"\
"\\0 2 1 4 3/"
def test_pretty_basic():
assert pretty( -Rational(1)/2 ) == '-1/2'
assert pretty( -Rational(13)/22 ) == \
"""\
-13 \n\
----\n\
22 \
"""
expr = oo
ascii_str = \
"""\
oo\
"""
ucode_str = \
u("""\
∞\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = (x**2)
ascii_str = \
"""\
2\n\
x \
"""
ucode_str = \
u("""\
2\n\
x \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = 1/x
ascii_str = \
"""\
1\n\
-\n\
x\
"""
ucode_str = \
u("""\
1\n\
─\n\
x\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
# not the same as 1/x
expr = x**-1.0
ascii_str = \
"""\
-1.0\n\
x \
"""
ucode_str = \
("""\
-1.0\n\
x \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
# see issue #2860
expr = Pow(S(2), -1.0, evaluate=False)
ascii_str = \
"""\
-1.0\n\
2 \
"""
ucode_str = \
("""\
-1.0\n\
2 \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = y*x**-2
ascii_str = \
"""\
y \n\
--\n\
2\n\
x \
"""
ucode_str = \
u("""\
y \n\
──\n\
2\n\
x \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
#see issue #14033
expr = x**Rational(1, 3)
ascii_str = \
"""\
1/3\n\
x \
"""
ucode_str = \
u("""\
1/3\n\
x \
""")
assert xpretty(expr, use_unicode=False, wrap_line=False,\
root_notation = False) == ascii_str
assert xpretty(expr, use_unicode=True, wrap_line=False,\
root_notation = False) == ucode_str
expr = x**Rational(-5, 2)
ascii_str = \
"""\
1 \n\
----\n\
5/2\n\
x \
"""
ucode_str = \
u("""\
1 \n\
────\n\
5/2\n\
x \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = (-2)**x
ascii_str = \
"""\
x\n\
(-2) \
"""
ucode_str = \
u("""\
x\n\
(-2) \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
# See issue 4923
expr = Pow(3, 1, evaluate=False)
ascii_str = \
"""\
1\n\
3 \
"""
ucode_str = \
u("""\
1\n\
3 \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = (x**2 + x + 1)
ascii_str_1 = \
"""\
2\n\
1 + x + x \
"""
ascii_str_2 = \
"""\
2 \n\
x + x + 1\
"""
ascii_str_3 = \
"""\
2 \n\
x + 1 + x\
"""
ucode_str_1 = \
u("""\
2\n\
1 + x + x \
""")
ucode_str_2 = \
u("""\
2 \n\
x + x + 1\
""")
ucode_str_3 = \
u("""\
2 \n\
x + 1 + x\
""")
assert pretty(expr) in [ascii_str_1, ascii_str_2, ascii_str_3]
assert upretty(expr) in [ucode_str_1, ucode_str_2, ucode_str_3]
expr = 1 - x
ascii_str_1 = \
"""\
1 - x\
"""
ascii_str_2 = \
"""\
-x + 1\
"""
ucode_str_1 = \
u("""\
1 - x\
""")
ucode_str_2 = \
u("""\
-x + 1\
""")
assert pretty(expr) in [ascii_str_1, ascii_str_2]
assert upretty(expr) in [ucode_str_1, ucode_str_2]
expr = 1 - 2*x
ascii_str_1 = \
"""\
1 - 2*x\
"""
ascii_str_2 = \
"""\
-2*x + 1\
"""
ucode_str_1 = \
u("""\
1 - 2⋅x\
""")
ucode_str_2 = \
u("""\
-2⋅x + 1\
""")
assert pretty(expr) in [ascii_str_1, ascii_str_2]
assert upretty(expr) in [ucode_str_1, ucode_str_2]
expr = x/y
ascii_str = \
"""\
x\n\
-\n\
y\
"""
ucode_str = \
u("""\
x\n\
─\n\
y\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = -x/y
ascii_str = \
"""\
-x \n\
---\n\
y \
"""
ucode_str = \
u("""\
-x \n\
───\n\
y \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = (x + 2)/y
ascii_str_1 = \
"""\
2 + x\n\
-----\n\
y \
"""
ascii_str_2 = \
"""\
x + 2\n\
-----\n\
y \
"""
ucode_str_1 = \
u("""\
2 + x\n\
─────\n\
y \
""")
ucode_str_2 = \
u("""\
x + 2\n\
─────\n\
y \
""")
assert pretty(expr) in [ascii_str_1, ascii_str_2]
assert upretty(expr) in [ucode_str_1, ucode_str_2]
expr = (1 + x)*y
ascii_str_1 = \
"""\
y*(1 + x)\
"""
ascii_str_2 = \
"""\
(1 + x)*y\
"""
ascii_str_3 = \
"""\
y*(x + 1)\
"""
ucode_str_1 = \
u("""\
y⋅(1 + x)\
""")
ucode_str_2 = \
u("""\
(1 + x)⋅y\
""")
ucode_str_3 = \
u("""\
y⋅(x + 1)\
""")
assert pretty(expr) in [ascii_str_1, ascii_str_2, ascii_str_3]
assert upretty(expr) in [ucode_str_1, ucode_str_2, ucode_str_3]
# Test for correct placement of the negative sign
expr = -5*x/(x + 10)
ascii_str_1 = \
"""\
-5*x \n\
------\n\
10 + x\
"""
ascii_str_2 = \
"""\
-5*x \n\
------\n\
x + 10\
"""
ucode_str_1 = \
u("""\
-5⋅x \n\
──────\n\
10 + x\
""")
ucode_str_2 = \
u("""\
-5⋅x \n\
──────\n\
x + 10\
""")
assert pretty(expr) in [ascii_str_1, ascii_str_2]
assert upretty(expr) in [ucode_str_1, ucode_str_2]
expr = -S.Half - 3*x
ascii_str = \
"""\
-3*x - 1/2\
"""
ucode_str = \
u("""\
-3⋅x - 1/2\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = S.Half - 3*x
ascii_str = \
"""\
1/2 - 3*x\
"""
ucode_str = \
u("""\
1/2 - 3⋅x\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = -S.Half - 3*x/2
ascii_str = \
"""\
3*x 1\n\
- --- - -\n\
2 2\
"""
ucode_str = \
u("""\
3⋅x 1\n\
- ─── - ─\n\
2 2\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = S.Half - 3*x/2
ascii_str = \
"""\
1 3*x\n\
- - ---\n\
2 2 \
"""
ucode_str = \
u("""\
1 3⋅x\n\
─ - ───\n\
2 2 \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_negative_fractions():
expr = -x/y
ascii_str =\
"""\
-x \n\
---\n\
y \
"""
ucode_str =\
u("""\
-x \n\
───\n\
y \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = -x*z/y
ascii_str =\
"""\
-x*z \n\
-----\n\
y \
"""
ucode_str =\
u("""\
-x⋅z \n\
─────\n\
y \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = x**2/y
ascii_str =\
"""\
2\n\
x \n\
--\n\
y \
"""
ucode_str =\
u("""\
2\n\
x \n\
──\n\
y \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = -x**2/y
ascii_str =\
"""\
2 \n\
-x \n\
----\n\
y \
"""
ucode_str =\
u("""\
2 \n\
-x \n\
────\n\
y \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = -x/(y*z)
ascii_str =\
"""\
-x \n\
---\n\
y*z\
"""
ucode_str =\
u("""\
-x \n\
───\n\
y⋅z\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = -a/y**2
ascii_str =\
"""\
-a \n\
---\n\
2\n\
y \
"""
ucode_str =\
u("""\
-a \n\
───\n\
2\n\
y \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = y**(-a/b)
ascii_str =\
"""\
-a \n\
---\n\
b \n\
y \
"""
ucode_str =\
u("""\
-a \n\
───\n\
b \n\
y \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = -1/y**2
ascii_str =\
"""\
-1 \n\
---\n\
2\n\
y \
"""
ucode_str =\
u("""\
-1 \n\
───\n\
2\n\
y \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = -10/b**2
ascii_str =\
"""\
-10 \n\
----\n\
2 \n\
b \
"""
ucode_str =\
u("""\
-10 \n\
────\n\
2 \n\
b \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Rational(-200, 37)
ascii_str =\
"""\
-200 \n\
-----\n\
37 \
"""
ucode_str =\
u("""\
-200 \n\
─────\n\
37 \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Mul(0, 1, evaluate=False)
assert pretty(expr) == "0*1"
assert upretty(expr) == "0⋅1"
expr = Mul(1, 0, evaluate=False)
assert pretty(expr) == "1*0"
assert upretty(expr) == "1⋅0"
expr = Mul(1, 1, evaluate=False)
assert pretty(expr) == "1*1"
assert upretty(expr) == "1⋅1"
expr = Mul(1, 1, 1, evaluate=False)
assert pretty(expr) == "1*1*1"
assert upretty(expr) == "1⋅1⋅1"
expr = Mul(1, 2, evaluate=False)
assert pretty(expr) == "1*2"
assert upretty(expr) == "1⋅2"
expr = Add(0, 1, evaluate=False)
assert pretty(expr) == "0 + 1"
assert upretty(expr) == "0 + 1"
expr = Mul(1, 1, 2, evaluate=False)
assert pretty(expr) == "1*1*2"
assert upretty(expr) == "1⋅1⋅2"
expr = Add(0, 0, 1, evaluate=False)
assert pretty(expr) == "0 + 0 + 1"
assert upretty(expr) == "0 + 0 + 1"
expr = Mul(1, -1, evaluate=False)
assert pretty(expr) == "1*(-1)"
assert upretty(expr) == "1⋅(-1)"
expr = Mul(1.0, x, evaluate=False)
assert pretty(expr) == "1.0*x"
assert upretty(expr) == "1.0⋅x"
expr = Mul(1, 1, 2, 3, x, evaluate=False)
assert pretty(expr) == "1*1*2*3*x"
assert upretty(expr) == "1⋅1⋅2⋅3⋅x"
expr = Mul(-1, 1, evaluate=False)
assert pretty(expr) == "-1*1"
assert upretty(expr) == "-1⋅1"
expr = Mul(4, 3, 2, 1, 0, y, x, evaluate=False)
assert pretty(expr) == "4*3*2*1*0*y*x"
assert upretty(expr) == "4⋅3⋅2⋅1⋅0⋅y⋅x"
expr = Mul(4, 3, 2, 1+z, 0, y, x, evaluate=False)
assert pretty(expr) == "4*3*2*(z + 1)*0*y*x"
assert upretty(expr) == "4⋅3⋅2⋅(z + 1)⋅0⋅y⋅x"
expr = Mul(Rational(2, 3), Rational(5, 7), evaluate=False)
assert pretty(expr) == "2/3*5/7"
assert upretty(expr) == "2/3⋅5/7"
def test_issue_5524():
assert pretty(-(-x + 5)*(-x - 2*sqrt(2) + 5) - (-y + 5)*(-y + 5)) == \
"""\
2 / ___ \\\n\
- (5 - y) + (x - 5)*\\-x - 2*\\/ 2 + 5/\
"""
assert upretty(-(-x + 5)*(-x - 2*sqrt(2) + 5) - (-y + 5)*(-y + 5)) == \
u("""\
2 \n\
- (5 - y) + (x - 5)⋅(-x - 2⋅√2 + 5)\
""")
def test_pretty_ordering():
assert pretty(x**2 + x + 1, order='lex') == \
"""\
2 \n\
x + x + 1\
"""
assert pretty(x**2 + x + 1, order='rev-lex') == \
"""\
2\n\
1 + x + x \
"""
assert pretty(1 - x, order='lex') == '-x + 1'
assert pretty(1 - x, order='rev-lex') == '1 - x'
assert pretty(1 - 2*x, order='lex') == '-2*x + 1'
assert pretty(1 - 2*x, order='rev-lex') == '1 - 2*x'
f = 2*x**4 + y**2 - x**2 + y**3
assert pretty(f, order=None) == \
"""\
4 2 3 2\n\
2*x - x + y + y \
"""
assert pretty(f, order='lex') == \
"""\
4 2 3 2\n\
2*x - x + y + y \
"""
assert pretty(f, order='rev-lex') == \
"""\
2 3 2 4\n\
y + y - x + 2*x \
"""
expr = x - x**3/6 + x**5/120 + O(x**6)
ascii_str = \
"""\
3 5 \n\
x x / 6\\\n\
x - -- + --- + O\\x /\n\
6 120 \
"""
ucode_str = \
u("""\
3 5 \n\
x x ⎛ 6⎞\n\
x - ── + ─── + O⎝x ⎠\n\
6 120 \
""")
assert pretty(expr, order=None) == ascii_str
assert upretty(expr, order=None) == ucode_str
assert pretty(expr, order='lex') == ascii_str
assert upretty(expr, order='lex') == ucode_str
assert pretty(expr, order='rev-lex') == ascii_str
assert upretty(expr, order='rev-lex') == ucode_str
def test_EulerGamma():
assert pretty(EulerGamma) == str(EulerGamma) == "EulerGamma"
assert upretty(EulerGamma) == u"γ"
def test_GoldenRatio():
assert pretty(GoldenRatio) == str(GoldenRatio) == "GoldenRatio"
assert upretty(GoldenRatio) == u"φ"
def test_pretty_relational():
expr = Eq(x, y)
ascii_str = \
"""\
x = y\
"""
ucode_str = \
u("""\
x = y\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Lt(x, y)
ascii_str = \
"""\
x < y\
"""
ucode_str = \
u("""\
x < y\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Gt(x, y)
ascii_str = \
"""\
x > y\
"""
ucode_str = \
u("""\
x > y\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Le(x, y)
ascii_str = \
"""\
x <= y\
"""
ucode_str = \
u("""\
x ≤ y\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Ge(x, y)
ascii_str = \
"""\
x >= y\
"""
ucode_str = \
u("""\
x ≥ y\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Ne(x/(y + 1), y**2)
ascii_str_1 = \
"""\
x 2\n\
----- != y \n\
1 + y \
"""
ascii_str_2 = \
"""\
x 2\n\
----- != y \n\
y + 1 \
"""
ucode_str_1 = \
u("""\
x 2\n\
───── ≠ y \n\
1 + y \
""")
ucode_str_2 = \
u("""\
x 2\n\
───── ≠ y \n\
y + 1 \
""")
assert pretty(expr) in [ascii_str_1, ascii_str_2]
assert upretty(expr) in [ucode_str_1, ucode_str_2]
def test_Assignment():
expr = Assignment(x, y)
ascii_str = \
"""\
x := y\
"""
ucode_str = \
u("""\
x := y\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_AugmentedAssignment():
expr = AddAugmentedAssignment(x, y)
ascii_str = \
"""\
x += y\
"""
ucode_str = \
u("""\
x += y\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = SubAugmentedAssignment(x, y)
ascii_str = \
"""\
x -= y\
"""
ucode_str = \
u("""\
x -= y\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = MulAugmentedAssignment(x, y)
ascii_str = \
"""\
x *= y\
"""
ucode_str = \
u("""\
x *= y\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = DivAugmentedAssignment(x, y)
ascii_str = \
"""\
x /= y\
"""
ucode_str = \
u("""\
x /= y\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = ModAugmentedAssignment(x, y)
ascii_str = \
"""\
x %= y\
"""
ucode_str = \
u("""\
x %= y\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_pretty_rational():
expr = y*x**-2
ascii_str = \
"""\
y \n\
--\n\
2\n\
x \
"""
ucode_str = \
u("""\
y \n\
──\n\
2\n\
x \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = y**Rational(3, 2) * x**Rational(-5, 2)
ascii_str = \
"""\
3/2\n\
y \n\
----\n\
5/2\n\
x \
"""
ucode_str = \
u("""\
3/2\n\
y \n\
────\n\
5/2\n\
x \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = sin(x)**3/tan(x)**2
ascii_str = \
"""\
3 \n\
sin (x)\n\
-------\n\
2 \n\
tan (x)\
"""
ucode_str = \
u("""\
3 \n\
sin (x)\n\
───────\n\
2 \n\
tan (x)\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_pretty_functions():
"""Tests for Abs, conjugate, exp, function braces, and factorial."""
expr = (2*x + exp(x))
ascii_str_1 = \
"""\
x\n\
2*x + e \
"""
ascii_str_2 = \
"""\
x \n\
e + 2*x\
"""
ucode_str_1 = \
u("""\
x\n\
2⋅x + ℯ \
""")
ucode_str_2 = \
u("""\
x \n\
ℯ + 2⋅x\
""")
assert pretty(expr) in [ascii_str_1, ascii_str_2]
assert upretty(expr) in [ucode_str_1, ucode_str_2]
expr = Abs(x)
ascii_str = \
"""\
|x|\
"""
ucode_str = \
u("""\
│x│\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Abs(x/(x**2 + 1))
ascii_str_1 = \
"""\
| x |\n\
|------|\n\
| 2|\n\
|1 + x |\
"""
ascii_str_2 = \
"""\
| x |\n\
|------|\n\
| 2 |\n\
|x + 1|\
"""
ucode_str_1 = \
u("""\
│ x │\n\
│──────│\n\
│ 2│\n\
│1 + x │\
""")
ucode_str_2 = \
u("""\
│ x │\n\
│──────│\n\
│ 2 │\n\
│x + 1│\
""")
assert pretty(expr) in [ascii_str_1, ascii_str_2]
assert upretty(expr) in [ucode_str_1, ucode_str_2]
expr = Abs(1 / (y - Abs(x)))
ascii_str = \
"""\
1 \n\
---------\n\
|y - |x||\
"""
ucode_str = \
u("""\
1 \n\
─────────\n\
│y - │x││\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
n = Symbol('n', integer=True)
expr = factorial(n)
ascii_str = \
"""\
n!\
"""
ucode_str = \
u("""\
n!\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = factorial(2*n)
ascii_str = \
"""\
(2*n)!\
"""
ucode_str = \
u("""\
(2⋅n)!\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = factorial(factorial(factorial(n)))
ascii_str = \
"""\
((n!)!)!\
"""
ucode_str = \
u("""\
((n!)!)!\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = factorial(n + 1)
ascii_str_1 = \
"""\
(1 + n)!\
"""
ascii_str_2 = \
"""\
(n + 1)!\
"""
ucode_str_1 = \
u("""\
(1 + n)!\
""")
ucode_str_2 = \
u("""\
(n + 1)!\
""")
assert pretty(expr) in [ascii_str_1, ascii_str_2]
assert upretty(expr) in [ucode_str_1, ucode_str_2]
expr = subfactorial(n)
ascii_str = \
"""\
!n\
"""
ucode_str = \
u("""\
!n\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = subfactorial(2*n)
ascii_str = \
"""\
!(2*n)\
"""
ucode_str = \
u("""\
!(2⋅n)\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
n = Symbol('n', integer=True)
expr = factorial2(n)
ascii_str = \
"""\
n!!\
"""
ucode_str = \
u("""\
n!!\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = factorial2(2*n)
ascii_str = \
"""\
(2*n)!!\
"""
ucode_str = \
u("""\
(2⋅n)!!\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = factorial2(factorial2(factorial2(n)))
ascii_str = \
"""\
((n!!)!!)!!\
"""
ucode_str = \
u("""\
((n!!)!!)!!\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = factorial2(n + 1)
ascii_str_1 = \
"""\
(1 + n)!!\
"""
ascii_str_2 = \
"""\
(n + 1)!!\
"""
ucode_str_1 = \
u("""\
(1 + n)!!\
""")
ucode_str_2 = \
u("""\
(n + 1)!!\
""")
assert pretty(expr) in [ascii_str_1, ascii_str_2]
assert upretty(expr) in [ucode_str_1, ucode_str_2]
expr = 2*binomial(n, k)
ascii_str = \
"""\
/n\\\n\
2*| |\n\
\\k/\
"""
ucode_str = \
u("""\
⎛n⎞\n\
2⋅⎜ ⎟\n\
⎝k⎠\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = 2*binomial(2*n, k)
ascii_str = \
"""\
/2*n\\\n\
2*| |\n\
\\ k /\
"""
ucode_str = \
u("""\
⎛2⋅n⎞\n\
2⋅⎜ ⎟\n\
⎝ k ⎠\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = 2*binomial(n**2, k)
ascii_str = \
"""\
/ 2\\\n\
|n |\n\
2*| |\n\
\\k /\
"""
ucode_str = \
u("""\
⎛ 2⎞\n\
⎜n ⎟\n\
2⋅⎜ ⎟\n\
⎝k ⎠\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = catalan(n)
ascii_str = \
"""\
C \n\
n\
"""
ucode_str = \
u("""\
C \n\
n\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = catalan(n)
ascii_str = \
"""\
C \n\
n\
"""
ucode_str = \
u("""\
C \n\
n\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = bell(n)
ascii_str = \
"""\
B \n\
n\
"""
ucode_str = \
u("""\
B \n\
n\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = bernoulli(n)
ascii_str = \
"""\
B \n\
n\
"""
ucode_str = \
u("""\
B \n\
n\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = bernoulli(n, x)
ascii_str = \
"""\
B (x)\n\
n \
"""
ucode_str = \
u("""\
B (x)\n\
n \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = fibonacci(n)
ascii_str = \
"""\
F \n\
n\
"""
ucode_str = \
u("""\
F \n\
n\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = lucas(n)
ascii_str = \
"""\
L \n\
n\
"""
ucode_str = \
u("""\
L \n\
n\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = tribonacci(n)
ascii_str = \
"""\
T \n\
n\
"""
ucode_str = \
u("""\
T \n\
n\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = stieltjes(n)
ascii_str = \
"""\
stieltjes \n\
n\
"""
ucode_str = \
u("""\
γ \n\
n\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = stieltjes(n, x)
ascii_str = \
"""\
stieltjes (x)\n\
n \
"""
ucode_str = \
u("""\
γ (x)\n\
n \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = mathieuc(x, y, z)
ascii_str = 'C(x, y, z)'
ucode_str = u('C(x, y, z)')
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = mathieus(x, y, z)
ascii_str = 'S(x, y, z)'
ucode_str = u('S(x, y, z)')
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = mathieucprime(x, y, z)
ascii_str = "C'(x, y, z)"
ucode_str = u("C'(x, y, z)")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = mathieusprime(x, y, z)
ascii_str = "S'(x, y, z)"
ucode_str = u("S'(x, y, z)")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = conjugate(x)
ascii_str = \
"""\
_\n\
x\
"""
ucode_str = \
u("""\
_\n\
x\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
f = Function('f')
expr = conjugate(f(x + 1))
ascii_str_1 = \
"""\
________\n\
f(1 + x)\
"""
ascii_str_2 = \
"""\
________\n\
f(x + 1)\
"""
ucode_str_1 = \
u("""\
________\n\
f(1 + x)\
""")
ucode_str_2 = \
u("""\
________\n\
f(x + 1)\
""")
assert pretty(expr) in [ascii_str_1, ascii_str_2]
assert upretty(expr) in [ucode_str_1, ucode_str_2]
expr = f(x)
ascii_str = \
"""\
f(x)\
"""
ucode_str = \
u("""\
f(x)\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = f(x, y)
ascii_str = \
"""\
f(x, y)\
"""
ucode_str = \
u("""\
f(x, y)\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = f(x/(y + 1), y)
ascii_str_1 = \
"""\
/ x \\\n\
f|-----, y|\n\
\\1 + y /\
"""
ascii_str_2 = \
"""\
/ x \\\n\
f|-----, y|\n\
\\y + 1 /\
"""
ucode_str_1 = \
u("""\
⎛ x ⎞\n\
f⎜─────, y⎟\n\
⎝1 + y ⎠\
""")
ucode_str_2 = \
u("""\
⎛ x ⎞\n\
f⎜─────, y⎟\n\
⎝y + 1 ⎠\
""")
assert pretty(expr) in [ascii_str_1, ascii_str_2]
assert upretty(expr) in [ucode_str_1, ucode_str_2]
expr = f(x**x**x**x**x**x)
ascii_str = \
"""\
/ / / / / x\\\\\\\\\\
| | | | \\x /||||
| | | \\x /|||
| | \\x /||
| \\x /|
f\\x /\
"""
ucode_str = \
u("""\
⎛ ⎛ ⎛ ⎛ ⎛ x⎞⎞⎞⎞⎞
⎜ ⎜ ⎜ ⎜ ⎝x ⎠⎟⎟⎟⎟
⎜ ⎜ ⎜ ⎝x ⎠⎟⎟⎟
⎜ ⎜ ⎝x ⎠⎟⎟
⎜ ⎝x ⎠⎟
f⎝x ⎠\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = sin(x)**2
ascii_str = \
"""\
2 \n\
sin (x)\
"""
ucode_str = \
u("""\
2 \n\
sin (x)\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = conjugate(a + b*I)
ascii_str = \
"""\
_ _\n\
a - I*b\
"""
ucode_str = \
u("""\
_ _\n\
a - ⅈ⋅b\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = conjugate(exp(a + b*I))
ascii_str = \
"""\
_ _\n\
a - I*b\n\
e \
"""
ucode_str = \
u("""\
_ _\n\
a - ⅈ⋅b\n\
ℯ \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = conjugate( f(1 + conjugate(f(x))) )
ascii_str_1 = \
"""\
___________\n\
/ ____\\\n\
f\\1 + f(x)/\
"""
ascii_str_2 = \
"""\
___________\n\
/____ \\\n\
f\\f(x) + 1/\
"""
ucode_str_1 = \
u("""\
___________\n\
⎛ ____⎞\n\
f⎝1 + f(x)⎠\
""")
ucode_str_2 = \
u("""\
___________\n\
⎛____ ⎞\n\
f⎝f(x) + 1⎠\
""")
assert pretty(expr) in [ascii_str_1, ascii_str_2]
assert upretty(expr) in [ucode_str_1, ucode_str_2]
expr = f(x/(y + 1), y)
ascii_str_1 = \
"""\
/ x \\\n\
f|-----, y|\n\
\\1 + y /\
"""
ascii_str_2 = \
"""\
/ x \\\n\
f|-----, y|\n\
\\y + 1 /\
"""
ucode_str_1 = \
u("""\
⎛ x ⎞\n\
f⎜─────, y⎟\n\
⎝1 + y ⎠\
""")
ucode_str_2 = \
u("""\
⎛ x ⎞\n\
f⎜─────, y⎟\n\
⎝y + 1 ⎠\
""")
assert pretty(expr) in [ascii_str_1, ascii_str_2]
assert upretty(expr) in [ucode_str_1, ucode_str_2]
expr = floor(1 / (y - floor(x)))
ascii_str = \
"""\
/ 1 \\\n\
floor|------------|\n\
\\y - floor(x)/\
"""
ucode_str = \
u("""\
⎢ 1 ⎥\n\
⎢───────⎥\n\
⎣y - ⌊x⌋⎦\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = ceiling(1 / (y - ceiling(x)))
ascii_str = \
"""\
/ 1 \\\n\
ceiling|--------------|\n\
\\y - ceiling(x)/\
"""
ucode_str = \
u("""\
⎡ 1 ⎤\n\
⎢───────⎥\n\
⎢y - ⌈x⌉⎥\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = euler(n)
ascii_str = \
"""\
E \n\
n\
"""
ucode_str = \
u("""\
E \n\
n\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = euler(1/(1 + 1/(1 + 1/n)))
ascii_str = \
"""\
E \n\
1 \n\
---------\n\
1 \n\
1 + -----\n\
1\n\
1 + -\n\
n\
"""
ucode_str = \
u("""\
E \n\
1 \n\
─────────\n\
1 \n\
1 + ─────\n\
1\n\
1 + ─\n\
n\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = euler(n, x)
ascii_str = \
"""\
E (x)\n\
n \
"""
ucode_str = \
u("""\
E (x)\n\
n \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = euler(n, x/2)
ascii_str = \
"""\
/x\\\n\
E |-|\n\
n\\2/\
"""
ucode_str = \
u("""\
⎛x⎞\n\
E ⎜─⎟\n\
n⎝2⎠\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_pretty_sqrt():
expr = sqrt(2)
ascii_str = \
"""\
___\n\
\\/ 2 \
"""
ucode_str = \
u"√2"
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = 2**Rational(1, 3)
ascii_str = \
"""\
3 ___\n\
\\/ 2 \
"""
ucode_str = \
u("""\
3 ___\n\
╲╱ 2 \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = 2**Rational(1, 1000)
ascii_str = \
"""\
1000___\n\
\\/ 2 \
"""
ucode_str = \
u("""\
1000___\n\
╲╱ 2 \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = sqrt(x**2 + 1)
ascii_str = \
"""\
________\n\
/ 2 \n\
\\/ x + 1 \
"""
ucode_str = \
u("""\
________\n\
╱ 2 \n\
╲╱ x + 1 \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = (1 + sqrt(5))**Rational(1, 3)
ascii_str = \
"""\
___________\n\
3 / ___ \n\
\\/ 1 + \\/ 5 \
"""
ucode_str = \
u("""\
3 ________\n\
╲╱ 1 + √5 \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = 2**(1/x)
ascii_str = \
"""\
x ___\n\
\\/ 2 \
"""
ucode_str = \
u("""\
x ___\n\
╲╱ 2 \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = sqrt(2 + pi)
ascii_str = \
"""\
________\n\
\\/ 2 + pi \
"""
ucode_str = \
u("""\
_______\n\
╲╱ 2 + π \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = (2 + (
1 + x**2)/(2 + x))**Rational(1, 4) + (1 + x**Rational(1, 1000))/sqrt(3 + x**2)
ascii_str = \
"""\
____________ \n\
/ 2 1000___ \n\
/ x + 1 \\/ x + 1\n\
4 / 2 + ------ + -----------\n\
\\/ x + 2 ________\n\
/ 2 \n\
\\/ x + 3 \
"""
ucode_str = \
u("""\
____________ \n\
╱ 2 1000___ \n\
╱ x + 1 ╲╱ x + 1\n\
4 ╱ 2 + ────── + ───────────\n\
╲╱ x + 2 ________\n\
╱ 2 \n\
╲╱ x + 3 \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_pretty_sqrt_char_knob():
# See PR #9234.
expr = sqrt(2)
ucode_str1 = \
u("""\
___\n\
╲╱ 2 \
""")
ucode_str2 = \
u"√2"
assert xpretty(expr, use_unicode=True,
use_unicode_sqrt_char=False) == ucode_str1
assert xpretty(expr, use_unicode=True,
use_unicode_sqrt_char=True) == ucode_str2
def test_pretty_sqrt_longsymbol_no_sqrt_char():
# Do not use unicode sqrt char for long symbols (see PR #9234).
expr = sqrt(Symbol('C1'))
ucode_str = \
u("""\
____\n\
╲╱ C₁ \
""")
assert upretty(expr) == ucode_str
def test_pretty_KroneckerDelta():
x, y = symbols("x, y")
expr = KroneckerDelta(x, y)
ascii_str = \
"""\
d \n\
x,y\
"""
ucode_str = \
u("""\
δ \n\
x,y\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_pretty_product():
n, m, k, l = symbols('n m k l')
f = symbols('f', cls=Function)
expr = Product(f((n/3)**2), (n, k**2, l))
unicode_str = \
u("""\
l \n\
─┬──────┬─ \n\
│ │ ⎛ 2⎞\n\
│ │ ⎜n ⎟\n\
│ │ f⎜──⎟\n\
│ │ ⎝9 ⎠\n\
│ │ \n\
2 \n\
n = k """)
ascii_str = \
"""\
l \n\
__________ \n\
| | / 2\\\n\
| | |n |\n\
| | f|--|\n\
| | \\9 /\n\
| | \n\
2 \n\
n = k """
expr = Product(f((n/3)**2), (n, k**2, l), (l, 1, m))
unicode_str = \
u("""\
m l \n\
─┬──────┬─ ─┬──────┬─ \n\
│ │ │ │ ⎛ 2⎞\n\
│ │ │ │ ⎜n ⎟\n\
│ │ │ │ f⎜──⎟\n\
│ │ │ │ ⎝9 ⎠\n\
│ │ │ │ \n\
l = 1 2 \n\
n = k """)
ascii_str = \
"""\
m l \n\
__________ __________ \n\
| | | | / 2\\\n\
| | | | |n |\n\
| | | | f|--|\n\
| | | | \\9 /\n\
| | | | \n\
l = 1 2 \n\
n = k """
assert pretty(expr) == ascii_str
assert upretty(expr) == unicode_str
def test_pretty_Lambda():
# S.IdentityFunction is a special case
expr = Lambda(y, y)
assert pretty(expr) == "x -> x"
assert upretty(expr) == u"x ↦ x"
expr = Lambda(x, x+1)
assert pretty(expr) == "x -> x + 1"
assert upretty(expr) == u"x ↦ x + 1"
expr = Lambda(x, x**2)
ascii_str = \
"""\
2\n\
x -> x \
"""
ucode_str = \
u("""\
2\n\
x ↦ x \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Lambda(x, x**2)**2
ascii_str = \
"""\
2
/ 2\\ \n\
\\x -> x / \
"""
ucode_str = \
u("""\
2
⎛ 2⎞ \n\
⎝x ↦ x ⎠ \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Lambda((x, y), x)
ascii_str = "(x, y) -> x"
ucode_str = u"(x, y) ↦ x"
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Lambda((x, y), x**2)
ascii_str = \
"""\
2\n\
(x, y) -> x \
"""
ucode_str = \
u("""\
2\n\
(x, y) ↦ x \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Lambda(((x, y),), x**2)
ascii_str = \
"""\
2\n\
((x, y),) -> x \
"""
ucode_str = \
u("""\
2\n\
((x, y),) ↦ x \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_pretty_order():
expr = O(1)
ascii_str = \
"""\
O(1)\
"""
ucode_str = \
u("""\
O(1)\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = O(1/x)
ascii_str = \
"""\
/1\\\n\
O|-|\n\
\\x/\
"""
ucode_str = \
u("""\
⎛1⎞\n\
O⎜─⎟\n\
⎝x⎠\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = O(x**2 + y**2)
ascii_str = \
"""\
/ 2 2 \\\n\
O\\x + y ; (x, y) -> (0, 0)/\
"""
ucode_str = \
u("""\
⎛ 2 2 ⎞\n\
O⎝x + y ; (x, y) → (0, 0)⎠\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = O(1, (x, oo))
ascii_str = \
"""\
O(1; x -> oo)\
"""
ucode_str = \
u("""\
O(1; x → ∞)\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = O(1/x, (x, oo))
ascii_str = \
"""\
/1 \\\n\
O|-; x -> oo|\n\
\\x /\
"""
ucode_str = \
u("""\
⎛1 ⎞\n\
O⎜─; x → ∞⎟\n\
⎝x ⎠\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = O(x**2 + y**2, (x, oo), (y, oo))
ascii_str = \
"""\
/ 2 2 \\\n\
O\\x + y ; (x, y) -> (oo, oo)/\
"""
ucode_str = \
u("""\
⎛ 2 2 ⎞\n\
O⎝x + y ; (x, y) → (∞, ∞)⎠\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_pretty_derivatives():
# Simple
expr = Derivative(log(x), x, evaluate=False)
ascii_str = \
"""\
d \n\
--(log(x))\n\
dx \
"""
ucode_str = \
u("""\
d \n\
──(log(x))\n\
dx \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Derivative(log(x), x, evaluate=False) + x
ascii_str_1 = \
"""\
d \n\
x + --(log(x))\n\
dx \
"""
ascii_str_2 = \
"""\
d \n\
--(log(x)) + x\n\
dx \
"""
ucode_str_1 = \
u("""\
d \n\
x + ──(log(x))\n\
dx \
""")
ucode_str_2 = \
u("""\
d \n\
──(log(x)) + x\n\
dx \
""")
assert pretty(expr) in [ascii_str_1, ascii_str_2]
assert upretty(expr) in [ucode_str_1, ucode_str_2]
# basic partial derivatives
expr = Derivative(log(x + y) + x, x)
ascii_str_1 = \
"""\
d \n\
--(log(x + y) + x)\n\
dx \
"""
ascii_str_2 = \
"""\
d \n\
--(x + log(x + y))\n\
dx \
"""
ucode_str_1 = \
u("""\
∂ \n\
──(log(x + y) + x)\n\
∂x \
""")
ucode_str_2 = \
u("""\
∂ \n\
──(x + log(x + y))\n\
∂x \
""")
assert pretty(expr) in [ascii_str_1, ascii_str_2]
assert upretty(expr) in [ucode_str_1, ucode_str_2], upretty(expr)
# Multiple symbols
expr = Derivative(log(x) + x**2, x, y)
ascii_str_1 = \
"""\
2 \n\
d / 2\\\n\
-----\\log(x) + x /\n\
dy dx \
"""
ascii_str_2 = \
"""\
2 \n\
d / 2 \\\n\
-----\\x + log(x)/\n\
dy dx \
"""
ucode_str_1 = \
u("""\
2 \n\
d ⎛ 2⎞\n\
─────⎝log(x) + x ⎠\n\
dy dx \
""")
ucode_str_2 = \
u("""\
2 \n\
d ⎛ 2 ⎞\n\
─────⎝x + log(x)⎠\n\
dy dx \
""")
assert pretty(expr) in [ascii_str_1, ascii_str_2]
assert upretty(expr) in [ucode_str_1, ucode_str_2]
expr = Derivative(2*x*y, y, x) + x**2
ascii_str_1 = \
"""\
2 \n\
d 2\n\
-----(2*x*y) + x \n\
dx dy \
"""
ascii_str_2 = \
"""\
2 \n\
2 d \n\
x + -----(2*x*y)\n\
dx dy \
"""
ucode_str_1 = \
u("""\
2 \n\
∂ 2\n\
─────(2⋅x⋅y) + x \n\
∂x ∂y \
""")
ucode_str_2 = \
u("""\
2 \n\
2 ∂ \n\
x + ─────(2⋅x⋅y)\n\
∂x ∂y \
""")
assert pretty(expr) in [ascii_str_1, ascii_str_2]
assert upretty(expr) in [ucode_str_1, ucode_str_2]
expr = Derivative(2*x*y, x, x)
ascii_str = \
"""\
2 \n\
d \n\
---(2*x*y)\n\
2 \n\
dx \
"""
ucode_str = \
u("""\
2 \n\
∂ \n\
───(2⋅x⋅y)\n\
2 \n\
∂x \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Derivative(2*x*y, x, 17)
ascii_str = \
"""\
17 \n\
d \n\
----(2*x*y)\n\
17 \n\
dx \
"""
ucode_str = \
u("""\
17 \n\
∂ \n\
────(2⋅x⋅y)\n\
17 \n\
∂x \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Derivative(2*x*y, x, x, y)
ascii_str = \
"""\
3 \n\
d \n\
------(2*x*y)\n\
2 \n\
dy dx \
"""
ucode_str = \
u("""\
3 \n\
∂ \n\
──────(2⋅x⋅y)\n\
2 \n\
∂y ∂x \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
# Greek letters
alpha = Symbol('alpha')
beta = Function('beta')
expr = beta(alpha).diff(alpha)
ascii_str = \
"""\
d \n\
------(beta(alpha))\n\
dalpha \
"""
ucode_str = \
u("""\
d \n\
──(β(α))\n\
dα \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Derivative(f(x), (x, n))
ascii_str = \
"""\
n \n\
d \n\
---(f(x))\n\
n \n\
dx \
"""
ucode_str = \
u("""\
n \n\
d \n\
───(f(x))\n\
n \n\
dx \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_pretty_integrals():
expr = Integral(log(x), x)
ascii_str = \
"""\
/ \n\
| \n\
| log(x) dx\n\
| \n\
/ \
"""
ucode_str = \
u("""\
⌠ \n\
⎮ log(x) dx\n\
⌡ \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Integral(x**2, x)
ascii_str = \
"""\
/ \n\
| \n\
| 2 \n\
| x dx\n\
| \n\
/ \
"""
ucode_str = \
u("""\
⌠ \n\
⎮ 2 \n\
⎮ x dx\n\
⌡ \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Integral((sin(x))**2 / (tan(x))**2)
ascii_str = \
"""\
/ \n\
| \n\
| 2 \n\
| sin (x) \n\
| ------- dx\n\
| 2 \n\
| tan (x) \n\
| \n\
/ \
"""
ucode_str = \
u("""\
⌠ \n\
⎮ 2 \n\
⎮ sin (x) \n\
⎮ ─────── dx\n\
⎮ 2 \n\
⎮ tan (x) \n\
⌡ \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Integral(x**(2**x), x)
ascii_str = \
"""\
/ \n\
| \n\
| / x\\ \n\
| \\2 / \n\
| x dx\n\
| \n\
/ \
"""
ucode_str = \
u("""\
⌠ \n\
⎮ ⎛ x⎞ \n\
⎮ ⎝2 ⎠ \n\
⎮ x dx\n\
⌡ \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Integral(x**2, (x, 1, 2))
ascii_str = \
"""\
2 \n\
/ \n\
| \n\
| 2 \n\
| x dx\n\
| \n\
/ \n\
1 \
"""
ucode_str = \
u("""\
2 \n\
⌠ \n\
⎮ 2 \n\
⎮ x dx\n\
⌡ \n\
1 \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Integral(x**2, (x, Rational(1, 2), 10))
ascii_str = \
"""\
10 \n\
/ \n\
| \n\
| 2 \n\
| x dx\n\
| \n\
/ \n\
1/2 \
"""
ucode_str = \
u("""\
10 \n\
⌠ \n\
⎮ 2 \n\
⎮ x dx\n\
⌡ \n\
1/2 \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Integral(x**2*y**2, x, y)
ascii_str = \
"""\
/ / \n\
| | \n\
| | 2 2 \n\
| | x *y dx dy\n\
| | \n\
/ / \
"""
ucode_str = \
u("""\
⌠ ⌠ \n\
⎮ ⎮ 2 2 \n\
⎮ ⎮ x ⋅y dx dy\n\
⌡ ⌡ \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Integral(sin(th)/cos(ph), (th, 0, pi), (ph, 0, 2*pi))
ascii_str = \
"""\
2*pi pi \n\
/ / \n\
| | \n\
| | sin(theta) \n\
| | ---------- d(theta) d(phi)\n\
| | cos(phi) \n\
| | \n\
/ / \n\
0 0 \
"""
ucode_str = \
u("""\
2⋅π π \n\
⌠ ⌠ \n\
⎮ ⎮ sin(θ) \n\
⎮ ⎮ ────── dθ dφ\n\
⎮ ⎮ cos(φ) \n\
⌡ ⌡ \n\
0 0 \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_pretty_matrix():
# Empty Matrix
expr = Matrix()
ascii_str = "[]"
unicode_str = "[]"
assert pretty(expr) == ascii_str
assert upretty(expr) == unicode_str
expr = Matrix(2, 0, lambda i, j: 0)
ascii_str = "[]"
unicode_str = "[]"
assert pretty(expr) == ascii_str
assert upretty(expr) == unicode_str
expr = Matrix(0, 2, lambda i, j: 0)
ascii_str = "[]"
unicode_str = "[]"
assert pretty(expr) == ascii_str
assert upretty(expr) == unicode_str
expr = Matrix([[x**2 + 1, 1], [y, x + y]])
ascii_str_1 = \
"""\
[ 2 ]
[1 + x 1 ]
[ ]
[ y x + y]\
"""
ascii_str_2 = \
"""\
[ 2 ]
[x + 1 1 ]
[ ]
[ y x + y]\
"""
ucode_str_1 = \
u("""\
⎡ 2 ⎤
⎢1 + x 1 ⎥
⎢ ⎥
⎣ y x + y⎦\
""")
ucode_str_2 = \
u("""\
⎡ 2 ⎤
⎢x + 1 1 ⎥
⎢ ⎥
⎣ y x + y⎦\
""")
assert pretty(expr) in [ascii_str_1, ascii_str_2]
assert upretty(expr) in [ucode_str_1, ucode_str_2]
expr = Matrix([[x/y, y, th], [0, exp(I*k*ph), 1]])
ascii_str = \
"""\
[x ]
[- y theta]
[y ]
[ ]
[ I*k*phi ]
[0 e 1 ]\
"""
ucode_str = \
u("""\
⎡x ⎤
⎢─ y θ⎥
⎢y ⎥
⎢ ⎥
⎢ ⅈ⋅k⋅φ ⎥
⎣0 ℯ 1⎦\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
unicode_str = \
u("""\
⎡v̇_msc_00 0 0 ⎤
⎢ ⎥
⎢ 0 v̇_msc_01 0 ⎥
⎢ ⎥
⎣ 0 0 v̇_msc_02⎦\
""")
expr = diag(*MatrixSymbol('vdot_msc',1,3))
assert upretty(expr) == unicode_str
def test_pretty_ndim_arrays():
x, y, z, w = symbols("x y z w")
for ArrayType in (ImmutableDenseNDimArray, ImmutableSparseNDimArray, MutableDenseNDimArray, MutableSparseNDimArray):
# Basic: scalar array
M = ArrayType(x)
assert pretty(M) == "x"
assert upretty(M) == "x"
M = ArrayType([[1/x, y], [z, w]])
M1 = ArrayType([1/x, y, z])
M2 = tensorproduct(M1, M)
M3 = tensorproduct(M, M)
ascii_str = \
"""\
[1 ]\n\
[- y]\n\
[x ]\n\
[ ]\n\
[z w]\
"""
ucode_str = \
u("""\
⎡1 ⎤\n\
⎢─ y⎥\n\
⎢x ⎥\n\
⎢ ⎥\n\
⎣z w⎦\
""")
assert pretty(M) == ascii_str
assert upretty(M) == ucode_str
ascii_str = \
"""\
[1 ]\n\
[- y z]\n\
[x ]\
"""
ucode_str = \
u("""\
⎡1 ⎤\n\
⎢─ y z⎥\n\
⎣x ⎦\
""")
assert pretty(M1) == ascii_str
assert upretty(M1) == ucode_str
ascii_str = \
"""\
[[1 y] ]\n\
[[-- -] [z ]]\n\
[[ 2 x] [ y 2 ] [- y*z]]\n\
[[x ] [ - y ] [x ]]\n\
[[ ] [ x ] [ ]]\n\
[[z w] [ ] [ 2 ]]\n\
[[- -] [y*z w*y] [z w*z]]\n\
[[x x] ]\
"""
ucode_str = \
u("""\
⎡⎡1 y⎤ ⎤\n\
⎢⎢── ─⎥ ⎡z ⎤⎥\n\
⎢⎢ 2 x⎥ ⎡ y 2 ⎤ ⎢─ y⋅z⎥⎥\n\
⎢⎢x ⎥ ⎢ ─ y ⎥ ⎢x ⎥⎥\n\
⎢⎢ ⎥ ⎢ x ⎥ ⎢ ⎥⎥\n\
⎢⎢z w⎥ ⎢ ⎥ ⎢ 2 ⎥⎥\n\
⎢⎢─ ─⎥ ⎣y⋅z w⋅y⎦ ⎣z w⋅z⎦⎥\n\
⎣⎣x x⎦ ⎦\
""")
assert pretty(M2) == ascii_str
assert upretty(M2) == ucode_str
ascii_str = \
"""\
[ [1 y] ]\n\
[ [-- -] ]\n\
[ [ 2 x] [ y 2 ]]\n\
[ [x ] [ - y ]]\n\
[ [ ] [ x ]]\n\
[ [z w] [ ]]\n\
[ [- -] [y*z w*y]]\n\
[ [x x] ]\n\
[ ]\n\
[[z ] [ w ]]\n\
[[- y*z] [ - w*y]]\n\
[[x ] [ x ]]\n\
[[ ] [ ]]\n\
[[ 2 ] [ 2 ]]\n\
[[z w*z] [w*z w ]]\
"""
ucode_str = \
u("""\
⎡ ⎡1 y⎤ ⎤\n\
⎢ ⎢── ─⎥ ⎥\n\
⎢ ⎢ 2 x⎥ ⎡ y 2 ⎤⎥\n\
⎢ ⎢x ⎥ ⎢ ─ y ⎥⎥\n\
⎢ ⎢ ⎥ ⎢ x ⎥⎥\n\
⎢ ⎢z w⎥ ⎢ ⎥⎥\n\
⎢ ⎢─ ─⎥ ⎣y⋅z w⋅y⎦⎥\n\
⎢ ⎣x x⎦ ⎥\n\
⎢ ⎥\n\
⎢⎡z ⎤ ⎡ w ⎤⎥\n\
⎢⎢─ y⋅z⎥ ⎢ ─ w⋅y⎥⎥\n\
⎢⎢x ⎥ ⎢ x ⎥⎥\n\
⎢⎢ ⎥ ⎢ ⎥⎥\n\
⎢⎢ 2 ⎥ ⎢ 2 ⎥⎥\n\
⎣⎣z w⋅z⎦ ⎣w⋅z w ⎦⎦\
""")
assert pretty(M3) == ascii_str
assert upretty(M3) == ucode_str
Mrow = ArrayType([[x, y, 1 / z]])
Mcolumn = ArrayType([[x], [y], [1 / z]])
Mcol2 = ArrayType([Mcolumn.tolist()])
ascii_str = \
"""\
[[ 1]]\n\
[[x y -]]\n\
[[ z]]\
"""
ucode_str = \
u("""\
⎡⎡ 1⎤⎤\n\
⎢⎢x y ─⎥⎥\n\
⎣⎣ z⎦⎦\
""")
assert pretty(Mrow) == ascii_str
assert upretty(Mrow) == ucode_str
ascii_str = \
"""\
[x]\n\
[ ]\n\
[y]\n\
[ ]\n\
[1]\n\
[-]\n\
[z]\
"""
ucode_str = \
u("""\
⎡x⎤\n\
⎢ ⎥\n\
⎢y⎥\n\
⎢ ⎥\n\
⎢1⎥\n\
⎢─⎥\n\
⎣z⎦\
""")
assert pretty(Mcolumn) == ascii_str
assert upretty(Mcolumn) == ucode_str
ascii_str = \
"""\
[[x]]\n\
[[ ]]\n\
[[y]]\n\
[[ ]]\n\
[[1]]\n\
[[-]]\n\
[[z]]\
"""
ucode_str = \
u("""\
⎡⎡x⎤⎤\n\
⎢⎢ ⎥⎥\n\
⎢⎢y⎥⎥\n\
⎢⎢ ⎥⎥\n\
⎢⎢1⎥⎥\n\
⎢⎢─⎥⎥\n\
⎣⎣z⎦⎦\
""")
assert pretty(Mcol2) == ascii_str
assert upretty(Mcol2) == ucode_str
def test_tensor_TensorProduct():
A = MatrixSymbol("A", 3, 3)
B = MatrixSymbol("B", 3, 3)
assert upretty(TensorProduct(A, B)) == "A\u2297B"
assert upretty(TensorProduct(A, B, A)) == "A\u2297B\u2297A"
def test_diffgeom_print_WedgeProduct():
from sympy.diffgeom.rn import R2
from sympy.diffgeom import WedgeProduct
wp = WedgeProduct(R2.dx, R2.dy)
assert upretty(wp) == u("ⅆ x∧ⅆ y")
def test_Adjoint():
X = MatrixSymbol('X', 2, 2)
Y = MatrixSymbol('Y', 2, 2)
assert pretty(Adjoint(X)) == " +\nX "
assert pretty(Adjoint(X + Y)) == " +\n(X + Y) "
assert pretty(Adjoint(X) + Adjoint(Y)) == " + +\nX + Y "
assert pretty(Adjoint(X*Y)) == " +\n(X*Y) "
assert pretty(Adjoint(Y)*Adjoint(X)) == " + +\nY *X "
assert pretty(Adjoint(X**2)) == " +\n/ 2\\ \n\\X / "
assert pretty(Adjoint(X)**2) == " 2\n/ +\\ \n\\X / "
assert pretty(Adjoint(Inverse(X))) == " +\n/ -1\\ \n\\X / "
assert pretty(Inverse(Adjoint(X))) == " -1\n/ +\\ \n\\X / "
assert pretty(Adjoint(Transpose(X))) == " +\n/ T\\ \n\\X / "
assert pretty(Transpose(Adjoint(X))) == " T\n/ +\\ \n\\X / "
assert upretty(Adjoint(X)) == u" †\nX "
assert upretty(Adjoint(X + Y)) == u" †\n(X + Y) "
assert upretty(Adjoint(X) + Adjoint(Y)) == u" † †\nX + Y "
assert upretty(Adjoint(X*Y)) == u" †\n(X⋅Y) "
assert upretty(Adjoint(Y)*Adjoint(X)) == u" † †\nY ⋅X "
assert upretty(Adjoint(X**2)) == \
u" †\n⎛ 2⎞ \n⎝X ⎠ "
assert upretty(Adjoint(X)**2) == \
u" 2\n⎛ †⎞ \n⎝X ⎠ "
assert upretty(Adjoint(Inverse(X))) == \
u" †\n⎛ -1⎞ \n⎝X ⎠ "
assert upretty(Inverse(Adjoint(X))) == \
u" -1\n⎛ †⎞ \n⎝X ⎠ "
assert upretty(Adjoint(Transpose(X))) == \
u" †\n⎛ T⎞ \n⎝X ⎠ "
assert upretty(Transpose(Adjoint(X))) == \
u" T\n⎛ †⎞ \n⎝X ⎠ "
def test_pretty_Trace_issue_9044():
X = Matrix([[1, 2], [3, 4]])
Y = Matrix([[2, 4], [6, 8]])
ascii_str_1 = \
"""\
/[1 2]\\
tr|[ ]|
\\[3 4]/\
"""
ucode_str_1 = \
u("""\
⎛⎡1 2⎤⎞
tr⎜⎢ ⎥⎟
⎝⎣3 4⎦⎠\
""")
ascii_str_2 = \
"""\
/[1 2]\\ /[2 4]\\
tr|[ ]| + tr|[ ]|
\\[3 4]/ \\[6 8]/\
"""
ucode_str_2 = \
u("""\
⎛⎡1 2⎤⎞ ⎛⎡2 4⎤⎞
tr⎜⎢ ⎥⎟ + tr⎜⎢ ⎥⎟
⎝⎣3 4⎦⎠ ⎝⎣6 8⎦⎠\
""")
assert pretty(Trace(X)) == ascii_str_1
assert upretty(Trace(X)) == ucode_str_1
assert pretty(Trace(X) + Trace(Y)) == ascii_str_2
assert upretty(Trace(X) + Trace(Y)) == ucode_str_2
def test_MatrixSlice():
n = Symbol('n', integer=True)
x, y, z, w, t, = symbols('x y z w t')
X = MatrixSymbol('X', n, n)
Y = MatrixSymbol('Y', 10, 10)
Z = MatrixSymbol('Z', 10, 10)
expr = MatrixSlice(X, (None, None, None), (None, None, None))
assert pretty(expr) == upretty(expr) == 'X[:, :]'
expr = X[x:x + 1, y:y + 1]
assert pretty(expr) == upretty(expr) == 'X[x:x + 1, y:y + 1]'
expr = X[x:x + 1:2, y:y + 1:2]
assert pretty(expr) == upretty(expr) == 'X[x:x + 1:2, y:y + 1:2]'
expr = X[:x, y:]
assert pretty(expr) == upretty(expr) == 'X[:x, y:]'
expr = X[:x, y:]
assert pretty(expr) == upretty(expr) == 'X[:x, y:]'
expr = X[x:, :y]
assert pretty(expr) == upretty(expr) == 'X[x:, :y]'
expr = X[x:y, z:w]
assert pretty(expr) == upretty(expr) == 'X[x:y, z:w]'
expr = X[x:y:t, w:t:x]
assert pretty(expr) == upretty(expr) == 'X[x:y:t, w:t:x]'
expr = X[x::y, t::w]
assert pretty(expr) == upretty(expr) == 'X[x::y, t::w]'
expr = X[:x:y, :t:w]
assert pretty(expr) == upretty(expr) == 'X[:x:y, :t:w]'
expr = X[::x, ::y]
assert pretty(expr) == upretty(expr) == 'X[::x, ::y]'
expr = MatrixSlice(X, (0, None, None), (0, None, None))
assert pretty(expr) == upretty(expr) == 'X[:, :]'
expr = MatrixSlice(X, (None, n, None), (None, n, None))
assert pretty(expr) == upretty(expr) == 'X[:, :]'
expr = MatrixSlice(X, (0, n, None), (0, n, None))
assert pretty(expr) == upretty(expr) == 'X[:, :]'
expr = MatrixSlice(X, (0, n, 2), (0, n, 2))
assert pretty(expr) == upretty(expr) == 'X[::2, ::2]'
expr = X[1:2:3, 4:5:6]
assert pretty(expr) == upretty(expr) == 'X[1:2:3, 4:5:6]'
expr = X[1:3:5, 4:6:8]
assert pretty(expr) == upretty(expr) == 'X[1:3:5, 4:6:8]'
expr = X[1:10:2]
assert pretty(expr) == upretty(expr) == 'X[1:10:2, :]'
expr = Y[:5, 1:9:2]
assert pretty(expr) == upretty(expr) == 'Y[:5, 1:9:2]'
expr = Y[:5, 1:10:2]
assert pretty(expr) == upretty(expr) == 'Y[:5, 1::2]'
expr = Y[5, :5:2]
assert pretty(expr) == upretty(expr) == 'Y[5:6, :5:2]'
expr = X[0:1, 0:1]
assert pretty(expr) == upretty(expr) == 'X[:1, :1]'
expr = X[0:1:2, 0:1:2]
assert pretty(expr) == upretty(expr) == 'X[:1:2, :1:2]'
expr = (Y + Z)[2:, 2:]
assert pretty(expr) == upretty(expr) == '(Y + Z)[2:, 2:]'
def test_MatrixExpressions():
n = Symbol('n', integer=True)
X = MatrixSymbol('X', n, n)
assert pretty(X) == upretty(X) == "X"
# Apply function elementwise (`ElementwiseApplyFunc`):
expr = (X.T*X).applyfunc(sin)
ascii_str = """\
/ T \\\n\
(d -> sin(d)).\\X *X/\
"""
ucode_str = u("""\
⎛ T ⎞\n\
(d ↦ sin(d))˳⎝X ⋅X⎠\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
lamda = Lambda(x, 1/x)
expr = (n*X).applyfunc(lamda)
ascii_str = """\
/ 1\\ \n\
|x -> -|.(n*X)\n\
\\ x/ \
"""
ucode_str = u("""\
⎛ 1⎞ \n\
⎜x ↦ ─⎟˳(n⋅X)\n\
⎝ x⎠ \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_pretty_dotproduct():
from sympy.matrices import Matrix, MatrixSymbol
from sympy.matrices.expressions.dotproduct import DotProduct
n = symbols("n", integer=True)
A = MatrixSymbol('A', n, 1)
B = MatrixSymbol('B', n, 1)
C = Matrix(1, 3, [1, 2, 3])
D = Matrix(1, 3, [1, 3, 4])
assert pretty(DotProduct(A, B)) == u"A*B"
assert pretty(DotProduct(C, D)) == u"[1 2 3]*[1 3 4]"
assert upretty(DotProduct(A, B)) == u"A⋅B"
assert upretty(DotProduct(C, D)) == u"[1 2 3]⋅[1 3 4]"
def test_pretty_piecewise():
expr = Piecewise((x, x < 1), (x**2, True))
ascii_str = \
"""\
/x for x < 1\n\
| \n\
< 2 \n\
|x otherwise\n\
\\ \
"""
ucode_str = \
u("""\
⎧x for x < 1\n\
⎪ \n\
⎨ 2 \n\
⎪x otherwise\n\
⎩ \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = -Piecewise((x, x < 1), (x**2, True))
ascii_str = \
"""\
//x for x < 1\\\n\
|| |\n\
-|< 2 |\n\
||x otherwise|\n\
\\\\ /\
"""
ucode_str = \
u("""\
⎛⎧x for x < 1⎞\n\
⎜⎪ ⎟\n\
-⎜⎨ 2 ⎟\n\
⎜⎪x otherwise⎟\n\
⎝⎩ ⎠\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = x + Piecewise((x, x > 0), (y, True)) + Piecewise((x/y, x < 2),
(y**2, x > 2), (1, True)) + 1
ascii_str = \
"""\
//x \\ \n\
||- for x < 2| \n\
||y | \n\
//x for x > 0\\ || | \n\
x + |< | + |< 2 | + 1\n\
\\\\y otherwise/ ||y for x > 2| \n\
|| | \n\
||1 otherwise| \n\
\\\\ / \
"""
ucode_str = \
u("""\
⎛⎧x ⎞ \n\
⎜⎪─ for x < 2⎟ \n\
⎜⎪y ⎟ \n\
⎛⎧x for x > 0⎞ ⎜⎪ ⎟ \n\
x + ⎜⎨ ⎟ + ⎜⎨ 2 ⎟ + 1\n\
⎝⎩y otherwise⎠ ⎜⎪y for x > 2⎟ \n\
⎜⎪ ⎟ \n\
⎜⎪1 otherwise⎟ \n\
⎝⎩ ⎠ \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = x - Piecewise((x, x > 0), (y, True)) + Piecewise((x/y, x < 2),
(y**2, x > 2), (1, True)) + 1
ascii_str = \
"""\
//x \\ \n\
||- for x < 2| \n\
||y | \n\
//x for x > 0\\ || | \n\
x - |< | + |< 2 | + 1\n\
\\\\y otherwise/ ||y for x > 2| \n\
|| | \n\
||1 otherwise| \n\
\\\\ / \
"""
ucode_str = \
u("""\
⎛⎧x ⎞ \n\
⎜⎪─ for x < 2⎟ \n\
⎜⎪y ⎟ \n\
⎛⎧x for x > 0⎞ ⎜⎪ ⎟ \n\
x - ⎜⎨ ⎟ + ⎜⎨ 2 ⎟ + 1\n\
⎝⎩y otherwise⎠ ⎜⎪y for x > 2⎟ \n\
⎜⎪ ⎟ \n\
⎜⎪1 otherwise⎟ \n\
⎝⎩ ⎠ \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = x*Piecewise((x, x > 0), (y, True))
ascii_str = \
"""\
//x for x > 0\\\n\
x*|< |\n\
\\\\y otherwise/\
"""
ucode_str = \
u("""\
⎛⎧x for x > 0⎞\n\
x⋅⎜⎨ ⎟\n\
⎝⎩y otherwise⎠\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Piecewise((x, x > 0), (y, True))*Piecewise((x/y, x < 2), (y**2, x >
2), (1, True))
ascii_str = \
"""\
//x \\\n\
||- for x < 2|\n\
||y |\n\
//x for x > 0\\ || |\n\
|< |*|< 2 |\n\
\\\\y otherwise/ ||y for x > 2|\n\
|| |\n\
||1 otherwise|\n\
\\\\ /\
"""
ucode_str = \
u("""\
⎛⎧x ⎞\n\
⎜⎪─ for x < 2⎟\n\
⎜⎪y ⎟\n\
⎛⎧x for x > 0⎞ ⎜⎪ ⎟\n\
⎜⎨ ⎟⋅⎜⎨ 2 ⎟\n\
⎝⎩y otherwise⎠ ⎜⎪y for x > 2⎟\n\
⎜⎪ ⎟\n\
⎜⎪1 otherwise⎟\n\
⎝⎩ ⎠\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = -Piecewise((x, x > 0), (y, True))*Piecewise((x/y, x < 2), (y**2, x
> 2), (1, True))
ascii_str = \
"""\
//x \\\n\
||- for x < 2|\n\
||y |\n\
//x for x > 0\\ || |\n\
-|< |*|< 2 |\n\
\\\\y otherwise/ ||y for x > 2|\n\
|| |\n\
||1 otherwise|\n\
\\\\ /\
"""
ucode_str = \
u("""\
⎛⎧x ⎞\n\
⎜⎪─ for x < 2⎟\n\
⎜⎪y ⎟\n\
⎛⎧x for x > 0⎞ ⎜⎪ ⎟\n\
-⎜⎨ ⎟⋅⎜⎨ 2 ⎟\n\
⎝⎩y otherwise⎠ ⎜⎪y for x > 2⎟\n\
⎜⎪ ⎟\n\
⎜⎪1 otherwise⎟\n\
⎝⎩ ⎠\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Piecewise((0, Abs(1/y) < 1), (1, Abs(y) < 1), (y*meijerg(((2, 1),
()), ((), (1, 0)), 1/y), True))
ascii_str = \
"""\
/ 1 \n\
| 0 for --- < 1\n\
| |y| \n\
| \n\
< 1 for |y| < 1\n\
| \n\
| __0, 2 /2, 1 | 1\\ \n\
|y*/__ | | -| otherwise \n\
\\ \\_|2, 2 \\ 1, 0 | y/ \
"""
ucode_str = \
u("""\
⎧ 1 \n\
⎪ 0 for ─── < 1\n\
⎪ │y│ \n\
⎪ \n\
⎨ 1 for │y│ < 1\n\
⎪ \n\
⎪ ╭─╮0, 2 ⎛2, 1 │ 1⎞ \n\
⎪y⋅│╶┐ ⎜ │ ─⎟ otherwise \n\
⎩ ╰─╯2, 2 ⎝ 1, 0 │ y⎠ \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
# XXX: We have to use evaluate=False here because Piecewise._eval_power
# denests the power.
expr = Pow(Piecewise((x, x > 0), (y, True)), 2, evaluate=False)
ascii_str = \
"""\
2\n\
//x for x > 0\\ \n\
|< | \n\
\\\\y otherwise/ \
"""
ucode_str = \
u("""\
2\n\
⎛⎧x for x > 0⎞ \n\
⎜⎨ ⎟ \n\
⎝⎩y otherwise⎠ \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_pretty_ITE():
expr = ITE(x, y, z)
assert pretty(expr) == (
'/y for x \n'
'< \n'
'\\z otherwise'
)
assert upretty(expr) == u("""\
⎧y for x \n\
⎨ \n\
⎩z otherwise\
""")
def test_pretty_seq():
expr = ()
ascii_str = \
"""\
()\
"""
ucode_str = \
u("""\
()\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = []
ascii_str = \
"""\
[]\
"""
ucode_str = \
u("""\
[]\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = {}
expr_2 = {}
ascii_str = \
"""\
{}\
"""
ucode_str = \
u("""\
{}\
""")
assert pretty(expr) == ascii_str
assert pretty(expr_2) == ascii_str
assert upretty(expr) == ucode_str
assert upretty(expr_2) == ucode_str
expr = (1/x,)
ascii_str = \
"""\
1 \n\
(-,)\n\
x \
"""
ucode_str = \
u("""\
⎛1 ⎞\n\
⎜─,⎟\n\
⎝x ⎠\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = [x**2, 1/x, x, y, sin(th)**2/cos(ph)**2]
ascii_str = \
"""\
2 \n\
2 1 sin (theta) \n\
[x , -, x, y, -----------]\n\
x 2 \n\
cos (phi) \
"""
ucode_str = \
u("""\
⎡ 2 ⎤\n\
⎢ 2 1 sin (θ)⎥\n\
⎢x , ─, x, y, ───────⎥\n\
⎢ x 2 ⎥\n\
⎣ cos (φ)⎦\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = (x**2, 1/x, x, y, sin(th)**2/cos(ph)**2)
ascii_str = \
"""\
2 \n\
2 1 sin (theta) \n\
(x , -, x, y, -----------)\n\
x 2 \n\
cos (phi) \
"""
ucode_str = \
u("""\
⎛ 2 ⎞\n\
⎜ 2 1 sin (θ)⎟\n\
⎜x , ─, x, y, ───────⎟\n\
⎜ x 2 ⎟\n\
⎝ cos (φ)⎠\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Tuple(x**2, 1/x, x, y, sin(th)**2/cos(ph)**2)
ascii_str = \
"""\
2 \n\
2 1 sin (theta) \n\
(x , -, x, y, -----------)\n\
x 2 \n\
cos (phi) \
"""
ucode_str = \
u("""\
⎛ 2 ⎞\n\
⎜ 2 1 sin (θ)⎟\n\
⎜x , ─, x, y, ───────⎟\n\
⎜ x 2 ⎟\n\
⎝ cos (φ)⎠\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = {x: sin(x)}
expr_2 = Dict({x: sin(x)})
ascii_str = \
"""\
{x: sin(x)}\
"""
ucode_str = \
u("""\
{x: sin(x)}\
""")
assert pretty(expr) == ascii_str
assert pretty(expr_2) == ascii_str
assert upretty(expr) == ucode_str
assert upretty(expr_2) == ucode_str
expr = {1/x: 1/y, x: sin(x)**2}
expr_2 = Dict({1/x: 1/y, x: sin(x)**2})
ascii_str = \
"""\
1 1 2 \n\
{-: -, x: sin (x)}\n\
x y \
"""
ucode_str = \
u("""\
⎧1 1 2 ⎫\n\
⎨─: ─, x: sin (x)⎬\n\
⎩x y ⎭\
""")
assert pretty(expr) == ascii_str
assert pretty(expr_2) == ascii_str
assert upretty(expr) == ucode_str
assert upretty(expr_2) == ucode_str
# There used to be a bug with pretty-printing sequences of even height.
expr = [x**2]
ascii_str = \
"""\
2 \n\
[x ]\
"""
ucode_str = \
u("""\
⎡ 2⎤\n\
⎣x ⎦\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = (x**2,)
ascii_str = \
"""\
2 \n\
(x ,)\
"""
ucode_str = \
u("""\
⎛ 2 ⎞\n\
⎝x ,⎠\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Tuple(x**2)
ascii_str = \
"""\
2 \n\
(x ,)\
"""
ucode_str = \
u("""\
⎛ 2 ⎞\n\
⎝x ,⎠\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = {x**2: 1}
expr_2 = Dict({x**2: 1})
ascii_str = \
"""\
2 \n\
{x : 1}\
"""
ucode_str = \
u("""\
⎧ 2 ⎫\n\
⎨x : 1⎬\n\
⎩ ⎭\
""")
assert pretty(expr) == ascii_str
assert pretty(expr_2) == ascii_str
assert upretty(expr) == ucode_str
assert upretty(expr_2) == ucode_str
def test_any_object_in_sequence():
# Cf. issue 5306
b1 = Basic()
b2 = Basic(Basic())
expr = [b2, b1]
assert pretty(expr) == "[Basic(Basic()), Basic()]"
assert upretty(expr) == u"[Basic(Basic()), Basic()]"
expr = {b2, b1}
assert pretty(expr) == "{Basic(), Basic(Basic())}"
assert upretty(expr) == u"{Basic(), Basic(Basic())}"
expr = {b2: b1, b1: b2}
expr2 = Dict({b2: b1, b1: b2})
assert pretty(expr) == "{Basic(): Basic(Basic()), Basic(Basic()): Basic()}"
assert pretty(
expr2) == "{Basic(): Basic(Basic()), Basic(Basic()): Basic()}"
assert upretty(
expr) == u"{Basic(): Basic(Basic()), Basic(Basic()): Basic()}"
assert upretty(
expr2) == u"{Basic(): Basic(Basic()), Basic(Basic()): Basic()}"
def test_print_builtin_set():
assert pretty(set()) == 'set()'
assert upretty(set()) == u'set()'
assert pretty(frozenset()) == 'frozenset()'
assert upretty(frozenset()) == u'frozenset()'
s1 = {1/x, x}
s2 = frozenset(s1)
assert pretty(s1) == \
"""\
1 \n\
{-, x}
x \
"""
assert upretty(s1) == \
u"""\
⎧1 ⎫
⎨─, x⎬
⎩x ⎭\
"""
assert pretty(s2) == \
"""\
1 \n\
frozenset({-, x})
x \
"""
assert upretty(s2) == \
u"""\
⎛⎧1 ⎫⎞
frozenset⎜⎨─, x⎬⎟
⎝⎩x ⎭⎠\
"""
def test_pretty_sets():
s = FiniteSet
assert pretty(s(*[x*y, x**2])) == \
"""\
2 \n\
{x , x*y}\
"""
assert pretty(s(*range(1, 6))) == "{1, 2, 3, 4, 5}"
assert pretty(s(*range(1, 13))) == "{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}"
assert pretty(set([x*y, x**2])) == \
"""\
2 \n\
{x , x*y}\
"""
assert pretty(set(range(1, 6))) == "{1, 2, 3, 4, 5}"
assert pretty(set(range(1, 13))) == \
"{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}"
assert pretty(frozenset([x*y, x**2])) == \
"""\
2 \n\
frozenset({x , x*y})\
"""
assert pretty(frozenset(range(1, 6))) == "frozenset({1, 2, 3, 4, 5})"
assert pretty(frozenset(range(1, 13))) == \
"frozenset({1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12})"
assert pretty(Range(0, 3, 1)) == '{0, 1, 2}'
ascii_str = '{0, 1, ..., 29}'
ucode_str = u'{0, 1, …, 29}'
assert pretty(Range(0, 30, 1)) == ascii_str
assert upretty(Range(0, 30, 1)) == ucode_str
ascii_str = '{30, 29, ..., 2}'
ucode_str = u('{30, 29, …, 2}')
assert pretty(Range(30, 1, -1)) == ascii_str
assert upretty(Range(30, 1, -1)) == ucode_str
ascii_str = '{0, 2, ...}'
ucode_str = u'{0, 2, …}'
assert pretty(Range(0, oo, 2)) == ascii_str
assert upretty(Range(0, oo, 2)) == ucode_str
ascii_str = '{..., 2, 0}'
ucode_str = u('{…, 2, 0}')
assert pretty(Range(oo, -2, -2)) == ascii_str
assert upretty(Range(oo, -2, -2)) == ucode_str
ascii_str = '{-2, -3, ...}'
ucode_str = u('{-2, -3, …}')
assert pretty(Range(-2, -oo, -1)) == ascii_str
assert upretty(Range(-2, -oo, -1)) == ucode_str
def test_pretty_SetExpr():
iv = Interval(1, 3)
se = SetExpr(iv)
ascii_str = "SetExpr([1, 3])"
ucode_str = u("SetExpr([1, 3])")
assert pretty(se) == ascii_str
assert upretty(se) == ucode_str
def test_pretty_ImageSet():
imgset = ImageSet(Lambda((x, y), x + y), {1, 2, 3}, {3, 4})
ascii_str = '{x + y | x in {1, 2, 3} , y in {3, 4}}'
ucode_str = u('{x + y | x ∊ {1, 2, 3} , y ∊ {3, 4}}')
assert pretty(imgset) == ascii_str
assert upretty(imgset) == ucode_str
imgset = ImageSet(Lambda(((x, y),), x + y), ProductSet({1, 2, 3}, {3, 4}))
ascii_str = '{x + y | (x, y) in {1, 2, 3} x {3, 4}}'
ucode_str = u('{x + y | (x, y) ∊ {1, 2, 3} × {3, 4}}')
assert pretty(imgset) == ascii_str
assert upretty(imgset) == ucode_str
imgset = ImageSet(Lambda(x, x**2), S.Naturals)
ascii_str = \
' 2 \n'\
'{x | x in Naturals}'
ucode_str = u('''\
⎧ 2 ⎫\n\
⎨x | x ∊ ℕ⎬\n\
⎩ ⎭''')
assert pretty(imgset) == ascii_str
assert upretty(imgset) == ucode_str
def test_pretty_ConditionSet():
from sympy import ConditionSet
ascii_str = '{x | x in (-oo, oo) and sin(x) = 0}'
ucode_str = u'{x | x ∊ ℝ ∧ (sin(x) = 0)}'
assert pretty(ConditionSet(x, Eq(sin(x), 0), S.Reals)) == ascii_str
assert upretty(ConditionSet(x, Eq(sin(x), 0), S.Reals)) == ucode_str
assert pretty(ConditionSet(x, Contains(x, S.Reals, evaluate=False), FiniteSet(1))) == '{1}'
assert upretty(ConditionSet(x, Contains(x, S.Reals, evaluate=False), FiniteSet(1))) == u'{1}'
assert pretty(ConditionSet(x, And(x > 1, x < -1), FiniteSet(1, 2, 3))) == "EmptySet"
assert upretty(ConditionSet(x, And(x > 1, x < -1), FiniteSet(1, 2, 3))) == u"∅"
assert pretty(ConditionSet(x, Or(x > 1, x < -1), FiniteSet(1, 2))) == '{2}'
assert upretty(ConditionSet(x, Or(x > 1, x < -1), FiniteSet(1, 2))) == u'{2}'
def test_pretty_ComplexRegion():
from sympy import ComplexRegion
ucode_str = u'{x + y⋅ⅈ | x, y ∊ [3, 5] × [4, 6]}'
assert upretty(ComplexRegion(Interval(3, 5)*Interval(4, 6))) == ucode_str
ucode_str = u'{r⋅(ⅈ⋅sin(θ) + cos(θ)) | r, θ ∊ [0, 1] × [0, 2⋅π)}'
assert upretty(ComplexRegion(Interval(0, 1)*Interval(0, 2*pi), polar=True)) == ucode_str
def test_pretty_Union_issue_10414():
a, b = Interval(2, 3), Interval(4, 7)
ucode_str = u'[2, 3] ∪ [4, 7]'
ascii_str = '[2, 3] U [4, 7]'
assert upretty(Union(a, b)) == ucode_str
assert pretty(Union(a, b)) == ascii_str
def test_pretty_Intersection_issue_10414():
x, y, z, w = symbols('x, y, z, w')
a, b = Interval(x, y), Interval(z, w)
ucode_str = u'[x, y] ∩ [z, w]'
ascii_str = '[x, y] n [z, w]'
assert upretty(Intersection(a, b)) == ucode_str
assert pretty(Intersection(a, b)) == ascii_str
def test_ProductSet_exponent():
ucode_str = ' 1\n[0, 1] '
assert upretty(Interval(0, 1)**1) == ucode_str
ucode_str = ' 2\n[0, 1] '
assert upretty(Interval(0, 1)**2) == ucode_str
def test_ProductSet_parenthesis():
ucode_str = u'([4, 7] × {1, 2}) ∪ ([2, 3] × [4, 7])'
a, b = Interval(2, 3), Interval(4, 7)
assert upretty(Union(a*b, b*FiniteSet(1, 2))) == ucode_str
def test_ProductSet_prod_char_issue_10413():
ascii_str = '[2, 3] x [4, 7]'
ucode_str = u'[2, 3] × [4, 7]'
a, b = Interval(2, 3), Interval(4, 7)
assert pretty(a*b) == ascii_str
assert upretty(a*b) == ucode_str
def test_pretty_sequences():
s1 = SeqFormula(a**2, (0, oo))
s2 = SeqPer((1, 2))
ascii_str = '[0, 1, 4, 9, ...]'
ucode_str = u'[0, 1, 4, 9, …]'
assert pretty(s1) == ascii_str
assert upretty(s1) == ucode_str
ascii_str = '[1, 2, 1, 2, ...]'
ucode_str = u'[1, 2, 1, 2, …]'
assert pretty(s2) == ascii_str
assert upretty(s2) == ucode_str
s3 = SeqFormula(a**2, (0, 2))
s4 = SeqPer((1, 2), (0, 2))
ascii_str = '[0, 1, 4]'
ucode_str = u'[0, 1, 4]'
assert pretty(s3) == ascii_str
assert upretty(s3) == ucode_str
ascii_str = '[1, 2, 1]'
ucode_str = u'[1, 2, 1]'
assert pretty(s4) == ascii_str
assert upretty(s4) == ucode_str
s5 = SeqFormula(a**2, (-oo, 0))
s6 = SeqPer((1, 2), (-oo, 0))
ascii_str = '[..., 9, 4, 1, 0]'
ucode_str = u'[…, 9, 4, 1, 0]'
assert pretty(s5) == ascii_str
assert upretty(s5) == ucode_str
ascii_str = '[..., 2, 1, 2, 1]'
ucode_str = u'[…, 2, 1, 2, 1]'
assert pretty(s6) == ascii_str
assert upretty(s6) == ucode_str
ascii_str = '[1, 3, 5, 11, ...]'
ucode_str = u'[1, 3, 5, 11, …]'
assert pretty(SeqAdd(s1, s2)) == ascii_str
assert upretty(SeqAdd(s1, s2)) == ucode_str
ascii_str = '[1, 3, 5]'
ucode_str = u'[1, 3, 5]'
assert pretty(SeqAdd(s3, s4)) == ascii_str
assert upretty(SeqAdd(s3, s4)) == ucode_str
ascii_str = '[..., 11, 5, 3, 1]'
ucode_str = u'[…, 11, 5, 3, 1]'
assert pretty(SeqAdd(s5, s6)) == ascii_str
assert upretty(SeqAdd(s5, s6)) == ucode_str
ascii_str = '[0, 2, 4, 18, ...]'
ucode_str = u'[0, 2, 4, 18, …]'
assert pretty(SeqMul(s1, s2)) == ascii_str
assert upretty(SeqMul(s1, s2)) == ucode_str
ascii_str = '[0, 2, 4]'
ucode_str = u'[0, 2, 4]'
assert pretty(SeqMul(s3, s4)) == ascii_str
assert upretty(SeqMul(s3, s4)) == ucode_str
ascii_str = '[..., 18, 4, 2, 0]'
ucode_str = u'[…, 18, 4, 2, 0]'
assert pretty(SeqMul(s5, s6)) == ascii_str
assert upretty(SeqMul(s5, s6)) == ucode_str
# Sequences with symbolic limits, issue 12629
s7 = SeqFormula(a**2, (a, 0, x))
raises(NotImplementedError, lambda: pretty(s7))
raises(NotImplementedError, lambda: upretty(s7))
b = Symbol('b')
s8 = SeqFormula(b*a**2, (a, 0, 2))
ascii_str = u'[0, b, 4*b]'
ucode_str = u'[0, b, 4⋅b]'
assert pretty(s8) == ascii_str
assert upretty(s8) == ucode_str
def test_pretty_FourierSeries():
f = fourier_series(x, (x, -pi, pi))
ascii_str = \
"""\
2*sin(3*x) \n\
2*sin(x) - sin(2*x) + ---------- + ...\n\
3 \
"""
ucode_str = \
u("""\
2⋅sin(3⋅x) \n\
2⋅sin(x) - sin(2⋅x) + ────────── + …\n\
3 \
""")
assert pretty(f) == ascii_str
assert upretty(f) == ucode_str
def test_pretty_FormalPowerSeries():
f = fps(log(1 + x))
ascii_str = \
"""\
oo \n\
____ \n\
\\ ` \n\
\\ -k k \n\
\\ -(-1) *x \n\
/ -----------\n\
/ k \n\
/___, \n\
k = 1 \
"""
ucode_str = \
u("""\
∞ \n\
____ \n\
╲ \n\
╲ -k k \n\
╲ -(-1) ⋅x \n\
╱ ───────────\n\
╱ k \n\
╱ \n\
‾‾‾‾ \n\
k = 1 \
""")
assert pretty(f) == ascii_str
assert upretty(f) == ucode_str
def test_pretty_limits():
expr = Limit(x, x, oo)
ascii_str = \
"""\
lim x\n\
x->oo \
"""
ucode_str = \
u("""\
lim x\n\
x─→∞ \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Limit(x**2, x, 0)
ascii_str = \
"""\
2\n\
lim x \n\
x->0+ \
"""
ucode_str = \
u("""\
2\n\
lim x \n\
x─→0⁺ \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Limit(1/x, x, 0)
ascii_str = \
"""\
1\n\
lim -\n\
x->0+x\
"""
ucode_str = \
u("""\
1\n\
lim ─\n\
x─→0⁺x\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Limit(sin(x)/x, x, 0)
ascii_str = \
"""\
/sin(x)\\\n\
lim |------|\n\
x->0+\\ x /\
"""
ucode_str = \
u("""\
⎛sin(x)⎞\n\
lim ⎜──────⎟\n\
x─→0⁺⎝ x ⎠\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Limit(sin(x)/x, x, 0, "-")
ascii_str = \
"""\
/sin(x)\\\n\
lim |------|\n\
x->0-\\ x /\
"""
ucode_str = \
u("""\
⎛sin(x)⎞\n\
lim ⎜──────⎟\n\
x─→0⁻⎝ x ⎠\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Limit(x + sin(x), x, 0)
ascii_str = \
"""\
lim (x + sin(x))\n\
x->0+ \
"""
ucode_str = \
u("""\
lim (x + sin(x))\n\
x─→0⁺ \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Limit(x, x, 0)**2
ascii_str = \
"""\
2\n\
/ lim x\\ \n\
\\x->0+ / \
"""
ucode_str = \
u("""\
2\n\
⎛ lim x⎞ \n\
⎝x─→0⁺ ⎠ \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Limit(x*Limit(y/2,y,0), x, 0)
ascii_str = \
"""\
/ /y\\\\\n\
lim |x* lim |-||\n\
x->0+\\ y->0+\\2//\
"""
ucode_str = \
u("""\
⎛ ⎛y⎞⎞\n\
lim ⎜x⋅ lim ⎜─⎟⎟\n\
x─→0⁺⎝ y─→0⁺⎝2⎠⎠\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = 2*Limit(x*Limit(y/2,y,0), x, 0)
ascii_str = \
"""\
/ /y\\\\\n\
2* lim |x* lim |-||\n\
x->0+\\ y->0+\\2//\
"""
ucode_str = \
u("""\
⎛ ⎛y⎞⎞\n\
2⋅ lim ⎜x⋅ lim ⎜─⎟⎟\n\
x─→0⁺⎝ y─→0⁺⎝2⎠⎠\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Limit(sin(x), x, 0, dir='+-')
ascii_str = \
"""\
lim sin(x)\n\
x->0 \
"""
ucode_str = \
u("""\
lim sin(x)\n\
x─→0 \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_pretty_ComplexRootOf():
expr = rootof(x**5 + 11*x - 2, 0)
ascii_str = \
"""\
/ 5 \\\n\
CRootOf\\x + 11*x - 2, 0/\
"""
ucode_str = \
u("""\
⎛ 5 ⎞\n\
CRootOf⎝x + 11⋅x - 2, 0⎠\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_pretty_RootSum():
expr = RootSum(x**5 + 11*x - 2, auto=False)
ascii_str = \
"""\
/ 5 \\\n\
RootSum\\x + 11*x - 2/\
"""
ucode_str = \
u("""\
⎛ 5 ⎞\n\
RootSum⎝x + 11⋅x - 2⎠\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = RootSum(x**5 + 11*x - 2, Lambda(z, exp(z)))
ascii_str = \
"""\
/ 5 z\\\n\
RootSum\\x + 11*x - 2, z -> e /\
"""
ucode_str = \
u("""\
⎛ 5 z⎞\n\
RootSum⎝x + 11⋅x - 2, z ↦ ℯ ⎠\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_GroebnerBasis():
expr = groebner([], x, y)
ascii_str = \
"""\
GroebnerBasis([], x, y, domain=ZZ, order=lex)\
"""
ucode_str = \
u("""\
GroebnerBasis([], x, y, domain=ℤ, order=lex)\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
F = [x**2 - 3*y - x + 1, y**2 - 2*x + y - 1]
expr = groebner(F, x, y, order='grlex')
ascii_str = \
"""\
/[ 2 2 ] \\\n\
GroebnerBasis\\[x - x - 3*y + 1, y - 2*x + y - 1], x, y, domain=ZZ, order=grlex/\
"""
ucode_str = \
u("""\
⎛⎡ 2 2 ⎤ ⎞\n\
GroebnerBasis⎝⎣x - x - 3⋅y + 1, y - 2⋅x + y - 1⎦, x, y, domain=ℤ, order=grlex⎠\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = expr.fglm('lex')
ascii_str = \
"""\
/[ 2 4 3 2 ] \\\n\
GroebnerBasis\\[2*x - y - y + 1, y + 2*y - 3*y - 16*y + 7], x, y, domain=ZZ, order=lex/\
"""
ucode_str = \
u("""\
⎛⎡ 2 4 3 2 ⎤ ⎞\n\
GroebnerBasis⎝⎣2⋅x - y - y + 1, y + 2⋅y - 3⋅y - 16⋅y + 7⎦, x, y, domain=ℤ, order=lex⎠\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_pretty_UniversalSet():
assert pretty(S.UniversalSet) == "UniversalSet"
assert upretty(S.UniversalSet) == u'𝕌'
def test_pretty_Boolean():
expr = Not(x, evaluate=False)
assert pretty(expr) == "Not(x)"
assert upretty(expr) == u"¬x"
expr = And(x, y)
assert pretty(expr) == "And(x, y)"
assert upretty(expr) == u"x ∧ y"
expr = Or(x, y)
assert pretty(expr) == "Or(x, y)"
assert upretty(expr) == u"x ∨ y"
syms = symbols('a:f')
expr = And(*syms)
assert pretty(expr) == "And(a, b, c, d, e, f)"
assert upretty(expr) == u"a ∧ b ∧ c ∧ d ∧ e ∧ f"
expr = Or(*syms)
assert pretty(expr) == "Or(a, b, c, d, e, f)"
assert upretty(expr) == u"a ∨ b ∨ c ∨ d ∨ e ∨ f"
expr = Xor(x, y, evaluate=False)
assert pretty(expr) == "Xor(x, y)"
assert upretty(expr) == u"x ⊻ y"
expr = Nand(x, y, evaluate=False)
assert pretty(expr) == "Nand(x, y)"
assert upretty(expr) == u"x ⊼ y"
expr = Nor(x, y, evaluate=False)
assert pretty(expr) == "Nor(x, y)"
assert upretty(expr) == u"x ⊽ y"
expr = Implies(x, y, evaluate=False)
assert pretty(expr) == "Implies(x, y)"
assert upretty(expr) == u"x → y"
# don't sort args
expr = Implies(y, x, evaluate=False)
assert pretty(expr) == "Implies(y, x)"
assert upretty(expr) == u"y → x"
expr = Equivalent(x, y, evaluate=False)
assert pretty(expr) == "Equivalent(x, y)"
assert upretty(expr) == u"x ⇔ y"
expr = Equivalent(y, x, evaluate=False)
assert pretty(expr) == "Equivalent(x, y)"
assert upretty(expr) == u"x ⇔ y"
def test_pretty_Domain():
expr = FF(23)
assert pretty(expr) == "GF(23)"
assert upretty(expr) == u"ℤ₂₃"
expr = ZZ
assert pretty(expr) == "ZZ"
assert upretty(expr) == u"ℤ"
expr = QQ
assert pretty(expr) == "QQ"
assert upretty(expr) == u"ℚ"
expr = RR
assert pretty(expr) == "RR"
assert upretty(expr) == u"ℝ"
expr = QQ[x]
assert pretty(expr) == "QQ[x]"
assert upretty(expr) == u"ℚ[x]"
expr = QQ[x, y]
assert pretty(expr) == "QQ[x, y]"
assert upretty(expr) == u"ℚ[x, y]"
expr = ZZ.frac_field(x)
assert pretty(expr) == "ZZ(x)"
assert upretty(expr) == u"ℤ(x)"
expr = ZZ.frac_field(x, y)
assert pretty(expr) == "ZZ(x, y)"
assert upretty(expr) == u"ℤ(x, y)"
expr = QQ.poly_ring(x, y, order=grlex)
assert pretty(expr) == "QQ[x, y, order=grlex]"
assert upretty(expr) == u"ℚ[x, y, order=grlex]"
expr = QQ.poly_ring(x, y, order=ilex)
assert pretty(expr) == "QQ[x, y, order=ilex]"
assert upretty(expr) == u"ℚ[x, y, order=ilex]"
def test_pretty_prec():
assert xpretty(S("0.3"), full_prec=True, wrap_line=False) == "0.300000000000000"
assert xpretty(S("0.3"), full_prec="auto", wrap_line=False) == "0.300000000000000"
assert xpretty(S("0.3"), full_prec=False, wrap_line=False) == "0.3"
assert xpretty(S("0.3")*x, full_prec=True, use_unicode=False, wrap_line=False) in [
"0.300000000000000*x",
"x*0.300000000000000"
]
assert xpretty(S("0.3")*x, full_prec="auto", use_unicode=False, wrap_line=False) in [
"0.3*x",
"x*0.3"
]
assert xpretty(S("0.3")*x, full_prec=False, use_unicode=False, wrap_line=False) in [
"0.3*x",
"x*0.3"
]
def test_pprint():
import sys
from sympy.core.compatibility import StringIO
fd = StringIO()
sso = sys.stdout
sys.stdout = fd
try:
pprint(pi, use_unicode=False, wrap_line=False)
finally:
sys.stdout = sso
assert fd.getvalue() == 'pi\n'
def test_pretty_class():
"""Test that the printer dispatcher correctly handles classes."""
class C:
pass # C has no .__class__ and this was causing problems
class D(object):
pass
assert pretty( C ) == str( C )
assert pretty( D ) == str( D )
def test_pretty_no_wrap_line():
huge_expr = 0
for i in range(20):
huge_expr += i*sin(i + x)
assert xpretty(huge_expr ).find('\n') != -1
assert xpretty(huge_expr, wrap_line=False).find('\n') == -1
def test_settings():
raises(TypeError, lambda: pretty(S(4), method="garbage"))
def test_pretty_sum():
from sympy.abc import x, a, b, k, m, n
expr = Sum(k**k, (k, 0, n))
ascii_str = \
"""\
n \n\
___ \n\
\\ ` \n\
\\ k\n\
/ k \n\
/__, \n\
k = 0 \
"""
ucode_str = \
u("""\
n \n\
___ \n\
╲ \n\
╲ k\n\
╱ k \n\
╱ \n\
‾‾‾ \n\
k = 0 \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Sum(k**k, (k, oo, n))
ascii_str = \
"""\
n \n\
___ \n\
\\ ` \n\
\\ k\n\
/ k \n\
/__, \n\
k = oo \
"""
ucode_str = \
u("""\
n \n\
___ \n\
╲ \n\
╲ k\n\
╱ k \n\
╱ \n\
‾‾‾ \n\
k = ∞ \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Sum(k**(Integral(x**n, (x, -oo, oo))), (k, 0, n**n))
ascii_str = \
"""\
n \n\
n \n\
______ \n\
\\ ` \n\
\\ oo \n\
\\ / \n\
\\ | \n\
\\ | n \n\
) | x dx\n\
/ | \n\
/ / \n\
/ -oo \n\
/ k \n\
/_____, \n\
k = 0 \
"""
ucode_str = \
u("""\
n \n\
n \n\
______ \n\
╲ \n\
╲ \n\
╲ ∞ \n\
╲ ⌠ \n\
╲ ⎮ n \n\
╱ ⎮ x dx\n\
╱ ⌡ \n\
╱ -∞ \n\
╱ k \n\
╱ \n\
‾‾‾‾‾‾ \n\
k = 0 \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Sum(k**(
Integral(x**n, (x, -oo, oo))), (k, 0, Integral(x**x, (x, -oo, oo))))
ascii_str = \
"""\
oo \n\
/ \n\
| \n\
| x \n\
| x dx \n\
| \n\
/ \n\
-oo \n\
______ \n\
\\ ` \n\
\\ oo \n\
\\ / \n\
\\ | \n\
\\ | n \n\
) | x dx\n\
/ | \n\
/ / \n\
/ -oo \n\
/ k \n\
/_____, \n\
k = 0 \
"""
ucode_str = \
u("""\
∞ \n\
⌠ \n\
⎮ x \n\
⎮ x dx \n\
⌡ \n\
-∞ \n\
______ \n\
╲ \n\
╲ \n\
╲ ∞ \n\
╲ ⌠ \n\
╲ ⎮ n \n\
╱ ⎮ x dx\n\
╱ ⌡ \n\
╱ -∞ \n\
╱ k \n\
╱ \n\
‾‾‾‾‾‾ \n\
k = 0 \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Sum(k**(Integral(x**n, (x, -oo, oo))), (
k, x + n + x**2 + n**2 + (x/n) + (1/x), Integral(x**x, (x, -oo, oo))))
ascii_str = \
"""\
oo \n\
/ \n\
| \n\
| x \n\
| x dx \n\
| \n\
/ \n\
-oo \n\
______ \n\
\\ ` \n\
\\ oo \n\
\\ / \n\
\\ | \n\
\\ | n \n\
) | x dx\n\
/ | \n\
/ / \n\
/ -oo \n\
/ k \n\
/_____, \n\
2 2 1 x \n\
k = n + n + x + x + - + - \n\
x n \
"""
ucode_str = \
u("""\
∞ \n\
⌠ \n\
⎮ x \n\
⎮ x dx \n\
⌡ \n\
-∞ \n\
______ \n\
╲ \n\
╲ \n\
╲ ∞ \n\
╲ ⌠ \n\
╲ ⎮ n \n\
╱ ⎮ x dx\n\
╱ ⌡ \n\
╱ -∞ \n\
╱ k \n\
╱ \n\
‾‾‾‾‾‾ \n\
2 2 1 x \n\
k = n + n + x + x + ─ + ─ \n\
x n \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Sum(k**(
Integral(x**n, (x, -oo, oo))), (k, 0, x + n + x**2 + n**2 + (x/n) + (1/x)))
ascii_str = \
"""\
2 2 1 x \n\
n + n + x + x + - + - \n\
x n \n\
______ \n\
\\ ` \n\
\\ oo \n\
\\ / \n\
\\ | \n\
\\ | n \n\
) | x dx\n\
/ | \n\
/ / \n\
/ -oo \n\
/ k \n\
/_____, \n\
k = 0 \
"""
ucode_str = \
u("""\
2 2 1 x \n\
n + n + x + x + ─ + ─ \n\
x n \n\
______ \n\
╲ \n\
╲ \n\
╲ ∞ \n\
╲ ⌠ \n\
╲ ⎮ n \n\
╱ ⎮ x dx\n\
╱ ⌡ \n\
╱ -∞ \n\
╱ k \n\
╱ \n\
‾‾‾‾‾‾ \n\
k = 0 \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Sum(x, (x, 0, oo))
ascii_str = \
"""\
oo \n\
__ \n\
\\ ` \n\
) x\n\
/_, \n\
x = 0 \
"""
ucode_str = \
u("""\
∞ \n\
___ \n\
╲ \n\
╲ \n\
╱ x\n\
╱ \n\
‾‾‾ \n\
x = 0 \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Sum(x**2, (x, 0, oo))
ascii_str = \
u("""\
oo \n\
___ \n\
\\ ` \n\
\\ 2\n\
/ x \n\
/__, \n\
x = 0 \
""")
ucode_str = \
u("""\
∞ \n\
___ \n\
╲ \n\
╲ 2\n\
╱ x \n\
╱ \n\
‾‾‾ \n\
x = 0 \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Sum(x/2, (x, 0, oo))
ascii_str = \
"""\
oo \n\
___ \n\
\\ ` \n\
\\ x\n\
) -\n\
/ 2\n\
/__, \n\
x = 0 \
"""
ucode_str = \
u("""\
∞ \n\
____ \n\
╲ \n\
╲ \n\
╲ x\n\
╱ ─\n\
╱ 2\n\
╱ \n\
‾‾‾‾ \n\
x = 0 \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Sum(x**3/2, (x, 0, oo))
ascii_str = \
"""\
oo \n\
____ \n\
\\ ` \n\
\\ 3\n\
\\ x \n\
/ --\n\
/ 2 \n\
/___, \n\
x = 0 \
"""
ucode_str = \
u("""\
∞ \n\
____ \n\
╲ \n\
╲ 3\n\
╲ x \n\
╱ ──\n\
╱ 2 \n\
╱ \n\
‾‾‾‾ \n\
x = 0 \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Sum((x**3*y**(x/2))**n, (x, 0, oo))
ascii_str = \
"""\
oo \n\
____ \n\
\\ ` \n\
\\ n\n\
\\ / x\\ \n\
) | -| \n\
/ | 3 2| \n\
/ \\x *y / \n\
/___, \n\
x = 0 \
"""
ucode_str = \
u("""\
∞ \n\
_____ \n\
╲ \n\
╲ \n\
╲ n\n\
╲ ⎛ x⎞ \n\
╱ ⎜ ─⎟ \n\
╱ ⎜ 3 2⎟ \n\
╱ ⎝x ⋅y ⎠ \n\
╱ \n\
‾‾‾‾‾ \n\
x = 0 \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Sum(1/x**2, (x, 0, oo))
ascii_str = \
"""\
oo \n\
____ \n\
\\ ` \n\
\\ 1 \n\
\\ --\n\
/ 2\n\
/ x \n\
/___, \n\
x = 0 \
"""
ucode_str = \
u("""\
∞ \n\
____ \n\
╲ \n\
╲ 1 \n\
╲ ──\n\
╱ 2\n\
╱ x \n\
╱ \n\
‾‾‾‾ \n\
x = 0 \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Sum(1/y**(a/b), (x, 0, oo))
ascii_str = \
"""\
oo \n\
____ \n\
\\ ` \n\
\\ -a \n\
\\ ---\n\
/ b \n\
/ y \n\
/___, \n\
x = 0 \
"""
ucode_str = \
u("""\
∞ \n\
____ \n\
╲ \n\
╲ -a \n\
╲ ───\n\
╱ b \n\
╱ y \n\
╱ \n\
‾‾‾‾ \n\
x = 0 \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Sum(1/y**(a/b), (x, 0, oo), (y, 1, 2))
ascii_str = \
"""\
2 oo \n\
____ ____ \n\
\\ ` \\ ` \n\
\\ \\ -a\n\
\\ \\ --\n\
/ / b \n\
/ / y \n\
/___, /___, \n\
y = 1 x = 0 \
"""
ucode_str = \
u("""\
2 ∞ \n\
____ ____ \n\
╲ ╲ \n\
╲ ╲ -a\n\
╲ ╲ ──\n\
╱ ╱ b \n\
╱ ╱ y \n\
╱ ╱ \n\
‾‾‾‾ ‾‾‾‾ \n\
y = 1 x = 0 \
""")
expr = Sum(1/(1 + 1/(
1 + 1/k)) + 1, (k, 111, 1 + 1/n), (k, 1/(1 + m), oo)) + 1/(1 + 1/k)
ascii_str = \
"""\
1 \n\
1 + - \n\
oo n \n\
_____ _____ \n\
\\ ` \\ ` \n\
\\ \\ / 1 \\ \n\
\\ \\ |1 + ---------| \n\
\\ \\ | 1 | 1 \n\
) ) | 1 + -----| + -----\n\
/ / | 1| 1\n\
/ / | 1 + -| 1 + -\n\
/ / \\ k/ k\n\
/____, /____, \n\
1 k = 111 \n\
k = ----- \n\
m + 1 \
"""
ucode_str = \
u("""\
1 \n\
1 + ─ \n\
∞ n \n\
______ ______ \n\
╲ ╲ \n\
╲ ╲ \n\
╲ ╲ ⎛ 1 ⎞ \n\
╲ ╲ ⎜1 + ─────────⎟ \n\
╲ ╲ ⎜ 1 ⎟ 1 \n\
╱ ╱ ⎜ 1 + ─────⎟ + ─────\n\
╱ ╱ ⎜ 1⎟ 1\n\
╱ ╱ ⎜ 1 + ─⎟ 1 + ─\n\
╱ ╱ ⎝ k⎠ k\n\
╱ ╱ \n\
‾‾‾‾‾‾ ‾‾‾‾‾‾ \n\
1 k = 111 \n\
k = ───── \n\
m + 1 \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_units():
expr = joule
ascii_str1 = \
"""\
2\n\
kilogram*meter \n\
---------------\n\
2 \n\
second \
"""
unicode_str1 = \
u("""\
2\n\
kilogram⋅meter \n\
───────────────\n\
2 \n\
second \
""")
ascii_str2 = \
"""\
2\n\
3*x*y*kilogram*meter \n\
---------------------\n\
2 \n\
second \
"""
unicode_str2 = \
u("""\
2\n\
3⋅x⋅y⋅kilogram⋅meter \n\
─────────────────────\n\
2 \n\
second \
""")
from sympy.physics.units import kg, m, s
assert upretty(expr) == u("joule")
assert pretty(expr) == "joule"
assert upretty(expr.convert_to(kg*m**2/s**2)) == unicode_str1
assert pretty(expr.convert_to(kg*m**2/s**2)) == ascii_str1
assert upretty(3*kg*x*m**2*y/s**2) == unicode_str2
assert pretty(3*kg*x*m**2*y/s**2) == ascii_str2
def test_pretty_Subs():
f = Function('f')
expr = Subs(f(x), x, ph**2)
ascii_str = \
"""\
(f(x))| 2\n\
|x=phi \
"""
unicode_str = \
u("""\
(f(x))│ 2\n\
│x=φ \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == unicode_str
expr = Subs(f(x).diff(x), x, 0)
ascii_str = \
"""\
/d \\| \n\
|--(f(x))|| \n\
\\dx /|x=0\
"""
unicode_str = \
u("""\
⎛d ⎞│ \n\
⎜──(f(x))⎟│ \n\
⎝dx ⎠│x=0\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == unicode_str
expr = Subs(f(x).diff(x)/y, (x, y), (0, Rational(1, 2)))
ascii_str = \
"""\
/d \\| \n\
|--(f(x))|| \n\
|dx || \n\
|--------|| \n\
\\ y /|x=0, y=1/2\
"""
unicode_str = \
u("""\
⎛d ⎞│ \n\
⎜──(f(x))⎟│ \n\
⎜dx ⎟│ \n\
⎜────────⎟│ \n\
⎝ y ⎠│x=0, y=1/2\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == unicode_str
def test_gammas():
assert upretty(lowergamma(x, y)) == u"γ(x, y)"
assert upretty(uppergamma(x, y)) == u"Γ(x, y)"
assert xpretty(gamma(x), use_unicode=True) == u'Γ(x)'
assert xpretty(gamma, use_unicode=True) == u'Γ'
assert xpretty(symbols('gamma', cls=Function)(x), use_unicode=True) == u'γ(x)'
assert xpretty(symbols('gamma', cls=Function), use_unicode=True) == u'γ'
def test_beta():
assert xpretty(beta(x,y), use_unicode=True) == u'Β(x, y)'
assert xpretty(beta(x,y), use_unicode=False) == u'B(x, y)'
assert xpretty(beta, use_unicode=True) == u'Β'
assert xpretty(beta, use_unicode=False) == u'B'
mybeta = Function('beta')
assert xpretty(mybeta(x), use_unicode=True) == u'β(x)'
assert xpretty(mybeta(x, y, z), use_unicode=False) == u'beta(x, y, z)'
assert xpretty(mybeta, use_unicode=True) == u'β'
# test that notation passes to subclasses of the same name only
def test_function_subclass_different_name():
class mygamma(gamma):
pass
assert xpretty(mygamma, use_unicode=True) == r"mygamma"
assert xpretty(mygamma(x), use_unicode=True) == r"mygamma(x)"
def test_SingularityFunction():
assert xpretty(SingularityFunction(x, 0, n), use_unicode=True) == (
"""\
n\n\
<x> \
""")
assert xpretty(SingularityFunction(x, 1, n), use_unicode=True) == (
"""\
n\n\
<x - 1> \
""")
assert xpretty(SingularityFunction(x, -1, n), use_unicode=True) == (
"""\
n\n\
<x + 1> \
""")
assert xpretty(SingularityFunction(x, a, n), use_unicode=True) == (
"""\
n\n\
<-a + x> \
""")
assert xpretty(SingularityFunction(x, y, n), use_unicode=True) == (
"""\
n\n\
<x - y> \
""")
assert xpretty(SingularityFunction(x, 0, n), use_unicode=False) == (
"""\
n\n\
<x> \
""")
assert xpretty(SingularityFunction(x, 1, n), use_unicode=False) == (
"""\
n\n\
<x - 1> \
""")
assert xpretty(SingularityFunction(x, -1, n), use_unicode=False) == (
"""\
n\n\
<x + 1> \
""")
assert xpretty(SingularityFunction(x, a, n), use_unicode=False) == (
"""\
n\n\
<-a + x> \
""")
assert xpretty(SingularityFunction(x, y, n), use_unicode=False) == (
"""\
n\n\
<x - y> \
""")
def test_deltas():
assert xpretty(DiracDelta(x), use_unicode=True) == u'δ(x)'
assert xpretty(DiracDelta(x, 1), use_unicode=True) == \
u("""\
(1) \n\
δ (x)\
""")
assert xpretty(x*DiracDelta(x, 1), use_unicode=True) == \
u("""\
(1) \n\
x⋅δ (x)\
""")
def test_hyper():
expr = hyper((), (), z)
ucode_str = \
u("""\
┌─ ⎛ │ ⎞\n\
├─ ⎜ │ z⎟\n\
0╵ 0 ⎝ │ ⎠\
""")
ascii_str = \
"""\
_ \n\
|_ / | \\\n\
| | | z|\n\
0 0 \\ | /\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = hyper((), (1,), x)
ucode_str = \
u("""\
┌─ ⎛ │ ⎞\n\
├─ ⎜ │ x⎟\n\
0╵ 1 ⎝1 │ ⎠\
""")
ascii_str = \
"""\
_ \n\
|_ / | \\\n\
| | | x|\n\
0 1 \\1 | /\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = hyper([2], [1], x)
ucode_str = \
u("""\
┌─ ⎛2 │ ⎞\n\
├─ ⎜ │ x⎟\n\
1╵ 1 ⎝1 │ ⎠\
""")
ascii_str = \
"""\
_ \n\
|_ /2 | \\\n\
| | | x|\n\
1 1 \\1 | /\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = hyper((pi/3, -2*k), (3, 4, 5, -3), x)
ucode_str = \
u("""\
⎛ π │ ⎞\n\
┌─ ⎜ ─, -2⋅k │ ⎟\n\
├─ ⎜ 3 │ x⎟\n\
2╵ 4 ⎜ │ ⎟\n\
⎝3, 4, 5, -3 │ ⎠\
""")
ascii_str = \
"""\
\n\
_ / pi | \\\n\
|_ | --, -2*k | |\n\
| | 3 | x|\n\
2 4 | | |\n\
\\3, 4, 5, -3 | /\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = hyper((pi, S('2/3'), -2*k), (3, 4, 5, -3), x**2)
ucode_str = \
u("""\
┌─ ⎛π, 2/3, -2⋅k │ 2⎞\n\
├─ ⎜ │ x ⎟\n\
3╵ 4 ⎝3, 4, 5, -3 │ ⎠\
""")
ascii_str = \
"""\
_ \n\
|_ /pi, 2/3, -2*k | 2\\\n\
| | | x |\n\
3 4 \\ 3, 4, 5, -3 | /\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = hyper([1, 2], [3, 4], 1/(1/(1/(1/x + 1) + 1) + 1))
ucode_str = \
u("""\
⎛ │ 1 ⎞\n\
⎜ │ ─────────────⎟\n\
⎜ │ 1 ⎟\n\
┌─ ⎜1, 2 │ 1 + ─────────⎟\n\
├─ ⎜ │ 1 ⎟\n\
2╵ 2 ⎜3, 4 │ 1 + ─────⎟\n\
⎜ │ 1⎟\n\
⎜ │ 1 + ─⎟\n\
⎝ │ x⎠\
""")
ascii_str = \
"""\
\n\
/ | 1 \\\n\
| | -------------|\n\
_ | | 1 |\n\
|_ |1, 2 | 1 + ---------|\n\
| | | 1 |\n\
2 2 |3, 4 | 1 + -----|\n\
| | 1|\n\
| | 1 + -|\n\
\\ | x/\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_meijerg():
expr = meijerg([pi, pi, x], [1], [0, 1], [1, 2, 3], z)
ucode_str = \
u("""\
╭─╮2, 3 ⎛π, π, x 1 │ ⎞\n\
│╶┐ ⎜ │ z⎟\n\
╰─╯4, 5 ⎝ 0, 1 1, 2, 3 │ ⎠\
""")
ascii_str = \
"""\
__2, 3 /pi, pi, x 1 | \\\n\
/__ | | z|\n\
\\_|4, 5 \\ 0, 1 1, 2, 3 | /\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = meijerg([1, pi/7], [2, pi, 5], [], [], z**2)
ucode_str = \
u("""\
⎛ π │ ⎞\n\
╭─╮0, 2 ⎜1, ─ 2, π, 5 │ 2⎟\n\
│╶┐ ⎜ 7 │ z ⎟\n\
╰─╯5, 0 ⎜ │ ⎟\n\
⎝ │ ⎠\
""")
ascii_str = \
"""\
/ pi | \\\n\
__0, 2 |1, -- 2, pi, 5 | 2|\n\
/__ | 7 | z |\n\
\\_|5, 0 | | |\n\
\\ | /\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
ucode_str = \
u("""\
╭─╮ 1, 10 ⎛1, 1, 1, 1, 1, 1, 1, 1, 1, 1 1 │ ⎞\n\
│╶┐ ⎜ │ z⎟\n\
╰─╯11, 2 ⎝ 1 1 │ ⎠\
""")
ascii_str = \
"""\
__ 1, 10 /1, 1, 1, 1, 1, 1, 1, 1, 1, 1 1 | \\\n\
/__ | | z|\n\
\\_|11, 2 \\ 1 1 | /\
"""
expr = meijerg([1]*10, [1], [1], [1], z)
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = meijerg([1, 2, ], [4, 3], [3], [4, 5], 1/(1/(1/(1/x + 1) + 1) + 1))
ucode_str = \
u("""\
⎛ │ 1 ⎞\n\
⎜ │ ─────────────⎟\n\
⎜ │ 1 ⎟\n\
╭─╮1, 2 ⎜1, 2 4, 3 │ 1 + ─────────⎟\n\
│╶┐ ⎜ │ 1 ⎟\n\
╰─╯4, 3 ⎜ 3 4, 5 │ 1 + ─────⎟\n\
⎜ │ 1⎟\n\
⎜ │ 1 + ─⎟\n\
⎝ │ x⎠\
""")
ascii_str = \
"""\
/ | 1 \\\n\
| | -------------|\n\
| | 1 |\n\
__1, 2 |1, 2 4, 3 | 1 + ---------|\n\
/__ | | 1 |\n\
\\_|4, 3 | 3 4, 5 | 1 + -----|\n\
| | 1|\n\
| | 1 + -|\n\
\\ | x/\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Integral(expr, x)
ucode_str = \
u("""\
⌠ \n\
⎮ ⎛ │ 1 ⎞ \n\
⎮ ⎜ │ ─────────────⎟ \n\
⎮ ⎜ │ 1 ⎟ \n\
⎮ ╭─╮1, 2 ⎜1, 2 4, 3 │ 1 + ─────────⎟ \n\
⎮ │╶┐ ⎜ │ 1 ⎟ dx\n\
⎮ ╰─╯4, 3 ⎜ 3 4, 5 │ 1 + ─────⎟ \n\
⎮ ⎜ │ 1⎟ \n\
⎮ ⎜ │ 1 + ─⎟ \n\
⎮ ⎝ │ x⎠ \n\
⌡ \
""")
ascii_str = \
"""\
/ \n\
| \n\
| / | 1 \\ \n\
| | | -------------| \n\
| | | 1 | \n\
| __1, 2 |1, 2 4, 3 | 1 + ---------| \n\
| /__ | | 1 | dx\n\
| \\_|4, 3 | 3 4, 5 | 1 + -----| \n\
| | | 1| \n\
| | | 1 + -| \n\
| \\ | x/ \n\
| \n\
/ \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_noncommutative():
A, B, C = symbols('A,B,C', commutative=False)
expr = A*B*C**-1
ascii_str = \
"""\
-1\n\
A*B*C \
"""
ucode_str = \
u("""\
-1\n\
A⋅B⋅C \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = C**-1*A*B
ascii_str = \
"""\
-1 \n\
C *A*B\
"""
ucode_str = \
u("""\
-1 \n\
C ⋅A⋅B\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = A*C**-1*B
ascii_str = \
"""\
-1 \n\
A*C *B\
"""
ucode_str = \
u("""\
-1 \n\
A⋅C ⋅B\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = A*C**-1*B/x
ascii_str = \
"""\
-1 \n\
A*C *B\n\
-------\n\
x \
"""
ucode_str = \
u("""\
-1 \n\
A⋅C ⋅B\n\
───────\n\
x \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_pretty_special_functions():
x, y = symbols("x y")
# atan2
expr = atan2(y/sqrt(200), sqrt(x))
ascii_str = \
"""\
/ ___ \\\n\
|\\/ 2 *y ___|\n\
atan2|-------, \\/ x |\n\
\\ 20 /\
"""
ucode_str = \
u("""\
⎛√2⋅y ⎞\n\
atan2⎜────, √x⎟\n\
⎝ 20 ⎠\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_pretty_geometry():
e = Segment((0, 1), (0, 2))
assert pretty(e) == 'Segment2D(Point2D(0, 1), Point2D(0, 2))'
e = Ray((1, 1), angle=4.02*pi)
assert pretty(e) == 'Ray2D(Point2D(1, 1), Point2D(2, tan(pi/50) + 1))'
def test_expint():
expr = Ei(x)
string = 'Ei(x)'
assert pretty(expr) == string
assert upretty(expr) == string
expr = expint(1, z)
ucode_str = u"E₁(z)"
ascii_str = "expint(1, z)"
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
assert pretty(Shi(x)) == 'Shi(x)'
assert pretty(Si(x)) == 'Si(x)'
assert pretty(Ci(x)) == 'Ci(x)'
assert pretty(Chi(x)) == 'Chi(x)'
assert upretty(Shi(x)) == 'Shi(x)'
assert upretty(Si(x)) == 'Si(x)'
assert upretty(Ci(x)) == 'Ci(x)'
assert upretty(Chi(x)) == 'Chi(x)'
def test_elliptic_functions():
ascii_str = \
"""\
/ 1 \\\n\
K|-----|\n\
\\z + 1/\
"""
ucode_str = \
u("""\
⎛ 1 ⎞\n\
K⎜─────⎟\n\
⎝z + 1⎠\
""")
expr = elliptic_k(1/(z + 1))
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
ascii_str = \
"""\
/ | 1 \\\n\
F|1|-----|\n\
\\ |z + 1/\
"""
ucode_str = \
u("""\
⎛ │ 1 ⎞\n\
F⎜1│─────⎟\n\
⎝ │z + 1⎠\
""")
expr = elliptic_f(1, 1/(1 + z))
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
ascii_str = \
"""\
/ 1 \\\n\
E|-----|\n\
\\z + 1/\
"""
ucode_str = \
u("""\
⎛ 1 ⎞\n\
E⎜─────⎟\n\
⎝z + 1⎠\
""")
expr = elliptic_e(1/(z + 1))
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
ascii_str = \
"""\
/ | 1 \\\n\
E|1|-----|\n\
\\ |z + 1/\
"""
ucode_str = \
u("""\
⎛ │ 1 ⎞\n\
E⎜1│─────⎟\n\
⎝ │z + 1⎠\
""")
expr = elliptic_e(1, 1/(1 + z))
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
ascii_str = \
"""\
/ |4\\\n\
Pi|3|-|\n\
\\ |x/\
"""
ucode_str = \
u("""\
⎛ │4⎞\n\
Π⎜3│─⎟\n\
⎝ │x⎠\
""")
expr = elliptic_pi(3, 4/x)
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
ascii_str = \
"""\
/ 4| \\\n\
Pi|3; -|6|\n\
\\ x| /\
"""
ucode_str = \
u("""\
⎛ 4│ ⎞\n\
Π⎜3; ─│6⎟\n\
⎝ x│ ⎠\
""")
expr = elliptic_pi(3, 4/x, 6)
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_RandomDomain():
from sympy.stats import Normal, Die, Exponential, pspace, where
X = Normal('x1', 0, 1)
assert upretty(where(X > 0)) == u"Domain: 0 < x₁ ∧ x₁ < ∞"
D = Die('d1', 6)
assert upretty(where(D > 4)) == u'Domain: d₁ = 5 ∨ d₁ = 6'
A = Exponential('a', 1)
B = Exponential('b', 1)
assert upretty(pspace(Tuple(A, B)).domain) == \
u'Domain: 0 ≤ a ∧ 0 ≤ b ∧ a < ∞ ∧ b < ∞'
def test_PrettyPoly():
F = QQ.frac_field(x, y)
R = QQ.poly_ring(x, y)
expr = F.convert(x/(x + y))
assert pretty(expr) == "x/(x + y)"
assert upretty(expr) == u"x/(x + y)"
expr = R.convert(x + y)
assert pretty(expr) == "x + y"
assert upretty(expr) == u"x + y"
def test_issue_6285():
assert pretty(Pow(2, -5, evaluate=False)) == '1 \n--\n 5\n2 '
assert pretty(Pow(x, (1/pi))) == 'pi___\n\\/ x '
def test_issue_6359():
assert pretty(Integral(x**2, x)**2) == \
"""\
2
/ / \\ \n\
| | | \n\
| | 2 | \n\
| | x dx| \n\
| | | \n\
\\/ / \
"""
assert upretty(Integral(x**2, x)**2) == \
u("""\
2
⎛⌠ ⎞ \n\
⎜⎮ 2 ⎟ \n\
⎜⎮ x dx⎟ \n\
⎝⌡ ⎠ \
""")
assert pretty(Sum(x**2, (x, 0, 1))**2) == \
"""\
2
/ 1 \\ \n\
| ___ | \n\
| \\ ` | \n\
| \\ 2| \n\
| / x | \n\
| /__, | \n\
\\x = 0 / \
"""
assert upretty(Sum(x**2, (x, 0, 1))**2) == \
u("""\
2
⎛ 1 ⎞ \n\
⎜ ___ ⎟ \n\
⎜ ╲ ⎟ \n\
⎜ ╲ 2⎟ \n\
⎜ ╱ x ⎟ \n\
⎜ ╱ ⎟ \n\
⎜ ‾‾‾ ⎟ \n\
⎝x = 0 ⎠ \
""")
assert pretty(Product(x**2, (x, 1, 2))**2) == \
"""\
2
/ 2 \\ \n\
|______ | \n\
| | | 2| \n\
| | | x | \n\
| | | | \n\
\\x = 1 / \
"""
assert upretty(Product(x**2, (x, 1, 2))**2) == \
u("""\
2
⎛ 2 ⎞ \n\
⎜─┬──┬─ ⎟ \n\
⎜ │ │ 2⎟ \n\
⎜ │ │ x ⎟ \n\
⎜ │ │ ⎟ \n\
⎝x = 1 ⎠ \
""")
f = Function('f')
assert pretty(Derivative(f(x), x)**2) == \
"""\
2
/d \\ \n\
|--(f(x))| \n\
\\dx / \
"""
assert upretty(Derivative(f(x), x)**2) == \
u("""\
2
⎛d ⎞ \n\
⎜──(f(x))⎟ \n\
⎝dx ⎠ \
""")
def test_issue_6739():
ascii_str = \
"""\
1 \n\
-----\n\
___\n\
\\/ x \
"""
ucode_str = \
u("""\
1 \n\
──\n\
√x\
""")
assert pretty(1/sqrt(x)) == ascii_str
assert upretty(1/sqrt(x)) == ucode_str
def test_complicated_symbol_unchanged():
for symb_name in ["dexpr2_d1tau", "dexpr2^d1tau"]:
assert pretty(Symbol(symb_name)) == symb_name
def test_categories():
from sympy.categories import (Object, IdentityMorphism,
NamedMorphism, Category, Diagram, DiagramGrid)
A1 = Object("A1")
A2 = Object("A2")
A3 = Object("A3")
f1 = NamedMorphism(A1, A2, "f1")
f2 = NamedMorphism(A2, A3, "f2")
id_A1 = IdentityMorphism(A1)
K1 = Category("K1")
assert pretty(A1) == "A1"
assert upretty(A1) == u"A₁"
assert pretty(f1) == "f1:A1-->A2"
assert upretty(f1) == u"f₁:A₁——▶A₂"
assert pretty(id_A1) == "id:A1-->A1"
assert upretty(id_A1) == u"id:A₁——▶A₁"
assert pretty(f2*f1) == "f2*f1:A1-->A3"
assert upretty(f2*f1) == u"f₂∘f₁:A₁——▶A₃"
assert pretty(K1) == "K1"
assert upretty(K1) == u"K₁"
# Test how diagrams are printed.
d = Diagram()
assert pretty(d) == "EmptySet"
assert upretty(d) == u"∅"
d = Diagram({f1: "unique", f2: S.EmptySet})
assert pretty(d) == "{f2*f1:A1-->A3: EmptySet, id:A1-->A1: " \
"EmptySet, id:A2-->A2: EmptySet, id:A3-->A3: " \
"EmptySet, f1:A1-->A2: {unique}, f2:A2-->A3: EmptySet}"
assert upretty(d) == u("{f₂∘f₁:A₁——▶A₃: ∅, id:A₁——▶A₁: ∅, " \
"id:A₂——▶A₂: ∅, id:A₃——▶A₃: ∅, f₁:A₁——▶A₂: {unique}, f₂:A₂——▶A₃: ∅}")
d = Diagram({f1: "unique", f2: S.EmptySet}, {f2 * f1: "unique"})
assert pretty(d) == "{f2*f1:A1-->A3: EmptySet, id:A1-->A1: " \
"EmptySet, id:A2-->A2: EmptySet, id:A3-->A3: " \
"EmptySet, f1:A1-->A2: {unique}, f2:A2-->A3: EmptySet}" \
" ==> {f2*f1:A1-->A3: {unique}}"
assert upretty(d) == u("{f₂∘f₁:A₁——▶A₃: ∅, id:A₁——▶A₁: ∅, id:A₂——▶A₂: " \
"∅, id:A₃——▶A₃: ∅, f₁:A₁——▶A₂: {unique}, f₂:A₂——▶A₃: ∅}" \
" ══▶ {f₂∘f₁:A₁——▶A₃: {unique}}")
grid = DiagramGrid(d)
assert pretty(grid) == "A1 A2\n \nA3 "
assert upretty(grid) == u"A₁ A₂\n \nA₃ "
def test_PrettyModules():
R = QQ.old_poly_ring(x, y)
F = R.free_module(2)
M = F.submodule([x, y], [1, x**2])
ucode_str = \
u("""\
2\n\
ℚ[x, y] \
""")
ascii_str = \
"""\
2\n\
QQ[x, y] \
"""
assert upretty(F) == ucode_str
assert pretty(F) == ascii_str
ucode_str = \
u("""\
╱ ⎡ 2⎤╲\n\
╲[x, y], ⎣1, x ⎦╱\
""")
ascii_str = \
"""\
2 \n\
<[x, y], [1, x ]>\
"""
assert upretty(M) == ucode_str
assert pretty(M) == ascii_str
I = R.ideal(x**2, y)
ucode_str = \
u("""\
╱ 2 ╲\n\
╲x , y╱\
""")
ascii_str = \
"""\
2 \n\
<x , y>\
"""
assert upretty(I) == ucode_str
assert pretty(I) == ascii_str
Q = F / M
ucode_str = \
u("""\
2 \n\
ℚ[x, y] \n\
─────────────────\n\
╱ ⎡ 2⎤╲\n\
╲[x, y], ⎣1, x ⎦╱\
""")
ascii_str = \
"""\
2 \n\
QQ[x, y] \n\
-----------------\n\
2 \n\
<[x, y], [1, x ]>\
"""
assert upretty(Q) == ucode_str
assert pretty(Q) == ascii_str
ucode_str = \
u("""\
╱⎡ 3⎤ ╲\n\
│⎢ x ⎥ ╱ ⎡ 2⎤╲ ╱ ⎡ 2⎤╲│\n\
│⎢1, ──⎥ + ╲[x, y], ⎣1, x ⎦╱, [2, y] + ╲[x, y], ⎣1, x ⎦╱│\n\
╲⎣ 2 ⎦ ╱\
""")
ascii_str = \
"""\
3 \n\
x 2 2 \n\
<[1, --] + <[x, y], [1, x ]>, [2, y] + <[x, y], [1, x ]>>\n\
2 \
"""
def test_QuotientRing():
R = QQ.old_poly_ring(x)/[x**2 + 1]
ucode_str = \
u("""\
ℚ[x] \n\
────────\n\
╱ 2 ╲\n\
╲x + 1╱\
""")
ascii_str = \
"""\
QQ[x] \n\
--------\n\
2 \n\
<x + 1>\
"""
assert upretty(R) == ucode_str
assert pretty(R) == ascii_str
ucode_str = \
u("""\
╱ 2 ╲\n\
1 + ╲x + 1╱\
""")
ascii_str = \
"""\
2 \n\
1 + <x + 1>\
"""
assert upretty(R.one) == ucode_str
assert pretty(R.one) == ascii_str
def test_Homomorphism():
from sympy.polys.agca import homomorphism
R = QQ.old_poly_ring(x)
expr = homomorphism(R.free_module(1), R.free_module(1), [0])
ucode_str = \
u("""\
1 1\n\
[0] : ℚ[x] ──> ℚ[x] \
""")
ascii_str = \
"""\
1 1\n\
[0] : QQ[x] --> QQ[x] \
"""
assert upretty(expr) == ucode_str
assert pretty(expr) == ascii_str
expr = homomorphism(R.free_module(2), R.free_module(2), [0, 0])
ucode_str = \
u("""\
⎡0 0⎤ 2 2\n\
⎢ ⎥ : ℚ[x] ──> ℚ[x] \n\
⎣0 0⎦ \
""")
ascii_str = \
"""\
[0 0] 2 2\n\
[ ] : QQ[x] --> QQ[x] \n\
[0 0] \
"""
assert upretty(expr) == ucode_str
assert pretty(expr) == ascii_str
expr = homomorphism(R.free_module(1), R.free_module(1) / [[x]], [0])
ucode_str = \
u("""\
1\n\
1 ℚ[x] \n\
[0] : ℚ[x] ──> ─────\n\
<[x]>\
""")
ascii_str = \
"""\
1\n\
1 QQ[x] \n\
[0] : QQ[x] --> ------\n\
<[x]> \
"""
assert upretty(expr) == ucode_str
assert pretty(expr) == ascii_str
def test_Tr():
A, B = symbols('A B', commutative=False)
t = Tr(A*B)
assert pretty(t) == r'Tr(A*B)'
assert upretty(t) == u'Tr(A⋅B)'
def test_pretty_Add():
eq = Mul(-2, x - 2, evaluate=False) + 5
assert pretty(eq) == '5 - 2*(x - 2)'
def test_issue_7179():
assert upretty(Not(Equivalent(x, y))) == u'x ⇎ y'
assert upretty(Not(Implies(x, y))) == u'x ↛ y'
def test_issue_7180():
assert upretty(Equivalent(x, y)) == u'x ⇔ y'
def test_pretty_Complement():
assert pretty(S.Reals - S.Naturals) == '(-oo, oo) \\ Naturals'
assert upretty(S.Reals - S.Naturals) == u'ℝ \\ ℕ'
assert pretty(S.Reals - S.Naturals0) == '(-oo, oo) \\ Naturals0'
assert upretty(S.Reals - S.Naturals0) == u'ℝ \\ ℕ₀'
def test_pretty_SymmetricDifference():
from sympy import SymmetricDifference, Interval
from sympy.testing.pytest import raises
assert upretty(SymmetricDifference(Interval(2,3), Interval(3,5), \
evaluate = False)) == u'[2, 3] ∆ [3, 5]'
with raises(NotImplementedError):
pretty(SymmetricDifference(Interval(2,3), Interval(3,5), evaluate = False))
def test_pretty_Contains():
assert pretty(Contains(x, S.Integers)) == 'Contains(x, Integers)'
assert upretty(Contains(x, S.Integers)) == u'x ∈ ℤ'
def test_issue_8292():
from sympy.core import sympify
e = sympify('((x+x**4)/(x-1))-(2*(x-1)**4/(x-1)**4)', evaluate=False)
ucode_str = \
u("""\
4 4 \n\
2⋅(x - 1) x + x\n\
- ────────── + ──────\n\
4 x - 1 \n\
(x - 1) \
""")
ascii_str = \
"""\
4 4 \n\
2*(x - 1) x + x\n\
- ---------- + ------\n\
4 x - 1 \n\
(x - 1) \
"""
assert pretty(e) == ascii_str
assert upretty(e) == ucode_str
def test_issue_4335():
y = Function('y')
expr = -y(x).diff(x)
ucode_str = \
u("""\
d \n\
-──(y(x))\n\
dx \
""")
ascii_str = \
"""\
d \n\
- --(y(x))\n\
dx \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_issue_8344():
from sympy.core import sympify
e = sympify('2*x*y**2/1**2 + 1', evaluate=False)
ucode_str = \
u("""\
2 \n\
2⋅x⋅y \n\
────── + 1\n\
2 \n\
1 \
""")
assert upretty(e) == ucode_str
def test_issue_6324():
x = Pow(2, 3, evaluate=False)
y = Pow(10, -2, evaluate=False)
e = Mul(x, y, evaluate=False)
ucode_str = \
u("""\
3\n\
2 \n\
───\n\
2\n\
10 \
""")
assert upretty(e) == ucode_str
def test_issue_7927():
e = sin(x/2)**cos(x/2)
ucode_str = \
u("""\
⎛x⎞\n\
cos⎜─⎟\n\
⎝2⎠\n\
⎛ ⎛x⎞⎞ \n\
⎜sin⎜─⎟⎟ \n\
⎝ ⎝2⎠⎠ \
""")
assert upretty(e) == ucode_str
e = sin(x)**(S(11)/13)
ucode_str = \
u("""\
11\n\
──\n\
13\n\
(sin(x)) \
""")
assert upretty(e) == ucode_str
def test_issue_6134():
from sympy.abc import lamda, t
phi = Function('phi')
e = lamda*x*Integral(phi(t)*pi*sin(pi*t), (t, 0, 1)) + lamda*x**2*Integral(phi(t)*2*pi*sin(2*pi*t), (t, 0, 1))
ucode_str = \
u("""\
1 1 \n\
2 ⌠ ⌠ \n\
λ⋅x ⋅⎮ 2⋅π⋅φ(t)⋅sin(2⋅π⋅t) dt + λ⋅x⋅⎮ π⋅φ(t)⋅sin(π⋅t) dt\n\
⌡ ⌡ \n\
0 0 \
""")
assert upretty(e) == ucode_str
def test_issue_9877():
ucode_str1 = u'(2, 3) ∪ ([1, 2] \\ {x})'
a, b, c = Interval(2, 3, True, True), Interval(1, 2), FiniteSet(x)
assert upretty(Union(a, Complement(b, c))) == ucode_str1
ucode_str2 = u'{x} ∩ {y} ∩ ({z} \\ [1, 2])'
d, e, f, g = FiniteSet(x), FiniteSet(y), FiniteSet(z), Interval(1, 2)
assert upretty(Intersection(d, e, Complement(f, g))) == ucode_str2
def test_issue_13651():
expr1 = c + Mul(-1, a + b, evaluate=False)
assert pretty(expr1) == 'c - (a + b)'
expr2 = c + Mul(-1, a - b + d, evaluate=False)
assert pretty(expr2) == 'c - (a - b + d)'
def test_pretty_primenu():
from sympy.ntheory.factor_ import primenu
ascii_str1 = "nu(n)"
ucode_str1 = u("ν(n)")
n = symbols('n', integer=True)
assert pretty(primenu(n)) == ascii_str1
assert upretty(primenu(n)) == ucode_str1
def test_pretty_primeomega():
from sympy.ntheory.factor_ import primeomega
ascii_str1 = "Omega(n)"
ucode_str1 = u("Ω(n)")
n = symbols('n', integer=True)
assert pretty(primeomega(n)) == ascii_str1
assert upretty(primeomega(n)) == ucode_str1
def test_pretty_Mod():
from sympy.core import Mod
ascii_str1 = "x mod 7"
ucode_str1 = u("x mod 7")
ascii_str2 = "(x + 1) mod 7"
ucode_str2 = u("(x + 1) mod 7")
ascii_str3 = "2*x mod 7"
ucode_str3 = u("2⋅x mod 7")
ascii_str4 = "(x mod 7) + 1"
ucode_str4 = u("(x mod 7) + 1")
ascii_str5 = "2*(x mod 7)"
ucode_str5 = u("2⋅(x mod 7)")
x = symbols('x', integer=True)
assert pretty(Mod(x, 7)) == ascii_str1
assert upretty(Mod(x, 7)) == ucode_str1
assert pretty(Mod(x + 1, 7)) == ascii_str2
assert upretty(Mod(x + 1, 7)) == ucode_str2
assert pretty(Mod(2 * x, 7)) == ascii_str3
assert upretty(Mod(2 * x, 7)) == ucode_str3
assert pretty(Mod(x, 7) + 1) == ascii_str4
assert upretty(Mod(x, 7) + 1) == ucode_str4
assert pretty(2 * Mod(x, 7)) == ascii_str5
assert upretty(2 * Mod(x, 7)) == ucode_str5
def test_issue_11801():
assert pretty(Symbol("")) == ""
assert upretty(Symbol("")) == ""
def test_pretty_UnevaluatedExpr():
x = symbols('x')
he = UnevaluatedExpr(1/x)
ucode_str = \
u("""\
1\n\
─\n\
x\
""")
assert upretty(he) == ucode_str
ucode_str = \
u("""\
2\n\
⎛1⎞ \n\
⎜─⎟ \n\
⎝x⎠ \
""")
assert upretty(he**2) == ucode_str
ucode_str = \
u("""\
1\n\
1 + ─\n\
x\
""")
assert upretty(he + 1) == ucode_str
ucode_str = \
u('''\
1\n\
x⋅─\n\
x\
''')
assert upretty(x*he) == ucode_str
def test_issue_10472():
M = (Matrix([[0, 0], [0, 0]]), Matrix([0, 0]))
ucode_str = \
u("""\
⎛⎡0 0⎤ ⎡0⎤⎞
⎜⎢ ⎥, ⎢ ⎥⎟
⎝⎣0 0⎦ ⎣0⎦⎠\
""")
assert upretty(M) == ucode_str
def test_MatrixElement_printing():
# test cases for issue #11821
A = MatrixSymbol("A", 1, 3)
B = MatrixSymbol("B", 1, 3)
C = MatrixSymbol("C", 1, 3)
ascii_str1 = "A_00"
ucode_str1 = u("A₀₀")
assert pretty(A[0, 0]) == ascii_str1
assert upretty(A[0, 0]) == ucode_str1
ascii_str1 = "3*A_00"
ucode_str1 = u("3⋅A₀₀")
assert pretty(3*A[0, 0]) == ascii_str1
assert upretty(3*A[0, 0]) == ucode_str1
ascii_str1 = "(-B + A)[0, 0]"
ucode_str1 = u("(-B + A)[0, 0]")
F = C[0, 0].subs(C, A - B)
assert pretty(F) == ascii_str1
assert upretty(F) == ucode_str1
def test_issue_12675():
from sympy.vector import CoordSys3D
x, y, t, j = symbols('x y t j')
e = CoordSys3D('e')
ucode_str = \
u("""\
⎛ t⎞ \n\
⎜⎛x⎞ ⎟ j_e\n\
⎜⎜─⎟ ⎟ \n\
⎝⎝y⎠ ⎠ \
""")
assert upretty((x/y)**t*e.j) == ucode_str
ucode_str = \
u("""\
⎛1⎞ \n\
⎜─⎟ j_e\n\
⎝y⎠ \
""")
assert upretty((1/y)*e.j) == ucode_str
def test_MatrixSymbol_printing():
# test cases for issue #14237
A = MatrixSymbol("A", 3, 3)
B = MatrixSymbol("B", 3, 3)
C = MatrixSymbol("C", 3, 3)
assert pretty(-A*B*C) == "-A*B*C"
assert pretty(A - B) == "-B + A"
assert pretty(A*B*C - A*B - B*C) == "-A*B -B*C + A*B*C"
# issue #14814
x = MatrixSymbol('x', n, n)
y = MatrixSymbol('y*', n, n)
assert pretty(x + y) == "x + y*"
ascii_str = \
"""\
2 \n\
-2*y* -a*x\
"""
assert pretty(-a*x + -2*y*y) == ascii_str
def test_degree_printing():
expr1 = 90*degree
assert pretty(expr1) == u'90°'
expr2 = x*degree
assert pretty(expr2) == u'x°'
expr3 = cos(x*degree + 90*degree)
assert pretty(expr3) == u'cos(x° + 90°)'
def test_vector_expr_pretty_printing():
A = CoordSys3D('A')
assert upretty(Cross(A.i, A.x*A.i+3*A.y*A.j)) == u("(i_A)×((x_A) i_A + (3⋅y_A) j_A)")
assert upretty(x*Cross(A.i, A.j)) == u('x⋅(i_A)×(j_A)')
assert upretty(Curl(A.x*A.i + 3*A.y*A.j)) == u("∇×((x_A) i_A + (3⋅y_A) j_A)")
assert upretty(Divergence(A.x*A.i + 3*A.y*A.j)) == u("∇⋅((x_A) i_A + (3⋅y_A) j_A)")
assert upretty(Dot(A.i, A.x*A.i+3*A.y*A.j)) == u("(i_A)⋅((x_A) i_A + (3⋅y_A) j_A)")
assert upretty(Gradient(A.x+3*A.y)) == u("∇(x_A + 3⋅y_A)")
assert upretty(Laplacian(A.x+3*A.y)) == u("∆(x_A + 3⋅y_A)")
# TODO: add support for ASCII pretty.
def test_pretty_print_tensor_expr():
L = TensorIndexType("L")
i, j, k = tensor_indices("i j k", L)
i0 = tensor_indices("i_0", L)
A, B, C, D = tensor_heads("A B C D", [L])
H = TensorHead("H", [L, L])
expr = -i
ascii_str = \
"""\
-i\
"""
ucode_str = \
u("""\
-i\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = A(i)
ascii_str = \
"""\
i\n\
A \n\
\
"""
ucode_str = \
u("""\
i\n\
A \n\
\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = A(i0)
ascii_str = \
"""\
i_0\n\
A \n\
\
"""
ucode_str = \
u("""\
i₀\n\
A \n\
\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = A(-i)
ascii_str = \
"""\
\n\
A \n\
i\
"""
ucode_str = \
u("""\
\n\
A \n\
i\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = -3*A(-i)
ascii_str = \
"""\
\n\
-3*A \n\
i\
"""
ucode_str = \
u("""\
\n\
-3⋅A \n\
i\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = H(i, -j)
ascii_str = \
"""\
i \n\
H \n\
j\
"""
ucode_str = \
u("""\
i \n\
H \n\
j\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = H(i, -i)
ascii_str = \
"""\
L_0 \n\
H \n\
L_0\
"""
ucode_str = \
u("""\
L₀ \n\
H \n\
L₀\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = H(i, -j)*A(j)*B(k)
ascii_str = \
"""\
i L_0 k\n\
H *A *B \n\
L_0 \
"""
ucode_str = \
u("""\
i L₀ k\n\
H ⋅A ⋅B \n\
L₀ \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = (1+x)*A(i)
ascii_str = \
"""\
i\n\
(x + 1)*A \n\
\
"""
ucode_str = \
u("""\
i\n\
(x + 1)⋅A \n\
\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = A(i) + 3*B(i)
ascii_str = \
"""\
i i\n\
3*B + A \n\
\
"""
ucode_str = \
u("""\
i i\n\
3⋅B + A \n\
\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_pretty_print_tensor_partial_deriv():
from sympy.tensor.toperators import PartialDerivative
from sympy.tensor.tensor import TensorIndexType, tensor_indices, TensorHead, tensor_heads
L = TensorIndexType("L")
i, j, k = tensor_indices("i j k", L)
A, B, C, D = tensor_heads("A B C D", [L])
H = TensorHead("H", [L, L])
expr = PartialDerivative(A(i), A(j))
ascii_str = \
"""\
d / i\\\n\
---|A |\n\
j\\ /\n\
dA \n\
\
"""
ucode_str = \
u("""\
∂ ⎛ i⎞\n\
───⎜A ⎟\n\
j⎝ ⎠\n\
∂A \n\
\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = A(i)*PartialDerivative(H(k, -i), A(j))
ascii_str = \
"""\
L_0 d / k \\\n\
A *---|H |\n\
j\\ L_0/\n\
dA \n\
\
"""
ucode_str = \
u("""\
L₀ ∂ ⎛ k ⎞\n\
A ⋅───⎜H ⎟\n\
j⎝ L₀⎠\n\
∂A \n\
\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = A(i)*PartialDerivative(B(k)*C(-i) + 3*H(k, -i), A(j))
ascii_str = \
"""\
L_0 d / k k \\\n\
A *---|3*H + B *C |\n\
j\\ L_0 L_0/\n\
dA \n\
\
"""
ucode_str = \
u("""\
L₀ ∂ ⎛ k k ⎞\n\
A ⋅───⎜3⋅H + B ⋅C ⎟\n\
j⎝ L₀ L₀⎠\n\
∂A \n\
\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = (A(i) + B(i))*PartialDerivative(C(j), D(j))
ascii_str = \
"""\
/ i i\\ d / L_0\\\n\
|A + B |*-----|C |\n\
\\ / L_0\\ /\n\
dD \n\
\
"""
ucode_str = \
u("""\
⎛ i i⎞ ∂ ⎛ L₀⎞\n\
⎜A + B ⎟⋅────⎜C ⎟\n\
⎝ ⎠ L₀⎝ ⎠\n\
∂D \n\
\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = (A(i) + B(i))*PartialDerivative(C(-i), D(j))
ascii_str = \
"""\
/ L_0 L_0\\ d / \\\n\
|A + B |*---|C |\n\
\\ / j\\ L_0/\n\
dD \n\
\
"""
ucode_str = \
u("""\
⎛ L₀ L₀⎞ ∂ ⎛ ⎞\n\
⎜A + B ⎟⋅───⎜C ⎟\n\
⎝ ⎠ j⎝ L₀⎠\n\
∂D \n\
\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = PartialDerivative(B(-i) + A(-i), A(-j), A(-n))
ucode_str = u("""\
2 \n\
∂ ⎛ ⎞\n\
───────⎜A + B ⎟\n\
⎝ i i⎠\n\
∂A ∂A \n\
n j \
""")
assert upretty(expr) == ucode_str
expr = PartialDerivative(3*A(-i), A(-j), A(-n))
ucode_str = u("""\
2 \n\
∂ ⎛ ⎞\n\
───────⎜3⋅A ⎟\n\
⎝ i⎠\n\
∂A ∂A \n\
n j \
""")
assert upretty(expr) == ucode_str
expr = TensorElement(H(i, j), {i:1})
ascii_str = \
"""\
i=1,j\n\
H \n\
\
"""
ucode_str = ascii_str
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = TensorElement(H(i, j), {i: 1, j: 1})
ascii_str = \
"""\
i=1,j=1\n\
H \n\
\
"""
ucode_str = ascii_str
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = TensorElement(H(i, j), {j: 1})
ascii_str = \
"""\
i,j=1\n\
H \n\
\
"""
ucode_str = ascii_str
expr = TensorElement(H(-i, j), {-i: 1})
ascii_str = \
"""\
j\n\
H \n\
i=1 \
"""
ucode_str = ascii_str
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_issue_15560():
a = MatrixSymbol('a', 1, 1)
e = pretty(a*(KroneckerProduct(a, a)))
result = 'a*(a x a)'
assert e == result
def test_print_lerchphi():
# Part of issue 6013
a = Symbol('a')
pretty(lerchphi(a, 1, 2))
uresult = u'Φ(a, 1, 2)'
aresult = 'lerchphi(a, 1, 2)'
assert pretty(lerchphi(a, 1, 2)) == aresult
assert upretty(lerchphi(a, 1, 2)) == uresult
def test_issue_15583():
N = mechanics.ReferenceFrame('N')
result = '(n_x, n_y, n_z)'
e = pretty((N.x, N.y, N.z))
assert e == result
def test_matrixSymbolBold():
# Issue 15871
def boldpretty(expr):
return xpretty(expr, use_unicode=True, wrap_line=False, mat_symbol_style="bold")
from sympy import trace
A = MatrixSymbol("A", 2, 2)
assert boldpretty(trace(A)) == u'tr(𝐀)'
A = MatrixSymbol("A", 3, 3)
B = MatrixSymbol("B", 3, 3)
C = MatrixSymbol("C", 3, 3)
assert boldpretty(-A) == u'-𝐀'
assert boldpretty(A - A*B - B) == u'-𝐁 -𝐀⋅𝐁 + 𝐀'
assert boldpretty(-A*B - A*B*C - B) == u'-𝐁 -𝐀⋅𝐁 -𝐀⋅𝐁⋅𝐂'
A = MatrixSymbol("Addot", 3, 3)
assert boldpretty(A) == u'𝐀̈'
omega = MatrixSymbol("omega", 3, 3)
assert boldpretty(omega) == u'ω'
omega = MatrixSymbol("omeganorm", 3, 3)
assert boldpretty(omega) == u'‖ω‖'
a = Symbol('alpha')
b = Symbol('b')
c = MatrixSymbol("c", 3, 1)
d = MatrixSymbol("d", 3, 1)
assert boldpretty(a*B*c+b*d) == u'b⋅𝐝 + α⋅𝐁⋅𝐜'
d = MatrixSymbol("delta", 3, 1)
B = MatrixSymbol("Beta", 3, 3)
assert boldpretty(a*B*c+b*d) == u'b⋅δ + α⋅Β⋅𝐜'
A = MatrixSymbol("A_2", 3, 3)
assert boldpretty(A) == u'𝐀₂'
def test_center_accent():
assert center_accent('a', u'\N{COMBINING TILDE}') == u'ã'
assert center_accent('aa', u'\N{COMBINING TILDE}') == u'aã'
assert center_accent('aaa', u'\N{COMBINING TILDE}') == u'aãa'
assert center_accent('aaaa', u'\N{COMBINING TILDE}') == u'aaãa'
assert center_accent('aaaaa', u'\N{COMBINING TILDE}') == u'aaãaa'
assert center_accent('abcdefg', u'\N{COMBINING FOUR DOTS ABOVE}') == u'abcd⃜efg'
def test_imaginary_unit():
from sympy import pretty # As it is redefined above
assert pretty(1 + I, use_unicode=False) == '1 + I'
assert pretty(1 + I, use_unicode=True) == u'1 + ⅈ'
assert pretty(1 + I, use_unicode=False, imaginary_unit='j') == '1 + I'
assert pretty(1 + I, use_unicode=True, imaginary_unit='j') == u'1 + ⅉ'
raises(TypeError, lambda: pretty(I, imaginary_unit=I))
raises(ValueError, lambda: pretty(I, imaginary_unit="kkk"))
def test_str_special_matrices():
from sympy.matrices import Identity, ZeroMatrix, OneMatrix
assert pretty(Identity(4)) == 'I'
assert upretty(Identity(4)) == u'𝕀'
assert pretty(ZeroMatrix(2, 2)) == '0'
assert upretty(ZeroMatrix(2, 2)) == u'𝟘'
assert pretty(OneMatrix(2, 2)) == '1'
assert upretty(OneMatrix(2, 2)) == u'𝟙'
def test_pretty_misc_functions():
assert pretty(LambertW(x)) == 'W(x)'
assert upretty(LambertW(x)) == u'W(x)'
assert pretty(LambertW(x, y)) == 'W(x, y)'
assert upretty(LambertW(x, y)) == u'W(x, y)'
assert pretty(airyai(x)) == 'Ai(x)'
assert upretty(airyai(x)) == u'Ai(x)'
assert pretty(airybi(x)) == 'Bi(x)'
assert upretty(airybi(x)) == u'Bi(x)'
assert pretty(airyaiprime(x)) == "Ai'(x)"
assert upretty(airyaiprime(x)) == u"Ai'(x)"
assert pretty(airybiprime(x)) == "Bi'(x)"
assert upretty(airybiprime(x)) == u"Bi'(x)"
assert pretty(fresnelc(x)) == 'C(x)'
assert upretty(fresnelc(x)) == u'C(x)'
assert pretty(fresnels(x)) == 'S(x)'
assert upretty(fresnels(x)) == u'S(x)'
assert pretty(Heaviside(x)) == 'Heaviside(x)'
assert upretty(Heaviside(x)) == u'θ(x)'
assert pretty(Heaviside(x, y)) == 'Heaviside(x, y)'
assert upretty(Heaviside(x, y)) == u'θ(x, y)'
assert pretty(dirichlet_eta(x)) == 'dirichlet_eta(x)'
assert upretty(dirichlet_eta(x)) == u'η(x)'
def test_hadamard_power():
m, n, p = symbols('m, n, p', integer=True)
A = MatrixSymbol('A', m, n)
B = MatrixSymbol('B', m, n)
# Testing printer:
expr = hadamard_power(A, n)
ascii_str = \
"""\
.n\n\
A \
"""
ucode_str = \
u("""\
∘n\n\
A \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = hadamard_power(A, 1+n)
ascii_str = \
"""\
.(n + 1)\n\
A \
"""
ucode_str = \
u("""\
∘(n + 1)\n\
A \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = hadamard_power(A*B.T, 1+n)
ascii_str = \
"""\
.(n + 1)\n\
/ T\\ \n\
\\A*B / \
"""
ucode_str = \
u("""\
∘(n + 1)\n\
⎛ T⎞ \n\
⎝A⋅B ⎠ \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_issue_17258():
n = Symbol('n', integer=True)
assert pretty(Sum(n, (n, -oo, 1))) == \
' 1 \n'\
' __ \n'\
' \\ ` \n'\
' ) n\n'\
' /_, \n'\
'n = -oo '
assert upretty(Sum(n, (n, -oo, 1))) == \
u("""\
1 \n\
___ \n\
╲ \n\
╲ \n\
╱ n\n\
╱ \n\
‾‾‾ \n\
n = -∞ \
""")
def test_is_combining():
line = u("v̇_m")
assert [is_combining(sym) for sym in line] == \
[False, True, False, False]
def test_issue_17857():
assert pretty(Range(-oo, oo)) == '{..., -1, 0, 1, ...}'
assert pretty(Range(oo, -oo, -1)) == '{..., 1, 0, -1, ...}'
def test_issue_18272():
x = Symbol('x')
n = Symbol('n')
assert upretty(ConditionSet(x, Eq(-x + exp(x), 0), S.Complexes)) == \
'⎧ ⎛ x ⎞⎫\n'\
'⎨x | x ∊ ℂ ∧ ⎝-x + ℯ = 0⎠⎬\n'\
'⎩ ⎭'
assert upretty(ConditionSet(x, Contains(n/2, Interval(0, oo)), FiniteSet(-n/2, n/2))) == \
'⎧ ⎧-n n⎫ ⎛n ⎞⎫\n'\
'⎨x | x ∊ ⎨───, ─⎬ ∧ ⎜─ ∈ [0, ∞)⎟⎬\n'\
'⎩ ⎩ 2 2⎭ ⎝2 ⎠⎭'
assert upretty(ConditionSet(x, Eq(Piecewise((1, x >= 3), (x/2 - 1/2, x >= 2), (1/2, x >= 1),
(x/2, True)) - 1/2, 0), Interval(0, 3))) == \
'⎧ ⎛⎛⎧ 1 for x ≥ 3⎞ ⎞⎫\n'\
'⎪ ⎜⎜⎪ ⎟ ⎟⎪\n'\
'⎪ ⎜⎜⎪x ⎟ ⎟⎪\n'\
'⎪ ⎜⎜⎪─ - 0.5 for x ≥ 2⎟ ⎟⎪\n'\
'⎪ ⎜⎜⎪2 ⎟ ⎟⎪\n'\
'⎨x | x ∊ [0, 3] ∧ ⎜⎜⎨ ⎟ - 0.5 = 0⎟⎬\n'\
'⎪ ⎜⎜⎪ 0.5 for x ≥ 1⎟ ⎟⎪\n'\
'⎪ ⎜⎜⎪ ⎟ ⎟⎪\n'\
'⎪ ⎜⎜⎪ x ⎟ ⎟⎪\n'\
'⎪ ⎜⎜⎪ ─ otherwise⎟ ⎟⎪\n'\
'⎩ ⎝⎝⎩ 2 ⎠ ⎠⎭'
def test_diffgeom():
from sympy.diffgeom import Manifold, Patch, CoordSystem, BaseScalarField
m = Manifold('M', 2)
assert pretty(m) == 'M'
p = Patch('P', m)
assert pretty(p) == "P"
rect = CoordSystem('rect', p)
assert pretty(rect) == "rect"
b = BaseScalarField(rect, 0)
assert pretty(b) == "rect_0"
|
2894476fb9691ca4c19ad366ecddf3e9753cf742ed03dcdbd9217173567f4dbf | from sympy import (S, symbols, I, atan, log, Poly, sqrt, simplify,
integrate, Rational, Dummy)
from sympy.integrals.rationaltools import ratint, ratint_logpart, log_to_atan
from sympy.abc import a, b, x, t
half = S.Half
def test_ratint():
assert ratint(S.Zero, x) == 0
assert ratint(S(7), x) == 7*x
assert ratint(x, x) == x**2/2
assert ratint(2*x, x) == x**2
assert ratint(-2*x, x) == -x**2
assert ratint(8*x**7 + 2*x + 1, x) == x**8 + x**2 + x
f = S.One
g = x + 1
assert ratint(f / g, x) == log(x + 1)
assert ratint((f, g), x) == log(x + 1)
f = x**3 - x
g = x - 1
assert ratint(f/g, x) == x**3/3 + x**2/2
f = x
g = (x - a)*(x + a)
assert ratint(f/g, x) == log(x**2 - a**2)/2
f = S.One
g = x**2 + 1
assert ratint(f/g, x, real=None) == atan(x)
assert ratint(f/g, x, real=True) == atan(x)
assert ratint(f/g, x, real=False) == I*log(x + I)/2 - I*log(x - I)/2
f = S(36)
g = x**5 - 2*x**4 - 2*x**3 + 4*x**2 + x - 2
assert ratint(f/g, x) == \
-4*log(x + 1) + 4*log(x - 2) + (12*x + 6)/(x**2 - 1)
f = x**4 - 3*x**2 + 6
g = x**6 - 5*x**4 + 5*x**2 + 4
assert ratint(f/g, x) == \
atan(x) + atan(x**3) + atan(x/2 - Rational(3, 2)*x**3 + S.Half*x**5)
f = x**7 - 24*x**4 - 4*x**2 + 8*x - 8
g = x**8 + 6*x**6 + 12*x**4 + 8*x**2
assert ratint(f/g, x) == \
(4 + 6*x + 8*x**2 + 3*x**3)/(4*x + 4*x**3 + x**5) + log(x)
assert ratint((x**3*f)/(x*g), x) == \
-(12 - 16*x + 6*x**2 - 14*x**3)/(4 + 4*x**2 + x**4) - \
5*sqrt(2)*atan(x*sqrt(2)/2) + S.Half*x**2 - 3*log(2 + x**2)
f = x**5 - x**4 + 4*x**3 + x**2 - x + 5
g = x**4 - 2*x**3 + 5*x**2 - 4*x + 4
assert ratint(f/g, x) == \
x + S.Half*x**2 + S.Half*log(2 - x + x**2) - (4*x - 9)/(14 - 7*x + 7*x**2) + \
13*sqrt(7)*atan(Rational(-1, 7)*sqrt(7) + 2*x*sqrt(7)/7)/49
assert ratint(1/(x**2 + x + 1), x) == \
2*sqrt(3)*atan(sqrt(3)/3 + 2*x*sqrt(3)/3)/3
assert ratint(1/(x**3 + 1), x) == \
-log(1 - x + x**2)/6 + log(1 + x)/3 + sqrt(3)*atan(-sqrt(3)
/3 + 2*x*sqrt(3)/3)/3
assert ratint(1/(x**2 + x + 1), x, real=False) == \
-I*3**half*log(half + x - half*I*3**half)/3 + \
I*3**half*log(half + x + half*I*3**half)/3
assert ratint(1/(x**3 + 1), x, real=False) == log(1 + x)/3 + \
(Rational(-1, 6) + I*3**half/6)*log(-half + x + I*3**half/2) + \
(Rational(-1, 6) - I*3**half/6)*log(-half + x - I*3**half/2)
# issue 4991
assert ratint(1/(x*(a + b*x)**3), x) == \
(3*a + 2*b*x)/(2*a**4 + 4*a**3*b*x + 2*a**2*b**2*x**2) + (
log(x) - log(a/b + x))/a**3
assert ratint(x/(1 - x**2), x) == -log(x**2 - 1)/2
assert ratint(-x/(1 - x**2), x) == log(x**2 - 1)/2
assert ratint((x/4 - 4/(1 - x)).diff(x), x) == x/4 + 4/(x - 1)
ans = atan(x)
assert ratint(1/(x**2 + 1), x, symbol=x) == ans
assert ratint(1/(x**2 + 1), x, symbol='x') == ans
assert ratint(1/(x**2 + 1), x, symbol=a) == ans
# this asserts that as_dummy must return a unique symbol
# even if the symbol is already a Dummy
d = Dummy()
assert ratint(1/(d**2 + 1), d, symbol=d) == atan(d)
def test_ratint_logpart():
assert ratint_logpart(x, x**2 - 9, x, t) == \
[(Poly(x**2 - 9, x), Poly(-2*t + 1, t))]
assert ratint_logpart(x**2, x**3 - 5, x, t) == \
[(Poly(x**3 - 5, x), Poly(-3*t + 1, t))]
def test_issue_5414():
assert ratint(1/(x**2 + 16), x) == atan(x/4)/4
def test_issue_5249():
assert ratint(
1/(x**2 + a**2), x) == (-I*log(-I*a + x)/2 + I*log(I*a + x)/2)/a
def test_issue_5817():
a, b, c = symbols('a,b,c', positive=True)
assert simplify(ratint(a/(b*c*x**2 + a**2 + b*a), x)) == \
sqrt(a)*atan(sqrt(
b)*sqrt(c)*x/(sqrt(a)*sqrt(a + b)))/(sqrt(b)*sqrt(c)*sqrt(a + b))
def test_issue_5981():
u = symbols('u')
assert integrate(1/(u**2 + 1)) == atan(u)
def test_issue_10488():
a,b,c,x = symbols('a b c x', real=True, positive=True)
assert integrate(x/(a*x+b),x) == x/a - b*log(a*x + b)/a**2
def test_issues_8246_12050_13501_14080():
a = symbols('a', nonzero=True)
assert integrate(a/(x**2 + a**2), x) == atan(x/a)
assert integrate(1/(x**2 + a**2), x) == atan(x/a)/a
assert integrate(1/(1 + a**2*x**2), x) == atan(a*x)/a
def test_issue_6308():
k, a0 = symbols('k a0', real=True)
assert integrate((x**2 + 1 - k**2)/(x**2 + 1 + a0**2), x) == \
x - (a0**2 + k**2)*atan(x/sqrt(a0**2 + 1))/sqrt(a0**2 + 1)
def test_issue_5907():
a = symbols('a', nonzero=True)
assert integrate(1/(x**2 + a**2)**2, x) == \
x/(2*a**4 + 2*a**2*x**2) + atan(x/a)/(2*a**3)
def test_log_to_atan():
f, g = (Poly(x + S.Half, x, domain='QQ'), Poly(sqrt(3)/2, x, domain='EX'))
fg_ans = 2*atan(2*sqrt(3)*x/3 + sqrt(3)/3)
assert log_to_atan(f, g) == fg_ans
assert log_to_atan(g, f) == -fg_ans
|
0968f8df4ae26deb0b89204f41ff28787f5ba5044765376752733701da8846b8 | from sympy import (
Symbol, Wild, sin, cos, exp, sqrt, pi, Function, Derivative,
Integer, Eq, symbols, Add, I, Float, log, Rational,
Lambda, atan2, cse, cot, tan, S, Tuple, Basic, Dict,
Piecewise, oo, Mul, factor, nsimplify, zoo, Subs, RootOf,
AccumBounds, Matrix, zeros, ZeroMatrix)
from sympy.core.basic import _aresame
from sympy.testing.pytest import XFAIL
from sympy.abc import a, x, y, z, t
def test_subs():
n3 = Rational(3)
e = x
e = e.subs(x, n3)
assert e == Rational(3)
e = 2*x
assert e == 2*x
e = e.subs(x, n3)
assert e == Rational(6)
def test_subs_Matrix():
z = zeros(2)
z1 = ZeroMatrix(2, 2)
assert (x*y).subs({x:z, y:0}) in [z, z1]
assert (x*y).subs({y:z, x:0}) == 0
assert (x*y).subs({y:z, x:0}, simultaneous=True) in [z, z1]
assert (x + y).subs({x: z, y: z}, simultaneous=True) in [z, z1]
assert (x + y).subs({x: z, y: z}) in [z, z1]
# Issue #15528
assert Mul(Matrix([[3]]), x).subs(x, 2.0) == Matrix([[6.0]])
# Does not raise a TypeError, see comment on the MatAdd postprocessor
assert Add(Matrix([[3]]), x).subs(x, 2.0) == Add(Matrix([[3]]), 2.0)
def test_subs_AccumBounds():
e = x
e = e.subs(x, AccumBounds(1, 3))
assert e == AccumBounds(1, 3)
e = 2*x
e = e.subs(x, AccumBounds(1, 3))
assert e == AccumBounds(2, 6)
e = x + x**2
e = e.subs(x, AccumBounds(-1, 1))
assert e == AccumBounds(-1, 2)
def test_trigonometric():
n3 = Rational(3)
e = (sin(x)**2).diff(x)
assert e == 2*sin(x)*cos(x)
e = e.subs(x, n3)
assert e == 2*cos(n3)*sin(n3)
e = (sin(x)**2).diff(x)
assert e == 2*sin(x)*cos(x)
e = e.subs(sin(x), cos(x))
assert e == 2*cos(x)**2
assert exp(pi).subs(exp, sin) == 0
assert cos(exp(pi)).subs(exp, sin) == 1
i = Symbol('i', integer=True)
zoo = S.ComplexInfinity
assert tan(x).subs(x, pi/2) is zoo
assert cot(x).subs(x, pi) is zoo
assert cot(i*x).subs(x, pi) is zoo
assert tan(i*x).subs(x, pi/2) == tan(i*pi/2)
assert tan(i*x).subs(x, pi/2).subs(i, 1) is zoo
o = Symbol('o', odd=True)
assert tan(o*x).subs(x, pi/2) == tan(o*pi/2)
def test_powers():
assert sqrt(1 - sqrt(x)).subs(x, 4) == I
assert (sqrt(1 - x**2)**3).subs(x, 2) == - 3*I*sqrt(3)
assert (x**Rational(1, 3)).subs(x, 27) == 3
assert (x**Rational(1, 3)).subs(x, -27) == 3*(-1)**Rational(1, 3)
assert ((-x)**Rational(1, 3)).subs(x, 27) == 3*(-1)**Rational(1, 3)
n = Symbol('n', negative=True)
assert (x**n).subs(x, 0) is S.ComplexInfinity
assert exp(-1).subs(S.Exp1, 0) is S.ComplexInfinity
assert (x**(4.0*y)).subs(x**(2.0*y), n) == n**2.0
assert (2**(x + 2)).subs(2, 3) == 3**(x + 3)
def test_logexppow(): # no eval()
x = Symbol('x', real=True)
w = Symbol('w')
e = (3**(1 + x) + 2**(1 + x))/(3**x + 2**x)
assert e.subs(2**x, w) != e
assert e.subs(exp(x*log(Rational(2))), w) != e
def test_bug():
x1 = Symbol('x1')
x2 = Symbol('x2')
y = x1*x2
assert y.subs(x1, Float(3.0)) == Float(3.0)*x2
def test_subbug1():
# see that they don't fail
(x**x).subs(x, 1)
(x**x).subs(x, 1.0)
def test_subbug2():
# Ensure this does not cause infinite recursion
assert Float(7.7).epsilon_eq(abs(x).subs(x, -7.7))
def test_dict_set():
a, b, c = map(Wild, 'abc')
f = 3*cos(4*x)
r = f.match(a*cos(b*x))
assert r == {a: 3, b: 4}
e = a/b*sin(b*x)
assert e.subs(r) == r[a]/r[b]*sin(r[b]*x)
assert e.subs(r) == 3*sin(4*x) / 4
s = set(r.items())
assert e.subs(s) == r[a]/r[b]*sin(r[b]*x)
assert e.subs(s) == 3*sin(4*x) / 4
assert e.subs(r) == r[a]/r[b]*sin(r[b]*x)
assert e.subs(r) == 3*sin(4*x) / 4
assert x.subs(Dict((x, 1))) == 1
def test_dict_ambigous(): # see issue 3566
f = x*exp(x)
g = z*exp(z)
df = {x: y, exp(x): y}
dg = {z: y, exp(z): y}
assert f.subs(df) == y**2
assert g.subs(dg) == y**2
# and this is how order can affect the result
assert f.subs(x, y).subs(exp(x), y) == y*exp(y)
assert f.subs(exp(x), y).subs(x, y) == y**2
# length of args and count_ops are the same so
# default_sort_key resolves ordering...if one
# doesn't want this result then an unordered
# sequence should not be used.
e = 1 + x*y
assert e.subs({x: y, y: 2}) == 5
# here, there are no obviously clashing keys or values
# but the results depend on the order
assert exp(x/2 + y).subs({exp(y + 1): 2, x: 2}) == exp(y + 1)
def test_deriv_sub_bug3():
f = Function('f')
pat = Derivative(f(x), x, x)
assert pat.subs(y, y**2) == Derivative(f(x), x, x)
assert pat.subs(y, y**2) != Derivative(f(x), x)
def test_equality_subs1():
f = Function('f')
eq = Eq(f(x)**2, x)
res = Eq(Integer(16), x)
assert eq.subs(f(x), 4) == res
def test_equality_subs2():
f = Function('f')
eq = Eq(f(x)**2, 16)
assert bool(eq.subs(f(x), 3)) is False
assert bool(eq.subs(f(x), 4)) is True
def test_issue_3742():
e = sqrt(x)*exp(y)
assert e.subs(sqrt(x), 1) == exp(y)
def test_subs_dict1():
assert (1 + x*y).subs(x, pi) == 1 + pi*y
assert (1 + x*y).subs({x: pi, y: 2}) == 1 + 2*pi
c2, c3, q1p, q2p, c1, s1, s2, s3 = symbols('c2 c3 q1p q2p c1 s1 s2 s3')
test = (c2**2*q2p*c3 + c1**2*s2**2*q2p*c3 + s1**2*s2**2*q2p*c3
- c1**2*q1p*c2*s3 - s1**2*q1p*c2*s3)
assert (test.subs({c1**2: 1 - s1**2, c2**2: 1 - s2**2, c3**3: 1 - s3**2})
== c3*q2p*(1 - s2**2) + c3*q2p*s2**2*(1 - s1**2)
- c2*q1p*s3*(1 - s1**2) + c3*q2p*s1**2*s2**2 - c2*q1p*s3*s1**2)
def test_mul():
x, y, z, a, b, c = symbols('x y z a b c')
A, B, C = symbols('A B C', commutative=0)
assert (x*y*z).subs(z*x, y) == y**2
assert (z*x).subs(1/x, z) == 1
assert (x*y/z).subs(1/z, a) == a*x*y
assert (x*y/z).subs(x/z, a) == a*y
assert (x*y/z).subs(y/z, a) == a*x
assert (x*y/z).subs(x/z, 1/a) == y/a
assert (x*y/z).subs(x, 1/a) == y/(z*a)
assert (2*x*y).subs(5*x*y, z) != z*Rational(2, 5)
assert (x*y*A).subs(x*y, a) == a*A
assert (x**2*y**(x*Rational(3, 2))).subs(x*y**(x/2), 2) == 4*y**(x/2)
assert (x*exp(x*2)).subs(x*exp(x), 2) == 2*exp(x)
assert ((x**(2*y))**3).subs(x**y, 2) == 64
assert (x*A*B).subs(x*A, y) == y*B
assert (x*y*(1 + x)*(1 + x*y)).subs(x*y, 2) == 6*(1 + x)
assert ((1 + A*B)*A*B).subs(A*B, x*A*B)
assert (x*a/z).subs(x/z, A) == a*A
assert (x**3*A).subs(x**2*A, a) == a*x
assert (x**2*A*B).subs(x**2*B, a) == a*A
assert (x**2*A*B).subs(x**2*A, a) == a*B
assert (b*A**3/(a**3*c**3)).subs(a**4*c**3*A**3/b**4, z) == \
b*A**3/(a**3*c**3)
assert (6*x).subs(2*x, y) == 3*y
assert (y*exp(x*Rational(3, 2))).subs(y*exp(x), 2) == 2*exp(x/2)
assert (y*exp(x*Rational(3, 2))).subs(y*exp(x), 2) == 2*exp(x/2)
assert (A**2*B*A**2*B*A**2).subs(A*B*A, C) == A*C**2*A
assert (x*A**3).subs(x*A, y) == y*A**2
assert (x**2*A**3).subs(x*A, y) == y**2*A
assert (x*A**3).subs(x*A, B) == B*A**2
assert (x*A*B*A*exp(x*A*B)).subs(x*A, B) == B**2*A*exp(B*B)
assert (x**2*A*B*A*exp(x*A*B)).subs(x*A, B) == B**3*exp(B**2)
assert (x**3*A*exp(x*A*B)*A*exp(x*A*B)).subs(x*A, B) == \
x*B*exp(B**2)*B*exp(B**2)
assert (x*A*B*C*A*B).subs(x*A*B, C) == C**2*A*B
assert (-I*a*b).subs(a*b, 2) == -2*I
# issue 6361
assert (-8*I*a).subs(-2*a, 1) == 4*I
assert (-I*a).subs(-a, 1) == I
# issue 6441
assert (4*x**2).subs(2*x, y) == y**2
assert (2*4*x**2).subs(2*x, y) == 2*y**2
assert (-x**3/9).subs(-x/3, z) == -z**2*x
assert (-x**3/9).subs(x/3, z) == -z**2*x
assert (-2*x**3/9).subs(x/3, z) == -2*x*z**2
assert (-2*x**3/9).subs(-x/3, z) == -2*x*z**2
assert (-2*x**3/9).subs(-2*x, z) == z*x**2/9
assert (-2*x**3/9).subs(2*x, z) == -z*x**2/9
assert (2*(3*x/5/7)**2).subs(3*x/5, z) == 2*(Rational(1, 7))**2*z**2
assert (4*x).subs(-2*x, z) == 4*x # try keep subs literal
def test_subs_simple():
a = symbols('a', commutative=True)
x = symbols('x', commutative=False)
assert (2*a).subs(1, 3) == 2*a
assert (2*a).subs(2, 3) == 3*a
assert (2*a).subs(a, 3) == 6
assert sin(2).subs(1, 3) == sin(2)
assert sin(2).subs(2, 3) == sin(3)
assert sin(a).subs(a, 3) == sin(3)
assert (2*x).subs(1, 3) == 2*x
assert (2*x).subs(2, 3) == 3*x
assert (2*x).subs(x, 3) == 6
assert sin(x).subs(x, 3) == sin(3)
def test_subs_constants():
a, b = symbols('a b', commutative=True)
x, y = symbols('x y', commutative=False)
assert (a*b).subs(2*a, 1) == a*b
assert (1.5*a*b).subs(a, 1) == 1.5*b
assert (2*a*b).subs(2*a, 1) == b
assert (2*a*b).subs(4*a, 1) == 2*a*b
assert (x*y).subs(2*x, 1) == x*y
assert (1.5*x*y).subs(x, 1) == 1.5*y
assert (2*x*y).subs(2*x, 1) == y
assert (2*x*y).subs(4*x, 1) == 2*x*y
def test_subs_commutative():
a, b, c, d, K = symbols('a b c d K', commutative=True)
assert (a*b).subs(a*b, K) == K
assert (a*b*a*b).subs(a*b, K) == K**2
assert (a*a*b*b).subs(a*b, K) == K**2
assert (a*b*c*d).subs(a*b*c, K) == d*K
assert (a*b**c).subs(a, K) == K*b**c
assert (a*b**c).subs(b, K) == a*K**c
assert (a*b**c).subs(c, K) == a*b**K
assert (a*b*c*b*a).subs(a*b, K) == c*K**2
assert (a**3*b**2*a).subs(a*b, K) == a**2*K**2
def test_subs_noncommutative():
w, x, y, z, L = symbols('w x y z L', commutative=False)
alpha = symbols('alpha', commutative=True)
someint = symbols('someint', commutative=True, integer=True)
assert (x*y).subs(x*y, L) == L
assert (w*y*x).subs(x*y, L) == w*y*x
assert (w*x*y*z).subs(x*y, L) == w*L*z
assert (x*y*x*y).subs(x*y, L) == L**2
assert (x*x*y).subs(x*y, L) == x*L
assert (x*x*y*y).subs(x*y, L) == x*L*y
assert (w*x*y).subs(x*y*z, L) == w*x*y
assert (x*y**z).subs(x, L) == L*y**z
assert (x*y**z).subs(y, L) == x*L**z
assert (x*y**z).subs(z, L) == x*y**L
assert (w*x*y*z*x*y).subs(x*y*z, L) == w*L*x*y
assert (w*x*y*y*w*x*x*y*x*y*y*x*y).subs(x*y, L) == w*L*y*w*x*L**2*y*L
# Check fractional power substitutions. It should not do
# substitutions that choose a value for noncommutative log,
# or inverses that don't already appear in the expressions.
assert (x*x*x).subs(x*x, L) == L*x
assert (x*x*x*y*x*x*x*x).subs(x*x, L) == L*x*y*L**2
for p in range(1, 5):
for k in range(10):
assert (y * x**k).subs(x**p, L) == y * L**(k//p) * x**(k % p)
assert (x**Rational(3, 2)).subs(x**S.Half, L) == x**Rational(3, 2)
assert (x**S.Half).subs(x**S.Half, L) == L
assert (x**Rational(-1, 2)).subs(x**S.Half, L) == x**Rational(-1, 2)
assert (x**Rational(-1, 2)).subs(x**Rational(-1, 2), L) == L
assert (x**(2*someint)).subs(x**someint, L) == L**2
assert (x**(2*someint + 3)).subs(x**someint, L) == L**2*x**3
assert (x**(3*someint + 3)).subs(x**someint, L) == L**3*x**3
assert (x**(3*someint)).subs(x**(2*someint), L) == L * x**someint
assert (x**(4*someint)).subs(x**(2*someint), L) == L**2
assert (x**(4*someint + 1)).subs(x**(2*someint), L) == L**2 * x
assert (x**(4*someint)).subs(x**(3*someint), L) == L * x**someint
assert (x**(4*someint + 1)).subs(x**(3*someint), L) == L * x**(someint + 1)
assert (x**(2*alpha)).subs(x**alpha, L) == x**(2*alpha)
assert (x**(2*alpha + 2)).subs(x**2, L) == x**(2*alpha + 2)
assert ((2*z)**alpha).subs(z**alpha, y) == (2*z)**alpha
assert (x**(2*someint*alpha)).subs(x**someint, L) == x**(2*someint*alpha)
assert (x**(2*someint + alpha)).subs(x**someint, L) == x**(2*someint + alpha)
# This could in principle be substituted, but is not currently
# because it requires recognizing that someint**2 is divisible by
# someint.
assert (x**(someint**2 + 3)).subs(x**someint, L) == x**(someint**2 + 3)
# alpha**z := exp(log(alpha) z) is usually well-defined
assert (4**z).subs(2**z, y) == y**2
# Negative powers
assert (x**(-1)).subs(x**3, L) == x**(-1)
assert (x**(-2)).subs(x**3, L) == x**(-2)
assert (x**(-3)).subs(x**3, L) == L**(-1)
assert (x**(-4)).subs(x**3, L) == L**(-1) * x**(-1)
assert (x**(-5)).subs(x**3, L) == L**(-1) * x**(-2)
assert (x**(-1)).subs(x**(-3), L) == x**(-1)
assert (x**(-2)).subs(x**(-3), L) == x**(-2)
assert (x**(-3)).subs(x**(-3), L) == L
assert (x**(-4)).subs(x**(-3), L) == L * x**(-1)
assert (x**(-5)).subs(x**(-3), L) == L * x**(-2)
assert (x**1).subs(x**(-3), L) == x
assert (x**2).subs(x**(-3), L) == x**2
assert (x**3).subs(x**(-3), L) == L**(-1)
assert (x**4).subs(x**(-3), L) == L**(-1) * x
assert (x**5).subs(x**(-3), L) == L**(-1) * x**2
def test_subs_basic_funcs():
a, b, c, d, K = symbols('a b c d K', commutative=True)
w, x, y, z, L = symbols('w x y z L', commutative=False)
assert (x + y).subs(x + y, L) == L
assert (x - y).subs(x - y, L) == L
assert (x/y).subs(x, L) == L/y
assert (x**y).subs(x, L) == L**y
assert (x**y).subs(y, L) == x**L
assert ((a - c)/b).subs(b, K) == (a - c)/K
assert (exp(x*y - z)).subs(x*y, L) == exp(L - z)
assert (a*exp(x*y - w*z) + b*exp(x*y + w*z)).subs(z, 0) == \
a*exp(x*y) + b*exp(x*y)
assert ((a - b)/(c*d - a*b)).subs(c*d - a*b, K) == (a - b)/K
assert (w*exp(a*b - c)*x*y/4).subs(x*y, L) == w*exp(a*b - c)*L/4
def test_subs_wild():
R, S, T, U = symbols('R S T U', cls=Wild)
assert (R*S).subs(R*S, T) == T
assert (S*R).subs(R*S, T) == T
assert (R + S).subs(R + S, T) == T
assert (R**S).subs(R, T) == T**S
assert (R**S).subs(S, T) == R**T
assert (R*S**T).subs(R, U) == U*S**T
assert (R*S**T).subs(S, U) == R*U**T
assert (R*S**T).subs(T, U) == R*S**U
def test_subs_mixed():
a, b, c, d, K = symbols('a b c d K', commutative=True)
w, x, y, z, L = symbols('w x y z L', commutative=False)
R, S, T, U = symbols('R S T U', cls=Wild)
assert (a*x*y).subs(x*y, L) == a*L
assert (a*b*x*y*x).subs(x*y, L) == a*b*L*x
assert (R*x*y*exp(x*y)).subs(x*y, L) == R*L*exp(L)
assert (a*x*y*y*x - x*y*z*exp(a*b)).subs(x*y, L) == a*L*y*x - L*z*exp(a*b)
e = c*y*x*y*x**(R*S - a*b) - T*(a*R*b*S)
assert e.subs(x*y, L).subs(a*b, K).subs(R*S, U) == \
c*y*L*x**(U - K) - T*(U*K)
def test_division():
a, b, c = symbols('a b c', commutative=True)
x, y, z = symbols('x y z', commutative=True)
assert (1/a).subs(a, c) == 1/c
assert (1/a**2).subs(a, c) == 1/c**2
assert (1/a**2).subs(a, -2) == Rational(1, 4)
assert (-(1/a**2)).subs(a, -2) == Rational(-1, 4)
assert (1/x).subs(x, z) == 1/z
assert (1/x**2).subs(x, z) == 1/z**2
assert (1/x**2).subs(x, -2) == Rational(1, 4)
assert (-(1/x**2)).subs(x, -2) == Rational(-1, 4)
#issue 5360
assert (1/x).subs(x, 0) == 1/S.Zero
def test_add():
a, b, c, d, x, y, t = symbols('a b c d x y t')
assert (a**2 - b - c).subs(a**2 - b, d) in [d - c, a**2 - b - c]
assert (a**2 - c).subs(a**2 - c, d) == d
assert (a**2 - b - c).subs(a**2 - c, d) in [d - b, a**2 - b - c]
assert (a**2 - x - c).subs(a**2 - c, d) in [d - x, a**2 - x - c]
assert (a**2 - b - sqrt(a)).subs(a**2 - sqrt(a), c) == c - b
assert (a + b + exp(a + b)).subs(a + b, c) == c + exp(c)
assert (c + b + exp(c + b)).subs(c + b, a) == a + exp(a)
assert (a + b + c + d).subs(b + c, x) == a + d + x
assert (a + b + c + d).subs(-b - c, x) == a + d - x
assert ((x + 1)*y).subs(x + 1, t) == t*y
assert ((-x - 1)*y).subs(x + 1, t) == -t*y
assert ((x - 1)*y).subs(x + 1, t) == y*(t - 2)
assert ((-x + 1)*y).subs(x + 1, t) == y*(-t + 2)
# this should work every time:
e = a**2 - b - c
assert e.subs(Add(*e.args[:2]), d) == d + e.args[2]
assert e.subs(a**2 - c, d) == d - b
# the fallback should recognize when a change has
# been made; while .1 == Rational(1, 10) they are not the same
# and the change should be made
assert (0.1 + a).subs(0.1, Rational(1, 10)) == Rational(1, 10) + a
e = (-x*(-y + 1) - y*(y - 1))
ans = (-x*(x) - y*(-x)).expand()
assert e.subs(-y + 1, x) == ans
#Test issue 18747
assert (exp(x) + cos(x)).subs(x, oo) == oo
assert Add(*[AccumBounds(-1, 1), oo]) == oo
assert Add(*[oo, AccumBounds(-1, 1)]) == oo
def test_subs_issue_4009():
assert (I*Symbol('a')).subs(1, 2) == I*Symbol('a')
def test_functions_subs():
f, g = symbols('f g', cls=Function)
l = Lambda((x, y), sin(x) + y)
assert (g(y, x) + cos(x)).subs(g, l) == sin(y) + x + cos(x)
assert (f(x)**2).subs(f, sin) == sin(x)**2
assert (f(x, y)).subs(f, log) == log(x, y)
assert (f(x, y)).subs(f, sin) == f(x, y)
assert (sin(x) + atan2(x, y)).subs([[atan2, f], [sin, g]]) == \
f(x, y) + g(x)
assert (g(f(x + y, x))).subs([[f, l], [g, exp]]) == exp(x + sin(x + y))
def test_derivative_subs():
f = Function('f')
g = Function('g')
assert Derivative(f(x), x).subs(f(x), y) != 0
# need xreplace to put the function back, see #13803
assert Derivative(f(x), x).subs(f(x), y).xreplace({y: f(x)}) == \
Derivative(f(x), x)
# issues 5085, 5037
assert cse(Derivative(f(x), x) + f(x))[1][0].has(Derivative)
assert cse(Derivative(f(x, y), x) +
Derivative(f(x, y), y))[1][0].has(Derivative)
eq = Derivative(g(x), g(x))
assert eq.subs(g, f) == Derivative(f(x), f(x))
assert eq.subs(g(x), f(x)) == Derivative(f(x), f(x))
assert eq.subs(g, cos) == Subs(Derivative(y, y), y, cos(x))
def test_derivative_subs2():
f_func, g_func = symbols('f g', cls=Function)
f, g = f_func(x, y, z), g_func(x, y, z)
assert Derivative(f, x, y).subs(Derivative(f, x, y), g) == g
assert Derivative(f, y, x).subs(Derivative(f, x, y), g) == g
assert Derivative(f, x, y).subs(Derivative(f, x), g) == Derivative(g, y)
assert Derivative(f, x, y).subs(Derivative(f, y), g) == Derivative(g, x)
assert (Derivative(f, x, y, z).subs(
Derivative(f, x, z), g) == Derivative(g, y))
assert (Derivative(f, x, y, z).subs(
Derivative(f, z, y), g) == Derivative(g, x))
assert (Derivative(f, x, y, z).subs(
Derivative(f, z, y, x), g) == g)
# Issue 9135
assert (Derivative(f, x, x, y).subs(
Derivative(f, y, y), g) == Derivative(f, x, x, y))
assert (Derivative(f, x, y, y, z).subs(
Derivative(f, x, y, y, y), g) == Derivative(f, x, y, y, z))
assert Derivative(f, x, y).subs(Derivative(f_func(x), x, y), g) == Derivative(f, x, y)
def test_derivative_subs3():
dex = Derivative(exp(x), x)
assert Derivative(dex, x).subs(dex, exp(x)) == dex
assert dex.subs(exp(x), dex) == Derivative(exp(x), x, x)
def test_issue_5284():
A, B = symbols('A B', commutative=False)
assert (x*A).subs(x**2*A, B) == x*A
assert (A**2).subs(A**3, B) == A**2
assert (A**6).subs(A**3, B) == B**2
def test_subs_iter():
assert x.subs(reversed([[x, y]])) == y
it = iter([[x, y]])
assert x.subs(it) == y
assert x.subs(Tuple((x, y))) == y
def test_subs_dict():
a, b, c, d, e = symbols('a b c d e')
assert (2*x + y + z).subs(dict(x=1, y=2)) == 4 + z
l = [(sin(x), 2), (x, 1)]
assert (sin(x)).subs(l) == \
(sin(x)).subs(dict(l)) == 2
assert sin(x).subs(reversed(l)) == sin(1)
expr = sin(2*x) + sqrt(sin(2*x))*cos(2*x)*sin(exp(x)*x)
reps = dict([
(sin(2*x), c),
(sqrt(sin(2*x)), a),
(cos(2*x), b),
(exp(x), e),
(x, d),
])
assert expr.subs(reps) == c + a*b*sin(d*e)
l = [(x, 3), (y, x**2)]
assert (x + y).subs(l) == 3 + x**2
assert (x + y).subs(reversed(l)) == 12
# If changes are made to convert lists into dictionaries and do
# a dictionary-lookup replacement, these tests will help to catch
# some logical errors that might occur
l = [(y, z + 2), (1 + z, 5), (z, 2)]
assert (y - 1 + 3*x).subs(l) == 5 + 3*x
l = [(y, z + 2), (z, 3)]
assert (y - 2).subs(l) == 3
def test_no_arith_subs_on_floats():
assert (x + 3).subs(x + 3, a) == a
assert (x + 3).subs(x + 2, a) == a + 1
assert (x + y + 3).subs(x + 3, a) == a + y
assert (x + y + 3).subs(x + 2, a) == a + y + 1
assert (x + 3.0).subs(x + 3.0, a) == a
assert (x + 3.0).subs(x + 2.0, a) == x + 3.0
assert (x + y + 3.0).subs(x + 3.0, a) == a + y
assert (x + y + 3.0).subs(x + 2.0, a) == x + y + 3.0
def test_issue_5651():
a, b, c, K = symbols('a b c K', commutative=True)
assert (a/(b*c)).subs(b*c, K) == a/K
assert (a/(b**2*c**3)).subs(b*c, K) == a/(c*K**2)
assert (1/(x*y)).subs(x*y, 2) == S.Half
assert ((1 + x*y)/(x*y)).subs(x*y, 1) == 2
assert (x*y*z).subs(x*y, 2) == 2*z
assert ((1 + x*y)/(x*y)/z).subs(x*y, 1) == 2/z
def test_issue_6075():
assert Tuple(1, True).subs(1, 2) == Tuple(2, True)
def test_issue_6079():
# since x + 2.0 == x + 2 we can't do a simple equality test
assert _aresame((x + 2.0).subs(2, 3), x + 2.0)
assert _aresame((x + 2.0).subs(2.0, 3), x + 3)
assert not _aresame(x + 2, x + 2.0)
assert not _aresame(Basic(cos, 1), Basic(cos, 1.))
assert _aresame(cos, cos)
assert not _aresame(1, S.One)
assert not _aresame(x, symbols('x', positive=True))
def test_issue_4680():
N = Symbol('N')
assert N.subs(dict(N=3)) == 3
def test_issue_6158():
assert (x - 1).subs(1, y) == x - y
assert (x - 1).subs(-1, y) == x + y
assert (x - oo).subs(oo, y) == x - y
assert (x - oo).subs(-oo, y) == x + y
def test_Function_subs():
f, g, h, i = symbols('f g h i', cls=Function)
p = Piecewise((g(f(x, y)), x < -1), (g(x), x <= 1))
assert p.subs(g, h) == Piecewise((h(f(x, y)), x < -1), (h(x), x <= 1))
assert (f(y) + g(x)).subs({f: h, g: i}) == i(x) + h(y)
def test_simultaneous_subs():
reps = {x: 0, y: 0}
assert (x/y).subs(reps) != (y/x).subs(reps)
assert (x/y).subs(reps, simultaneous=True) == \
(y/x).subs(reps, simultaneous=True)
reps = reps.items()
assert (x/y).subs(reps) != (y/x).subs(reps)
assert (x/y).subs(reps, simultaneous=True) == \
(y/x).subs(reps, simultaneous=True)
assert Derivative(x, y, z).subs(reps, simultaneous=True) == \
Subs(Derivative(0, y, z), y, 0)
def test_issue_6419_6421():
assert (1/(1 + x/y)).subs(x/y, x) == 1/(1 + x)
assert (-2*I).subs(2*I, x) == -x
assert (-I*x).subs(I*x, x) == -x
assert (-3*I*y**4).subs(3*I*y**2, x) == -x*y**2
def test_issue_6559():
assert (-12*x + y).subs(-x, 1) == 12 + y
# though this involves cse it generated a failure in Mul._eval_subs
x0, x1 = symbols('x0 x1')
e = -log(-12*sqrt(2) + 17)/24 - log(-2*sqrt(2) + 3)/12 + sqrt(2)/3
# XXX modify cse so x1 is eliminated and x0 = -sqrt(2)?
assert cse(e) == (
[(x0, sqrt(2))], [x0/3 - log(-12*x0 + 17)/24 - log(-2*x0 + 3)/12])
def test_issue_5261():
x = symbols('x', real=True)
e = I*x
assert exp(e).subs(exp(x), y) == y**I
assert (2**e).subs(2**x, y) == y**I
eq = (-2)**e
assert eq.subs((-2)**x, y) == eq
def test_issue_6923():
assert (-2*x*sqrt(2)).subs(2*x, y) == -sqrt(2)*y
def test_2arg_hack():
N = Symbol('N', commutative=False)
ans = Mul(2, y + 1, evaluate=False)
assert (2*x*(y + 1)).subs(x, 1, hack2=True) == ans
assert (2*(y + 1 + N)).subs(N, 0, hack2=True) == ans
@XFAIL
def test_mul2():
"""When this fails, remove things labelled "2-arg hack"
1) remove special handling in the fallback of subs that
was added in the same commit as this test
2) remove the special handling in Mul.flatten
"""
assert (2*(x + 1)).is_Mul
def test_noncommutative_subs():
x,y = symbols('x,y', commutative=False)
assert (x*y*x).subs([(x, x*y), (y, x)], simultaneous=True) == (x*y*x**2*y)
def test_issue_2877():
f = Float(2.0)
assert (x + f).subs({f: 2}) == x + 2
def r(a, b, c):
return factor(a*x**2 + b*x + c)
e = r(5.0/6, 10, 5)
assert nsimplify(e) == 5*x**2/6 + 10*x + 5
def test_issue_5910():
t = Symbol('t')
assert (1/(1 - t)).subs(t, 1) is zoo
n = t
d = t - 1
assert (n/d).subs(t, 1) is zoo
assert (-n/-d).subs(t, 1) is zoo
def test_issue_5217():
s = Symbol('s')
z = (1 - 2*x*x)
w = (1 + 2*x*x)
q = 2*x*x*2*y*y
sub = {2*x*x: s}
assert w.subs(sub) == 1 + s
assert z.subs(sub) == 1 - s
assert q == 4*x**2*y**2
assert q.subs(sub) == 2*y**2*s
def test_issue_10829():
assert (4**x).subs(2**x, y) == y**2
assert (9**x).subs(3**x, y) == y**2
def test_pow_eval_subs_no_cache():
# Tests pull request 9376 is working
from sympy.core.cache import clear_cache
s = 1/sqrt(x**2)
# This bug only appeared when the cache was turned off.
# We need to approximate running this test without the cache.
# This creates approximately the same situation.
clear_cache()
# This used to fail with a wrong result.
# It incorrectly returned 1/sqrt(x**2) before this pull request.
result = s.subs(sqrt(x**2), y)
assert result == 1/y
def test_RootOf_issue_10092():
x = Symbol('x', real=True)
eq = x**3 - 17*x**2 + 81*x - 118
r = RootOf(eq, 0)
assert (x < r).subs(x, r) is S.false
def test_issue_8886():
from sympy.physics.mechanics import ReferenceFrame as R
# if something can't be sympified we assume that it
# doesn't play well with SymPy and disallow the
# substitution
v = R('A').x
assert x.subs(x, v) == x
assert v.subs(v, x) == v
assert v.__eq__(x) is False
def test_issue_12657():
# treat -oo like the atom that it is
reps = [(-oo, 1), (oo, 2)]
assert (x < -oo).subs(reps) == (x < 1)
assert (x < -oo).subs(list(reversed(reps))) == (x < 1)
reps = [(-oo, 2), (oo, 1)]
assert (x < oo).subs(reps) == (x < 1)
assert (x < oo).subs(list(reversed(reps))) == (x < 1)
def test_recurse_Application_args():
F = Lambda((x, y), exp(2*x + 3*y))
f = Function('f')
A = f(x, f(x, x))
C = F(x, F(x, x))
assert A.subs(f, F) == A.replace(f, F) == C
def test_Subs_subs():
assert Subs(x*y, x, x).subs(x, y) == Subs(x*y, x, y)
assert Subs(x*y, x, x + 1).subs(x, y) == \
Subs(x*y, x, y + 1)
assert Subs(x*y, y, x + 1).subs(x, y) == \
Subs(y**2, y, y + 1)
a = Subs(x*y*z, (y, x, z), (x + 1, x + z, x))
b = Subs(x*y*z, (y, x, z), (x + 1, y + z, y))
assert a.subs(x, y) == b and \
a.doit().subs(x, y) == a.subs(x, y).doit()
f = Function('f')
g = Function('g')
assert Subs(2*f(x, y) + g(x), f(x, y), 1).subs(y, 2) == Subs(
2*f(x, y) + g(x), (f(x, y), y), (1, 2))
def test_issue_13333():
eq = 1/x
assert eq.subs(dict(x='1/2')) == 2
assert eq.subs(dict(x='(1/2)')) == 2
def test_issue_15234():
x, y = symbols('x y', real=True)
p = 6*x**5 + x**4 - 4*x**3 + 4*x**2 - 2*x + 3
p_subbed = 6*x**5 - 4*x**3 - 2*x + y**4 + 4*y**2 + 3
assert p.subs([(x**i, y**i) for i in [2, 4]]) == p_subbed
x, y = symbols('x y', complex=True)
p = 6*x**5 + x**4 - 4*x**3 + 4*x**2 - 2*x + 3
p_subbed = 6*x**5 - 4*x**3 - 2*x + y**4 + 4*y**2 + 3
assert p.subs([(x**i, y**i) for i in [2, 4]]) == p_subbed
def test_issue_6976():
x, y = symbols('x y')
assert (sqrt(x)**3 + sqrt(x) + x + x**2).subs(sqrt(x), y) == \
y**4 + y**3 + y**2 + y
assert (x**4 + x**3 + x**2 + x + sqrt(x)).subs(x**2, y) == \
sqrt(x) + x**3 + x + y**2 + y
assert x.subs(x**3, y) == x
assert x.subs(x**Rational(1, 3), y) == y**3
# More substitutions are possible with nonnegative symbols
x, y = symbols('x y', nonnegative=True)
assert (x**4 + x**3 + x**2 + x + sqrt(x)).subs(x**2, y) == \
y**Rational(1, 4) + y**Rational(3, 2) + sqrt(y) + y**2 + y
assert x.subs(x**3, y) == y**Rational(1, 3)
def test_issue_11746():
assert (1/x).subs(x**2, 1) == 1/x
assert (1/(x**3)).subs(x**2, 1) == x**(-3)
assert (1/(x**4)).subs(x**2, 1) == 1
assert (1/(x**3)).subs(x**4, 1) == x**(-3)
assert (1/(y**5)).subs(x**5, 1) == y**(-5)
def test_issue_17823():
from sympy.physics.mechanics import dynamicsymbols
q1, q2 = dynamicsymbols('q1, q2')
expr = q1.diff().diff()**2*q1 + q1.diff()*q2.diff()
reps={q1: a, q1.diff(): a*x*y, q1.diff().diff(): z}
assert expr.subs(reps) == a*x*y*Derivative(q2, t) + a*z**2
def test_issue_19326():
x, y = [i(t) for i in map(Function, 'xy')]
assert (x*y).subs({x: 1 + x, y: x}) == (1 + x)*x
|
ab7c157ce9247834a3ce9cc59b77be53782f68c9e7486a392390b14ba64047dd | from sympy import (Add, Basic, Expr, S, Symbol, Wild, Float, Integer, Rational, I,
sin, cos, tan, exp, log, nan, oo, sqrt, symbols, Integral, sympify,
WildFunction, Poly, Function, Derivative, Number, pi, NumberSymbol, zoo,
Piecewise, Mul, Pow, nsimplify, ratsimp, trigsimp, radsimp, powsimp,
simplify, together, collect, factorial, apart, combsimp, factor, refine,
cancel, Tuple, default_sort_key, DiracDelta, gamma, Dummy, Sum, E,
exp_polar, expand, diff, O, Heaviside, Si, Max, UnevaluatedExpr,
integrate, gammasimp, Gt)
from sympy.core.expr import ExprBuilder, unchanged
from sympy.core.function import AppliedUndef
from sympy.physics.secondquant import FockState
from sympy.physics.units import meter
from sympy.testing.pytest import raises, XFAIL
from sympy.abc import a, b, c, n, t, u, x, y, z
class DummyNumber:
"""
Minimal implementation of a number that works with SymPy.
If one has a Number class (e.g. Sage Integer, or some other custom class)
that one wants to work well with SymPy, one has to implement at least the
methods of this class DummyNumber, resp. its subclasses I5 and F1_1.
Basically, one just needs to implement either __int__() or __float__() and
then one needs to make sure that the class works with Python integers and
with itself.
"""
def __radd__(self, a):
if isinstance(a, (int, float)):
return a + self.number
return NotImplemented
def __truediv__(a, b):
return a.__div__(b)
def __rtruediv__(a, b):
return a.__rdiv__(b)
def __add__(self, a):
if isinstance(a, (int, float, DummyNumber)):
return self.number + a
return NotImplemented
def __rsub__(self, a):
if isinstance(a, (int, float)):
return a - self.number
return NotImplemented
def __sub__(self, a):
if isinstance(a, (int, float, DummyNumber)):
return self.number - a
return NotImplemented
def __rmul__(self, a):
if isinstance(a, (int, float)):
return a * self.number
return NotImplemented
def __mul__(self, a):
if isinstance(a, (int, float, DummyNumber)):
return self.number * a
return NotImplemented
def __rdiv__(self, a):
if isinstance(a, (int, float)):
return a / self.number
return NotImplemented
def __div__(self, a):
if isinstance(a, (int, float, DummyNumber)):
return self.number / a
return NotImplemented
def __rpow__(self, a):
if isinstance(a, (int, float)):
return a ** self.number
return NotImplemented
def __pow__(self, a):
if isinstance(a, (int, float, DummyNumber)):
return self.number ** a
return NotImplemented
def __pos__(self):
return self.number
def __neg__(self):
return - self.number
class I5(DummyNumber):
number = 5
def __int__(self):
return self.number
class F1_1(DummyNumber):
number = 1.1
def __float__(self):
return self.number
i5 = I5()
f1_1 = F1_1()
# basic sympy objects
basic_objs = [
Rational(2),
Float("1.3"),
x,
y,
pow(x, y)*y,
]
# all supported objects
all_objs = basic_objs + [
5,
5.5,
i5,
f1_1
]
def dotest(s):
for xo in all_objs:
for yo in all_objs:
s(xo, yo)
return True
def test_basic():
def j(a, b):
x = a
x = +a
x = -a
x = a + b
x = a - b
x = a*b
x = a/b
x = a**b
del x
assert dotest(j)
def test_ibasic():
def s(a, b):
x = a
x += b
x = a
x -= b
x = a
x *= b
x = a
x /= b
assert dotest(s)
class NonBasic:
'''This class represents an object that knows how to implement binary
operations like +, -, etc with Expr but is not a subclass of Basic itself.
The NonExpr subclass below does subclass Basic but not Expr.
For both NonBasic and NonExpr it should be possible for them to override
Expr.__add__ etc because Expr.__add__ should be returning NotImplemented
for non Expr classes. Otherwise Expr.__add__ would create meaningless
objects like Add(Integer(1), FiniteSet(2)) and it wouldn't be possible for
other classes to override these operations when interacting with Expr.
'''
def __add__(self, other):
return SpecialOp('+', self, other)
def __radd__(self, other):
return SpecialOp('+', other, self)
def __sub__(self, other):
return SpecialOp('-', self, other)
def __rsub__(self, other):
return SpecialOp('-', other, self)
def __mul__(self, other):
return SpecialOp('*', self, other)
def __rmul__(self, other):
return SpecialOp('*', other, self)
def __div__(self, other):
return SpecialOp('/', self, other)
def __rdiv__(self, other):
return SpecialOp('/', other, self)
__truediv__ = __div__
__rtruediv__ = __rdiv__
def __floordiv__(self, other):
return SpecialOp('//', self, other)
def __rfloordiv__(self, other):
return SpecialOp('//', other, self)
def __mod__(self, other):
return SpecialOp('%', self, other)
def __rmod__(self, other):
return SpecialOp('%', other, self)
def __divmod__(self, other):
return SpecialOp('divmod', self, other)
def __rdivmod__(self, other):
return SpecialOp('divmod', other, self)
def __pow__(self, other):
return SpecialOp('**', self, other)
def __rpow__(self, other):
return SpecialOp('**', other, self)
def __lt__(self, other):
return SpecialOp('<', self, other)
def __gt__(self, other):
return SpecialOp('>', self, other)
def __le__(self, other):
return SpecialOp('<=', self, other)
def __ge__(self, other):
return SpecialOp('>=', self, other)
class NonExpr(Basic, NonBasic):
'''Like NonBasic above except this is a subclass of Basic but not Expr'''
pass
class SpecialOp(Basic):
'''Represents the results of operations with NonBasic and NonExpr'''
def __new__(cls, op, arg1, arg2):
return Basic.__new__(cls, op, arg1, arg2)
class NonArithmetic(Basic):
'''Represents a Basic subclass that does not support arithmetic operations'''
pass
def test_cooperative_operations():
'''Tests that Expr uses binary operations cooperatively.
In particular it should be possible for non-Expr classes to override
binary operators like +, - etc when used with Expr instances. This should
work for non-Expr classes whether they are Basic subclasses or not. Also
non-Expr classes that do not define binary operators with Expr should give
TypeError.
'''
# A bunch of instances of Expr subclasses
exprs = [
Expr(),
S.Zero,
S.One,
S.Infinity,
S.NegativeInfinity,
S.ComplexInfinity,
S.Half,
Float(0.5),
Integer(2),
Symbol('x'),
Mul(2, Symbol('x')),
Add(2, Symbol('x')),
Pow(2, Symbol('x')),
]
for e in exprs:
# Test that these classes can override arithmetic operations in
# combination with various Expr types.
for ne in [NonBasic(), NonExpr()]:
results = [
(ne + e, ('+', ne, e)),
(e + ne, ('+', e, ne)),
(ne - e, ('-', ne, e)),
(e - ne, ('-', e, ne)),
(ne * e, ('*', ne, e)),
(e * ne, ('*', e, ne)),
(ne / e, ('/', ne, e)),
(e / ne, ('/', e, ne)),
(ne // e, ('//', ne, e)),
(e // ne, ('//', e, ne)),
(ne % e, ('%', ne, e)),
(e % ne, ('%', e, ne)),
(divmod(ne, e), ('divmod', ne, e)),
(divmod(e, ne), ('divmod', e, ne)),
(ne ** e, ('**', ne, e)),
(e ** ne, ('**', e, ne)),
(e < ne, ('>', ne, e)),
(ne < e, ('<', ne, e)),
(e > ne, ('<', ne, e)),
(ne > e, ('>', ne, e)),
(e <= ne, ('>=', ne, e)),
(ne <= e, ('<=', ne, e)),
(e >= ne, ('<=', ne, e)),
(ne >= e, ('>=', ne, e)),
]
for res, args in results:
assert type(res) is SpecialOp and res.args == args
# These classes do not support binary operators with Expr. Every
# operation should raise in combination with any of the Expr types.
for na in [NonArithmetic(), object()]:
raises(TypeError, lambda : e + na)
raises(TypeError, lambda : na + e)
raises(TypeError, lambda : e - na)
raises(TypeError, lambda : na - e)
raises(TypeError, lambda : e * na)
raises(TypeError, lambda : na * e)
raises(TypeError, lambda : e / na)
raises(TypeError, lambda : na / e)
raises(TypeError, lambda : e // na)
raises(TypeError, lambda : na // e)
raises(TypeError, lambda : e % na)
raises(TypeError, lambda : na % e)
raises(TypeError, lambda : divmod(e, na))
raises(TypeError, lambda : divmod(na, e))
raises(TypeError, lambda : e ** na)
raises(TypeError, lambda : na ** e)
raises(TypeError, lambda : e > na)
raises(TypeError, lambda : na > e)
raises(TypeError, lambda : e < na)
raises(TypeError, lambda : na < e)
raises(TypeError, lambda : e >= na)
raises(TypeError, lambda : na >= e)
raises(TypeError, lambda : e <= na)
raises(TypeError, lambda : na <= e)
def test_relational():
from sympy import Lt
assert (pi < 3) is S.false
assert (pi <= 3) is S.false
assert (pi > 3) is S.true
assert (pi >= 3) is S.true
assert (-pi < 3) is S.true
assert (-pi <= 3) is S.true
assert (-pi > 3) is S.false
assert (-pi >= 3) is S.false
r = Symbol('r', real=True)
assert (r - 2 < r - 3) is S.false
assert Lt(x + I, x + I + 2).func == Lt # issue 8288
def test_relational_assumptions():
from sympy import Lt, Gt, Le, Ge
m1 = Symbol("m1", nonnegative=False)
m2 = Symbol("m2", positive=False)
m3 = Symbol("m3", nonpositive=False)
m4 = Symbol("m4", negative=False)
assert (m1 < 0) == Lt(m1, 0)
assert (m2 <= 0) == Le(m2, 0)
assert (m3 > 0) == Gt(m3, 0)
assert (m4 >= 0) == Ge(m4, 0)
m1 = Symbol("m1", nonnegative=False, real=True)
m2 = Symbol("m2", positive=False, real=True)
m3 = Symbol("m3", nonpositive=False, real=True)
m4 = Symbol("m4", negative=False, real=True)
assert (m1 < 0) is S.true
assert (m2 <= 0) is S.true
assert (m3 > 0) is S.true
assert (m4 >= 0) is S.true
m1 = Symbol("m1", negative=True)
m2 = Symbol("m2", nonpositive=True)
m3 = Symbol("m3", positive=True)
m4 = Symbol("m4", nonnegative=True)
assert (m1 < 0) is S.true
assert (m2 <= 0) is S.true
assert (m3 > 0) is S.true
assert (m4 >= 0) is S.true
m1 = Symbol("m1", negative=False, real=True)
m2 = Symbol("m2", nonpositive=False, real=True)
m3 = Symbol("m3", positive=False, real=True)
m4 = Symbol("m4", nonnegative=False, real=True)
assert (m1 < 0) is S.false
assert (m2 <= 0) is S.false
assert (m3 > 0) is S.false
assert (m4 >= 0) is S.false
# See https://github.com/sympy/sympy/issues/17708
#def test_relational_noncommutative():
# from sympy import Lt, Gt, Le, Ge
# A, B = symbols('A,B', commutative=False)
# assert (A < B) == Lt(A, B)
# assert (A <= B) == Le(A, B)
# assert (A > B) == Gt(A, B)
# assert (A >= B) == Ge(A, B)
def test_basic_nostr():
for obj in basic_objs:
raises(TypeError, lambda: obj + '1')
raises(TypeError, lambda: obj - '1')
if obj == 2:
assert obj * '1' == '11'
else:
raises(TypeError, lambda: obj * '1')
raises(TypeError, lambda: obj / '1')
raises(TypeError, lambda: obj ** '1')
def test_series_expansion_for_uniform_order():
assert (1/x + y + x).series(x, 0, 0) == 1/x + O(1, x)
assert (1/x + y + x).series(x, 0, 1) == 1/x + y + O(x)
assert (1/x + 1 + x).series(x, 0, 0) == 1/x + O(1, x)
assert (1/x + 1 + x).series(x, 0, 1) == 1/x + 1 + O(x)
assert (1/x + x).series(x, 0, 0) == 1/x + O(1, x)
assert (1/x + y + y*x + x).series(x, 0, 0) == 1/x + O(1, x)
assert (1/x + y + y*x + x).series(x, 0, 1) == 1/x + y + O(x)
def test_leadterm():
assert (3 + 2*x**(log(3)/log(2) - 1)).leadterm(x) == (3, 0)
assert (1/x**2 + 1 + x + x**2).leadterm(x)[1] == -2
assert (1/x + 1 + x + x**2).leadterm(x)[1] == -1
assert (x**2 + 1/x).leadterm(x)[1] == -1
assert (1 + x**2).leadterm(x)[1] == 0
assert (x + 1).leadterm(x)[1] == 0
assert (x + x**2).leadterm(x)[1] == 1
assert (x**2).leadterm(x)[1] == 2
def test_as_leading_term():
assert (3 + 2*x**(log(3)/log(2) - 1)).as_leading_term(x) == 3
assert (1/x**2 + 1 + x + x**2).as_leading_term(x) == 1/x**2
assert (1/x + 1 + x + x**2).as_leading_term(x) == 1/x
assert (x**2 + 1/x).as_leading_term(x) == 1/x
assert (1 + x**2).as_leading_term(x) == 1
assert (x + 1).as_leading_term(x) == 1
assert (x + x**2).as_leading_term(x) == x
assert (x**2).as_leading_term(x) == x**2
assert (x + oo).as_leading_term(x) is oo
raises(ValueError, lambda: (x + 1).as_leading_term(1))
def test_leadterm2():
assert (x*cos(1)*cos(1 + sin(1)) + sin(1 + sin(1))).leadterm(x) == \
(sin(1 + sin(1)), 0)
def test_leadterm3():
assert (y + z + x).leadterm(x) == (y + z, 0)
def test_as_leading_term2():
assert (x*cos(1)*cos(1 + sin(1)) + sin(1 + sin(1))).as_leading_term(x) == \
sin(1 + sin(1))
def test_as_leading_term3():
assert (2 + pi + x).as_leading_term(x) == 2 + pi
assert (2*x + pi*x + x**2).as_leading_term(x) == (2 + pi)*x
def test_as_leading_term4():
# see issue 6843
n = Symbol('n', integer=True, positive=True)
r = -n**3/(2*n**2 + 4*n + 2) - n**2/(n**2 + 2*n + 1) + \
n**2/(n + 1) - n/(2*n**2 + 4*n + 2) + n/(n*x + x) + 2*n/(n + 1) - \
1 + 1/(n*x + x) + 1/(n + 1) - 1/x
assert r.as_leading_term(x).cancel() == n/2
def test_as_leading_term_stub():
class foo(Function):
pass
assert foo(1/x).as_leading_term(x) == foo(1/x)
assert foo(1).as_leading_term(x) == foo(1)
raises(NotImplementedError, lambda: foo(x).as_leading_term(x))
def test_as_leading_term_deriv_integral():
# related to issue 11313
assert Derivative(x ** 3, x).as_leading_term(x) == 3*x**2
assert Derivative(x ** 3, y).as_leading_term(x) == 0
assert Integral(x ** 3, x).as_leading_term(x) == x**4/4
assert Integral(x ** 3, y).as_leading_term(x) == y*x**3
assert Derivative(exp(x), x).as_leading_term(x) == 1
assert Derivative(log(x), x).as_leading_term(x) == (1/x).as_leading_term(x)
def test_atoms():
assert x.atoms() == {x}
assert (1 + x).atoms() == {x, S.One}
assert (1 + 2*cos(x)).atoms(Symbol) == {x}
assert (1 + 2*cos(x)).atoms(Symbol, Number) == {S.One, S(2), x}
assert (2*(x**(y**x))).atoms() == {S(2), x, y}
assert S.Half.atoms() == {S.Half}
assert S.Half.atoms(Symbol) == set()
assert sin(oo).atoms(oo) == set()
assert Poly(0, x).atoms() == {S.Zero, x}
assert Poly(1, x).atoms() == {S.One, x}
assert Poly(x, x).atoms() == {x}
assert Poly(x, x, y).atoms() == {x, y}
assert Poly(x + y, x, y).atoms() == {x, y}
assert Poly(x + y, x, y, z).atoms() == {x, y, z}
assert Poly(x + y*t, x, y, z).atoms() == {t, x, y, z}
assert (I*pi).atoms(NumberSymbol) == {pi}
assert (I*pi).atoms(NumberSymbol, I) == \
(I*pi).atoms(I, NumberSymbol) == {pi, I}
assert exp(exp(x)).atoms(exp) == {exp(exp(x)), exp(x)}
assert (1 + x*(2 + y) + exp(3 + z)).atoms(Add) == \
{1 + x*(2 + y) + exp(3 + z), 2 + y, 3 + z}
# issue 6132
f = Function('f')
e = (f(x) + sin(x) + 2)
assert e.atoms(AppliedUndef) == \
{f(x)}
assert e.atoms(AppliedUndef, Function) == \
{f(x), sin(x)}
assert e.atoms(Function) == \
{f(x), sin(x)}
assert e.atoms(AppliedUndef, Number) == \
{f(x), S(2)}
assert e.atoms(Function, Number) == \
{S(2), sin(x), f(x)}
def test_is_polynomial():
k = Symbol('k', nonnegative=True, integer=True)
assert Rational(2).is_polynomial(x, y, z) is True
assert (S.Pi).is_polynomial(x, y, z) is True
assert x.is_polynomial(x) is True
assert x.is_polynomial(y) is True
assert (x**2).is_polynomial(x) is True
assert (x**2).is_polynomial(y) is True
assert (x**(-2)).is_polynomial(x) is False
assert (x**(-2)).is_polynomial(y) is True
assert (2**x).is_polynomial(x) is False
assert (2**x).is_polynomial(y) is True
assert (x**k).is_polynomial(x) is False
assert (x**k).is_polynomial(k) is False
assert (x**x).is_polynomial(x) is False
assert (k**k).is_polynomial(k) is False
assert (k**x).is_polynomial(k) is False
assert (x**(-k)).is_polynomial(x) is False
assert ((2*x)**k).is_polynomial(x) is False
assert (x**2 + 3*x - 8).is_polynomial(x) is True
assert (x**2 + 3*x - 8).is_polynomial(y) is True
assert (x**2 + 3*x - 8).is_polynomial() is True
assert sqrt(x).is_polynomial(x) is False
assert (sqrt(x)**3).is_polynomial(x) is False
assert (x**2 + 3*x*sqrt(y) - 8).is_polynomial(x) is True
assert (x**2 + 3*x*sqrt(y) - 8).is_polynomial(y) is False
assert ((x**2)*(y**2) + x*(y**2) + y*x + exp(2)).is_polynomial() is True
assert ((x**2)*(y**2) + x*(y**2) + y*x + exp(x)).is_polynomial() is False
assert (
(x**2)*(y**2) + x*(y**2) + y*x + exp(2)).is_polynomial(x, y) is True
assert (
(x**2)*(y**2) + x*(y**2) + y*x + exp(x)).is_polynomial(x, y) is False
def test_is_rational_function():
assert Integer(1).is_rational_function() is True
assert Integer(1).is_rational_function(x) is True
assert Rational(17, 54).is_rational_function() is True
assert Rational(17, 54).is_rational_function(x) is True
assert (12/x).is_rational_function() is True
assert (12/x).is_rational_function(x) is True
assert (x/y).is_rational_function() is True
assert (x/y).is_rational_function(x) is True
assert (x/y).is_rational_function(x, y) is True
assert (x**2 + 1/x/y).is_rational_function() is True
assert (x**2 + 1/x/y).is_rational_function(x) is True
assert (x**2 + 1/x/y).is_rational_function(x, y) is True
assert (sin(y)/x).is_rational_function() is False
assert (sin(y)/x).is_rational_function(y) is False
assert (sin(y)/x).is_rational_function(x) is True
assert (sin(y)/x).is_rational_function(x, y) is False
assert (S.NaN).is_rational_function() is False
assert (S.Infinity).is_rational_function() is False
assert (S.NegativeInfinity).is_rational_function() is False
assert (S.ComplexInfinity).is_rational_function() is False
def test_is_meromorphic():
f = a/x**2 + b + x + c*x**2
assert f.is_meromorphic(x, 0) is True
assert f.is_meromorphic(x, 1) is True
assert f.is_meromorphic(x, zoo) is True
g = 3 + 2*x**(log(3)/log(2) - 1)
assert g.is_meromorphic(x, 0) is False
assert g.is_meromorphic(x, 1) is True
assert g.is_meromorphic(x, zoo) is False
n = Symbol('n', integer=True)
h = sin(1/x)**n*x
assert h.is_meromorphic(x, 0) is False
assert h.is_meromorphic(x, 1) is True
assert h.is_meromorphic(x, zoo) is False
e = log(x)**pi
assert e.is_meromorphic(x, 0) is False
assert e.is_meromorphic(x, 1) is False
assert e.is_meromorphic(x, 2) is True
assert e.is_meromorphic(x, zoo) is False
assert (log(x)**a).is_meromorphic(x, 0) is False
assert (log(x)**a).is_meromorphic(x, 1) is False
assert (a**log(x)).is_meromorphic(x, 0) is None
assert (3**log(x)).is_meromorphic(x, 0) is False
assert (3**log(x)).is_meromorphic(x, 1) is True
def test_is_algebraic_expr():
assert sqrt(3).is_algebraic_expr(x) is True
assert sqrt(3).is_algebraic_expr() is True
eq = ((1 + x**2)/(1 - y**2))**(S.One/3)
assert eq.is_algebraic_expr(x) is True
assert eq.is_algebraic_expr(y) is True
assert (sqrt(x) + y**(S(2)/3)).is_algebraic_expr(x) is True
assert (sqrt(x) + y**(S(2)/3)).is_algebraic_expr(y) is True
assert (sqrt(x) + y**(S(2)/3)).is_algebraic_expr() is True
assert (cos(y)/sqrt(x)).is_algebraic_expr() is False
assert (cos(y)/sqrt(x)).is_algebraic_expr(x) is True
assert (cos(y)/sqrt(x)).is_algebraic_expr(y) is False
assert (cos(y)/sqrt(x)).is_algebraic_expr(x, y) is False
def test_SAGE1():
#see https://github.com/sympy/sympy/issues/3346
class MyInt:
def _sympy_(self):
return Integer(5)
m = MyInt()
e = Rational(2)*m
assert e == 10
raises(TypeError, lambda: Rational(2)*MyInt)
def test_SAGE2():
class MyInt:
def __int__(self):
return 5
assert sympify(MyInt()) == 5
e = Rational(2)*MyInt()
assert e == 10
raises(TypeError, lambda: Rational(2)*MyInt)
def test_SAGE3():
class MySymbol:
def __rmul__(self, other):
return ('mys', other, self)
o = MySymbol()
e = x*o
assert e == ('mys', x, o)
def test_len():
e = x*y
assert len(e.args) == 2
e = x + y + z
assert len(e.args) == 3
def test_doit():
a = Integral(x**2, x)
assert isinstance(a.doit(), Integral) is False
assert isinstance(a.doit(integrals=True), Integral) is False
assert isinstance(a.doit(integrals=False), Integral) is True
assert (2*Integral(x, x)).doit() == x**2
def test_attribute_error():
raises(AttributeError, lambda: x.cos())
raises(AttributeError, lambda: x.sin())
raises(AttributeError, lambda: x.exp())
def test_args():
assert (x*y).args in ((x, y), (y, x))
assert (x + y).args in ((x, y), (y, x))
assert (x*y + 1).args in ((x*y, 1), (1, x*y))
assert sin(x*y).args == (x*y,)
assert sin(x*y).args[0] == x*y
assert (x**y).args == (x, y)
assert (x**y).args[0] == x
assert (x**y).args[1] == y
def test_noncommutative_expand_issue_3757():
A, B, C = symbols('A,B,C', commutative=False)
assert A*B - B*A != 0
assert (A*(A + B)*B).expand() == A**2*B + A*B**2
assert (A*(A + B + C)*B).expand() == A**2*B + A*B**2 + A*C*B
def test_as_numer_denom():
a, b, c = symbols('a, b, c')
assert nan.as_numer_denom() == (nan, 1)
assert oo.as_numer_denom() == (oo, 1)
assert (-oo).as_numer_denom() == (-oo, 1)
assert zoo.as_numer_denom() == (zoo, 1)
assert (-zoo).as_numer_denom() == (zoo, 1)
assert x.as_numer_denom() == (x, 1)
assert (1/x).as_numer_denom() == (1, x)
assert (x/y).as_numer_denom() == (x, y)
assert (x/2).as_numer_denom() == (x, 2)
assert (x*y/z).as_numer_denom() == (x*y, z)
assert (x/(y*z)).as_numer_denom() == (x, y*z)
assert S.Half.as_numer_denom() == (1, 2)
assert (1/y**2).as_numer_denom() == (1, y**2)
assert (x/y**2).as_numer_denom() == (x, y**2)
assert ((x**2 + 1)/y).as_numer_denom() == (x**2 + 1, y)
assert (x*(y + 1)/y**7).as_numer_denom() == (x*(y + 1), y**7)
assert (x**-2).as_numer_denom() == (1, x**2)
assert (a/x + b/2/x + c/3/x).as_numer_denom() == \
(6*a + 3*b + 2*c, 6*x)
assert (a/x + b/2/x + c/3/y).as_numer_denom() == \
(2*c*x + y*(6*a + 3*b), 6*x*y)
assert (a/x + b/2/x + c/.5/x).as_numer_denom() == \
(2*a + b + 4.0*c, 2*x)
# this should take no more than a few seconds
assert int(log(Add(*[Dummy()/i/x for i in range(1, 705)]
).as_numer_denom()[1]/x).n(4)) == 705
for i in [S.Infinity, S.NegativeInfinity, S.ComplexInfinity]:
assert (i + x/3).as_numer_denom() == \
(x + i, 3)
assert (S.Infinity + x/3 + y/4).as_numer_denom() == \
(4*x + 3*y + S.Infinity, 12)
assert (oo*x + zoo*y).as_numer_denom() == \
(zoo*y + oo*x, 1)
A, B, C = symbols('A,B,C', commutative=False)
assert (A*B*C**-1).as_numer_denom() == (A*B*C**-1, 1)
assert (A*B*C**-1/x).as_numer_denom() == (A*B*C**-1, x)
assert (C**-1*A*B).as_numer_denom() == (C**-1*A*B, 1)
assert (C**-1*A*B/x).as_numer_denom() == (C**-1*A*B, x)
assert ((A*B*C)**-1).as_numer_denom() == ((A*B*C)**-1, 1)
assert ((A*B*C)**-1/x).as_numer_denom() == ((A*B*C)**-1, x)
def test_trunc():
import math
x, y = symbols('x y')
assert math.trunc(2) == 2
assert math.trunc(4.57) == 4
assert math.trunc(-5.79) == -5
assert math.trunc(pi) == 3
assert math.trunc(log(7)) == 1
assert math.trunc(exp(5)) == 148
assert math.trunc(cos(pi)) == -1
assert math.trunc(sin(5)) == 0
raises(TypeError, lambda: math.trunc(x))
raises(TypeError, lambda: math.trunc(x + y**2))
raises(TypeError, lambda: math.trunc(oo))
def test_as_independent():
assert S.Zero.as_independent(x, as_Add=True) == (0, 0)
assert S.Zero.as_independent(x, as_Add=False) == (0, 0)
assert (2*x*sin(x) + y + x).as_independent(x) == (y, x + 2*x*sin(x))
assert (2*x*sin(x) + y + x).as_independent(y) == (x + 2*x*sin(x), y)
assert (2*x*sin(x) + y + x).as_independent(x, y) == (0, y + x + 2*x*sin(x))
assert (x*sin(x)*cos(y)).as_independent(x) == (cos(y), x*sin(x))
assert (x*sin(x)*cos(y)).as_independent(y) == (x*sin(x), cos(y))
assert (x*sin(x)*cos(y)).as_independent(x, y) == (1, x*sin(x)*cos(y))
assert (sin(x)).as_independent(x) == (1, sin(x))
assert (sin(x)).as_independent(y) == (sin(x), 1)
assert (2*sin(x)).as_independent(x) == (2, sin(x))
assert (2*sin(x)).as_independent(y) == (2*sin(x), 1)
# issue 4903 = 1766b
n1, n2, n3 = symbols('n1 n2 n3', commutative=False)
assert (n1 + n1*n2).as_independent(n2) == (n1, n1*n2)
assert (n2*n1 + n1*n2).as_independent(n2) == (0, n1*n2 + n2*n1)
assert (n1*n2*n1).as_independent(n2) == (n1, n2*n1)
assert (n1*n2*n1).as_independent(n1) == (1, n1*n2*n1)
assert (3*x).as_independent(x, as_Add=True) == (0, 3*x)
assert (3*x).as_independent(x, as_Add=False) == (3, x)
assert (3 + x).as_independent(x, as_Add=True) == (3, x)
assert (3 + x).as_independent(x, as_Add=False) == (1, 3 + x)
# issue 5479
assert (3*x).as_independent(Symbol) == (3, x)
# issue 5648
assert (n1*x*y).as_independent(x) == (n1*y, x)
assert ((x + n1)*(x - y)).as_independent(x) == (1, (x + n1)*(x - y))
assert ((x + n1)*(x - y)).as_independent(y) == (x + n1, x - y)
assert (DiracDelta(x - n1)*DiracDelta(x - y)).as_independent(x) \
== (1, DiracDelta(x - n1)*DiracDelta(x - y))
assert (x*y*n1*n2*n3).as_independent(n2) == (x*y*n1, n2*n3)
assert (x*y*n1*n2*n3).as_independent(n1) == (x*y, n1*n2*n3)
assert (x*y*n1*n2*n3).as_independent(n3) == (x*y*n1*n2, n3)
assert (DiracDelta(x - n1)*DiracDelta(y - n1)*DiracDelta(x - n2)).as_independent(y) == \
(DiracDelta(x - n1)*DiracDelta(x - n2), DiracDelta(y - n1))
# issue 5784
assert (x + Integral(x, (x, 1, 2))).as_independent(x, strict=True) == \
(Integral(x, (x, 1, 2)), x)
eq = Add(x, -x, 2, -3, evaluate=False)
assert eq.as_independent(x) == (-1, Add(x, -x, evaluate=False))
eq = Mul(x, 1/x, 2, -3, evaluate=False)
eq.as_independent(x) == (-6, Mul(x, 1/x, evaluate=False))
assert (x*y).as_independent(z, as_Add=True) == (x*y, 0)
@XFAIL
def test_call_2():
# TODO UndefinedFunction does not subclass Expr
f = Function('f')
assert (2*f)(x) == 2*f(x)
def test_replace():
f = log(sin(x)) + tan(sin(x**2))
assert f.replace(sin, cos) == log(cos(x)) + tan(cos(x**2))
assert f.replace(
sin, lambda a: sin(2*a)) == log(sin(2*x)) + tan(sin(2*x**2))
a = Wild('a')
b = Wild('b')
assert f.replace(sin(a), cos(a)) == log(cos(x)) + tan(cos(x**2))
assert f.replace(
sin(a), lambda a: sin(2*a)) == log(sin(2*x)) + tan(sin(2*x**2))
# test exact
assert (2*x).replace(a*x + b, b - a, exact=True) == 2*x
assert (2*x).replace(a*x + b, b - a) == 2*x
assert (2*x).replace(a*x + b, b - a, exact=False) == 2/x
assert (2*x).replace(a*x + b, lambda a, b: b - a, exact=True) == 2*x
assert (2*x).replace(a*x + b, lambda a, b: b - a) == 2*x
assert (2*x).replace(a*x + b, lambda a, b: b - a, exact=False) == 2/x
g = 2*sin(x**3)
assert g.replace(
lambda expr: expr.is_Number, lambda expr: expr**2) == 4*sin(x**9)
assert cos(x).replace(cos, sin, map=True) == (sin(x), {cos(x): sin(x)})
assert sin(x).replace(cos, sin) == sin(x)
cond, func = lambda x: x.is_Mul, lambda x: 2*x
assert (x*y).replace(cond, func, map=True) == (2*x*y, {x*y: 2*x*y})
assert (x*(1 + x*y)).replace(cond, func, map=True) == \
(2*x*(2*x*y + 1), {x*(2*x*y + 1): 2*x*(2*x*y + 1), x*y: 2*x*y})
assert (y*sin(x)).replace(sin, lambda expr: sin(expr)/y, map=True) == \
(sin(x), {sin(x): sin(x)/y})
# if not simultaneous then y*sin(x) -> y*sin(x)/y = sin(x) -> sin(x)/y
assert (y*sin(x)).replace(sin, lambda expr: sin(expr)/y,
simultaneous=False) == sin(x)/y
assert (x**2 + O(x**3)).replace(Pow, lambda b, e: b**e/e
) == x**2/2 + O(x**3)
assert (x**2 + O(x**3)).replace(Pow, lambda b, e: b**e/e,
simultaneous=False) == x**2/2 + O(x**3)
assert (x*(x*y + 3)).replace(lambda x: x.is_Mul, lambda x: 2 + x) == \
x*(x*y + 5) + 2
e = (x*y + 1)*(2*x*y + 1) + 1
assert e.replace(cond, func, map=True) == (
2*((2*x*y + 1)*(4*x*y + 1)) + 1,
{2*x*y: 4*x*y, x*y: 2*x*y, (2*x*y + 1)*(4*x*y + 1):
2*((2*x*y + 1)*(4*x*y + 1))})
assert x.replace(x, y) == y
assert (x + 1).replace(1, 2) == x + 2
# https://groups.google.com/forum/#!topic/sympy/8wCgeC95tz0
n1, n2, n3 = symbols('n1:4', commutative=False)
f = Function('f')
assert (n1*f(n2)).replace(f, lambda x: x) == n1*n2
assert (n3*f(n2)).replace(f, lambda x: x) == n3*n2
# issue 16725
assert S.Zero.replace(Wild('x'), 1) == 1
# let the user override the default decision of False
assert S.Zero.replace(Wild('x'), 1, exact=True) == 0
def test_find():
expr = (x + y + 2 + sin(3*x))
assert expr.find(lambda u: u.is_Integer) == {S(2), S(3)}
assert expr.find(lambda u: u.is_Symbol) == {x, y}
assert expr.find(lambda u: u.is_Integer, group=True) == {S(2): 1, S(3): 1}
assert expr.find(lambda u: u.is_Symbol, group=True) == {x: 2, y: 1}
assert expr.find(Integer) == {S(2), S(3)}
assert expr.find(Symbol) == {x, y}
assert expr.find(Integer, group=True) == {S(2): 1, S(3): 1}
assert expr.find(Symbol, group=True) == {x: 2, y: 1}
a = Wild('a')
expr = sin(sin(x)) + sin(x) + cos(x) + x
assert expr.find(lambda u: type(u) is sin) == {sin(x), sin(sin(x))}
assert expr.find(
lambda u: type(u) is sin, group=True) == {sin(x): 2, sin(sin(x)): 1}
assert expr.find(sin(a)) == {sin(x), sin(sin(x))}
assert expr.find(sin(a), group=True) == {sin(x): 2, sin(sin(x)): 1}
assert expr.find(sin) == {sin(x), sin(sin(x))}
assert expr.find(sin, group=True) == {sin(x): 2, sin(sin(x)): 1}
def test_count():
expr = (x + y + 2 + sin(3*x))
assert expr.count(lambda u: u.is_Integer) == 2
assert expr.count(lambda u: u.is_Symbol) == 3
assert expr.count(Integer) == 2
assert expr.count(Symbol) == 3
assert expr.count(2) == 1
a = Wild('a')
assert expr.count(sin) == 1
assert expr.count(sin(a)) == 1
assert expr.count(lambda u: type(u) is sin) == 1
f = Function('f')
assert f(x).count(f(x)) == 1
assert f(x).diff(x).count(f(x)) == 1
assert f(x).diff(x).count(x) == 2
def test_has_basics():
f = Function('f')
g = Function('g')
p = Wild('p')
assert sin(x).has(x)
assert sin(x).has(sin)
assert not sin(x).has(y)
assert not sin(x).has(cos)
assert f(x).has(x)
assert f(x).has(f)
assert not f(x).has(y)
assert not f(x).has(g)
assert f(x).diff(x).has(x)
assert f(x).diff(x).has(f)
assert f(x).diff(x).has(Derivative)
assert not f(x).diff(x).has(y)
assert not f(x).diff(x).has(g)
assert not f(x).diff(x).has(sin)
assert (x**2).has(Symbol)
assert not (x**2).has(Wild)
assert (2*p).has(Wild)
assert not x.has()
def test_has_multiple():
f = x**2*y + sin(2**t + log(z))
assert f.has(x)
assert f.has(y)
assert f.has(z)
assert f.has(t)
assert not f.has(u)
assert f.has(x, y, z, t)
assert f.has(x, y, z, t, u)
i = Integer(4400)
assert not i.has(x)
assert (i*x**i).has(x)
assert not (i*y**i).has(x)
assert (i*y**i).has(x, y)
assert not (i*y**i).has(x, z)
def test_has_piecewise():
f = (x*y + 3/y)**(3 + 2)
g = Function('g')
h = Function('h')
p = Piecewise((g(x), x < -1), (1, x <= 1), (f, True))
assert p.has(x)
assert p.has(y)
assert not p.has(z)
assert p.has(1)
assert p.has(3)
assert not p.has(4)
assert p.has(f)
assert p.has(g)
assert not p.has(h)
def test_has_iterative():
A, B, C = symbols('A,B,C', commutative=False)
f = x*gamma(x)*sin(x)*exp(x*y)*A*B*C*cos(x*A*B)
assert f.has(x)
assert f.has(x*y)
assert f.has(x*sin(x))
assert not f.has(x*sin(y))
assert f.has(x*A)
assert f.has(x*A*B)
assert not f.has(x*A*C)
assert f.has(x*A*B*C)
assert not f.has(x*A*C*B)
assert f.has(x*sin(x)*A*B*C)
assert not f.has(x*sin(x)*A*C*B)
assert not f.has(x*sin(y)*A*B*C)
assert f.has(x*gamma(x))
assert not f.has(x + sin(x))
assert (x & y & z).has(x & z)
def test_has_integrals():
f = Integral(x**2 + sin(x*y*z), (x, 0, x + y + z))
assert f.has(x + y)
assert f.has(x + z)
assert f.has(y + z)
assert f.has(x*y)
assert f.has(x*z)
assert f.has(y*z)
assert not f.has(2*x + y)
assert not f.has(2*x*y)
def test_has_tuple():
f = Function('f')
g = Function('g')
h = Function('h')
assert Tuple(x, y).has(x)
assert not Tuple(x, y).has(z)
assert Tuple(f(x), g(x)).has(x)
assert not Tuple(f(x), g(x)).has(y)
assert Tuple(f(x), g(x)).has(f)
assert Tuple(f(x), g(x)).has(f(x))
assert not Tuple(f, g).has(x)
assert Tuple(f, g).has(f)
assert not Tuple(f, g).has(h)
assert Tuple(True).has(True) is True # .has(1) will also be True
def test_has_units():
from sympy.physics.units import m, s
assert (x*m/s).has(x)
assert (x*m/s).has(y, z) is False
def test_has_polys():
poly = Poly(x**2 + x*y*sin(z), x, y, t)
assert poly.has(x)
assert poly.has(x, y, z)
assert poly.has(x, y, z, t)
def test_has_physics():
assert FockState((x, y)).has(x)
def test_as_poly_as_expr():
f = x**2 + 2*x*y
assert f.as_poly().as_expr() == f
assert f.as_poly(x, y).as_expr() == f
assert (f + sin(x)).as_poly(x, y) is None
p = Poly(f, x, y)
assert p.as_poly() == p
raises(AttributeError, lambda: Tuple(x, x).as_poly(x))
raises(AttributeError, lambda: Tuple(x ** 2, x, y).as_poly(x))
def test_nonzero():
assert bool(S.Zero) is False
assert bool(S.One) is True
assert bool(x) is True
assert bool(x + y) is True
assert bool(x - x) is False
assert bool(x*y) is True
assert bool(x*1) is True
assert bool(x*0) is False
def test_is_number():
assert Float(3.14).is_number is True
assert Integer(737).is_number is True
assert Rational(3, 2).is_number is True
assert Rational(8).is_number is True
assert x.is_number is False
assert (2*x).is_number is False
assert (x + y).is_number is False
assert log(2).is_number is True
assert log(x).is_number is False
assert (2 + log(2)).is_number is True
assert (8 + log(2)).is_number is True
assert (2 + log(x)).is_number is False
assert (8 + log(2) + x).is_number is False
assert (1 + x**2/x - x).is_number is True
assert Tuple(Integer(1)).is_number is False
assert Add(2, x).is_number is False
assert Mul(3, 4).is_number is True
assert Pow(log(2), 2).is_number is True
assert oo.is_number is True
g = WildFunction('g')
assert g.is_number is False
assert (2*g).is_number is False
assert (x**2).subs(x, 3).is_number is True
# test extensibility of .is_number
# on subinstances of Basic
class A(Basic):
pass
a = A()
assert a.is_number is False
def test_as_coeff_add():
assert S(2).as_coeff_add() == (2, ())
assert S(3.0).as_coeff_add() == (0, (S(3.0),))
assert S(-3.0).as_coeff_add() == (0, (S(-3.0),))
assert x.as_coeff_add() == (0, (x,))
assert (x - 1).as_coeff_add() == (-1, (x,))
assert (x + 1).as_coeff_add() == (1, (x,))
assert (x + 2).as_coeff_add() == (2, (x,))
assert (x + y).as_coeff_add(y) == (x, (y,))
assert (3*x).as_coeff_add(y) == (3*x, ())
# don't do expansion
e = (x + y)**2
assert e.as_coeff_add(y) == (0, (e,))
def test_as_coeff_mul():
assert S(2).as_coeff_mul() == (2, ())
assert S(3.0).as_coeff_mul() == (1, (S(3.0),))
assert S(-3.0).as_coeff_mul() == (-1, (S(3.0),))
assert S(-3.0).as_coeff_mul(rational=False) == (-S(3.0), ())
assert x.as_coeff_mul() == (1, (x,))
assert (-x).as_coeff_mul() == (-1, (x,))
assert (2*x).as_coeff_mul() == (2, (x,))
assert (x*y).as_coeff_mul(y) == (x, (y,))
assert (3 + x).as_coeff_mul() == (1, (3 + x,))
assert (3 + x).as_coeff_mul(y) == (3 + x, ())
# don't do expansion
e = exp(x + y)
assert e.as_coeff_mul(y) == (1, (e,))
e = 2**(x + y)
assert e.as_coeff_mul(y) == (1, (e,))
assert (1.1*x).as_coeff_mul(rational=False) == (1.1, (x,))
assert (1.1*x).as_coeff_mul() == (1, (1.1, x))
assert (-oo*x).as_coeff_mul(rational=True) == (-1, (oo, x))
def test_as_coeff_exponent():
assert (3*x**4).as_coeff_exponent(x) == (3, 4)
assert (2*x**3).as_coeff_exponent(x) == (2, 3)
assert (4*x**2).as_coeff_exponent(x) == (4, 2)
assert (6*x**1).as_coeff_exponent(x) == (6, 1)
assert (3*x**0).as_coeff_exponent(x) == (3, 0)
assert (2*x**0).as_coeff_exponent(x) == (2, 0)
assert (1*x**0).as_coeff_exponent(x) == (1, 0)
assert (0*x**0).as_coeff_exponent(x) == (0, 0)
assert (-1*x**0).as_coeff_exponent(x) == (-1, 0)
assert (-2*x**0).as_coeff_exponent(x) == (-2, 0)
assert (2*x**3 + pi*x**3).as_coeff_exponent(x) == (2 + pi, 3)
assert (x*log(2)/(2*x + pi*x)).as_coeff_exponent(x) == \
(log(2)/(2 + pi), 0)
# issue 4784
D = Derivative
f = Function('f')
fx = D(f(x), x)
assert fx.as_coeff_exponent(f(x)) == (fx, 0)
def test_extractions():
assert ((x*y)**3).extract_multiplicatively(x**2 * y) == x*y**2
assert ((x*y)**3).extract_multiplicatively(x**4 * y) is None
assert (2*x).extract_multiplicatively(2) == x
assert (2*x).extract_multiplicatively(3) is None
assert (2*x).extract_multiplicatively(-1) is None
assert (S.Half*x).extract_multiplicatively(3) == x/6
assert (sqrt(x)).extract_multiplicatively(x) is None
assert (sqrt(x)).extract_multiplicatively(1/x) is None
assert x.extract_multiplicatively(-x) is None
assert (-2 - 4*I).extract_multiplicatively(-2) == 1 + 2*I
assert (-2 - 4*I).extract_multiplicatively(3) is None
assert (-2*x - 4*y - 8).extract_multiplicatively(-2) == x + 2*y + 4
assert (-2*x*y - 4*x**2*y).extract_multiplicatively(-2*y) == 2*x**2 + x
assert (2*x*y + 4*x**2*y).extract_multiplicatively(2*y) == 2*x**2 + x
assert (-4*y**2*x).extract_multiplicatively(-3*y) is None
assert (2*x).extract_multiplicatively(1) == 2*x
assert (-oo).extract_multiplicatively(5) is -oo
assert (oo).extract_multiplicatively(5) is oo
assert ((x*y)**3).extract_additively(1) is None
assert (x + 1).extract_additively(x) == 1
assert (x + 1).extract_additively(2*x) is None
assert (x + 1).extract_additively(-x) is None
assert (-x + 1).extract_additively(2*x) is None
assert (2*x + 3).extract_additively(x) == x + 3
assert (2*x + 3).extract_additively(2) == 2*x + 1
assert (2*x + 3).extract_additively(3) == 2*x
assert (2*x + 3).extract_additively(-2) is None
assert (2*x + 3).extract_additively(3*x) is None
assert (2*x + 3).extract_additively(2*x) == 3
assert x.extract_additively(0) == x
assert S(2).extract_additively(x) is None
assert S(2.).extract_additively(2) is S.Zero
assert S(2*x + 3).extract_additively(x + 1) == x + 2
assert S(2*x + 3).extract_additively(y + 1) is None
assert S(2*x - 3).extract_additively(x + 1) is None
assert S(2*x - 3).extract_additively(y + z) is None
assert ((a + 1)*x*4 + y).extract_additively(x).expand() == \
4*a*x + 3*x + y
assert ((a + 1)*x*4 + 3*y).extract_additively(x + 2*y).expand() == \
4*a*x + 3*x + y
assert (y*(x + 1)).extract_additively(x + 1) is None
assert ((y + 1)*(x + 1) + 3).extract_additively(x + 1) == \
y*(x + 1) + 3
assert ((x + y)*(x + 1) + x + y + 3).extract_additively(x + y) == \
x*(x + y) + 3
assert (x + y + 2*((x + y)*(x + 1)) + 3).extract_additively((x + y)*(x + 1)) == \
x + y + (x + 1)*(x + y) + 3
assert ((y + 1)*(x + 2*y + 1) + 3).extract_additively(y + 1) == \
(x + 2*y)*(y + 1) + 3
n = Symbol("n", integer=True)
assert (Integer(-3)).could_extract_minus_sign() is True
assert (-n*x + x).could_extract_minus_sign() != \
(n*x - x).could_extract_minus_sign()
assert (x - y).could_extract_minus_sign() != \
(-x + y).could_extract_minus_sign()
assert (1 - x - y).could_extract_minus_sign() is True
assert (1 - x + y).could_extract_minus_sign() is False
assert ((-x - x*y)/y).could_extract_minus_sign() is True
assert (-(x + x*y)/y).could_extract_minus_sign() is True
assert ((x + x*y)/(-y)).could_extract_minus_sign() is True
assert ((x + x*y)/y).could_extract_minus_sign() is False
assert (x*(-x - x**3)).could_extract_minus_sign() is True
assert ((-x - y)/(x + y)).could_extract_minus_sign() is True
class sign_invariant(Function, Expr):
nargs = 1
def __neg__(self):
return self
foo = sign_invariant(x)
assert foo == -foo
assert foo.could_extract_minus_sign() is False
# The results of each of these will vary on different machines, e.g.
# the first one might be False and the other (then) is true or vice versa,
# so both are included.
assert ((-x - y)/(x - y)).could_extract_minus_sign() is False or \
((-x - y)/(y - x)).could_extract_minus_sign() is False
assert (x - y).could_extract_minus_sign() is False
assert (-x + y).could_extract_minus_sign() is True
# check that result is canonical
eq = (3*x + 15*y).extract_multiplicatively(3)
assert eq.args == eq.func(*eq.args).args
def test_nan_extractions():
for r in (1, 0, I, nan):
assert nan.extract_additively(r) is None
assert nan.extract_multiplicatively(r) is None
def test_coeff():
assert (x + 1).coeff(x + 1) == 1
assert (3*x).coeff(0) == 0
assert (z*(1 + x)*x**2).coeff(1 + x) == z*x**2
assert (1 + 2*x*x**(1 + x)).coeff(x*x**(1 + x)) == 2
assert (1 + 2*x**(y + z)).coeff(x**(y + z)) == 2
assert (3 + 2*x + 4*x**2).coeff(1) == 0
assert (3 + 2*x + 4*x**2).coeff(-1) == 0
assert (3 + 2*x + 4*x**2).coeff(x) == 2
assert (3 + 2*x + 4*x**2).coeff(x**2) == 4
assert (3 + 2*x + 4*x**2).coeff(x**3) == 0
assert (-x/8 + x*y).coeff(x) == Rational(-1, 8) + y
assert (-x/8 + x*y).coeff(-x) == S.One/8
assert (4*x).coeff(2*x) == 0
assert (2*x).coeff(2*x) == 1
assert (-oo*x).coeff(x*oo) == -1
assert (10*x).coeff(x, 0) == 0
assert (10*x).coeff(10*x, 0) == 0
n1, n2 = symbols('n1 n2', commutative=False)
assert (n1*n2).coeff(n1) == 1
assert (n1*n2).coeff(n2) == n1
assert (n1*n2 + x*n1).coeff(n1) == 1 # 1*n1*(n2+x)
assert (n2*n1 + x*n1).coeff(n1) == n2 + x
assert (n2*n1 + x*n1**2).coeff(n1) == n2
assert (n1**x).coeff(n1) == 0
assert (n1*n2 + n2*n1).coeff(n1) == 0
assert (2*(n1 + n2)*n2).coeff(n1 + n2, right=1) == n2
assert (2*(n1 + n2)*n2).coeff(n1 + n2, right=0) == 2
f = Function('f')
assert (2*f(x) + 3*f(x).diff(x)).coeff(f(x)) == 2
expr = z*(x + y)**2
expr2 = z*(x + y)**2 + z*(2*x + 2*y)**2
assert expr.coeff(z) == (x + y)**2
assert expr.coeff(x + y) == 0
assert expr2.coeff(z) == (x + y)**2 + (2*x + 2*y)**2
assert (x + y + 3*z).coeff(1) == x + y
assert (-x + 2*y).coeff(-1) == x
assert (x - 2*y).coeff(-1) == 2*y
assert (3 + 2*x + 4*x**2).coeff(1) == 0
assert (-x - 2*y).coeff(2) == -y
assert (x + sqrt(2)*x).coeff(sqrt(2)) == x
assert (3 + 2*x + 4*x**2).coeff(x) == 2
assert (3 + 2*x + 4*x**2).coeff(x**2) == 4
assert (3 + 2*x + 4*x**2).coeff(x**3) == 0
assert (z*(x + y)**2).coeff((x + y)**2) == z
assert (z*(x + y)**2).coeff(x + y) == 0
assert (2 + 2*x + (x + 1)*y).coeff(x + 1) == y
assert (x + 2*y + 3).coeff(1) == x
assert (x + 2*y + 3).coeff(x, 0) == 2*y + 3
assert (x**2 + 2*y + 3*x).coeff(x**2, 0) == 2*y + 3*x
assert x.coeff(0, 0) == 0
assert x.coeff(x, 0) == 0
n, m, o, l = symbols('n m o l', commutative=False)
assert n.coeff(n) == 1
assert y.coeff(n) == 0
assert (3*n).coeff(n) == 3
assert (2 + n).coeff(x*m) == 0
assert (2*x*n*m).coeff(x) == 2*n*m
assert (2 + n).coeff(x*m*n + y) == 0
assert (2*x*n*m).coeff(3*n) == 0
assert (n*m + m*n*m).coeff(n) == 1 + m
assert (n*m + m*n*m).coeff(n, right=True) == m # = (1 + m)*n*m
assert (n*m + m*n).coeff(n) == 0
assert (n*m + o*m*n).coeff(m*n) == o
assert (n*m + o*m*n).coeff(m*n, right=1) == 1
assert (n*m + n*m*n).coeff(n*m, right=1) == 1 + n # = n*m*(n + 1)
assert (x*y).coeff(z, 0) == x*y
def test_coeff2():
r, kappa = symbols('r, kappa')
psi = Function("psi")
g = 1/r**2 * (2*r*psi(r).diff(r, 1) + r**2 * psi(r).diff(r, 2))
g = g.expand()
assert g.coeff(psi(r).diff(r)) == 2/r
def test_coeff2_0():
r, kappa = symbols('r, kappa')
psi = Function("psi")
g = 1/r**2 * (2*r*psi(r).diff(r, 1) + r**2 * psi(r).diff(r, 2))
g = g.expand()
assert g.coeff(psi(r).diff(r, 2)) == 1
def test_coeff_expand():
expr = z*(x + y)**2
expr2 = z*(x + y)**2 + z*(2*x + 2*y)**2
assert expr.coeff(z) == (x + y)**2
assert expr2.coeff(z) == (x + y)**2 + (2*x + 2*y)**2
def test_integrate():
assert x.integrate(x) == x**2/2
assert x.integrate((x, 0, 1)) == S.Half
def test_as_base_exp():
assert x.as_base_exp() == (x, S.One)
assert (x*y*z).as_base_exp() == (x*y*z, S.One)
assert (x + y + z).as_base_exp() == (x + y + z, S.One)
assert ((x + y)**z).as_base_exp() == (x + y, z)
def test_issue_4963():
assert hasattr(Mul(x, y), "is_commutative")
assert hasattr(Mul(x, y, evaluate=False), "is_commutative")
assert hasattr(Pow(x, y), "is_commutative")
assert hasattr(Pow(x, y, evaluate=False), "is_commutative")
expr = Mul(Pow(2, 2, evaluate=False), 3, evaluate=False) + 1
assert hasattr(expr, "is_commutative")
def test_action_verbs():
assert nsimplify(1/(exp(3*pi*x/5) + 1)) == \
(1/(exp(3*pi*x/5) + 1)).nsimplify()
assert ratsimp(1/x + 1/y) == (1/x + 1/y).ratsimp()
assert trigsimp(log(x), deep=True) == (log(x)).trigsimp(deep=True)
assert radsimp(1/(2 + sqrt(2))) == (1/(2 + sqrt(2))).radsimp()
assert radsimp(1/(a + b*sqrt(c)), symbolic=False) == \
(1/(a + b*sqrt(c))).radsimp(symbolic=False)
assert powsimp(x**y*x**z*y**z, combine='all') == \
(x**y*x**z*y**z).powsimp(combine='all')
assert (x**t*y**t).powsimp(force=True) == (x*y)**t
assert simplify(x**y*x**z*y**z) == (x**y*x**z*y**z).simplify()
assert together(1/x + 1/y) == (1/x + 1/y).together()
assert collect(a*x**2 + b*x**2 + a*x - b*x + c, x) == \
(a*x**2 + b*x**2 + a*x - b*x + c).collect(x)
assert apart(y/(y + 2)/(y + 1), y) == (y/(y + 2)/(y + 1)).apart(y)
assert combsimp(y/(x + 2)/(x + 1)) == (y/(x + 2)/(x + 1)).combsimp()
assert gammasimp(gamma(x)/gamma(x-5)) == (gamma(x)/gamma(x-5)).gammasimp()
assert factor(x**2 + 5*x + 6) == (x**2 + 5*x + 6).factor()
assert refine(sqrt(x**2)) == sqrt(x**2).refine()
assert cancel((x**2 + 5*x + 6)/(x + 2)) == ((x**2 + 5*x + 6)/(x + 2)).cancel()
def test_as_powers_dict():
assert x.as_powers_dict() == {x: 1}
assert (x**y*z).as_powers_dict() == {x: y, z: 1}
assert Mul(2, 2, evaluate=False).as_powers_dict() == {S(2): S(2)}
assert (x*y).as_powers_dict()[z] == 0
assert (x + y).as_powers_dict()[z] == 0
def test_as_coefficients_dict():
check = [S.One, x, y, x*y, 1]
assert [Add(3*x, 2*x, y, 3).as_coefficients_dict()[i] for i in check] == \
[3, 5, 1, 0, 3]
assert [Add(3*x, 2*x, y, 3, evaluate=False).as_coefficients_dict()[i]
for i in check] == [3, 5, 1, 0, 3]
assert [(3*x*y).as_coefficients_dict()[i] for i in check] == \
[0, 0, 0, 3, 0]
assert [(3.0*x*y).as_coefficients_dict()[i] for i in check] == \
[0, 0, 0, 3.0, 0]
assert (3.0*x*y).as_coefficients_dict()[3.0*x*y] == 0
def test_args_cnc():
A = symbols('A', commutative=False)
assert (x + A).args_cnc() == \
[[], [x + A]]
assert (x + a).args_cnc() == \
[[a + x], []]
assert (x*a).args_cnc() == \
[[a, x], []]
assert (x*y*A*(A + 1)).args_cnc(cset=True) == \
[{x, y}, [A, 1 + A]]
assert Mul(x, x, evaluate=False).args_cnc(cset=True, warn=False) == \
[{x}, []]
assert Mul(x, x**2, evaluate=False).args_cnc(cset=True, warn=False) == \
[{x, x**2}, []]
raises(ValueError, lambda: Mul(x, x, evaluate=False).args_cnc(cset=True))
assert Mul(x, y, x, evaluate=False).args_cnc() == \
[[x, y, x], []]
# always split -1 from leading number
assert (-1.*x).args_cnc() == [[-1, 1.0, x], []]
def test_new_rawargs():
n = Symbol('n', commutative=False)
a = x + n
assert a.is_commutative is False
assert a._new_rawargs(x).is_commutative
assert a._new_rawargs(x, y).is_commutative
assert a._new_rawargs(x, n).is_commutative is False
assert a._new_rawargs(x, y, n).is_commutative is False
m = x*n
assert m.is_commutative is False
assert m._new_rawargs(x).is_commutative
assert m._new_rawargs(n).is_commutative is False
assert m._new_rawargs(x, y).is_commutative
assert m._new_rawargs(x, n).is_commutative is False
assert m._new_rawargs(x, y, n).is_commutative is False
assert m._new_rawargs(x, n, reeval=False).is_commutative is False
assert m._new_rawargs(S.One) is S.One
def test_issue_5226():
assert Add(evaluate=False) == 0
assert Mul(evaluate=False) == 1
assert Mul(x + y, evaluate=False).is_Add
def test_free_symbols():
# free_symbols should return the free symbols of an object
assert S.One.free_symbols == set()
assert x.free_symbols == {x}
assert Integral(x, (x, 1, y)).free_symbols == {y}
assert (-Integral(x, (x, 1, y))).free_symbols == {y}
assert meter.free_symbols == set()
assert (meter**x).free_symbols == {x}
def test_issue_5300():
x = Symbol('x', commutative=False)
assert x*sqrt(2)/sqrt(6) == x*sqrt(3)/3
def test_floordiv():
from sympy.functions.elementary.integers import floor
assert x // y == floor(x / y)
def test_as_coeff_Mul():
assert S.Zero.as_coeff_Mul() == (S.One, S.Zero)
assert Integer(3).as_coeff_Mul() == (Integer(3), Integer(1))
assert Rational(3, 4).as_coeff_Mul() == (Rational(3, 4), Integer(1))
assert Float(5.0).as_coeff_Mul() == (Float(5.0), Integer(1))
assert (Integer(3)*x).as_coeff_Mul() == (Integer(3), x)
assert (Rational(3, 4)*x).as_coeff_Mul() == (Rational(3, 4), x)
assert (Float(5.0)*x).as_coeff_Mul() == (Float(5.0), x)
assert (Integer(3)*x*y).as_coeff_Mul() == (Integer(3), x*y)
assert (Rational(3, 4)*x*y).as_coeff_Mul() == (Rational(3, 4), x*y)
assert (Float(5.0)*x*y).as_coeff_Mul() == (Float(5.0), x*y)
assert (x).as_coeff_Mul() == (S.One, x)
assert (x*y).as_coeff_Mul() == (S.One, x*y)
assert (-oo*x).as_coeff_Mul(rational=True) == (-1, oo*x)
def test_as_coeff_Add():
assert Integer(3).as_coeff_Add() == (Integer(3), Integer(0))
assert Rational(3, 4).as_coeff_Add() == (Rational(3, 4), Integer(0))
assert Float(5.0).as_coeff_Add() == (Float(5.0), Integer(0))
assert (Integer(3) + x).as_coeff_Add() == (Integer(3), x)
assert (Rational(3, 4) + x).as_coeff_Add() == (Rational(3, 4), x)
assert (Float(5.0) + x).as_coeff_Add() == (Float(5.0), x)
assert (Float(5.0) + x).as_coeff_Add(rational=True) == (0, Float(5.0) + x)
assert (Integer(3) + x + y).as_coeff_Add() == (Integer(3), x + y)
assert (Rational(3, 4) + x + y).as_coeff_Add() == (Rational(3, 4), x + y)
assert (Float(5.0) + x + y).as_coeff_Add() == (Float(5.0), x + y)
assert (x).as_coeff_Add() == (S.Zero, x)
assert (x*y).as_coeff_Add() == (S.Zero, x*y)
def test_expr_sorting():
f, g = symbols('f,g', cls=Function)
exprs = [1/x**2, 1/x, sqrt(sqrt(x)), sqrt(x), x, sqrt(x)**3, x**2]
assert sorted(exprs, key=default_sort_key) == exprs
exprs = [x, 2*x, 2*x**2, 2*x**3, x**n, 2*x**n, sin(x), sin(x)**n,
sin(x**2), cos(x), cos(x**2), tan(x)]
assert sorted(exprs, key=default_sort_key) == exprs
exprs = [x + 1, x**2 + x + 1, x**3 + x**2 + x + 1]
assert sorted(exprs, key=default_sort_key) == exprs
exprs = [S(4), x - 3*I/2, x + 3*I/2, x - 4*I + 1, x + 4*I + 1]
assert sorted(exprs, key=default_sort_key) == exprs
exprs = [f(1), f(2), f(3), f(1, 2, 3), g(1), g(2), g(3), g(1, 2, 3)]
assert sorted(exprs, key=default_sort_key) == exprs
exprs = [f(x), g(x), exp(x), sin(x), cos(x), factorial(x)]
assert sorted(exprs, key=default_sort_key) == exprs
exprs = [Tuple(x, y), Tuple(x, z), Tuple(x, y, z)]
assert sorted(exprs, key=default_sort_key) == exprs
exprs = [[3], [1, 2]]
assert sorted(exprs, key=default_sort_key) == exprs
exprs = [[1, 2], [2, 3]]
assert sorted(exprs, key=default_sort_key) == exprs
exprs = [[1, 2], [1, 2, 3]]
assert sorted(exprs, key=default_sort_key) == exprs
exprs = [{x: -y}, {x: y}]
assert sorted(exprs, key=default_sort_key) == exprs
exprs = [{1}, {1, 2}]
assert sorted(exprs, key=default_sort_key) == exprs
a, b = exprs = [Dummy('x'), Dummy('x')]
assert sorted([b, a], key=default_sort_key) == exprs
def test_as_ordered_factors():
f, g = symbols('f,g', cls=Function)
assert x.as_ordered_factors() == [x]
assert (2*x*x**n*sin(x)*cos(x)).as_ordered_factors() \
== [Integer(2), x, x**n, sin(x), cos(x)]
args = [f(1), f(2), f(3), f(1, 2, 3), g(1), g(2), g(3), g(1, 2, 3)]
expr = Mul(*args)
assert expr.as_ordered_factors() == args
A, B = symbols('A,B', commutative=False)
assert (A*B).as_ordered_factors() == [A, B]
assert (B*A).as_ordered_factors() == [B, A]
def test_as_ordered_terms():
f, g = symbols('f,g', cls=Function)
assert x.as_ordered_terms() == [x]
assert (sin(x)**2*cos(x) + sin(x)*cos(x)**2 + 1).as_ordered_terms() \
== [sin(x)**2*cos(x), sin(x)*cos(x)**2, 1]
args = [f(1), f(2), f(3), f(1, 2, 3), g(1), g(2), g(3), g(1, 2, 3)]
expr = Add(*args)
assert expr.as_ordered_terms() == args
assert (1 + 4*sqrt(3)*pi*x).as_ordered_terms() == [4*pi*x*sqrt(3), 1]
assert ( 2 + 3*I).as_ordered_terms() == [2, 3*I]
assert (-2 + 3*I).as_ordered_terms() == [-2, 3*I]
assert ( 2 - 3*I).as_ordered_terms() == [2, -3*I]
assert (-2 - 3*I).as_ordered_terms() == [-2, -3*I]
assert ( 4 + 3*I).as_ordered_terms() == [4, 3*I]
assert (-4 + 3*I).as_ordered_terms() == [-4, 3*I]
assert ( 4 - 3*I).as_ordered_terms() == [4, -3*I]
assert (-4 - 3*I).as_ordered_terms() == [-4, -3*I]
f = x**2*y**2 + x*y**4 + y + 2
assert f.as_ordered_terms(order="lex") == [x**2*y**2, x*y**4, y, 2]
assert f.as_ordered_terms(order="grlex") == [x*y**4, x**2*y**2, y, 2]
assert f.as_ordered_terms(order="rev-lex") == [2, y, x*y**4, x**2*y**2]
assert f.as_ordered_terms(order="rev-grlex") == [2, y, x**2*y**2, x*y**4]
k = symbols('k')
assert k.as_ordered_terms(data=True) == ([(k, ((1.0, 0.0), (1,), ()))], [k])
def test_sort_key_atomic_expr():
from sympy.physics.units import m, s
assert sorted([-m, s], key=lambda arg: arg.sort_key()) == [-m, s]
def test_eval_interval():
assert exp(x)._eval_interval(*Tuple(x, 0, 1)) == exp(1) - exp(0)
# issue 4199
# first subs and limit gives NaN
a = x/y
assert a._eval_interval(x, S.Zero, oo)._eval_interval(y, oo, S.Zero) is S.NaN
# second subs and limit gives NaN
assert a._eval_interval(x, S.Zero, oo)._eval_interval(y, S.Zero, oo) is S.NaN
# difference gives S.NaN
a = x - y
assert a._eval_interval(x, S.One, oo)._eval_interval(y, oo, S.One) is S.NaN
raises(ValueError, lambda: x._eval_interval(x, None, None))
a = -y*Heaviside(x - y)
assert a._eval_interval(x, -oo, oo) == -y
assert a._eval_interval(x, oo, -oo) == y
def test_eval_interval_zoo():
# Test that limit is used when zoo is returned
assert Si(1/x)._eval_interval(x, S.Zero, S.One) == -pi/2 + Si(1)
def test_primitive():
assert (3*(x + 1)**2).primitive() == (3, (x + 1)**2)
assert (6*x + 2).primitive() == (2, 3*x + 1)
assert (x/2 + 3).primitive() == (S.Half, x + 6)
eq = (6*x + 2)*(x/2 + 3)
assert eq.primitive()[0] == 1
eq = (2 + 2*x)**2
assert eq.primitive()[0] == 1
assert (4.0*x).primitive() == (1, 4.0*x)
assert (4.0*x + y/2).primitive() == (S.Half, 8.0*x + y)
assert (-2*x).primitive() == (2, -x)
assert Add(5*z/7, 0.5*x, 3*y/2, evaluate=False).primitive() == \
(S.One/14, 7.0*x + 21*y + 10*z)
for i in [S.Infinity, S.NegativeInfinity, S.ComplexInfinity]:
assert (i + x/3).primitive() == \
(S.One/3, i + x)
assert (S.Infinity + 2*x/3 + 4*y/7).primitive() == \
(S.One/21, 14*x + 12*y + oo)
assert S.Zero.primitive() == (S.One, S.Zero)
def test_issue_5843():
a = 1 + x
assert (2*a).extract_multiplicatively(a) == 2
assert (4*a).extract_multiplicatively(2*a) == 2
assert ((3*a)*(2*a)).extract_multiplicatively(a) == 6*a
def test_is_constant():
from sympy.solvers.solvers import checksol
Sum(x, (x, 1, 10)).is_constant() is True
Sum(x, (x, 1, n)).is_constant() is False
Sum(x, (x, 1, n)).is_constant(y) is True
Sum(x, (x, 1, n)).is_constant(n) is False
Sum(x, (x, 1, n)).is_constant(x) is True
eq = a*cos(x)**2 + a*sin(x)**2 - a
eq.is_constant() is True
assert eq.subs({x: pi, a: 2}) == eq.subs({x: pi, a: 3}) == 0
assert x.is_constant() is False
assert x.is_constant(y) is True
assert checksol(x, x, Sum(x, (x, 1, n))) is False
assert checksol(x, x, Sum(x, (x, 1, n))) is False
f = Function('f')
assert f(1).is_constant
assert checksol(x, x, f(x)) is False
assert Pow(x, S.Zero, evaluate=False).is_constant() is True # == 1
assert Pow(S.Zero, x, evaluate=False).is_constant() is False # == 0 or 1
assert (2**x).is_constant() is False
assert Pow(S(2), S(3), evaluate=False).is_constant() is True
z1, z2 = symbols('z1 z2', zero=True)
assert (z1 + 2*z2).is_constant() is True
assert meter.is_constant() is True
assert (3*meter).is_constant() is True
assert (x*meter).is_constant() is False
def test_equals():
assert (-3 - sqrt(5) + (-sqrt(10)/2 - sqrt(2)/2)**2).equals(0)
assert (x**2 - 1).equals((x + 1)*(x - 1))
assert (cos(x)**2 + sin(x)**2).equals(1)
assert (a*cos(x)**2 + a*sin(x)**2).equals(a)
r = sqrt(2)
assert (-1/(r + r*x) + 1/r/(1 + x)).equals(0)
assert factorial(x + 1).equals((x + 1)*factorial(x))
assert sqrt(3).equals(2*sqrt(3)) is False
assert (sqrt(5)*sqrt(3)).equals(sqrt(3)) is False
assert (sqrt(5) + sqrt(3)).equals(0) is False
assert (sqrt(5) + pi).equals(0) is False
assert meter.equals(0) is False
assert (3*meter**2).equals(0) is False
eq = -(-1)**(S(3)/4)*6**(S.One/4) + (-6)**(S.One/4)*I
if eq != 0: # if canonicalization makes this zero, skip the test
assert eq.equals(0)
assert sqrt(x).equals(0) is False
# from integrate(x*sqrt(1 + 2*x), x);
# diff is zero only when assumptions allow
i = 2*sqrt(2)*x**(S(5)/2)*(1 + 1/(2*x))**(S(5)/2)/5 + \
2*sqrt(2)*x**(S(3)/2)*(1 + 1/(2*x))**(S(5)/2)/(-6 - 3/x)
ans = sqrt(2*x + 1)*(6*x**2 + x - 1)/15
diff = i - ans
assert diff.equals(0) is False
assert diff.subs(x, Rational(-1, 2)/2) == 7*sqrt(2)/120
# there are regions for x for which the expression is True, for
# example, when x < -1/2 or x > 0 the expression is zero
p = Symbol('p', positive=True)
assert diff.subs(x, p).equals(0) is True
assert diff.subs(x, -1).equals(0) is True
# prove via minimal_polynomial or self-consistency
eq = sqrt(1 + sqrt(3)) + sqrt(3 + 3*sqrt(3)) - sqrt(10 + 6*sqrt(3))
assert eq.equals(0)
q = 3**Rational(1, 3) + 3
p = expand(q**3)**Rational(1, 3)
assert (p - q).equals(0)
# issue 6829
# eq = q*x + q/4 + x**4 + x**3 + 2*x**2 - S.One/3
# z = eq.subs(x, solve(eq, x)[0])
q = symbols('q')
z = (q*(-sqrt(-2*(-(q - S(7)/8)**S(2)/8 - S(2197)/13824)**(S.One/3) -
S(13)/12)/2 - sqrt((2*q - S(7)/4)/sqrt(-2*(-(q - S(7)/8)**S(2)/8 -
S(2197)/13824)**(S.One/3) - S(13)/12) + 2*(-(q - S(7)/8)**S(2)/8 -
S(2197)/13824)**(S.One/3) - S(13)/6)/2 - S.One/4) + q/4 + (-sqrt(-2*(-(q
- S(7)/8)**S(2)/8 - S(2197)/13824)**(S.One/3) - S(13)/12)/2 - sqrt((2*q
- S(7)/4)/sqrt(-2*(-(q - S(7)/8)**S(2)/8 - S(2197)/13824)**(S.One/3) -
S(13)/12) + 2*(-(q - S(7)/8)**S(2)/8 - S(2197)/13824)**(S.One/3) -
S(13)/6)/2 - S.One/4)**4 + (-sqrt(-2*(-(q - S(7)/8)**S(2)/8 -
S(2197)/13824)**(S.One/3) - S(13)/12)/2 - sqrt((2*q -
S(7)/4)/sqrt(-2*(-(q - S(7)/8)**S(2)/8 - S(2197)/13824)**(S.One/3) -
S(13)/12) + 2*(-(q - S(7)/8)**S(2)/8 - S(2197)/13824)**(S.One/3) -
S(13)/6)/2 - S.One/4)**3 + 2*(-sqrt(-2*(-(q - S(7)/8)**S(2)/8 -
S(2197)/13824)**(S.One/3) - S(13)/12)/2 - sqrt((2*q -
S(7)/4)/sqrt(-2*(-(q - S(7)/8)**S(2)/8 - S(2197)/13824)**(S.One/3) -
S(13)/12) + 2*(-(q - S(7)/8)**S(2)/8 - S(2197)/13824)**(S.One/3) -
S(13)/6)/2 - S.One/4)**2 - Rational(1, 3))
assert z.equals(0)
def test_random():
from sympy import posify, lucas
assert posify(x)[0]._random() is not None
assert lucas(n)._random(2, -2, 0, -1, 1) is None
# issue 8662
assert Piecewise((Max(x, y), z))._random() is None
def test_round():
from sympy.abc import x
assert str(Float('0.1249999').round(2)) == '0.12'
d20 = 12345678901234567890
ans = S(d20).round(2)
assert ans.is_Integer and ans == d20
ans = S(d20).round(-2)
assert ans.is_Integer and ans == 12345678901234567900
assert str(S('1/7').round(4)) == '0.1429'
assert str(S('.[12345]').round(4)) == '0.1235'
assert str(S('.1349').round(2)) == '0.13'
n = S(12345)
ans = n.round()
assert ans.is_Integer
assert ans == n
ans = n.round(1)
assert ans.is_Integer
assert ans == n
ans = n.round(4)
assert ans.is_Integer
assert ans == n
assert n.round(-1) == 12340
r = Float(str(n)).round(-4)
assert r == 10000
assert n.round(-5) == 0
assert str((pi + sqrt(2)).round(2)) == '4.56'
assert (10*(pi + sqrt(2))).round(-1) == 50
raises(TypeError, lambda: round(x + 2, 2))
assert str(S(2.3).round(1)) == '2.3'
# rounding in SymPy (as in Decimal) should be
# exact for the given precision; we check here
# that when a 5 follows the last digit that
# the rounded digit will be even.
for i in range(-99, 100):
# construct a decimal that ends in 5, e.g. 123 -> 0.1235
s = str(abs(i))
p = len(s) # we are going to round to the last digit of i
n = '0.%s5' % s # put a 5 after i's digits
j = p + 2 # 2 for '0.'
if i < 0: # 1 for '-'
j += 1
n = '-' + n
v = str(Float(n).round(p))[:j] # pertinent digits
if v.endswith('.'):
continue # it ends with 0 which is even
L = int(v[-1]) # last digit
assert L % 2 == 0, (n, '->', v)
assert (Float(.3, 3) + 2*pi).round() == 7
assert (Float(.3, 3) + 2*pi*100).round() == 629
assert (pi + 2*E*I).round() == 3 + 5*I
# don't let request for extra precision give more than
# what is known (in this case, only 3 digits)
assert str((Float(.03, 3) + 2*pi/100).round(5)) == '0.0928'
assert str((Float(.03, 3) + 2*pi/100).round(4)) == '0.0928'
assert S.Zero.round() == 0
a = (Add(1, Float('1.' + '9'*27, ''), evaluate=0))
assert a.round(10) == Float('3.0000000000', '')
assert a.round(25) == Float('3.0000000000000000000000000', '')
assert a.round(26) == Float('3.00000000000000000000000000', '')
assert a.round(27) == Float('2.999999999999999999999999999', '')
assert a.round(30) == Float('2.999999999999999999999999999', '')
raises(TypeError, lambda: x.round())
f = Function('f')
raises(TypeError, lambda: f(1).round())
# exact magnitude of 10
assert str(S.One.round()) == '1'
assert str(S(100).round()) == '100'
# applied to real and imaginary portions
assert (2*pi + E*I).round() == 6 + 3*I
assert (2*pi + I/10).round() == 6
assert (pi/10 + 2*I).round() == 2*I
# the lhs re and im parts are Float with dps of 2
# and those on the right have dps of 15 so they won't compare
# equal unless we use string or compare components (which will
# then coerce the floats to the same precision) or re-create
# the floats
assert str((pi/10 + E*I).round(2)) == '0.31 + 2.72*I'
assert str((pi/10 + E*I).round(2).as_real_imag()) == '(0.31, 2.72)'
assert str((pi/10 + E*I).round(2)) == '0.31 + 2.72*I'
# issue 6914
assert (I**(I + 3)).round(3) == Float('-0.208', '')*I
# issue 8720
assert S(-123.6).round() == -124
assert S(-1.5).round() == -2
assert S(-100.5).round() == -100
assert S(-1.5 - 10.5*I).round() == -2 - 10*I
# issue 7961
assert str(S(0.006).round(2)) == '0.01'
assert str(S(0.00106).round(4)) == '0.0011'
# issue 8147
assert S.NaN.round() is S.NaN
assert S.Infinity.round() is S.Infinity
assert S.NegativeInfinity.round() is S.NegativeInfinity
assert S.ComplexInfinity.round() is S.ComplexInfinity
# check that types match
for i in range(2):
f = float(i)
# 2 args
assert all(type(round(i, p)) is int for p in (-1, 0, 1))
assert all(S(i).round(p).is_Integer for p in (-1, 0, 1))
assert all(type(round(f, p)) is float for p in (-1, 0, 1))
assert all(S(f).round(p).is_Float for p in (-1, 0, 1))
# 1 arg (p is None)
assert type(round(i)) is int
assert S(i).round().is_Integer
assert type(round(f)) is int
assert S(f).round().is_Integer
def test_held_expression_UnevaluatedExpr():
x = symbols("x")
he = UnevaluatedExpr(1/x)
e1 = x*he
assert isinstance(e1, Mul)
assert e1.args == (x, he)
assert e1.doit() == 1
assert UnevaluatedExpr(Derivative(x, x)).doit(deep=False
) == Derivative(x, x)
assert UnevaluatedExpr(Derivative(x, x)).doit() == 1
xx = Mul(x, x, evaluate=False)
assert xx != x**2
ue2 = UnevaluatedExpr(xx)
assert isinstance(ue2, UnevaluatedExpr)
assert ue2.args == (xx,)
assert ue2.doit() == x**2
assert ue2.doit(deep=False) == xx
x2 = UnevaluatedExpr(2)*2
assert type(x2) is Mul
assert x2.args == (2, UnevaluatedExpr(2))
def test_round_exception_nostr():
# Don't use the string form of the expression in the round exception, as
# it's too slow
s = Symbol('bad')
try:
s.round()
except TypeError as e:
assert 'bad' not in str(e)
else:
# Did not raise
raise AssertionError("Did not raise")
def test_extract_branch_factor():
assert exp_polar(2.0*I*pi).extract_branch_factor() == (1, 1)
def test_identity_removal():
assert Add.make_args(x + 0) == (x,)
assert Mul.make_args(x*1) == (x,)
def test_float_0():
assert Float(0.0) + 1 == Float(1.0)
@XFAIL
def test_float_0_fail():
assert Float(0.0)*x == Float(0.0)
assert (x + Float(0.0)).is_Add
def test_issue_6325():
ans = (b**2 + z**2 - (b*(a + b*t) + z*(c + t*z))**2/(
(a + b*t)**2 + (c + t*z)**2))/sqrt((a + b*t)**2 + (c + t*z)**2)
e = sqrt((a + b*t)**2 + (c + z*t)**2)
assert diff(e, t, 2) == ans
e.diff(t, 2) == ans
assert diff(e, t, 2, simplify=False) != ans
def test_issue_7426():
f1 = a % c
f2 = x % z
assert f1.equals(f2) is None
def test_issue_11122():
x = Symbol('x', extended_positive=False)
assert unchanged(Gt, x, 0) # (x > 0)
# (x > 0) should remain unevaluated after PR #16956
x = Symbol('x', positive=False, real=True)
assert (x > 0) is S.false
def test_issue_10651():
x = Symbol('x', real=True)
e1 = (-1 + x)/(1 - x)
e3 = (4*x**2 - 4)/((1 - x)*(1 + x))
e4 = 1/(cos(x)**2) - (tan(x))**2
x = Symbol('x', positive=True)
e5 = (1 + x)/x
assert e1.is_constant() is None
assert e3.is_constant() is None
assert e4.is_constant() is None
assert e5.is_constant() is False
def test_issue_10161():
x = symbols('x', real=True)
assert x*abs(x)*abs(x) == x**3
def test_issue_10755():
x = symbols('x')
raises(TypeError, lambda: int(log(x)))
raises(TypeError, lambda: log(x).round(2))
def test_issue_11877():
x = symbols('x')
assert integrate(log(S.Half - x), (x, 0, S.Half)) == Rational(-1, 2) -log(2)/2
def test_normal():
x = symbols('x')
e = Mul(S.Half, 1 + x, evaluate=False)
assert e.normal() == e
def test_expr():
x = symbols('x')
raises(TypeError, lambda: tan(x).series(x, 2, oo, "+"))
def test_ExprBuilder():
eb = ExprBuilder(Mul)
eb.args.extend([x, x])
assert eb.build() == x**2
def test_non_string_equality():
# Expressions should not compare equal to strings
x = symbols('x')
one = sympify(1)
assert (x == 'x') is False
assert (x != 'x') is True
assert (one == '1') is False
assert (one != '1') is True
assert (x + 1 == 'x + 1') is False
assert (x + 1 != 'x + 1') is True
# Make sure == doesn't try to convert the resulting expression to a string
# (e.g., by calling sympify() instead of _sympify())
class BadRepr:
def __repr__(self):
raise RuntimeError
assert (x == BadRepr()) is False
assert (x != BadRepr()) is True
|
08a66522fa189a1e81c1e7210a68d0cdd7b1a887b74f73f05299807f20c5c4c5 | """Test whether all elements of cls.args are instances of Basic. """
# NOTE: keep tests sorted by (module, class name) key. If a class can't
# be instantiated, add it here anyway with @SKIP("abstract class) (see
# e.g. Function).
import os
import re
from sympy import (Basic, S, symbols, sqrt, sin, oo, Interval, exp, Lambda, pi,
Eq, log, Function, Rational)
from sympy.testing.pytest import XFAIL, SKIP
x, y, z = symbols('x,y,z')
def test_all_classes_are_tested():
this = os.path.split(__file__)[0]
path = os.path.join(this, os.pardir, os.pardir)
sympy_path = os.path.abspath(path)
prefix = os.path.split(sympy_path)[0] + os.sep
re_cls = re.compile(r"^class ([A-Za-z][A-Za-z0-9_]*)\s*\(", re.MULTILINE)
modules = {}
for root, dirs, files in os.walk(sympy_path):
module = root.replace(prefix, "").replace(os.sep, ".")
for file in files:
if file.startswith(("_", "test_", "bench_")):
continue
if not file.endswith(".py"):
continue
with open(os.path.join(root, file), "r", encoding='utf-8') as f:
text = f.read()
submodule = module + '.' + file[:-3]
names = re_cls.findall(text)
if not names:
continue
try:
mod = __import__(submodule, fromlist=names)
except ImportError:
continue
def is_Basic(name):
cls = getattr(mod, name)
if hasattr(cls, '_sympy_deprecated_func'):
cls = cls._sympy_deprecated_func
return issubclass(cls, Basic)
names = list(filter(is_Basic, names))
if names:
modules[submodule] = names
ns = globals()
failed = []
for module, names in modules.items():
mod = module.replace('.', '__')
for name in names:
test = 'test_' + mod + '__' + name
if test not in ns:
failed.append(module + '.' + name)
assert not failed, "Missing classes: %s. Please add tests for these to sympy/core/tests/test_args.py." % ", ".join(failed)
def _test_args(obj):
all_basic = all(isinstance(arg, Basic) for arg in obj.args)
# Ideally obj.func(*obj.args) would always recreate the object, but for
# now, we only require it for objects with non-empty .args
recreatable = not obj.args or obj.func(*obj.args) == obj
return all_basic and recreatable
def test_sympy__assumptions__assume__AppliedPredicate():
from sympy.assumptions.assume import AppliedPredicate, Predicate
from sympy import Q
assert _test_args(AppliedPredicate(Predicate("test"), 2))
assert _test_args(Q.is_true(True))
def test_sympy__assumptions__assume__Predicate():
from sympy.assumptions.assume import Predicate
assert _test_args(Predicate("test"))
def test_sympy__assumptions__sathandlers__UnevaluatedOnFree():
from sympy.assumptions.sathandlers import UnevaluatedOnFree
from sympy import Q
assert _test_args(UnevaluatedOnFree(Q.positive))
def test_sympy__assumptions__sathandlers__AllArgs():
from sympy.assumptions.sathandlers import AllArgs
from sympy import Q
assert _test_args(AllArgs(Q.positive))
def test_sympy__assumptions__sathandlers__AnyArgs():
from sympy.assumptions.sathandlers import AnyArgs
from sympy import Q
assert _test_args(AnyArgs(Q.positive))
def test_sympy__assumptions__sathandlers__ExactlyOneArg():
from sympy.assumptions.sathandlers import ExactlyOneArg
from sympy import Q
assert _test_args(ExactlyOneArg(Q.positive))
def test_sympy__assumptions__sathandlers__CheckOldAssump():
from sympy.assumptions.sathandlers import CheckOldAssump
from sympy import Q
assert _test_args(CheckOldAssump(Q.positive))
def test_sympy__assumptions__sathandlers__CheckIsPrime():
from sympy.assumptions.sathandlers import CheckIsPrime
from sympy import Q
# Input must be a number
assert _test_args(CheckIsPrime(Q.positive))
@SKIP("abstract Class")
def test_sympy__codegen__ast__AssignmentBase():
from sympy.codegen.ast import AssignmentBase
assert _test_args(AssignmentBase(x, 1))
@SKIP("abstract Class")
def test_sympy__codegen__ast__AugmentedAssignment():
from sympy.codegen.ast import AugmentedAssignment
assert _test_args(AugmentedAssignment(x, 1))
def test_sympy__codegen__ast__AddAugmentedAssignment():
from sympy.codegen.ast import AddAugmentedAssignment
assert _test_args(AddAugmentedAssignment(x, 1))
def test_sympy__codegen__ast__SubAugmentedAssignment():
from sympy.codegen.ast import SubAugmentedAssignment
assert _test_args(SubAugmentedAssignment(x, 1))
def test_sympy__codegen__ast__MulAugmentedAssignment():
from sympy.codegen.ast import MulAugmentedAssignment
assert _test_args(MulAugmentedAssignment(x, 1))
def test_sympy__codegen__ast__DivAugmentedAssignment():
from sympy.codegen.ast import DivAugmentedAssignment
assert _test_args(DivAugmentedAssignment(x, 1))
def test_sympy__codegen__ast__ModAugmentedAssignment():
from sympy.codegen.ast import ModAugmentedAssignment
assert _test_args(ModAugmentedAssignment(x, 1))
def test_sympy__codegen__ast__CodeBlock():
from sympy.codegen.ast import CodeBlock, Assignment
assert _test_args(CodeBlock(Assignment(x, 1), Assignment(y, 2)))
def test_sympy__codegen__ast__For():
from sympy.codegen.ast import For, CodeBlock, AddAugmentedAssignment
from sympy import Range
assert _test_args(For(x, Range(10), CodeBlock(AddAugmentedAssignment(y, 1))))
def test_sympy__codegen__ast__Token():
from sympy.codegen.ast import Token
assert _test_args(Token())
def test_sympy__codegen__ast__ContinueToken():
from sympy.codegen.ast import ContinueToken
assert _test_args(ContinueToken())
def test_sympy__codegen__ast__BreakToken():
from sympy.codegen.ast import BreakToken
assert _test_args(BreakToken())
def test_sympy__codegen__ast__NoneToken():
from sympy.codegen.ast import NoneToken
assert _test_args(NoneToken())
def test_sympy__codegen__ast__String():
from sympy.codegen.ast import String
assert _test_args(String('foobar'))
def test_sympy__codegen__ast__QuotedString():
from sympy.codegen.ast import QuotedString
assert _test_args(QuotedString('foobar'))
def test_sympy__codegen__ast__Comment():
from sympy.codegen.ast import Comment
assert _test_args(Comment('this is a comment'))
def test_sympy__codegen__ast__Node():
from sympy.codegen.ast import Node
assert _test_args(Node())
assert _test_args(Node(attrs={1, 2, 3}))
def test_sympy__codegen__ast__Type():
from sympy.codegen.ast import Type
assert _test_args(Type('float128'))
def test_sympy__codegen__ast__IntBaseType():
from sympy.codegen.ast import IntBaseType
assert _test_args(IntBaseType('bigint'))
def test_sympy__codegen__ast___SizedIntType():
from sympy.codegen.ast import _SizedIntType
assert _test_args(_SizedIntType('int128', 128))
def test_sympy__codegen__ast__SignedIntType():
from sympy.codegen.ast import SignedIntType
assert _test_args(SignedIntType('int128_with_sign', 128))
def test_sympy__codegen__ast__UnsignedIntType():
from sympy.codegen.ast import UnsignedIntType
assert _test_args(UnsignedIntType('unt128', 128))
def test_sympy__codegen__ast__FloatBaseType():
from sympy.codegen.ast import FloatBaseType
assert _test_args(FloatBaseType('positive_real'))
def test_sympy__codegen__ast__FloatType():
from sympy.codegen.ast import FloatType
assert _test_args(FloatType('float242', 242, nmant=142, nexp=99))
def test_sympy__codegen__ast__ComplexBaseType():
from sympy.codegen.ast import ComplexBaseType
assert _test_args(ComplexBaseType('positive_cmplx'))
def test_sympy__codegen__ast__ComplexType():
from sympy.codegen.ast import ComplexType
assert _test_args(ComplexType('complex42', 42, nmant=15, nexp=5))
def test_sympy__codegen__ast__Attribute():
from sympy.codegen.ast import Attribute
assert _test_args(Attribute('noexcept'))
def test_sympy__codegen__ast__Variable():
from sympy.codegen.ast import Variable, Type, value_const
assert _test_args(Variable(x))
assert _test_args(Variable(y, Type('float32'), {value_const}))
assert _test_args(Variable(z, type=Type('float64')))
def test_sympy__codegen__ast__Pointer():
from sympy.codegen.ast import Pointer, Type, pointer_const
assert _test_args(Pointer(x))
assert _test_args(Pointer(y, type=Type('float32')))
assert _test_args(Pointer(z, Type('float64'), {pointer_const}))
def test_sympy__codegen__ast__Declaration():
from sympy.codegen.ast import Declaration, Variable, Type
vx = Variable(x, type=Type('float'))
assert _test_args(Declaration(vx))
def test_sympy__codegen__ast__While():
from sympy.codegen.ast import While, AddAugmentedAssignment
assert _test_args(While(abs(x) < 1, [AddAugmentedAssignment(x, -1)]))
def test_sympy__codegen__ast__Scope():
from sympy.codegen.ast import Scope, AddAugmentedAssignment
assert _test_args(Scope([AddAugmentedAssignment(x, -1)]))
def test_sympy__codegen__ast__Stream():
from sympy.codegen.ast import Stream
assert _test_args(Stream('stdin'))
def test_sympy__codegen__ast__Print():
from sympy.codegen.ast import Print
assert _test_args(Print([x, y]))
assert _test_args(Print([x, y], "%d %d"))
def test_sympy__codegen__ast__FunctionPrototype():
from sympy.codegen.ast import FunctionPrototype, real, Declaration, Variable
inp_x = Declaration(Variable(x, type=real))
assert _test_args(FunctionPrototype(real, 'pwer', [inp_x]))
def test_sympy__codegen__ast__FunctionDefinition():
from sympy.codegen.ast import FunctionDefinition, real, Declaration, Variable, Assignment
inp_x = Declaration(Variable(x, type=real))
assert _test_args(FunctionDefinition(real, 'pwer', [inp_x], [Assignment(x, x**2)]))
def test_sympy__codegen__ast__Return():
from sympy.codegen.ast import Return
assert _test_args(Return(x))
def test_sympy__codegen__ast__FunctionCall():
from sympy.codegen.ast import FunctionCall
assert _test_args(FunctionCall('pwer', [x]))
def test_sympy__codegen__ast__Element():
from sympy.codegen.ast import Element
assert _test_args(Element('x', range(3)))
def test_sympy__codegen__cnodes__CommaOperator():
from sympy.codegen.cnodes import CommaOperator
assert _test_args(CommaOperator(1, 2))
def test_sympy__codegen__cnodes__goto():
from sympy.codegen.cnodes import goto
assert _test_args(goto('early_exit'))
def test_sympy__codegen__cnodes__Label():
from sympy.codegen.cnodes import Label
assert _test_args(Label('early_exit'))
def test_sympy__codegen__cnodes__PreDecrement():
from sympy.codegen.cnodes import PreDecrement
assert _test_args(PreDecrement(x))
def test_sympy__codegen__cnodes__PostDecrement():
from sympy.codegen.cnodes import PostDecrement
assert _test_args(PostDecrement(x))
def test_sympy__codegen__cnodes__PreIncrement():
from sympy.codegen.cnodes import PreIncrement
assert _test_args(PreIncrement(x))
def test_sympy__codegen__cnodes__PostIncrement():
from sympy.codegen.cnodes import PostIncrement
assert _test_args(PostIncrement(x))
def test_sympy__codegen__cnodes__struct():
from sympy.codegen.ast import real, Variable
from sympy.codegen.cnodes import struct
assert _test_args(struct(declarations=[
Variable(x, type=real),
Variable(y, type=real)
]))
def test_sympy__codegen__cnodes__union():
from sympy.codegen.ast import float32, int32, Variable
from sympy.codegen.cnodes import union
assert _test_args(union(declarations=[
Variable(x, type=float32),
Variable(y, type=int32)
]))
def test_sympy__codegen__cxxnodes__using():
from sympy.codegen.cxxnodes import using
assert _test_args(using('std::vector'))
assert _test_args(using('std::vector', 'vec'))
def test_sympy__codegen__fnodes__Program():
from sympy.codegen.fnodes import Program
assert _test_args(Program('foobar', []))
def test_sympy__codegen__fnodes__Module():
from sympy.codegen.fnodes import Module
assert _test_args(Module('foobar', [], []))
def test_sympy__codegen__fnodes__Subroutine():
from sympy.codegen.fnodes import Subroutine
x = symbols('x', real=True)
assert _test_args(Subroutine('foo', [x], []))
def test_sympy__codegen__fnodes__GoTo():
from sympy.codegen.fnodes import GoTo
assert _test_args(GoTo([10]))
assert _test_args(GoTo([10, 20], x > 1))
def test_sympy__codegen__fnodes__FortranReturn():
from sympy.codegen.fnodes import FortranReturn
assert _test_args(FortranReturn(10))
def test_sympy__codegen__fnodes__Extent():
from sympy.codegen.fnodes import Extent
assert _test_args(Extent())
assert _test_args(Extent(None))
assert _test_args(Extent(':'))
assert _test_args(Extent(-3, 4))
assert _test_args(Extent(x, y))
def test_sympy__codegen__fnodes__use_rename():
from sympy.codegen.fnodes import use_rename
assert _test_args(use_rename('loc', 'glob'))
def test_sympy__codegen__fnodes__use():
from sympy.codegen.fnodes import use
assert _test_args(use('modfoo', only='bar'))
def test_sympy__codegen__fnodes__SubroutineCall():
from sympy.codegen.fnodes import SubroutineCall
assert _test_args(SubroutineCall('foo', ['bar', 'baz']))
def test_sympy__codegen__fnodes__Do():
from sympy.codegen.fnodes import Do
assert _test_args(Do([], 'i', 1, 42))
def test_sympy__codegen__fnodes__ImpliedDoLoop():
from sympy.codegen.fnodes import ImpliedDoLoop
assert _test_args(ImpliedDoLoop('i', 'i', 1, 42))
def test_sympy__codegen__fnodes__ArrayConstructor():
from sympy.codegen.fnodes import ArrayConstructor
assert _test_args(ArrayConstructor([1, 2, 3]))
from sympy.codegen.fnodes import ImpliedDoLoop
idl = ImpliedDoLoop('i', 'i', 1, 42)
assert _test_args(ArrayConstructor([1, idl, 3]))
def test_sympy__codegen__fnodes__sum_():
from sympy.codegen.fnodes import sum_
assert _test_args(sum_('arr'))
def test_sympy__codegen__fnodes__product_():
from sympy.codegen.fnodes import product_
assert _test_args(product_('arr'))
@XFAIL
def test_sympy__combinatorics__graycode__GrayCode():
from sympy.combinatorics.graycode import GrayCode
# an integer is given and returned from GrayCode as the arg
assert _test_args(GrayCode(3, start='100'))
assert _test_args(GrayCode(3, rank=1))
def test_sympy__combinatorics__subsets__Subset():
from sympy.combinatorics.subsets import Subset
assert _test_args(Subset([0, 1], [0, 1, 2, 3]))
assert _test_args(Subset(['c', 'd'], ['a', 'b', 'c', 'd']))
def test_sympy__combinatorics__permutations__Permutation():
from sympy.combinatorics.permutations import Permutation
assert _test_args(Permutation([0, 1, 2, 3]))
def test_sympy__combinatorics__permutations__AppliedPermutation():
from sympy.combinatorics.permutations import Permutation
from sympy.combinatorics.permutations import AppliedPermutation
p = Permutation([0, 1, 2, 3])
assert _test_args(AppliedPermutation(p, 1))
def test_sympy__combinatorics__perm_groups__PermutationGroup():
from sympy.combinatorics.permutations import Permutation
from sympy.combinatorics.perm_groups import PermutationGroup
assert _test_args(PermutationGroup([Permutation([0, 1])]))
def test_sympy__combinatorics__polyhedron__Polyhedron():
from sympy.combinatorics.permutations import Permutation
from sympy.combinatorics.polyhedron import Polyhedron
from sympy.abc import w, x, y, z
pgroup = [Permutation([[0, 1, 2], [3]]),
Permutation([[0, 1, 3], [2]]),
Permutation([[0, 2, 3], [1]]),
Permutation([[1, 2, 3], [0]]),
Permutation([[0, 1], [2, 3]]),
Permutation([[0, 2], [1, 3]]),
Permutation([[0, 3], [1, 2]]),
Permutation([[0, 1, 2, 3]])]
corners = [w, x, y, z]
faces = [(w, x, y), (w, y, z), (w, z, x), (x, y, z)]
assert _test_args(Polyhedron(corners, faces, pgroup))
@XFAIL
def test_sympy__combinatorics__prufer__Prufer():
from sympy.combinatorics.prufer import Prufer
assert _test_args(Prufer([[0, 1], [0, 2], [0, 3]], 4))
def test_sympy__combinatorics__partitions__Partition():
from sympy.combinatorics.partitions import Partition
assert _test_args(Partition([1]))
@XFAIL
def test_sympy__combinatorics__partitions__IntegerPartition():
from sympy.combinatorics.partitions import IntegerPartition
assert _test_args(IntegerPartition([1]))
def test_sympy__concrete__products__Product():
from sympy.concrete.products import Product
assert _test_args(Product(x, (x, 0, 10)))
assert _test_args(Product(x, (x, 0, y), (y, 0, 10)))
@SKIP("abstract Class")
def test_sympy__concrete__expr_with_limits__ExprWithLimits():
from sympy.concrete.expr_with_limits import ExprWithLimits
assert _test_args(ExprWithLimits(x, (x, 0, 10)))
assert _test_args(ExprWithLimits(x*y, (x, 0, 10.),(y,1.,3)))
@SKIP("abstract Class")
def test_sympy__concrete__expr_with_limits__AddWithLimits():
from sympy.concrete.expr_with_limits import AddWithLimits
assert _test_args(AddWithLimits(x, (x, 0, 10)))
assert _test_args(AddWithLimits(x*y, (x, 0, 10),(y,1,3)))
@SKIP("abstract Class")
def test_sympy__concrete__expr_with_intlimits__ExprWithIntLimits():
from sympy.concrete.expr_with_intlimits import ExprWithIntLimits
assert _test_args(ExprWithIntLimits(x, (x, 0, 10)))
assert _test_args(ExprWithIntLimits(x*y, (x, 0, 10),(y,1,3)))
def test_sympy__concrete__summations__Sum():
from sympy.concrete.summations import Sum
assert _test_args(Sum(x, (x, 0, 10)))
assert _test_args(Sum(x, (x, 0, y), (y, 0, 10)))
def test_sympy__core__add__Add():
from sympy.core.add import Add
assert _test_args(Add(x, y, z, 2))
def test_sympy__core__basic__Atom():
from sympy.core.basic import Atom
assert _test_args(Atom())
def test_sympy__core__basic__Basic():
from sympy.core.basic import Basic
assert _test_args(Basic())
def test_sympy__core__containers__Dict():
from sympy.core.containers import Dict
assert _test_args(Dict({x: y, y: z}))
def test_sympy__core__containers__Tuple():
from sympy.core.containers import Tuple
assert _test_args(Tuple(x, y, z, 2))
def test_sympy__core__expr__AtomicExpr():
from sympy.core.expr import AtomicExpr
assert _test_args(AtomicExpr())
def test_sympy__core__expr__Expr():
from sympy.core.expr import Expr
assert _test_args(Expr())
def test_sympy__core__expr__UnevaluatedExpr():
from sympy.core.expr import UnevaluatedExpr
from sympy.abc import x
assert _test_args(UnevaluatedExpr(x))
def test_sympy__core__function__Application():
from sympy.core.function import Application
assert _test_args(Application(1, 2, 3))
def test_sympy__core__function__AppliedUndef():
from sympy.core.function import AppliedUndef
assert _test_args(AppliedUndef(1, 2, 3))
def test_sympy__core__function__Derivative():
from sympy.core.function import Derivative
assert _test_args(Derivative(2, x, y, 3))
@SKIP("abstract class")
def test_sympy__core__function__Function():
pass
def test_sympy__core__function__Lambda():
assert _test_args(Lambda((x, y), x + y + z))
def test_sympy__core__function__Subs():
from sympy.core.function import Subs
assert _test_args(Subs(x + y, x, 2))
def test_sympy__core__function__WildFunction():
from sympy.core.function import WildFunction
assert _test_args(WildFunction('f'))
def test_sympy__core__mod__Mod():
from sympy.core.mod import Mod
assert _test_args(Mod(x, 2))
def test_sympy__core__mul__Mul():
from sympy.core.mul import Mul
assert _test_args(Mul(2, x, y, z))
def test_sympy__core__numbers__Catalan():
from sympy.core.numbers import Catalan
assert _test_args(Catalan())
def test_sympy__core__numbers__ComplexInfinity():
from sympy.core.numbers import ComplexInfinity
assert _test_args(ComplexInfinity())
def test_sympy__core__numbers__EulerGamma():
from sympy.core.numbers import EulerGamma
assert _test_args(EulerGamma())
def test_sympy__core__numbers__Exp1():
from sympy.core.numbers import Exp1
assert _test_args(Exp1())
def test_sympy__core__numbers__Float():
from sympy.core.numbers import Float
assert _test_args(Float(1.23))
def test_sympy__core__numbers__GoldenRatio():
from sympy.core.numbers import GoldenRatio
assert _test_args(GoldenRatio())
def test_sympy__core__numbers__TribonacciConstant():
from sympy.core.numbers import TribonacciConstant
assert _test_args(TribonacciConstant())
def test_sympy__core__numbers__Half():
from sympy.core.numbers import Half
assert _test_args(Half())
def test_sympy__core__numbers__ImaginaryUnit():
from sympy.core.numbers import ImaginaryUnit
assert _test_args(ImaginaryUnit())
def test_sympy__core__numbers__Infinity():
from sympy.core.numbers import Infinity
assert _test_args(Infinity())
def test_sympy__core__numbers__Integer():
from sympy.core.numbers import Integer
assert _test_args(Integer(7))
@SKIP("abstract class")
def test_sympy__core__numbers__IntegerConstant():
pass
def test_sympy__core__numbers__NaN():
from sympy.core.numbers import NaN
assert _test_args(NaN())
def test_sympy__core__numbers__NegativeInfinity():
from sympy.core.numbers import NegativeInfinity
assert _test_args(NegativeInfinity())
def test_sympy__core__numbers__NegativeOne():
from sympy.core.numbers import NegativeOne
assert _test_args(NegativeOne())
def test_sympy__core__numbers__Number():
from sympy.core.numbers import Number
assert _test_args(Number(1, 7))
def test_sympy__core__numbers__NumberSymbol():
from sympy.core.numbers import NumberSymbol
assert _test_args(NumberSymbol())
def test_sympy__core__numbers__One():
from sympy.core.numbers import One
assert _test_args(One())
def test_sympy__core__numbers__Pi():
from sympy.core.numbers import Pi
assert _test_args(Pi())
def test_sympy__core__numbers__Rational():
from sympy.core.numbers import Rational
assert _test_args(Rational(1, 7))
@SKIP("abstract class")
def test_sympy__core__numbers__RationalConstant():
pass
def test_sympy__core__numbers__Zero():
from sympy.core.numbers import Zero
assert _test_args(Zero())
@SKIP("abstract class")
def test_sympy__core__operations__AssocOp():
pass
@SKIP("abstract class")
def test_sympy__core__operations__LatticeOp():
pass
def test_sympy__core__power__Pow():
from sympy.core.power import Pow
assert _test_args(Pow(x, 2))
def test_sympy__algebras__quaternion__Quaternion():
from sympy.algebras.quaternion import Quaternion
assert _test_args(Quaternion(x, 1, 2, 3))
def test_sympy__core__relational__Equality():
from sympy.core.relational import Equality
assert _test_args(Equality(x, 2))
def test_sympy__core__relational__GreaterThan():
from sympy.core.relational import GreaterThan
assert _test_args(GreaterThan(x, 2))
def test_sympy__core__relational__LessThan():
from sympy.core.relational import LessThan
assert _test_args(LessThan(x, 2))
@SKIP("abstract class")
def test_sympy__core__relational__Relational():
pass
def test_sympy__core__relational__StrictGreaterThan():
from sympy.core.relational import StrictGreaterThan
assert _test_args(StrictGreaterThan(x, 2))
def test_sympy__core__relational__StrictLessThan():
from sympy.core.relational import StrictLessThan
assert _test_args(StrictLessThan(x, 2))
def test_sympy__core__relational__Unequality():
from sympy.core.relational import Unequality
assert _test_args(Unequality(x, 2))
def test_sympy__sandbox__indexed_integrals__IndexedIntegral():
from sympy.tensor import IndexedBase, Idx
from sympy.sandbox.indexed_integrals import IndexedIntegral
A = IndexedBase('A')
i, j = symbols('i j', integer=True)
a1, a2 = symbols('a1:3', cls=Idx)
assert _test_args(IndexedIntegral(A[a1], A[a2]))
assert _test_args(IndexedIntegral(A[i], A[j]))
def test_sympy__calculus__util__AccumulationBounds():
from sympy.calculus.util import AccumulationBounds
assert _test_args(AccumulationBounds(0, 1))
def test_sympy__sets__ordinals__OmegaPower():
from sympy.sets.ordinals import OmegaPower
assert _test_args(OmegaPower(1, 1))
def test_sympy__sets__ordinals__Ordinal():
from sympy.sets.ordinals import Ordinal, OmegaPower
assert _test_args(Ordinal(OmegaPower(2, 1)))
def test_sympy__sets__ordinals__OrdinalOmega():
from sympy.sets.ordinals import OrdinalOmega
assert _test_args(OrdinalOmega())
def test_sympy__sets__ordinals__OrdinalZero():
from sympy.sets.ordinals import OrdinalZero
assert _test_args(OrdinalZero())
def test_sympy__sets__powerset__PowerSet():
from sympy.sets.powerset import PowerSet
from sympy.core.singleton import S
assert _test_args(PowerSet(S.EmptySet))
def test_sympy__sets__sets__EmptySet():
from sympy.sets.sets import EmptySet
assert _test_args(EmptySet())
def test_sympy__sets__sets__UniversalSet():
from sympy.sets.sets import UniversalSet
assert _test_args(UniversalSet())
def test_sympy__sets__sets__FiniteSet():
from sympy.sets.sets import FiniteSet
assert _test_args(FiniteSet(x, y, z))
def test_sympy__sets__sets__Interval():
from sympy.sets.sets import Interval
assert _test_args(Interval(0, 1))
def test_sympy__sets__sets__ProductSet():
from sympy.sets.sets import ProductSet, Interval
assert _test_args(ProductSet(Interval(0, 1), Interval(0, 1)))
@SKIP("does it make sense to test this?")
def test_sympy__sets__sets__Set():
from sympy.sets.sets import Set
assert _test_args(Set())
def test_sympy__sets__sets__Intersection():
from sympy.sets.sets import Intersection, Interval
from sympy.core.symbol import Symbol
x = Symbol('x')
y = Symbol('y')
S = Intersection(Interval(0, x), Interval(y, 1))
assert isinstance(S, Intersection)
assert _test_args(S)
def test_sympy__sets__sets__Union():
from sympy.sets.sets import Union, Interval
assert _test_args(Union(Interval(0, 1), Interval(2, 3)))
def test_sympy__sets__sets__Complement():
from sympy.sets.sets import Complement
assert _test_args(Complement(Interval(0, 2), Interval(0, 1)))
def test_sympy__sets__sets__SymmetricDifference():
from sympy.sets.sets import FiniteSet, SymmetricDifference
assert _test_args(SymmetricDifference(FiniteSet(1, 2, 3), \
FiniteSet(2, 3, 4)))
def test_sympy__sets__sets__DisjointUnion():
from sympy.sets.sets import FiniteSet, DisjointUnion
assert _test_args(DisjointUnion(FiniteSet(1, 2, 3), \
FiniteSet(2, 3, 4)))
def test_sympy__core__trace__Tr():
from sympy.core.trace import Tr
a, b = symbols('a b')
assert _test_args(Tr(a + b))
def test_sympy__sets__setexpr__SetExpr():
from sympy.sets.setexpr import SetExpr
assert _test_args(SetExpr(Interval(0, 1)))
def test_sympy__sets__fancysets__Rationals():
from sympy.sets.fancysets import Rationals
assert _test_args(Rationals())
def test_sympy__sets__fancysets__Naturals():
from sympy.sets.fancysets import Naturals
assert _test_args(Naturals())
def test_sympy__sets__fancysets__Naturals0():
from sympy.sets.fancysets import Naturals0
assert _test_args(Naturals0())
def test_sympy__sets__fancysets__Integers():
from sympy.sets.fancysets import Integers
assert _test_args(Integers())
def test_sympy__sets__fancysets__Reals():
from sympy.sets.fancysets import Reals
assert _test_args(Reals())
def test_sympy__sets__fancysets__Complexes():
from sympy.sets.fancysets import Complexes
assert _test_args(Complexes())
def test_sympy__sets__fancysets__ComplexRegion():
from sympy.sets.fancysets import ComplexRegion
from sympy import S
from sympy.sets import Interval
a = Interval(0, 1)
b = Interval(2, 3)
theta = Interval(0, 2*S.Pi)
assert _test_args(ComplexRegion(a*b))
assert _test_args(ComplexRegion(a*theta, polar=True))
def test_sympy__sets__fancysets__CartesianComplexRegion():
from sympy.sets.fancysets import CartesianComplexRegion
from sympy.sets import Interval
a = Interval(0, 1)
b = Interval(2, 3)
assert _test_args(CartesianComplexRegion(a*b))
def test_sympy__sets__fancysets__PolarComplexRegion():
from sympy.sets.fancysets import PolarComplexRegion
from sympy import S
from sympy.sets import Interval
a = Interval(0, 1)
theta = Interval(0, 2*S.Pi)
assert _test_args(PolarComplexRegion(a*theta))
def test_sympy__sets__fancysets__ImageSet():
from sympy.sets.fancysets import ImageSet
from sympy import S, Symbol
x = Symbol('x')
assert _test_args(ImageSet(Lambda(x, x**2), S.Naturals))
def test_sympy__sets__fancysets__Range():
from sympy.sets.fancysets import Range
assert _test_args(Range(1, 5, 1))
def test_sympy__sets__conditionset__ConditionSet():
from sympy.sets.conditionset import ConditionSet
from sympy import S, Symbol
x = Symbol('x')
assert _test_args(ConditionSet(x, Eq(x**2, 1), S.Reals))
def test_sympy__sets__contains__Contains():
from sympy.sets.fancysets import Range
from sympy.sets.contains import Contains
assert _test_args(Contains(x, Range(0, 10, 2)))
# STATS
from sympy.stats.crv_types import NormalDistribution
nd = NormalDistribution(0, 1)
from sympy.stats.frv_types import DieDistribution
die = DieDistribution(6)
def test_sympy__stats__crv__ContinuousDomain():
from sympy.stats.crv import ContinuousDomain
assert _test_args(ContinuousDomain({x}, Interval(-oo, oo)))
def test_sympy__stats__crv__SingleContinuousDomain():
from sympy.stats.crv import SingleContinuousDomain
assert _test_args(SingleContinuousDomain(x, Interval(-oo, oo)))
def test_sympy__stats__crv__ProductContinuousDomain():
from sympy.stats.crv import SingleContinuousDomain, ProductContinuousDomain
D = SingleContinuousDomain(x, Interval(-oo, oo))
E = SingleContinuousDomain(y, Interval(0, oo))
assert _test_args(ProductContinuousDomain(D, E))
def test_sympy__stats__crv__ConditionalContinuousDomain():
from sympy.stats.crv import (SingleContinuousDomain,
ConditionalContinuousDomain)
D = SingleContinuousDomain(x, Interval(-oo, oo))
assert _test_args(ConditionalContinuousDomain(D, x > 0))
def test_sympy__stats__crv__ContinuousPSpace():
from sympy.stats.crv import ContinuousPSpace, SingleContinuousDomain
D = SingleContinuousDomain(x, Interval(-oo, oo))
assert _test_args(ContinuousPSpace(D, nd))
def test_sympy__stats__crv__SingleContinuousPSpace():
from sympy.stats.crv import SingleContinuousPSpace
assert _test_args(SingleContinuousPSpace(x, nd))
@SKIP("abstract class")
def test_sympy__stats__crv__SingleContinuousDistribution():
pass
def test_sympy__stats__drv__SingleDiscreteDomain():
from sympy.stats.drv import SingleDiscreteDomain
assert _test_args(SingleDiscreteDomain(x, S.Naturals))
def test_sympy__stats__drv__ProductDiscreteDomain():
from sympy.stats.drv import SingleDiscreteDomain, ProductDiscreteDomain
X = SingleDiscreteDomain(x, S.Naturals)
Y = SingleDiscreteDomain(y, S.Integers)
assert _test_args(ProductDiscreteDomain(X, Y))
def test_sympy__stats__drv__SingleDiscretePSpace():
from sympy.stats.drv import SingleDiscretePSpace
from sympy.stats.drv_types import PoissonDistribution
assert _test_args(SingleDiscretePSpace(x, PoissonDistribution(1)))
def test_sympy__stats__drv__DiscretePSpace():
from sympy.stats.drv import DiscretePSpace, SingleDiscreteDomain
density = Lambda(x, 2**(-x))
domain = SingleDiscreteDomain(x, S.Naturals)
assert _test_args(DiscretePSpace(domain, density))
def test_sympy__stats__drv__ConditionalDiscreteDomain():
from sympy.stats.drv import ConditionalDiscreteDomain, SingleDiscreteDomain
X = SingleDiscreteDomain(x, S.Naturals0)
assert _test_args(ConditionalDiscreteDomain(X, x > 2))
def test_sympy__stats__joint_rv__JointPSpace():
from sympy.stats.joint_rv import JointPSpace, JointDistribution
assert _test_args(JointPSpace('X', JointDistribution(1)))
def test_sympy__stats__joint_rv__JointRandomSymbol():
from sympy.stats.joint_rv import JointRandomSymbol
assert _test_args(JointRandomSymbol(x))
def test_sympy__stats__joint_rv__JointDistributionHandmade():
from sympy import Indexed
from sympy.stats.joint_rv import JointDistributionHandmade
x1, x2 = (Indexed('x', i) for i in (1, 2))
assert _test_args(JointDistributionHandmade(x1 + x2, S.Reals**2))
def test_sympy__stats__joint_rv__MarginalDistribution():
from sympy.stats.rv import RandomSymbol
from sympy.stats.joint_rv import MarginalDistribution
r = RandomSymbol(S('r'))
assert _test_args(MarginalDistribution(r, (r,)))
def test_sympy__stats__joint_rv__CompoundDistribution():
from sympy.stats.joint_rv import CompoundDistribution
from sympy.stats.drv_types import PoissonDistribution
r = PoissonDistribution(x)
assert _test_args(CompoundDistribution(PoissonDistribution(r)))
@SKIP("abstract class")
def test_sympy__stats__drv__SingleDiscreteDistribution():
pass
@SKIP("abstract class")
def test_sympy__stats__drv__DiscreteDistribution():
pass
@SKIP("abstract class")
def test_sympy__stats__drv__DiscreteDomain():
pass
def test_sympy__stats__rv__RandomDomain():
from sympy.stats.rv import RandomDomain
from sympy.sets.sets import FiniteSet
assert _test_args(RandomDomain(FiniteSet(x), FiniteSet(1, 2, 3)))
def test_sympy__stats__rv__SingleDomain():
from sympy.stats.rv import SingleDomain
from sympy.sets.sets import FiniteSet
assert _test_args(SingleDomain(x, FiniteSet(1, 2, 3)))
def test_sympy__stats__rv__ConditionalDomain():
from sympy.stats.rv import ConditionalDomain, RandomDomain
from sympy.sets.sets import FiniteSet
D = RandomDomain(FiniteSet(x), FiniteSet(1, 2))
assert _test_args(ConditionalDomain(D, x > 1))
def test_sympy__stats__rv__PSpace():
from sympy.stats.rv import PSpace, RandomDomain
from sympy import FiniteSet
D = RandomDomain(FiniteSet(x), FiniteSet(1, 2, 3, 4, 5, 6))
assert _test_args(PSpace(D, die))
@SKIP("abstract Class")
def test_sympy__stats__rv__SinglePSpace():
pass
def test_sympy__stats__rv__RandomSymbol():
from sympy.stats.rv import RandomSymbol
from sympy.stats.crv import SingleContinuousPSpace
A = SingleContinuousPSpace(x, nd)
assert _test_args(RandomSymbol(x, A))
@SKIP("abstract Class")
def test_sympy__stats__rv__ProductPSpace():
pass
def test_sympy__stats__rv__IndependentProductPSpace():
from sympy.stats.rv import IndependentProductPSpace
from sympy.stats.crv import SingleContinuousPSpace
A = SingleContinuousPSpace(x, nd)
B = SingleContinuousPSpace(y, nd)
assert _test_args(IndependentProductPSpace(A, B))
def test_sympy__stats__rv__ProductDomain():
from sympy.stats.rv import ProductDomain, SingleDomain
D = SingleDomain(x, Interval(-oo, oo))
E = SingleDomain(y, Interval(0, oo))
assert _test_args(ProductDomain(D, E))
def test_sympy__stats__symbolic_probability__Probability():
from sympy.stats.symbolic_probability import Probability
from sympy.stats import Normal
X = Normal('X', 0, 1)
assert _test_args(Probability(X > 0))
def test_sympy__stats__symbolic_probability__Expectation():
from sympy.stats.symbolic_probability import Expectation
from sympy.stats import Normal
X = Normal('X', 0, 1)
assert _test_args(Expectation(X > 0))
def test_sympy__stats__symbolic_probability__Covariance():
from sympy.stats.symbolic_probability import Covariance
from sympy.stats import Normal
X = Normal('X', 0, 1)
Y = Normal('Y', 0, 3)
assert _test_args(Covariance(X, Y))
def test_sympy__stats__symbolic_probability__Variance():
from sympy.stats.symbolic_probability import Variance
from sympy.stats import Normal
X = Normal('X', 0, 1)
assert _test_args(Variance(X))
def test_sympy__stats__frv_types__DiscreteUniformDistribution():
from sympy.stats.frv_types import DiscreteUniformDistribution
from sympy.core.containers import Tuple
assert _test_args(DiscreteUniformDistribution(Tuple(*list(range(6)))))
def test_sympy__stats__frv_types__DieDistribution():
assert _test_args(die)
def test_sympy__stats__frv_types__BernoulliDistribution():
from sympy.stats.frv_types import BernoulliDistribution
assert _test_args(BernoulliDistribution(S.Half, 0, 1))
def test_sympy__stats__frv_types__BinomialDistribution():
from sympy.stats.frv_types import BinomialDistribution
assert _test_args(BinomialDistribution(5, S.Half, 1, 0))
def test_sympy__stats__frv_types__BetaBinomialDistribution():
from sympy.stats.frv_types import BetaBinomialDistribution
assert _test_args(BetaBinomialDistribution(5, 1, 1))
def test_sympy__stats__frv_types__HypergeometricDistribution():
from sympy.stats.frv_types import HypergeometricDistribution
assert _test_args(HypergeometricDistribution(10, 5, 3))
def test_sympy__stats__frv_types__RademacherDistribution():
from sympy.stats.frv_types import RademacherDistribution
assert _test_args(RademacherDistribution())
def test_sympy__stats__frv__FiniteDomain():
from sympy.stats.frv import FiniteDomain
assert _test_args(FiniteDomain({(x, 1), (x, 2)})) # x can be 1 or 2
def test_sympy__stats__frv__SingleFiniteDomain():
from sympy.stats.frv import SingleFiniteDomain
assert _test_args(SingleFiniteDomain(x, {1, 2})) # x can be 1 or 2
def test_sympy__stats__frv__ProductFiniteDomain():
from sympy.stats.frv import SingleFiniteDomain, ProductFiniteDomain
xd = SingleFiniteDomain(x, {1, 2})
yd = SingleFiniteDomain(y, {1, 2})
assert _test_args(ProductFiniteDomain(xd, yd))
def test_sympy__stats__frv__ConditionalFiniteDomain():
from sympy.stats.frv import SingleFiniteDomain, ConditionalFiniteDomain
xd = SingleFiniteDomain(x, {1, 2})
assert _test_args(ConditionalFiniteDomain(xd, x > 1))
def test_sympy__stats__frv__FinitePSpace():
from sympy.stats.frv import FinitePSpace, SingleFiniteDomain
xd = SingleFiniteDomain(x, {1, 2, 3, 4, 5, 6})
assert _test_args(FinitePSpace(xd, {(x, 1): S.Half, (x, 2): S.Half}))
xd = SingleFiniteDomain(x, {1, 2})
assert _test_args(FinitePSpace(xd, {(x, 1): S.Half, (x, 2): S.Half}))
def test_sympy__stats__frv__SingleFinitePSpace():
from sympy.stats.frv import SingleFinitePSpace
from sympy import Symbol
assert _test_args(SingleFinitePSpace(Symbol('x'), die))
def test_sympy__stats__frv__ProductFinitePSpace():
from sympy.stats.frv import SingleFinitePSpace, ProductFinitePSpace
from sympy import Symbol
xp = SingleFinitePSpace(Symbol('x'), die)
yp = SingleFinitePSpace(Symbol('y'), die)
assert _test_args(ProductFinitePSpace(xp, yp))
@SKIP("abstract class")
def test_sympy__stats__frv__SingleFiniteDistribution():
pass
@SKIP("abstract class")
def test_sympy__stats__crv__ContinuousDistribution():
pass
def test_sympy__stats__frv_types__FiniteDistributionHandmade():
from sympy.stats.frv_types import FiniteDistributionHandmade
from sympy import Dict
assert _test_args(FiniteDistributionHandmade(Dict({1: 1})))
def test_sympy__stats__crv_types__ContinuousDistributionHandmade():
from sympy.stats.crv_types import ContinuousDistributionHandmade
from sympy import Interval, Lambda
from sympy.abc import x
assert _test_args(ContinuousDistributionHandmade(Lambda(x, 2*x),
Interval(0, 1)))
def test_sympy__stats__drv_types__DiscreteDistributionHandmade():
from sympy.stats.drv_types import DiscreteDistributionHandmade
from sympy import Lambda, FiniteSet
from sympy.abc import x
assert _test_args(DiscreteDistributionHandmade(Lambda(x, Rational(1, 10)),
FiniteSet(*range(10))))
def test_sympy__stats__rv__Density():
from sympy.stats.rv import Density
from sympy.stats.crv_types import Normal
assert _test_args(Density(Normal('x', 0, 1)))
def test_sympy__stats__crv_types__ArcsinDistribution():
from sympy.stats.crv_types import ArcsinDistribution
assert _test_args(ArcsinDistribution(0, 1))
def test_sympy__stats__crv_types__BeniniDistribution():
from sympy.stats.crv_types import BeniniDistribution
assert _test_args(BeniniDistribution(1, 1, 1))
def test_sympy__stats__crv_types__BetaDistribution():
from sympy.stats.crv_types import BetaDistribution
assert _test_args(BetaDistribution(1, 1))
def test_sympy__stats__crv_types__BetaNoncentralDistribution():
from sympy.stats.crv_types import BetaNoncentralDistribution
assert _test_args(BetaNoncentralDistribution(1, 1, 1))
def test_sympy__stats__crv_types__BetaPrimeDistribution():
from sympy.stats.crv_types import BetaPrimeDistribution
assert _test_args(BetaPrimeDistribution(1, 1))
def test_sympy__stats__crv_types__BoundedParetoDistribution():
from sympy.stats.crv_types import BoundedParetoDistribution
assert _test_args(BoundedParetoDistribution(1, 1, 2))
def test_sympy__stats__crv_types__CauchyDistribution():
from sympy.stats.crv_types import CauchyDistribution
assert _test_args(CauchyDistribution(0, 1))
def test_sympy__stats__crv_types__ChiDistribution():
from sympy.stats.crv_types import ChiDistribution
assert _test_args(ChiDistribution(1))
def test_sympy__stats__crv_types__ChiNoncentralDistribution():
from sympy.stats.crv_types import ChiNoncentralDistribution
assert _test_args(ChiNoncentralDistribution(1,1))
def test_sympy__stats__crv_types__ChiSquaredDistribution():
from sympy.stats.crv_types import ChiSquaredDistribution
assert _test_args(ChiSquaredDistribution(1))
def test_sympy__stats__crv_types__DagumDistribution():
from sympy.stats.crv_types import DagumDistribution
assert _test_args(DagumDistribution(1, 1, 1))
def test_sympy__stats__crv_types__ExGaussianDistribution():
from sympy.stats.crv_types import ExGaussianDistribution
assert _test_args(ExGaussianDistribution(1, 1, 1))
def test_sympy__stats__crv_types__ExponentialDistribution():
from sympy.stats.crv_types import ExponentialDistribution
assert _test_args(ExponentialDistribution(1))
def test_sympy__stats__crv_types__ExponentialPowerDistribution():
from sympy.stats.crv_types import ExponentialPowerDistribution
assert _test_args(ExponentialPowerDistribution(0, 1, 1))
def test_sympy__stats__crv_types__FDistributionDistribution():
from sympy.stats.crv_types import FDistributionDistribution
assert _test_args(FDistributionDistribution(1, 1))
def test_sympy__stats__crv_types__FisherZDistribution():
from sympy.stats.crv_types import FisherZDistribution
assert _test_args(FisherZDistribution(1, 1))
def test_sympy__stats__crv_types__FrechetDistribution():
from sympy.stats.crv_types import FrechetDistribution
assert _test_args(FrechetDistribution(1, 1, 1))
def test_sympy__stats__crv_types__GammaInverseDistribution():
from sympy.stats.crv_types import GammaInverseDistribution
assert _test_args(GammaInverseDistribution(1, 1))
def test_sympy__stats__crv_types__GammaDistribution():
from sympy.stats.crv_types import GammaDistribution
assert _test_args(GammaDistribution(1, 1))
def test_sympy__stats__crv_types__GumbelDistribution():
from sympy.stats.crv_types import GumbelDistribution
assert _test_args(GumbelDistribution(1, 1, False))
def test_sympy__stats__crv_types__GompertzDistribution():
from sympy.stats.crv_types import GompertzDistribution
assert _test_args(GompertzDistribution(1, 1))
def test_sympy__stats__crv_types__KumaraswamyDistribution():
from sympy.stats.crv_types import KumaraswamyDistribution
assert _test_args(KumaraswamyDistribution(1, 1))
def test_sympy__stats__crv_types__LaplaceDistribution():
from sympy.stats.crv_types import LaplaceDistribution
assert _test_args(LaplaceDistribution(0, 1))
def test_sympy__stats__crv_types__LevyDistribution():
from sympy.stats.crv_types import LevyDistribution
assert _test_args(LevyDistribution(0, 1))
def test_sympy__stats__crv_types__LogisticDistribution():
from sympy.stats.crv_types import LogisticDistribution
assert _test_args(LogisticDistribution(0, 1))
def test_sympy__stats__crv_types__LogLogisticDistribution():
from sympy.stats.crv_types import LogLogisticDistribution
assert _test_args(LogLogisticDistribution(1, 1))
def test_sympy__stats__crv_types__LogNormalDistribution():
from sympy.stats.crv_types import LogNormalDistribution
assert _test_args(LogNormalDistribution(0, 1))
def test_sympy__stats__crv_types__LomaxDistribution():
from sympy.stats.crv_types import LomaxDistribution
assert _test_args(LomaxDistribution(1, 2))
def test_sympy__stats__crv_types__MaxwellDistribution():
from sympy.stats.crv_types import MaxwellDistribution
assert _test_args(MaxwellDistribution(1))
def test_sympy__stats__crv_types__MoyalDistribution():
from sympy.stats.crv_types import MoyalDistribution
assert _test_args(MoyalDistribution(1,2))
def test_sympy__stats__crv_types__NakagamiDistribution():
from sympy.stats.crv_types import NakagamiDistribution
assert _test_args(NakagamiDistribution(1, 1))
def test_sympy__stats__crv_types__NormalDistribution():
from sympy.stats.crv_types import NormalDistribution
assert _test_args(NormalDistribution(0, 1))
def test_sympy__stats__crv_types__GaussianInverseDistribution():
from sympy.stats.crv_types import GaussianInverseDistribution
assert _test_args(GaussianInverseDistribution(1, 1))
def test_sympy__stats__crv_types__ParetoDistribution():
from sympy.stats.crv_types import ParetoDistribution
assert _test_args(ParetoDistribution(1, 1))
def test_sympy__stats__crv_types__PowerFunctionDistribution():
from sympy.stats.crv_types import PowerFunctionDistribution
assert _test_args(PowerFunctionDistribution(2,0,1))
def test_sympy__stats__crv_types__QuadraticUDistribution():
from sympy.stats.crv_types import QuadraticUDistribution
assert _test_args(QuadraticUDistribution(1, 2))
def test_sympy__stats__crv_types__RaisedCosineDistribution():
from sympy.stats.crv_types import RaisedCosineDistribution
assert _test_args(RaisedCosineDistribution(1, 1))
def test_sympy__stats__crv_types__RayleighDistribution():
from sympy.stats.crv_types import RayleighDistribution
assert _test_args(RayleighDistribution(1))
def test_sympy__stats__crv_types__ReciprocalDistribution():
from sympy.stats.crv_types import ReciprocalDistribution
assert _test_args(ReciprocalDistribution(5, 30))
def test_sympy__stats__crv_types__ShiftedGompertzDistribution():
from sympy.stats.crv_types import ShiftedGompertzDistribution
assert _test_args(ShiftedGompertzDistribution(1, 1))
def test_sympy__stats__crv_types__StudentTDistribution():
from sympy.stats.crv_types import StudentTDistribution
assert _test_args(StudentTDistribution(1))
def test_sympy__stats__crv_types__TrapezoidalDistribution():
from sympy.stats.crv_types import TrapezoidalDistribution
assert _test_args(TrapezoidalDistribution(1, 2, 3, 4))
def test_sympy__stats__crv_types__TriangularDistribution():
from sympy.stats.crv_types import TriangularDistribution
assert _test_args(TriangularDistribution(-1, 0, 1))
def test_sympy__stats__crv_types__UniformDistribution():
from sympy.stats.crv_types import UniformDistribution
assert _test_args(UniformDistribution(0, 1))
def test_sympy__stats__crv_types__UniformSumDistribution():
from sympy.stats.crv_types import UniformSumDistribution
assert _test_args(UniformSumDistribution(1))
def test_sympy__stats__crv_types__VonMisesDistribution():
from sympy.stats.crv_types import VonMisesDistribution
assert _test_args(VonMisesDistribution(1, 1))
def test_sympy__stats__crv_types__WeibullDistribution():
from sympy.stats.crv_types import WeibullDistribution
assert _test_args(WeibullDistribution(1, 1))
def test_sympy__stats__crv_types__WignerSemicircleDistribution():
from sympy.stats.crv_types import WignerSemicircleDistribution
assert _test_args(WignerSemicircleDistribution(1))
def test_sympy__stats__drv_types__GeometricDistribution():
from sympy.stats.drv_types import GeometricDistribution
assert _test_args(GeometricDistribution(.5))
def test_sympy__stats__drv_types__HermiteDistribution():
from sympy.stats.drv_types import HermiteDistribution
assert _test_args(HermiteDistribution(1, 2))
def test_sympy__stats__drv_types__LogarithmicDistribution():
from sympy.stats.drv_types import LogarithmicDistribution
assert _test_args(LogarithmicDistribution(.5))
def test_sympy__stats__drv_types__NegativeBinomialDistribution():
from sympy.stats.drv_types import NegativeBinomialDistribution
assert _test_args(NegativeBinomialDistribution(.5, .5))
def test_sympy__stats__drv_types__PoissonDistribution():
from sympy.stats.drv_types import PoissonDistribution
assert _test_args(PoissonDistribution(1))
def test_sympy__stats__drv_types__SkellamDistribution():
from sympy.stats.drv_types import SkellamDistribution
assert _test_args(SkellamDistribution(1, 1))
def test_sympy__stats__drv_types__YuleSimonDistribution():
from sympy.stats.drv_types import YuleSimonDistribution
assert _test_args(YuleSimonDistribution(.5))
def test_sympy__stats__drv_types__ZetaDistribution():
from sympy.stats.drv_types import ZetaDistribution
assert _test_args(ZetaDistribution(1.5))
def test_sympy__stats__joint_rv__JointDistribution():
from sympy.stats.joint_rv import JointDistribution
assert _test_args(JointDistribution(1, 2, 3, 4))
def test_sympy__stats__joint_rv_types__MultivariateNormalDistribution():
from sympy.stats.joint_rv_types import MultivariateNormalDistribution
assert _test_args(
MultivariateNormalDistribution([0, 1], [[1, 0],[0, 1]]))
def test_sympy__stats__joint_rv_types__MultivariateLaplaceDistribution():
from sympy.stats.joint_rv_types import MultivariateLaplaceDistribution
assert _test_args(MultivariateLaplaceDistribution([0, 1], [[1, 0],[0, 1]]))
def test_sympy__stats__joint_rv_types__MultivariateTDistribution():
from sympy.stats.joint_rv_types import MultivariateTDistribution
assert _test_args(MultivariateTDistribution([0, 1], [[1, 0],[0, 1]], 1))
def test_sympy__stats__joint_rv_types__NormalGammaDistribution():
from sympy.stats.joint_rv_types import NormalGammaDistribution
assert _test_args(NormalGammaDistribution(1, 2, 3, 4))
def test_sympy__stats__joint_rv_types__GeneralizedMultivariateLogGammaDistribution():
from sympy.stats.joint_rv_types import GeneralizedMultivariateLogGammaDistribution
v, l, mu = (4, [1, 2, 3, 4], [1, 2, 3, 4])
assert _test_args(GeneralizedMultivariateLogGammaDistribution(S.Half, v, l, mu))
def test_sympy__stats__joint_rv_types__MultivariateBetaDistribution():
from sympy.stats.joint_rv_types import MultivariateBetaDistribution
assert _test_args(MultivariateBetaDistribution([1, 2, 3]))
def test_sympy__stats__joint_rv_types__MultivariateEwensDistribution():
from sympy.stats.joint_rv_types import MultivariateEwensDistribution
assert _test_args(MultivariateEwensDistribution(5, 1))
def test_sympy__stats__joint_rv_types__MultinomialDistribution():
from sympy.stats.joint_rv_types import MultinomialDistribution
assert _test_args(MultinomialDistribution(5, [0.5, 0.1, 0.3]))
def test_sympy__stats__joint_rv_types__NegativeMultinomialDistribution():
from sympy.stats.joint_rv_types import NegativeMultinomialDistribution
assert _test_args(NegativeMultinomialDistribution(5, [0.5, 0.1, 0.3]))
def test_sympy__stats__rv__RandomIndexedSymbol():
from sympy.stats.rv import RandomIndexedSymbol, pspace
from sympy.stats.stochastic_process_types import DiscreteMarkovChain
X = DiscreteMarkovChain("X")
assert _test_args(RandomIndexedSymbol(X[0].symbol, pspace(X[0])))
def test_sympy__stats__rv__RandomMatrixSymbol():
from sympy.stats.rv import RandomMatrixSymbol
from sympy.stats.random_matrix import RandomMatrixPSpace
pspace = RandomMatrixPSpace('P')
assert _test_args(RandomMatrixSymbol('M', 3, 3, pspace))
def test_sympy__stats__stochastic_process__StochasticPSpace():
from sympy.stats.stochastic_process import StochasticPSpace
from sympy.stats.stochastic_process_types import StochasticProcess
from sympy.stats.frv_types import BernoulliDistribution
assert _test_args(StochasticPSpace("Y", StochasticProcess("Y", [1, 2, 3]), BernoulliDistribution(S.Half, 1, 0)))
def test_sympy__stats__stochastic_process_types__StochasticProcess():
from sympy.stats.stochastic_process_types import StochasticProcess
assert _test_args(StochasticProcess("Y", [1, 2, 3]))
def test_sympy__stats__stochastic_process_types__MarkovProcess():
from sympy.stats.stochastic_process_types import MarkovProcess
assert _test_args(MarkovProcess("Y", [1, 2, 3]))
def test_sympy__stats__stochastic_process_types__DiscreteTimeStochasticProcess():
from sympy.stats.stochastic_process_types import DiscreteTimeStochasticProcess
assert _test_args(DiscreteTimeStochasticProcess("Y", [1, 2, 3]))
def test_sympy__stats__stochastic_process_types__ContinuousTimeStochasticProcess():
from sympy.stats.stochastic_process_types import ContinuousTimeStochasticProcess
assert _test_args(ContinuousTimeStochasticProcess("Y", [1, 2, 3]))
def test_sympy__stats__stochastic_process_types__TransitionMatrixOf():
from sympy.stats.stochastic_process_types import TransitionMatrixOf, DiscreteMarkovChain
from sympy import MatrixSymbol
DMC = DiscreteMarkovChain("Y")
assert _test_args(TransitionMatrixOf(DMC, MatrixSymbol('T', 3, 3)))
def test_sympy__stats__stochastic_process_types__GeneratorMatrixOf():
from sympy.stats.stochastic_process_types import GeneratorMatrixOf, ContinuousMarkovChain
from sympy import MatrixSymbol
DMC = ContinuousMarkovChain("Y")
assert _test_args(GeneratorMatrixOf(DMC, MatrixSymbol('T', 3, 3)))
def test_sympy__stats__stochastic_process_types__StochasticStateSpaceOf():
from sympy.stats.stochastic_process_types import StochasticStateSpaceOf, DiscreteMarkovChain
DMC = DiscreteMarkovChain("Y")
assert _test_args(StochasticStateSpaceOf(DMC, [0, 1, 2]))
def test_sympy__stats__stochastic_process_types__DiscreteMarkovChain():
from sympy.stats.stochastic_process_types import DiscreteMarkovChain
from sympy import MatrixSymbol
assert _test_args(DiscreteMarkovChain("Y", [0, 1, 2], MatrixSymbol('T', 3, 3)))
def test_sympy__stats__stochastic_process_types__ContinuousMarkovChain():
from sympy.stats.stochastic_process_types import ContinuousMarkovChain
from sympy import MatrixSymbol
assert _test_args(ContinuousMarkovChain("Y", [0, 1, 2], MatrixSymbol('T', 3, 3)))
def test_sympy__stats__stochastic_process_types__BernoulliProcess():
from sympy.stats.stochastic_process_types import BernoulliProcess
assert _test_args(BernoulliProcess("B", 0.5, 1, 0))
def test_sympy__stats__random_matrix__RandomMatrixPSpace():
from sympy.stats.random_matrix import RandomMatrixPSpace
from sympy.stats.random_matrix_models import RandomMatrixEnsemble
assert _test_args(RandomMatrixPSpace('P', RandomMatrixEnsemble('R', 3)))
def test_sympy__stats__random_matrix_models__RandomMatrixEnsemble():
from sympy.stats.random_matrix_models import RandomMatrixEnsemble
assert _test_args(RandomMatrixEnsemble('R', 3))
def test_sympy__stats__random_matrix_models__GaussianEnsemble():
from sympy.stats.random_matrix_models import GaussianEnsemble
assert _test_args(GaussianEnsemble('G', 3))
def test_sympy__stats__random_matrix_models__GaussianUnitaryEnsemble():
from sympy.stats import GaussianUnitaryEnsemble
assert _test_args(GaussianUnitaryEnsemble('U', 3))
def test_sympy__stats__random_matrix_models__GaussianOrthogonalEnsemble():
from sympy.stats import GaussianOrthogonalEnsemble
assert _test_args(GaussianOrthogonalEnsemble('U', 3))
def test_sympy__stats__random_matrix_models__GaussianSymplecticEnsemble():
from sympy.stats import GaussianSymplecticEnsemble
assert _test_args(GaussianSymplecticEnsemble('U', 3))
def test_sympy__stats__random_matrix_models__CircularEnsemble():
from sympy.stats import CircularEnsemble
assert _test_args(CircularEnsemble('C', 3))
def test_sympy__stats__random_matrix_models__CircularUnitaryEnsemble():
from sympy.stats import CircularUnitaryEnsemble
assert _test_args(CircularUnitaryEnsemble('U', 3))
def test_sympy__stats__random_matrix_models__CircularOrthogonalEnsemble():
from sympy.stats import CircularOrthogonalEnsemble
assert _test_args(CircularOrthogonalEnsemble('O', 3))
def test_sympy__stats__random_matrix_models__CircularSymplecticEnsemble():
from sympy.stats import CircularSymplecticEnsemble
assert _test_args(CircularSymplecticEnsemble('S', 3))
def test_sympy__core__symbol__Dummy():
from sympy.core.symbol import Dummy
assert _test_args(Dummy('t'))
def test_sympy__core__symbol__Symbol():
from sympy.core.symbol import Symbol
assert _test_args(Symbol('t'))
def test_sympy__core__symbol__Wild():
from sympy.core.symbol import Wild
assert _test_args(Wild('x', exclude=[x]))
@SKIP("abstract class")
def test_sympy__functions__combinatorial__factorials__CombinatorialFunction():
pass
def test_sympy__functions__combinatorial__factorials__FallingFactorial():
from sympy.functions.combinatorial.factorials import FallingFactorial
assert _test_args(FallingFactorial(2, x))
def test_sympy__functions__combinatorial__factorials__MultiFactorial():
from sympy.functions.combinatorial.factorials import MultiFactorial
assert _test_args(MultiFactorial(x))
def test_sympy__functions__combinatorial__factorials__RisingFactorial():
from sympy.functions.combinatorial.factorials import RisingFactorial
assert _test_args(RisingFactorial(2, x))
def test_sympy__functions__combinatorial__factorials__binomial():
from sympy.functions.combinatorial.factorials import binomial
assert _test_args(binomial(2, x))
def test_sympy__functions__combinatorial__factorials__subfactorial():
from sympy.functions.combinatorial.factorials import subfactorial
assert _test_args(subfactorial(1))
def test_sympy__functions__combinatorial__factorials__factorial():
from sympy.functions.combinatorial.factorials import factorial
assert _test_args(factorial(x))
def test_sympy__functions__combinatorial__factorials__factorial2():
from sympy.functions.combinatorial.factorials import factorial2
assert _test_args(factorial2(x))
def test_sympy__functions__combinatorial__numbers__bell():
from sympy.functions.combinatorial.numbers import bell
assert _test_args(bell(x, y))
def test_sympy__functions__combinatorial__numbers__bernoulli():
from sympy.functions.combinatorial.numbers import bernoulli
assert _test_args(bernoulli(x))
def test_sympy__functions__combinatorial__numbers__catalan():
from sympy.functions.combinatorial.numbers import catalan
assert _test_args(catalan(x))
def test_sympy__functions__combinatorial__numbers__genocchi():
from sympy.functions.combinatorial.numbers import genocchi
assert _test_args(genocchi(x))
def test_sympy__functions__combinatorial__numbers__euler():
from sympy.functions.combinatorial.numbers import euler
assert _test_args(euler(x))
def test_sympy__functions__combinatorial__numbers__carmichael():
from sympy.functions.combinatorial.numbers import carmichael
assert _test_args(carmichael(x))
def test_sympy__functions__combinatorial__numbers__fibonacci():
from sympy.functions.combinatorial.numbers import fibonacci
assert _test_args(fibonacci(x))
def test_sympy__functions__combinatorial__numbers__tribonacci():
from sympy.functions.combinatorial.numbers import tribonacci
assert _test_args(tribonacci(x))
def test_sympy__functions__combinatorial__numbers__harmonic():
from sympy.functions.combinatorial.numbers import harmonic
assert _test_args(harmonic(x, 2))
def test_sympy__functions__combinatorial__numbers__lucas():
from sympy.functions.combinatorial.numbers import lucas
assert _test_args(lucas(x))
def test_sympy__functions__combinatorial__numbers__partition():
from sympy.core.symbol import Symbol
from sympy.functions.combinatorial.numbers import partition
assert _test_args(partition(Symbol('a', integer=True)))
def test_sympy__functions__elementary__complexes__Abs():
from sympy.functions.elementary.complexes import Abs
assert _test_args(Abs(x))
def test_sympy__functions__elementary__complexes__adjoint():
from sympy.functions.elementary.complexes import adjoint
assert _test_args(adjoint(x))
def test_sympy__functions__elementary__complexes__arg():
from sympy.functions.elementary.complexes import arg
assert _test_args(arg(x))
def test_sympy__functions__elementary__complexes__conjugate():
from sympy.functions.elementary.complexes import conjugate
assert _test_args(conjugate(x))
def test_sympy__functions__elementary__complexes__im():
from sympy.functions.elementary.complexes import im
assert _test_args(im(x))
def test_sympy__functions__elementary__complexes__re():
from sympy.functions.elementary.complexes import re
assert _test_args(re(x))
def test_sympy__functions__elementary__complexes__sign():
from sympy.functions.elementary.complexes import sign
assert _test_args(sign(x))
def test_sympy__functions__elementary__complexes__polar_lift():
from sympy.functions.elementary.complexes import polar_lift
assert _test_args(polar_lift(x))
def test_sympy__functions__elementary__complexes__periodic_argument():
from sympy.functions.elementary.complexes import periodic_argument
assert _test_args(periodic_argument(x, y))
def test_sympy__functions__elementary__complexes__principal_branch():
from sympy.functions.elementary.complexes import principal_branch
assert _test_args(principal_branch(x, y))
def test_sympy__functions__elementary__complexes__transpose():
from sympy.functions.elementary.complexes import transpose
assert _test_args(transpose(x))
def test_sympy__functions__elementary__exponential__LambertW():
from sympy.functions.elementary.exponential import LambertW
assert _test_args(LambertW(2))
@SKIP("abstract class")
def test_sympy__functions__elementary__exponential__ExpBase():
pass
def test_sympy__functions__elementary__exponential__exp():
from sympy.functions.elementary.exponential import exp
assert _test_args(exp(2))
def test_sympy__functions__elementary__exponential__exp_polar():
from sympy.functions.elementary.exponential import exp_polar
assert _test_args(exp_polar(2))
def test_sympy__functions__elementary__exponential__log():
from sympy.functions.elementary.exponential import log
assert _test_args(log(2))
@SKIP("abstract class")
def test_sympy__functions__elementary__hyperbolic__HyperbolicFunction():
pass
@SKIP("abstract class")
def test_sympy__functions__elementary__hyperbolic__ReciprocalHyperbolicFunction():
pass
@SKIP("abstract class")
def test_sympy__functions__elementary__hyperbolic__InverseHyperbolicFunction():
pass
def test_sympy__functions__elementary__hyperbolic__acosh():
from sympy.functions.elementary.hyperbolic import acosh
assert _test_args(acosh(2))
def test_sympy__functions__elementary__hyperbolic__acoth():
from sympy.functions.elementary.hyperbolic import acoth
assert _test_args(acoth(2))
def test_sympy__functions__elementary__hyperbolic__asinh():
from sympy.functions.elementary.hyperbolic import asinh
assert _test_args(asinh(2))
def test_sympy__functions__elementary__hyperbolic__atanh():
from sympy.functions.elementary.hyperbolic import atanh
assert _test_args(atanh(2))
def test_sympy__functions__elementary__hyperbolic__asech():
from sympy.functions.elementary.hyperbolic import asech
assert _test_args(asech(2))
def test_sympy__functions__elementary__hyperbolic__acsch():
from sympy.functions.elementary.hyperbolic import acsch
assert _test_args(acsch(2))
def test_sympy__functions__elementary__hyperbolic__cosh():
from sympy.functions.elementary.hyperbolic import cosh
assert _test_args(cosh(2))
def test_sympy__functions__elementary__hyperbolic__coth():
from sympy.functions.elementary.hyperbolic import coth
assert _test_args(coth(2))
def test_sympy__functions__elementary__hyperbolic__csch():
from sympy.functions.elementary.hyperbolic import csch
assert _test_args(csch(2))
def test_sympy__functions__elementary__hyperbolic__sech():
from sympy.functions.elementary.hyperbolic import sech
assert _test_args(sech(2))
def test_sympy__functions__elementary__hyperbolic__sinh():
from sympy.functions.elementary.hyperbolic import sinh
assert _test_args(sinh(2))
def test_sympy__functions__elementary__hyperbolic__tanh():
from sympy.functions.elementary.hyperbolic import tanh
assert _test_args(tanh(2))
@SKIP("does this work at all?")
def test_sympy__functions__elementary__integers__RoundFunction():
from sympy.functions.elementary.integers import RoundFunction
assert _test_args(RoundFunction())
def test_sympy__functions__elementary__integers__ceiling():
from sympy.functions.elementary.integers import ceiling
assert _test_args(ceiling(x))
def test_sympy__functions__elementary__integers__floor():
from sympy.functions.elementary.integers import floor
assert _test_args(floor(x))
def test_sympy__functions__elementary__integers__frac():
from sympy.functions.elementary.integers import frac
assert _test_args(frac(x))
def test_sympy__functions__elementary__miscellaneous__IdentityFunction():
from sympy.functions.elementary.miscellaneous import IdentityFunction
assert _test_args(IdentityFunction())
def test_sympy__functions__elementary__miscellaneous__Max():
from sympy.functions.elementary.miscellaneous import Max
assert _test_args(Max(x, 2))
def test_sympy__functions__elementary__miscellaneous__Min():
from sympy.functions.elementary.miscellaneous import Min
assert _test_args(Min(x, 2))
@SKIP("abstract class")
def test_sympy__functions__elementary__miscellaneous__MinMaxBase():
pass
def test_sympy__functions__elementary__piecewise__ExprCondPair():
from sympy.functions.elementary.piecewise import ExprCondPair
assert _test_args(ExprCondPair(1, True))
def test_sympy__functions__elementary__piecewise__Piecewise():
from sympy.functions.elementary.piecewise import Piecewise
assert _test_args(Piecewise((1, x >= 0), (0, True)))
@SKIP("abstract class")
def test_sympy__functions__elementary__trigonometric__TrigonometricFunction():
pass
@SKIP("abstract class")
def test_sympy__functions__elementary__trigonometric__ReciprocalTrigonometricFunction():
pass
@SKIP("abstract class")
def test_sympy__functions__elementary__trigonometric__InverseTrigonometricFunction():
pass
def test_sympy__functions__elementary__trigonometric__acos():
from sympy.functions.elementary.trigonometric import acos
assert _test_args(acos(2))
def test_sympy__functions__elementary__trigonometric__acot():
from sympy.functions.elementary.trigonometric import acot
assert _test_args(acot(2))
def test_sympy__functions__elementary__trigonometric__asin():
from sympy.functions.elementary.trigonometric import asin
assert _test_args(asin(2))
def test_sympy__functions__elementary__trigonometric__asec():
from sympy.functions.elementary.trigonometric import asec
assert _test_args(asec(2))
def test_sympy__functions__elementary__trigonometric__acsc():
from sympy.functions.elementary.trigonometric import acsc
assert _test_args(acsc(2))
def test_sympy__functions__elementary__trigonometric__atan():
from sympy.functions.elementary.trigonometric import atan
assert _test_args(atan(2))
def test_sympy__functions__elementary__trigonometric__atan2():
from sympy.functions.elementary.trigonometric import atan2
assert _test_args(atan2(2, 3))
def test_sympy__functions__elementary__trigonometric__cos():
from sympy.functions.elementary.trigonometric import cos
assert _test_args(cos(2))
def test_sympy__functions__elementary__trigonometric__csc():
from sympy.functions.elementary.trigonometric import csc
assert _test_args(csc(2))
def test_sympy__functions__elementary__trigonometric__cot():
from sympy.functions.elementary.trigonometric import cot
assert _test_args(cot(2))
def test_sympy__functions__elementary__trigonometric__sin():
assert _test_args(sin(2))
def test_sympy__functions__elementary__trigonometric__sinc():
from sympy.functions.elementary.trigonometric import sinc
assert _test_args(sinc(2))
def test_sympy__functions__elementary__trigonometric__sec():
from sympy.functions.elementary.trigonometric import sec
assert _test_args(sec(2))
def test_sympy__functions__elementary__trigonometric__tan():
from sympy.functions.elementary.trigonometric import tan
assert _test_args(tan(2))
@SKIP("abstract class")
def test_sympy__functions__special__bessel__BesselBase():
pass
@SKIP("abstract class")
def test_sympy__functions__special__bessel__SphericalBesselBase():
pass
@SKIP("abstract class")
def test_sympy__functions__special__bessel__SphericalHankelBase():
pass
def test_sympy__functions__special__bessel__besseli():
from sympy.functions.special.bessel import besseli
assert _test_args(besseli(x, 1))
def test_sympy__functions__special__bessel__besselj():
from sympy.functions.special.bessel import besselj
assert _test_args(besselj(x, 1))
def test_sympy__functions__special__bessel__besselk():
from sympy.functions.special.bessel import besselk
assert _test_args(besselk(x, 1))
def test_sympy__functions__special__bessel__bessely():
from sympy.functions.special.bessel import bessely
assert _test_args(bessely(x, 1))
def test_sympy__functions__special__bessel__hankel1():
from sympy.functions.special.bessel import hankel1
assert _test_args(hankel1(x, 1))
def test_sympy__functions__special__bessel__hankel2():
from sympy.functions.special.bessel import hankel2
assert _test_args(hankel2(x, 1))
def test_sympy__functions__special__bessel__jn():
from sympy.functions.special.bessel import jn
assert _test_args(jn(0, x))
def test_sympy__functions__special__bessel__yn():
from sympy.functions.special.bessel import yn
assert _test_args(yn(0, x))
def test_sympy__functions__special__bessel__hn1():
from sympy.functions.special.bessel import hn1
assert _test_args(hn1(0, x))
def test_sympy__functions__special__bessel__hn2():
from sympy.functions.special.bessel import hn2
assert _test_args(hn2(0, x))
def test_sympy__functions__special__bessel__AiryBase():
pass
def test_sympy__functions__special__bessel__airyai():
from sympy.functions.special.bessel import airyai
assert _test_args(airyai(2))
def test_sympy__functions__special__bessel__airybi():
from sympy.functions.special.bessel import airybi
assert _test_args(airybi(2))
def test_sympy__functions__special__bessel__airyaiprime():
from sympy.functions.special.bessel import airyaiprime
assert _test_args(airyaiprime(2))
def test_sympy__functions__special__bessel__airybiprime():
from sympy.functions.special.bessel import airybiprime
assert _test_args(airybiprime(2))
def test_sympy__functions__special__bessel__marcumq():
from sympy.functions.special.bessel import marcumq
assert _test_args(marcumq(x, y, z))
def test_sympy__functions__special__elliptic_integrals__elliptic_k():
from sympy.functions.special.elliptic_integrals import elliptic_k as K
assert _test_args(K(x))
def test_sympy__functions__special__elliptic_integrals__elliptic_f():
from sympy.functions.special.elliptic_integrals import elliptic_f as F
assert _test_args(F(x, y))
def test_sympy__functions__special__elliptic_integrals__elliptic_e():
from sympy.functions.special.elliptic_integrals import elliptic_e as E
assert _test_args(E(x))
assert _test_args(E(x, y))
def test_sympy__functions__special__elliptic_integrals__elliptic_pi():
from sympy.functions.special.elliptic_integrals import elliptic_pi as P
assert _test_args(P(x, y))
assert _test_args(P(x, y, z))
def test_sympy__functions__special__delta_functions__DiracDelta():
from sympy.functions.special.delta_functions import DiracDelta
assert _test_args(DiracDelta(x, 1))
def test_sympy__functions__special__singularity_functions__SingularityFunction():
from sympy.functions.special.singularity_functions import SingularityFunction
assert _test_args(SingularityFunction(x, y, z))
def test_sympy__functions__special__delta_functions__Heaviside():
from sympy.functions.special.delta_functions import Heaviside
assert _test_args(Heaviside(x))
def test_sympy__functions__special__error_functions__erf():
from sympy.functions.special.error_functions import erf
assert _test_args(erf(2))
def test_sympy__functions__special__error_functions__erfc():
from sympy.functions.special.error_functions import erfc
assert _test_args(erfc(2))
def test_sympy__functions__special__error_functions__erfi():
from sympy.functions.special.error_functions import erfi
assert _test_args(erfi(2))
def test_sympy__functions__special__error_functions__erf2():
from sympy.functions.special.error_functions import erf2
assert _test_args(erf2(2, 3))
def test_sympy__functions__special__error_functions__erfinv():
from sympy.functions.special.error_functions import erfinv
assert _test_args(erfinv(2))
def test_sympy__functions__special__error_functions__erfcinv():
from sympy.functions.special.error_functions import erfcinv
assert _test_args(erfcinv(2))
def test_sympy__functions__special__error_functions__erf2inv():
from sympy.functions.special.error_functions import erf2inv
assert _test_args(erf2inv(2, 3))
@SKIP("abstract class")
def test_sympy__functions__special__error_functions__FresnelIntegral():
pass
def test_sympy__functions__special__error_functions__fresnels():
from sympy.functions.special.error_functions import fresnels
assert _test_args(fresnels(2))
def test_sympy__functions__special__error_functions__fresnelc():
from sympy.functions.special.error_functions import fresnelc
assert _test_args(fresnelc(2))
def test_sympy__functions__special__error_functions__erfs():
from sympy.functions.special.error_functions import _erfs
assert _test_args(_erfs(2))
def test_sympy__functions__special__error_functions__Ei():
from sympy.functions.special.error_functions import Ei
assert _test_args(Ei(2))
def test_sympy__functions__special__error_functions__li():
from sympy.functions.special.error_functions import li
assert _test_args(li(2))
def test_sympy__functions__special__error_functions__Li():
from sympy.functions.special.error_functions import Li
assert _test_args(Li(2))
@SKIP("abstract class")
def test_sympy__functions__special__error_functions__TrigonometricIntegral():
pass
def test_sympy__functions__special__error_functions__Si():
from sympy.functions.special.error_functions import Si
assert _test_args(Si(2))
def test_sympy__functions__special__error_functions__Ci():
from sympy.functions.special.error_functions import Ci
assert _test_args(Ci(2))
def test_sympy__functions__special__error_functions__Shi():
from sympy.functions.special.error_functions import Shi
assert _test_args(Shi(2))
def test_sympy__functions__special__error_functions__Chi():
from sympy.functions.special.error_functions import Chi
assert _test_args(Chi(2))
def test_sympy__functions__special__error_functions__expint():
from sympy.functions.special.error_functions import expint
assert _test_args(expint(y, x))
def test_sympy__functions__special__gamma_functions__gamma():
from sympy.functions.special.gamma_functions import gamma
assert _test_args(gamma(x))
def test_sympy__functions__special__gamma_functions__loggamma():
from sympy.functions.special.gamma_functions import loggamma
assert _test_args(loggamma(2))
def test_sympy__functions__special__gamma_functions__lowergamma():
from sympy.functions.special.gamma_functions import lowergamma
assert _test_args(lowergamma(x, 2))
def test_sympy__functions__special__gamma_functions__polygamma():
from sympy.functions.special.gamma_functions import polygamma
assert _test_args(polygamma(x, 2))
def test_sympy__functions__special__gamma_functions__digamma():
from sympy.functions.special.gamma_functions import digamma
assert _test_args(digamma(x))
def test_sympy__functions__special__gamma_functions__trigamma():
from sympy.functions.special.gamma_functions import trigamma
assert _test_args(trigamma(x))
def test_sympy__functions__special__gamma_functions__uppergamma():
from sympy.functions.special.gamma_functions import uppergamma
assert _test_args(uppergamma(x, 2))
def test_sympy__functions__special__gamma_functions__multigamma():
from sympy.functions.special.gamma_functions import multigamma
assert _test_args(multigamma(x, 1))
def test_sympy__functions__special__beta_functions__beta():
from sympy.functions.special.beta_functions import beta
assert _test_args(beta(x, x))
def test_sympy__functions__special__mathieu_functions__MathieuBase():
pass
def test_sympy__functions__special__mathieu_functions__mathieus():
from sympy.functions.special.mathieu_functions import mathieus
assert _test_args(mathieus(1, 1, 1))
def test_sympy__functions__special__mathieu_functions__mathieuc():
from sympy.functions.special.mathieu_functions import mathieuc
assert _test_args(mathieuc(1, 1, 1))
def test_sympy__functions__special__mathieu_functions__mathieusprime():
from sympy.functions.special.mathieu_functions import mathieusprime
assert _test_args(mathieusprime(1, 1, 1))
def test_sympy__functions__special__mathieu_functions__mathieucprime():
from sympy.functions.special.mathieu_functions import mathieucprime
assert _test_args(mathieucprime(1, 1, 1))
@SKIP("abstract class")
def test_sympy__functions__special__hyper__TupleParametersBase():
pass
@SKIP("abstract class")
def test_sympy__functions__special__hyper__TupleArg():
pass
def test_sympy__functions__special__hyper__hyper():
from sympy.functions.special.hyper import hyper
assert _test_args(hyper([1, 2, 3], [4, 5], x))
def test_sympy__functions__special__hyper__meijerg():
from sympy.functions.special.hyper import meijerg
assert _test_args(meijerg([1, 2, 3], [4, 5], [6], [], x))
@SKIP("abstract class")
def test_sympy__functions__special__hyper__HyperRep():
pass
def test_sympy__functions__special__hyper__HyperRep_power1():
from sympy.functions.special.hyper import HyperRep_power1
assert _test_args(HyperRep_power1(x, y))
def test_sympy__functions__special__hyper__HyperRep_power2():
from sympy.functions.special.hyper import HyperRep_power2
assert _test_args(HyperRep_power2(x, y))
def test_sympy__functions__special__hyper__HyperRep_log1():
from sympy.functions.special.hyper import HyperRep_log1
assert _test_args(HyperRep_log1(x))
def test_sympy__functions__special__hyper__HyperRep_atanh():
from sympy.functions.special.hyper import HyperRep_atanh
assert _test_args(HyperRep_atanh(x))
def test_sympy__functions__special__hyper__HyperRep_asin1():
from sympy.functions.special.hyper import HyperRep_asin1
assert _test_args(HyperRep_asin1(x))
def test_sympy__functions__special__hyper__HyperRep_asin2():
from sympy.functions.special.hyper import HyperRep_asin2
assert _test_args(HyperRep_asin2(x))
def test_sympy__functions__special__hyper__HyperRep_sqrts1():
from sympy.functions.special.hyper import HyperRep_sqrts1
assert _test_args(HyperRep_sqrts1(x, y))
def test_sympy__functions__special__hyper__HyperRep_sqrts2():
from sympy.functions.special.hyper import HyperRep_sqrts2
assert _test_args(HyperRep_sqrts2(x, y))
def test_sympy__functions__special__hyper__HyperRep_log2():
from sympy.functions.special.hyper import HyperRep_log2
assert _test_args(HyperRep_log2(x))
def test_sympy__functions__special__hyper__HyperRep_cosasin():
from sympy.functions.special.hyper import HyperRep_cosasin
assert _test_args(HyperRep_cosasin(x, y))
def test_sympy__functions__special__hyper__HyperRep_sinasin():
from sympy.functions.special.hyper import HyperRep_sinasin
assert _test_args(HyperRep_sinasin(x, y))
def test_sympy__functions__special__hyper__appellf1():
from sympy.functions.special.hyper import appellf1
a, b1, b2, c, x, y = symbols('a b1 b2 c x y')
assert _test_args(appellf1(a, b1, b2, c, x, y))
@SKIP("abstract class")
def test_sympy__functions__special__polynomials__OrthogonalPolynomial():
pass
def test_sympy__functions__special__polynomials__jacobi():
from sympy.functions.special.polynomials import jacobi
assert _test_args(jacobi(x, 2, 2, 2))
def test_sympy__functions__special__polynomials__gegenbauer():
from sympy.functions.special.polynomials import gegenbauer
assert _test_args(gegenbauer(x, 2, 2))
def test_sympy__functions__special__polynomials__chebyshevt():
from sympy.functions.special.polynomials import chebyshevt
assert _test_args(chebyshevt(x, 2))
def test_sympy__functions__special__polynomials__chebyshevt_root():
from sympy.functions.special.polynomials import chebyshevt_root
assert _test_args(chebyshevt_root(3, 2))
def test_sympy__functions__special__polynomials__chebyshevu():
from sympy.functions.special.polynomials import chebyshevu
assert _test_args(chebyshevu(x, 2))
def test_sympy__functions__special__polynomials__chebyshevu_root():
from sympy.functions.special.polynomials import chebyshevu_root
assert _test_args(chebyshevu_root(3, 2))
def test_sympy__functions__special__polynomials__hermite():
from sympy.functions.special.polynomials import hermite
assert _test_args(hermite(x, 2))
def test_sympy__functions__special__polynomials__legendre():
from sympy.functions.special.polynomials import legendre
assert _test_args(legendre(x, 2))
def test_sympy__functions__special__polynomials__assoc_legendre():
from sympy.functions.special.polynomials import assoc_legendre
assert _test_args(assoc_legendre(x, 0, y))
def test_sympy__functions__special__polynomials__laguerre():
from sympy.functions.special.polynomials import laguerre
assert _test_args(laguerre(x, 2))
def test_sympy__functions__special__polynomials__assoc_laguerre():
from sympy.functions.special.polynomials import assoc_laguerre
assert _test_args(assoc_laguerre(x, 0, y))
def test_sympy__functions__special__spherical_harmonics__Ynm():
from sympy.functions.special.spherical_harmonics import Ynm
assert _test_args(Ynm(1, 1, x, y))
def test_sympy__functions__special__spherical_harmonics__Znm():
from sympy.functions.special.spherical_harmonics import Znm
assert _test_args(Znm(1, 1, x, y))
def test_sympy__functions__special__tensor_functions__LeviCivita():
from sympy.functions.special.tensor_functions import LeviCivita
assert _test_args(LeviCivita(x, y, 2))
def test_sympy__functions__special__tensor_functions__KroneckerDelta():
from sympy.functions.special.tensor_functions import KroneckerDelta
assert _test_args(KroneckerDelta(x, y))
def test_sympy__functions__special__zeta_functions__dirichlet_eta():
from sympy.functions.special.zeta_functions import dirichlet_eta
assert _test_args(dirichlet_eta(x))
def test_sympy__functions__special__zeta_functions__zeta():
from sympy.functions.special.zeta_functions import zeta
assert _test_args(zeta(101))
def test_sympy__functions__special__zeta_functions__lerchphi():
from sympy.functions.special.zeta_functions import lerchphi
assert _test_args(lerchphi(x, y, z))
def test_sympy__functions__special__zeta_functions__polylog():
from sympy.functions.special.zeta_functions import polylog
assert _test_args(polylog(x, y))
def test_sympy__functions__special__zeta_functions__stieltjes():
from sympy.functions.special.zeta_functions import stieltjes
assert _test_args(stieltjes(x, y))
def test_sympy__integrals__integrals__Integral():
from sympy.integrals.integrals import Integral
assert _test_args(Integral(2, (x, 0, 1)))
def test_sympy__integrals__risch__NonElementaryIntegral():
from sympy.integrals.risch import NonElementaryIntegral
assert _test_args(NonElementaryIntegral(exp(-x**2), x))
@SKIP("abstract class")
def test_sympy__integrals__transforms__IntegralTransform():
pass
def test_sympy__integrals__transforms__MellinTransform():
from sympy.integrals.transforms import MellinTransform
assert _test_args(MellinTransform(2, x, y))
def test_sympy__integrals__transforms__InverseMellinTransform():
from sympy.integrals.transforms import InverseMellinTransform
assert _test_args(InverseMellinTransform(2, x, y, 0, 1))
def test_sympy__integrals__transforms__LaplaceTransform():
from sympy.integrals.transforms import LaplaceTransform
assert _test_args(LaplaceTransform(2, x, y))
def test_sympy__integrals__transforms__InverseLaplaceTransform():
from sympy.integrals.transforms import InverseLaplaceTransform
assert _test_args(InverseLaplaceTransform(2, x, y, 0))
@SKIP("abstract class")
def test_sympy__integrals__transforms__FourierTypeTransform():
pass
def test_sympy__integrals__transforms__InverseFourierTransform():
from sympy.integrals.transforms import InverseFourierTransform
assert _test_args(InverseFourierTransform(2, x, y))
def test_sympy__integrals__transforms__FourierTransform():
from sympy.integrals.transforms import FourierTransform
assert _test_args(FourierTransform(2, x, y))
@SKIP("abstract class")
def test_sympy__integrals__transforms__SineCosineTypeTransform():
pass
def test_sympy__integrals__transforms__InverseSineTransform():
from sympy.integrals.transforms import InverseSineTransform
assert _test_args(InverseSineTransform(2, x, y))
def test_sympy__integrals__transforms__SineTransform():
from sympy.integrals.transforms import SineTransform
assert _test_args(SineTransform(2, x, y))
def test_sympy__integrals__transforms__InverseCosineTransform():
from sympy.integrals.transforms import InverseCosineTransform
assert _test_args(InverseCosineTransform(2, x, y))
def test_sympy__integrals__transforms__CosineTransform():
from sympy.integrals.transforms import CosineTransform
assert _test_args(CosineTransform(2, x, y))
@SKIP("abstract class")
def test_sympy__integrals__transforms__HankelTypeTransform():
pass
def test_sympy__integrals__transforms__InverseHankelTransform():
from sympy.integrals.transforms import InverseHankelTransform
assert _test_args(InverseHankelTransform(2, x, y, 0))
def test_sympy__integrals__transforms__HankelTransform():
from sympy.integrals.transforms import HankelTransform
assert _test_args(HankelTransform(2, x, y, 0))
@XFAIL
def test_sympy__liealgebras__cartan_type__CartanType_generator():
from sympy.liealgebras.cartan_type import CartanType_generator
assert _test_args(CartanType_generator("A2"))
@XFAIL
def test_sympy__liealgebras__cartan_type__Standard_Cartan():
from sympy.liealgebras.cartan_type import Standard_Cartan
assert _test_args(Standard_Cartan("A", 2))
@XFAIL
def test_sympy__liealgebras__weyl_group__WeylGroup():
from sympy.liealgebras.weyl_group import WeylGroup
assert _test_args(WeylGroup("B4"))
@XFAIL
def test_sympy__liealgebras__root_system__RootSystem():
from sympy.liealgebras.root_system import RootSystem
assert _test_args(RootSystem("A2"))
@XFAIL
def test_sympy__liealgebras__type_a__TypeA():
from sympy.liealgebras.type_a import TypeA
assert _test_args(TypeA(2))
@XFAIL
def test_sympy__liealgebras__type_b__TypeB():
from sympy.liealgebras.type_b import TypeB
assert _test_args(TypeB(4))
@XFAIL
def test_sympy__liealgebras__type_c__TypeC():
from sympy.liealgebras.type_c import TypeC
assert _test_args(TypeC(4))
@XFAIL
def test_sympy__liealgebras__type_d__TypeD():
from sympy.liealgebras.type_d import TypeD
assert _test_args(TypeD(4))
@XFAIL
def test_sympy__liealgebras__type_e__TypeE():
from sympy.liealgebras.type_e import TypeE
assert _test_args(TypeE(6))
@XFAIL
def test_sympy__liealgebras__type_f__TypeF():
from sympy.liealgebras.type_f import TypeF
assert _test_args(TypeF(4))
@XFAIL
def test_sympy__liealgebras__type_g__TypeG():
from sympy.liealgebras.type_g import TypeG
assert _test_args(TypeG(2))
def test_sympy__logic__boolalg__And():
from sympy.logic.boolalg import And
assert _test_args(And(x, y, 1))
@SKIP("abstract class")
def test_sympy__logic__boolalg__Boolean():
pass
def test_sympy__logic__boolalg__BooleanFunction():
from sympy.logic.boolalg import BooleanFunction
assert _test_args(BooleanFunction(1, 2, 3))
@SKIP("abstract class")
def test_sympy__logic__boolalg__BooleanAtom():
pass
def test_sympy__logic__boolalg__BooleanTrue():
from sympy.logic.boolalg import true
assert _test_args(true)
def test_sympy__logic__boolalg__BooleanFalse():
from sympy.logic.boolalg import false
assert _test_args(false)
def test_sympy__logic__boolalg__Equivalent():
from sympy.logic.boolalg import Equivalent
assert _test_args(Equivalent(x, 2))
def test_sympy__logic__boolalg__ITE():
from sympy.logic.boolalg import ITE
assert _test_args(ITE(x, y, 1))
def test_sympy__logic__boolalg__Implies():
from sympy.logic.boolalg import Implies
assert _test_args(Implies(x, y))
def test_sympy__logic__boolalg__Nand():
from sympy.logic.boolalg import Nand
assert _test_args(Nand(x, y, 1))
def test_sympy__logic__boolalg__Nor():
from sympy.logic.boolalg import Nor
assert _test_args(Nor(x, y))
def test_sympy__logic__boolalg__Not():
from sympy.logic.boolalg import Not
assert _test_args(Not(x))
def test_sympy__logic__boolalg__Or():
from sympy.logic.boolalg import Or
assert _test_args(Or(x, y))
def test_sympy__logic__boolalg__Xor():
from sympy.logic.boolalg import Xor
assert _test_args(Xor(x, y, 2))
def test_sympy__logic__boolalg__Xnor():
from sympy.logic.boolalg import Xnor
assert _test_args(Xnor(x, y, 2))
def test_sympy__matrices__matrices__DeferredVector():
from sympy.matrices.matrices import DeferredVector
assert _test_args(DeferredVector("X"))
@SKIP("abstract class")
def test_sympy__matrices__expressions__matexpr__MatrixBase():
pass
def test_sympy__matrices__immutable__ImmutableDenseMatrix():
from sympy.matrices.immutable import ImmutableDenseMatrix
m = ImmutableDenseMatrix([[1, 2], [3, 4]])
assert _test_args(m)
assert _test_args(Basic(*list(m)))
m = ImmutableDenseMatrix(1, 1, [1])
assert _test_args(m)
assert _test_args(Basic(*list(m)))
m = ImmutableDenseMatrix(2, 2, lambda i, j: 1)
assert m[0, 0] is S.One
m = ImmutableDenseMatrix(2, 2, lambda i, j: 1/(1 + i) + 1/(1 + j))
assert m[1, 1] is S.One # true div. will give 1.0 if i,j not sympified
assert _test_args(m)
assert _test_args(Basic(*list(m)))
def test_sympy__matrices__immutable__ImmutableSparseMatrix():
from sympy.matrices.immutable import ImmutableSparseMatrix
m = ImmutableSparseMatrix([[1, 2], [3, 4]])
assert _test_args(m)
assert _test_args(Basic(*list(m)))
m = ImmutableSparseMatrix(1, 1, {(0, 0): 1})
assert _test_args(m)
assert _test_args(Basic(*list(m)))
m = ImmutableSparseMatrix(1, 1, [1])
assert _test_args(m)
assert _test_args(Basic(*list(m)))
m = ImmutableSparseMatrix(2, 2, lambda i, j: 1)
assert m[0, 0] is S.One
m = ImmutableSparseMatrix(2, 2, lambda i, j: 1/(1 + i) + 1/(1 + j))
assert m[1, 1] is S.One # true div. will give 1.0 if i,j not sympified
assert _test_args(m)
assert _test_args(Basic(*list(m)))
def test_sympy__matrices__expressions__slice__MatrixSlice():
from sympy.matrices.expressions.slice import MatrixSlice
from sympy.matrices.expressions import MatrixSymbol
X = MatrixSymbol('X', 4, 4)
assert _test_args(MatrixSlice(X, (0, 2), (0, 2)))
def test_sympy__matrices__expressions__applyfunc__ElementwiseApplyFunction():
from sympy.matrices.expressions.applyfunc import ElementwiseApplyFunction
from sympy.matrices.expressions import MatrixSymbol
X = MatrixSymbol("X", x, x)
func = Lambda(x, x**2)
assert _test_args(ElementwiseApplyFunction(func, X))
def test_sympy__matrices__expressions__blockmatrix__BlockDiagMatrix():
from sympy.matrices.expressions.blockmatrix import BlockDiagMatrix
from sympy.matrices.expressions import MatrixSymbol
X = MatrixSymbol('X', x, x)
Y = MatrixSymbol('Y', y, y)
assert _test_args(BlockDiagMatrix(X, Y))
def test_sympy__matrices__expressions__blockmatrix__BlockMatrix():
from sympy.matrices.expressions.blockmatrix import BlockMatrix
from sympy.matrices.expressions import MatrixSymbol, ZeroMatrix
X = MatrixSymbol('X', x, x)
Y = MatrixSymbol('Y', y, y)
Z = MatrixSymbol('Z', x, y)
O = ZeroMatrix(y, x)
assert _test_args(BlockMatrix([[X, Z], [O, Y]]))
def test_sympy__matrices__expressions__inverse__Inverse():
from sympy.matrices.expressions.inverse import Inverse
from sympy.matrices.expressions import MatrixSymbol
assert _test_args(Inverse(MatrixSymbol('A', 3, 3)))
def test_sympy__matrices__expressions__matadd__MatAdd():
from sympy.matrices.expressions.matadd import MatAdd
from sympy.matrices.expressions import MatrixSymbol
X = MatrixSymbol('X', x, y)
Y = MatrixSymbol('Y', x, y)
assert _test_args(MatAdd(X, Y))
def test_sympy__matrices__expressions__matexpr__Identity():
from sympy.matrices.expressions.matexpr import Identity
assert _test_args(Identity(3))
def test_sympy__matrices__expressions__matexpr__GenericIdentity():
from sympy.matrices.expressions.matexpr import GenericIdentity
assert _test_args(GenericIdentity())
@SKIP("abstract class")
def test_sympy__matrices__expressions__matexpr__MatrixExpr():
pass
def test_sympy__matrices__expressions__matexpr__MatrixElement():
from sympy.matrices.expressions.matexpr import MatrixSymbol, MatrixElement
from sympy import S
assert _test_args(MatrixElement(MatrixSymbol('A', 3, 5), S(2), S(3)))
def test_sympy__matrices__expressions__matexpr__MatrixSymbol():
from sympy.matrices.expressions.matexpr import MatrixSymbol
assert _test_args(MatrixSymbol('A', 3, 5))
def test_sympy__matrices__expressions__matexpr__ZeroMatrix():
from sympy.matrices.expressions.matexpr import ZeroMatrix
assert _test_args(ZeroMatrix(3, 5))
def test_sympy__matrices__expressions__matexpr__OneMatrix():
from sympy.matrices.expressions.matexpr import OneMatrix
assert _test_args(OneMatrix(3, 5))
def test_sympy__matrices__expressions__matexpr__GenericZeroMatrix():
from sympy.matrices.expressions.matexpr import GenericZeroMatrix
assert _test_args(GenericZeroMatrix())
def test_sympy__matrices__expressions__matmul__MatMul():
from sympy.matrices.expressions.matmul import MatMul
from sympy.matrices.expressions import MatrixSymbol
X = MatrixSymbol('X', x, y)
Y = MatrixSymbol('Y', y, x)
assert _test_args(MatMul(X, Y))
def test_sympy__matrices__expressions__dotproduct__DotProduct():
from sympy.matrices.expressions.dotproduct import DotProduct
from sympy.matrices.expressions import MatrixSymbol
X = MatrixSymbol('X', x, 1)
Y = MatrixSymbol('Y', x, 1)
assert _test_args(DotProduct(X, Y))
def test_sympy__matrices__expressions__diagonal__DiagonalMatrix():
from sympy.matrices.expressions.diagonal import DiagonalMatrix
from sympy.matrices.expressions import MatrixSymbol
x = MatrixSymbol('x', 10, 1)
assert _test_args(DiagonalMatrix(x))
def test_sympy__matrices__expressions__diagonal__DiagonalOf():
from sympy.matrices.expressions.diagonal import DiagonalOf
from sympy.matrices.expressions import MatrixSymbol
X = MatrixSymbol('x', 10, 10)
assert _test_args(DiagonalOf(X))
def test_sympy__matrices__expressions__diagonal__DiagMatrix():
from sympy.matrices.expressions.diagonal import DiagMatrix
from sympy.matrices.expressions import MatrixSymbol
x = MatrixSymbol('x', 10, 1)
assert _test_args(DiagMatrix(x))
def test_sympy__matrices__expressions__hadamard__HadamardProduct():
from sympy.matrices.expressions.hadamard import HadamardProduct
from sympy.matrices.expressions import MatrixSymbol
X = MatrixSymbol('X', x, y)
Y = MatrixSymbol('Y', x, y)
assert _test_args(HadamardProduct(X, Y))
def test_sympy__matrices__expressions__hadamard__HadamardPower():
from sympy.matrices.expressions.hadamard import HadamardPower
from sympy.matrices.expressions import MatrixSymbol
from sympy import Symbol
X = MatrixSymbol('X', x, y)
n = Symbol("n")
assert _test_args(HadamardPower(X, n))
def test_sympy__matrices__expressions__kronecker__KroneckerProduct():
from sympy.matrices.expressions.kronecker import KroneckerProduct
from sympy.matrices.expressions import MatrixSymbol
X = MatrixSymbol('X', x, y)
Y = MatrixSymbol('Y', x, y)
assert _test_args(KroneckerProduct(X, Y))
def test_sympy__matrices__expressions__matpow__MatPow():
from sympy.matrices.expressions.matpow import MatPow
from sympy.matrices.expressions import MatrixSymbol
X = MatrixSymbol('X', x, x)
assert _test_args(MatPow(X, 2))
def test_sympy__matrices__expressions__transpose__Transpose():
from sympy.matrices.expressions.transpose import Transpose
from sympy.matrices.expressions import MatrixSymbol
assert _test_args(Transpose(MatrixSymbol('A', 3, 5)))
def test_sympy__matrices__expressions__adjoint__Adjoint():
from sympy.matrices.expressions.adjoint import Adjoint
from sympy.matrices.expressions import MatrixSymbol
assert _test_args(Adjoint(MatrixSymbol('A', 3, 5)))
def test_sympy__matrices__expressions__trace__Trace():
from sympy.matrices.expressions.trace import Trace
from sympy.matrices.expressions import MatrixSymbol
assert _test_args(Trace(MatrixSymbol('A', 3, 3)))
def test_sympy__matrices__expressions__determinant__Determinant():
from sympy.matrices.expressions.determinant import Determinant
from sympy.matrices.expressions import MatrixSymbol
assert _test_args(Determinant(MatrixSymbol('A', 3, 3)))
def test_sympy__matrices__expressions__funcmatrix__FunctionMatrix():
from sympy.matrices.expressions.funcmatrix import FunctionMatrix
from sympy import symbols
i, j = symbols('i,j')
assert _test_args(FunctionMatrix(3, 3, Lambda((i, j), i - j) ))
def test_sympy__matrices__expressions__fourier__DFT():
from sympy.matrices.expressions.fourier import DFT
from sympy import S
assert _test_args(DFT(S(2)))
def test_sympy__matrices__expressions__fourier__IDFT():
from sympy.matrices.expressions.fourier import IDFT
from sympy import S
assert _test_args(IDFT(S(2)))
from sympy.matrices.expressions import MatrixSymbol
X = MatrixSymbol('X', 10, 10)
def test_sympy__matrices__expressions__factorizations__LofLU():
from sympy.matrices.expressions.factorizations import LofLU
assert _test_args(LofLU(X))
def test_sympy__matrices__expressions__factorizations__UofLU():
from sympy.matrices.expressions.factorizations import UofLU
assert _test_args(UofLU(X))
def test_sympy__matrices__expressions__factorizations__QofQR():
from sympy.matrices.expressions.factorizations import QofQR
assert _test_args(QofQR(X))
def test_sympy__matrices__expressions__factorizations__RofQR():
from sympy.matrices.expressions.factorizations import RofQR
assert _test_args(RofQR(X))
def test_sympy__matrices__expressions__factorizations__LofCholesky():
from sympy.matrices.expressions.factorizations import LofCholesky
assert _test_args(LofCholesky(X))
def test_sympy__matrices__expressions__factorizations__UofCholesky():
from sympy.matrices.expressions.factorizations import UofCholesky
assert _test_args(UofCholesky(X))
def test_sympy__matrices__expressions__factorizations__EigenVectors():
from sympy.matrices.expressions.factorizations import EigenVectors
assert _test_args(EigenVectors(X))
def test_sympy__matrices__expressions__factorizations__EigenValues():
from sympy.matrices.expressions.factorizations import EigenValues
assert _test_args(EigenValues(X))
def test_sympy__matrices__expressions__factorizations__UofSVD():
from sympy.matrices.expressions.factorizations import UofSVD
assert _test_args(UofSVD(X))
def test_sympy__matrices__expressions__factorizations__VofSVD():
from sympy.matrices.expressions.factorizations import VofSVD
assert _test_args(VofSVD(X))
def test_sympy__matrices__expressions__factorizations__SofSVD():
from sympy.matrices.expressions.factorizations import SofSVD
assert _test_args(SofSVD(X))
@SKIP("abstract class")
def test_sympy__matrices__expressions__factorizations__Factorization():
pass
def test_sympy__matrices__expressions__permutation__PermutationMatrix():
from sympy.combinatorics import Permutation
from sympy.matrices.expressions.permutation import PermutationMatrix
assert _test_args(PermutationMatrix(Permutation([2, 0, 1])))
def test_sympy__matrices__expressions__permutation__MatrixPermute():
from sympy.combinatorics import Permutation
from sympy.matrices.expressions.matexpr import MatrixSymbol
from sympy.matrices.expressions.permutation import MatrixPermute
A = MatrixSymbol('A', 3, 3)
assert _test_args(MatrixPermute(A, Permutation([2, 0, 1])))
def test_sympy__matrices__expressions__companion__CompanionMatrix():
from sympy.core.symbol import Symbol
from sympy.matrices.expressions.companion import CompanionMatrix
from sympy.polys.polytools import Poly
x = Symbol('x')
p = Poly([1, 2, 3], x)
assert _test_args(CompanionMatrix(p))
def test_sympy__physics__vector__frame__CoordinateSym():
from sympy.physics.vector import CoordinateSym
from sympy.physics.vector import ReferenceFrame
assert _test_args(CoordinateSym('R_x', ReferenceFrame('R'), 0))
def test_sympy__physics__paulialgebra__Pauli():
from sympy.physics.paulialgebra import Pauli
assert _test_args(Pauli(1))
def test_sympy__physics__quantum__anticommutator__AntiCommutator():
from sympy.physics.quantum.anticommutator import AntiCommutator
assert _test_args(AntiCommutator(x, y))
def test_sympy__physics__quantum__cartesian__PositionBra3D():
from sympy.physics.quantum.cartesian import PositionBra3D
assert _test_args(PositionBra3D(x, y, z))
def test_sympy__physics__quantum__cartesian__PositionKet3D():
from sympy.physics.quantum.cartesian import PositionKet3D
assert _test_args(PositionKet3D(x, y, z))
def test_sympy__physics__quantum__cartesian__PositionState3D():
from sympy.physics.quantum.cartesian import PositionState3D
assert _test_args(PositionState3D(x, y, z))
def test_sympy__physics__quantum__cartesian__PxBra():
from sympy.physics.quantum.cartesian import PxBra
assert _test_args(PxBra(x, y, z))
def test_sympy__physics__quantum__cartesian__PxKet():
from sympy.physics.quantum.cartesian import PxKet
assert _test_args(PxKet(x, y, z))
def test_sympy__physics__quantum__cartesian__PxOp():
from sympy.physics.quantum.cartesian import PxOp
assert _test_args(PxOp(x, y, z))
def test_sympy__physics__quantum__cartesian__XBra():
from sympy.physics.quantum.cartesian import XBra
assert _test_args(XBra(x))
def test_sympy__physics__quantum__cartesian__XKet():
from sympy.physics.quantum.cartesian import XKet
assert _test_args(XKet(x))
def test_sympy__physics__quantum__cartesian__XOp():
from sympy.physics.quantum.cartesian import XOp
assert _test_args(XOp(x))
def test_sympy__physics__quantum__cartesian__YOp():
from sympy.physics.quantum.cartesian import YOp
assert _test_args(YOp(x))
def test_sympy__physics__quantum__cartesian__ZOp():
from sympy.physics.quantum.cartesian import ZOp
assert _test_args(ZOp(x))
def test_sympy__physics__quantum__cg__CG():
from sympy.physics.quantum.cg import CG
from sympy import S
assert _test_args(CG(Rational(3, 2), Rational(3, 2), S.Half, Rational(-1, 2), 1, 1))
def test_sympy__physics__quantum__cg__Wigner3j():
from sympy.physics.quantum.cg import Wigner3j
assert _test_args(Wigner3j(6, 0, 4, 0, 2, 0))
def test_sympy__physics__quantum__cg__Wigner6j():
from sympy.physics.quantum.cg import Wigner6j
assert _test_args(Wigner6j(1, 2, 3, 2, 1, 2))
def test_sympy__physics__quantum__cg__Wigner9j():
from sympy.physics.quantum.cg import Wigner9j
assert _test_args(Wigner9j(2, 1, 1, Rational(3, 2), S.Half, 1, S.Half, S.Half, 0))
def test_sympy__physics__quantum__circuitplot__Mz():
from sympy.physics.quantum.circuitplot import Mz
assert _test_args(Mz(0))
def test_sympy__physics__quantum__circuitplot__Mx():
from sympy.physics.quantum.circuitplot import Mx
assert _test_args(Mx(0))
def test_sympy__physics__quantum__commutator__Commutator():
from sympy.physics.quantum.commutator import Commutator
A, B = symbols('A,B', commutative=False)
assert _test_args(Commutator(A, B))
def test_sympy__physics__quantum__constants__HBar():
from sympy.physics.quantum.constants import HBar
assert _test_args(HBar())
def test_sympy__physics__quantum__dagger__Dagger():
from sympy.physics.quantum.dagger import Dagger
from sympy.physics.quantum.state import Ket
assert _test_args(Dagger(Dagger(Ket('psi'))))
def test_sympy__physics__quantum__gate__CGate():
from sympy.physics.quantum.gate import CGate, Gate
assert _test_args(CGate((0, 1), Gate(2)))
def test_sympy__physics__quantum__gate__CGateS():
from sympy.physics.quantum.gate import CGateS, Gate
assert _test_args(CGateS((0, 1), Gate(2)))
def test_sympy__physics__quantum__gate__CNotGate():
from sympy.physics.quantum.gate import CNotGate
assert _test_args(CNotGate(0, 1))
def test_sympy__physics__quantum__gate__Gate():
from sympy.physics.quantum.gate import Gate
assert _test_args(Gate(0))
def test_sympy__physics__quantum__gate__HadamardGate():
from sympy.physics.quantum.gate import HadamardGate
assert _test_args(HadamardGate(0))
def test_sympy__physics__quantum__gate__IdentityGate():
from sympy.physics.quantum.gate import IdentityGate
assert _test_args(IdentityGate(0))
def test_sympy__physics__quantum__gate__OneQubitGate():
from sympy.physics.quantum.gate import OneQubitGate
assert _test_args(OneQubitGate(0))
def test_sympy__physics__quantum__gate__PhaseGate():
from sympy.physics.quantum.gate import PhaseGate
assert _test_args(PhaseGate(0))
def test_sympy__physics__quantum__gate__SwapGate():
from sympy.physics.quantum.gate import SwapGate
assert _test_args(SwapGate(0, 1))
def test_sympy__physics__quantum__gate__TGate():
from sympy.physics.quantum.gate import TGate
assert _test_args(TGate(0))
def test_sympy__physics__quantum__gate__TwoQubitGate():
from sympy.physics.quantum.gate import TwoQubitGate
assert _test_args(TwoQubitGate(0))
def test_sympy__physics__quantum__gate__UGate():
from sympy.physics.quantum.gate import UGate
from sympy.matrices.immutable import ImmutableDenseMatrix
from sympy import Integer, Tuple
assert _test_args(
UGate(Tuple(Integer(1)), ImmutableDenseMatrix([[1, 0], [0, 2]])))
def test_sympy__physics__quantum__gate__XGate():
from sympy.physics.quantum.gate import XGate
assert _test_args(XGate(0))
def test_sympy__physics__quantum__gate__YGate():
from sympy.physics.quantum.gate import YGate
assert _test_args(YGate(0))
def test_sympy__physics__quantum__gate__ZGate():
from sympy.physics.quantum.gate import ZGate
assert _test_args(ZGate(0))
@SKIP("TODO: sympy.physics")
def test_sympy__physics__quantum__grover__OracleGate():
from sympy.physics.quantum.grover import OracleGate
assert _test_args(OracleGate())
def test_sympy__physics__quantum__grover__WGate():
from sympy.physics.quantum.grover import WGate
assert _test_args(WGate(1))
def test_sympy__physics__quantum__hilbert__ComplexSpace():
from sympy.physics.quantum.hilbert import ComplexSpace
assert _test_args(ComplexSpace(x))
def test_sympy__physics__quantum__hilbert__DirectSumHilbertSpace():
from sympy.physics.quantum.hilbert import DirectSumHilbertSpace, ComplexSpace, FockSpace
c = ComplexSpace(2)
f = FockSpace()
assert _test_args(DirectSumHilbertSpace(c, f))
def test_sympy__physics__quantum__hilbert__FockSpace():
from sympy.physics.quantum.hilbert import FockSpace
assert _test_args(FockSpace())
def test_sympy__physics__quantum__hilbert__HilbertSpace():
from sympy.physics.quantum.hilbert import HilbertSpace
assert _test_args(HilbertSpace())
def test_sympy__physics__quantum__hilbert__L2():
from sympy.physics.quantum.hilbert import L2
from sympy import oo, Interval
assert _test_args(L2(Interval(0, oo)))
def test_sympy__physics__quantum__hilbert__TensorPowerHilbertSpace():
from sympy.physics.quantum.hilbert import TensorPowerHilbertSpace, FockSpace
f = FockSpace()
assert _test_args(TensorPowerHilbertSpace(f, 2))
def test_sympy__physics__quantum__hilbert__TensorProductHilbertSpace():
from sympy.physics.quantum.hilbert import TensorProductHilbertSpace, FockSpace, ComplexSpace
c = ComplexSpace(2)
f = FockSpace()
assert _test_args(TensorProductHilbertSpace(f, c))
def test_sympy__physics__quantum__innerproduct__InnerProduct():
from sympy.physics.quantum import Bra, Ket, InnerProduct
b = Bra('b')
k = Ket('k')
assert _test_args(InnerProduct(b, k))
def test_sympy__physics__quantum__operator__DifferentialOperator():
from sympy.physics.quantum.operator import DifferentialOperator
from sympy import Derivative, Function
f = Function('f')
assert _test_args(DifferentialOperator(1/x*Derivative(f(x), x), f(x)))
def test_sympy__physics__quantum__operator__HermitianOperator():
from sympy.physics.quantum.operator import HermitianOperator
assert _test_args(HermitianOperator('H'))
def test_sympy__physics__quantum__operator__IdentityOperator():
from sympy.physics.quantum.operator import IdentityOperator
assert _test_args(IdentityOperator(5))
def test_sympy__physics__quantum__operator__Operator():
from sympy.physics.quantum.operator import Operator
assert _test_args(Operator('A'))
def test_sympy__physics__quantum__operator__OuterProduct():
from sympy.physics.quantum.operator import OuterProduct
from sympy.physics.quantum import Ket, Bra
b = Bra('b')
k = Ket('k')
assert _test_args(OuterProduct(k, b))
def test_sympy__physics__quantum__operator__UnitaryOperator():
from sympy.physics.quantum.operator import UnitaryOperator
assert _test_args(UnitaryOperator('U'))
def test_sympy__physics__quantum__piab__PIABBra():
from sympy.physics.quantum.piab import PIABBra
assert _test_args(PIABBra('B'))
def test_sympy__physics__quantum__boson__BosonOp():
from sympy.physics.quantum.boson import BosonOp
assert _test_args(BosonOp('a'))
assert _test_args(BosonOp('a', False))
def test_sympy__physics__quantum__boson__BosonFockKet():
from sympy.physics.quantum.boson import BosonFockKet
assert _test_args(BosonFockKet(1))
def test_sympy__physics__quantum__boson__BosonFockBra():
from sympy.physics.quantum.boson import BosonFockBra
assert _test_args(BosonFockBra(1))
def test_sympy__physics__quantum__boson__BosonCoherentKet():
from sympy.physics.quantum.boson import BosonCoherentKet
assert _test_args(BosonCoherentKet(1))
def test_sympy__physics__quantum__boson__BosonCoherentBra():
from sympy.physics.quantum.boson import BosonCoherentBra
assert _test_args(BosonCoherentBra(1))
def test_sympy__physics__quantum__fermion__FermionOp():
from sympy.physics.quantum.fermion import FermionOp
assert _test_args(FermionOp('c'))
assert _test_args(FermionOp('c', False))
def test_sympy__physics__quantum__fermion__FermionFockKet():
from sympy.physics.quantum.fermion import FermionFockKet
assert _test_args(FermionFockKet(1))
def test_sympy__physics__quantum__fermion__FermionFockBra():
from sympy.physics.quantum.fermion import FermionFockBra
assert _test_args(FermionFockBra(1))
def test_sympy__physics__quantum__pauli__SigmaOpBase():
from sympy.physics.quantum.pauli import SigmaOpBase
assert _test_args(SigmaOpBase())
def test_sympy__physics__quantum__pauli__SigmaX():
from sympy.physics.quantum.pauli import SigmaX
assert _test_args(SigmaX())
def test_sympy__physics__quantum__pauli__SigmaY():
from sympy.physics.quantum.pauli import SigmaY
assert _test_args(SigmaY())
def test_sympy__physics__quantum__pauli__SigmaZ():
from sympy.physics.quantum.pauli import SigmaZ
assert _test_args(SigmaZ())
def test_sympy__physics__quantum__pauli__SigmaMinus():
from sympy.physics.quantum.pauli import SigmaMinus
assert _test_args(SigmaMinus())
def test_sympy__physics__quantum__pauli__SigmaPlus():
from sympy.physics.quantum.pauli import SigmaPlus
assert _test_args(SigmaPlus())
def test_sympy__physics__quantum__pauli__SigmaZKet():
from sympy.physics.quantum.pauli import SigmaZKet
assert _test_args(SigmaZKet(0))
def test_sympy__physics__quantum__pauli__SigmaZBra():
from sympy.physics.quantum.pauli import SigmaZBra
assert _test_args(SigmaZBra(0))
def test_sympy__physics__quantum__piab__PIABHamiltonian():
from sympy.physics.quantum.piab import PIABHamiltonian
assert _test_args(PIABHamiltonian('P'))
def test_sympy__physics__quantum__piab__PIABKet():
from sympy.physics.quantum.piab import PIABKet
assert _test_args(PIABKet('K'))
def test_sympy__physics__quantum__qexpr__QExpr():
from sympy.physics.quantum.qexpr import QExpr
assert _test_args(QExpr(0))
def test_sympy__physics__quantum__qft__Fourier():
from sympy.physics.quantum.qft import Fourier
assert _test_args(Fourier(0, 1))
def test_sympy__physics__quantum__qft__IQFT():
from sympy.physics.quantum.qft import IQFT
assert _test_args(IQFT(0, 1))
def test_sympy__physics__quantum__qft__QFT():
from sympy.physics.quantum.qft import QFT
assert _test_args(QFT(0, 1))
def test_sympy__physics__quantum__qft__RkGate():
from sympy.physics.quantum.qft import RkGate
assert _test_args(RkGate(0, 1))
def test_sympy__physics__quantum__qubit__IntQubit():
from sympy.physics.quantum.qubit import IntQubit
assert _test_args(IntQubit(0))
def test_sympy__physics__quantum__qubit__IntQubitBra():
from sympy.physics.quantum.qubit import IntQubitBra
assert _test_args(IntQubitBra(0))
def test_sympy__physics__quantum__qubit__IntQubitState():
from sympy.physics.quantum.qubit import IntQubitState, QubitState
assert _test_args(IntQubitState(QubitState(0, 1)))
def test_sympy__physics__quantum__qubit__Qubit():
from sympy.physics.quantum.qubit import Qubit
assert _test_args(Qubit(0, 0, 0))
def test_sympy__physics__quantum__qubit__QubitBra():
from sympy.physics.quantum.qubit import QubitBra
assert _test_args(QubitBra('1', 0))
def test_sympy__physics__quantum__qubit__QubitState():
from sympy.physics.quantum.qubit import QubitState
assert _test_args(QubitState(0, 1))
def test_sympy__physics__quantum__density__Density():
from sympy.physics.quantum.density import Density
from sympy.physics.quantum.state import Ket
assert _test_args(Density([Ket(0), 0.5], [Ket(1), 0.5]))
@SKIP("TODO: sympy.physics.quantum.shor: Cmod Not Implemented")
def test_sympy__physics__quantum__shor__CMod():
from sympy.physics.quantum.shor import CMod
assert _test_args(CMod())
def test_sympy__physics__quantum__spin__CoupledSpinState():
from sympy.physics.quantum.spin import CoupledSpinState
assert _test_args(CoupledSpinState(1, 0, (1, 1)))
assert _test_args(CoupledSpinState(1, 0, (1, S.Half, S.Half)))
assert _test_args(CoupledSpinState(
1, 0, (1, S.Half, S.Half), ((2, 3, S.Half), (1, 2, 1)) ))
j, m, j1, j2, j3, j12, x = symbols('j m j1:4 j12 x')
assert CoupledSpinState(
j, m, (j1, j2, j3)).subs(j2, x) == CoupledSpinState(j, m, (j1, x, j3))
assert CoupledSpinState(j, m, (j1, j2, j3), ((1, 3, j12), (1, 2, j)) ).subs(j12, x) == \
CoupledSpinState(j, m, (j1, j2, j3), ((1, 3, x), (1, 2, j)) )
def test_sympy__physics__quantum__spin__J2Op():
from sympy.physics.quantum.spin import J2Op
assert _test_args(J2Op('J'))
def test_sympy__physics__quantum__spin__JminusOp():
from sympy.physics.quantum.spin import JminusOp
assert _test_args(JminusOp('J'))
def test_sympy__physics__quantum__spin__JplusOp():
from sympy.physics.quantum.spin import JplusOp
assert _test_args(JplusOp('J'))
def test_sympy__physics__quantum__spin__JxBra():
from sympy.physics.quantum.spin import JxBra
assert _test_args(JxBra(1, 0))
def test_sympy__physics__quantum__spin__JxBraCoupled():
from sympy.physics.quantum.spin import JxBraCoupled
assert _test_args(JxBraCoupled(1, 0, (1, 1)))
def test_sympy__physics__quantum__spin__JxKet():
from sympy.physics.quantum.spin import JxKet
assert _test_args(JxKet(1, 0))
def test_sympy__physics__quantum__spin__JxKetCoupled():
from sympy.physics.quantum.spin import JxKetCoupled
assert _test_args(JxKetCoupled(1, 0, (1, 1)))
def test_sympy__physics__quantum__spin__JxOp():
from sympy.physics.quantum.spin import JxOp
assert _test_args(JxOp('J'))
def test_sympy__physics__quantum__spin__JyBra():
from sympy.physics.quantum.spin import JyBra
assert _test_args(JyBra(1, 0))
def test_sympy__physics__quantum__spin__JyBraCoupled():
from sympy.physics.quantum.spin import JyBraCoupled
assert _test_args(JyBraCoupled(1, 0, (1, 1)))
def test_sympy__physics__quantum__spin__JyKet():
from sympy.physics.quantum.spin import JyKet
assert _test_args(JyKet(1, 0))
def test_sympy__physics__quantum__spin__JyKetCoupled():
from sympy.physics.quantum.spin import JyKetCoupled
assert _test_args(JyKetCoupled(1, 0, (1, 1)))
def test_sympy__physics__quantum__spin__JyOp():
from sympy.physics.quantum.spin import JyOp
assert _test_args(JyOp('J'))
def test_sympy__physics__quantum__spin__JzBra():
from sympy.physics.quantum.spin import JzBra
assert _test_args(JzBra(1, 0))
def test_sympy__physics__quantum__spin__JzBraCoupled():
from sympy.physics.quantum.spin import JzBraCoupled
assert _test_args(JzBraCoupled(1, 0, (1, 1)))
def test_sympy__physics__quantum__spin__JzKet():
from sympy.physics.quantum.spin import JzKet
assert _test_args(JzKet(1, 0))
def test_sympy__physics__quantum__spin__JzKetCoupled():
from sympy.physics.quantum.spin import JzKetCoupled
assert _test_args(JzKetCoupled(1, 0, (1, 1)))
def test_sympy__physics__quantum__spin__JzOp():
from sympy.physics.quantum.spin import JzOp
assert _test_args(JzOp('J'))
def test_sympy__physics__quantum__spin__Rotation():
from sympy.physics.quantum.spin import Rotation
assert _test_args(Rotation(pi, 0, pi/2))
def test_sympy__physics__quantum__spin__SpinState():
from sympy.physics.quantum.spin import SpinState
assert _test_args(SpinState(1, 0))
def test_sympy__physics__quantum__spin__WignerD():
from sympy.physics.quantum.spin import WignerD
assert _test_args(WignerD(0, 1, 2, 3, 4, 5))
def test_sympy__physics__quantum__state__Bra():
from sympy.physics.quantum.state import Bra
assert _test_args(Bra(0))
def test_sympy__physics__quantum__state__BraBase():
from sympy.physics.quantum.state import BraBase
assert _test_args(BraBase(0))
def test_sympy__physics__quantum__state__Ket():
from sympy.physics.quantum.state import Ket
assert _test_args(Ket(0))
def test_sympy__physics__quantum__state__KetBase():
from sympy.physics.quantum.state import KetBase
assert _test_args(KetBase(0))
def test_sympy__physics__quantum__state__State():
from sympy.physics.quantum.state import State
assert _test_args(State(0))
def test_sympy__physics__quantum__state__StateBase():
from sympy.physics.quantum.state import StateBase
assert _test_args(StateBase(0))
def test_sympy__physics__quantum__state__OrthogonalBra():
from sympy.physics.quantum.state import OrthogonalBra
assert _test_args(OrthogonalBra(0))
def test_sympy__physics__quantum__state__OrthogonalKet():
from sympy.physics.quantum.state import OrthogonalKet
assert _test_args(OrthogonalKet(0))
def test_sympy__physics__quantum__state__OrthogonalState():
from sympy.physics.quantum.state import OrthogonalState
assert _test_args(OrthogonalState(0))
def test_sympy__physics__quantum__state__TimeDepBra():
from sympy.physics.quantum.state import TimeDepBra
assert _test_args(TimeDepBra('psi', 't'))
def test_sympy__physics__quantum__state__TimeDepKet():
from sympy.physics.quantum.state import TimeDepKet
assert _test_args(TimeDepKet('psi', 't'))
def test_sympy__physics__quantum__state__TimeDepState():
from sympy.physics.quantum.state import TimeDepState
assert _test_args(TimeDepState('psi', 't'))
def test_sympy__physics__quantum__state__Wavefunction():
from sympy.physics.quantum.state import Wavefunction
from sympy.functions import sin
from sympy import Piecewise
n = 1
L = 1
g = Piecewise((0, x < 0), (0, x > L), (sqrt(2//L)*sin(n*pi*x/L), True))
assert _test_args(Wavefunction(g, x))
def test_sympy__physics__quantum__tensorproduct__TensorProduct():
from sympy.physics.quantum.tensorproduct import TensorProduct
assert _test_args(TensorProduct(x, y))
def test_sympy__physics__quantum__identitysearch__GateIdentity():
from sympy.physics.quantum.gate import X
from sympy.physics.quantum.identitysearch import GateIdentity
assert _test_args(GateIdentity(X(0), X(0)))
def test_sympy__physics__quantum__sho1d__SHOOp():
from sympy.physics.quantum.sho1d import SHOOp
assert _test_args(SHOOp('a'))
def test_sympy__physics__quantum__sho1d__RaisingOp():
from sympy.physics.quantum.sho1d import RaisingOp
assert _test_args(RaisingOp('a'))
def test_sympy__physics__quantum__sho1d__LoweringOp():
from sympy.physics.quantum.sho1d import LoweringOp
assert _test_args(LoweringOp('a'))
def test_sympy__physics__quantum__sho1d__NumberOp():
from sympy.physics.quantum.sho1d import NumberOp
assert _test_args(NumberOp('N'))
def test_sympy__physics__quantum__sho1d__Hamiltonian():
from sympy.physics.quantum.sho1d import Hamiltonian
assert _test_args(Hamiltonian('H'))
def test_sympy__physics__quantum__sho1d__SHOState():
from sympy.physics.quantum.sho1d import SHOState
assert _test_args(SHOState(0))
def test_sympy__physics__quantum__sho1d__SHOKet():
from sympy.physics.quantum.sho1d import SHOKet
assert _test_args(SHOKet(0))
def test_sympy__physics__quantum__sho1d__SHOBra():
from sympy.physics.quantum.sho1d import SHOBra
assert _test_args(SHOBra(0))
def test_sympy__physics__secondquant__AnnihilateBoson():
from sympy.physics.secondquant import AnnihilateBoson
assert _test_args(AnnihilateBoson(0))
def test_sympy__physics__secondquant__AnnihilateFermion():
from sympy.physics.secondquant import AnnihilateFermion
assert _test_args(AnnihilateFermion(0))
@SKIP("abstract class")
def test_sympy__physics__secondquant__Annihilator():
pass
def test_sympy__physics__secondquant__AntiSymmetricTensor():
from sympy.physics.secondquant import AntiSymmetricTensor
i, j = symbols('i j', below_fermi=True)
a, b = symbols('a b', above_fermi=True)
assert _test_args(AntiSymmetricTensor('v', (a, i), (b, j)))
def test_sympy__physics__secondquant__BosonState():
from sympy.physics.secondquant import BosonState
assert _test_args(BosonState((0, 1)))
@SKIP("abstract class")
def test_sympy__physics__secondquant__BosonicOperator():
pass
def test_sympy__physics__secondquant__Commutator():
from sympy.physics.secondquant import Commutator
assert _test_args(Commutator(x, y))
def test_sympy__physics__secondquant__CreateBoson():
from sympy.physics.secondquant import CreateBoson
assert _test_args(CreateBoson(0))
def test_sympy__physics__secondquant__CreateFermion():
from sympy.physics.secondquant import CreateFermion
assert _test_args(CreateFermion(0))
@SKIP("abstract class")
def test_sympy__physics__secondquant__Creator():
pass
def test_sympy__physics__secondquant__Dagger():
from sympy.physics.secondquant import Dagger
from sympy import I
assert _test_args(Dagger(2*I))
def test_sympy__physics__secondquant__FermionState():
from sympy.physics.secondquant import FermionState
assert _test_args(FermionState((0, 1)))
def test_sympy__physics__secondquant__FermionicOperator():
from sympy.physics.secondquant import FermionicOperator
assert _test_args(FermionicOperator(0))
def test_sympy__physics__secondquant__FockState():
from sympy.physics.secondquant import FockState
assert _test_args(FockState((0, 1)))
def test_sympy__physics__secondquant__FockStateBosonBra():
from sympy.physics.secondquant import FockStateBosonBra
assert _test_args(FockStateBosonBra((0, 1)))
def test_sympy__physics__secondquant__FockStateBosonKet():
from sympy.physics.secondquant import FockStateBosonKet
assert _test_args(FockStateBosonKet((0, 1)))
def test_sympy__physics__secondquant__FockStateBra():
from sympy.physics.secondquant import FockStateBra
assert _test_args(FockStateBra((0, 1)))
def test_sympy__physics__secondquant__FockStateFermionBra():
from sympy.physics.secondquant import FockStateFermionBra
assert _test_args(FockStateFermionBra((0, 1)))
def test_sympy__physics__secondquant__FockStateFermionKet():
from sympy.physics.secondquant import FockStateFermionKet
assert _test_args(FockStateFermionKet((0, 1)))
def test_sympy__physics__secondquant__FockStateKet():
from sympy.physics.secondquant import FockStateKet
assert _test_args(FockStateKet((0, 1)))
def test_sympy__physics__secondquant__InnerProduct():
from sympy.physics.secondquant import InnerProduct
from sympy.physics.secondquant import FockStateKet, FockStateBra
assert _test_args(InnerProduct(FockStateBra((0, 1)), FockStateKet((0, 1))))
def test_sympy__physics__secondquant__NO():
from sympy.physics.secondquant import NO, F, Fd
assert _test_args(NO(Fd(x)*F(y)))
def test_sympy__physics__secondquant__PermutationOperator():
from sympy.physics.secondquant import PermutationOperator
assert _test_args(PermutationOperator(0, 1))
def test_sympy__physics__secondquant__SqOperator():
from sympy.physics.secondquant import SqOperator
assert _test_args(SqOperator(0))
def test_sympy__physics__secondquant__TensorSymbol():
from sympy.physics.secondquant import TensorSymbol
assert _test_args(TensorSymbol(x))
def test_sympy__physics__units__dimensions__Dimension():
from sympy.physics.units.dimensions import Dimension
assert _test_args(Dimension("length", "L"))
def test_sympy__physics__units__dimensions__DimensionSystem():
from sympy.physics.units.dimensions import DimensionSystem
from sympy.physics.units.definitions.dimension_definitions import length, time, velocity
assert _test_args(DimensionSystem((length, time), (velocity,)))
def test_sympy__physics__units__quantities__Quantity():
from sympy.physics.units.quantities import Quantity
assert _test_args(Quantity("dam"))
def test_sympy__physics__units__prefixes__Prefix():
from sympy.physics.units.prefixes import Prefix
assert _test_args(Prefix('kilo', 'k', 3))
def test_sympy__core__numbers__AlgebraicNumber():
from sympy.core.numbers import AlgebraicNumber
assert _test_args(AlgebraicNumber(sqrt(2), [1, 2, 3]))
def test_sympy__polys__polytools__GroebnerBasis():
from sympy.polys.polytools import GroebnerBasis
assert _test_args(GroebnerBasis([x, y, z], x, y, z))
def test_sympy__polys__polytools__Poly():
from sympy.polys.polytools import Poly
assert _test_args(Poly(2, x, y))
def test_sympy__polys__polytools__PurePoly():
from sympy.polys.polytools import PurePoly
assert _test_args(PurePoly(2, x, y))
@SKIP('abstract class')
def test_sympy__polys__rootoftools__RootOf():
pass
def test_sympy__polys__rootoftools__ComplexRootOf():
from sympy.polys.rootoftools import ComplexRootOf
assert _test_args(ComplexRootOf(x**3 + x + 1, 0))
def test_sympy__polys__rootoftools__RootSum():
from sympy.polys.rootoftools import RootSum
assert _test_args(RootSum(x**3 + x + 1, sin))
def test_sympy__series__limits__Limit():
from sympy.series.limits import Limit
assert _test_args(Limit(x, x, 0, dir='-'))
def test_sympy__series__order__Order():
from sympy.series.order import Order
assert _test_args(Order(1, x, y))
@SKIP('Abstract Class')
def test_sympy__series__sequences__SeqBase():
pass
def test_sympy__series__sequences__EmptySequence():
# Need to imort the instance from series not the class from
# series.sequence
from sympy.series import EmptySequence
assert _test_args(EmptySequence)
@SKIP('Abstract Class')
def test_sympy__series__sequences__SeqExpr():
pass
def test_sympy__series__sequences__SeqPer():
from sympy.series.sequences import SeqPer
assert _test_args(SeqPer((1, 2, 3), (0, 10)))
def test_sympy__series__sequences__SeqFormula():
from sympy.series.sequences import SeqFormula
assert _test_args(SeqFormula(x**2, (0, 10)))
def test_sympy__series__sequences__RecursiveSeq():
from sympy.series.sequences import RecursiveSeq
y = Function("y")
n = symbols("n")
assert _test_args(RecursiveSeq(y(n - 1) + y(n - 2), y(n), n, (0, 1)))
assert _test_args(RecursiveSeq(y(n - 1) + y(n - 2), y(n), n))
def test_sympy__series__sequences__SeqExprOp():
from sympy.series.sequences import SeqExprOp, sequence
s1 = sequence((1, 2, 3))
s2 = sequence(x**2)
assert _test_args(SeqExprOp(s1, s2))
def test_sympy__series__sequences__SeqAdd():
from sympy.series.sequences import SeqAdd, sequence
s1 = sequence((1, 2, 3))
s2 = sequence(x**2)
assert _test_args(SeqAdd(s1, s2))
def test_sympy__series__sequences__SeqMul():
from sympy.series.sequences import SeqMul, sequence
s1 = sequence((1, 2, 3))
s2 = sequence(x**2)
assert _test_args(SeqMul(s1, s2))
@SKIP('Abstract Class')
def test_sympy__series__series_class__SeriesBase():
pass
def test_sympy__series__fourier__FourierSeries():
from sympy.series.fourier import fourier_series
assert _test_args(fourier_series(x, (x, -pi, pi)))
def test_sympy__series__fourier__FiniteFourierSeries():
from sympy.series.fourier import fourier_series
assert _test_args(fourier_series(sin(pi*x), (x, -1, 1)))
def test_sympy__series__formal__FormalPowerSeries():
from sympy.series.formal import fps
assert _test_args(fps(log(1 + x), x))
def test_sympy__series__formal__Coeff():
from sympy.series.formal import fps
assert _test_args(fps(x**2 + x + 1, x))
@SKIP('Abstract Class')
def test_sympy__series__formal__FiniteFormalPowerSeries():
pass
def test_sympy__series__formal__FormalPowerSeriesProduct():
from sympy.series.formal import fps
f1, f2 = fps(sin(x)), fps(exp(x))
assert _test_args(f1.product(f2, x))
def test_sympy__series__formal__FormalPowerSeriesCompose():
from sympy.series.formal import fps
f1, f2 = fps(exp(x)), fps(sin(x))
assert _test_args(f1.compose(f2, x))
def test_sympy__series__formal__FormalPowerSeriesInverse():
from sympy.series.formal import fps
f1 = fps(exp(x))
assert _test_args(f1.inverse(x))
def test_sympy__simplify__hyperexpand__Hyper_Function():
from sympy.simplify.hyperexpand import Hyper_Function
assert _test_args(Hyper_Function([2], [1]))
def test_sympy__simplify__hyperexpand__G_Function():
from sympy.simplify.hyperexpand import G_Function
assert _test_args(G_Function([2], [1], [], []))
@SKIP("abstract class")
def test_sympy__tensor__array__ndim_array__ImmutableNDimArray():
pass
def test_sympy__tensor__array__dense_ndim_array__ImmutableDenseNDimArray():
from sympy.tensor.array.dense_ndim_array import ImmutableDenseNDimArray
densarr = ImmutableDenseNDimArray(range(10, 34), (2, 3, 4))
assert _test_args(densarr)
def test_sympy__tensor__array__sparse_ndim_array__ImmutableSparseNDimArray():
from sympy.tensor.array.sparse_ndim_array import ImmutableSparseNDimArray
sparr = ImmutableSparseNDimArray(range(10, 34), (2, 3, 4))
assert _test_args(sparr)
def test_sympy__tensor__array__array_comprehension__ArrayComprehension():
from sympy.tensor.array.array_comprehension import ArrayComprehension
arrcom = ArrayComprehension(x, (x, 1, 5))
assert _test_args(arrcom)
def test_sympy__tensor__array__array_comprehension__ArrayComprehensionMap():
from sympy.tensor.array.array_comprehension import ArrayComprehensionMap
arrcomma = ArrayComprehensionMap(lambda: 0, (x, 1, 5))
assert _test_args(arrcomma)
def test_sympy__tensor__array__arrayop__Flatten():
from sympy.tensor.array.arrayop import Flatten
from sympy.tensor.array.dense_ndim_array import ImmutableDenseNDimArray
fla = Flatten(ImmutableDenseNDimArray(range(24)).reshape(2, 3, 4))
assert _test_args(fla)
def test_sympy__tensor__functions__TensorProduct():
from sympy.tensor.functions import TensorProduct
A = MatrixSymbol('A', 3, 3)
B = MatrixSymbol('B', 3, 3)
tp = TensorProduct(A, B)
assert _test_args(tp)
def test_sympy__tensor__indexed__Idx():
from sympy.tensor.indexed import Idx
assert _test_args(Idx('test'))
assert _test_args(Idx(1, (0, 10)))
def test_sympy__tensor__indexed__Indexed():
from sympy.tensor.indexed import Indexed, Idx
assert _test_args(Indexed('A', Idx('i'), Idx('j')))
def test_sympy__tensor__indexed__IndexedBase():
from sympy.tensor.indexed import IndexedBase
assert _test_args(IndexedBase('A', shape=(x, y)))
assert _test_args(IndexedBase('A', 1))
assert _test_args(IndexedBase('A')[0, 1])
def test_sympy__tensor__tensor__TensorIndexType():
from sympy.tensor.tensor import TensorIndexType
assert _test_args(TensorIndexType('Lorentz'))
@SKIP("deprecated class")
def test_sympy__tensor__tensor__TensorType():
pass
def test_sympy__tensor__tensor__TensorSymmetry():
from sympy.tensor.tensor import TensorSymmetry, get_symmetric_group_sgs
assert _test_args(TensorSymmetry(get_symmetric_group_sgs(2)))
def test_sympy__tensor__tensor__TensorHead():
from sympy.tensor.tensor import TensorIndexType, TensorSymmetry, get_symmetric_group_sgs, TensorHead
Lorentz = TensorIndexType('Lorentz', dummy_name='L')
sym = TensorSymmetry(get_symmetric_group_sgs(1))
assert _test_args(TensorHead('p', [Lorentz], sym, 0))
def test_sympy__tensor__tensor__TensorIndex():
from sympy.tensor.tensor import TensorIndexType, TensorIndex
Lorentz = TensorIndexType('Lorentz', dummy_name='L')
assert _test_args(TensorIndex('i', Lorentz))
@SKIP("abstract class")
def test_sympy__tensor__tensor__TensExpr():
pass
def test_sympy__tensor__tensor__TensAdd():
from sympy.tensor.tensor import TensorIndexType, TensorSymmetry, get_symmetric_group_sgs, tensor_indices, TensAdd, tensor_heads
Lorentz = TensorIndexType('Lorentz', dummy_name='L')
a, b = tensor_indices('a,b', Lorentz)
sym = TensorSymmetry(get_symmetric_group_sgs(1))
p, q = tensor_heads('p,q', [Lorentz], sym)
t1 = p(a)
t2 = q(a)
assert _test_args(TensAdd(t1, t2))
def test_sympy__tensor__tensor__Tensor():
from sympy.tensor.tensor import TensorIndexType, TensorSymmetry, get_symmetric_group_sgs, tensor_indices, TensorHead
Lorentz = TensorIndexType('Lorentz', dummy_name='L')
a, b = tensor_indices('a,b', Lorentz)
sym = TensorSymmetry(get_symmetric_group_sgs(1))
p = TensorHead('p', [Lorentz], sym)
assert _test_args(p(a))
def test_sympy__tensor__tensor__TensMul():
from sympy.tensor.tensor import TensorIndexType, TensorSymmetry, get_symmetric_group_sgs, tensor_indices, tensor_heads
Lorentz = TensorIndexType('Lorentz', dummy_name='L')
a, b = tensor_indices('a,b', Lorentz)
sym = TensorSymmetry(get_symmetric_group_sgs(1))
p, q = tensor_heads('p, q', [Lorentz], sym)
assert _test_args(3*p(a)*q(b))
def test_sympy__tensor__tensor__TensorElement():
from sympy.tensor.tensor import TensorIndexType, TensorHead, TensorElement
L = TensorIndexType("L")
A = TensorHead("A", [L, L])
telem = TensorElement(A(x, y), {x: 1})
assert _test_args(telem)
def test_sympy__tensor__toperators__PartialDerivative():
from sympy.tensor.tensor import TensorIndexType, tensor_indices, TensorHead
from sympy.tensor.toperators import PartialDerivative
Lorentz = TensorIndexType('Lorentz', dummy_name='L')
a, b = tensor_indices('a,b', Lorentz)
A = TensorHead("A", [Lorentz])
assert _test_args(PartialDerivative(A(a), A(b)))
def test_as_coeff_add():
assert (7, (3*x, 4*x**2)) == (7 + 3*x + 4*x**2).as_coeff_add()
def test_sympy__geometry__curve__Curve():
from sympy.geometry.curve import Curve
assert _test_args(Curve((x, 1), (x, 0, 1)))
def test_sympy__geometry__point__Point():
from sympy.geometry.point import Point
assert _test_args(Point(0, 1))
def test_sympy__geometry__point__Point2D():
from sympy.geometry.point import Point2D
assert _test_args(Point2D(0, 1))
def test_sympy__geometry__point__Point3D():
from sympy.geometry.point import Point3D
assert _test_args(Point3D(0, 1, 2))
def test_sympy__geometry__ellipse__Ellipse():
from sympy.geometry.ellipse import Ellipse
assert _test_args(Ellipse((0, 1), 2, 3))
def test_sympy__geometry__ellipse__Circle():
from sympy.geometry.ellipse import Circle
assert _test_args(Circle((0, 1), 2))
def test_sympy__geometry__parabola__Parabola():
from sympy.geometry.parabola import Parabola
from sympy.geometry.line import Line
assert _test_args(Parabola((0, 0), Line((2, 3), (4, 3))))
@SKIP("abstract class")
def test_sympy__geometry__line__LinearEntity():
pass
def test_sympy__geometry__line__Line():
from sympy.geometry.line import Line
assert _test_args(Line((0, 1), (2, 3)))
def test_sympy__geometry__line__Ray():
from sympy.geometry.line import Ray
assert _test_args(Ray((0, 1), (2, 3)))
def test_sympy__geometry__line__Segment():
from sympy.geometry.line import Segment
assert _test_args(Segment((0, 1), (2, 3)))
@SKIP("abstract class")
def test_sympy__geometry__line__LinearEntity2D():
pass
def test_sympy__geometry__line__Line2D():
from sympy.geometry.line import Line2D
assert _test_args(Line2D((0, 1), (2, 3)))
def test_sympy__geometry__line__Ray2D():
from sympy.geometry.line import Ray2D
assert _test_args(Ray2D((0, 1), (2, 3)))
def test_sympy__geometry__line__Segment2D():
from sympy.geometry.line import Segment2D
assert _test_args(Segment2D((0, 1), (2, 3)))
@SKIP("abstract class")
def test_sympy__geometry__line__LinearEntity3D():
pass
def test_sympy__geometry__line__Line3D():
from sympy.geometry.line import Line3D
assert _test_args(Line3D((0, 1, 1), (2, 3, 4)))
def test_sympy__geometry__line__Segment3D():
from sympy.geometry.line import Segment3D
assert _test_args(Segment3D((0, 1, 1), (2, 3, 4)))
def test_sympy__geometry__line__Ray3D():
from sympy.geometry.line import Ray3D
assert _test_args(Ray3D((0, 1, 1), (2, 3, 4)))
def test_sympy__geometry__plane__Plane():
from sympy.geometry.plane import Plane
assert _test_args(Plane((1, 1, 1), (-3, 4, -2), (1, 2, 3)))
def test_sympy__geometry__polygon__Polygon():
from sympy.geometry.polygon import Polygon
assert _test_args(Polygon((0, 1), (2, 3), (4, 5), (6, 7)))
def test_sympy__geometry__polygon__RegularPolygon():
from sympy.geometry.polygon import RegularPolygon
assert _test_args(RegularPolygon((0, 1), 2, 3, 4))
def test_sympy__geometry__polygon__Triangle():
from sympy.geometry.polygon import Triangle
assert _test_args(Triangle((0, 1), (2, 3), (4, 5)))
def test_sympy__geometry__entity__GeometryEntity():
from sympy.geometry.entity import GeometryEntity
from sympy.geometry.point import Point
assert _test_args(GeometryEntity(Point(1, 0), 1, [1, 2]))
@SKIP("abstract class")
def test_sympy__geometry__entity__GeometrySet():
pass
def test_sympy__diffgeom__diffgeom__Manifold():
from sympy.diffgeom import Manifold
assert _test_args(Manifold('name', 3))
def test_sympy__diffgeom__diffgeom__Patch():
from sympy.diffgeom import Manifold, Patch
assert _test_args(Patch('name', Manifold('name', 3)))
def test_sympy__diffgeom__diffgeom__CoordSystem():
from sympy.diffgeom import Manifold, Patch, CoordSystem
assert _test_args(CoordSystem('name', Patch('name', Manifold('name', 3))))
@XFAIL
def test_sympy__diffgeom__diffgeom__Point():
from sympy.diffgeom import Manifold, Patch, CoordSystem, Point
assert _test_args(Point(
CoordSystem('name', Patch('name', Manifold('name', 3))), [x, y]))
def test_sympy__diffgeom__diffgeom__BaseScalarField():
from sympy.diffgeom import Manifold, Patch, CoordSystem, BaseScalarField
cs = CoordSystem('name', Patch('name', Manifold('name', 3)))
assert _test_args(BaseScalarField(cs, 0))
def test_sympy__diffgeom__diffgeom__BaseVectorField():
from sympy.diffgeom import Manifold, Patch, CoordSystem, BaseVectorField
cs = CoordSystem('name', Patch('name', Manifold('name', 3)))
assert _test_args(BaseVectorField(cs, 0))
def test_sympy__diffgeom__diffgeom__Differential():
from sympy.diffgeom import Manifold, Patch, CoordSystem, BaseScalarField, Differential
cs = CoordSystem('name', Patch('name', Manifold('name', 3)))
assert _test_args(Differential(BaseScalarField(cs, 0)))
def test_sympy__diffgeom__diffgeom__Commutator():
from sympy.diffgeom import Manifold, Patch, CoordSystem, BaseVectorField, Commutator
cs = CoordSystem('name', Patch('name', Manifold('name', 3)))
cs1 = CoordSystem('name1', Patch('name', Manifold('name', 3)))
v = BaseVectorField(cs, 0)
v1 = BaseVectorField(cs1, 0)
assert _test_args(Commutator(v, v1))
def test_sympy__diffgeom__diffgeom__TensorProduct():
from sympy.diffgeom import Manifold, Patch, CoordSystem, BaseScalarField, Differential, TensorProduct
cs = CoordSystem('name', Patch('name', Manifold('name', 3)))
d = Differential(BaseScalarField(cs, 0))
assert _test_args(TensorProduct(d, d))
def test_sympy__diffgeom__diffgeom__WedgeProduct():
from sympy.diffgeom import Manifold, Patch, CoordSystem, BaseScalarField, Differential, WedgeProduct
cs = CoordSystem('name', Patch('name', Manifold('name', 3)))
d = Differential(BaseScalarField(cs, 0))
d1 = Differential(BaseScalarField(cs, 1))
assert _test_args(WedgeProduct(d, d1))
def test_sympy__diffgeom__diffgeom__LieDerivative():
from sympy.diffgeom import Manifold, Patch, CoordSystem, BaseScalarField, Differential, BaseVectorField, LieDerivative
cs = CoordSystem('name', Patch('name', Manifold('name', 3)))
d = Differential(BaseScalarField(cs, 0))
v = BaseVectorField(cs, 0)
assert _test_args(LieDerivative(v, d))
@XFAIL
def test_sympy__diffgeom__diffgeom__BaseCovarDerivativeOp():
from sympy.diffgeom import Manifold, Patch, CoordSystem, BaseCovarDerivativeOp
cs = CoordSystem('name', Patch('name', Manifold('name', 3)))
assert _test_args(BaseCovarDerivativeOp(cs, 0, [[[0, ]*3, ]*3, ]*3))
def test_sympy__diffgeom__diffgeom__CovarDerivativeOp():
from sympy.diffgeom import Manifold, Patch, CoordSystem, BaseVectorField, CovarDerivativeOp
cs = CoordSystem('name', Patch('name', Manifold('name', 3)))
v = BaseVectorField(cs, 0)
_test_args(CovarDerivativeOp(v, [[[0, ]*3, ]*3, ]*3))
def test_sympy__categories__baseclasses__Class():
from sympy.categories.baseclasses import Class
assert _test_args(Class())
def test_sympy__categories__baseclasses__Object():
from sympy.categories import Object
assert _test_args(Object("A"))
@XFAIL
def test_sympy__categories__baseclasses__Morphism():
from sympy.categories import Object, Morphism
assert _test_args(Morphism(Object("A"), Object("B")))
def test_sympy__categories__baseclasses__IdentityMorphism():
from sympy.categories import Object, IdentityMorphism
assert _test_args(IdentityMorphism(Object("A")))
def test_sympy__categories__baseclasses__NamedMorphism():
from sympy.categories import Object, NamedMorphism
assert _test_args(NamedMorphism(Object("A"), Object("B"), "f"))
def test_sympy__categories__baseclasses__CompositeMorphism():
from sympy.categories import Object, NamedMorphism, CompositeMorphism
A = Object("A")
B = Object("B")
C = Object("C")
f = NamedMorphism(A, B, "f")
g = NamedMorphism(B, C, "g")
assert _test_args(CompositeMorphism(f, g))
def test_sympy__categories__baseclasses__Diagram():
from sympy.categories import Object, NamedMorphism, Diagram
A = Object("A")
B = Object("B")
f = NamedMorphism(A, B, "f")
d = Diagram([f])
assert _test_args(d)
def test_sympy__categories__baseclasses__Category():
from sympy.categories import Object, NamedMorphism, Diagram, Category
A = Object("A")
B = Object("B")
C = Object("C")
f = NamedMorphism(A, B, "f")
g = NamedMorphism(B, C, "g")
d1 = Diagram([f, g])
d2 = Diagram([f])
K = Category("K", commutative_diagrams=[d1, d2])
assert _test_args(K)
def test_sympy__ntheory__factor___totient():
from sympy.ntheory.factor_ import totient
k = symbols('k', integer=True)
t = totient(k)
assert _test_args(t)
def test_sympy__ntheory__factor___reduced_totient():
from sympy.ntheory.factor_ import reduced_totient
k = symbols('k', integer=True)
t = reduced_totient(k)
assert _test_args(t)
def test_sympy__ntheory__factor___divisor_sigma():
from sympy.ntheory.factor_ import divisor_sigma
k = symbols('k', integer=True)
n = symbols('n', integer=True)
t = divisor_sigma(n, k)
assert _test_args(t)
def test_sympy__ntheory__factor___udivisor_sigma():
from sympy.ntheory.factor_ import udivisor_sigma
k = symbols('k', integer=True)
n = symbols('n', integer=True)
t = udivisor_sigma(n, k)
assert _test_args(t)
def test_sympy__ntheory__factor___primenu():
from sympy.ntheory.factor_ import primenu
n = symbols('n', integer=True)
t = primenu(n)
assert _test_args(t)
def test_sympy__ntheory__factor___primeomega():
from sympy.ntheory.factor_ import primeomega
n = symbols('n', integer=True)
t = primeomega(n)
assert _test_args(t)
def test_sympy__ntheory__residue_ntheory__mobius():
from sympy.ntheory import mobius
assert _test_args(mobius(2))
def test_sympy__ntheory__generate__primepi():
from sympy.ntheory import primepi
n = symbols('n')
t = primepi(n)
assert _test_args(t)
def test_sympy__physics__optics__waves__TWave():
from sympy.physics.optics import TWave
A, f, phi = symbols('A, f, phi')
assert _test_args(TWave(A, f, phi))
def test_sympy__physics__optics__gaussopt__BeamParameter():
from sympy.physics.optics import BeamParameter
assert _test_args(BeamParameter(530e-9, 1, w=1e-3))
def test_sympy__physics__optics__medium__Medium():
from sympy.physics.optics import Medium
assert _test_args(Medium('m'))
def test_sympy__codegen__array_utils__CodegenArrayContraction():
from sympy.codegen.array_utils import CodegenArrayContraction
from sympy import IndexedBase
A = symbols("A", cls=IndexedBase)
assert _test_args(CodegenArrayContraction(A, (0, 1)))
def test_sympy__codegen__array_utils__CodegenArrayDiagonal():
from sympy.codegen.array_utils import CodegenArrayDiagonal
from sympy import IndexedBase
A = symbols("A", cls=IndexedBase)
assert _test_args(CodegenArrayDiagonal(A, (0, 1)))
def test_sympy__codegen__array_utils__CodegenArrayTensorProduct():
from sympy.codegen.array_utils import CodegenArrayTensorProduct
from sympy import IndexedBase
A, B = symbols("A B", cls=IndexedBase)
assert _test_args(CodegenArrayTensorProduct(A, B))
def test_sympy__codegen__array_utils__CodegenArrayElementwiseAdd():
from sympy.codegen.array_utils import CodegenArrayElementwiseAdd
from sympy import IndexedBase
A, B = symbols("A B", cls=IndexedBase)
assert _test_args(CodegenArrayElementwiseAdd(A, B))
def test_sympy__codegen__array_utils__CodegenArrayPermuteDims():
from sympy.codegen.array_utils import CodegenArrayPermuteDims
from sympy import IndexedBase
A = symbols("A", cls=IndexedBase)
assert _test_args(CodegenArrayPermuteDims(A, (1, 0)))
def test_sympy__codegen__ast__Assignment():
from sympy.codegen.ast import Assignment
assert _test_args(Assignment(x, y))
def test_sympy__codegen__cfunctions__expm1():
from sympy.codegen.cfunctions import expm1
assert _test_args(expm1(x))
def test_sympy__codegen__cfunctions__log1p():
from sympy.codegen.cfunctions import log1p
assert _test_args(log1p(x))
def test_sympy__codegen__cfunctions__exp2():
from sympy.codegen.cfunctions import exp2
assert _test_args(exp2(x))
def test_sympy__codegen__cfunctions__log2():
from sympy.codegen.cfunctions import log2
assert _test_args(log2(x))
def test_sympy__codegen__cfunctions__fma():
from sympy.codegen.cfunctions import fma
assert _test_args(fma(x, y, z))
def test_sympy__codegen__cfunctions__log10():
from sympy.codegen.cfunctions import log10
assert _test_args(log10(x))
def test_sympy__codegen__cfunctions__Sqrt():
from sympy.codegen.cfunctions import Sqrt
assert _test_args(Sqrt(x))
def test_sympy__codegen__cfunctions__Cbrt():
from sympy.codegen.cfunctions import Cbrt
assert _test_args(Cbrt(x))
def test_sympy__codegen__cfunctions__hypot():
from sympy.codegen.cfunctions import hypot
assert _test_args(hypot(x, y))
def test_sympy__codegen__fnodes__FFunction():
from sympy.codegen.fnodes import FFunction
assert _test_args(FFunction('f'))
def test_sympy__codegen__fnodes__F95Function():
from sympy.codegen.fnodes import F95Function
assert _test_args(F95Function('f'))
def test_sympy__codegen__fnodes__isign():
from sympy.codegen.fnodes import isign
assert _test_args(isign(1, x))
def test_sympy__codegen__fnodes__dsign():
from sympy.codegen.fnodes import dsign
assert _test_args(dsign(1, x))
def test_sympy__codegen__fnodes__cmplx():
from sympy.codegen.fnodes import cmplx
assert _test_args(cmplx(x, y))
def test_sympy__codegen__fnodes__kind():
from sympy.codegen.fnodes import kind
assert _test_args(kind(x))
def test_sympy__codegen__fnodes__merge():
from sympy.codegen.fnodes import merge
assert _test_args(merge(1, 2, Eq(x, 0)))
def test_sympy__codegen__fnodes___literal():
from sympy.codegen.fnodes import _literal
assert _test_args(_literal(1))
def test_sympy__codegen__fnodes__literal_sp():
from sympy.codegen.fnodes import literal_sp
assert _test_args(literal_sp(1))
def test_sympy__codegen__fnodes__literal_dp():
from sympy.codegen.fnodes import literal_dp
assert _test_args(literal_dp(1))
def test_sympy__codegen__matrix_nodes__MatrixSolve():
from sympy.matrices import MatrixSymbol
from sympy.codegen.matrix_nodes import MatrixSolve
A = MatrixSymbol('A', 3, 3)
v = MatrixSymbol('x', 3, 1)
assert _test_args(MatrixSolve(A, v))
def test_sympy__vector__coordsysrect__CoordSys3D():
from sympy.vector.coordsysrect import CoordSys3D
assert _test_args(CoordSys3D('C'))
def test_sympy__vector__point__Point():
from sympy.vector.point import Point
assert _test_args(Point('P'))
def test_sympy__vector__basisdependent__BasisDependent():
#from sympy.vector.basisdependent import BasisDependent
#These classes have been created to maintain an OOP hierarchy
#for Vectors and Dyadics. Are NOT meant to be initialized
pass
def test_sympy__vector__basisdependent__BasisDependentMul():
#from sympy.vector.basisdependent import BasisDependentMul
#These classes have been created to maintain an OOP hierarchy
#for Vectors and Dyadics. Are NOT meant to be initialized
pass
def test_sympy__vector__basisdependent__BasisDependentAdd():
#from sympy.vector.basisdependent import BasisDependentAdd
#These classes have been created to maintain an OOP hierarchy
#for Vectors and Dyadics. Are NOT meant to be initialized
pass
def test_sympy__vector__basisdependent__BasisDependentZero():
#from sympy.vector.basisdependent import BasisDependentZero
#These classes have been created to maintain an OOP hierarchy
#for Vectors and Dyadics. Are NOT meant to be initialized
pass
def test_sympy__vector__vector__BaseVector():
from sympy.vector.vector import BaseVector
from sympy.vector.coordsysrect import CoordSys3D
C = CoordSys3D('C')
assert _test_args(BaseVector(0, C, ' ', ' '))
def test_sympy__vector__vector__VectorAdd():
from sympy.vector.vector import VectorAdd, VectorMul
from sympy.vector.coordsysrect import CoordSys3D
C = CoordSys3D('C')
from sympy.abc import a, b, c, x, y, z
v1 = a*C.i + b*C.j + c*C.k
v2 = x*C.i + y*C.j + z*C.k
assert _test_args(VectorAdd(v1, v2))
assert _test_args(VectorMul(x, v1))
def test_sympy__vector__vector__VectorMul():
from sympy.vector.vector import VectorMul
from sympy.vector.coordsysrect import CoordSys3D
C = CoordSys3D('C')
from sympy.abc import a
assert _test_args(VectorMul(a, C.i))
def test_sympy__vector__vector__VectorZero():
from sympy.vector.vector import VectorZero
assert _test_args(VectorZero())
def test_sympy__vector__vector__Vector():
#from sympy.vector.vector import Vector
#Vector is never to be initialized using args
pass
def test_sympy__vector__vector__Cross():
from sympy.vector.vector import Cross
from sympy.vector.coordsysrect import CoordSys3D
C = CoordSys3D('C')
_test_args(Cross(C.i, C.j))
def test_sympy__vector__vector__Dot():
from sympy.vector.vector import Dot
from sympy.vector.coordsysrect import CoordSys3D
C = CoordSys3D('C')
_test_args(Dot(C.i, C.j))
def test_sympy__vector__dyadic__Dyadic():
#from sympy.vector.dyadic import Dyadic
#Dyadic is never to be initialized using args
pass
def test_sympy__vector__dyadic__BaseDyadic():
from sympy.vector.dyadic import BaseDyadic
from sympy.vector.coordsysrect import CoordSys3D
C = CoordSys3D('C')
assert _test_args(BaseDyadic(C.i, C.j))
def test_sympy__vector__dyadic__DyadicMul():
from sympy.vector.dyadic import BaseDyadic, DyadicMul
from sympy.vector.coordsysrect import CoordSys3D
C = CoordSys3D('C')
assert _test_args(DyadicMul(3, BaseDyadic(C.i, C.j)))
def test_sympy__vector__dyadic__DyadicAdd():
from sympy.vector.dyadic import BaseDyadic, DyadicAdd
from sympy.vector.coordsysrect import CoordSys3D
C = CoordSys3D('C')
assert _test_args(2 * DyadicAdd(BaseDyadic(C.i, C.i),
BaseDyadic(C.i, C.j)))
def test_sympy__vector__dyadic__DyadicZero():
from sympy.vector.dyadic import DyadicZero
assert _test_args(DyadicZero())
def test_sympy__vector__deloperator__Del():
from sympy.vector.deloperator import Del
assert _test_args(Del())
def test_sympy__vector__operators__Curl():
from sympy.vector.operators import Curl
from sympy.vector.coordsysrect import CoordSys3D
C = CoordSys3D('C')
assert _test_args(Curl(C.i))
def test_sympy__vector__operators__Laplacian():
from sympy.vector.operators import Laplacian
from sympy.vector.coordsysrect import CoordSys3D
C = CoordSys3D('C')
assert _test_args(Laplacian(C.i))
def test_sympy__vector__operators__Divergence():
from sympy.vector.operators import Divergence
from sympy.vector.coordsysrect import CoordSys3D
C = CoordSys3D('C')
assert _test_args(Divergence(C.i))
def test_sympy__vector__operators__Gradient():
from sympy.vector.operators import Gradient
from sympy.vector.coordsysrect import CoordSys3D
C = CoordSys3D('C')
assert _test_args(Gradient(C.x))
def test_sympy__vector__orienters__Orienter():
#from sympy.vector.orienters import Orienter
#Not to be initialized
pass
def test_sympy__vector__orienters__ThreeAngleOrienter():
#from sympy.vector.orienters import ThreeAngleOrienter
#Not to be initialized
pass
def test_sympy__vector__orienters__AxisOrienter():
from sympy.vector.orienters import AxisOrienter
from sympy.vector.coordsysrect import CoordSys3D
C = CoordSys3D('C')
assert _test_args(AxisOrienter(x, C.i))
def test_sympy__vector__orienters__BodyOrienter():
from sympy.vector.orienters import BodyOrienter
assert _test_args(BodyOrienter(x, y, z, '123'))
def test_sympy__vector__orienters__SpaceOrienter():
from sympy.vector.orienters import SpaceOrienter
assert _test_args(SpaceOrienter(x, y, z, '123'))
def test_sympy__vector__orienters__QuaternionOrienter():
from sympy.vector.orienters import QuaternionOrienter
a, b, c, d = symbols('a b c d')
assert _test_args(QuaternionOrienter(a, b, c, d))
def test_sympy__vector__scalar__BaseScalar():
from sympy.vector.scalar import BaseScalar
from sympy.vector.coordsysrect import CoordSys3D
C = CoordSys3D('C')
assert _test_args(BaseScalar(0, C, ' ', ' '))
def test_sympy__physics__wigner__Wigner3j():
from sympy.physics.wigner import Wigner3j
assert _test_args(Wigner3j(0, 0, 0, 0, 0, 0))
def test_sympy__integrals__rubi__symbol__matchpyWC():
from sympy.integrals.rubi.symbol import matchpyWC
assert _test_args(matchpyWC(1, True, 'a'))
def test_sympy__integrals__rubi__utility_function__rubi_unevaluated_expr():
from sympy.integrals.rubi.utility_function import rubi_unevaluated_expr
a = symbols('a')
assert _test_args(rubi_unevaluated_expr(a))
def test_sympy__integrals__rubi__utility_function__rubi_exp():
from sympy.integrals.rubi.utility_function import rubi_exp
assert _test_args(rubi_exp(5))
def test_sympy__integrals__rubi__utility_function__rubi_log():
from sympy.integrals.rubi.utility_function import rubi_log
assert _test_args(rubi_log(5))
def test_sympy__integrals__rubi__utility_function__Int():
from sympy.integrals.rubi.utility_function import Int
assert _test_args(Int(5, x))
def test_sympy__integrals__rubi__utility_function__Util_Coefficient():
from sympy.integrals.rubi.utility_function import Util_Coefficient
a, x = symbols('a x')
assert _test_args(Util_Coefficient(a, x))
def test_sympy__integrals__rubi__utility_function__Gamma():
from sympy.integrals.rubi.utility_function import Gamma
assert _test_args(Gamma(5))
def test_sympy__integrals__rubi__utility_function__Util_Part():
from sympy.integrals.rubi.utility_function import Util_Part
a, b = symbols('a b')
assert _test_args(Util_Part(a + b, 0))
def test_sympy__integrals__rubi__utility_function__PolyGamma():
from sympy.integrals.rubi.utility_function import PolyGamma
assert _test_args(PolyGamma(1, 1))
def test_sympy__integrals__rubi__utility_function__ProductLog():
from sympy.integrals.rubi.utility_function import ProductLog
assert _test_args(ProductLog(1))
def test_sympy__combinatorics__schur_number__SchurNumber():
from sympy.combinatorics.schur_number import SchurNumber
assert _test_args(SchurNumber(1))
def test_sympy__combinatorics__perm_groups__SymmetricPermutationGroup():
from sympy.combinatorics.perm_groups import SymmetricPermutationGroup
assert _test_args(SymmetricPermutationGroup(5))
def test_sympy__combinatorics__perm_groups__Coset():
from sympy.combinatorics.permutations import Permutation
from sympy.combinatorics.perm_groups import PermutationGroup, Coset
a = Permutation(1, 2)
b = Permutation(0, 1)
G = PermutationGroup([a, b])
assert _test_args(Coset(a, G))
|
0d9d7e274f9d9e6d6a781aa7989e2d84e5c9c293eb9fe89f526480e69e9cef44 | from sympy import (Symbol, Wild, GreaterThan, LessThan, StrictGreaterThan,
StrictLessThan, pi, I, Rational, sympify, symbols, Dummy)
from sympy.core.symbol import uniquely_named_symbol, _symbol
from sympy.testing.pytest import raises
from sympy.core.symbol import disambiguate
def test_Symbol():
a = Symbol("a")
x1 = Symbol("x")
x2 = Symbol("x")
xdummy1 = Dummy("x")
xdummy2 = Dummy("x")
assert a != x1
assert a != x2
assert x1 == x2
assert x1 != xdummy1
assert xdummy1 != xdummy2
assert Symbol("x") == Symbol("x")
assert Dummy("x") != Dummy("x")
d = symbols('d', cls=Dummy)
assert isinstance(d, Dummy)
c, d = symbols('c,d', cls=Dummy)
assert isinstance(c, Dummy)
assert isinstance(d, Dummy)
raises(TypeError, lambda: Symbol())
def test_Dummy():
assert Dummy() != Dummy()
def test_Dummy_force_dummy_index():
raises(AssertionError, lambda: Dummy(dummy_index=1))
assert Dummy('d', dummy_index=2) == Dummy('d', dummy_index=2)
assert Dummy('d1', dummy_index=2) != Dummy('d2', dummy_index=2)
d1 = Dummy('d', dummy_index=3)
d2 = Dummy('d')
# might fail if d1 were created with dummy_index >= 10**6
assert d1 != d2
d3 = Dummy('d', dummy_index=3)
assert d1 == d3
assert Dummy()._count == Dummy('d', dummy_index=3)._count
def test_lt_gt():
from sympy import sympify as S
x, y = Symbol('x'), Symbol('y')
assert (x >= y) == GreaterThan(x, y)
assert (x >= 0) == GreaterThan(x, 0)
assert (x <= y) == LessThan(x, y)
assert (x <= 0) == LessThan(x, 0)
assert (0 <= x) == GreaterThan(x, 0)
assert (0 >= x) == LessThan(x, 0)
assert (S(0) >= x) == GreaterThan(0, x)
assert (S(0) <= x) == LessThan(0, x)
assert (x > y) == StrictGreaterThan(x, y)
assert (x > 0) == StrictGreaterThan(x, 0)
assert (x < y) == StrictLessThan(x, y)
assert (x < 0) == StrictLessThan(x, 0)
assert (0 < x) == StrictGreaterThan(x, 0)
assert (0 > x) == StrictLessThan(x, 0)
assert (S(0) > x) == StrictGreaterThan(0, x)
assert (S(0) < x) == StrictLessThan(0, x)
e = x**2 + 4*x + 1
assert (e >= 0) == GreaterThan(e, 0)
assert (0 <= e) == GreaterThan(e, 0)
assert (e > 0) == StrictGreaterThan(e, 0)
assert (0 < e) == StrictGreaterThan(e, 0)
assert (e <= 0) == LessThan(e, 0)
assert (0 >= e) == LessThan(e, 0)
assert (e < 0) == StrictLessThan(e, 0)
assert (0 > e) == StrictLessThan(e, 0)
assert (S(0) >= e) == GreaterThan(0, e)
assert (S(0) <= e) == LessThan(0, e)
assert (S(0) < e) == StrictLessThan(0, e)
assert (S(0) > e) == StrictGreaterThan(0, e)
def test_no_len():
# there should be no len for numbers
x = Symbol('x')
raises(TypeError, lambda: len(x))
def test_ineq_unequal():
S = sympify
x, y, z = symbols('x,y,z')
e = (
S(-1) >= x, S(-1) >= y, S(-1) >= z,
S(-1) > x, S(-1) > y, S(-1) > z,
S(-1) <= x, S(-1) <= y, S(-1) <= z,
S(-1) < x, S(-1) < y, S(-1) < z,
S(0) >= x, S(0) >= y, S(0) >= z,
S(0) > x, S(0) > y, S(0) > z,
S(0) <= x, S(0) <= y, S(0) <= z,
S(0) < x, S(0) < y, S(0) < z,
S('3/7') >= x, S('3/7') >= y, S('3/7') >= z,
S('3/7') > x, S('3/7') > y, S('3/7') > z,
S('3/7') <= x, S('3/7') <= y, S('3/7') <= z,
S('3/7') < x, S('3/7') < y, S('3/7') < z,
S(1.5) >= x, S(1.5) >= y, S(1.5) >= z,
S(1.5) > x, S(1.5) > y, S(1.5) > z,
S(1.5) <= x, S(1.5) <= y, S(1.5) <= z,
S(1.5) < x, S(1.5) < y, S(1.5) < z,
S(2) >= x, S(2) >= y, S(2) >= z,
S(2) > x, S(2) > y, S(2) > z,
S(2) <= x, S(2) <= y, S(2) <= z,
S(2) < x, S(2) < y, S(2) < z,
x >= -1, y >= -1, z >= -1,
x > -1, y > -1, z > -1,
x <= -1, y <= -1, z <= -1,
x < -1, y < -1, z < -1,
x >= 0, y >= 0, z >= 0,
x > 0, y > 0, z > 0,
x <= 0, y <= 0, z <= 0,
x < 0, y < 0, z < 0,
x >= 1.5, y >= 1.5, z >= 1.5,
x > 1.5, y > 1.5, z > 1.5,
x <= 1.5, y <= 1.5, z <= 1.5,
x < 1.5, y < 1.5, z < 1.5,
x >= 2, y >= 2, z >= 2,
x > 2, y > 2, z > 2,
x <= 2, y <= 2, z <= 2,
x < 2, y < 2, z < 2,
x >= y, x >= z, y >= x, y >= z, z >= x, z >= y,
x > y, x > z, y > x, y > z, z > x, z > y,
x <= y, x <= z, y <= x, y <= z, z <= x, z <= y,
x < y, x < z, y < x, y < z, z < x, z < y,
x - pi >= y + z, y - pi >= x + z, z - pi >= x + y,
x - pi > y + z, y - pi > x + z, z - pi > x + y,
x - pi <= y + z, y - pi <= x + z, z - pi <= x + y,
x - pi < y + z, y - pi < x + z, z - pi < x + y,
True, False
)
left_e = e[:-1]
for i, e1 in enumerate( left_e ):
for e2 in e[i + 1:]:
assert e1 != e2
def test_Wild_properties():
# these tests only include Atoms
x = Symbol("x")
y = Symbol("y")
p = Symbol("p", positive=True)
k = Symbol("k", integer=True)
n = Symbol("n", integer=True, positive=True)
given_patterns = [ x, y, p, k, -k, n, -n, sympify(-3), sympify(3),
pi, Rational(3, 2), I ]
integerp = lambda k: k.is_integer
positivep = lambda k: k.is_positive
symbolp = lambda k: k.is_Symbol
realp = lambda k: k.is_extended_real
S = Wild("S", properties=[symbolp])
R = Wild("R", properties=[realp])
Y = Wild("Y", exclude=[x, p, k, n])
P = Wild("P", properties=[positivep])
K = Wild("K", properties=[integerp])
N = Wild("N", properties=[positivep, integerp])
given_wildcards = [ S, R, Y, P, K, N ]
goodmatch = {
S: (x, y, p, k, n),
R: (p, k, -k, n, -n, -3, 3, pi, Rational(3, 2)),
Y: (y, -3, 3, pi, Rational(3, 2), I ),
P: (p, n, 3, pi, Rational(3, 2)),
K: (k, -k, n, -n, -3, 3),
N: (n, 3)}
for A in given_wildcards:
for pat in given_patterns:
d = pat.match(A)
if pat in goodmatch[A]:
assert d[A] in goodmatch[A]
else:
assert d is None
def test_symbols():
x = Symbol('x')
y = Symbol('y')
z = Symbol('z')
assert symbols('x') == x
assert symbols('x ') == x
assert symbols(' x ') == x
assert symbols('x,') == (x,)
assert symbols('x, ') == (x,)
assert symbols('x ,') == (x,)
assert symbols('x , y') == (x, y)
assert symbols('x,y,z') == (x, y, z)
assert symbols('x y z') == (x, y, z)
assert symbols('x,y,z,') == (x, y, z)
assert symbols('x y z ') == (x, y, z)
xyz = Symbol('xyz')
abc = Symbol('abc')
assert symbols('xyz') == xyz
assert symbols('xyz,') == (xyz,)
assert symbols('xyz,abc') == (xyz, abc)
assert symbols(('xyz',)) == (xyz,)
assert symbols(('xyz,',)) == ((xyz,),)
assert symbols(('x,y,z,',)) == ((x, y, z),)
assert symbols(('xyz', 'abc')) == (xyz, abc)
assert symbols(('xyz,abc',)) == ((xyz, abc),)
assert symbols(('xyz,abc', 'x,y,z')) == ((xyz, abc), (x, y, z))
assert symbols(('x', 'y', 'z')) == (x, y, z)
assert symbols(['x', 'y', 'z']) == [x, y, z]
assert symbols({'x', 'y', 'z'}) == {x, y, z}
raises(ValueError, lambda: symbols(''))
raises(ValueError, lambda: symbols(','))
raises(ValueError, lambda: symbols('x,,y,,z'))
raises(ValueError, lambda: symbols(('x', '', 'y', '', 'z')))
a, b = symbols('x,y', real=True)
assert a.is_real and b.is_real
x0 = Symbol('x0')
x1 = Symbol('x1')
x2 = Symbol('x2')
y0 = Symbol('y0')
y1 = Symbol('y1')
assert symbols('x0:0') == ()
assert symbols('x0:1') == (x0,)
assert symbols('x0:2') == (x0, x1)
assert symbols('x0:3') == (x0, x1, x2)
assert symbols('x:0') == ()
assert symbols('x:1') == (x0,)
assert symbols('x:2') == (x0, x1)
assert symbols('x:3') == (x0, x1, x2)
assert symbols('x1:1') == ()
assert symbols('x1:2') == (x1,)
assert symbols('x1:3') == (x1, x2)
assert symbols('x1:3,x,y,z') == (x1, x2, x, y, z)
assert symbols('x:3,y:2') == (x0, x1, x2, y0, y1)
assert symbols(('x:3', 'y:2')) == ((x0, x1, x2), (y0, y1))
a = Symbol('a')
b = Symbol('b')
c = Symbol('c')
d = Symbol('d')
assert symbols('x:z') == (x, y, z)
assert symbols('a:d,x:z') == (a, b, c, d, x, y, z)
assert symbols(('a:d', 'x:z')) == ((a, b, c, d), (x, y, z))
aa = Symbol('aa')
ab = Symbol('ab')
ac = Symbol('ac')
ad = Symbol('ad')
assert symbols('aa:d') == (aa, ab, ac, ad)
assert symbols('aa:d,x:z') == (aa, ab, ac, ad, x, y, z)
assert symbols(('aa:d','x:z')) == ((aa, ab, ac, ad), (x, y, z))
# issue 6675
def sym(s):
return str(symbols(s))
assert sym('a0:4') == '(a0, a1, a2, a3)'
assert sym('a2:4,b1:3') == '(a2, a3, b1, b2)'
assert sym('a1(2:4)') == '(a12, a13)'
assert sym('a0:2.0:2') == '(a0.0, a0.1, a1.0, a1.1)'
assert sym('aa:cz') == '(aaz, abz, acz)'
assert sym('aa:c0:2') == '(aa0, aa1, ab0, ab1, ac0, ac1)'
assert sym('aa:ba:b') == '(aaa, aab, aba, abb)'
assert sym('a:3b') == '(a0b, a1b, a2b)'
assert sym('a-1:3b') == '(a-1b, a-2b)'
assert sym(r'a:2\,:2' + chr(0)) == '(a0,0%s, a0,1%s, a1,0%s, a1,1%s)' % (
(chr(0),)*4)
assert sym('x(:a:3)') == '(x(a0), x(a1), x(a2))'
assert sym('x(:c):1') == '(xa0, xb0, xc0)'
assert sym('x((:a)):3') == '(x(a)0, x(a)1, x(a)2)'
assert sym('x(:a:3') == '(x(a0, x(a1, x(a2)'
assert sym(':2') == '(0, 1)'
assert sym(':b') == '(a, b)'
assert sym(':b:2') == '(a0, a1, b0, b1)'
assert sym(':2:2') == '(00, 01, 10, 11)'
assert sym(':b:b') == '(aa, ab, ba, bb)'
raises(ValueError, lambda: symbols(':'))
raises(ValueError, lambda: symbols('a:'))
raises(ValueError, lambda: symbols('::'))
raises(ValueError, lambda: symbols('a::'))
raises(ValueError, lambda: symbols(':a:'))
raises(ValueError, lambda: symbols('::a'))
def test_symbols_become_functions_issue_3539():
from sympy.abc import alpha, phi, beta, t
raises(TypeError, lambda: beta(2))
raises(TypeError, lambda: beta(2.5))
raises(TypeError, lambda: phi(2.5))
raises(TypeError, lambda: alpha(2.5))
raises(TypeError, lambda: phi(t))
def test_unicode():
xu = Symbol('x')
x = Symbol('x')
assert x == xu
raises(TypeError, lambda: Symbol(1))
def testuniquely_named_symbol_and__symbol():
F = uniquely_named_symbol
x = Symbol('x')
assert F(x) == x
assert F('x') == x
assert str(F('x', x)) == 'x0'
assert str(F('x', (x + 1, 1/x))) == 'x0'
_x = Symbol('x', real=True)
assert F(('x', _x)) == _x
assert F((x, _x)) == _x
assert F('x', real=True).is_real
y = Symbol('y')
assert F(('x', y), real=True).is_real
r = Symbol('x', real=True)
assert F(('x', r)).is_real
assert F(('x', r), real=False).is_real
assert F('x1', Symbol('x1'),
compare=lambda i: str(i).rstrip('1')).name == 'x1'
assert F('x1', Symbol('x1'),
modify=lambda i: i + '_').name == 'x1_'
assert _symbol(x, _x) == x
def test_disambiguate():
x, y, y_1, _x, x_1, x_2 = symbols('x y y_1 _x x_1 x_2')
t1 = Dummy('y'), _x, Dummy('x'), Dummy('x')
t2 = Dummy('x'), Dummy('x')
t3 = Dummy('x'), Dummy('y')
t4 = x, Dummy('x')
t5 = Symbol('x', integer=True), x, Symbol('x_1')
assert disambiguate(*t1) == (y, x_2, x, x_1)
assert disambiguate(*t2) == (x, x_1)
assert disambiguate(*t3) == (x, y)
assert disambiguate(*t4) == (x_1, x)
assert disambiguate(*t5) == (t5[0], x_2, x_1)
assert disambiguate(*t5)[0] != x # assumptions are retained
t6 = _x, Dummy('x')/y
t7 = y*Dummy('y'), y
assert disambiguate(*t6) == (x_1, x/y)
assert disambiguate(*t7) == (y*y_1, y_1)
assert disambiguate(Dummy('x_1'), Dummy('x_1')
) == (x_1, Symbol('x_1_1'))
|
ea3e57bb1d2d34157784be7989e7499445b1bcbf00c1830b98dc2360e2b55c6c | from sympy import symbols, sin, exp, cos, Derivative, Integral, Basic, \
count_ops, S, And, I, pi, Eq, Or, Not, Xor, Nand, Nor, Implies, \
Equivalent, MatrixSymbol, Symbol, ITE, Rel, Rational, Sum
from sympy.core.containers import Tuple
x, y, z = symbols('x,y,z')
a, b, c = symbols('a,b,c')
def test_count_ops_non_visual():
def count(val):
return count_ops(val, visual=False)
assert count(x) == 0
assert count(x) is not S.Zero
assert count(x + y) == 1
assert count(x + y) is not S.One
assert count(x + y*x + 2*y) == 4
assert count({x + y: x}) == 1
assert count({x + y: S(2) + x}) is not S.One
assert count(x < y) == 1
assert count(Or(x,y)) == 1
assert count(And(x,y)) == 1
assert count(Not(x)) == 1
assert count(Nor(x,y)) == 2
assert count(Nand(x,y)) == 2
assert count(Xor(x,y)) == 1
assert count(Implies(x,y)) == 1
assert count(Equivalent(x,y)) == 1
assert count(ITE(x,y,z)) == 1
assert count(ITE(True,x,y)) == 0
def test_count_ops_visual():
ADD, MUL, POW, SIN, COS, EXP, AND, D, G, M = symbols(
'Add Mul Pow sin cos exp And Derivative Integral Sum'.upper())
DIV, SUB, NEG = symbols('DIV SUB NEG')
LT, LE, GT, GE, EQ, NE = symbols('LT LE GT GE EQ NE')
NOT, OR, AND, XOR, IMPLIES, EQUIVALENT, _ITE, BASIC, TUPLE = symbols(
'Not Or And Xor Implies Equivalent ITE Basic Tuple'.upper())
def count(val):
return count_ops(val, visual=True)
assert count(7) is S.Zero
assert count(S(7)) is S.Zero
assert count(-1) == NEG
assert count(-2) == NEG
assert count(S(2)/3) == DIV
assert count(Rational(2, 3)) == DIV
assert count(pi/3) == DIV
assert count(-pi/3) == DIV + NEG
assert count(I - 1) == SUB
assert count(1 - I) == SUB
assert count(1 - 2*I) == SUB + MUL
assert count(x) is S.Zero
assert count(-x) == NEG
assert count(-2*x/3) == NEG + DIV + MUL
assert count(Rational(-2, 3)*x) == NEG + DIV + MUL
assert count(1/x) == DIV
assert count(1/(x*y)) == DIV + MUL
assert count(-1/x) == NEG + DIV
assert count(-2/x) == NEG + DIV
assert count(x/y) == DIV
assert count(-x/y) == NEG + DIV
assert count(x**2) == POW
assert count(-x**2) == POW + NEG
assert count(-2*x**2) == POW + MUL + NEG
assert count(x + pi/3) == ADD + DIV
assert count(x + S.One/3) == ADD + DIV
assert count(x + Rational(1, 3)) == ADD + DIV
assert count(x + y) == ADD
assert count(x - y) == SUB
assert count(y - x) == SUB
assert count(-1/(x - y)) == DIV + NEG + SUB
assert count(-1/(y - x)) == DIV + NEG + SUB
assert count(1 + x**y) == ADD + POW
assert count(1 + x + y) == 2*ADD
assert count(1 + x + y + z) == 3*ADD
assert count(1 + x**y + 2*x*y + y**2) == 3*ADD + 2*POW + 2*MUL
assert count(2*z + y + x + 1) == 3*ADD + MUL
assert count(2*z + y**17 + x + 1) == 3*ADD + MUL + POW
assert count(2*z + y**17 + x + sin(x)) == 3*ADD + POW + MUL + SIN
assert count(2*z + y**17 + x + sin(x**2)) == 3*ADD + MUL + 2*POW + SIN
assert count(2*z + y**17 + x + sin(
x**2) + exp(cos(x))) == 4*ADD + MUL + 2*POW + EXP + COS + SIN
assert count(Derivative(x, x)) == D
assert count(Integral(x, x) + 2*x/(1 + x)) == G + DIV + MUL + 2*ADD
assert count(Sum(x, (x, 1, x + 1)) + 2*x/(1 + x)) == M + DIV + MUL + 3*ADD
assert count(Basic()) is S.Zero
assert count({x + 1: sin(x)}) == ADD + SIN
assert count([x + 1, sin(x) + y, None]) == ADD + SIN + ADD
assert count({x + 1: sin(x), y: cos(x) + 1}) == SIN + COS + 2*ADD
assert count({}) is S.Zero
assert count([x + 1, sin(x)*y, None]) == SIN + ADD + MUL
assert count([]) is S.Zero
assert count(Basic()) == 0
assert count(Basic(Basic(),Basic(x,x+y))) == ADD + 2*BASIC
assert count(Basic(x, x + y)) == ADD + BASIC
assert [count(Rel(x, y, op)) for op in '< <= > >= == <> !='.split()
] == [LT, LE, GT, GE, EQ, NE, NE]
assert count(Or(x, y)) == OR
assert count(And(x, y)) == AND
assert count(Or(x, Or(y, And(z, a)))) == AND + OR
assert count(Nor(x, y)) == NOT + OR
assert count(Nand(x, y)) == NOT + AND
assert count(Xor(x, y)) == XOR
assert count(Implies(x, y)) == IMPLIES
assert count(Equivalent(x, y)) == EQUIVALENT
assert count(ITE(x, y, z)) == _ITE
assert count([Or(x, y), And(x, y), Basic(x + y)]
) == ADD + AND + BASIC + OR
assert count(Basic(Tuple(x))) == BASIC + TUPLE
#It checks that TUPLE is counted as an operation.
assert count(Eq(x + y, S(2))) == ADD + EQ
def test_issue_9324():
def count(val):
return count_ops(val, visual=False)
M = MatrixSymbol('M', 10, 10)
assert count(M[0, 0]) == 0
assert count(2 * M[0, 0] + M[5, 7]) == 2
P = MatrixSymbol('P', 3, 3)
Q = MatrixSymbol('Q', 3, 3)
assert count(P + Q) == 1
m = Symbol('m', integer=True)
n = Symbol('n', integer=True)
M = MatrixSymbol('M', m + n, m * m)
assert count(M[0, 1]) == 2
|
0081ba2b1f1a25c5eb2c3cb72f4b034470c4f9f770c699d7cf90c8ab213ca933 | from sympy import (Lambda, Symbol, Function, Derivative, Subs, sqrt,
log, exp, Rational, Float, sin, cos, acos, diff, I, re, im,
E, expand, pi, O, Sum, S, polygamma, loggamma, expint,
Tuple, Dummy, Eq, Expr, symbols, nfloat, Piecewise, Indexed,
Matrix, Basic, Dict, oo, zoo, nan, Pow)
from sympy.core.basic import _aresame
from sympy.core.cache import clear_cache
from sympy.core.expr import unchanged
from sympy.core.function import (PoleError, _mexpand, arity,
BadSignatureError, BadArgumentsError)
from sympy.core.sympify import sympify
from sympy.matrices import MutableMatrix, ImmutableMatrix
from sympy.sets.sets import FiniteSet
from sympy.solvers.solveset import solveset
from sympy.tensor.array import NDimArray
from sympy.utilities.iterables import subsets, variations
from sympy.testing.pytest import XFAIL, raises, warns_deprecated_sympy
from sympy.abc import t, w, x, y, z
f, g, h = symbols('f g h', cls=Function)
_xi_1, _xi_2, _xi_3 = [Dummy() for i in range(3)]
def test_f_expand_complex():
x = Symbol('x', real=True)
assert f(x).expand(complex=True) == I*im(f(x)) + re(f(x))
assert exp(x).expand(complex=True) == exp(x)
assert exp(I*x).expand(complex=True) == cos(x) + I*sin(x)
assert exp(z).expand(complex=True) == cos(im(z))*exp(re(z)) + \
I*sin(im(z))*exp(re(z))
def test_bug1():
e = sqrt(-log(w))
assert e.subs(log(w), -x) == sqrt(x)
e = sqrt(-5*log(w))
assert e.subs(log(w), -x) == sqrt(5*x)
def test_general_function():
nu = Function('nu')
e = nu(x)
edx = e.diff(x)
edy = e.diff(y)
edxdx = e.diff(x).diff(x)
edxdy = e.diff(x).diff(y)
assert e == nu(x)
assert edx != nu(x)
assert edx == diff(nu(x), x)
assert edy == 0
assert edxdx == diff(diff(nu(x), x), x)
assert edxdy == 0
def test_general_function_nullary():
nu = Function('nu')
e = nu()
edx = e.diff(x)
edxdx = e.diff(x).diff(x)
assert e == nu()
assert edx != nu()
assert edx == 0
assert edxdx == 0
def test_derivative_subs_bug():
e = diff(g(x), x)
assert e.subs(g(x), f(x)) != e
assert e.subs(g(x), f(x)) == Derivative(f(x), x)
assert e.subs(g(x), -f(x)) == Derivative(-f(x), x)
assert e.subs(x, y) == Derivative(g(y), y)
def test_derivative_subs_self_bug():
d = diff(f(x), x)
assert d.subs(d, y) == y
def test_derivative_linearity():
assert diff(-f(x), x) == -diff(f(x), x)
assert diff(8*f(x), x) == 8*diff(f(x), x)
assert diff(8*f(x), x) != 7*diff(f(x), x)
assert diff(8*f(x)*x, x) == 8*f(x) + 8*x*diff(f(x), x)
assert diff(8*f(x)*y*x, x).expand() == 8*y*f(x) + 8*y*x*diff(f(x), x)
def test_derivative_evaluate():
assert Derivative(sin(x), x) != diff(sin(x), x)
assert Derivative(sin(x), x).doit() == diff(sin(x), x)
assert Derivative(Derivative(f(x), x), x) == diff(f(x), x, x)
assert Derivative(sin(x), x, 0) == sin(x)
assert Derivative(sin(x), (x, y), (x, -y)) == sin(x)
def test_diff_symbols():
assert diff(f(x, y, z), x, y, z) == Derivative(f(x, y, z), x, y, z)
assert diff(f(x, y, z), x, x, x) == Derivative(f(x, y, z), x, x, x) == Derivative(f(x, y, z), (x, 3))
assert diff(f(x, y, z), x, 3) == Derivative(f(x, y, z), x, 3)
# issue 5028
assert [diff(-z + x/y, sym) for sym in (z, x, y)] == [-1, 1/y, -x/y**2]
assert diff(f(x, y, z), x, y, z, 2) == Derivative(f(x, y, z), x, y, z, z)
assert diff(f(x, y, z), x, y, z, 2, evaluate=False) == \
Derivative(f(x, y, z), x, y, z, z)
assert Derivative(f(x, y, z), x, y, z)._eval_derivative(z) == \
Derivative(f(x, y, z), x, y, z, z)
assert Derivative(Derivative(f(x, y, z), x), y)._eval_derivative(z) == \
Derivative(f(x, y, z), x, y, z)
raises(TypeError, lambda: cos(x).diff((x, y)).variables)
assert cos(x).diff((x, y))._wrt_variables == [x]
def test_Function():
class myfunc(Function):
@classmethod
def eval(cls): # zero args
return
assert myfunc.nargs == FiniteSet(0)
assert myfunc().nargs == FiniteSet(0)
raises(TypeError, lambda: myfunc(x).nargs)
class myfunc(Function):
@classmethod
def eval(cls, x): # one arg
return
assert myfunc.nargs == FiniteSet(1)
assert myfunc(x).nargs == FiniteSet(1)
raises(TypeError, lambda: myfunc(x, y).nargs)
class myfunc(Function):
@classmethod
def eval(cls, *x): # star args
return
assert myfunc.nargs == S.Naturals0
assert myfunc(x).nargs == S.Naturals0
def test_nargs():
f = Function('f')
assert f.nargs == S.Naturals0
assert f(1).nargs == S.Naturals0
assert Function('f', nargs=2)(1, 2).nargs == FiniteSet(2)
assert sin.nargs == FiniteSet(1)
assert sin(2).nargs == FiniteSet(1)
assert log.nargs == FiniteSet(1, 2)
assert log(2).nargs == FiniteSet(1, 2)
assert Function('f', nargs=2).nargs == FiniteSet(2)
assert Function('f', nargs=0).nargs == FiniteSet(0)
assert Function('f', nargs=(0, 1)).nargs == FiniteSet(0, 1)
assert Function('f', nargs=None).nargs == S.Naturals0
raises(ValueError, lambda: Function('f', nargs=()))
def test_nargs_inheritance():
class f1(Function):
nargs = 2
class f2(f1):
pass
class f3(f2):
pass
class f4(f3):
nargs = 1,2
class f5(f4):
pass
class f6(f5):
pass
class f7(f6):
nargs=None
class f8(f7):
pass
class f9(f8):
pass
class f10(f9):
nargs = 1
class f11(f10):
pass
assert f1.nargs == FiniteSet(2)
assert f2.nargs == FiniteSet(2)
assert f3.nargs == FiniteSet(2)
assert f4.nargs == FiniteSet(1, 2)
assert f5.nargs == FiniteSet(1, 2)
assert f6.nargs == FiniteSet(1, 2)
assert f7.nargs == S.Naturals0
assert f8.nargs == S.Naturals0
assert f9.nargs == S.Naturals0
assert f10.nargs == FiniteSet(1)
assert f11.nargs == FiniteSet(1)
def test_arity():
f = lambda x, y: 1
assert arity(f) == 2
def f(x, y, z=None):
pass
assert arity(f) == (2, 3)
assert arity(lambda *x: x) is None
assert arity(log) == (1, 2)
def test_Lambda():
e = Lambda(x, x**2)
assert e(4) == 16
assert e(x) == x**2
assert e(y) == y**2
assert Lambda((), 42)() == 42
assert unchanged(Lambda, (), 42)
assert Lambda((), 42) != Lambda((), 43)
assert Lambda((), f(x))() == f(x)
assert Lambda((), 42).nargs == FiniteSet(0)
assert unchanged(Lambda, (x,), x**2)
assert Lambda(x, x**2) == Lambda((x,), x**2)
assert Lambda(x, x**2) != Lambda(x, x**2 + 1)
assert Lambda((x, y), x**y) != Lambda((y, x), y**x)
assert Lambda((x, y), x**y) != Lambda((x, y), y**x)
assert Lambda((x, y), x**y)(x, y) == x**y
assert Lambda((x, y), x**y)(3, 3) == 3**3
assert Lambda((x, y), x**y)(x, 3) == x**3
assert Lambda((x, y), x**y)(3, y) == 3**y
assert Lambda(x, f(x))(x) == f(x)
assert Lambda(x, x**2)(e(x)) == x**4
assert e(e(x)) == x**4
x1, x2 = (Indexed('x', i) for i in (1, 2))
assert Lambda((x1, x2), x1 + x2)(x, y) == x + y
assert Lambda((x, y), x + y).nargs == FiniteSet(2)
p = x, y, z, t
assert Lambda(p, t*(x + y + z))(*p) == t * (x + y + z)
eq = Lambda(x, 2*x) + Lambda(y, 2*y)
assert eq != 2*Lambda(x, 2*x)
assert eq.as_dummy() == 2*Lambda(x, 2*x).as_dummy()
assert Lambda(x, 2*x) not in [ Lambda(x, x) ]
raises(BadSignatureError, lambda: Lambda(1, x))
assert Lambda(x, 1)(1) is S.One
raises(BadSignatureError, lambda: Lambda((x, x), x + 2))
raises(BadSignatureError, lambda: Lambda(((x, x), y), x))
raises(BadSignatureError, lambda: Lambda(((y, x), x), x))
raises(BadSignatureError, lambda: Lambda(((y, 1), 2), x))
with warns_deprecated_sympy():
assert Lambda([x, y], x+y) == Lambda((x, y), x+y)
flam = Lambda( ((x, y),) , x + y)
assert flam((2, 3)) == 5
flam = Lambda( ((x, y), z) , x + y + z)
assert flam((2, 3), 1) == 6
flam = Lambda( (((x,y),z),) , x+y+z)
assert flam( ((2,3),1) ) == 6
raises(BadArgumentsError, lambda: flam(1, 2, 3))
flam = Lambda( (x,), (x, x))
assert flam(1,) == (1, 1)
assert flam((1,)) == ((1,), (1,))
flam = Lambda( ((x,),) , (x, x))
raises(BadArgumentsError, lambda: flam(1))
assert flam((1,)) == (1, 1)
# Previously TypeError was raised so this is potentially needed for
# backwards compatibility.
assert issubclass(BadSignatureError, TypeError)
assert issubclass(BadArgumentsError, TypeError)
# These are tested to see they don't raise:
hash(Lambda(x, 2*x))
hash(Lambda(x, x)) # IdentityFunction subclass
def test_IdentityFunction():
assert Lambda(x, x) is Lambda(y, y) is S.IdentityFunction
assert Lambda(x, 2*x) is not S.IdentityFunction
assert Lambda((x, y), x) is not S.IdentityFunction
def test_Lambda_symbols():
assert Lambda(x, 2*x).free_symbols == set()
assert Lambda(x, x*y).free_symbols == {y}
assert Lambda((), 42).free_symbols == set()
assert Lambda((), x*y).free_symbols == {x,y}
def test_functionclas_symbols():
assert f.free_symbols == set()
def test_Lambda_arguments():
raises(TypeError, lambda: Lambda(x, 2*x)(x, y))
raises(TypeError, lambda: Lambda((x, y), x + y)(x))
raises(TypeError, lambda: Lambda((), 42)(x))
def test_Lambda_equality():
assert Lambda((x, y), 2*x) == Lambda((x, y), 2*x)
# these, of course, should never be equal
assert Lambda(x, 2*x) != Lambda((x, y), 2*x)
assert Lambda(x, 2*x) != 2*x
# But it is tempting to want expressions that differ only
# in bound symbols to compare the same. But this is not what
# Python's `==` is intended to do; two objects that compare
# as equal means that they are indistibguishable and cache to the
# same value. We wouldn't want to expression that are
# mathematically the same but written in different variables to be
# interchanged else what is the point of allowing for different
# variable names?
assert Lambda(x, 2*x) != Lambda(y, 2*y)
def test_Subs():
assert Subs(1, (), ()) is S.One
# check null subs influence on hashing
assert Subs(x, y, z) != Subs(x, y, 1)
# neutral subs works
assert Subs(x, x, 1).subs(x, y).has(y)
# self mapping var/point
assert Subs(Derivative(f(x), (x, 2)), x, x).doit() == f(x).diff(x, x)
assert Subs(x, x, 0).has(x) # it's a structural answer
assert not Subs(x, x, 0).free_symbols
assert Subs(Subs(x + y, x, 2), y, 1) == Subs(x + y, (x, y), (2, 1))
assert Subs(x, (x,), (0,)) == Subs(x, x, 0)
assert Subs(x, x, 0) == Subs(y, y, 0)
assert Subs(x, x, 0).subs(x, 1) == Subs(x, x, 0)
assert Subs(y, x, 0).subs(y, 1) == Subs(1, x, 0)
assert Subs(f(x), x, 0).doit() == f(0)
assert Subs(f(x**2), x**2, 0).doit() == f(0)
assert Subs(f(x, y, z), (x, y, z), (0, 1, 1)) != \
Subs(f(x, y, z), (x, y, z), (0, 0, 1))
assert Subs(x, y, 2).subs(x, y).doit() == 2
assert Subs(f(x, y), (x, y, z), (0, 1, 1)) != \
Subs(f(x, y) + z, (x, y, z), (0, 1, 0))
assert Subs(f(x, y), (x, y), (0, 1)).doit() == f(0, 1)
assert Subs(Subs(f(x, y), x, 0), y, 1).doit() == f(0, 1)
raises(ValueError, lambda: Subs(f(x, y), (x, y), (0, 0, 1)))
raises(ValueError, lambda: Subs(f(x, y), (x, x, y), (0, 0, 1)))
assert len(Subs(f(x, y), (x, y), (0, 1)).variables) == 2
assert Subs(f(x, y), (x, y), (0, 1)).point == Tuple(0, 1)
assert Subs(f(x), x, 0) == Subs(f(y), y, 0)
assert Subs(f(x, y), (x, y), (0, 1)) == Subs(f(x, y), (y, x), (1, 0))
assert Subs(f(x)*y, (x, y), (0, 1)) == Subs(f(y)*x, (y, x), (0, 1))
assert Subs(f(x)*y, (x, y), (1, 1)) == Subs(f(y)*x, (x, y), (1, 1))
assert Subs(f(x), x, 0).subs(x, 1).doit() == f(0)
assert Subs(f(x), x, y).subs(y, 0) == Subs(f(x), x, 0)
assert Subs(y*f(x), x, y).subs(y, 2) == Subs(2*f(x), x, 2)
assert (2 * Subs(f(x), x, 0)).subs(Subs(f(x), x, 0), y) == 2*y
assert Subs(f(x), x, 0).free_symbols == set()
assert Subs(f(x, y), x, z).free_symbols == {y, z}
assert Subs(f(x).diff(x), x, 0).doit(), Subs(f(x).diff(x), x, 0)
assert Subs(1 + f(x).diff(x), x, 0).doit(), 1 + Subs(f(x).diff(x), x, 0)
assert Subs(y*f(x, y).diff(x), (x, y), (0, 2)).doit() == \
2*Subs(Derivative(f(x, 2), x), x, 0)
assert Subs(y**2*f(x), x, 0).diff(y) == 2*y*f(0)
e = Subs(y**2*f(x), x, y)
assert e.diff(y) == e.doit().diff(y) == y**2*Derivative(f(y), y) + 2*y*f(y)
assert Subs(f(x), x, 0) + Subs(f(x), x, 0) == 2*Subs(f(x), x, 0)
e1 = Subs(z*f(x), x, 1)
e2 = Subs(z*f(y), y, 1)
assert e1 + e2 == 2*e1
assert e1.__hash__() == e2.__hash__()
assert Subs(z*f(x + 1), x, 1) not in [ e1, e2 ]
assert Derivative(f(x), x).subs(x, g(x)) == Derivative(f(g(x)), g(x))
assert Derivative(f(x), x).subs(x, x + y) == Subs(Derivative(f(x), x),
x, x + y)
assert Subs(f(x)*cos(y) + z, (x, y), (0, pi/3)).n(2) == \
Subs(f(x)*cos(y) + z, (x, y), (0, pi/3)).evalf(2) == \
z + Rational('1/2').n(2)*f(0)
assert f(x).diff(x).subs(x, 0).subs(x, y) == f(x).diff(x).subs(x, 0)
assert (x*f(x).diff(x).subs(x, 0)).subs(x, y) == y*f(x).diff(x).subs(x, 0)
assert Subs(Derivative(g(x)**2, g(x), x), g(x), exp(x)
).doit() == 2*exp(x)
assert Subs(Derivative(g(x)**2, g(x), x), g(x), exp(x)
).doit(deep=False) == 2*Derivative(exp(x), x)
assert Derivative(f(x, g(x)), x).doit() == Derivative(
f(x, g(x)), g(x))*Derivative(g(x), x) + Subs(Derivative(
f(y, g(x)), y), y, x)
def test_doitdoit():
done = Derivative(f(x, g(x)), x, g(x)).doit()
assert done == done.doit()
@XFAIL
def test_Subs2():
# this reflects a limitation of subs(), probably won't fix
assert Subs(f(x), x**2, x).doit() == f(sqrt(x))
def test_expand_function():
assert expand(x + y) == x + y
assert expand(x + y, complex=True) == I*im(x) + I*im(y) + re(x) + re(y)
assert expand((x + y)**11, modulus=11) == x**11 + y**11
def test_function_comparable():
assert sin(x).is_comparable is False
assert cos(x).is_comparable is False
assert sin(Float('0.1')).is_comparable is True
assert cos(Float('0.1')).is_comparable is True
assert sin(E).is_comparable is True
assert cos(E).is_comparable is True
assert sin(Rational(1, 3)).is_comparable is True
assert cos(Rational(1, 3)).is_comparable is True
def test_function_comparable_infinities():
assert sin(oo).is_comparable is False
assert sin(-oo).is_comparable is False
assert sin(zoo).is_comparable is False
assert sin(nan).is_comparable is False
def test_deriv1():
# These all require derivatives evaluated at a point (issue 4719) to work.
# See issue 4624
assert f(2*x).diff(x) == 2*Subs(Derivative(f(x), x), x, 2*x)
assert (f(x)**3).diff(x) == 3*f(x)**2*f(x).diff(x)
assert (f(2*x)**3).diff(x) == 6*f(2*x)**2*Subs(
Derivative(f(x), x), x, 2*x)
assert f(2 + x).diff(x) == Subs(Derivative(f(x), x), x, x + 2)
assert f(2 + 3*x).diff(x) == 3*Subs(
Derivative(f(x), x), x, 3*x + 2)
assert f(3*sin(x)).diff(x) == 3*cos(x)*Subs(
Derivative(f(x), x), x, 3*sin(x))
# See issue 8510
assert f(x, x + z).diff(x) == (
Subs(Derivative(f(y, x + z), y), y, x) +
Subs(Derivative(f(x, y), y), y, x + z))
assert f(x, x**2).diff(x) == (
2*x*Subs(Derivative(f(x, y), y), y, x**2) +
Subs(Derivative(f(y, x**2), y), y, x))
# but Subs is not always necessary
assert f(x, g(y)).diff(g(y)) == Derivative(f(x, g(y)), g(y))
def test_deriv2():
assert (x**3).diff(x) == 3*x**2
assert (x**3).diff(x, evaluate=False) != 3*x**2
assert (x**3).diff(x, evaluate=False) == Derivative(x**3, x)
assert diff(x**3, x) == 3*x**2
assert diff(x**3, x, evaluate=False) != 3*x**2
assert diff(x**3, x, evaluate=False) == Derivative(x**3, x)
def test_func_deriv():
assert f(x).diff(x) == Derivative(f(x), x)
# issue 4534
assert f(x, y).diff(x, y) - f(x, y).diff(y, x) == 0
assert Derivative(f(x, y), x, y).args[1:] == ((x, 1), (y, 1))
assert Derivative(f(x, y), y, x).args[1:] == ((y, 1), (x, 1))
assert (Derivative(f(x, y), x, y) - Derivative(f(x, y), y, x)).doit() == 0
def test_suppressed_evaluation():
a = sin(0, evaluate=False)
assert a != 0
assert a.func is sin
assert a.args == (0,)
def test_function_evalf():
def eq(a, b, eps):
return abs(a - b) < eps
assert eq(sin(1).evalf(15), Float("0.841470984807897"), 1e-13)
assert eq(
sin(2).evalf(25), Float("0.9092974268256816953960199", 25), 1e-23)
assert eq(sin(1 + I).evalf(
15), Float("1.29845758141598") + Float("0.634963914784736")*I, 1e-13)
assert eq(exp(1 + I).evalf(15), Float(
"1.46869393991588") + Float("2.28735528717884239")*I, 1e-13)
assert eq(exp(-0.5 + 1.5*I).evalf(15), Float(
"0.0429042815937374") + Float("0.605011292285002")*I, 1e-13)
assert eq(log(pi + sqrt(2)*I).evalf(
15), Float("1.23699044022052") + Float("0.422985442737893")*I, 1e-13)
assert eq(cos(100).evalf(15), Float("0.86231887228768"), 1e-13)
def test_extensibility_eval():
class MyFunc(Function):
@classmethod
def eval(cls, *args):
return (0, 0, 0)
assert MyFunc(0) == (0, 0, 0)
def test_function_non_commutative():
x = Symbol('x', commutative=False)
assert f(x).is_commutative is False
assert sin(x).is_commutative is False
assert exp(x).is_commutative is False
assert log(x).is_commutative is False
assert f(x).is_complex is False
assert sin(x).is_complex is False
assert exp(x).is_complex is False
assert log(x).is_complex is False
def test_function_complex():
x = Symbol('x', complex=True)
xzf = Symbol('x', complex=True, zero=False)
assert f(x).is_commutative is True
assert sin(x).is_commutative is True
assert exp(x).is_commutative is True
assert log(x).is_commutative is True
assert f(x).is_complex is None
assert sin(x).is_complex is True
assert exp(x).is_complex is True
assert log(x).is_complex is None
assert log(xzf).is_complex is True
def test_function__eval_nseries():
n = Symbol('n')
assert sin(x)._eval_nseries(x, 2, None) == x + O(x**2)
assert sin(x + 1)._eval_nseries(x, 2, None) == x*cos(1) + sin(1) + O(x**2)
assert sin(pi*(1 - x))._eval_nseries(x, 2, None) == pi*x + O(x**2)
assert acos(1 - x**2)._eval_nseries(x, 2, None) == sqrt(2)*x + O(x**2)
assert polygamma(n, x + 1)._eval_nseries(x, 2, None) == \
polygamma(n, 1) + polygamma(n + 1, 1)*x + O(x**2)
raises(PoleError, lambda: sin(1/x)._eval_nseries(x, 2, None))
assert acos(1 - x)._eval_nseries(x, 2, None) == sqrt(2)*sqrt(x) + O(x)
assert acos(1 + x)._eval_nseries(x, 2, None) == sqrt(2)*I*sqrt(x) + O(x) # XXX: wrong, branch cuts
assert loggamma(1/x)._eval_nseries(x, 0, None) == \
log(x)/2 - log(x)/x - 1/x + O(1, x)
assert loggamma(log(1/x)).nseries(x, n=1, logx=y) == loggamma(-y)
# issue 6725:
assert expint(Rational(3, 2), -x)._eval_nseries(x, 5, None) == \
2 - 2*sqrt(pi)*I*sqrt(x) - 2*x - x**2/3 - x**3/15 - x**4/84 + O(x**5)
assert sin(sqrt(x))._eval_nseries(x, 3, None) == \
sqrt(x) - x**Rational(3, 2)/6 + x**Rational(5, 2)/120 + O(x**3)
def test_doit():
n = Symbol('n', integer=True)
f = Sum(2 * n * x, (n, 1, 3))
d = Derivative(f, x)
assert d.doit() == 12
assert d.doit(deep=False) == Sum(2*n, (n, 1, 3))
def test_evalf_default():
from sympy.functions.special.gamma_functions import polygamma
assert type(sin(4.0)) == Float
assert type(re(sin(I + 1.0))) == Float
assert type(im(sin(I + 1.0))) == Float
assert type(sin(4)) == sin
assert type(polygamma(2.0, 4.0)) == Float
assert type(sin(Rational(1, 4))) == sin
def test_issue_5399():
args = [x, y, S(2), S.Half]
def ok(a):
"""Return True if the input args for diff are ok"""
if not a:
return False
if a[0].is_Symbol is False:
return False
s_at = [i for i in range(len(a)) if a[i].is_Symbol]
n_at = [i for i in range(len(a)) if not a[i].is_Symbol]
# every symbol is followed by symbol or int
# every number is followed by a symbol
return (all(a[i + 1].is_Symbol or a[i + 1].is_Integer
for i in s_at if i + 1 < len(a)) and
all(a[i + 1].is_Symbol
for i in n_at if i + 1 < len(a)))
eq = x**10*y**8
for a in subsets(args):
for v in variations(a, len(a)):
if ok(v):
eq.diff(*v) # does not raise
else:
raises(ValueError, lambda: eq.diff(*v))
def test_derivative_numerically():
from random import random
z0 = random() + I*random()
assert abs(Derivative(sin(x), x).doit_numerically(z0) - cos(z0)) < 1e-15
def test_fdiff_argument_index_error():
from sympy.core.function import ArgumentIndexError
class myfunc(Function):
nargs = 1 # define since there is no eval routine
def fdiff(self, idx):
raise ArgumentIndexError
mf = myfunc(x)
assert mf.diff(x) == Derivative(mf, x)
raises(TypeError, lambda: myfunc(x, x))
def test_deriv_wrt_function():
x = f(t)
xd = diff(x, t)
xdd = diff(xd, t)
y = g(t)
yd = diff(y, t)
assert diff(x, t) == xd
assert diff(2 * x + 4, t) == 2 * xd
assert diff(2 * x + 4 + y, t) == 2 * xd + yd
assert diff(2 * x + 4 + y * x, t) == 2 * xd + x * yd + xd * y
assert diff(2 * x + 4 + y * x, x) == 2 + y
assert (diff(4 * x**2 + 3 * x + x * y, t) == 3 * xd + x * yd + xd * y +
8 * x * xd)
assert (diff(4 * x**2 + 3 * xd + x * y, t) == 3 * xdd + x * yd + xd * y +
8 * x * xd)
assert diff(4 * x**2 + 3 * xd + x * y, xd) == 3
assert diff(4 * x**2 + 3 * xd + x * y, xdd) == 0
assert diff(sin(x), t) == xd * cos(x)
assert diff(exp(x), t) == xd * exp(x)
assert diff(sqrt(x), t) == xd / (2 * sqrt(x))
def test_diff_wrt_value():
assert Expr()._diff_wrt is False
assert x._diff_wrt is True
assert f(x)._diff_wrt is True
assert Derivative(f(x), x)._diff_wrt is True
assert Derivative(x**2, x)._diff_wrt is False
def test_diff_wrt():
fx = f(x)
dfx = diff(f(x), x)
ddfx = diff(f(x), x, x)
assert diff(sin(fx) + fx**2, fx) == cos(fx) + 2*fx
assert diff(sin(dfx) + dfx**2, dfx) == cos(dfx) + 2*dfx
assert diff(sin(ddfx) + ddfx**2, ddfx) == cos(ddfx) + 2*ddfx
assert diff(fx**2, dfx) == 0
assert diff(fx**2, ddfx) == 0
assert diff(dfx**2, fx) == 0
assert diff(dfx**2, ddfx) == 0
assert diff(ddfx**2, dfx) == 0
assert diff(fx*dfx*ddfx, fx) == dfx*ddfx
assert diff(fx*dfx*ddfx, dfx) == fx*ddfx
assert diff(fx*dfx*ddfx, ddfx) == fx*dfx
assert diff(f(x), x).diff(f(x)) == 0
assert (sin(f(x)) - cos(diff(f(x), x))).diff(f(x)) == cos(f(x))
assert diff(sin(fx), fx, x) == diff(sin(fx), x, fx)
# Chain rule cases
assert f(g(x)).diff(x) == \
Derivative(g(x), x)*Derivative(f(g(x)), g(x))
assert diff(f(g(x), h(y)), x) == \
Derivative(g(x), x)*Derivative(f(g(x), h(y)), g(x))
assert diff(f(g(x), h(x)), x) == (
Derivative(f(g(x), h(x)), g(x))*Derivative(g(x), x) +
Derivative(f(g(x), h(x)), h(x))*Derivative(h(x), x))
assert f(
sin(x)).diff(x) == cos(x)*Subs(Derivative(f(x), x), x, sin(x))
assert diff(f(g(x)), g(x)) == Derivative(f(g(x)), g(x))
def test_diff_wrt_func_subs():
assert f(g(x)).diff(x).subs(g, Lambda(x, 2*x)).doit() == f(2*x).diff(x)
def test_subs_in_derivative():
expr = sin(x*exp(y))
u = Function('u')
v = Function('v')
assert Derivative(expr, y).subs(expr, y) == Derivative(y, y)
assert Derivative(expr, y).subs(y, x).doit() == \
Derivative(expr, y).doit().subs(y, x)
assert Derivative(f(x, y), y).subs(y, x) == Subs(Derivative(f(x, y), y), y, x)
assert Derivative(f(x, y), y).subs(x, y) == Subs(Derivative(f(x, y), y), x, y)
assert Derivative(f(x, y), y).subs(y, g(x, y)) == Subs(Derivative(f(x, y), y), y, g(x, y)).doit()
assert Derivative(f(x, y), y).subs(x, g(x, y)) == Subs(Derivative(f(x, y), y), x, g(x, y))
assert Derivative(f(x, y), g(y)).subs(x, g(x, y)) == Derivative(f(g(x, y), y), g(y))
assert Derivative(f(u(x), h(y)), h(y)).subs(h(y), g(x, y)) == \
Subs(Derivative(f(u(x), h(y)), h(y)), h(y), g(x, y)).doit()
assert Derivative(f(x, y), y).subs(y, z) == Derivative(f(x, z), z)
assert Derivative(f(x, y), y).subs(y, g(y)) == Derivative(f(x, g(y)), g(y))
assert Derivative(f(g(x), h(y)), h(y)).subs(h(y), u(y)) == \
Derivative(f(g(x), u(y)), u(y))
assert Derivative(f(x, f(x, x)), f(x, x)).subs(
f, Lambda((x, y), x + y)) == Subs(
Derivative(z + x, z), z, 2*x)
assert Subs(Derivative(f(f(x)), x), f, cos).doit() == sin(x)*sin(cos(x))
assert Subs(Derivative(f(f(x)), f(x)), f, cos).doit() == -sin(cos(x))
# Issue 13791. No comparison (it's a long formula) but this used to raise an exception.
assert isinstance(v(x, y, u(x, y)).diff(y).diff(x).diff(y), Expr)
# This is also related to issues 13791 and 13795; issue 15190
F = Lambda((x, y), exp(2*x + 3*y))
abstract = f(x, f(x, x)).diff(x, 2)
concrete = F(x, F(x, x)).diff(x, 2)
assert (abstract.subs(f, F).doit() - concrete).simplify() == 0
# don't introduce a new symbol if not necessary
assert x in f(x).diff(x).subs(x, 0).atoms()
# case (4)
assert Derivative(f(x,f(x,y)), x, y).subs(x, g(y)
) == Subs(Derivative(f(x, f(x, y)), x, y), x, g(y))
assert Derivative(f(x, x), x).subs(x, 0
) == Subs(Derivative(f(x, x), x), x, 0)
# issue 15194
assert Derivative(f(y, g(x)), (x, z)).subs(z, x
) == Derivative(f(y, g(x)), (x, x))
df = f(x).diff(x)
assert df.subs(df, 1) is S.One
assert df.diff(df) is S.One
dxy = Derivative(f(x, y), x, y)
dyx = Derivative(f(x, y), y, x)
assert dxy.subs(Derivative(f(x, y), y, x), 1) is S.One
assert dxy.diff(dyx) is S.One
assert Derivative(f(x, y), x, 2, y, 3).subs(
dyx, g(x, y)) == Derivative(g(x, y), x, 1, y, 2)
assert Derivative(f(x, x - y), y).subs(x, x + y) == Subs(
Derivative(f(x, x - y), y), x, x + y)
def test_diff_wrt_not_allowed():
# issue 7027 included
for wrt in (
cos(x), re(x), x**2, x*y, 1 + x,
Derivative(cos(x), x), Derivative(f(f(x)), x)):
raises(ValueError, lambda: diff(f(x), wrt))
# if we don't differentiate wrt then don't raise error
assert diff(exp(x*y), x*y, 0) == exp(x*y)
def test_klein_gordon_lagrangian():
m = Symbol('m')
phi = f(x, t)
L = -(diff(phi, t)**2 - diff(phi, x)**2 - m**2*phi**2)/2
eqna = Eq(
diff(L, phi) - diff(L, diff(phi, x), x) - diff(L, diff(phi, t), t), 0)
eqnb = Eq(diff(phi, t, t) - diff(phi, x, x) + m**2*phi, 0)
assert eqna == eqnb
def test_sho_lagrangian():
m = Symbol('m')
k = Symbol('k')
x = f(t)
L = m*diff(x, t)**2/2 - k*x**2/2
eqna = Eq(diff(L, x), diff(L, diff(x, t), t))
eqnb = Eq(-k*x, m*diff(x, t, t))
assert eqna == eqnb
assert diff(L, x, t) == diff(L, t, x)
assert diff(L, diff(x, t), t) == m*diff(x, t, 2)
assert diff(L, t, diff(x, t)) == -k*x + m*diff(x, t, 2)
def test_straight_line():
F = f(x)
Fd = F.diff(x)
L = sqrt(1 + Fd**2)
assert diff(L, F) == 0
assert diff(L, Fd) == Fd/sqrt(1 + Fd**2)
def test_sort_variable():
vsort = Derivative._sort_variable_count
def vsort0(*v, **kw):
reverse = kw.get('reverse', False)
return [i[0] for i in vsort([(i, 0) for i in (
reversed(v) if reverse else v)])]
for R in range(2):
assert vsort0(y, x, reverse=R) == [x, y]
assert vsort0(f(x), x, reverse=R) == [x, f(x)]
assert vsort0(f(y), f(x), reverse=R) == [f(x), f(y)]
assert vsort0(g(x), f(y), reverse=R) == [f(y), g(x)]
assert vsort0(f(x, y), f(x), reverse=R) == [f(x), f(x, y)]
fx = f(x).diff(x)
assert vsort0(fx, y, reverse=R) == [y, fx]
fy = f(y).diff(y)
assert vsort0(fy, fx, reverse=R) == [fx, fy]
fxx = fx.diff(x)
assert vsort0(fxx, fx, reverse=R) == [fx, fxx]
assert vsort0(Basic(x), f(x), reverse=R) == [f(x), Basic(x)]
assert vsort0(Basic(y), Basic(x), reverse=R) == [Basic(x), Basic(y)]
assert vsort0(Basic(y, z), Basic(x), reverse=R) == [
Basic(x), Basic(y, z)]
assert vsort0(fx, x, reverse=R) == [
x, fx] if R else [fx, x]
assert vsort0(Basic(x), x, reverse=R) == [
x, Basic(x)] if R else [Basic(x), x]
assert vsort0(Basic(f(x)), f(x), reverse=R) == [
f(x), Basic(f(x))] if R else [Basic(f(x)), f(x)]
assert vsort0(Basic(x, z), Basic(x), reverse=R) == [
Basic(x), Basic(x, z)] if R else [Basic(x, z), Basic(x)]
assert vsort([]) == []
assert _aresame(vsort([(x, 1)]), [Tuple(x, 1)])
assert vsort([(x, y), (x, z)]) == [(x, y + z)]
assert vsort([(y, 1), (x, 1 + y)]) == [(x, 1 + y), (y, 1)]
# coverage complete; legacy tests below
assert vsort([(x, 3), (y, 2), (z, 1)]) == [(x, 3), (y, 2), (z, 1)]
assert vsort([(h(x), 1), (g(x), 1), (f(x), 1)]) == [
(f(x), 1), (g(x), 1), (h(x), 1)]
assert vsort([(z, 1), (y, 2), (x, 3), (h(x), 1), (g(x), 1),
(f(x), 1)]) == [(x, 3), (y, 2), (z, 1), (f(x), 1), (g(x), 1),
(h(x), 1)]
assert vsort([(x, 1), (f(x), 1), (y, 1), (f(y), 1)]) == [(x, 1),
(y, 1), (f(x), 1), (f(y), 1)]
assert vsort([(y, 1), (x, 2), (g(x), 1), (f(x), 1), (z, 1),
(h(x), 1), (y, 2), (x, 1)]) == [(x, 3), (y, 3), (z, 1),
(f(x), 1), (g(x), 1), (h(x), 1)]
assert vsort([(z, 1), (y, 1), (f(x), 1), (x, 1), (f(x), 1),
(g(x), 1)]) == [(x, 1), (y, 1), (z, 1), (f(x), 2), (g(x), 1)]
assert vsort([(z, 1), (y, 2), (f(x), 1), (x, 2), (f(x), 2),
(g(x), 1), (z, 2), (z, 1), (y, 1), (x, 1)]) == [(x, 3), (y, 3),
(z, 4), (f(x), 3), (g(x), 1)]
assert vsort(((y, 2), (x, 1), (y, 1), (x, 1))) == [(x, 2), (y, 3)]
assert isinstance(vsort([(x, 3), (y, 2), (z, 1)])[0], Tuple)
assert vsort([(x, 1), (f(x), 1), (x, 1)]) == [(x, 2), (f(x), 1)]
assert vsort([(y, 2), (x, 3), (z, 1)]) == [(x, 3), (y, 2), (z, 1)]
assert vsort([(h(y), 1), (g(x), 1), (f(x), 1)]) == [
(f(x), 1), (g(x), 1), (h(y), 1)]
assert vsort([(x, 1), (y, 1), (x, 1)]) == [(x, 2), (y, 1)]
assert vsort([(f(x), 1), (f(y), 1), (f(x), 1)]) == [
(f(x), 2), (f(y), 1)]
dfx = f(x).diff(x)
self = [(dfx, 1), (x, 1)]
assert vsort(self) == self
assert vsort([
(dfx, 1), (y, 1), (f(x), 1), (x, 1), (f(y), 1), (x, 1)]) == [
(y, 1), (f(x), 1), (f(y), 1), (dfx, 1), (x, 2)]
dfy = f(y).diff(y)
assert vsort([(dfy, 1), (dfx, 1)]) == [(dfx, 1), (dfy, 1)]
d2fx = dfx.diff(x)
assert vsort([(d2fx, 1), (dfx, 1)]) == [(dfx, 1), (d2fx, 1)]
def test_multiple_derivative():
# Issue #15007
assert f(x, y).diff(y, y, x, y, x
) == Derivative(f(x, y), (x, 2), (y, 3))
def test_unhandled():
class MyExpr(Expr):
def _eval_derivative(self, s):
if not s.name.startswith('xi'):
return self
else:
return None
eq = MyExpr(f(x), y, z)
assert diff(eq, x, y, f(x), z) == Derivative(eq, f(x))
assert diff(eq, f(x), x) == Derivative(eq, f(x))
assert f(x, y).diff(x,(y, z)) == Derivative(f(x, y), x, (y, z))
assert f(x, y).diff(x,(y, 0)) == Derivative(f(x, y), x)
def test_nfloat():
from sympy.core.basic import _aresame
from sympy.polys.rootoftools import rootof
x = Symbol("x")
eq = x**Rational(4, 3) + 4*x**(S.One/3)/3
assert _aresame(nfloat(eq), x**Rational(4, 3) + (4.0/3)*x**(S.One/3))
assert _aresame(nfloat(eq, exponent=True), x**(4.0/3) + (4.0/3)*x**(1.0/3))
eq = x**Rational(4, 3) + 4*x**(x/3)/3
assert _aresame(nfloat(eq), x**Rational(4, 3) + (4.0/3)*x**(x/3))
big = 12345678901234567890
# specify precision to match value used in nfloat
Float_big = Float(big, 15)
assert _aresame(nfloat(big), Float_big)
assert _aresame(nfloat(big*x), Float_big*x)
assert _aresame(nfloat(x**big, exponent=True), x**Float_big)
assert nfloat(cos(x + sqrt(2))) == cos(x + nfloat(sqrt(2)))
# issue 6342
f = S('x*lamda + lamda**3*(x/2 + 1/2) + lamda**2 + 1/4')
assert not any(a.free_symbols for a in solveset(f.subs(x, -0.139)))
# issue 6632
assert nfloat(-100000*sqrt(2500000001) + 5000000001) == \
9.99999999800000e-11
# issue 7122
eq = cos(3*x**4 + y)*rootof(x**5 + 3*x**3 + 1, 0)
assert str(nfloat(eq, exponent=False, n=1)) == '-0.7*cos(3.0*x**4 + y)'
# issue 10933
for ti in (dict, Dict):
d = ti({S.Half: S.Half})
n = nfloat(d)
assert isinstance(n, ti)
assert _aresame(list(n.items()).pop(), (S.Half, Float(.5)))
for ti in (dict, Dict):
d = ti({S.Half: S.Half})
n = nfloat(d, dkeys=True)
assert isinstance(n, ti)
assert _aresame(list(n.items()).pop(), (Float(.5), Float(.5)))
d = [S.Half]
n = nfloat(d)
assert type(n) is list
assert _aresame(n[0], Float(.5))
assert _aresame(nfloat(Eq(x, S.Half)).rhs, Float(.5))
assert _aresame(nfloat(S(True)), S(True))
assert _aresame(nfloat(Tuple(S.Half))[0], Float(.5))
assert nfloat(Eq((3 - I)**2/2 + I, 0)) == S.false
# pass along kwargs
assert nfloat([{S.Half: x}], dkeys=True) == [{Float(0.5): x}]
# Issue 17706
A = MutableMatrix([[1, 2], [3, 4]])
B = MutableMatrix(
[[Float('1.0', precision=53), Float('2.0', precision=53)],
[Float('3.0', precision=53), Float('4.0', precision=53)]])
assert _aresame(nfloat(A), B)
A = ImmutableMatrix([[1, 2], [3, 4]])
B = ImmutableMatrix(
[[Float('1.0', precision=53), Float('2.0', precision=53)],
[Float('3.0', precision=53), Float('4.0', precision=53)]])
assert _aresame(nfloat(A), B)
def test_issue_7068():
from sympy.abc import a, b
f = Function('f')
y1 = Dummy('y')
y2 = Dummy('y')
func1 = f(a + y1 * b)
func2 = f(a + y2 * b)
func1_y = func1.diff(y1)
func2_y = func2.diff(y2)
assert func1_y != func2_y
z1 = Subs(f(a), a, y1)
z2 = Subs(f(a), a, y2)
assert z1 != z2
def test_issue_7231():
from sympy.abc import a
ans1 = f(x).series(x, a)
res = (f(a) + (-a + x)*Subs(Derivative(f(y), y), y, a) +
(-a + x)**2*Subs(Derivative(f(y), y, y), y, a)/2 +
(-a + x)**3*Subs(Derivative(f(y), y, y, y),
y, a)/6 +
(-a + x)**4*Subs(Derivative(f(y), y, y, y, y),
y, a)/24 +
(-a + x)**5*Subs(Derivative(f(y), y, y, y, y, y),
y, a)/120 + O((-a + x)**6, (x, a)))
assert res == ans1
ans2 = f(x).series(x, a)
assert res == ans2
def test_issue_7687():
from sympy.core.function import Function
from sympy.abc import x
f = Function('f')(x)
ff = Function('f')(x)
match_with_cache = ff.matches(f)
assert isinstance(f, type(ff))
clear_cache()
ff = Function('f')(x)
assert isinstance(f, type(ff))
assert match_with_cache == ff.matches(f)
def test_issue_7688():
from sympy.core.function import Function, UndefinedFunction
f = Function('f') # actually an UndefinedFunction
clear_cache()
class A(UndefinedFunction):
pass
a = A('f')
assert isinstance(a, type(f))
def test_mexpand():
from sympy.abc import x
assert _mexpand(None) is None
assert _mexpand(1) is S.One
assert _mexpand(x*(x + 1)**2) == (x*(x + 1)**2).expand()
def test_issue_8469():
# This should not take forever to run
N = 40
def g(w, theta):
return 1/(1+exp(w-theta))
ws = symbols(['w%i'%i for i in range(N)])
import functools
expr = functools.reduce(g, ws)
assert isinstance(expr, Pow)
def test_issue_12996():
# foo=True imitates the sort of arguments that Derivative can get
# from Integral when it passes doit to the expression
assert Derivative(im(x), x).doit(foo=True) == Derivative(im(x), x)
def test_should_evalf():
# This should not take forever to run (see #8506)
assert isinstance(sin((1.0 + 1.0*I)**10000 + 1), sin)
def test_Derivative_as_finite_difference():
# Central 1st derivative at gridpoint
x, h = symbols('x h', real=True)
dfdx = f(x).diff(x)
assert (dfdx.as_finite_difference([x-2, x-1, x, x+1, x+2]) -
(S.One/12*(f(x-2)-f(x+2)) + Rational(2, 3)*(f(x+1)-f(x-1)))).simplify() == 0
# Central 1st derivative "half-way"
assert (dfdx.as_finite_difference() -
(f(x + S.Half)-f(x - S.Half))).simplify() == 0
assert (dfdx.as_finite_difference(h) -
(f(x + h/S(2))-f(x - h/S(2)))/h).simplify() == 0
assert (dfdx.as_finite_difference([x - 3*h, x-h, x+h, x + 3*h]) -
(S(9)/(8*2*h)*(f(x+h) - f(x-h)) +
S.One/(24*2*h)*(f(x - 3*h) - f(x + 3*h)))).simplify() == 0
# One sided 1st derivative at gridpoint
assert (dfdx.as_finite_difference([0, 1, 2], 0) -
(Rational(-3, 2)*f(0) + 2*f(1) - f(2)/2)).simplify() == 0
assert (dfdx.as_finite_difference([x, x+h], x) -
(f(x+h) - f(x))/h).simplify() == 0
assert (dfdx.as_finite_difference([x-h, x, x+h], x-h) -
(-S(3)/(2*h)*f(x-h) + 2/h*f(x) -
S.One/(2*h)*f(x+h))).simplify() == 0
# One sided 1st derivative "half-way"
assert (dfdx.as_finite_difference([x-h, x+h, x + 3*h, x + 5*h, x + 7*h])
- 1/(2*h)*(-S(11)/(12)*f(x-h) + S(17)/(24)*f(x+h)
+ Rational(3, 8)*f(x + 3*h) - Rational(5, 24)*f(x + 5*h)
+ S.One/24*f(x + 7*h))).simplify() == 0
d2fdx2 = f(x).diff(x, 2)
# Central 2nd derivative at gridpoint
assert (d2fdx2.as_finite_difference([x-h, x, x+h]) -
h**-2 * (f(x-h) + f(x+h) - 2*f(x))).simplify() == 0
assert (d2fdx2.as_finite_difference([x - 2*h, x-h, x, x+h, x + 2*h]) -
h**-2 * (Rational(-1, 12)*(f(x - 2*h) + f(x + 2*h)) +
Rational(4, 3)*(f(x+h) + f(x-h)) - Rational(5, 2)*f(x))).simplify() == 0
# Central 2nd derivative "half-way"
assert (d2fdx2.as_finite_difference([x - 3*h, x-h, x+h, x + 3*h]) -
(2*h)**-2 * (S.Half*(f(x - 3*h) + f(x + 3*h)) -
S.Half*(f(x+h) + f(x-h)))).simplify() == 0
# One sided 2nd derivative at gridpoint
assert (d2fdx2.as_finite_difference([x, x+h, x + 2*h, x + 3*h]) -
h**-2 * (2*f(x) - 5*f(x+h) +
4*f(x+2*h) - f(x+3*h))).simplify() == 0
# One sided 2nd derivative at "half-way"
assert (d2fdx2.as_finite_difference([x-h, x+h, x + 3*h, x + 5*h]) -
(2*h)**-2 * (Rational(3, 2)*f(x-h) - Rational(7, 2)*f(x+h) + Rational(5, 2)*f(x + 3*h) -
S.Half*f(x + 5*h))).simplify() == 0
d3fdx3 = f(x).diff(x, 3)
# Central 3rd derivative at gridpoint
assert (d3fdx3.as_finite_difference() -
(-f(x - Rational(3, 2)) + 3*f(x - S.Half) -
3*f(x + S.Half) + f(x + Rational(3, 2)))).simplify() == 0
assert (d3fdx3.as_finite_difference(
[x - 3*h, x - 2*h, x-h, x, x+h, x + 2*h, x + 3*h]) -
h**-3 * (S.One/8*(f(x - 3*h) - f(x + 3*h)) - f(x - 2*h) +
f(x + 2*h) + Rational(13, 8)*(f(x-h) - f(x+h)))).simplify() == 0
# Central 3rd derivative at "half-way"
assert (d3fdx3.as_finite_difference([x - 3*h, x-h, x+h, x + 3*h]) -
(2*h)**-3 * (f(x + 3*h)-f(x - 3*h) +
3*(f(x-h)-f(x+h)))).simplify() == 0
# One sided 3rd derivative at gridpoint
assert (d3fdx3.as_finite_difference([x, x+h, x + 2*h, x + 3*h]) -
h**-3 * (f(x + 3*h)-f(x) + 3*(f(x+h)-f(x + 2*h)))).simplify() == 0
# One sided 3rd derivative at "half-way"
assert (d3fdx3.as_finite_difference([x-h, x+h, x + 3*h, x + 5*h]) -
(2*h)**-3 * (f(x + 5*h)-f(x-h) +
3*(f(x+h)-f(x + 3*h)))).simplify() == 0
# issue 11007
y = Symbol('y', real=True)
d2fdxdy = f(x, y).diff(x, y)
ref0 = Derivative(f(x + S.Half, y), y) - Derivative(f(x - S.Half, y), y)
assert (d2fdxdy.as_finite_difference(wrt=x) - ref0).simplify() == 0
half = S.Half
xm, xp, ym, yp = x-half, x+half, y-half, y+half
ref2 = f(xm, ym) + f(xp, yp) - f(xp, ym) - f(xm, yp)
assert (d2fdxdy.as_finite_difference() - ref2).simplify() == 0
def test_issue_11159():
# Tests Application._eval_subs
expr1 = E
expr0 = expr1 * expr1
expr1 = expr0.subs(expr1,expr0)
assert expr0 == expr1
def test_issue_12005():
e1 = Subs(Derivative(f(x), x), x, x)
assert e1.diff(x) == Derivative(f(x), x, x)
e2 = Subs(Derivative(f(x), x), x, x**2 + 1)
assert e2.diff(x) == 2*x*Subs(Derivative(f(x), x, x), x, x**2 + 1)
e3 = Subs(Derivative(f(x) + y**2 - y, y), y, y**2)
assert e3.diff(y) == 4*y
e4 = Subs(Derivative(f(x + y), y), y, (x**2))
assert e4.diff(y) is S.Zero
e5 = Subs(Derivative(f(x), x), (y, z), (y, z))
assert e5.diff(x) == Derivative(f(x), x, x)
assert f(g(x)).diff(g(x), g(x)) == Derivative(f(g(x)), g(x), g(x))
def test_issue_13843():
x = symbols('x')
f = Function('f')
m, n = symbols('m n', integer=True)
assert Derivative(Derivative(f(x), (x, m)), (x, n)) == Derivative(f(x), (x, m + n))
assert Derivative(Derivative(f(x), (x, m+5)), (x, n+3)) == Derivative(f(x), (x, m + n + 8))
assert Derivative(f(x), (x, n)).doit() == Derivative(f(x), (x, n))
def test_order_could_be_zero():
x, y = symbols('x, y')
n = symbols('n', integer=True, nonnegative=True)
m = symbols('m', integer=True, positive=True)
assert diff(y, (x, n)) == Piecewise((y, Eq(n, 0)), (0, True))
assert diff(y, (x, n + 1)) is S.Zero
assert diff(y, (x, m)) is S.Zero
def test_undefined_function_eq():
f = Function('f')
f2 = Function('f')
g = Function('g')
f_real = Function('f', is_real=True)
# This test may only be meaningful if the cache is turned off
assert f == f2
assert hash(f) == hash(f2)
assert f == f
assert f != g
assert f != f_real
def test_function_assumptions():
x = Symbol('x')
f = Function('f')
f_real = Function('f', real=True)
f_real1 = Function('f', real=1)
f_real_inherit = Function(Symbol('f', real=True))
assert f_real == f_real1 # assumptions are sanitized
assert f != f_real
assert f(x) != f_real(x)
assert f(x).is_real is None
assert f_real(x).is_real is True
assert f_real_inherit(x).is_real is True and f_real_inherit.name == 'f'
# Can also do it this way, but it won't be equal to f_real because of the
# way UndefinedFunction.__new__ works. Any non-recognized assumptions
# are just added literally as something which is used in the hash
f_real2 = Function('f', is_real=True)
assert f_real2(x).is_real is True
def test_undef_fcn_float_issue_6938():
f = Function('ceil')
assert not f(0.3).is_number
f = Function('sin')
assert not f(0.3).is_number
assert not f(pi).evalf().is_number
x = Symbol('x')
assert not f(x).evalf(subs={x:1.2}).is_number
def test_undefined_function_eval():
# Issue 15170. Make sure UndefinedFunction with eval defined works
# properly. The issue there was that the hash was determined before _nargs
# was set, which is included in the hash, hence changing the hash. The
# class is added to sympy.core.core.all_classes before the hash is
# changed, meaning "temp in all_classes" would fail, causing sympify(temp(t))
# to give a new class. We will eventually remove all_classes, but make
# sure this continues to work.
fdiff = lambda self, argindex=1: cos(self.args[argindex - 1])
eval = classmethod(lambda cls, t: None)
_imp_ = classmethod(lambda cls, t: sin(t))
temp = Function('temp', fdiff=fdiff, eval=eval, _imp_=_imp_)
expr = temp(t)
assert sympify(expr) == expr
assert type(sympify(expr)).fdiff.__name__ == "<lambda>"
assert expr.diff(t) == cos(t)
def test_issue_15241():
F = f(x)
Fx = F.diff(x)
assert (F + x*Fx).diff(x, Fx) == 2
assert (F + x*Fx).diff(Fx, x) == 1
assert (x*F + x*Fx*F).diff(F, x) == x*Fx.diff(x) + Fx + 1
assert (x*F + x*Fx*F).diff(x, F) == x*Fx.diff(x) + Fx + 1
y = f(x)
G = f(y)
Gy = G.diff(y)
assert (G + y*Gy).diff(y, Gy) == 2
assert (G + y*Gy).diff(Gy, y) == 1
assert (y*G + y*Gy*G).diff(G, y) == y*Gy.diff(y) + Gy + 1
assert (y*G + y*Gy*G).diff(y, G) == y*Gy.diff(y) + Gy + 1
def test_issue_15226():
assert Subs(Derivative(f(y), x, y), y, g(x)).doit() != 0
def test_issue_7027():
for wrt in (cos(x), re(x), Derivative(cos(x), x)):
raises(ValueError, lambda: diff(f(x), wrt))
def test_derivative_quick_exit():
assert f(x).diff(y) == 0
assert f(x).diff(y, f(x)) == 0
assert f(x).diff(x, f(y)) == 0
assert f(f(x)).diff(x, f(x), f(y)) == 0
assert f(f(x)).diff(x, f(x), y) == 0
assert f(x).diff(g(x)) == 0
assert f(x).diff(x, f(x).diff(x)) == 1
df = f(x).diff(x)
assert f(x).diff(df) == 0
dg = g(x).diff(x)
assert dg.diff(df).doit() == 0
def test_issue_15084_13166():
eq = f(x, g(x))
assert eq.diff((g(x), y)) == Derivative(f(x, g(x)), (g(x), y))
# issue 13166
assert eq.diff(x, 2).doit() == (
(Derivative(f(x, g(x)), (g(x), 2))*Derivative(g(x), x) +
Subs(Derivative(f(x, _xi_2), _xi_2, x), _xi_2, g(x)))*Derivative(g(x),
x) + Derivative(f(x, g(x)), g(x))*Derivative(g(x), (x, 2)) +
Derivative(g(x), x)*Subs(Derivative(f(_xi_1, g(x)), _xi_1, g(x)),
_xi_1, x) + Subs(Derivative(f(_xi_1, g(x)), (_xi_1, 2)), _xi_1, x))
# issue 6681
assert diff(f(x, t, g(x, t)), x).doit() == (
Derivative(f(x, t, g(x, t)), g(x, t))*Derivative(g(x, t), x) +
Subs(Derivative(f(_xi_1, t, g(x, t)), _xi_1), _xi_1, x))
# make sure the order doesn't matter when using diff
assert eq.diff(x, g(x)) == eq.diff(g(x), x)
def test_negative_counts():
# issue 13873
raises(ValueError, lambda: sin(x).diff(x, -1))
def test_Derivative__new__():
raises(TypeError, lambda: f(x).diff((x, 2), 0))
assert f(x, y).diff([(x, y), 0]) == f(x, y)
assert f(x, y).diff([(x, y), 1]) == NDimArray([
Derivative(f(x, y), x), Derivative(f(x, y), y)])
assert f(x,y).diff(y, (x, z), y, x) == Derivative(
f(x, y), (x, z + 1), (y, 2))
assert Matrix([x]).diff(x, 2) == Matrix([0]) # is_zero exit
def test_issue_14719_10150():
class V(Expr):
_diff_wrt = True
is_scalar = False
assert V().diff(V()) == Derivative(V(), V())
assert (2*V()).diff(V()) == 2*Derivative(V(), V())
class X(Expr):
_diff_wrt = True
assert X().diff(X()) == 1
assert (2*X()).diff(X()) == 2
def test_noncommutative_issue_15131():
x = Symbol('x', commutative=False)
t = Symbol('t', commutative=False)
fx = Function('Fx', commutative=False)(x)
ft = Function('Ft', commutative=False)(t)
A = Symbol('A', commutative=False)
eq = fx * A * ft
eqdt = eq.diff(t)
assert eqdt.args[-1] == ft.diff(t)
def test_Subs_Derivative():
a = Derivative(f(g(x), h(x)), g(x), h(x),x)
b = Derivative(Derivative(f(g(x), h(x)), g(x), h(x)),x)
c = f(g(x), h(x)).diff(g(x), h(x), x)
d = f(g(x), h(x)).diff(g(x), h(x)).diff(x)
e = Derivative(f(g(x), h(x)), x)
eqs = (a, b, c, d, e)
subs = lambda arg: arg.subs(f, Lambda((x, y), exp(x + y))
).subs(g(x), 1/x).subs(h(x), x**3)
ans = 3*x**2*exp(1/x)*exp(x**3) - exp(1/x)*exp(x**3)/x**2
assert all(subs(i).doit().expand() == ans for i in eqs)
assert all(subs(i.doit()).doit().expand() == ans for i in eqs)
def test_issue_15360():
f = Function('f')
assert f.name == 'f'
def test_issue_15947():
assert f._diff_wrt is False
raises(TypeError, lambda: f(f))
raises(TypeError, lambda: f(x).diff(f))
def test_Derivative_free_symbols():
f = Function('f')
n = Symbol('n', integer=True, positive=True)
assert diff(f(x), (x, n)).free_symbols == {n, x}
def test_issue_10503():
f = exp(x**3)*cos(x**6)
assert f.series(x, 0, 14) == 1 + x**3 + x**6/2 + x**9/6 - 11*x**12/24 + O(x**14)
|
ea01947cae9447c0a665e2123388358741305d854929584ed3257e06f681a6a3 | """This tests sympy/core/basic.py with (ideally) no reference to subclasses
of Basic or Atom."""
import collections
import sys
from sympy.core.basic import (Basic, Atom, preorder_traversal, as_Basic,
_atomic, _aresame)
from sympy.core.singleton import S
from sympy.core.symbol import symbols, Symbol, Dummy
from sympy.core.sympify import SympifyError
from sympy.core.function import Function, Lambda
from sympy.core.compatibility import default_sort_key
from sympy import sin, Q, cos, gamma, Tuple, Integral, Sum
from sympy.functions.elementary.exponential import exp
from sympy.testing.pytest import raises
from sympy.core import I, pi
b1 = Basic()
b2 = Basic(b1)
b3 = Basic(b2)
b21 = Basic(b2, b1)
def test__aresame():
assert not _aresame(Basic([]), Basic())
assert not _aresame(Basic([]), Basic(()))
assert not _aresame(Basic(2), Basic(2.))
def test_structure():
assert b21.args == (b2, b1)
assert b21.func(*b21.args) == b21
assert bool(b1)
def test_equality():
instances = [b1, b2, b3, b21, Basic(b1, b1, b1), Basic]
for i, b_i in enumerate(instances):
for j, b_j in enumerate(instances):
assert (b_i == b_j) == (i == j)
assert (b_i != b_j) == (i != j)
assert Basic() != []
assert not(Basic() == [])
assert Basic() != 0
assert not(Basic() == 0)
class Foo:
"""
Class that is unaware of Basic, and relies on both classes returning
the NotImplemented singleton for equivalence to evaluate to False.
"""
b = Basic()
foo = Foo()
assert b != foo
assert foo != b
assert not b == foo
assert not foo == b
class Bar:
"""
Class that considers itself equal to any instance of Basic, and relies
on Basic returning the NotImplemented singleton in order to achieve
a symmetric equivalence relation.
"""
def __eq__(self, other):
if isinstance(other, Basic):
return True
return NotImplemented
def __ne__(self, other):
return not self == other
bar = Bar()
assert b == bar
assert bar == b
assert not b != bar
assert not bar != b
def test_matches_basic():
instances = [Basic(b1, b1, b2), Basic(b1, b2, b1), Basic(b2, b1, b1),
Basic(b1, b2), Basic(b2, b1), b2, b1]
for i, b_i in enumerate(instances):
for j, b_j in enumerate(instances):
if i == j:
assert b_i.matches(b_j) == {}
else:
assert b_i.matches(b_j) is None
assert b1.match(b1) == {}
def test_has():
assert b21.has(b1)
assert b21.has(b3, b1)
assert b21.has(Basic)
assert not b1.has(b21, b3)
assert not b21.has()
raises(SympifyError, lambda: Symbol("x").has("x"))
def test_subs():
assert b21.subs(b2, b1) == Basic(b1, b1)
assert b21.subs(b2, b21) == Basic(b21, b1)
assert b3.subs(b2, b1) == b2
assert b21.subs([(b2, b1), (b1, b2)]) == Basic(b2, b2)
assert b21.subs({b1: b2, b2: b1}) == Basic(b2, b2)
if sys.version_info >= (3, 4):
assert b21.subs(collections.ChainMap({b1: b2}, {b2: b1})) == Basic(b2, b2)
assert b21.subs(collections.OrderedDict([(b2, b1), (b1, b2)])) == Basic(b2, b2)
raises(ValueError, lambda: b21.subs('bad arg'))
raises(ValueError, lambda: b21.subs(b1, b2, b3))
# dict(b1=foo) creates a string 'b1' but leaves foo unchanged; subs
# will convert the first to a symbol but will raise an error if foo
# cannot be sympified; sympification is strict if foo is not string
raises(ValueError, lambda: b21.subs(b1='bad arg'))
assert Symbol("text").subs({"text": b1}) == b1
assert Symbol("s").subs({"s": 1}) == 1
def test_subs_with_unicode_symbols():
expr = Symbol('var1')
replaced = expr.subs('var1', 'x')
assert replaced.name == 'x'
replaced = expr.subs('var1', 'x')
assert replaced.name == 'x'
def test_atoms():
assert b21.atoms() == {Basic()}
def test_free_symbols_empty():
assert b21.free_symbols == set()
def test_doit():
assert b21.doit() == b21
assert b21.doit(deep=False) == b21
def test_S():
assert repr(S) == 'S'
def test_xreplace():
assert b21.xreplace({b2: b1}) == Basic(b1, b1)
assert b21.xreplace({b2: b21}) == Basic(b21, b1)
assert b3.xreplace({b2: b1}) == b2
assert Basic(b1, b2).xreplace({b1: b2, b2: b1}) == Basic(b2, b1)
assert Atom(b1).xreplace({b1: b2}) == Atom(b1)
assert Atom(b1).xreplace({Atom(b1): b2}) == b2
raises(TypeError, lambda: b1.xreplace())
raises(TypeError, lambda: b1.xreplace([b1, b2]))
for f in (exp, Function('f')):
assert f.xreplace({}) == f
assert f.xreplace({}, hack2=True) == f
assert f.xreplace({f: b1}) == b1
assert f.xreplace({f: b1}, hack2=True) == b1
def test_preorder_traversal():
expr = Basic(b21, b3)
assert list(
preorder_traversal(expr)) == [expr, b21, b2, b1, b1, b3, b2, b1]
assert list(preorder_traversal(('abc', ('d', 'ef')))) == [
('abc', ('d', 'ef')), 'abc', ('d', 'ef'), 'd', 'ef']
result = []
pt = preorder_traversal(expr)
for i in pt:
result.append(i)
if i == b2:
pt.skip()
assert result == [expr, b21, b2, b1, b3, b2]
w, x, y, z = symbols('w:z')
expr = z + w*(x + y)
assert list(preorder_traversal([expr], keys=default_sort_key)) == \
[[w*(x + y) + z], w*(x + y) + z, z, w*(x + y), w, x + y, x, y]
assert list(preorder_traversal((x + y)*z, keys=True)) == \
[z*(x + y), z, x + y, x, y]
def test_sorted_args():
x = symbols('x')
assert b21._sorted_args == b21.args
raises(AttributeError, lambda: x._sorted_args)
def test_call():
x, y = symbols('x y')
# See the long history of this in issues 5026 and 5105.
raises(TypeError, lambda: sin(x)({ x : 1, sin(x) : 2}))
raises(TypeError, lambda: sin(x)(1))
# No effect as there are no callables
assert sin(x).rcall(1) == sin(x)
assert (1 + sin(x)).rcall(1) == 1 + sin(x)
# Effect in the pressence of callables
l = Lambda(x, 2*x)
assert (l + x).rcall(y) == 2*y + x
assert (x**l).rcall(2) == x**4
# TODO UndefinedFunction does not subclass Expr
#f = Function('f')
#assert (2*f)(x) == 2*f(x)
assert (Q.real & Q.positive).rcall(x) == Q.real(x) & Q.positive(x)
def test_rewrite():
x, y, z = symbols('x y z')
a, b = symbols('a b')
f1 = sin(x) + cos(x)
assert f1.rewrite(cos,exp) == exp(I*x)/2 + sin(x) + exp(-I*x)/2
assert f1.rewrite([cos],sin) == sin(x) + sin(x + pi/2, evaluate=False)
f2 = sin(x) + cos(y)/gamma(z)
assert f2.rewrite(sin,exp) == -I*(exp(I*x) - exp(-I*x))/2 + cos(y)/gamma(z)
assert f1.rewrite() == f1
def test_literal_evalf_is_number_is_zero_is_comparable():
from sympy.integrals.integrals import Integral
from sympy.core.symbol import symbols
from sympy.core.function import Function
from sympy.functions.elementary.trigonometric import cos, sin
x = symbols('x')
f = Function('f')
# issue 5033
assert f.is_number is False
# issue 6646
assert f(1).is_number is False
i = Integral(0, (x, x, x))
# expressions that are symbolically 0 can be difficult to prove
# so in case there is some easy way to know if something is 0
# it should appear in the is_zero property for that object;
# if is_zero is true evalf should always be able to compute that
# zero
assert i.n() == 0
assert i.is_zero
assert i.is_number is False
assert i.evalf(2, strict=False) == 0
# issue 10268
n = sin(1)**2 + cos(1)**2 - 1
assert n.is_comparable is False
assert n.n(2).is_comparable is False
assert n.n(2).n(2).is_comparable
def test_as_Basic():
assert as_Basic(1) is S.One
assert as_Basic(()) == Tuple()
raises(TypeError, lambda: as_Basic([]))
def test_atomic():
g, h = map(Function, 'gh')
x = symbols('x')
assert _atomic(g(x + h(x))) == {g(x + h(x))}
assert _atomic(g(x + h(x)), recursive=True) == {h(x), x, g(x + h(x))}
assert _atomic(1) == set()
assert _atomic(Basic(1,2)) == {Basic(1, 2)}
def test_as_dummy():
u, v, x, y, z, _0, _1 = symbols('u v x y z _0 _1')
assert Lambda(x, x + 1).as_dummy() == Lambda(_0, _0 + 1)
assert Lambda(x, x + _0).as_dummy() == Lambda(_1, _0 + _1)
eq = (1 + Sum(x, (x, 1, x)))
ans = 1 + Sum(_0, (_0, 1, x))
once = eq.as_dummy()
assert once == ans
twice = once.as_dummy()
assert twice == ans
assert Integral(x + _0, (x, x + 1), (_0, 1, 2)
).as_dummy() == Integral(_0 + _1, (_0, x + 1), (_1, 1, 2))
for T in (Symbol, Dummy):
d = T('x', real=True)
D = d.as_dummy()
assert D != d and D.func == Dummy and D.is_real is None
assert Dummy().as_dummy().is_commutative
assert Dummy(commutative=False).as_dummy().is_commutative is False
def test_canonical_variables():
x, i0, i1 = symbols('x _:2')
assert Integral(x, (x, x + 1)).canonical_variables == {x: i0}
assert Integral(x, (x, x + 1), (i0, 1, 2)).canonical_variables == {
x: i0, i0: i1}
assert Integral(x, (x, x + i0)).canonical_variables == {x: i1}
def test_replace_exceptions():
from sympy import Wild
x, y = symbols('x y')
e = (x**2 + x*y)
raises(TypeError, lambda: e.replace(sin, 2))
b = Wild('b')
c = Wild('c')
raises(TypeError, lambda: e.replace(b*c, c.is_real))
raises(TypeError, lambda: e.replace(b.is_real, 1))
raises(TypeError, lambda: e.replace(lambda d: d.is_Number, 1))
|
a2bc0448076aed3b560b9596c3a41f97f546afec8ba2c39b99e0e60c9c098fa3 | from sympy.abc import x, y
from sympy.core.parameters import evaluate
from sympy.core import Mul, Add, Pow, S
from sympy import sqrt, oo
def test_add():
with evaluate(False):
p = oo - oo
assert isinstance(p, Add) and p.args == (oo, -oo)
p = 5 - oo
assert isinstance(p, Add) and p.args == (-oo, 5)
p = oo - 5
assert isinstance(p, Add) and p.args == (oo, -5)
p = oo + 5
assert isinstance(p, Add) and p.args == (oo, 5)
p = 5 + oo
assert isinstance(p, Add) and p.args == (oo, 5)
p = -oo + 5
assert isinstance(p, Add) and p.args == (-oo, 5)
p = -5 - oo
assert isinstance(p, Add) and p.args == (-oo, -5)
with evaluate(False):
expr = x + x
assert isinstance(expr, Add)
assert expr.args == (x, x)
with evaluate(True):
assert (x + x).args == (2, x)
assert (x + x).args == (x, x)
assert isinstance(x + x, Mul)
with evaluate(False):
assert S.One + 1 == Add(1, 1)
assert 1 + S.One == Add(1, 1)
assert S(4) - 3 == Add(4, -3)
assert -3 + S(4) == Add(4, -3)
assert S(2) * 4 == Mul(2, 4)
assert 4 * S(2) == Mul(2, 4)
assert S(6) / 3 == Mul(6, Pow(3, -1))
assert S.One / 3 * 6 == Mul(S.One / 3, 6)
assert 9 ** S(2) == Pow(9, 2)
assert S(2) ** 9 == Pow(2, 9)
assert S(2) / 2 == Mul(2, Pow(2, -1))
assert S.One / 2 * 2 == Mul(S.One / 2, 2)
assert S(2) / 3 + 1 == Add(S(2) / 3, 1)
assert 1 + S(2) / 3 == Add(1, S(2) / 3)
assert S(4) / 7 - 3 == Add(S(4) / 7, -3)
assert -3 + S(4) / 7 == Add(-3, S(4) / 7)
assert S(2) / 4 * 4 == Mul(S(2) / 4, 4)
assert 4 * (S(2) / 4) == Mul(4, S(2) / 4)
assert S(6) / 3 == Mul(6, Pow(3, -1))
assert S.One / 3 * 6 == Mul(S.One / 3, 6)
assert S.One / 3 + sqrt(3) == Add(S.One / 3, sqrt(3))
assert sqrt(3) + S.One / 3 == Add(sqrt(3), S.One / 3)
assert S.One / 2 * 10.333 == Mul(S.One / 2, 10.333)
assert 10.333 * (S.One / 2) == Mul(10.333, S.One / 2)
assert sqrt(2) * sqrt(2) == Mul(sqrt(2), sqrt(2))
assert S.One / 2 + x == Add(S.One / 2, x)
assert x + S.One / 2 == Add(x, S.One / 2)
assert S.One / x * x == Mul(S.One / x, x)
assert x * (S.One / x) == Mul(x, Pow(x, -1))
assert S.One / 3 == Pow(3, -1)
assert S.One / x == Pow(x, -1)
assert 1 / S(3) == Pow(3, -1)
assert 1 / x == Pow(x, -1)
def test_nested():
with evaluate(False):
expr = (x + x) + (y + y)
assert expr.args == ((x + x), (y + y))
assert expr.args[0].args == (x, x)
|
45ea77ead581b3f62f43781a4594575702f74c8af6ba808a1db49501a00efe09 | from sympy.core import (
Basic, Rational, Symbol, S, Float, Integer, Mul, Number, Pow,
Expr, I, nan, pi, symbols, oo, zoo, N)
from sympy.core.tests.test_evalf import NS
from sympy.core.function import expand_multinomial
from sympy.functions.elementary.miscellaneous import sqrt, cbrt
from sympy.functions.elementary.exponential import exp, log
from sympy.functions.special.error_functions import erf
from sympy.functions.elementary.trigonometric import (
sin, cos, tan, sec, csc, sinh, cosh, tanh, atan)
from sympy.polys import Poly
from sympy.series.order import O
from sympy.sets import FiniteSet
from sympy.core.expr import unchanged
from sympy.testing.pytest import warns_deprecated_sympy
def test_rational():
a = Rational(1, 5)
r = sqrt(5)/5
assert sqrt(a) == r
assert 2*sqrt(a) == 2*r
r = a*a**S.Half
assert a**Rational(3, 2) == r
assert 2*a**Rational(3, 2) == 2*r
r = a**5*a**Rational(2, 3)
assert a**Rational(17, 3) == r
assert 2 * a**Rational(17, 3) == 2*r
def test_large_rational():
e = (Rational(123712**12 - 1, 7) + Rational(1, 7))**Rational(1, 3)
assert e == 234232585392159195136 * (Rational(1, 7)**Rational(1, 3))
def test_negative_real():
def feq(a, b):
return abs(a - b) < 1E-10
assert feq(S.One / Float(-0.5), -Integer(2))
def test_expand():
x = Symbol('x')
assert (2**(-1 - x)).expand() == S.Half*2**(-x)
def test_issue_3449():
#test if powers are simplified correctly
#see also issue 3995
x = Symbol('x')
assert ((x**Rational(1, 3))**Rational(2)) == x**Rational(2, 3)
assert (
(x**Rational(3))**Rational(2, 5)) == (x**Rational(3))**Rational(2, 5)
a = Symbol('a', real=True)
b = Symbol('b', real=True)
assert (a**2)**b == (abs(a)**b)**2
assert sqrt(1/a) != 1/sqrt(a) # e.g. for a = -1
assert (a**3)**Rational(1, 3) != a
assert (x**a)**b != x**(a*b) # e.g. x = -1, a=2, b=1/2
assert (x**.5)**b == x**(.5*b)
assert (x**.5)**.5 == x**.25
assert (x**2.5)**.5 != x**1.25 # e.g. for x = 5*I
k = Symbol('k', integer=True)
m = Symbol('m', integer=True)
assert (x**k)**m == x**(k*m)
assert Number(5)**Rational(2, 3) == Number(25)**Rational(1, 3)
assert (x**.5)**2 == x**1.0
assert (x**2)**k == (x**k)**2 == x**(2*k)
a = Symbol('a', positive=True)
assert (a**3)**Rational(2, 5) == a**Rational(6, 5)
assert (a**2)**b == (a**b)**2
assert (a**Rational(2, 3))**x == a**(x*Rational(2, 3)) != (a**x)**Rational(2, 3)
def test_issue_3866():
assert --sqrt(sqrt(5) - 1) == sqrt(sqrt(5) - 1)
def test_negative_one():
x = Symbol('x', complex=True)
y = Symbol('y', complex=True)
assert 1/x**y == x**(-y)
def test_issue_4362():
neg = Symbol('neg', negative=True)
nonneg = Symbol('nonneg', nonnegative=True)
any = Symbol('any')
num, den = sqrt(1/neg).as_numer_denom()
assert num == sqrt(-1)
assert den == sqrt(-neg)
num, den = sqrt(1/nonneg).as_numer_denom()
assert num == 1
assert den == sqrt(nonneg)
num, den = sqrt(1/any).as_numer_denom()
assert num == sqrt(1/any)
assert den == 1
def eqn(num, den, pow):
return (num/den)**pow
npos = 1
nneg = -1
dpos = 2 - sqrt(3)
dneg = 1 - sqrt(3)
assert dpos > 0 and dneg < 0 and npos > 0 and nneg < 0
# pos or neg integer
eq = eqn(npos, dpos, 2)
assert eq.is_Pow and eq.as_numer_denom() == (1, dpos**2)
eq = eqn(npos, dneg, 2)
assert eq.is_Pow and eq.as_numer_denom() == (1, dneg**2)
eq = eqn(nneg, dpos, 2)
assert eq.is_Pow and eq.as_numer_denom() == (1, dpos**2)
eq = eqn(nneg, dneg, 2)
assert eq.is_Pow and eq.as_numer_denom() == (1, dneg**2)
eq = eqn(npos, dpos, -2)
assert eq.is_Pow and eq.as_numer_denom() == (dpos**2, 1)
eq = eqn(npos, dneg, -2)
assert eq.is_Pow and eq.as_numer_denom() == (dneg**2, 1)
eq = eqn(nneg, dpos, -2)
assert eq.is_Pow and eq.as_numer_denom() == (dpos**2, 1)
eq = eqn(nneg, dneg, -2)
assert eq.is_Pow and eq.as_numer_denom() == (dneg**2, 1)
# pos or neg rational
pow = S.Half
eq = eqn(npos, dpos, pow)
assert eq.is_Pow and eq.as_numer_denom() == (npos**pow, dpos**pow)
eq = eqn(npos, dneg, pow)
assert eq.is_Pow is False and eq.as_numer_denom() == ((-npos)**pow, (-dneg)**pow)
eq = eqn(nneg, dpos, pow)
assert not eq.is_Pow or eq.as_numer_denom() == (nneg**pow, dpos**pow)
eq = eqn(nneg, dneg, pow)
assert eq.is_Pow and eq.as_numer_denom() == ((-nneg)**pow, (-dneg)**pow)
eq = eqn(npos, dpos, -pow)
assert eq.is_Pow and eq.as_numer_denom() == (dpos**pow, npos**pow)
eq = eqn(npos, dneg, -pow)
assert eq.is_Pow is False and eq.as_numer_denom() == (-(-npos)**pow*(-dneg)**pow, npos)
eq = eqn(nneg, dpos, -pow)
assert not eq.is_Pow or eq.as_numer_denom() == (dpos**pow, nneg**pow)
eq = eqn(nneg, dneg, -pow)
assert eq.is_Pow and eq.as_numer_denom() == ((-dneg)**pow, (-nneg)**pow)
# unknown exponent
pow = 2*any
eq = eqn(npos, dpos, pow)
assert eq.is_Pow and eq.as_numer_denom() == (npos**pow, dpos**pow)
eq = eqn(npos, dneg, pow)
assert eq.is_Pow and eq.as_numer_denom() == ((-npos)**pow, (-dneg)**pow)
eq = eqn(nneg, dpos, pow)
assert eq.is_Pow and eq.as_numer_denom() == (nneg**pow, dpos**pow)
eq = eqn(nneg, dneg, pow)
assert eq.is_Pow and eq.as_numer_denom() == ((-nneg)**pow, (-dneg)**pow)
eq = eqn(npos, dpos, -pow)
assert eq.as_numer_denom() == (dpos**pow, npos**pow)
eq = eqn(npos, dneg, -pow)
assert eq.is_Pow and eq.as_numer_denom() == ((-dneg)**pow, (-npos)**pow)
eq = eqn(nneg, dpos, -pow)
assert eq.is_Pow and eq.as_numer_denom() == (dpos**pow, nneg**pow)
eq = eqn(nneg, dneg, -pow)
assert eq.is_Pow and eq.as_numer_denom() == ((-dneg)**pow, (-nneg)**pow)
x = Symbol('x')
y = Symbol('y')
assert ((1/(1 + x/3))**(-S.One)).as_numer_denom() == (3 + x, 3)
notp = Symbol('notp', positive=False) # not positive does not imply real
b = ((1 + x/notp)**-2)
assert (b**(-y)).as_numer_denom() == (1, b**y)
assert (b**(-S.One)).as_numer_denom() == ((notp + x)**2, notp**2)
nonp = Symbol('nonp', nonpositive=True)
assert (((1 + x/nonp)**-2)**(-S.One)).as_numer_denom() == ((-nonp -
x)**2, nonp**2)
n = Symbol('n', negative=True)
assert (x**n).as_numer_denom() == (1, x**-n)
assert sqrt(1/n).as_numer_denom() == (S.ImaginaryUnit, sqrt(-n))
n = Symbol('0 or neg', nonpositive=True)
# if x and n are split up without negating each term and n is negative
# then the answer might be wrong; if n is 0 it won't matter since
# 1/oo and 1/zoo are both zero as is sqrt(0)/sqrt(-x) unless x is also
# zero (in which case the negative sign doesn't matter):
# 1/sqrt(1/-1) = -I but sqrt(-1)/sqrt(1) = I
assert (1/sqrt(x/n)).as_numer_denom() == (sqrt(-n), sqrt(-x))
c = Symbol('c', complex=True)
e = sqrt(1/c)
assert e.as_numer_denom() == (e, 1)
i = Symbol('i', integer=True)
assert ((1 + x/y)**i).as_numer_denom() == ((x + y)**i, y**i)
def test_Pow_Expr_args():
x = Symbol('x')
bases = [Basic(), Poly(x, x), FiniteSet(x)]
for base in bases:
with warns_deprecated_sympy():
Pow(base, S.One)
def test_Pow_signs():
"""Cf. issues 4595 and 5250"""
x = Symbol('x')
y = Symbol('y')
n = Symbol('n', even=True)
assert (3 - y)**2 != (y - 3)**2
assert (3 - y)**n != (y - 3)**n
assert (-3 + y - x)**2 != (3 - y + x)**2
assert (y - 3)**3 != -(3 - y)**3
def test_power_with_noncommutative_mul_as_base():
x = Symbol('x', commutative=False)
y = Symbol('y', commutative=False)
assert not (x*y)**3 == x**3*y**3
assert (2*x*y)**3 == 8*(x*y)**3
def test_power_rewrite_exp():
assert (I**I).rewrite(exp) == exp(-pi/2)
expr = (2 + 3*I)**(4 + 5*I)
assert expr.rewrite(exp) == exp((4 + 5*I)*(log(sqrt(13)) + I*atan(Rational(3, 2))))
assert expr.rewrite(exp).expand() == \
169*exp(5*I*log(13)/2)*exp(4*I*atan(Rational(3, 2)))*exp(-5*atan(Rational(3, 2)))
assert ((6 + 7*I)**5).rewrite(exp) == 7225*sqrt(85)*exp(5*I*atan(Rational(7, 6)))
expr = 5**(6 + 7*I)
assert expr.rewrite(exp) == exp((6 + 7*I)*log(5))
assert expr.rewrite(exp).expand() == 15625*exp(7*I*log(5))
assert Pow(123, 789, evaluate=False).rewrite(exp) == 123**789
assert (1**I).rewrite(exp) == 1**I
assert (0**I).rewrite(exp) == 0**I
expr = (-2)**(2 + 5*I)
assert expr.rewrite(exp) == exp((2 + 5*I)*(log(2) + I*pi))
assert expr.rewrite(exp).expand() == 4*exp(-5*pi)*exp(5*I*log(2))
assert ((-2)**S(-5)).rewrite(exp) == (-2)**S(-5)
x, y = symbols('x y')
assert (x**y).rewrite(exp) == exp(y*log(x))
assert (7**x).rewrite(exp) == exp(x*log(7), evaluate=False)
assert ((2 + 3*I)**x).rewrite(exp) == exp(x*(log(sqrt(13)) + I*atan(Rational(3, 2))))
assert (y**(5 + 6*I)).rewrite(exp) == exp(log(y)*(5 + 6*I))
assert all((1/func(x)).rewrite(exp) == 1/(func(x).rewrite(exp)) for func in
(sin, cos, tan, sec, csc, sinh, cosh, tanh))
def test_zero():
x = Symbol('x')
y = Symbol('y')
assert 0**x != 0
assert 0**(2*x) == 0**x
assert 0**(1.0*x) == 0**x
assert 0**(2.0*x) == 0**x
assert (0**(2 - x)).as_base_exp() == (0, 2 - x)
assert 0**(x - 2) != S.Infinity**(2 - x)
assert 0**(2*x*y) == 0**(x*y)
assert 0**(-2*x*y) == S.ComplexInfinity**(x*y)
def test_pow_as_base_exp():
x = Symbol('x')
assert (S.Infinity**(2 - x)).as_base_exp() == (S.Infinity, 2 - x)
assert (S.Infinity**(x - 2)).as_base_exp() == (S.Infinity, x - 2)
p = S.Half**x
assert p.base, p.exp == p.as_base_exp() == (S(2), -x)
# issue 8344:
assert Pow(1, 2, evaluate=False).as_base_exp() == (S.One, S(2))
def test_issue_6100_12942_4473():
x = Symbol('x')
y = Symbol('y')
assert x**1.0 != x
assert x != x**1.0
assert True != x**1.0
assert x**1.0 is not True
assert x is not True
assert x*y != (x*y)**1.0
# Pow != Symbol
assert (x**1.0)**1.0 != x
assert (x**1.0)**2.0 != x**2
b = Expr()
assert Pow(b, 1.0, evaluate=False) != b
# if the following gets distributed as a Mul (x**1.0*y**1.0 then
# __eq__ methods could be added to Symbol and Pow to detect the
# power-of-1.0 case.
assert ((x*y)**1.0).func is Pow
def test_issue_6208():
from sympy import root, Rational
I = S.ImaginaryUnit
assert sqrt(33**(I*Rational(9, 10))) == -33**(I*Rational(9, 20))
assert root((6*I)**(2*I), 3).as_base_exp()[1] == Rational(1, 3) # != 2*I/3
assert root((6*I)**(I/3), 3).as_base_exp()[1] == I/9
assert sqrt(exp(3*I)) == exp(I*Rational(3, 2))
assert sqrt(-sqrt(3)*(1 + 2*I)) == sqrt(sqrt(3))*sqrt(-1 - 2*I)
assert sqrt(exp(5*I)) == -exp(I*Rational(5, 2))
assert root(exp(5*I), 3).exp == Rational(1, 3)
def test_issue_6990():
x = Symbol('x')
a = Symbol('a')
b = Symbol('b')
assert (sqrt(a + b*x + x**2)).series(x, 0, 3).removeO() == \
b*x/(2*sqrt(a)) + x**2*(1/(2*sqrt(a)) - \
b**2/(8*a**Rational(3, 2))) + sqrt(a)
def test_issue_6068():
x = Symbol('x')
assert sqrt(sin(x)).series(x, 0, 7) == \
sqrt(x) - x**Rational(5, 2)/12 + x**Rational(9, 2)/1440 - \
x**Rational(13, 2)/24192 + O(x**7)
assert sqrt(sin(x)).series(x, 0, 9) == \
sqrt(x) - x**Rational(5, 2)/12 + x**Rational(9, 2)/1440 - \
x**Rational(13, 2)/24192 - 67*x**Rational(17, 2)/29030400 + O(x**9)
assert sqrt(sin(x**3)).series(x, 0, 19) == \
x**Rational(3, 2) - x**Rational(15, 2)/12 + x**Rational(27, 2)/1440 + O(x**19)
assert sqrt(sin(x**3)).series(x, 0, 20) == \
x**Rational(3, 2) - x**Rational(15, 2)/12 + x**Rational(27, 2)/1440 - \
x**Rational(39, 2)/24192 + O(x**20)
def test_issue_6782():
x = Symbol('x')
assert sqrt(sin(x**3)).series(x, 0, 7) == x**Rational(3, 2) + O(x**7)
assert sqrt(sin(x**4)).series(x, 0, 3) == x**2 + O(x**3)
def test_issue_6653():
x = Symbol('x')
assert (1 / sqrt(1 + sin(x**2))).series(x, 0, 3) == 1 - x**2/2 + O(x**3)
def test_issue_6429():
x = Symbol('x')
c = Symbol('c')
f = (c**2 + x)**(0.5)
assert f.series(x, x0=0, n=1) == (c**2)**0.5 + O(x)
assert f.taylor_term(0, x) == (c**2)**0.5
assert f.taylor_term(1, x) == 0.5*x*(c**2)**(-0.5)
assert f.taylor_term(2, x) == -0.125*x**2*(c**2)**(-1.5)
def test_issue_7638():
f = pi/log(sqrt(2))
assert ((1 + I)**(I*f/2))**0.3 == (1 + I)**(0.15*I*f)
# if 1/3 -> 1.0/3 this should fail since it cannot be shown that the
# sign will be +/-1; for the previous "small arg" case, it didn't matter
# that this could not be proved
assert (1 + I)**(4*I*f) == ((1 + I)**(12*I*f))**Rational(1, 3)
assert (((1 + I)**(I*(1 + 7*f)))**Rational(1, 3)).exp == Rational(1, 3)
r = symbols('r', real=True)
assert sqrt(r**2) == abs(r)
assert cbrt(r**3) != r
assert sqrt(Pow(2*I, 5*S.Half)) != (2*I)**Rational(5, 4)
p = symbols('p', positive=True)
assert cbrt(p**2) == p**Rational(2, 3)
assert NS(((0.2 + 0.7*I)**(0.7 + 1.0*I))**(0.5 - 0.1*I), 1) == '0.4 + 0.2*I'
assert sqrt(1/(1 + I)) == sqrt(1 - I)/sqrt(2) # or 1/sqrt(1 + I)
e = 1/(1 - sqrt(2))
assert sqrt(e) == I/sqrt(-1 + sqrt(2))
assert e**Rational(-1, 2) == -I*sqrt(-1 + sqrt(2))
assert sqrt((cos(1)**2 + sin(1)**2 - 1)**(3 + I)).exp in [S.Half,
Rational(3, 2) + I/2]
assert sqrt(r**Rational(4, 3)) != r**Rational(2, 3)
assert sqrt((p + I)**Rational(4, 3)) == (p + I)**Rational(2, 3)
assert sqrt((p - p**2*I)**2) == p - p**2*I
assert sqrt((p + r*I)**2) != p + r*I
e = (1 + I/5)
assert sqrt(e**5) == e**(5*S.Half)
assert sqrt(e**6) == e**3
assert sqrt((1 + I*r)**6) != (1 + I*r)**3
def test_issue_8582():
assert 1**oo is nan
assert 1**(-oo) is nan
assert 1**zoo is nan
assert 1**(oo + I) is nan
assert 1**(1 + I*oo) is nan
assert 1**(oo + I*oo) is nan
def test_issue_8650():
n = Symbol('n', integer=True, nonnegative=True)
assert (n**n).is_positive is True
x = 5*n + 5
assert (x**(5*(n + 1))).is_positive is True
def test_issue_13914():
b = Symbol('b')
assert (-1)**zoo is nan
assert 2**zoo is nan
assert (S.Half)**(1 + zoo) is nan
assert I**(zoo + I) is nan
assert b**(I + zoo) is nan
def test_better_sqrt():
n = Symbol('n', integer=True, nonnegative=True)
assert sqrt(3 + 4*I) == 2 + I
assert sqrt(3 - 4*I) == 2 - I
assert sqrt(-3 - 4*I) == 1 - 2*I
assert sqrt(-3 + 4*I) == 1 + 2*I
assert sqrt(32 + 24*I) == 6 + 2*I
assert sqrt(32 - 24*I) == 6 - 2*I
assert sqrt(-32 - 24*I) == 2 - 6*I
assert sqrt(-32 + 24*I) == 2 + 6*I
# triple (3, 4, 5):
# parity of 3 matches parity of 5 and
# den, 4, is a square
assert sqrt((3 + 4*I)/4) == 1 + I/2
# triple (8, 15, 17)
# parity of 8 doesn't match parity of 17 but
# den/2, 8/2, is a square
assert sqrt((8 + 15*I)/8) == (5 + 3*I)/4
# handle the denominator
assert sqrt((3 - 4*I)/25) == (2 - I)/5
assert sqrt((3 - 4*I)/26) == (2 - I)/sqrt(26)
# mul
# issue #12739
assert sqrt((3 + 4*I)/(3 - 4*I)) == (3 + 4*I)/5
assert sqrt(2/(3 + 4*I)) == sqrt(2)/5*(2 - I)
assert sqrt(n/(3 + 4*I)).subs(n, 2) == sqrt(2)/5*(2 - I)
assert sqrt(-2/(3 + 4*I)) == sqrt(2)/5*(1 + 2*I)
assert sqrt(-n/(3 + 4*I)).subs(n, 2) == sqrt(2)/5*(1 + 2*I)
# power
assert sqrt(1/(3 + I*4)) == (2 - I)/5
assert sqrt(1/(3 - I)) == sqrt(10)*sqrt(3 + I)/10
# symbolic
i = symbols('i', imaginary=True)
assert sqrt(3/i) == Mul(sqrt(3), 1/sqrt(i), evaluate=False)
# multiples of 1/2; don't make this too automatic
assert sqrt(3 + 4*I)**3 == (2 + I)**3
assert Pow(3 + 4*I, Rational(3, 2)) == 2 + 11*I
assert Pow(6 + 8*I, Rational(3, 2)) == 2*sqrt(2)*(2 + 11*I)
n, d = (3 + 4*I), (3 - 4*I)**3
a = n/d
assert a.args == (1/d, n)
eq = sqrt(a)
assert eq.args == (a, S.Half)
assert expand_multinomial(eq) == sqrt((-117 + 44*I)*(3 + 4*I))/125
assert eq.expand() == (7 - 24*I)/125
# issue 12775
# pos im part
assert sqrt(2*I) == (1 + I)
assert sqrt(2*9*I) == Mul(3, 1 + I, evaluate=False)
assert Pow(2*I, 3*S.Half) == (1 + I)**3
# neg im part
assert sqrt(-I/2) == Mul(S.Half, 1 - I, evaluate=False)
# fractional im part
assert Pow(Rational(-9, 2)*I, Rational(3, 2)) == 27*(1 - I)**3/8
def test_issue_2993():
x = Symbol('x')
assert str((2.3*x - 4)**0.3) == '1.5157165665104*(0.575*x - 1)**0.3'
assert str((2.3*x + 4)**0.3) == '1.5157165665104*(0.575*x + 1)**0.3'
assert str((-2.3*x + 4)**0.3) == '1.5157165665104*(1 - 0.575*x)**0.3'
assert str((-2.3*x - 4)**0.3) == '1.5157165665104*(-0.575*x - 1)**0.3'
assert str((2.3*x - 2)**0.3) == '1.28386201800527*(x - 0.869565217391304)**0.3'
assert str((-2.3*x - 2)**0.3) == '1.28386201800527*(-x - 0.869565217391304)**0.3'
assert str((-2.3*x + 2)**0.3) == '1.28386201800527*(0.869565217391304 - x)**0.3'
assert str((2.3*x + 2)**0.3) == '1.28386201800527*(x + 0.869565217391304)**0.3'
assert str((2.3*x - 4)**Rational(1, 3)) == '2**(2/3)*(0.575*x - 1)**(1/3)'
eq = (2.3*x + 4)
assert eq**2 == 16*(0.575*x + 1)**2
assert (1/eq).args == (eq, -1) # don't change trivial power
# issue 17735
q=.5*exp(x) - .5*exp(-x) + 0.1
assert int((q**2).subs(x, 1)) == 1
# issue 17756
y = Symbol('y')
assert len(sqrt(x/(x + y)**2 + Float('0.008', 30)).subs(y, pi.n(25)).atoms(Float)) == 2
# issue 17756
a, b, c, d, e, f, g = symbols('a:g')
expr = sqrt(1 + a*(c**4 + g*d - 2*g*e - f*(-g + d))**2/
(c**3*b**2*(d - 3*e + 2*f)**2))/2
r = [
(a, N('0.0170992456333788667034850458615', 30)),
(b, N('0.0966594956075474769169134801223', 30)),
(c, N('0.390911862903463913632151616184', 30)),
(d, N('0.152812084558656566271750185933', 30)),
(e, N('0.137562344465103337106561623432', 30)),
(f, N('0.174259178881496659302933610355', 30)),
(g, N('0.220745448491223779615401870086', 30))]
tru = expr.n(30, subs=dict(r))
seq = expr.subs(r)
# although `tru` is the right way to evaluate
# expr with numerical values, `seq` will have
# significant loss of precision if extraction of
# the largest coefficient of a power's base's terms
# is done improperly
assert seq == tru
def test_issue_17450():
assert (erf(cosh(1)**7)**I).is_real is None
assert (erf(cosh(1)**7)**I).is_imaginary is False
assert (Pow(exp(1+sqrt(2)), ((1-sqrt(2))*I*pi), evaluate=False)).is_real is None
assert ((-10)**(10*I*pi/3)).is_real is False
assert ((-5)**(4*I*pi)).is_real is False
def test_issue_18190():
assert sqrt(1 / tan(1 + I)) == 1 / sqrt(tan(1 + I))
def test_issue_14815():
x = Symbol('x', real=True)
assert sqrt(x).is_extended_negative is False
x = Symbol('x', real=False)
assert sqrt(x).is_extended_negative is None
x = Symbol('x', complex=True)
assert sqrt(x).is_extended_negative is False
x = Symbol('x', extended_real=True)
assert sqrt(x).is_extended_negative is False
assert sqrt(zoo, evaluate=False).is_extended_negative is None
assert sqrt(nan, evaluate=False).is_extended_negative is None
def test_issue_18509():
assert unchanged(Mul, oo, 1/pi**oo)
assert (1/pi**oo).is_extended_positive == False
def test_issue_18762():
e, p = symbols('e p')
g0 = sqrt(1 + e**2 - 2*e*cos(p))
assert len(g0.series(e, 1, 3).args) == 4
|
8f6c120b9a84cc0e45c27768031cd029d4f2f660f3b14fce36aebe3310d0650d | from sympy import (Symbol, exp, Integer, Float, sin, cos, log, Poly, Lambda,
Function, I, S, sqrt, srepr, Rational, Tuple, Matrix, Interval, Add, Mul,
Pow, Or, true, false, Abs, pi, Range, Xor)
from sympy.abc import x, y
from sympy.core.sympify import (sympify, _sympify, SympifyError, kernS,
CantSympify)
from sympy.core.decorators import _sympifyit
from sympy.external import import_module
from sympy.testing.pytest import raises, XFAIL, skip, warns_deprecated_sympy
from sympy.utilities.decorator import conserve_mpmath_dps
from sympy.geometry import Point, Line
from sympy.functions.combinatorial.factorials import factorial, factorial2
from sympy.abc import _clash, _clash1, _clash2
from sympy.core.compatibility import exec_, HAS_GMPY
from sympy.sets import FiniteSet, EmptySet
from sympy.tensor.array.dense_ndim_array import ImmutableDenseNDimArray
import mpmath
from collections import defaultdict, OrderedDict
from mpmath.rational import mpq
numpy = import_module('numpy')
def test_issue_3538():
v = sympify("exp(x)")
assert v == exp(x)
assert type(v) == type(exp(x))
assert str(type(v)) == str(type(exp(x)))
def test_sympify1():
assert sympify("x") == Symbol("x")
assert sympify(" x") == Symbol("x")
assert sympify(" x ") == Symbol("x")
# issue 4877
n1 = S.Half
assert sympify('--.5') == n1
assert sympify('-1/2') == -n1
assert sympify('-+--.5') == -n1
assert sympify('-.[3]') == Rational(-1, 3)
assert sympify('.[3]') == Rational(1, 3)
assert sympify('+.[3]') == Rational(1, 3)
assert sympify('+0.[3]*10**-2') == Rational(1, 300)
assert sympify('.[052631578947368421]') == Rational(1, 19)
assert sympify('.0[526315789473684210]') == Rational(1, 19)
assert sympify('.034[56]') == Rational(1711, 49500)
# options to make reals into rationals
assert sympify('1.22[345]', rational=True) == \
1 + Rational(22, 100) + Rational(345, 99900)
assert sympify('2/2.6', rational=True) == Rational(10, 13)
assert sympify('2.6/2', rational=True) == Rational(13, 10)
assert sympify('2.6e2/17', rational=True) == Rational(260, 17)
assert sympify('2.6e+2/17', rational=True) == Rational(260, 17)
assert sympify('2.6e-2/17', rational=True) == Rational(26, 17000)
assert sympify('2.1+3/4', rational=True) == \
Rational(21, 10) + Rational(3, 4)
assert sympify('2.234456', rational=True) == Rational(279307, 125000)
assert sympify('2.234456e23', rational=True) == 223445600000000000000000
assert sympify('2.234456e-23', rational=True) == \
Rational(279307, 12500000000000000000000000000)
assert sympify('-2.234456e-23', rational=True) == \
Rational(-279307, 12500000000000000000000000000)
assert sympify('12345678901/17', rational=True) == \
Rational(12345678901, 17)
assert sympify('1/.3 + x', rational=True) == Rational(10, 3) + x
# make sure longs in fractions work
assert sympify('222222222222/11111111111') == \
Rational(222222222222, 11111111111)
# ... even if they come from repetend notation
assert sympify('1/.2[123456789012]') == Rational(333333333333, 70781892967)
# ... or from high precision reals
assert sympify('.1234567890123456', rational=True) == \
Rational(19290123283179, 156250000000000)
def test_sympify_Fraction():
try:
import fractions
except ImportError:
pass
else:
value = sympify(fractions.Fraction(101, 127))
assert value == Rational(101, 127) and type(value) is Rational
def test_sympify_gmpy():
if HAS_GMPY:
if HAS_GMPY == 2:
import gmpy2 as gmpy
elif HAS_GMPY == 1:
import gmpy
value = sympify(gmpy.mpz(1000001))
assert value == Integer(1000001) and type(value) is Integer
value = sympify(gmpy.mpq(101, 127))
assert value == Rational(101, 127) and type(value) is Rational
@conserve_mpmath_dps
def test_sympify_mpmath():
value = sympify(mpmath.mpf(1.0))
assert value == Float(1.0) and type(value) is Float
mpmath.mp.dps = 12
assert sympify(
mpmath.pi).epsilon_eq(Float("3.14159265359"), Float("1e-12")) == True
assert sympify(
mpmath.pi).epsilon_eq(Float("3.14159265359"), Float("1e-13")) == False
mpmath.mp.dps = 6
assert sympify(
mpmath.pi).epsilon_eq(Float("3.14159"), Float("1e-5")) == True
assert sympify(
mpmath.pi).epsilon_eq(Float("3.14159"), Float("1e-6")) == False
assert sympify(mpmath.mpc(1.0 + 2.0j)) == Float(1.0) + Float(2.0)*I
assert sympify(mpq(1, 2)) == S.Half
def test_sympify2():
class A:
def _sympy_(self):
return Symbol("x")**3
a = A()
assert _sympify(a) == x**3
assert sympify(a) == x**3
assert a == x**3
def test_sympify3():
assert sympify("x**3") == x**3
assert sympify("x^3") == x**3
assert sympify("1/2") == Integer(1)/2
raises(SympifyError, lambda: _sympify('x**3'))
raises(SympifyError, lambda: _sympify('1/2'))
def test_sympify_keywords():
raises(SympifyError, lambda: sympify('if'))
raises(SympifyError, lambda: sympify('for'))
raises(SympifyError, lambda: sympify('while'))
raises(SympifyError, lambda: sympify('lambda'))
def test_sympify_float():
assert sympify("1e-64") != 0
assert sympify("1e-20000") != 0
def test_sympify_bool():
assert sympify(True) is true
assert sympify(False) is false
def test_sympyify_iterables():
ans = [Rational(3, 10), Rational(1, 5)]
assert sympify(['.3', '.2'], rational=True) == ans
assert sympify(dict(x=0, y=1)) == {x: 0, y: 1}
assert sympify(['1', '2', ['3', '4']]) == [S(1), S(2), [S(3), S(4)]]
@XFAIL
def test_issue_16772():
# because there is a converter for tuple, the
# args are only sympified without the flags being passed
# along; list, on the other hand, is not converted
# with a converter so its args are traversed later
ans = [Rational(3, 10), Rational(1, 5)]
assert sympify(tuple(['.3', '.2']), rational=True) == Tuple(*ans)
def test_issue_16859():
class no(float, CantSympify):
pass
raises(SympifyError, lambda: sympify(no(1.2)))
def test_sympify4():
class A:
def _sympy_(self):
return Symbol("x")
a = A()
assert _sympify(a)**3 == x**3
assert sympify(a)**3 == x**3
assert a == x
def test_sympify_text():
assert sympify('some') == Symbol('some')
assert sympify('core') == Symbol('core')
assert sympify('True') is True
assert sympify('False') is False
assert sympify('Poly') == Poly
assert sympify('sin') == sin
def test_sympify_function():
assert sympify('factor(x**2-1, x)') == -(1 - x)*(x + 1)
assert sympify('sin(pi/2)*cos(pi)') == -Integer(1)
def test_sympify_poly():
p = Poly(x**2 + x + 1, x)
assert _sympify(p) is p
assert sympify(p) is p
def test_sympify_factorial():
assert sympify('x!') == factorial(x)
assert sympify('(x+1)!') == factorial(x + 1)
assert sympify('(1 + y*(x + 1))!') == factorial(1 + y*(x + 1))
assert sympify('(1 + y*(x + 1)!)^2') == (1 + y*factorial(x + 1))**2
assert sympify('y*x!') == y*factorial(x)
assert sympify('x!!') == factorial2(x)
assert sympify('(x+1)!!') == factorial2(x + 1)
assert sympify('(1 + y*(x + 1))!!') == factorial2(1 + y*(x + 1))
assert sympify('(1 + y*(x + 1)!!)^2') == (1 + y*factorial2(x + 1))**2
assert sympify('y*x!!') == y*factorial2(x)
assert sympify('factorial2(x)!') == factorial(factorial2(x))
raises(SympifyError, lambda: sympify("+!!"))
raises(SympifyError, lambda: sympify(")!!"))
raises(SympifyError, lambda: sympify("!"))
raises(SympifyError, lambda: sympify("(!)"))
raises(SympifyError, lambda: sympify("x!!!"))
def test_sage():
# how to effectivelly test for the _sage_() method without having SAGE
# installed?
assert hasattr(x, "_sage_")
assert hasattr(Integer(3), "_sage_")
assert hasattr(sin(x), "_sage_")
assert hasattr(cos(x), "_sage_")
assert hasattr(x**2, "_sage_")
assert hasattr(x + y, "_sage_")
assert hasattr(exp(x), "_sage_")
assert hasattr(log(x), "_sage_")
def test_issue_3595():
assert sympify("a_") == Symbol("a_")
assert sympify("_a") == Symbol("_a")
def test_lambda():
x = Symbol('x')
assert sympify('lambda: 1') == Lambda((), 1)
assert sympify('lambda x: x') == Lambda(x, x)
assert sympify('lambda x: 2*x') == Lambda(x, 2*x)
assert sympify('lambda x, y: 2*x+y') == Lambda((x, y), 2*x + y)
def test_lambda_raises():
raises(SympifyError, lambda: sympify("lambda *args: args")) # args argument error
raises(SympifyError, lambda: sympify("lambda **kwargs: kwargs[0]")) # kwargs argument error
raises(SympifyError, lambda: sympify("lambda x = 1: x")) # Keyword argument error
with raises(SympifyError):
_sympify('lambda: 1')
def test_sympify_raises():
raises(SympifyError, lambda: sympify("fx)"))
class A:
def __str__(self):
return 'x'
with warns_deprecated_sympy():
assert sympify(A()) == Symbol('x')
def test__sympify():
x = Symbol('x')
f = Function('f')
# positive _sympify
assert _sympify(x) is x
assert _sympify(f) is f
assert _sympify(1) == Integer(1)
assert _sympify(0.5) == Float("0.5")
assert _sympify(1 + 1j) == 1.0 + I*1.0
class A:
def _sympy_(self):
return Integer(5)
a = A()
assert _sympify(a) == Integer(5)
# negative _sympify
raises(SympifyError, lambda: _sympify('1'))
raises(SympifyError, lambda: _sympify([1, 2, 3]))
def test_sympifyit():
x = Symbol('x')
y = Symbol('y')
@_sympifyit('b', NotImplemented)
def add(a, b):
return a + b
assert add(x, 1) == x + 1
assert add(x, 0.5) == x + Float('0.5')
assert add(x, y) == x + y
assert add(x, '1') == NotImplemented
@_sympifyit('b')
def add_raises(a, b):
return a + b
assert add_raises(x, 1) == x + 1
assert add_raises(x, 0.5) == x + Float('0.5')
assert add_raises(x, y) == x + y
raises(SympifyError, lambda: add_raises(x, '1'))
def test_int_float():
class F1_1:
def __float__(self):
return 1.1
class F1_1b:
"""
This class is still a float, even though it also implements __int__().
"""
def __float__(self):
return 1.1
def __int__(self):
return 1
class F1_1c:
"""
This class is still a float, because it implements _sympy_()
"""
def __float__(self):
return 1.1
def __int__(self):
return 1
def _sympy_(self):
return Float(1.1)
class I5:
def __int__(self):
return 5
class I5b:
"""
This class implements both __int__() and __float__(), so it will be
treated as Float in SymPy. One could change this behavior, by using
float(a) == int(a), but deciding that integer-valued floats represent
exact numbers is arbitrary and often not correct, so we do not do it.
If, in the future, we decide to do it anyway, the tests for I5b need to
be changed.
"""
def __float__(self):
return 5.0
def __int__(self):
return 5
class I5c:
"""
This class implements both __int__() and __float__(), but also
a _sympy_() method, so it will be Integer.
"""
def __float__(self):
return 5.0
def __int__(self):
return 5
def _sympy_(self):
return Integer(5)
i5 = I5()
i5b = I5b()
i5c = I5c()
f1_1 = F1_1()
f1_1b = F1_1b()
f1_1c = F1_1c()
assert sympify(i5) == 5
assert isinstance(sympify(i5), Integer)
assert sympify(i5b) == 5
assert isinstance(sympify(i5b), Float)
assert sympify(i5c) == 5
assert isinstance(sympify(i5c), Integer)
assert abs(sympify(f1_1) - 1.1) < 1e-5
assert abs(sympify(f1_1b) - 1.1) < 1e-5
assert abs(sympify(f1_1c) - 1.1) < 1e-5
assert _sympify(i5) == 5
assert isinstance(_sympify(i5), Integer)
assert _sympify(i5b) == 5
assert isinstance(_sympify(i5b), Float)
assert _sympify(i5c) == 5
assert isinstance(_sympify(i5c), Integer)
assert abs(_sympify(f1_1) - 1.1) < 1e-5
assert abs(_sympify(f1_1b) - 1.1) < 1e-5
assert abs(_sympify(f1_1c) - 1.1) < 1e-5
def test_evaluate_false():
cases = {
'2 + 3': Add(2, 3, evaluate=False),
'2**2 / 3': Mul(Pow(2, 2, evaluate=False), Pow(3, -1, evaluate=False), evaluate=False),
'2 + 3 * 5': Add(2, Mul(3, 5, evaluate=False), evaluate=False),
'2 - 3 * 5': Add(2, Mul(-1, Mul(3, 5,evaluate=False), evaluate=False), evaluate=False),
'1 / 3': Mul(1, Pow(3, -1, evaluate=False), evaluate=False),
'True | False': Or(True, False, evaluate=False),
'1 + 2 + 3 + 5*3 + integrate(x)': Add(1, 2, 3, Mul(5, 3, evaluate=False), x**2/2, evaluate=False),
'2 * 4 * 6 + 8': Add(Mul(2, 4, 6, evaluate=False), 8, evaluate=False),
'2 - 8 / 4': Add(2, Mul(-1, Mul(8, Pow(4, -1, evaluate=False), evaluate=False), evaluate=False), evaluate=False),
'2 - 2**2': Add(2, Mul(-1, Pow(2, 2, evaluate=False), evaluate=False), evaluate=False),
}
for case, result in cases.items():
assert sympify(case, evaluate=False) == result
def test_issue_4133():
a = sympify('Integer(4)')
assert a == Integer(4)
assert a.is_Integer
def test_issue_3982():
a = [3, 2.0]
assert sympify(a) == [Integer(3), Float(2.0)]
assert sympify(tuple(a)) == Tuple(Integer(3), Float(2.0))
assert sympify(set(a)) == FiniteSet(Integer(3), Float(2.0))
def test_S_sympify():
assert S(1)/2 == sympify(1)/2
assert (-2)**(S(1)/2) == sqrt(2)*I
def test_issue_4788():
assert srepr(S(1.0 + 0J)) == srepr(S(1.0)) == srepr(Float(1.0))
def test_issue_4798_None():
assert S(None) is None
def test_issue_3218():
assert sympify("x+\ny") == x + y
def test_issue_4988_builtins():
C = Symbol('C')
vars = {'C': C}
exp1 = sympify('C')
assert exp1 == C # Make sure it did not get mixed up with sympy.C
exp2 = sympify('C', vars)
assert exp2 == C # Make sure it did not get mixed up with sympy.C
def test_geometry():
p = sympify(Point(0, 1))
assert p == Point(0, 1) and isinstance(p, Point)
L = sympify(Line(p, (1, 0)))
assert L == Line((0, 1), (1, 0)) and isinstance(L, Line)
def test_kernS():
s = '-1 - 2*(-(-x + 1/x)/(x*(x - 1/x)**2) - 1/(x*(x - 1/x)))'
# when 1497 is fixed, this no longer should pass: the expression
# should be unchanged
assert -1 - 2*(-(-x + 1/x)/(x*(x - 1/x)**2) - 1/(x*(x - 1/x))) == -1
# sympification should not allow the constant to enter a Mul
# or else the structure can change dramatically
ss = kernS(s)
assert ss != -1 and ss.simplify() == -1
s = '-1 - 2*(-(-x + 1/x)/(x*(x - 1/x)**2) - 1/(x*(x - 1/x)))'.replace(
'x', '_kern')
ss = kernS(s)
assert ss != -1 and ss.simplify() == -1
# issue 6687
assert (kernS('Interval(-1,-2 - 4*(-3))')
== Interval(-1, Add(-2, Mul(12, 1, evaluate=False), evaluate=False)))
assert kernS('_kern') == Symbol('_kern')
assert kernS('E**-(x)') == exp(-x)
e = 2*(x + y)*y
assert kernS(['2*(x + y)*y', ('2*(x + y)*y',)]) == [e, (e,)]
assert kernS('-(2*sin(x)**2 + 2*sin(x)*cos(x))*y/2') == \
-y*(2*sin(x)**2 + 2*sin(x)*cos(x))/2
# issue 15132
assert kernS('(1 - x)/(1 - x*(1-y))') == kernS('(1-x)/(1-(1-y)*x)')
assert kernS('(1-2**-(4+1)*(1-y)*x)') == (1 - x*(1 - y)/32)
assert kernS('(1-2**(4+1)*(1-y)*x)') == (1 - 32*x*(1 - y))
assert kernS('(1-2.*(1-y)*x)') == 1 - 2.*x*(1 - y)
one = kernS('x - (x - 1)')
assert one != 1 and one.expand() == 1
def test_issue_6540_6552():
assert S('[[1/3,2], (2/5,)]') == [[Rational(1, 3), 2], (Rational(2, 5),)]
assert S('[[2/6,2], (2/4,)]') == [[Rational(1, 3), 2], (S.Half,)]
assert S('[[[2*(1)]]]') == [[[2]]]
assert S('Matrix([2*(1)])') == Matrix([2])
def test_issue_6046():
assert str(S("Q & C", locals=_clash1)) == 'C & Q'
assert str(S('pi(x)', locals=_clash2)) == 'pi(x)'
assert str(S('pi(C, Q)', locals=_clash)) == 'pi(C, Q)'
locals = {}
exec_("from sympy.abc import Q, C", locals)
assert str(S('C&Q', locals)) == 'C & Q'
def test_issue_8821_highprec_from_str():
s = str(pi.evalf(128))
p = sympify(s)
assert Abs(sin(p)) < 1e-127
def test_issue_10295():
if not numpy:
skip("numpy not installed.")
A = numpy.array([[1, 3, -1],
[0, 1, 7]])
sA = S(A)
assert sA.shape == (2, 3)
for (ri, ci), val in numpy.ndenumerate(A):
assert sA[ri, ci] == val
B = numpy.array([-7, x, 3*y**2])
sB = S(B)
assert sB.shape == (3,)
assert B[0] == sB[0] == -7
assert B[1] == sB[1] == x
assert B[2] == sB[2] == 3*y**2
C = numpy.arange(0, 24)
C.resize(2,3,4)
sC = S(C)
assert sC[0, 0, 0].is_integer
assert sC[0, 0, 0] == 0
a1 = numpy.array([1, 2, 3])
a2 = numpy.array([i for i in range(24)])
a2.resize(2, 4, 3)
assert sympify(a1) == ImmutableDenseNDimArray([1, 2, 3])
assert sympify(a2) == ImmutableDenseNDimArray([i for i in range(24)], (2, 4, 3))
def test_Range():
# Only works in Python 3 where range returns a range type
assert sympify(range(10)) == Range(10)
assert _sympify(range(10)) == Range(10)
def test_sympify_set():
n = Symbol('n')
assert sympify({n}) == FiniteSet(n)
assert sympify(set()) == EmptySet
def test_sympify_numpy():
if not numpy:
skip('numpy not installed. Abort numpy tests.')
np = numpy
def equal(x, y):
return x == y and type(x) == type(y)
assert sympify(np.bool_(1)) is S(True)
try:
assert equal(
sympify(np.int_(1234567891234567891)), S(1234567891234567891))
assert equal(
sympify(np.intp(1234567891234567891)), S(1234567891234567891))
except OverflowError:
# May fail on 32-bit systems: Python int too large to convert to C long
pass
assert equal(sympify(np.intc(1234567891)), S(1234567891))
assert equal(sympify(np.int8(-123)), S(-123))
assert equal(sympify(np.int16(-12345)), S(-12345))
assert equal(sympify(np.int32(-1234567891)), S(-1234567891))
assert equal(
sympify(np.int64(-1234567891234567891)), S(-1234567891234567891))
assert equal(sympify(np.uint8(123)), S(123))
assert equal(sympify(np.uint16(12345)), S(12345))
assert equal(sympify(np.uint32(1234567891)), S(1234567891))
assert equal(
sympify(np.uint64(1234567891234567891)), S(1234567891234567891))
assert equal(sympify(np.float32(1.123456)), Float(1.123456, precision=24))
assert equal(sympify(np.float64(1.1234567891234)),
Float(1.1234567891234, precision=53))
assert equal(sympify(np.longdouble(1.123456789)),
Float(1.123456789, precision=80))
assert equal(sympify(np.complex64(1 + 2j)), S(1.0 + 2.0*I))
assert equal(sympify(np.complex128(1 + 2j)), S(1.0 + 2.0*I))
assert equal(sympify(np.longcomplex(1 + 2j)), S(1.0 + 2.0*I))
#float96 does not exist on all platforms
if hasattr(np, 'float96'):
assert equal(sympify(np.float96(1.123456789)),
Float(1.123456789, precision=80))
#float128 does not exist on all platforms
if hasattr(np, 'float128'):
assert equal(sympify(np.float128(1.123456789123)),
Float(1.123456789123, precision=80))
@XFAIL
def test_sympify_rational_numbers_set():
ans = [Rational(3, 10), Rational(1, 5)]
assert sympify({'.3', '.2'}, rational=True) == FiniteSet(*ans)
def test_issue_13924():
if not numpy:
skip("numpy not installed.")
a = sympify(numpy.array([1]))
assert isinstance(a, ImmutableDenseNDimArray)
assert a[0] == 1
def test_numpy_sympify_args():
# Issue 15098. Make sure sympify args work with numpy types (like numpy.str_)
if not numpy:
skip("numpy not installed.")
a = sympify(numpy.str_('a'))
assert type(a) is Symbol
assert a == Symbol('a')
class CustomSymbol(Symbol):
pass
a = sympify(numpy.str_('a'), {"Symbol": CustomSymbol})
assert isinstance(a, CustomSymbol)
a = sympify(numpy.str_('x^y'))
assert a == x**y
a = sympify(numpy.str_('x^y'), convert_xor=False)
assert a == Xor(x, y)
raises(SympifyError, lambda: sympify(numpy.str_('x'), strict=True))
a = sympify(numpy.str_('1.1'))
assert isinstance(a, Float)
assert a == 1.1
a = sympify(numpy.str_('1.1'), rational=True)
assert isinstance(a, Rational)
assert a == Rational(11, 10)
a = sympify(numpy.str_('x + x'))
assert isinstance(a, Mul)
assert a == 2*x
a = sympify(numpy.str_('x + x'), evaluate=False)
assert isinstance(a, Add)
assert a == Add(x, x, evaluate=False)
def test_issue_5939():
a = Symbol('a')
b = Symbol('b')
assert sympify('''a+\nb''') == a + b
def test_issue_16759():
d = sympify({.5: 1})
assert S.Half not in d
assert Float(.5) in d
assert d[.5] is S.One
d = sympify(OrderedDict({.5: 1}))
assert S.Half not in d
assert Float(.5) in d
assert d[.5] is S.One
d = sympify(defaultdict(int, {.5: 1}))
assert S.Half not in d
assert Float(.5) in d
assert d[.5] is S.One
def test_issue_17811():
a = Function('a')
assert sympify('a(x)*5', evaluate=False) == Mul(a(x), 5, evaluate=False)
def test_issue_14706():
if not numpy:
skip("numpy not installed.")
z1 = numpy.zeros((1, 1), dtype=numpy.float)
z2 = numpy.zeros((2, 2), dtype=numpy.float)
z3 = numpy.zeros((), dtype=numpy.float)
y1 = numpy.ones((1, 1), dtype=numpy.float)
y2 = numpy.ones((2, 2), dtype=numpy.float)
y3 = numpy.ones((), dtype=numpy.float)
assert numpy.all(x + z1 == numpy.full((1, 1), x))
assert numpy.all(x + z2 == numpy.full((2, 2), x))
assert numpy.all(z1 + x == numpy.full((1, 1), x))
assert numpy.all(z2 + x == numpy.full((2, 2), x))
for z in [z3,
numpy.int(0),
numpy.float(0),
numpy.complex(0)]:
assert x + z == x
assert z + x == x
assert isinstance(x + z, Symbol)
assert isinstance(z + x, Symbol)
# If these tests fail, then it means that numpy has finally
# fixed the issue of scalar conversion for rank>0 arrays
# which is mentioned in numpy/numpy#10404. In that case,
# some changes have to be made in sympify.py.
# Note: For future reference, for anyone who takes up this
# issue when numpy has finally fixed their side of the problem,
# the changes for this temporary fix were introduced in PR 18651
assert numpy.all(x + y1 == numpy.full((1, 1), x + 1.0))
assert numpy.all(x + y2 == numpy.full((2, 2), x + 1.0))
assert numpy.all(y1 + x == numpy.full((1, 1), x + 1.0))
assert numpy.all(y2 + x == numpy.full((2, 2), x + 1.0))
for y_ in [y3,
numpy.int(1),
numpy.float(1),
numpy.complex(1)]:
assert x + y_ == y_ + x
assert isinstance(x + y_, Add)
assert isinstance(y_ + x, Add)
assert x + numpy.array(x) == 2 * x
assert x + numpy.array([x]) == numpy.array([2*x], dtype=object)
assert sympify(numpy.array([1])) == ImmutableDenseNDimArray([1], 1)
assert sympify(numpy.array([[[1]]])) == ImmutableDenseNDimArray([1], (1, 1, 1))
assert sympify(z1) == ImmutableDenseNDimArray([0], (1, 1))
assert sympify(z2) == ImmutableDenseNDimArray([0, 0, 0, 0], (2, 2))
assert sympify(z3) == ImmutableDenseNDimArray([0], ())
assert sympify(z3, strict=True) == 0.0
raises(SympifyError, lambda: sympify(numpy.array([1]), strict=True))
raises(SympifyError, lambda: sympify(z1, strict=True))
raises(SympifyError, lambda: sympify(z2, strict=True))
|
3dc70e940359f8aec71e93b531a572ff405909afaf5c80c782438cb9ccb0235d | from sympy import (abc, Add, cos, collect, Derivative, diff, exp, Float, Function,
I, Integer, log, Mul, oo, Poly, Rational, S, sin, sqrt, Symbol, symbols,
Wild, pi, meijerg, Sum
)
from sympy.testing.pytest import XFAIL
def test_symbol():
x = Symbol('x')
a, b, c, p, q = map(Wild, 'abcpq')
e = x
assert e.match(x) == {}
assert e.matches(x) == {}
assert e.match(a) == {a: x}
e = Rational(5)
assert e.match(c) == {c: 5}
assert e.match(e) == {}
assert e.match(e + 1) is None
def test_add():
x, y, a, b, c = map(Symbol, 'xyabc')
p, q, r = map(Wild, 'pqr')
e = a + b
assert e.match(p + b) == {p: a}
assert e.match(p + a) == {p: b}
e = 1 + b
assert e.match(p + b) == {p: 1}
e = a + b + c
assert e.match(a + p + c) == {p: b}
assert e.match(b + p + c) == {p: a}
e = a + b + c + x
assert e.match(a + p + x + c) == {p: b}
assert e.match(b + p + c + x) == {p: a}
assert e.match(b) is None
assert e.match(b + p) == {p: a + c + x}
assert e.match(a + p + c) == {p: b + x}
assert e.match(b + p + c) == {p: a + x}
e = 4*x + 5
assert e.match(4*x + p) == {p: 5}
assert e.match(3*x + p) == {p: x + 5}
assert e.match(p*x + 5) == {p: 4}
def test_power():
x, y, a, b, c = map(Symbol, 'xyabc')
p, q, r = map(Wild, 'pqr')
e = (x + y)**a
assert e.match(p**q) == {p: x + y, q: a}
assert e.match(p**p) is None
e = (x + y)**(x + y)
assert e.match(p**p) == {p: x + y}
assert e.match(p**q) == {p: x + y, q: x + y}
e = (2*x)**2
assert e.match(p*q**r) == {p: 4, q: x, r: 2}
e = Integer(1)
assert e.match(x**p) == {p: 0}
def test_match_exclude():
x = Symbol('x')
y = Symbol('y')
p = Wild("p")
q = Wild("q")
r = Wild("r")
e = Rational(6)
assert e.match(2*p) == {p: 3}
e = 3/(4*x + 5)
assert e.match(3/(p*x + q)) == {p: 4, q: 5}
e = 3/(4*x + 5)
assert e.match(p/(q*x + r)) == {p: 3, q: 4, r: 5}
e = 2/(x + 1)
assert e.match(p/(q*x + r)) == {p: 2, q: 1, r: 1}
e = 1/(x + 1)
assert e.match(p/(q*x + r)) == {p: 1, q: 1, r: 1}
e = 4*x + 5
assert e.match(p*x + q) == {p: 4, q: 5}
e = 4*x + 5*y + 6
assert e.match(p*x + q*y + r) == {p: 4, q: 5, r: 6}
a = Wild('a', exclude=[x])
e = 3*x
assert e.match(p*x) == {p: 3}
assert e.match(a*x) == {a: 3}
e = 3*x**2
assert e.match(p*x) == {p: 3*x}
assert e.match(a*x) is None
e = 3*x + 3 + 6/x
assert e.match(p*x**2 + p*x + 2*p) == {p: 3/x}
assert e.match(a*x**2 + a*x + 2*a) is None
def test_mul():
x, y, a, b, c = map(Symbol, 'xyabc')
p, q = map(Wild, 'pq')
e = 4*x
assert e.match(p*x) == {p: 4}
assert e.match(p*y) is None
assert e.match(e + p*y) == {p: 0}
e = a*x*b*c
assert e.match(p*x) == {p: a*b*c}
assert e.match(c*p*x) == {p: a*b}
e = (a + b)*(a + c)
assert e.match((p + b)*(p + c)) == {p: a}
e = x
assert e.match(p*x) == {p: 1}
e = exp(x)
assert e.match(x**p*exp(x*q)) == {p: 0, q: 1}
e = I*Poly(x, x)
assert e.match(I*p) == {p: x}
def test_mul_noncommutative():
x, y = symbols('x y')
A, B, C = symbols('A B C', commutative=False)
u, v = symbols('u v', cls=Wild)
w, z = symbols('w z', cls=Wild, commutative=False)
assert (u*v).matches(x) in ({v: x, u: 1}, {u: x, v: 1})
assert (u*v).matches(x*y) in ({v: y, u: x}, {u: y, v: x})
assert (u*v).matches(A) is None
assert (u*v).matches(A*B) is None
assert (u*v).matches(x*A) is None
assert (u*v).matches(x*y*A) is None
assert (u*v).matches(x*A*B) is None
assert (u*v).matches(x*y*A*B) is None
assert (v*w).matches(x) is None
assert (v*w).matches(x*y) is None
assert (v*w).matches(A) == {w: A, v: 1}
assert (v*w).matches(A*B) == {w: A*B, v: 1}
assert (v*w).matches(x*A) == {w: A, v: x}
assert (v*w).matches(x*y*A) == {w: A, v: x*y}
assert (v*w).matches(x*A*B) == {w: A*B, v: x}
assert (v*w).matches(x*y*A*B) == {w: A*B, v: x*y}
assert (v*w).matches(-x) is None
assert (v*w).matches(-x*y) is None
assert (v*w).matches(-A) == {w: A, v: -1}
assert (v*w).matches(-A*B) == {w: A*B, v: -1}
assert (v*w).matches(-x*A) == {w: A, v: -x}
assert (v*w).matches(-x*y*A) == {w: A, v: -x*y}
assert (v*w).matches(-x*A*B) == {w: A*B, v: -x}
assert (v*w).matches(-x*y*A*B) == {w: A*B, v: -x*y}
assert (w*z).matches(x) is None
assert (w*z).matches(x*y) is None
assert (w*z).matches(A) is None
assert (w*z).matches(A*B) == {w: A, z: B}
assert (w*z).matches(B*A) == {w: B, z: A}
assert (w*z).matches(A*B*C) in [{w: A, z: B*C}, {w: A*B, z: C}]
assert (w*z).matches(x*A) is None
assert (w*z).matches(x*y*A) is None
assert (w*z).matches(x*A*B) is None
assert (w*z).matches(x*y*A*B) is None
assert (w*A).matches(A) is None
assert (A*w*B).matches(A*B) is None
assert (u*w*z).matches(x) is None
assert (u*w*z).matches(x*y) is None
assert (u*w*z).matches(A) is None
assert (u*w*z).matches(A*B) == {u: 1, w: A, z: B}
assert (u*w*z).matches(B*A) == {u: 1, w: B, z: A}
assert (u*w*z).matches(x*A) is None
assert (u*w*z).matches(x*y*A) is None
assert (u*w*z).matches(x*A*B) == {u: x, w: A, z: B}
assert (u*w*z).matches(x*B*A) == {u: x, w: B, z: A}
assert (u*w*z).matches(x*y*A*B) == {u: x*y, w: A, z: B}
assert (u*w*z).matches(x*y*B*A) == {u: x*y, w: B, z: A}
assert (u*A).matches(x*A) == {u: x}
assert (u*A).matches(x*A*B) is None
assert (u*B).matches(x*A) is None
assert (u*A*B).matches(x*A*B) == {u: x}
assert (u*A*B).matches(x*B*A) is None
assert (u*A*B).matches(x*A) is None
assert (u*w*A).matches(x*A*B) is None
assert (u*w*B).matches(x*A*B) == {u: x, w: A}
assert (u*v*A*B).matches(x*A*B) in [{u: x, v: 1}, {v: x, u: 1}]
assert (u*v*A*B).matches(x*B*A) is None
assert (u*v*A*B).matches(u*v*A*C) is None
def test_mul_noncommutative_mismatch():
A, B, C = symbols('A B C', commutative=False)
w = symbols('w', cls=Wild, commutative=False)
assert (w*B*w).matches(A*B*A) == {w: A}
assert (w*B*w).matches(A*C*B*A*C) == {w: A*C}
assert (w*B*w).matches(A*C*B*A*B) is None
assert (w*B*w).matches(A*B*C) is None
assert (w*w*C).matches(A*B*C) is None
def test_mul_noncommutative_pow():
A, B, C = symbols('A B C', commutative=False)
w = symbols('w', cls=Wild, commutative=False)
assert (A*B*w).matches(A*B**2) == {w: B}
assert (A*(B**2)*w*(B**3)).matches(A*B**8) == {w: B**3}
assert (A*B*w*C).matches(A*(B**4)*C) == {w: B**3}
assert (A*B*(w**(-1))).matches(A*B*(C**(-1))) == {w: C}
assert (A*(B*w)**(-1)*C).matches(A*(B*C)**(-1)*C) == {w: C}
assert ((w**2)*B*C).matches((A**2)*B*C) == {w: A}
assert ((w**2)*B*(w**3)).matches((A**2)*B*(A**3)) == {w: A}
assert ((w**2)*B*(w**4)).matches((A**2)*B*(A**2)) is None
def test_complex():
a, b, c = map(Symbol, 'abc')
x, y = map(Wild, 'xy')
assert (1 + I).match(x + I) == {x: 1}
assert (a + I).match(x + I) == {x: a}
assert (2*I).match(x*I) == {x: 2}
assert (a*I).match(x*I) == {x: a}
assert (a*I).match(x*y) == {x: I, y: a}
assert (2*I).match(x*y) == {x: 2, y: I}
assert (a + b*I).match(x + y*I) == {x: a, y: b}
def test_functions():
from sympy.core.function import WildFunction
x = Symbol('x')
g = WildFunction('g')
p = Wild('p')
q = Wild('q')
f = cos(5*x)
notf = x
assert f.match(p*cos(q*x)) == {p: 1, q: 5}
assert f.match(p*g) == {p: 1, g: cos(5*x)}
assert notf.match(g) is None
@XFAIL
def test_functions_X1():
from sympy.core.function import WildFunction
x = Symbol('x')
g = WildFunction('g')
p = Wild('p')
q = Wild('q')
f = cos(5*x)
assert f.match(p*g(q*x)) == {p: 1, g: cos, q: 5}
def test_interface():
x, y = map(Symbol, 'xy')
p, q = map(Wild, 'pq')
assert (x + 1).match(p + 1) == {p: x}
assert (x*3).match(p*3) == {p: x}
assert (x**3).match(p**3) == {p: x}
assert (x*cos(y)).match(p*cos(q)) == {p: x, q: y}
assert (x*y).match(p*q) in [{p:x, q:y}, {p:y, q:x}]
assert (x + y).match(p + q) in [{p:x, q:y}, {p:y, q:x}]
assert (x*y + 1).match(p*q) in [{p:1, q:1 + x*y}, {p:1 + x*y, q:1}]
def test_derivative1():
x, y = map(Symbol, 'xy')
p, q = map(Wild, 'pq')
f = Function('f', nargs=1)
fd = Derivative(f(x), x)
assert fd.match(p) == {p: fd}
assert (fd + 1).match(p + 1) == {p: fd}
assert (fd).match(fd) == {}
assert (3*fd).match(p*fd) is not None
assert (3*fd - 1).match(p*fd + q) == {p: 3, q: -1}
def test_derivative_bug1():
f = Function("f")
x = Symbol("x")
a = Wild("a", exclude=[f, x])
b = Wild("b", exclude=[f])
pattern = a * Derivative(f(x), x, x) + b
expr = Derivative(f(x), x) + x**2
d1 = {b: x**2}
d2 = pattern.xreplace(d1).matches(expr, d1)
assert d2 is None
def test_derivative2():
f = Function("f")
x = Symbol("x")
a = Wild("a", exclude=[f, x])
b = Wild("b", exclude=[f])
e = Derivative(f(x), x)
assert e.match(Derivative(f(x), x)) == {}
assert e.match(Derivative(f(x), x, x)) is None
e = Derivative(f(x), x, x)
assert e.match(Derivative(f(x), x)) is None
assert e.match(Derivative(f(x), x, x)) == {}
e = Derivative(f(x), x) + x**2
assert e.match(a*Derivative(f(x), x) + b) == {a: 1, b: x**2}
assert e.match(a*Derivative(f(x), x, x) + b) is None
e = Derivative(f(x), x, x) + x**2
assert e.match(a*Derivative(f(x), x) + b) is None
assert e.match(a*Derivative(f(x), x, x) + b) == {a: 1, b: x**2}
def test_match_deriv_bug1():
n = Function('n')
l = Function('l')
x = Symbol('x')
p = Wild('p')
e = diff(l(x), x)/x - diff(diff(n(x), x), x)/2 - \
diff(n(x), x)**2/4 + diff(n(x), x)*diff(l(x), x)/4
e = e.subs(n(x), -l(x)).doit()
t = x*exp(-l(x))
t2 = t.diff(x, x)/t
assert e.match( (p*t2).expand() ) == {p: Rational(-1, 2)}
def test_match_bug2():
x, y = map(Symbol, 'xy')
p, q, r = map(Wild, 'pqr')
res = (x + y).match(p + q + r)
assert (p + q + r).subs(res) == x + y
def test_match_bug3():
x, a, b = map(Symbol, 'xab')
p = Wild('p')
assert (b*x*exp(a*x)).match(x*exp(p*x)) is None
def test_match_bug4():
x = Symbol('x')
p = Wild('p')
e = x
assert e.match(-p*x) == {p: -1}
def test_match_bug5():
x = Symbol('x')
p = Wild('p')
e = -x
assert e.match(-p*x) == {p: 1}
def test_match_bug6():
x = Symbol('x')
p = Wild('p')
e = x
assert e.match(3*p*x) == {p: Rational(1)/3}
def test_match_polynomial():
x = Symbol('x')
a = Wild('a', exclude=[x])
b = Wild('b', exclude=[x])
c = Wild('c', exclude=[x])
d = Wild('d', exclude=[x])
eq = 4*x**3 + 3*x**2 + 2*x + 1
pattern = a*x**3 + b*x**2 + c*x + d
assert eq.match(pattern) == {a: 4, b: 3, c: 2, d: 1}
assert (eq - 3*x**2).match(pattern) == {a: 4, b: 0, c: 2, d: 1}
assert (x + sqrt(2) + 3).match(a + b*x + c*x**2) == \
{b: 1, a: sqrt(2) + 3, c: 0}
def test_exclude():
x, y, a = map(Symbol, 'xya')
p = Wild('p', exclude=[1, x])
q = Wild('q')
r = Wild('r', exclude=[sin, y])
assert sin(x).match(r) is None
assert cos(y).match(r) is None
e = 3*x**2 + y*x + a
assert e.match(p*x**2 + q*x + r) == {p: 3, q: y, r: a}
e = x + 1
assert e.match(x + p) is None
assert e.match(p + 1) is None
assert e.match(x + 1 + p) == {p: 0}
e = cos(x) + 5*sin(y)
assert e.match(r) is None
assert e.match(cos(y) + r) is None
assert e.match(r + p*sin(q)) == {r: cos(x), p: 5, q: y}
def test_floats():
a, b = map(Wild, 'ab')
e = cos(0.12345, evaluate=False)**2
r = e.match(a*cos(b)**2)
assert r == {a: 1, b: Float(0.12345)}
def test_Derivative_bug1():
f = Function("f")
x = abc.x
a = Wild("a", exclude=[f(x)])
b = Wild("b", exclude=[f(x)])
eq = f(x).diff(x)
assert eq.match(a*Derivative(f(x), x) + b) == {a: 1, b: 0}
def test_match_wild_wild():
p = Wild('p')
q = Wild('q')
r = Wild('r')
assert p.match(q + r) in [ {q: p, r: 0}, {q: 0, r: p} ]
assert p.match(q*r) in [ {q: p, r: 1}, {q: 1, r: p} ]
p = Wild('p')
q = Wild('q', exclude=[p])
r = Wild('r')
assert p.match(q + r) == {q: 0, r: p}
assert p.match(q*r) == {q: 1, r: p}
p = Wild('p')
q = Wild('q', exclude=[p])
r = Wild('r', exclude=[p])
assert p.match(q + r) is None
assert p.match(q*r) is None
def test__combine_inverse():
x, y = symbols("x y")
assert Mul._combine_inverse(x*I*y, x*I) == y
assert Mul._combine_inverse(x*x**(1 + y), x**(1 + y)) == x
assert Mul._combine_inverse(x*I*y, y*I) == x
assert Mul._combine_inverse(oo*I*y, y*I) is oo
assert Mul._combine_inverse(oo*I*y, oo*I) == y
assert Mul._combine_inverse(oo*I*y, oo*I) == y
assert Mul._combine_inverse(oo*y, -oo) == -y
assert Mul._combine_inverse(-oo*y, oo) == -y
assert Add._combine_inverse(oo, oo) is S.Zero
assert Add._combine_inverse(oo*I, oo*I) is S.Zero
assert Add._combine_inverse(x*oo, x*oo) is S.Zero
assert Add._combine_inverse(-x*oo, -x*oo) is S.Zero
assert Add._combine_inverse((x - oo)*(x + oo), -oo)
def test_issue_3773():
x = symbols('x')
z, phi, r = symbols('z phi r')
c, A, B, N = symbols('c A B N', cls=Wild)
l = Wild('l', exclude=(0,))
eq = z * sin(2*phi) * r**7
matcher = c * sin(phi*N)**l * r**A * log(r)**B
assert eq.match(matcher) == {c: z, l: 1, N: 2, A: 7, B: 0}
assert (-eq).match(matcher) == {c: -z, l: 1, N: 2, A: 7, B: 0}
assert (x*eq).match(matcher) == {c: x*z, l: 1, N: 2, A: 7, B: 0}
assert (-7*x*eq).match(matcher) == {c: -7*x*z, l: 1, N: 2, A: 7, B: 0}
matcher = c*sin(phi*N)**l * r**A
assert eq.match(matcher) == {c: z, l: 1, N: 2, A: 7}
assert (-eq).match(matcher) == {c: -z, l: 1, N: 2, A: 7}
assert (x*eq).match(matcher) == {c: x*z, l: 1, N: 2, A: 7}
assert (-7*x*eq).match(matcher) == {c: -7*x*z, l: 1, N: 2, A: 7}
def test_issue_3883():
from sympy.abc import gamma, mu, x
f = (-gamma * (x - mu)**2 - log(gamma) + log(2*pi))/2
a, b, c = symbols('a b c', cls=Wild, exclude=(gamma,))
assert f.match(a * log(gamma) + b * gamma + c) == \
{a: Rational(-1, 2), b: -(mu - x)**2/2, c: log(2*pi)/2}
assert f.expand().collect(gamma).match(a * log(gamma) + b * gamma + c) == \
{a: Rational(-1, 2), b: (-(x - mu)**2/2).expand(), c: (log(2*pi)/2).expand()}
g1 = Wild('g1', exclude=[gamma])
g2 = Wild('g2', exclude=[gamma])
g3 = Wild('g3', exclude=[gamma])
assert f.expand().match(g1 * log(gamma) + g2 * gamma + g3) == \
{g3: log(2)/2 + log(pi)/2, g1: Rational(-1, 2), g2: -mu**2/2 + mu*x - x**2/2}
def test_issue_4418():
x = Symbol('x')
a, b, c = symbols('a b c', cls=Wild, exclude=(x,))
f, g = symbols('f g', cls=Function)
eq = diff(g(x)*f(x).diff(x), x)
assert eq.match(
g(x).diff(x)*f(x).diff(x) + g(x)*f(x).diff(x, x) + c) == {c: 0}
assert eq.match(a*g(x).diff(
x)*f(x).diff(x) + b*g(x)*f(x).diff(x, x) + c) == {a: 1, b: 1, c: 0}
def test_issue_4700():
f = Function('f')
x = Symbol('x')
a, b = symbols('a b', cls=Wild, exclude=(f(x),))
p = a*f(x) + b
eq1 = sin(x)
eq2 = f(x) + sin(x)
eq3 = f(x) + x + sin(x)
eq4 = x + sin(x)
assert eq1.match(p) == {a: 0, b: sin(x)}
assert eq2.match(p) == {a: 1, b: sin(x)}
assert eq3.match(p) == {a: 1, b: x + sin(x)}
assert eq4.match(p) == {a: 0, b: x + sin(x)}
def test_issue_5168():
a, b, c = symbols('a b c', cls=Wild)
x = Symbol('x')
f = Function('f')
assert x.match(a) == {a: x}
assert x.match(a*f(x)**c) == {a: x, c: 0}
assert x.match(a*b) == {a: 1, b: x}
assert x.match(a*b*f(x)**c) == {a: 1, b: x, c: 0}
assert (-x).match(a) == {a: -x}
assert (-x).match(a*f(x)**c) == {a: -x, c: 0}
assert (-x).match(a*b) == {a: -1, b: x}
assert (-x).match(a*b*f(x)**c) == {a: -1, b: x, c: 0}
assert (2*x).match(a) == {a: 2*x}
assert (2*x).match(a*f(x)**c) == {a: 2*x, c: 0}
assert (2*x).match(a*b) == {a: 2, b: x}
assert (2*x).match(a*b*f(x)**c) == {a: 2, b: x, c: 0}
assert (-2*x).match(a) == {a: -2*x}
assert (-2*x).match(a*f(x)**c) == {a: -2*x, c: 0}
assert (-2*x).match(a*b) == {a: -2, b: x}
assert (-2*x).match(a*b*f(x)**c) == {a: -2, b: x, c: 0}
def test_issue_4559():
x = Symbol('x')
e = Symbol('e')
w = Wild('w', exclude=[x])
y = Wild('y')
# this is as it should be
assert (3/x).match(w/y) == {w: 3, y: x}
assert (3*x).match(w*y) == {w: 3, y: x}
assert (x/3).match(y/w) == {w: 3, y: x}
assert (3*x).match(y/w) == {w: S.One/3, y: x}
assert (3*x).match(y/w) == {w: Rational(1, 3), y: x}
# these could be allowed to fail
assert (x/3).match(w/y) == {w: S.One/3, y: 1/x}
assert (3*x).match(w/y) == {w: 3, y: 1/x}
assert (3/x).match(w*y) == {w: 3, y: 1/x}
# Note that solve will give
# multiple roots but match only gives one:
#
# >>> solve(x**r-y**2,y)
# [-x**(r/2), x**(r/2)]
r = Symbol('r', rational=True)
assert (x**r).match(y**2) == {y: x**(r/2)}
assert (x**e).match(y**2) == {y: sqrt(x**e)}
# since (x**i = y) -> x = y**(1/i) where i is an integer
# the following should also be valid as long as y is not
# zero when i is negative.
a = Wild('a')
e = S.Zero
assert e.match(a) == {a: e}
assert e.match(1/a) is None
assert e.match(a**.3) is None
e = S(3)
assert e.match(1/a) == {a: 1/e}
assert e.match(1/a**2) == {a: 1/sqrt(e)}
e = pi
assert e.match(1/a) == {a: 1/e}
assert e.match(1/a**2) == {a: 1/sqrt(e)}
assert (-e).match(sqrt(a)) is None
assert (-e).match(a**2) == {a: I*sqrt(pi)}
# The pattern matcher doesn't know how to handle (x - a)**2 == (a - x)**2. To
# avoid ambiguity in actual applications, don't put a coefficient (including a
# minus sign) in front of a wild.
@XFAIL
def test_issue_4883():
a = Wild('a')
x = Symbol('x')
e = [i**2 for i in (x - 2, 2 - x)]
p = [i**2 for i in (x - a, a- x)]
for eq in e:
for pat in p:
assert eq.match(pat) == {a: 2}
def test_issue_4319():
x, y = symbols('x y')
p = -x*(S.One/8 - y)
ans = {S.Zero, y - S.One/8}
def ok(pat):
assert set(p.match(pat).values()) == ans
ok(Wild("coeff", exclude=[x])*x + Wild("rest"))
ok(Wild("w", exclude=[x])*x + Wild("rest"))
ok(Wild("coeff", exclude=[x])*x + Wild("rest"))
ok(Wild("w", exclude=[x])*x + Wild("rest"))
ok(Wild("e", exclude=[x])*x + Wild("rest"))
ok(Wild("ress", exclude=[x])*x + Wild("rest"))
ok(Wild("resu", exclude=[x])*x + Wild("rest"))
def test_issue_3778():
p, c, q = symbols('p c q', cls=Wild)
x = Symbol('x')
assert (sin(x)**2).match(sin(p)*sin(q)*c) == {q: x, c: 1, p: x}
assert (2*sin(x)).match(sin(p) + sin(q) + c) == {q: x, c: 0, p: x}
def test_issue_6103():
x = Symbol('x')
a = Wild('a')
assert (-I*x*oo).match(I*a*oo) == {a: -x}
def test_issue_3539():
a = Wild('a')
x = Symbol('x')
assert (x - 2).match(a - x) is None
assert (6/x).match(a*x) is None
assert (6/x**2).match(a/x) == {a: 6/x}
def test_gh_issue_2711():
x = Symbol('x')
f = meijerg(((), ()), ((0,), ()), x)
a = Wild('a')
b = Wild('b')
assert f.find(a) == {(S.Zero,), ((), ()), ((S.Zero,), ()), x, S.Zero,
(), meijerg(((), ()), ((S.Zero,), ()), x)}
assert f.find(a + b) == \
{meijerg(((), ()), ((S.Zero,), ()), x), x, S.Zero}
assert f.find(a**2) == {meijerg(((), ()), ((S.Zero,), ()), x), x}
def test_match_issue_17397():
f = Function("f")
x = Symbol("x")
a3 = Wild('a3', exclude=[f(x), f(x).diff(x), f(x).diff(x, 2)])
b3 = Wild('b3', exclude=[f(x), f(x).diff(x), f(x).diff(x, 2)])
c3 = Wild('c3', exclude=[f(x), f(x).diff(x), f(x).diff(x, 2)])
deq = a3*(f(x).diff(x, 2)) + b3*f(x).diff(x) + c3*f(x)
eq = (x-2)**2*(f(x).diff(x, 2)) + (x-2)*(f(x).diff(x)) + ((x-2)**2 - 4)*f(x)
r = collect(eq, [f(x).diff(x, 2), f(x).diff(x), f(x)]).match(deq)
assert r == {a3: (x - 2)**2, c3: (x - 2)**2 - 4, b3: x - 2}
eq =x*f(x) + x*Derivative(f(x), (x, 2)) - 4*f(x) + Derivative(f(x), x) \
- 4*Derivative(f(x), (x, 2)) - 2*Derivative(f(x), x)/x + 4*Derivative(f(x), (x, 2))/x
r = collect(eq, [f(x).diff(x, 2), f(x).diff(x), f(x)]).match(deq)
assert r == {a3: x - 4 + 4/x, b3: 1 - 2/x, c3: x - 4}
def test_match_terms():
X, Y = map(Wild, "XY")
x, y, z = symbols('x y z')
assert (5*y - x).match(5*X - Y) == {X: y, Y: x}
# 15907
assert (x + (y - 1)*z).match(x + X*z) == {X: y - 1}
def test_match_bound():
V, W = map(Wild, "VW")
x, y = symbols('x y')
assert Sum(x, (x, 1, 2)).match(Sum(y, (y, 1, W))) == {W: 2}
assert Sum(x, (x, 1, 2)).match(Sum(V, (V, 1, W))) == {W: 2, V:x}
assert Sum(x, (x, 1, 2)).match(Sum(V, (V, 1, 2))) == {V:x}
|
7a7586e2b7fe40167199f77b24b404415ebd57b21b6e27216555aebff9dd7817 | from sympy import (Basic, Symbol, sin, cos, atan, exp, sqrt, Rational,
Float, re, pi, sympify, Add, Mul, Pow, Mod, I, log, S, Max, symbols,
oo, zoo, Integer, sign, im, nan, Dummy, factorial, comp, floor, Poly,
FiniteSet
)
from sympy.core.parameters import distribute
from sympy.core.expr import unchanged
from sympy.utilities.iterables import cartes
from sympy.testing.pytest import XFAIL, raises, warns_deprecated_sympy
from sympy.testing.randtest import verify_numerically
a, c, x, y, z = symbols('a,c,x,y,z')
b = Symbol("b", positive=True)
def same_and_same_prec(a, b):
# stricter matching for Floats
return a == b and a._prec == b._prec
def test_bug1():
assert re(x) != x
x.series(x, 0, 1)
assert re(x) != x
def test_Symbol():
e = a*b
assert e == a*b
assert a*b*b == a*b**2
assert a*b*b + c == c + a*b**2
assert a*b*b - c == -c + a*b**2
x = Symbol('x', complex=True, real=False)
assert x.is_imaginary is None # could be I or 1 + I
x = Symbol('x', complex=True, imaginary=False)
assert x.is_real is None # could be 1 or 1 + I
x = Symbol('x', real=True)
assert x.is_complex
x = Symbol('x', imaginary=True)
assert x.is_complex
x = Symbol('x', real=False, imaginary=False)
assert x.is_complex is None # might be a non-number
def test_arit0():
p = Rational(5)
e = a*b
assert e == a*b
e = a*b + b*a
assert e == 2*a*b
e = a*b + b*a + a*b + p*b*a
assert e == 8*a*b
e = a*b + b*a + a*b + p*b*a + a
assert e == a + 8*a*b
e = a + a
assert e == 2*a
e = a + b + a
assert e == b + 2*a
e = a + b*b + a + b*b
assert e == 2*a + 2*b**2
e = a + Rational(2) + b*b + a + b*b + p
assert e == 7 + 2*a + 2*b**2
e = (a + b*b + a + b*b)*p
assert e == 5*(2*a + 2*b**2)
e = (a*b*c + c*b*a + b*a*c)*p
assert e == 15*a*b*c
e = (a*b*c + c*b*a + b*a*c)*p - Rational(15)*a*b*c
assert e == Rational(0)
e = Rational(50)*(a - a)
assert e == Rational(0)
e = b*a - b - a*b + b
assert e == Rational(0)
e = a*b + c**p
assert e == a*b + c**5
e = a/b
assert e == a*b**(-1)
e = a*2*2
assert e == 4*a
e = 2 + a*2/2
assert e == 2 + a
e = 2 - a - 2
assert e == -a
e = 2*a*2
assert e == 4*a
e = 2/a/2
assert e == a**(-1)
e = 2**a**2
assert e == 2**(a**2)
e = -(1 + a)
assert e == -1 - a
e = S.Half*(1 + a)
assert e == S.Half + a/2
def test_div():
e = a/b
assert e == a*b**(-1)
e = a/b + c/2
assert e == a*b**(-1) + Rational(1)/2*c
e = (1 - b)/(b - 1)
assert e == (1 + -b)*((-1) + b)**(-1)
def test_pow():
n1 = Rational(1)
n2 = Rational(2)
n5 = Rational(5)
e = a*a
assert e == a**2
e = a*a*a
assert e == a**3
e = a*a*a*a**Rational(6)
assert e == a**9
e = a*a*a*a**Rational(6) - a**Rational(9)
assert e == Rational(0)
e = a**(b - b)
assert e == Rational(1)
e = (a + Rational(1) - a)**b
assert e == Rational(1)
e = (a + b + c)**n2
assert e == (a + b + c)**2
assert e.expand() == 2*b*c + 2*a*c + 2*a*b + a**2 + c**2 + b**2
e = (a + b)**n2
assert e == (a + b)**2
assert e.expand() == 2*a*b + a**2 + b**2
e = (a + b)**(n1/n2)
assert e == sqrt(a + b)
assert e.expand() == sqrt(a + b)
n = n5**(n1/n2)
assert n == sqrt(5)
e = n*a*b - n*b*a
assert e == Rational(0)
e = n*a*b + n*b*a
assert e == 2*a*b*sqrt(5)
assert e.diff(a) == 2*b*sqrt(5)
assert e.diff(a) == 2*b*sqrt(5)
e = a/b**2
assert e == a*b**(-2)
assert sqrt(2*(1 + sqrt(2))) == (2*(1 + 2**S.Half))**S.Half
x = Symbol('x')
y = Symbol('y')
assert ((x*y)**3).expand() == y**3 * x**3
assert ((x*y)**-3).expand() == y**-3 * x**-3
assert (x**5*(3*x)**(3)).expand() == 27 * x**8
assert (x**5*(-3*x)**(3)).expand() == -27 * x**8
assert (x**5*(3*x)**(-3)).expand() == x**2 * Rational(1, 27)
assert (x**5*(-3*x)**(-3)).expand() == x**2 * Rational(-1, 27)
# expand_power_exp
assert (x**(y**(x + exp(x + y)) + z)).expand(deep=False) == \
x**z*x**(y**(x + exp(x + y)))
assert (x**(y**(x + exp(x + y)) + z)).expand() == \
x**z*x**(y**x*y**(exp(x)*exp(y)))
n = Symbol('n', even=False)
k = Symbol('k', even=True)
o = Symbol('o', odd=True)
assert unchanged(Pow, -1, x)
assert unchanged(Pow, -1, n)
assert (-2)**k == 2**k
assert (-1)**k == 1
assert (-1)**o == -1
def test_pow2():
# x**(2*y) is always (x**y)**2 but is only (x**2)**y if
# x.is_positive or y.is_integer
# let x = 1 to see why the following are not true.
assert (-x)**Rational(2, 3) != x**Rational(2, 3)
assert (-x)**Rational(5, 7) != -x**Rational(5, 7)
assert ((-x)**2)**Rational(1, 3) != ((-x)**Rational(1, 3))**2
assert sqrt(x**2) != x
def test_pow3():
assert sqrt(2)**3 == 2 * sqrt(2)
assert sqrt(2)**3 == sqrt(8)
def test_mod_pow():
for s, t, u, v in [(4, 13, 497, 445), (4, -3, 497, 365),
(3.2, 2.1, 1.9, 0.1031015682350942), (S(3)/2, 5, S(5)/6, S(3)/32)]:
assert pow(S(s), t, u) == v
assert pow(S(s), S(t), u) == v
assert pow(S(s), t, S(u)) == v
assert pow(S(s), S(t), S(u)) == v
assert pow(S(2), S(10000000000), S(3)) == 1
assert pow(x, y, z) == x**y%z
raises(TypeError, lambda: pow(S(4), "13", 497))
raises(TypeError, lambda: pow(S(4), 13, "497"))
def test_pow_E():
assert 2**(y/log(2)) == S.Exp1**y
assert 2**(y/log(2)/3) == S.Exp1**(y/3)
assert 3**(1/log(-3)) != S.Exp1
assert (3 + 2*I)**(1/(log(-3 - 2*I) + I*pi)) == S.Exp1
assert (4 + 2*I)**(1/(log(-4 - 2*I) + I*pi)) == S.Exp1
assert (3 + 2*I)**(1/(log(-3 - 2*I, 3)/2 + I*pi/log(3)/2)) == 9
assert (3 + 2*I)**(1/(log(3 + 2*I, 3)/2)) == 9
# every time tests are run they will affirm with a different random
# value that this identity holds
while 1:
b = x._random()
r, i = b.as_real_imag()
if i:
break
assert verify_numerically(b**(1/(log(-b) + sign(i)*I*pi).n()), S.Exp1)
def test_pow_issue_3516():
assert 4**Rational(1, 4) == sqrt(2)
def test_pow_im():
for m in (-2, -1, 2):
for d in (3, 4, 5):
b = m*I
for i in range(1, 4*d + 1):
e = Rational(i, d)
assert (b**e - b.n()**e.n()).n(2, chop=1e-10) == 0
e = Rational(7, 3)
assert (2*x*I)**e == 4*2**Rational(1, 3)*(I*x)**e # same as Wolfram Alpha
im = symbols('im', imaginary=True)
assert (2*im*I)**e == 4*2**Rational(1, 3)*(I*im)**e
args = [I, I, I, I, 2]
e = Rational(1, 3)
ans = 2**e
assert Mul(*args, evaluate=False)**e == ans
assert Mul(*args)**e == ans
args = [I, I, I, 2]
e = Rational(1, 3)
ans = 2**e*(-I)**e
assert Mul(*args, evaluate=False)**e == ans
assert Mul(*args)**e == ans
args.append(-3)
ans = (6*I)**e
assert Mul(*args, evaluate=False)**e == ans
assert Mul(*args)**e == ans
args.append(-1)
ans = (-6*I)**e
assert Mul(*args, evaluate=False)**e == ans
assert Mul(*args)**e == ans
args = [I, I, 2]
e = Rational(1, 3)
ans = (-2)**e
assert Mul(*args, evaluate=False)**e == ans
assert Mul(*args)**e == ans
args.append(-3)
ans = (6)**e
assert Mul(*args, evaluate=False)**e == ans
assert Mul(*args)**e == ans
args.append(-1)
ans = (-6)**e
assert Mul(*args, evaluate=False)**e == ans
assert Mul(*args)**e == ans
assert Mul(Pow(-1, Rational(3, 2), evaluate=False), I, I) == I
assert Mul(I*Pow(I, S.Half, evaluate=False)) == sqrt(I)*I
def test_real_mul():
assert Float(0) * pi * x == 0
assert set((Float(1) * pi * x).args) == {Float(1), pi, x}
def test_ncmul():
A = Symbol("A", commutative=False)
B = Symbol("B", commutative=False)
C = Symbol("C", commutative=False)
assert A*B != B*A
assert A*B*C != C*B*A
assert A*b*B*3*C == 3*b*A*B*C
assert A*b*B*3*C != 3*b*B*A*C
assert A*b*B*3*C == 3*A*B*C*b
assert A + B == B + A
assert (A + B)*C != C*(A + B)
assert C*(A + B)*C != C*C*(A + B)
assert A*A == A**2
assert (A + B)*(A + B) == (A + B)**2
assert A**-1 * A == 1
assert A/A == 1
assert A/(A**2) == 1/A
assert A/(1 + A) == A/(1 + A)
assert set((A + B + 2*(A + B)).args) == \
{A, B, 2*(A + B)}
def test_mul_add_identity():
m = Mul(1, 2)
assert isinstance(m, Rational) and m.p == 2 and m.q == 1
m = Mul(1, 2, evaluate=False)
assert isinstance(m, Mul) and m.args == (1, 2)
m = Mul(0, 1)
assert m is S.Zero
m = Mul(0, 1, evaluate=False)
assert isinstance(m, Mul) and m.args == (0, 1)
m = Add(0, 1)
assert m is S.One
m = Add(0, 1, evaluate=False)
assert isinstance(m, Add) and m.args == (0, 1)
def test_ncpow():
x = Symbol('x', commutative=False)
y = Symbol('y', commutative=False)
z = Symbol('z', commutative=False)
a = Symbol('a')
b = Symbol('b')
c = Symbol('c')
assert (x**2)*(y**2) != (y**2)*(x**2)
assert (x**-2)*y != y*(x**2)
assert 2**x*2**y != 2**(x + y)
assert 2**x*2**y*2**z != 2**(x + y + z)
assert 2**x*2**(2*x) == 2**(3*x)
assert 2**x*2**(2*x)*2**x == 2**(4*x)
assert exp(x)*exp(y) != exp(y)*exp(x)
assert exp(x)*exp(y)*exp(z) != exp(y)*exp(x)*exp(z)
assert exp(x)*exp(y)*exp(z) != exp(x + y + z)
assert x**a*x**b != x**(a + b)
assert x**a*x**b*x**c != x**(a + b + c)
assert x**3*x**4 == x**7
assert x**3*x**4*x**2 == x**9
assert x**a*x**(4*a) == x**(5*a)
assert x**a*x**(4*a)*x**a == x**(6*a)
def test_powerbug():
x = Symbol("x")
assert x**1 != (-x)**1
assert x**2 == (-x)**2
assert x**3 != (-x)**3
assert x**4 == (-x)**4
assert x**5 != (-x)**5
assert x**6 == (-x)**6
assert x**128 == (-x)**128
assert x**129 != (-x)**129
assert (2*x)**2 == (-2*x)**2
def test_Mul_doesnt_expand_exp():
x = Symbol('x')
y = Symbol('y')
assert unchanged(Mul, exp(x), exp(y))
assert unchanged(Mul, 2**x, 2**y)
assert x**2*x**3 == x**5
assert 2**x*3**x == 6**x
assert x**(y)*x**(2*y) == x**(3*y)
assert sqrt(2)*sqrt(2) == 2
assert 2**x*2**(2*x) == 2**(3*x)
assert sqrt(2)*2**Rational(1, 4)*5**Rational(3, 4) == 10**Rational(3, 4)
assert (x**(-log(5)/log(3))*x)/(x*x**( - log(5)/log(3))) == sympify(1)
def test_Mul_is_integer():
k = Symbol('k', integer=True)
n = Symbol('n', integer=True)
nr = Symbol('nr', rational=False)
nz = Symbol('nz', integer=True, zero=False)
nze = Symbol('nze', even=True, zero=False)
e = Symbol('e', even=True)
o = Symbol('o', odd=True)
i2 = Symbol('2', prime=True, even=True)
assert (k/3).is_integer is None
assert (nz/3).is_integer is None
assert (nr/3).is_integer is False
assert (x*k*n).is_integer is None
assert (e/o).is_integer is None
assert (o/e).is_integer is False
assert (o/i2).is_integer is False
assert Mul(o, 1/o, evaluate=False).is_integer is True
assert Mul(k, 1/k, evaluate=False).is_integer is None
assert Mul(nze, 1/nze, evaluate=False).is_integer is True
assert Mul(2., S.Half, evaluate=False).is_integer is False
s = 2**2**2**Pow(2, 1000, evaluate=False)
m = Mul(s, s, evaluate=False)
assert m.is_integer
def test_Add_Mul_is_integer():
x = Symbol('x')
k = Symbol('k', integer=True)
n = Symbol('n', integer=True)
nk = Symbol('nk', integer=False)
nr = Symbol('nr', rational=False)
nz = Symbol('nz', integer=True, zero=False)
assert (-nk).is_integer is None
assert (-nr).is_integer is False
assert (2*k).is_integer is True
assert (-k).is_integer is True
assert (k + nk).is_integer is False
assert (k + n).is_integer is True
assert (k + x).is_integer is None
assert (k + n*x).is_integer is None
assert (k + n/3).is_integer is None
assert (k + nz/3).is_integer is None
assert (k + nr/3).is_integer is False
assert ((1 + sqrt(3))*(-sqrt(3) + 1)).is_integer is not False
assert (1 + (1 + sqrt(3))*(-sqrt(3) + 1)).is_integer is not False
def test_Add_Mul_is_finite():
x = Symbol('x', extended_real=True, finite=False)
assert sin(x).is_finite is True
assert (x*sin(x)).is_finite is None
assert (x*atan(x)).is_finite is False
assert (1024*sin(x)).is_finite is True
assert (sin(x)*exp(x)).is_finite is None
assert (sin(x)*cos(x)).is_finite is True
assert (x*sin(x)*exp(x)).is_finite is None
assert (sin(x) - 67).is_finite is True
assert (sin(x) + exp(x)).is_finite is not True
assert (1 + x).is_finite is False
assert (1 + x**2 + (1 + x)*(1 - x)).is_finite is None
assert (sqrt(2)*(1 + x)).is_finite is False
assert (sqrt(2)*(1 + x)*(1 - x)).is_finite is False
def test_Mul_is_even_odd():
x = Symbol('x', integer=True)
y = Symbol('y', integer=True)
k = Symbol('k', odd=True)
n = Symbol('n', odd=True)
m = Symbol('m', even=True)
assert (2*x).is_even is True
assert (2*x).is_odd is False
assert (3*x).is_even is None
assert (3*x).is_odd is None
assert (k/3).is_integer is None
assert (k/3).is_even is None
assert (k/3).is_odd is None
assert (2*n).is_even is True
assert (2*n).is_odd is False
assert (2*m).is_even is True
assert (2*m).is_odd is False
assert (-n).is_even is False
assert (-n).is_odd is True
assert (k*n).is_even is False
assert (k*n).is_odd is True
assert (k*m).is_even is True
assert (k*m).is_odd is False
assert (k*n*m).is_even is True
assert (k*n*m).is_odd is False
assert (k*m*x).is_even is True
assert (k*m*x).is_odd is False
# issue 6791:
assert (x/2).is_integer is None
assert (k/2).is_integer is False
assert (m/2).is_integer is True
assert (x*y).is_even is None
assert (x*x).is_even is None
assert (x*(x + k)).is_even is True
assert (x*(x + m)).is_even is None
assert (x*y).is_odd is None
assert (x*x).is_odd is None
assert (x*(x + k)).is_odd is False
assert (x*(x + m)).is_odd is None
@XFAIL
def test_evenness_in_ternary_integer_product_with_odd():
# Tests that oddness inference is independent of term ordering.
# Term ordering at the point of testing depends on SymPy's symbol order, so
# we try to force a different order by modifying symbol names.
x = Symbol('x', integer=True)
y = Symbol('y', integer=True)
k = Symbol('k', odd=True)
assert (x*y*(y + k)).is_even is True
assert (y*x*(x + k)).is_even is True
def test_evenness_in_ternary_integer_product_with_even():
x = Symbol('x', integer=True)
y = Symbol('y', integer=True)
m = Symbol('m', even=True)
assert (x*y*(y + m)).is_even is None
@XFAIL
def test_oddness_in_ternary_integer_product_with_odd():
# Tests that oddness inference is independent of term ordering.
# Term ordering at the point of testing depends on SymPy's symbol order, so
# we try to force a different order by modifying symbol names.
x = Symbol('x', integer=True)
y = Symbol('y', integer=True)
k = Symbol('k', odd=True)
assert (x*y*(y + k)).is_odd is False
assert (y*x*(x + k)).is_odd is False
def test_oddness_in_ternary_integer_product_with_even():
x = Symbol('x', integer=True)
y = Symbol('y', integer=True)
m = Symbol('m', even=True)
assert (x*y*(y + m)).is_odd is None
def test_Mul_is_rational():
x = Symbol('x')
n = Symbol('n', integer=True)
m = Symbol('m', integer=True, nonzero=True)
assert (n/m).is_rational is True
assert (x/pi).is_rational is None
assert (x/n).is_rational is None
assert (m/pi).is_rational is False
r = Symbol('r', rational=True)
assert (pi*r).is_rational is None
# issue 8008
z = Symbol('z', zero=True)
i = Symbol('i', imaginary=True)
assert (z*i).is_rational is True
bi = Symbol('i', imaginary=True, finite=True)
assert (z*bi).is_zero is True
def test_Add_is_rational():
x = Symbol('x')
n = Symbol('n', rational=True)
m = Symbol('m', rational=True)
assert (n + m).is_rational is True
assert (x + pi).is_rational is None
assert (x + n).is_rational is None
assert (n + pi).is_rational is False
def test_Add_is_even_odd():
x = Symbol('x', integer=True)
k = Symbol('k', odd=True)
n = Symbol('n', odd=True)
m = Symbol('m', even=True)
assert (k + 7).is_even is True
assert (k + 7).is_odd is False
assert (-k + 7).is_even is True
assert (-k + 7).is_odd is False
assert (k - 12).is_even is False
assert (k - 12).is_odd is True
assert (-k - 12).is_even is False
assert (-k - 12).is_odd is True
assert (k + n).is_even is True
assert (k + n).is_odd is False
assert (k + m).is_even is False
assert (k + m).is_odd is True
assert (k + n + m).is_even is True
assert (k + n + m).is_odd is False
assert (k + n + x + m).is_even is None
assert (k + n + x + m).is_odd is None
def test_Mul_is_negative_positive():
x = Symbol('x', real=True)
y = Symbol('y', extended_real=False, complex=True)
z = Symbol('z', zero=True)
e = 2*z
assert e.is_Mul and e.is_positive is False and e.is_negative is False
neg = Symbol('neg', negative=True)
pos = Symbol('pos', positive=True)
nneg = Symbol('nneg', nonnegative=True)
npos = Symbol('npos', nonpositive=True)
assert neg.is_negative is True
assert (-neg).is_negative is False
assert (2*neg).is_negative is True
assert (2*pos)._eval_is_extended_negative() is False
assert (2*pos).is_negative is False
assert pos.is_negative is False
assert (-pos).is_negative is True
assert (2*pos).is_negative is False
assert (pos*neg).is_negative is True
assert (2*pos*neg).is_negative is True
assert (-pos*neg).is_negative is False
assert (pos*neg*y).is_negative is False # y.is_real=F; !real -> !neg
assert nneg.is_negative is False
assert (-nneg).is_negative is None
assert (2*nneg).is_negative is False
assert npos.is_negative is None
assert (-npos).is_negative is False
assert (2*npos).is_negative is None
assert (nneg*npos).is_negative is None
assert (neg*nneg).is_negative is None
assert (neg*npos).is_negative is False
assert (pos*nneg).is_negative is False
assert (pos*npos).is_negative is None
assert (npos*neg*nneg).is_negative is False
assert (npos*pos*nneg).is_negative is None
assert (-npos*neg*nneg).is_negative is None
assert (-npos*pos*nneg).is_negative is False
assert (17*npos*neg*nneg).is_negative is False
assert (17*npos*pos*nneg).is_negative is None
assert (neg*npos*pos*nneg).is_negative is False
assert (x*neg).is_negative is None
assert (nneg*npos*pos*x*neg).is_negative is None
assert neg.is_positive is False
assert (-neg).is_positive is True
assert (2*neg).is_positive is False
assert pos.is_positive is True
assert (-pos).is_positive is False
assert (2*pos).is_positive is True
assert (pos*neg).is_positive is False
assert (2*pos*neg).is_positive is False
assert (-pos*neg).is_positive is True
assert (-pos*neg*y).is_positive is False # y.is_real=F; !real -> !neg
assert nneg.is_positive is None
assert (-nneg).is_positive is False
assert (2*nneg).is_positive is None
assert npos.is_positive is False
assert (-npos).is_positive is None
assert (2*npos).is_positive is False
assert (nneg*npos).is_positive is False
assert (neg*nneg).is_positive is False
assert (neg*npos).is_positive is None
assert (pos*nneg).is_positive is None
assert (pos*npos).is_positive is False
assert (npos*neg*nneg).is_positive is None
assert (npos*pos*nneg).is_positive is False
assert (-npos*neg*nneg).is_positive is False
assert (-npos*pos*nneg).is_positive is None
assert (17*npos*neg*nneg).is_positive is None
assert (17*npos*pos*nneg).is_positive is False
assert (neg*npos*pos*nneg).is_positive is None
assert (x*neg).is_positive is None
assert (nneg*npos*pos*x*neg).is_positive is None
def test_Mul_is_negative_positive_2():
a = Symbol('a', nonnegative=True)
b = Symbol('b', nonnegative=True)
c = Symbol('c', nonpositive=True)
d = Symbol('d', nonpositive=True)
assert (a*b).is_nonnegative is True
assert (a*b).is_negative is False
assert (a*b).is_zero is None
assert (a*b).is_positive is None
assert (c*d).is_nonnegative is True
assert (c*d).is_negative is False
assert (c*d).is_zero is None
assert (c*d).is_positive is None
assert (a*c).is_nonpositive is True
assert (a*c).is_positive is False
assert (a*c).is_zero is None
assert (a*c).is_negative is None
def test_Mul_is_nonpositive_nonnegative():
x = Symbol('x', real=True)
k = Symbol('k', negative=True)
n = Symbol('n', positive=True)
u = Symbol('u', nonnegative=True)
v = Symbol('v', nonpositive=True)
assert k.is_nonpositive is True
assert (-k).is_nonpositive is False
assert (2*k).is_nonpositive is True
assert n.is_nonpositive is False
assert (-n).is_nonpositive is True
assert (2*n).is_nonpositive is False
assert (n*k).is_nonpositive is True
assert (2*n*k).is_nonpositive is True
assert (-n*k).is_nonpositive is False
assert u.is_nonpositive is None
assert (-u).is_nonpositive is True
assert (2*u).is_nonpositive is None
assert v.is_nonpositive is True
assert (-v).is_nonpositive is None
assert (2*v).is_nonpositive is True
assert (u*v).is_nonpositive is True
assert (k*u).is_nonpositive is True
assert (k*v).is_nonpositive is None
assert (n*u).is_nonpositive is None
assert (n*v).is_nonpositive is True
assert (v*k*u).is_nonpositive is None
assert (v*n*u).is_nonpositive is True
assert (-v*k*u).is_nonpositive is True
assert (-v*n*u).is_nonpositive is None
assert (17*v*k*u).is_nonpositive is None
assert (17*v*n*u).is_nonpositive is True
assert (k*v*n*u).is_nonpositive is None
assert (x*k).is_nonpositive is None
assert (u*v*n*x*k).is_nonpositive is None
assert k.is_nonnegative is False
assert (-k).is_nonnegative is True
assert (2*k).is_nonnegative is False
assert n.is_nonnegative is True
assert (-n).is_nonnegative is False
assert (2*n).is_nonnegative is True
assert (n*k).is_nonnegative is False
assert (2*n*k).is_nonnegative is False
assert (-n*k).is_nonnegative is True
assert u.is_nonnegative is True
assert (-u).is_nonnegative is None
assert (2*u).is_nonnegative is True
assert v.is_nonnegative is None
assert (-v).is_nonnegative is True
assert (2*v).is_nonnegative is None
assert (u*v).is_nonnegative is None
assert (k*u).is_nonnegative is None
assert (k*v).is_nonnegative is True
assert (n*u).is_nonnegative is True
assert (n*v).is_nonnegative is None
assert (v*k*u).is_nonnegative is True
assert (v*n*u).is_nonnegative is None
assert (-v*k*u).is_nonnegative is None
assert (-v*n*u).is_nonnegative is True
assert (17*v*k*u).is_nonnegative is True
assert (17*v*n*u).is_nonnegative is None
assert (k*v*n*u).is_nonnegative is True
assert (x*k).is_nonnegative is None
assert (u*v*n*x*k).is_nonnegative is None
def test_Add_is_negative_positive():
x = Symbol('x', real=True)
k = Symbol('k', negative=True)
n = Symbol('n', positive=True)
u = Symbol('u', nonnegative=True)
v = Symbol('v', nonpositive=True)
assert (k - 2).is_negative is True
assert (k + 17).is_negative is None
assert (-k - 5).is_negative is None
assert (-k + 123).is_negative is False
assert (k - n).is_negative is True
assert (k + n).is_negative is None
assert (-k - n).is_negative is None
assert (-k + n).is_negative is False
assert (k - n - 2).is_negative is True
assert (k + n + 17).is_negative is None
assert (-k - n - 5).is_negative is None
assert (-k + n + 123).is_negative is False
assert (-2*k + 123*n + 17).is_negative is False
assert (k + u).is_negative is None
assert (k + v).is_negative is True
assert (n + u).is_negative is False
assert (n + v).is_negative is None
assert (u - v).is_negative is False
assert (u + v).is_negative is None
assert (-u - v).is_negative is None
assert (-u + v).is_negative is None
assert (u - v + n + 2).is_negative is False
assert (u + v + n + 2).is_negative is None
assert (-u - v + n + 2).is_negative is None
assert (-u + v + n + 2).is_negative is None
assert (k + x).is_negative is None
assert (k + x - n).is_negative is None
assert (k - 2).is_positive is False
assert (k + 17).is_positive is None
assert (-k - 5).is_positive is None
assert (-k + 123).is_positive is True
assert (k - n).is_positive is False
assert (k + n).is_positive is None
assert (-k - n).is_positive is None
assert (-k + n).is_positive is True
assert (k - n - 2).is_positive is False
assert (k + n + 17).is_positive is None
assert (-k - n - 5).is_positive is None
assert (-k + n + 123).is_positive is True
assert (-2*k + 123*n + 17).is_positive is True
assert (k + u).is_positive is None
assert (k + v).is_positive is False
assert (n + u).is_positive is True
assert (n + v).is_positive is None
assert (u - v).is_positive is None
assert (u + v).is_positive is None
assert (-u - v).is_positive is None
assert (-u + v).is_positive is False
assert (u - v - n - 2).is_positive is None
assert (u + v - n - 2).is_positive is None
assert (-u - v - n - 2).is_positive is None
assert (-u + v - n - 2).is_positive is False
assert (n + x).is_positive is None
assert (n + x - k).is_positive is None
z = (-3 - sqrt(5) + (-sqrt(10)/2 - sqrt(2)/2)**2)
assert z.is_zero
z = sqrt(1 + sqrt(3)) + sqrt(3 + 3*sqrt(3)) - sqrt(10 + 6*sqrt(3))
assert z.is_zero
def test_Add_is_nonpositive_nonnegative():
x = Symbol('x', real=True)
k = Symbol('k', negative=True)
n = Symbol('n', positive=True)
u = Symbol('u', nonnegative=True)
v = Symbol('v', nonpositive=True)
assert (u - 2).is_nonpositive is None
assert (u + 17).is_nonpositive is False
assert (-u - 5).is_nonpositive is True
assert (-u + 123).is_nonpositive is None
assert (u - v).is_nonpositive is None
assert (u + v).is_nonpositive is None
assert (-u - v).is_nonpositive is None
assert (-u + v).is_nonpositive is True
assert (u - v - 2).is_nonpositive is None
assert (u + v + 17).is_nonpositive is None
assert (-u - v - 5).is_nonpositive is None
assert (-u + v - 123).is_nonpositive is True
assert (-2*u + 123*v - 17).is_nonpositive is True
assert (k + u).is_nonpositive is None
assert (k + v).is_nonpositive is True
assert (n + u).is_nonpositive is False
assert (n + v).is_nonpositive is None
assert (k - n).is_nonpositive is True
assert (k + n).is_nonpositive is None
assert (-k - n).is_nonpositive is None
assert (-k + n).is_nonpositive is False
assert (k - n + u + 2).is_nonpositive is None
assert (k + n + u + 2).is_nonpositive is None
assert (-k - n + u + 2).is_nonpositive is None
assert (-k + n + u + 2).is_nonpositive is False
assert (u + x).is_nonpositive is None
assert (v - x - n).is_nonpositive is None
assert (u - 2).is_nonnegative is None
assert (u + 17).is_nonnegative is True
assert (-u - 5).is_nonnegative is False
assert (-u + 123).is_nonnegative is None
assert (u - v).is_nonnegative is True
assert (u + v).is_nonnegative is None
assert (-u - v).is_nonnegative is None
assert (-u + v).is_nonnegative is None
assert (u - v + 2).is_nonnegative is True
assert (u + v + 17).is_nonnegative is None
assert (-u - v - 5).is_nonnegative is None
assert (-u + v - 123).is_nonnegative is False
assert (2*u - 123*v + 17).is_nonnegative is True
assert (k + u).is_nonnegative is None
assert (k + v).is_nonnegative is False
assert (n + u).is_nonnegative is True
assert (n + v).is_nonnegative is None
assert (k - n).is_nonnegative is False
assert (k + n).is_nonnegative is None
assert (-k - n).is_nonnegative is None
assert (-k + n).is_nonnegative is True
assert (k - n - u - 2).is_nonnegative is False
assert (k + n - u - 2).is_nonnegative is None
assert (-k - n - u - 2).is_nonnegative is None
assert (-k + n - u - 2).is_nonnegative is None
assert (u - x).is_nonnegative is None
assert (v + x + n).is_nonnegative is None
def test_Pow_is_integer():
x = Symbol('x')
k = Symbol('k', integer=True)
n = Symbol('n', integer=True, nonnegative=True)
m = Symbol('m', integer=True, positive=True)
assert (k**2).is_integer is True
assert (k**(-2)).is_integer is None
assert ((m + 1)**(-2)).is_integer is False
assert (m**(-1)).is_integer is None # issue 8580
assert (2**k).is_integer is None
assert (2**(-k)).is_integer is None
assert (2**n).is_integer is True
assert (2**(-n)).is_integer is None
assert (2**m).is_integer is True
assert (2**(-m)).is_integer is False
assert (x**2).is_integer is None
assert (2**x).is_integer is None
assert (k**n).is_integer is True
assert (k**(-n)).is_integer is None
assert (k**x).is_integer is None
assert (x**k).is_integer is None
assert (k**(n*m)).is_integer is True
assert (k**(-n*m)).is_integer is None
assert sqrt(3).is_integer is False
assert sqrt(.3).is_integer is False
assert Pow(3, 2, evaluate=False).is_integer is True
assert Pow(3, 0, evaluate=False).is_integer is True
assert Pow(3, -2, evaluate=False).is_integer is False
assert Pow(S.Half, 3, evaluate=False).is_integer is False
# decided by re-evaluating
assert Pow(3, S.Half, evaluate=False).is_integer is False
assert Pow(3, S.Half, evaluate=False).is_integer is False
assert Pow(4, S.Half, evaluate=False).is_integer is True
assert Pow(S.Half, -2, evaluate=False).is_integer is True
assert ((-1)**k).is_integer
# issue 8641
x = Symbol('x', real=True, integer=False)
assert (x**2).is_integer is None
# issue 10458
x = Symbol('x', positive=True)
assert (1/(x + 1)).is_integer is False
assert (1/(-x - 1)).is_integer is False
def test_Pow_is_real():
x = Symbol('x', real=True)
y = Symbol('y', real=True, positive=True)
assert (x**2).is_real is True
assert (x**3).is_real is True
assert (x**x).is_real is None
assert (y**x).is_real is True
assert (x**Rational(1, 3)).is_real is None
assert (y**Rational(1, 3)).is_real is True
assert sqrt(-1 - sqrt(2)).is_real is False
i = Symbol('i', imaginary=True)
assert (i**i).is_real is None
assert (I**i).is_extended_real is True
assert ((-I)**i).is_extended_real is True
assert (2**i).is_real is None # (2**(pi/log(2) * I)) is real, 2**I is not
assert (2**I).is_real is False
assert (2**-I).is_real is False
assert (i**2).is_extended_real is True
assert (i**3).is_extended_real is False
assert (i**x).is_real is None # could be (-I)**(2/3)
e = Symbol('e', even=True)
o = Symbol('o', odd=True)
k = Symbol('k', integer=True)
assert (i**e).is_extended_real is True
assert (i**o).is_extended_real is False
assert (i**k).is_real is None
assert (i**(4*k)).is_extended_real is True
x = Symbol("x", nonnegative=True)
y = Symbol("y", nonnegative=True)
assert im(x**y).expand(complex=True) is S.Zero
assert (x**y).is_real is True
i = Symbol('i', imaginary=True)
assert (exp(i)**I).is_extended_real is True
assert log(exp(i)).is_imaginary is None # i could be 2*pi*I
c = Symbol('c', complex=True)
assert log(c).is_real is None # c could be 0 or 2, too
assert log(exp(c)).is_real is None # log(0), log(E), ...
n = Symbol('n', negative=False)
assert log(n).is_real is None
n = Symbol('n', nonnegative=True)
assert log(n).is_real is None
assert sqrt(-I).is_real is False # issue 7843
i = Symbol('i', integer=True)
assert (1/(i-1)).is_real is None
assert (1/(i-1)).is_extended_real is None
def test_real_Pow():
k = Symbol('k', integer=True, nonzero=True)
assert (k**(I*pi/log(k))).is_real
def test_Pow_is_finite():
xe = Symbol('xe', extended_real=True)
xr = Symbol('xr', real=True)
p = Symbol('p', positive=True)
n = Symbol('n', negative=True)
i = Symbol('i', integer=True)
assert (xe**2).is_finite is None # xe could be oo
assert (xr**2).is_finite is True
assert (xe**xe).is_finite is None
assert (xr**xe).is_finite is None
assert (xe**xr).is_finite is None
# FIXME: The line below should be True rather than None
# assert (xr**xr).is_finite is True
assert (xr**xr).is_finite is None
assert (p**xe).is_finite is None
assert (p**xr).is_finite is True
assert (n**xe).is_finite is None
assert (n**xr).is_finite is True
assert (sin(xe)**2).is_finite is True
assert (sin(xr)**2).is_finite is True
assert (sin(xe)**xe).is_finite is None # xe, xr could be -pi
assert (sin(xr)**xr).is_finite is None
# FIXME: Should the line below be True rather than None?
assert (sin(xe)**exp(xe)).is_finite is None
assert (sin(xr)**exp(xr)).is_finite is True
assert (1/sin(xe)).is_finite is None # if zero, no, otherwise yes
assert (1/sin(xr)).is_finite is None
assert (1/exp(xe)).is_finite is None # xe could be -oo
assert (1/exp(xr)).is_finite is True
assert (1/S.Pi).is_finite is True
assert (1/(i-1)).is_finite is None
def test_Pow_is_even_odd():
x = Symbol('x')
k = Symbol('k', even=True)
n = Symbol('n', odd=True)
m = Symbol('m', integer=True, nonnegative=True)
p = Symbol('p', integer=True, positive=True)
assert ((-1)**n).is_odd
assert ((-1)**k).is_odd
assert ((-1)**(m - p)).is_odd
assert (k**2).is_even is True
assert (n**2).is_even is False
assert (2**k).is_even is None
assert (x**2).is_even is None
assert (k**m).is_even is None
assert (n**m).is_even is False
assert (k**p).is_even is True
assert (n**p).is_even is False
assert (m**k).is_even is None
assert (p**k).is_even is None
assert (m**n).is_even is None
assert (p**n).is_even is None
assert (k**x).is_even is None
assert (n**x).is_even is None
assert (k**2).is_odd is False
assert (n**2).is_odd is True
assert (3**k).is_odd is None
assert (k**m).is_odd is None
assert (n**m).is_odd is True
assert (k**p).is_odd is False
assert (n**p).is_odd is True
assert (m**k).is_odd is None
assert (p**k).is_odd is None
assert (m**n).is_odd is None
assert (p**n).is_odd is None
assert (k**x).is_odd is None
assert (n**x).is_odd is None
def test_Pow_is_negative_positive():
r = Symbol('r', real=True)
k = Symbol('k', integer=True, positive=True)
n = Symbol('n', even=True)
m = Symbol('m', odd=True)
x = Symbol('x')
assert (2**r).is_positive is True
assert ((-2)**r).is_positive is None
assert ((-2)**n).is_positive is True
assert ((-2)**m).is_positive is False
assert (k**2).is_positive is True
assert (k**(-2)).is_positive is True
assert (k**r).is_positive is True
assert ((-k)**r).is_positive is None
assert ((-k)**n).is_positive is True
assert ((-k)**m).is_positive is False
assert (2**r).is_negative is False
assert ((-2)**r).is_negative is None
assert ((-2)**n).is_negative is False
assert ((-2)**m).is_negative is True
assert (k**2).is_negative is False
assert (k**(-2)).is_negative is False
assert (k**r).is_negative is False
assert ((-k)**r).is_negative is None
assert ((-k)**n).is_negative is False
assert ((-k)**m).is_negative is True
assert (2**x).is_positive is None
assert (2**x).is_negative is None
def test_Pow_is_zero():
z = Symbol('z', zero=True)
e = z**2
assert e.is_zero
assert e.is_positive is False
assert e.is_negative is False
assert Pow(0, 0, evaluate=False).is_zero is False
assert Pow(0, 3, evaluate=False).is_zero
assert Pow(0, oo, evaluate=False).is_zero
assert Pow(0, -3, evaluate=False).is_zero is False
assert Pow(0, -oo, evaluate=False).is_zero is False
assert Pow(2, 2, evaluate=False).is_zero is False
a = Symbol('a', zero=False)
assert Pow(a, 3).is_zero is False # issue 7965
assert Pow(2, oo, evaluate=False).is_zero is False
assert Pow(2, -oo, evaluate=False).is_zero
assert Pow(S.Half, oo, evaluate=False).is_zero
assert Pow(S.Half, -oo, evaluate=False).is_zero is False
# All combinations of real/complex base/exponent
h = S.Half
T = True
F = False
N = None
pow_iszero = [
['**', 0, h, 1, 2, -h, -1,-2,-2*I,-I/2,I/2,1+I,oo,-oo,zoo],
[ 0, F, T, T, T, F, F, F, F, F, F, N, T, F, N],
[ h, F, F, F, F, F, F, F, F, F, F, F, T, F, N],
[ 1, F, F, F, F, F, F, F, F, F, F, F, F, F, N],
[ 2, F, F, F, F, F, F, F, F, F, F, F, F, T, N],
[ -h, F, F, F, F, F, F, F, F, F, F, F, T, F, N],
[ -1, F, F, F, F, F, F, F, F, F, F, F, F, F, N],
[ -2, F, F, F, F, F, F, F, F, F, F, F, F, T, N],
[-2*I, F, F, F, F, F, F, F, F, F, F, F, F, T, N],
[-I/2, F, F, F, F, F, F, F, F, F, F, F, T, F, N],
[ I/2, F, F, F, F, F, F, F, F, F, F, F, T, F, N],
[ 1+I, F, F, F, F, F, F, F, F, F, F, F, F, T, N],
[ oo, F, F, F, F, T, T, T, F, F, F, F, F, T, N],
[ -oo, F, F, F, F, T, T, T, F, F, F, F, F, T, N],
[ zoo, F, F, F, F, T, T, T, N, N, N, N, F, T, N]
]
def test_table(table):
n = len(table[0])
for row in range(1, n):
base = table[row][0]
for col in range(1, n):
exp = table[0][col]
is_zero = table[row][col]
# The actual test here:
assert Pow(base, exp, evaluate=False).is_zero is is_zero
test_table(pow_iszero)
# A zero symbol...
zo, zo2 = symbols('zo, zo2', zero=True)
# All combinations of finite symbols
zf, zf2 = symbols('zf, zf2', finite=True)
wf, wf2 = symbols('wf, wf2', nonzero=True)
xf, xf2 = symbols('xf, xf2', real=True)
yf, yf2 = symbols('yf, yf2', nonzero=True)
af, af2 = symbols('af, af2', positive=True)
bf, bf2 = symbols('bf, bf2', nonnegative=True)
cf, cf2 = symbols('cf, cf2', negative=True)
df, df2 = symbols('df, df2', nonpositive=True)
# Without finiteness:
zi, zi2 = symbols('zi, zi2')
wi, wi2 = symbols('wi, wi2', zero=False)
xi, xi2 = symbols('xi, xi2', extended_real=True)
yi, yi2 = symbols('yi, yi2', zero=False, extended_real=True)
ai, ai2 = symbols('ai, ai2', extended_positive=True)
bi, bi2 = symbols('bi, bi2', extended_nonnegative=True)
ci, ci2 = symbols('ci, ci2', extended_negative=True)
di, di2 = symbols('di, di2', extended_nonpositive=True)
pow_iszero_sym = [
['**',zo,wf,yf,af,cf,zf,xf,bf,df,zi,wi,xi,yi,ai,bi,ci,di],
[ zo2, F, N, N, T, F, N, N, N, F, N, N, N, N, T, N, F, F],
[ wf2, F, F, F, F, F, F, F, F, F, N, N, N, N, N, N, N, N],
[ yf2, F, F, F, F, F, F, F, F, F, N, N, N, N, N, N, N, N],
[ af2, F, F, F, F, F, F, F, F, F, N, N, N, N, N, N, N, N],
[ cf2, F, F, F, F, F, F, F, F, F, N, N, N, N, N, N, N, N],
[ zf2, N, N, N, N, F, N, N, N, N, N, N, N, N, N, N, N, N],
[ xf2, N, N, N, N, F, N, N, N, N, N, N, N, N, N, N, N, N],
[ bf2, N, N, N, N, F, N, N, N, N, N, N, N, N, N, N, N, N],
[ df2, N, N, N, N, F, N, N, N, N, N, N, N, N, N, N, N, N],
[ zi2, N, N, N, N, N, N, N, N, N, N, N, N, N, N, N, N, N],
[ wi2, F, N, N, F, N, N, N, F, N, N, N, N, N, N, N, N, N],
[ xi2, N, N, N, N, N, N, N, N, N, N, N, N, N, N, N, N, N],
[ yi2, F, N, N, F, N, N, N, F, N, N, N, N, N, N, N, N, N],
[ ai2, F, N, N, F, N, N, N, F, N, N, N, N, N, N, N, N, N],
[ bi2, N, N, N, N, N, N, N, N, N, N, N, N, N, N, N, N, N],
[ ci2, F, N, N, F, N, N, N, F, N, N, N, N, N, N, N, N, N],
[ di2, N, N, N, N, N, N, N, N, N, N, N, N, N, N, N, N, N]
]
test_table(pow_iszero_sym)
# In some cases (x**x).is_zero is different from (x**y).is_zero even if y
# has the same assumptions as x.
assert (zo ** zo).is_zero is False
assert (wf ** wf).is_zero is False
assert (yf ** yf).is_zero is False
assert (af ** af).is_zero is False
assert (cf ** cf).is_zero is False
assert (zf ** zf).is_zero is None
assert (xf ** xf).is_zero is None
assert (bf ** bf).is_zero is False # None in table
assert (df ** df).is_zero is None
assert (zi ** zi).is_zero is None
assert (wi ** wi).is_zero is None
assert (xi ** xi).is_zero is None
assert (yi ** yi).is_zero is None
assert (ai ** ai).is_zero is False # None in table
assert (bi ** bi).is_zero is False # None in table
assert (ci ** ci).is_zero is None
assert (di ** di).is_zero is None
def test_Pow_is_nonpositive_nonnegative():
x = Symbol('x', real=True)
k = Symbol('k', integer=True, nonnegative=True)
l = Symbol('l', integer=True, positive=True)
n = Symbol('n', even=True)
m = Symbol('m', odd=True)
assert (x**(4*k)).is_nonnegative is True
assert (2**x).is_nonnegative is True
assert ((-2)**x).is_nonnegative is None
assert ((-2)**n).is_nonnegative is True
assert ((-2)**m).is_nonnegative is False
assert (k**2).is_nonnegative is True
assert (k**(-2)).is_nonnegative is None
assert (k**k).is_nonnegative is True
assert (k**x).is_nonnegative is None # NOTE (0**x).is_real = U
assert (l**x).is_nonnegative is True
assert (l**x).is_positive is True
assert ((-k)**x).is_nonnegative is None
assert ((-k)**m).is_nonnegative is None
assert (2**x).is_nonpositive is False
assert ((-2)**x).is_nonpositive is None
assert ((-2)**n).is_nonpositive is False
assert ((-2)**m).is_nonpositive is True
assert (k**2).is_nonpositive is None
assert (k**(-2)).is_nonpositive is None
assert (k**x).is_nonpositive is None
assert ((-k)**x).is_nonpositive is None
assert ((-k)**n).is_nonpositive is None
assert (x**2).is_nonnegative is True
i = symbols('i', imaginary=True)
assert (i**2).is_nonpositive is True
assert (i**4).is_nonpositive is False
assert (i**3).is_nonpositive is False
assert (I**i).is_nonnegative is True
assert (exp(I)**i).is_nonnegative is True
assert ((-l)**n).is_nonnegative is True
assert ((-l)**m).is_nonpositive is True
assert ((-k)**n).is_nonnegative is None
assert ((-k)**m).is_nonpositive is None
def test_Mul_is_imaginary_real():
r = Symbol('r', real=True)
p = Symbol('p', positive=True)
i1 = Symbol('i1', imaginary=True)
i2 = Symbol('i2', imaginary=True)
x = Symbol('x')
assert I.is_imaginary is True
assert I.is_real is False
assert (-I).is_imaginary is True
assert (-I).is_real is False
assert (3*I).is_imaginary is True
assert (3*I).is_real is False
assert (I*I).is_imaginary is False
assert (I*I).is_real is True
e = (p + p*I)
j = Symbol('j', integer=True, zero=False)
assert (e**j).is_real is None
assert (e**(2*j)).is_real is None
assert (e**j).is_imaginary is None
assert (e**(2*j)).is_imaginary is None
assert (e**-1).is_imaginary is False
assert (e**2).is_imaginary
assert (e**3).is_imaginary is False
assert (e**4).is_imaginary is False
assert (e**5).is_imaginary is False
assert (e**-1).is_real is False
assert (e**2).is_real is False
assert (e**3).is_real is False
assert (e**4).is_real is True
assert (e**5).is_real is False
assert (e**3).is_complex
assert (r*i1).is_imaginary is None
assert (r*i1).is_real is None
assert (x*i1).is_imaginary is None
assert (x*i1).is_real is None
assert (i1*i2).is_imaginary is False
assert (i1*i2).is_real is True
assert (r*i1*i2).is_imaginary is False
assert (r*i1*i2).is_real is True
# Github's issue 5874:
nr = Symbol('nr', real=False, complex=True) # e.g. I or 1 + I
a = Symbol('a', real=True, nonzero=True)
b = Symbol('b', real=True)
assert (i1*nr).is_real is None
assert (a*nr).is_real is False
assert (b*nr).is_real is None
ni = Symbol('ni', imaginary=False, complex=True) # e.g. 2 or 1 + I
a = Symbol('a', real=True, nonzero=True)
b = Symbol('b', real=True)
assert (i1*ni).is_real is False
assert (a*ni).is_real is None
assert (b*ni).is_real is None
def test_Mul_hermitian_antihermitian():
a = Symbol('a', hermitian=True, zero=False)
b = Symbol('b', hermitian=True)
c = Symbol('c', hermitian=False)
d = Symbol('d', antihermitian=True)
e1 = Mul(a, b, c, evaluate=False)
e2 = Mul(b, a, c, evaluate=False)
e3 = Mul(a, b, c, d, evaluate=False)
e4 = Mul(b, a, c, d, evaluate=False)
e5 = Mul(a, c, evaluate=False)
e6 = Mul(a, c, d, evaluate=False)
assert e1.is_hermitian is None
assert e2.is_hermitian is None
assert e1.is_antihermitian is None
assert e2.is_antihermitian is None
assert e3.is_antihermitian is None
assert e4.is_antihermitian is None
assert e5.is_antihermitian is None
assert e6.is_antihermitian is None
def test_Add_is_comparable():
assert (x + y).is_comparable is False
assert (x + 1).is_comparable is False
assert (Rational(1, 3) - sqrt(8)).is_comparable is True
def test_Mul_is_comparable():
assert (x*y).is_comparable is False
assert (x*2).is_comparable is False
assert (sqrt(2)*Rational(1, 3)).is_comparable is True
def test_Pow_is_comparable():
assert (x**y).is_comparable is False
assert (x**2).is_comparable is False
assert (sqrt(Rational(1, 3))).is_comparable is True
def test_Add_is_positive_2():
e = Rational(1, 3) - sqrt(8)
assert e.is_positive is False
assert e.is_negative is True
e = pi - 1
assert e.is_positive is True
assert e.is_negative is False
def test_Add_is_irrational():
i = Symbol('i', irrational=True)
assert i.is_irrational is True
assert i.is_rational is False
assert (i + 1).is_irrational is True
assert (i + 1).is_rational is False
def test_Mul_is_irrational():
expr = Mul(1, 2, 3, evaluate=False)
assert expr.is_irrational is False
expr = Mul(1, I, I, evaluate=False)
assert expr.is_rational is None # I * I = -1 but *no evaluation allowed*
# sqrt(2) * I * I = -sqrt(2) is irrational but
# this can't be determined without evaluating the
# expression and the eval_is routines shouldn't do that
expr = Mul(sqrt(2), I, I, evaluate=False)
assert expr.is_irrational is None
def test_issue_3531():
# https://github.com/sympy/sympy/issues/3531
# https://github.com/sympy/sympy/pull/18116
class MightyNumeric(tuple):
def __rdiv__(self, other):
return "something"
def __rtruediv__(self, other):
return "something"
assert sympify(1)/MightyNumeric((1, 2)) == "something"
def test_issue_3531b():
class Foo:
def __init__(self):
self.field = 1.0
def __mul__(self, other):
self.field = self.field * other
def __rmul__(self, other):
self.field = other * self.field
f = Foo()
x = Symbol("x")
assert f*x == x*f
def test_bug3():
a = Symbol("a")
b = Symbol("b", positive=True)
e = 2*a + b
f = b + 2*a
assert e == f
def test_suppressed_evaluation():
a = Add(0, 3, 2, evaluate=False)
b = Mul(1, 3, 2, evaluate=False)
c = Pow(3, 2, evaluate=False)
assert a != 6
assert a.func is Add
assert a.args == (0, 3, 2)
assert b != 6
assert b.func is Mul
assert b.args == (1, 3, 2)
assert c != 9
assert c.func is Pow
assert c.args == (3, 2)
def test_AssocOp_doit():
a = Add(x,x, evaluate=False)
b = Mul(y,y, evaluate=False)
c = Add(b,b, evaluate=False)
d = Mul(a,a, evaluate=False)
assert c.doit(deep=False).func == Mul
assert c.doit(deep=False).args == (2,y,y)
assert c.doit().func == Mul
assert c.doit().args == (2, Pow(y,2))
assert d.doit(deep=False).func == Pow
assert d.doit(deep=False).args == (a, 2*S.One)
assert d.doit().func == Mul
assert d.doit().args == (4*S.One, Pow(x,2))
def test_Add_Mul_Expr_args():
nonexpr = [Basic(), Poly(x, x), FiniteSet(x)]
for typ in [Add, Mul]:
for obj in nonexpr:
with warns_deprecated_sympy():
typ(obj, 1)
def test_Add_as_coeff_mul():
# issue 5524. These should all be (1, self)
assert (x + 1).as_coeff_mul() == (1, (x + 1,))
assert (x + 2).as_coeff_mul() == (1, (x + 2,))
assert (x + 3).as_coeff_mul() == (1, (x + 3,))
assert (x - 1).as_coeff_mul() == (1, (x - 1,))
assert (x - 2).as_coeff_mul() == (1, (x - 2,))
assert (x - 3).as_coeff_mul() == (1, (x - 3,))
n = Symbol('n', integer=True)
assert (n + 1).as_coeff_mul() == (1, (n + 1,))
assert (n + 2).as_coeff_mul() == (1, (n + 2,))
assert (n + 3).as_coeff_mul() == (1, (n + 3,))
assert (n - 1).as_coeff_mul() == (1, (n - 1,))
assert (n - 2).as_coeff_mul() == (1, (n - 2,))
assert (n - 3).as_coeff_mul() == (1, (n - 3,))
def test_Pow_as_coeff_mul_doesnt_expand():
assert exp(x + y).as_coeff_mul() == (1, (exp(x + y),))
assert exp(x + exp(x + y)) != exp(x + exp(x)*exp(y))
def test_issue_3514_18626():
assert sqrt(S.Half) * sqrt(6) == 2 * sqrt(3)/2
assert S.Half*sqrt(6)*sqrt(2) == sqrt(3)
assert sqrt(6)/2*sqrt(2) == sqrt(3)
assert sqrt(6)*sqrt(2)/2 == sqrt(3)
assert sqrt(8)**Rational(2, 3) == 2
def test_make_args():
assert Add.make_args(x) == (x,)
assert Mul.make_args(x) == (x,)
assert Add.make_args(x*y*z) == (x*y*z,)
assert Mul.make_args(x*y*z) == (x*y*z).args
assert Add.make_args(x + y + z) == (x + y + z).args
assert Mul.make_args(x + y + z) == (x + y + z,)
assert Add.make_args((x + y)**z) == ((x + y)**z,)
assert Mul.make_args((x + y)**z) == ((x + y)**z,)
def test_issue_5126():
assert (-2)**x*(-3)**x != 6**x
i = Symbol('i', integer=1)
assert (-2)**i*(-3)**i == 6**i
def test_Rational_as_content_primitive():
c, p = S.One, S.Zero
assert (c*p).as_content_primitive() == (c, p)
c, p = S.Half, S.One
assert (c*p).as_content_primitive() == (c, p)
def test_Add_as_content_primitive():
assert (x + 2).as_content_primitive() == (1, x + 2)
assert (3*x + 2).as_content_primitive() == (1, 3*x + 2)
assert (3*x + 3).as_content_primitive() == (3, x + 1)
assert (3*x + 6).as_content_primitive() == (3, x + 2)
assert (3*x + 2*y).as_content_primitive() == (1, 3*x + 2*y)
assert (3*x + 3*y).as_content_primitive() == (3, x + y)
assert (3*x + 6*y).as_content_primitive() == (3, x + 2*y)
assert (3/x + 2*x*y*z**2).as_content_primitive() == (1, 3/x + 2*x*y*z**2)
assert (3/x + 3*x*y*z**2).as_content_primitive() == (3, 1/x + x*y*z**2)
assert (3/x + 6*x*y*z**2).as_content_primitive() == (3, 1/x + 2*x*y*z**2)
assert (2*x/3 + 4*y/9).as_content_primitive() == \
(Rational(2, 9), 3*x + 2*y)
assert (2*x/3 + 2.5*y).as_content_primitive() == \
(Rational(1, 3), 2*x + 7.5*y)
# the coefficient may sort to a position other than 0
p = 3 + x + y
assert (2*p).expand().as_content_primitive() == (2, p)
assert (2.0*p).expand().as_content_primitive() == (1, 2.*p)
p *= -1
assert (2*p).expand().as_content_primitive() == (2, p)
def test_Mul_as_content_primitive():
assert (2*x).as_content_primitive() == (2, x)
assert (x*(2 + 2*x)).as_content_primitive() == (2, x*(1 + x))
assert (x*(2 + 2*y)*(3*x + 3)**2).as_content_primitive() == \
(18, x*(1 + y)*(x + 1)**2)
assert ((2 + 2*x)**2*(3 + 6*x) + S.Half).as_content_primitive() == \
(S.Half, 24*(x + 1)**2*(2*x + 1) + 1)
def test_Pow_as_content_primitive():
assert (x**y).as_content_primitive() == (1, x**y)
assert ((2*x + 2)**y).as_content_primitive() == \
(1, (Mul(2, (x + 1), evaluate=False))**y)
assert ((2*x + 2)**3).as_content_primitive() == (8, (x + 1)**3)
def test_issue_5460():
u = Mul(2, (1 + x), evaluate=False)
assert (2 + u).args == (2, u)
def test_product_irrational():
from sympy import I, pi
assert (I*pi).is_irrational is False
# The following used to be deduced from the above bug:
assert (I*pi).is_positive is False
def test_issue_5919():
assert (x/(y*(1 + y))).expand() == x/(y**2 + y)
def test_Mod():
assert Mod(x, 1).func is Mod
assert pi % pi is S.Zero
assert Mod(5, 3) == 2
assert Mod(-5, 3) == 1
assert Mod(5, -3) == -1
assert Mod(-5, -3) == -2
assert type(Mod(3.2, 2, evaluate=False)) == Mod
assert 5 % x == Mod(5, x)
assert x % 5 == Mod(x, 5)
assert x % y == Mod(x, y)
assert (x % y).subs({x: 5, y: 3}) == 2
assert Mod(nan, 1) is nan
assert Mod(1, nan) is nan
assert Mod(nan, nan) is nan
Mod(0, x) == 0
with raises(ZeroDivisionError):
Mod(x, 0)
k = Symbol('k', integer=True)
m = Symbol('m', integer=True, positive=True)
assert (x**m % x).func is Mod
assert (k**(-m) % k).func is Mod
assert k**m % k == 0
assert (-2*k)**m % k == 0
# Float handling
point3 = Float(3.3) % 1
assert (x - 3.3) % 1 == Mod(1.*x + 1 - point3, 1)
assert Mod(-3.3, 1) == 1 - point3
assert Mod(0.7, 1) == Float(0.7)
e = Mod(1.3, 1)
assert comp(e, .3) and e.is_Float
e = Mod(1.3, .7)
assert comp(e, .6) and e.is_Float
e = Mod(1.3, Rational(7, 10))
assert comp(e, .6) and e.is_Float
e = Mod(Rational(13, 10), 0.7)
assert comp(e, .6) and e.is_Float
e = Mod(Rational(13, 10), Rational(7, 10))
assert comp(e, .6) and e.is_Rational
# check that sign is right
r2 = sqrt(2)
r3 = sqrt(3)
for i in [-r3, -r2, r2, r3]:
for j in [-r3, -r2, r2, r3]:
assert verify_numerically(i % j, i.n() % j.n())
for _x in range(4):
for _y in range(9):
reps = [(x, _x), (y, _y)]
assert Mod(3*x + y, 9).subs(reps) == (3*_x + _y) % 9
# denesting
t = Symbol('t', real=True)
assert Mod(Mod(x, t), t) == Mod(x, t)
assert Mod(-Mod(x, t), t) == Mod(-x, t)
assert Mod(Mod(x, 2*t), t) == Mod(x, t)
assert Mod(-Mod(x, 2*t), t) == Mod(-x, t)
assert Mod(Mod(x, t), 2*t) == Mod(x, t)
assert Mod(-Mod(x, t), -2*t) == -Mod(x, t)
for i in [-4, -2, 2, 4]:
for j in [-4, -2, 2, 4]:
for k in range(4):
assert Mod(Mod(x, i), j).subs({x: k}) == (k % i) % j
assert Mod(-Mod(x, i), j).subs({x: k}) == -(k % i) % j
# known difference
assert Mod(5*sqrt(2), sqrt(5)) == 5*sqrt(2) - 3*sqrt(5)
p = symbols('p', positive=True)
assert Mod(2, p + 3) == 2
assert Mod(-2, p + 3) == p + 1
assert Mod(2, -p - 3) == -p - 1
assert Mod(-2, -p - 3) == -2
assert Mod(p + 5, p + 3) == 2
assert Mod(-p - 5, p + 3) == p + 1
assert Mod(p + 5, -p - 3) == -p - 1
assert Mod(-p - 5, -p - 3) == -2
assert Mod(p + 1, p - 1).func is Mod
# handling sums
assert (x + 3) % 1 == Mod(x, 1)
assert (x + 3.0) % 1 == Mod(1.*x, 1)
assert (x - S(33)/10) % 1 == Mod(x + S(7)/10, 1)
a = Mod(.6*x + y, .3*y)
b = Mod(0.1*y + 0.6*x, 0.3*y)
# Test that a, b are equal, with 1e-14 accuracy in coefficients
eps = 1e-14
assert abs((a.args[0] - b.args[0]).subs({x: 1, y: 1})) < eps
assert abs((a.args[1] - b.args[1]).subs({x: 1, y: 1})) < eps
assert (x + 1) % x == 1 % x
assert (x + y) % x == y % x
assert (x + y + 2) % x == (y + 2) % x
assert (a + 3*x + 1) % (2*x) == Mod(a + x + 1, 2*x)
assert (12*x + 18*y) % (3*x) == 3*Mod(6*y, x)
# gcd extraction
assert (-3*x) % (-2*y) == -Mod(3*x, 2*y)
assert (.6*pi) % (.3*x*pi) == 0.3*pi*Mod(2, x)
assert (.6*pi) % (.31*x*pi) == pi*Mod(0.6, 0.31*x)
assert (6*pi) % (.3*x*pi) == 0.3*pi*Mod(20, x)
assert (6*pi) % (.31*x*pi) == pi*Mod(6, 0.31*x)
assert (6*pi) % (.42*x*pi) == pi*Mod(6, 0.42*x)
assert (12*x) % (2*y) == 2*Mod(6*x, y)
assert (12*x) % (3*5*y) == 3*Mod(4*x, 5*y)
assert (12*x) % (15*x*y) == 3*x*Mod(4, 5*y)
assert (-2*pi) % (3*pi) == pi
assert (2*x + 2) % (x + 1) == 0
assert (x*(x + 1)) % (x + 1) == (x + 1)*Mod(x, 1)
assert Mod(5.0*x, 0.1*y) == 0.1*Mod(50*x, y)
i = Symbol('i', integer=True)
assert (3*i*x) % (2*i*y) == i*Mod(3*x, 2*y)
assert Mod(4*i, 4) == 0
# issue 8677
n = Symbol('n', integer=True, positive=True)
assert factorial(n) % n == 0
assert factorial(n + 2) % n == 0
assert (factorial(n + 4) % (n + 5)).func is Mod
# Wilson's theorem
factorial(18042, evaluate=False) % 18043 == 18042
p = Symbol('n', prime=True)
factorial(p - 1) % p == p - 1
factorial(p - 1) % -p == -1
(factorial(3, evaluate=False) % 4).doit() == 2
n = Symbol('n', composite=True, odd=True)
factorial(n - 1) % n == 0
# symbolic with known parity
n = Symbol('n', even=True)
assert Mod(n, 2) == 0
n = Symbol('n', odd=True)
assert Mod(n, 2) == 1
# issue 10963
assert (x**6000%400).args[1] == 400
#issue 13543
assert Mod(Mod(x + 1, 2) + 1 , 2) == Mod(x,2)
assert Mod(Mod(x + 2, 4)*(x + 4), 4) == Mod(x*(x + 2), 4)
assert Mod(Mod(x + 2, 4)*4, 4) == 0
# issue 15493
i, j = symbols('i j', integer=True, positive=True)
assert Mod(3*i, 2) == Mod(i, 2)
assert Mod(8*i/j, 4) == 4*Mod(2*i/j, 1)
assert Mod(8*i, 4) == 0
# rewrite
assert Mod(x, y).rewrite(floor) == x - y*floor(x/y)
assert ((x - Mod(x, y))/y).rewrite(floor) == floor(x/y)
def test_Mod_Pow():
# modular exponentiation
assert isinstance(Mod(Pow(2, 2, evaluate=False), 3), Integer)
assert Mod(Pow(4, 13, evaluate=False), 497) == Mod(Pow(4, 13), 497)
assert Mod(Pow(2, 10000000000, evaluate=False), 3) == 1
assert Mod(Pow(32131231232, 9**10**6, evaluate=False),10**12) == \
pow(32131231232,9**10**6,10**12)
assert Mod(Pow(33284959323, 123**999, evaluate=False),11**13) == \
pow(33284959323,123**999,11**13)
assert Mod(Pow(78789849597, 333**555, evaluate=False),12**9) == \
pow(78789849597,333**555,12**9)
# modular nested exponentiation
expr = Pow(2, 2, evaluate=False)
expr = Pow(2, expr, evaluate=False)
assert Mod(expr, 3**10) == 16
expr = Pow(2, expr, evaluate=False)
assert Mod(expr, 3**10) == 6487
expr = Pow(2, expr, evaluate=False)
assert Mod(expr, 3**10) == 32191
expr = Pow(2, expr, evaluate=False)
assert Mod(expr, 3**10) == 18016
expr = Pow(2, expr, evaluate=False)
assert Mod(expr, 3**10) == 5137
expr = Pow(2, 2, evaluate=False)
expr = Pow(expr, 2, evaluate=False)
assert Mod(expr, 3**10) == 16
expr = Pow(expr, 2, evaluate=False)
assert Mod(expr, 3**10) == 256
expr = Pow(expr, 2, evaluate=False)
assert Mod(expr, 3**10) == 6487
expr = Pow(expr, 2, evaluate=False)
assert Mod(expr, 3**10) == 38281
expr = Pow(expr, 2, evaluate=False)
assert Mod(expr, 3**10) == 15928
expr = Pow(2, 2, evaluate=False)
expr = Pow(expr, expr, evaluate=False)
assert Mod(expr, 3**10) == 256
expr = Pow(expr, expr, evaluate=False)
assert Mod(expr, 3**10) == 9229
expr = Pow(expr, expr, evaluate=False)
assert Mod(expr, 3**10) == 25708
expr = Pow(expr, expr, evaluate=False)
assert Mod(expr, 3**10) == 26608
expr = Pow(expr, expr, evaluate=False)
# XXX This used to fail in a nondeterministic way because of overflow
# error.
assert Mod(expr, 3**10) == 1966
def test_Mod_is_integer():
p = Symbol('p', integer=True)
q1 = Symbol('q1', integer=True)
q2 = Symbol('q2', integer=True, nonzero=True)
assert Mod(x, y).is_integer is None
assert Mod(p, q1).is_integer is None
assert Mod(x, q2).is_integer is None
assert Mod(p, q2).is_integer
def test_Mod_is_nonposneg():
n = Symbol('n', integer=True)
k = Symbol('k', integer=True, positive=True)
assert (n%3).is_nonnegative
assert Mod(n, -3).is_nonpositive
assert Mod(n, k).is_nonnegative
assert Mod(n, -k).is_nonpositive
assert Mod(k, n).is_nonnegative is None
def test_issue_6001():
A = Symbol("A", commutative=False)
eq = A + A**2
# it doesn't matter whether it's True or False; they should
# just all be the same
assert (
eq.is_commutative ==
(eq + 1).is_commutative ==
(A + 1).is_commutative)
B = Symbol("B", commutative=False)
# Although commutative terms could cancel we return True
# meaning "there are non-commutative symbols; aftersubstitution
# that definition can change, e.g. (A*B).subs(B,A**-1) -> 1
assert (sqrt(2)*A).is_commutative is False
assert (sqrt(2)*A*B).is_commutative is False
def test_polar():
from sympy import polar_lift
p = Symbol('p', polar=True)
x = Symbol('x')
assert p.is_polar
assert x.is_polar is None
assert S.One.is_polar is None
assert (p**x).is_polar is True
assert (x**p).is_polar is None
assert ((2*p)**x).is_polar is True
assert (2*p).is_polar is True
assert (-2*p).is_polar is not True
assert (polar_lift(-2)*p).is_polar is True
q = Symbol('q', polar=True)
assert (p*q)**2 == p**2 * q**2
assert (2*q)**2 == 4 * q**2
assert ((p*q)**x).expand() == p**x * q**x
def test_issue_6040():
a, b = Pow(1, 2, evaluate=False), S.One
assert a != b
assert b != a
assert not (a == b)
assert not (b == a)
def test_issue_6082():
# Comparison is symmetric
assert Basic.compare(Max(x, 1), Max(x, 2)) == \
- Basic.compare(Max(x, 2), Max(x, 1))
# Equal expressions compare equal
assert Basic.compare(Max(x, 1), Max(x, 1)) == 0
# Basic subtypes (such as Max) compare different than standard types
assert Basic.compare(Max(1, x), frozenset((1, x))) != 0
def test_issue_6077():
assert x**2.0/x == x**1.0
assert x/x**2.0 == x**-1.0
assert x*x**2.0 == x**3.0
assert x**1.5*x**2.5 == x**4.0
assert 2**(2.0*x)/2**x == 2**(1.0*x)
assert 2**x/2**(2.0*x) == 2**(-1.0*x)
assert 2**x*2**(2.0*x) == 2**(3.0*x)
assert 2**(1.5*x)*2**(2.5*x) == 2**(4.0*x)
def test_mul_flatten_oo():
p = symbols('p', positive=True)
n, m = symbols('n,m', negative=True)
x_im = symbols('x_im', imaginary=True)
assert n*oo is -oo
assert n*m*oo is oo
assert p*oo is oo
assert x_im*oo != I*oo # i could be +/- 3*I -> +/-oo
def test_add_flatten():
# see https://github.com/sympy/sympy/issues/2633#issuecomment-29545524
a = oo + I*oo
b = oo - I*oo
assert a + b is nan
assert a - b is nan
# FIXME: This evaluates as:
# >>> 1/a
# 0*(oo + oo*I)
# which should not simplify to 0. Should be fixed in Pow.eval
#assert (1/a).simplify() == (1/b).simplify() == 0
a = Pow(2, 3, evaluate=False)
assert a + a == 16
def test_issue_5160_6087_6089_6090():
# issue 6087
assert ((-2*x*y**y)**3.2).n(2) == (2**3.2*(-x*y**y)**3.2).n(2)
# issue 6089
A, B, C = symbols('A,B,C', commutative=False)
assert (2.*B*C)**3 == 8.0*(B*C)**3
assert (-2.*B*C)**3 == -8.0*(B*C)**3
assert (-2*B*C)**2 == 4*(B*C)**2
# issue 5160
assert sqrt(-1.0*x) == 1.0*sqrt(-x)
assert sqrt(1.0*x) == 1.0*sqrt(x)
# issue 6090
assert (-2*x*y*A*B)**2 == 4*x**2*y**2*(A*B)**2
def test_float_int_round():
assert int(float(sqrt(10))) == int(sqrt(10))
assert int(pi**1000) % 10 == 2
assert int(Float('1.123456789012345678901234567890e20', '')) == \
int(112345678901234567890)
assert int(Float('1.123456789012345678901234567890e25', '')) == \
int(11234567890123456789012345)
# decimal forces float so it's not an exact integer ending in 000000
assert int(Float('1.123456789012345678901234567890e35', '')) == \
112345678901234567890123456789000192
assert int(Float('123456789012345678901234567890e5', '')) == \
12345678901234567890123456789000000
assert Integer(Float('1.123456789012345678901234567890e20', '')) == \
112345678901234567890
assert Integer(Float('1.123456789012345678901234567890e25', '')) == \
11234567890123456789012345
# decimal forces float so it's not an exact integer ending in 000000
assert Integer(Float('1.123456789012345678901234567890e35', '')) == \
112345678901234567890123456789000192
assert Integer(Float('123456789012345678901234567890e5', '')) == \
12345678901234567890123456789000000
assert same_and_same_prec(Float('123000e-2',''), Float('1230.00', ''))
assert same_and_same_prec(Float('123000e2',''), Float('12300000', ''))
assert int(1 + Rational('.9999999999999999999999999')) == 1
assert int(pi/1e20) == 0
assert int(1 + pi/1e20) == 1
assert int(Add(1.2, -2, evaluate=False)) == int(1.2 - 2)
assert int(Add(1.2, +2, evaluate=False)) == int(1.2 + 2)
assert int(Add(1 + Float('.99999999999999999', ''), evaluate=False)) == 1
raises(TypeError, lambda: float(x))
raises(TypeError, lambda: float(sqrt(-1)))
assert int(12345678901234567890 + cos(1)**2 + sin(1)**2) == \
12345678901234567891
def test_issue_6611a():
assert Mul.flatten([3**Rational(1, 3),
Pow(-Rational(1, 9), Rational(2, 3), evaluate=False)]) == \
([Rational(1, 3), (-1)**Rational(2, 3)], [], None)
def test_denest_add_mul():
# when working with evaluated expressions make sure they denest
eq = x + 1
eq = Add(eq, 2, evaluate=False)
eq = Add(eq, 2, evaluate=False)
assert Add(*eq.args) == x + 5
eq = x*2
eq = Mul(eq, 2, evaluate=False)
eq = Mul(eq, 2, evaluate=False)
assert Mul(*eq.args) == 8*x
# but don't let them denest unecessarily
eq = Mul(-2, x - 2, evaluate=False)
assert 2*eq == Mul(-4, x - 2, evaluate=False)
assert -eq == Mul(2, x - 2, evaluate=False)
def test_mul_coeff():
# It is important that all Numbers be removed from the seq;
# This can be tricky when powers combine to produce those numbers
p = exp(I*pi/3)
assert p**2*x*p*y*p*x*p**2 == x**2*y
def test_mul_zero_detection():
nz = Dummy(real=True, zero=False)
r = Dummy(extended_real=True)
c = Dummy(real=False, complex=True)
c2 = Dummy(real=False, complex=True)
i = Dummy(imaginary=True)
e = nz*r*c
assert e.is_imaginary is None
assert e.is_extended_real is None
e = nz*c
assert e.is_imaginary is None
assert e.is_extended_real is False
e = nz*i*c
assert e.is_imaginary is False
assert e.is_extended_real is None
# check for more than one complex; it is important to use
# uniquely named Symbols to ensure that two factors appear
# e.g. if the symbols have the same name they just become
# a single factor, a power.
e = nz*i*c*c2
assert e.is_imaginary is None
assert e.is_extended_real is None
# _eval_is_extended_real and _eval_is_zero both employ trapping of the
# zero value so args should be tested in both directions and
# TO AVOID GETTING THE CACHED RESULT, Dummy MUST BE USED
# real is unknown
def test(z, b, e):
if z.is_zero and b.is_finite:
assert e.is_extended_real and e.is_zero
else:
assert e.is_extended_real is None
if b.is_finite:
if z.is_zero:
assert e.is_zero
else:
assert e.is_zero is None
elif b.is_finite is False:
if z.is_zero is None:
assert e.is_zero is None
else:
assert e.is_zero is False
for iz, ib in cartes(*[[True, False, None]]*2):
z = Dummy('z', nonzero=iz)
b = Dummy('f', finite=ib)
e = Mul(z, b, evaluate=False)
test(z, b, e)
z = Dummy('nz', nonzero=iz)
b = Dummy('f', finite=ib)
e = Mul(b, z, evaluate=False)
test(z, b, e)
# real is True
def test(z, b, e):
if z.is_zero and not b.is_finite:
assert e.is_extended_real is None
else:
assert e.is_extended_real is True
for iz, ib in cartes(*[[True, False, None]]*2):
z = Dummy('z', nonzero=iz, extended_real=True)
b = Dummy('b', finite=ib, extended_real=True)
e = Mul(z, b, evaluate=False)
test(z, b, e)
z = Dummy('z', nonzero=iz, extended_real=True)
b = Dummy('b', finite=ib, extended_real=True)
e = Mul(b, z, evaluate=False)
test(z, b, e)
def test_Mul_with_zero_infinite():
zer = Dummy(zero=True)
inf = Dummy(finite=False)
e = Mul(zer, inf, evaluate=False)
assert e.is_extended_positive is None
assert e.is_hermitian is None
e = Mul(inf, zer, evaluate=False)
assert e.is_extended_positive is None
assert e.is_hermitian is None
def test_Mul_does_not_cancel_infinities():
a, b = symbols('a b')
assert ((zoo + 3*a)/(3*a + zoo)) is nan
assert ((b - oo)/(b - oo)) is nan
# issue 13904
expr = (1/(a+b) + 1/(a-b))/(1/(a+b) - 1/(a-b))
assert expr.subs(b, a) is nan
def test_Mul_does_not_distribute_infinity():
a, b = symbols('a b')
assert ((1 + I)*oo).is_Mul
assert ((a + b)*(-oo)).is_Mul
assert ((a + 1)*zoo).is_Mul
assert ((1 + I)*oo).is_finite is False
z = (1 + I)*oo
assert ((1 - I)*z).expand() is oo
def test_issue_8247_8354():
from sympy import tan
z = sqrt(1 + sqrt(3)) + sqrt(3 + 3*sqrt(3)) - sqrt(10 + 6*sqrt(3))
assert z.is_positive is False # it's 0
z = S('''-2**(1/3)*(3*sqrt(93) + 29)**2 - 4*(3*sqrt(93) + 29)**(4/3) +
12*sqrt(93)*(3*sqrt(93) + 29)**(1/3) + 116*(3*sqrt(93) + 29)**(1/3) +
174*2**(1/3)*sqrt(93) + 1678*2**(1/3)''')
assert z.is_positive is False # it's 0
z = 2*(-3*tan(19*pi/90) + sqrt(3))*cos(11*pi/90)*cos(19*pi/90) - \
sqrt(3)*(-3 + 4*cos(19*pi/90)**2)
assert z.is_positive is not True # it's zero and it shouldn't hang
z = S('''9*(3*sqrt(93) + 29)**(2/3)*((3*sqrt(93) +
29)**(1/3)*(-2**(2/3)*(3*sqrt(93) + 29)**(1/3) - 2) - 2*2**(1/3))**3 +
72*(3*sqrt(93) + 29)**(2/3)*(81*sqrt(93) + 783) + (162*sqrt(93) +
1566)*((3*sqrt(93) + 29)**(1/3)*(-2**(2/3)*(3*sqrt(93) + 29)**(1/3) -
2) - 2*2**(1/3))**2''')
assert z.is_positive is False # it's 0 (and a single _mexpand isn't enough)
def test_Add_is_zero():
x, y = symbols('x y', zero=True)
assert (x + y).is_zero
# Issue 15873
e = -2*I + (1 + I)**2
assert e.is_zero is None
def test_issue_14392():
assert (sin(zoo)**2).as_real_imag() == (nan, nan)
def test_divmod():
assert divmod(x, y) == (x//y, x % y)
assert divmod(x, 3) == (x//3, x % 3)
assert divmod(3, x) == (3//x, 3 % x)
def test__neg__():
assert -(x*y) == -x*y
assert -(-x*y) == x*y
assert -(1.*x) == -1.*x
assert -(-1.*x) == 1.*x
assert -(2.*x) == -2.*x
assert -(-2.*x) == 2.*x
with distribute(False):
eq = -(x + y)
assert eq.is_Mul and eq.args == (-1, x + y)
def test_issue_18507():
assert Mul(zoo, zoo, 0) is nan
|
37a7860943a0eb4d366f5662e5ac4e23122ac4452c2add1de4322e1963bbc12d | """Tests that the IPython printing module is properly loaded. """
from sympy.interactive.session import init_ipython_session
from sympy.external import import_module
from sympy.testing.pytest import raises
# run_cell was added in IPython 0.11
ipython = import_module("IPython", min_module_version="0.11")
# disable tests if ipython is not present
if not ipython:
disabled = True
def test_ipythonprinting():
# Initialize and setup IPython session
app = init_ipython_session()
app.run_cell("ip = get_ipython()")
app.run_cell("inst = ip.instance()")
app.run_cell("format = inst.display_formatter.format")
app.run_cell("from sympy import Symbol")
# Printing without printing extension
app.run_cell("a = format(Symbol('pi'))")
app.run_cell("a2 = format(Symbol('pi')**2)")
# Deal with API change starting at IPython 1.0
if int(ipython.__version__.split(".")[0]) < 1:
assert app.user_ns['a']['text/plain'] == "pi"
assert app.user_ns['a2']['text/plain'] == "pi**2"
else:
assert app.user_ns['a'][0]['text/plain'] == "pi"
assert app.user_ns['a2'][0]['text/plain'] == "pi**2"
# Load printing extension
app.run_cell("from sympy import init_printing")
app.run_cell("init_printing()")
# Printing with printing extension
app.run_cell("a = format(Symbol('pi'))")
app.run_cell("a2 = format(Symbol('pi')**2)")
# Deal with API change starting at IPython 1.0
if int(ipython.__version__.split(".")[0]) < 1:
assert app.user_ns['a']['text/plain'] in ('\N{GREEK SMALL LETTER PI}', 'pi')
assert app.user_ns['a2']['text/plain'] in (' 2\n\N{GREEK SMALL LETTER PI} ', ' 2\npi ')
else:
assert app.user_ns['a'][0]['text/plain'] in ('\N{GREEK SMALL LETTER PI}', 'pi')
assert app.user_ns['a2'][0]['text/plain'] in (' 2\n\N{GREEK SMALL LETTER PI} ', ' 2\npi ')
def test_print_builtin_option():
# Initialize and setup IPython session
app = init_ipython_session()
app.run_cell("ip = get_ipython()")
app.run_cell("inst = ip.instance()")
app.run_cell("format = inst.display_formatter.format")
app.run_cell("from sympy import Symbol")
app.run_cell("from sympy import init_printing")
app.run_cell("a = format({Symbol('pi'): 3.14, Symbol('n_i'): 3})")
# Deal with API change starting at IPython 1.0
if int(ipython.__version__.split(".")[0]) < 1:
text = app.user_ns['a']['text/plain']
raises(KeyError, lambda: app.user_ns['a']['text/latex'])
else:
text = app.user_ns['a'][0]['text/plain']
raises(KeyError, lambda: app.user_ns['a'][0]['text/latex'])
# Note : Unicode of Python2 is equivalent to str in Python3. In Python 3 we have one
# text type: str which holds Unicode data and two byte types bytes and bytearray.
# XXX: How can we make this ignore the terminal width? This test fails if
# the terminal is too narrow.
assert text in ("{pi: 3.14, n_i: 3}",
'{n\N{LATIN SUBSCRIPT SMALL LETTER I}: 3, \N{GREEK SMALL LETTER PI}: 3.14}',
"{n_i: 3, pi: 3.14}",
'{\N{GREEK SMALL LETTER PI}: 3.14, n\N{LATIN SUBSCRIPT SMALL LETTER I}: 3}')
# If we enable the default printing, then the dictionary's should render
# as a LaTeX version of the whole dict: ${\pi: 3.14, n_i: 3}$
app.run_cell("inst.display_formatter.formatters['text/latex'].enabled = True")
app.run_cell("init_printing(use_latex=True)")
app.run_cell("a = format({Symbol('pi'): 3.14, Symbol('n_i'): 3})")
# Deal with API change starting at IPython 1.0
if int(ipython.__version__.split(".")[0]) < 1:
text = app.user_ns['a']['text/plain']
latex = app.user_ns['a']['text/latex']
else:
text = app.user_ns['a'][0]['text/plain']
latex = app.user_ns['a'][0]['text/latex']
assert text in ("{pi: 3.14, n_i: 3}",
'{n\N{LATIN SUBSCRIPT SMALL LETTER I}: 3, \N{GREEK SMALL LETTER PI}: 3.14}',
"{n_i: 3, pi: 3.14}",
'{\N{GREEK SMALL LETTER PI}: 3.14, n\N{LATIN SUBSCRIPT SMALL LETTER I}: 3}')
assert latex == r'$\displaystyle \left\{ n_{i} : 3, \ \pi : 3.14\right\}$'
# Objects with an _latex overload should also be handled by our tuple
# printer.
app.run_cell("""\
class WithOverload:
def _latex(self, printer):
return r"\\LaTeX"
""")
app.run_cell("a = format((WithOverload(),))")
# Deal with API change starting at IPython 1.0
if int(ipython.__version__.split(".")[0]) < 1:
latex = app.user_ns['a']['text/latex']
else:
latex = app.user_ns['a'][0]['text/latex']
assert latex == r'$\displaystyle \left( \LaTeX,\right)$'
app.run_cell("inst.display_formatter.formatters['text/latex'].enabled = True")
app.run_cell("init_printing(use_latex=True, print_builtin=False)")
app.run_cell("a = format({Symbol('pi'): 3.14, Symbol('n_i'): 3})")
# Deal with API change starting at IPython 1.0
if int(ipython.__version__.split(".")[0]) < 1:
text = app.user_ns['a']['text/plain']
raises(KeyError, lambda: app.user_ns['a']['text/latex'])
else:
text = app.user_ns['a'][0]['text/plain']
raises(KeyError, lambda: app.user_ns['a'][0]['text/latex'])
# Note : In Python 3 we have one text type: str which holds Unicode data
# and two byte types bytes and bytearray.
# Python 3.3.3 + IPython 0.13.2 gives: '{n_i: 3, pi: 3.14}'
# Python 3.3.3 + IPython 1.1.0 gives: '{n_i: 3, pi: 3.14}'
assert text in ("{pi: 3.14, n_i: 3}", "{n_i: 3, pi: 3.14}")
def test_builtin_containers():
# Initialize and setup IPython session
app = init_ipython_session()
app.run_cell("ip = get_ipython()")
app.run_cell("inst = ip.instance()")
app.run_cell("format = inst.display_formatter.format")
app.run_cell("inst.display_formatter.formatters['text/latex'].enabled = True")
app.run_cell("from sympy import init_printing, Matrix")
app.run_cell('init_printing(use_latex=True, use_unicode=False)')
# Make sure containers that shouldn't pretty print don't.
app.run_cell('a = format((True, False))')
app.run_cell('import sys')
app.run_cell('b = format(sys.flags)')
app.run_cell('c = format((Matrix([1, 2]),))')
# Deal with API change starting at IPython 1.0
if int(ipython.__version__.split(".")[0]) < 1:
assert app.user_ns['a']['text/plain'] == '(True, False)'
assert 'text/latex' not in app.user_ns['a']
assert app.user_ns['b']['text/plain'][:10] == 'sys.flags('
assert 'text/latex' not in app.user_ns['b']
assert app.user_ns['c']['text/plain'] == \
"""\
[1] \n\
([ ],)
[2] \
"""
assert app.user_ns['c']['text/latex'] == '$\\displaystyle \\left( \\left[\\begin{matrix}1\\\\2\\end{matrix}\\right],\\right)$'
else:
assert app.user_ns['a'][0]['text/plain'] == '(True, False)'
assert 'text/latex' not in app.user_ns['a'][0]
assert app.user_ns['b'][0]['text/plain'][:10] == 'sys.flags('
assert 'text/latex' not in app.user_ns['b'][0]
assert app.user_ns['c'][0]['text/plain'] == \
"""\
[1] \n\
([ ],)
[2] \
"""
assert app.user_ns['c'][0]['text/latex'] == '$\\displaystyle \\left( \\left[\\begin{matrix}1\\\\2\\end{matrix}\\right],\\right)$'
def test_matplotlib_bad_latex():
# Initialize and setup IPython session
app = init_ipython_session()
app.run_cell("import IPython")
app.run_cell("ip = get_ipython()")
app.run_cell("inst = ip.instance()")
app.run_cell("format = inst.display_formatter.format")
app.run_cell("from sympy import init_printing, Matrix")
app.run_cell("init_printing(use_latex='matplotlib')")
# The png formatter is not enabled by default in this context
app.run_cell("inst.display_formatter.formatters['image/png'].enabled = True")
# Make sure no warnings are raised by IPython
app.run_cell("import warnings")
# IPython.core.formatters.FormatterWarning was introduced in IPython 2.0
if int(ipython.__version__.split(".")[0]) < 2:
app.run_cell("warnings.simplefilter('error')")
else:
app.run_cell("warnings.simplefilter('error', IPython.core.formatters.FormatterWarning)")
# This should not raise an exception
app.run_cell("a = format(Matrix([1, 2, 3]))")
# issue 9799
app.run_cell("from sympy import Piecewise, Symbol, Eq")
app.run_cell("x = Symbol('x'); pw = format(Piecewise((1, Eq(x, 0)), (0, True)))")
|
8d6f125b0531912b5d457690b218e49d73886ebbfaba8be46049f2848b9ccf4e | """Domains of Gaussian type."""
from sympy.core.basic import Basic
from sympy.core.numbers import I
from sympy.polys.polyerrors import CoercionFailed
from sympy.polys.domains import ZZ, QQ
from sympy.polys.domains.domainelement import DomainElement
from sympy.polys.domains.field import Field
from sympy.polys.domains.ring import Ring
class GaussianElement(DomainElement):
"""Base class for elements of Gaussian type domains."""
base = None # base ring
_parent = None
__slots__ = ('x', 'y')
def __init__(self, x, y):
conv = self.base.convert
self.x = conv(x)
self.y = conv(y)
def parent(self):
return self._parent
def __hash__(self):
return hash((self.x, self.y))
def __eq__(self, other):
if isinstance(other, self.__class__):
return self.x == other.x and self.y == other.y
else:
return NotImplemented
def __neg__(self):
return self.__class__(-self.x, -self.y)
def __repr__(self):
return "%s(%s, %s)" % (type(self).__name__, self.x, self.y)
def __str__(self):
return str(self._parent.to_sympy(self))
@classmethod
def _get_xy(cls, other):
if not isinstance(other, cls):
try:
other = cls._parent.convert(other)
except CoercionFailed:
return None, None
return other.x, other.y
def __add__(self, other):
x, y = self._get_xy(other)
if x is not None:
return self.__class__(self.x + x, self.y + y)
else:
return NotImplemented
__radd__ = __add__
def __sub__(self, other):
x, y = self._get_xy(other)
if x is not None:
return self.__class__(self.x - x, self.y - y)
else:
return NotImplemented
def __rsub__(self, other):
x, y = self._get_xy(other)
if x is not None:
return self.__class__(x - self.x, y - self.y)
else:
return NotImplemented
def __mul__(self, other):
x, y = self._get_xy(other)
if x is not None:
return self.__class__(self.x*x - self.y*y,
self.x*y + self.y*x)
__rmul__ = __mul__
def __bool__(self):
return bool(self.x) or bool(self.y)
__nonzero__ = __bool__ # for Python 2
def quadrant(self):
"""Return quadrant index 0-3.
0 is included in quadrant 0.
"""
if self.y > 0:
return 0 if self.x > 0 else 1
elif self.y < 0:
return 2 if self.x < 0 else 3
else:
return 0 if self.x >= 0 else 2
class GaussianInteger(GaussianElement):
base = ZZ
def __truediv__(self, other):
"""Return a Gaussian rational."""
return QQ_I.convert(self)/other
__div__ = __truediv__
def __rtruediv__(self, other):
return other/QQ_I.convert(self)
__rdiv__ = __rtruediv__
def __divmod__(self, other):
if not other:
raise ZeroDivisionError('divmod({}, 0)'.format(self))
x, y = self._get_xy(other)
if x is None:
return NotImplemented
# multiply self and other by x - I*y
# self/other == (a + I*b)/c
a, b = self.x*x + self.y*y, -self.x*y + self.y*x
c = x*x + y*y
# find integers qx and qy such that
# |a - qx*c| <= c/2 and |b - qy*c| <= c/2
qx = (2*a + c) // (2*c) # -c <= 2*a - qx*2*c < c
qy = (2*b + c) // (2*c)
q = GaussianInteger(qx, qy)
# |self/other - q| < 1 since
# |a/c - qx|**2 + |b/c - qy|**2 <= 1/4 + 1/4 < 1
return q, self - q*other # |r| < |other|
def __rdivmod__(self, other):
try:
other = self._parent.convert(other)
except CoercionFailed:
return NotImplemented
else:
return other.__divmod__(self)
def __floordiv__(self, other):
qr = self.__divmod__(other)
return qr if qr is NotImplemented else qr[0]
def __rfloordiv__(self, other):
qr = self.__rdivmod__(other)
return qr if qr is NotImplemented else qr[0]
def __mod__(self, other):
qr = self.__divmod__(other)
return qr if qr is NotImplemented else qr[1]
def __rmod__(self, other):
qr = self.__rdivmod__(other)
return qr if qr is NotImplemented else qr[1]
class GaussianRational(GaussianElement):
base = QQ
def __truediv__(self, other):
if not other:
raise ZeroDivisionError('{} / 0'.format(self))
x, y = self._get_xy(other)
if x is None:
return NotImplemented
c = x*x + y*y
return GaussianRational((self.x*x + self.y*y)/c,
(-self.x*y + self.y*x)/c)
__floordiv__ = __div__ = __truediv__
def __rtruediv__(self, other):
try:
other = self._parent.convert(other)
except CoercionFailed:
return NotImplemented
else:
return other.__truediv__(self)
__rfloordiv__ = __rdiv__ = __rtruediv__
def __mod__(self, other):
try:
other = self._parent.convert(other)
except CoercionFailed:
return NotImplemented
if not other:
raise ZeroDivisionError('{} % 0'.format(self))
else:
return self._parent.zero # XXX always 0?
def __rmod__(self, other):
try:
other = self._parent.convert(other)
except CoercionFailed:
return NotImplemented
else:
return other.__mod__(self)
def __divmod__(self, other):
return self.__truediv__(other), self.__mod__(other)
def __rdivmod__(self, other):
return self.__rtruediv__(other), self.__rmod__(other)
class GaussianDomain():
"""Base class for Gaussian domains."""
base = None # base domain, ZZ or QQ
has_assoc_Ring = True
has_assoc_Field = True
def to_sympy(self, a):
"""Convert ``a`` to a SymPy object. """
conv = self.base.to_sympy
return conv(a.x) + I*conv(a.y)
def from_sympy(self, a):
"""Convert a SymPy object to ``self.dtype``."""
r, b = a.as_coeff_Add()
x = self.base.from_sympy(r) # may raise CoercionFailed
if not b:
return self.new(x, 0)
r, b = b.as_coeff_Mul()
y = self.base.from_sympy(r)
if b is I:
return self.new(x, y)
else:
raise CoercionFailed("{} is not Gaussian".format(a))
def convert(self, element):
"""Convert ``element`` to ``self.dtype``.
Raises CoercionFailed on failure.
"""
if isinstance(element, self.dtype):
return element
elif isinstance(element, GaussianElement):
return self.new(element.x, element.y)
elif isinstance(element, Basic):
return self.from_sympy(element)
else: # convertible to base type or failure
return self.new(element, 0)
class GaussianIntegerRing(GaussianDomain, Ring):
"""Ring of Gaussian integers."""
base = ZZ
dtype = GaussianInteger
zero = dtype(0, 0)
one = dtype(1, 0)
imag_unit = dtype(0, 1)
units = (one, imag_unit, -one, -imag_unit) # powers of i
rep = 'ZZ_I'
def __init__(self): # override Domain.__init__
"""For constructing ZZ_I."""
def get_ring(self):
"""Returns a ring associated with ``self``. """
return self
def get_field(self):
"""Returns a field associated with ``self``. """
return QQ_I
def normalize(self, d, *args):
"""Return first quadrant element associated with ``d``.
Also multiply the other arguments by the same power of i.
"""
unit = self.units[-d.quadrant()] # - for inverse power
d *= unit
args = tuple(a*unit for a in args)
return (d,) + args if args else d
def gcd(self, a, b):
while b:
a, b = b, a % b
return self.normalize(a)
ZZ_I = GaussianInteger._parent = GaussianIntegerRing()
class GaussianRationalField(GaussianDomain, Field):
"""Field of Gaussian rational numbers."""
base = QQ
dtype = GaussianRational
zero = dtype(0, 0)
one = dtype(1, 0)
rep = 'QQ_I'
def __init__(self): # override Domain.__init__
"""For constructing QQ_I."""
def get_ring(self):
"""Returns a ring associated with ``self``. """
return ZZ_I
def get_field(self):
"""Returns a field associated with ``self``. """
return self
QQ_I = GaussianRational._parent = GaussianRationalField()
|
0f0042697b84511743b9acedd00d3971412b74b3b2bf48ef07069a67c4566aba | """Tests for algorithms for partial fraction decomposition of rational
functions. """
from sympy.polys.partfrac import (
apart_undetermined_coeffs,
apart,
apart_list, assemble_partfrac_list
)
from sympy import (S, Poly, E, pi, I, Matrix, Eq, RootSum, Lambda,
Symbol, Dummy, factor, together, sqrt, Expr, Rational)
from sympy.testing.pytest import raises, XFAIL
from sympy.abc import x, y, a, b, c
def test_apart():
assert apart(1) == 1
assert apart(1, x) == 1
f, g = (x**2 + 1)/(x + 1), 2/(x + 1) + x - 1
assert apart(f, full=False) == g
assert apart(f, full=True) == g
f, g = 1/(x + 2)/(x + 1), 1/(1 + x) - 1/(2 + x)
assert apart(f, full=False) == g
assert apart(f, full=True) == g
f, g = 1/(x + 1)/(x + 5), -1/(5 + x)/4 + 1/(1 + x)/4
assert apart(f, full=False) == g
assert apart(f, full=True) == g
assert apart((E*x + 2)/(x - pi)*(x - 1), x) == \
2 - E + E*pi + E*x + (E*pi + 2)*(pi - 1)/(x - pi)
assert apart(Eq((x**2 + 1)/(x + 1), x), x) == Eq(x - 1 + 2/(x + 1), x)
assert apart(x/2, y) == x/2
f, g = (x+y)/(2*x - y), Rational(3, 2)*y/((2*x - y)) + S.Half
assert apart(f, x, full=False) == g
assert apart(f, x, full=True) == g
f, g = (x+y)/(2*x - y), 3*x/(2*x - y) - 1
assert apart(f, y, full=False) == g
assert apart(f, y, full=True) == g
raises(NotImplementedError, lambda: apart(1/(x + 1)/(y + 2)))
def test_apart_matrix():
M = Matrix(2, 2, lambda i, j: 1/(x + i + 1)/(x + j))
assert apart(M) == Matrix([
[1/x - 1/(x + 1), (x + 1)**(-2)],
[1/(2*x) - (S.Half)/(x + 2), 1/(x + 1) - 1/(x + 2)],
])
def test_apart_symbolic():
f = a*x**4 + (2*b + 2*a*c)*x**3 + (4*b*c - a**2 + a*c**2)*x**2 + \
(-2*a*b + 2*b*c**2)*x - b**2
g = a**2*x**4 + (2*a*b + 2*c*a**2)*x**3 + (4*a*b*c + b**2 +
a**2*c**2)*x**2 + (2*c*b**2 + 2*a*b*c**2)*x + b**2*c**2
assert apart(f/g, x) == 1/a - 1/(x + c)**2 - b**2/(a*(a*x + b)**2)
assert apart(1/((x + a)*(x + b)*(x + c)), x) == \
1/((a - c)*(b - c)*(c + x)) - 1/((a - b)*(b - c)*(b + x)) + \
1/((a - b)*(a - c)*(a + x))
def test_apart_extension():
f = 2/(x**2 + 1)
g = I/(x + I) - I/(x - I)
assert apart(f, extension=I) == g
assert apart(f, gaussian=True) == g
f = x/((x - 2)*(x + I))
assert factor(together(apart(f)).expand()) == f
# https://github.com/sympy/sympy/issues/18531
from sympy.core import Mul
def mul2(expr):
# 2-arg mul hack...
return Mul(2, expr, evaluate=False)
f = ((x**2 + 1)**3/((x - 1)**2*(x + 1)**2*(-x**2 + 2*x + 1)*(x**2 + 2*x - 1)))
g = (1/mul2(x - sqrt(2) + 1)
- 1/mul2(x - sqrt(2) - 1)
+ 1/mul2(x + 1 + sqrt(2))
- 1/mul2(x - 1 + sqrt(2))
+ 1/mul2((x + 1)**2)
+ 1/mul2((x - 1)**2))
assert apart(f, x, extension={sqrt(2)}) == g
def test_apart_full():
f = 1/(x**2 + 1)
assert apart(f, full=False) == f
assert apart(f, full=True).dummy_eq(
-RootSum(x**2 + 1, Lambda(a, a/(x - a)), auto=False)/2)
f = 1/(x**3 + x + 1)
assert apart(f, full=False) == f
assert apart(f, full=True).dummy_eq(
RootSum(x**3 + x + 1,
Lambda(a, (a**2*Rational(6, 31) - a*Rational(9, 31) + Rational(4, 31))/(x - a)), auto=False))
f = 1/(x**5 + 1)
assert apart(f, full=False) == \
(Rational(-1, 5))*((x**3 - 2*x**2 + 3*x - 4)/(x**4 - x**3 + x**2 -
x + 1)) + (Rational(1, 5))/(x + 1)
assert apart(f, full=True).dummy_eq(
-RootSum(x**4 - x**3 + x**2 - x + 1,
Lambda(a, a/(x - a)), auto=False)/5 + (Rational(1, 5))/(x + 1))
def test_apart_undetermined_coeffs():
p = Poly(2*x - 3)
q = Poly(x**9 - x**8 - x**6 + x**5 - 2*x**2 + 3*x - 1)
r = (-x**7 - x**6 - x**5 + 4)/(x**8 - x**5 - 2*x + 1) + 1/(x - 1)
assert apart_undetermined_coeffs(p, q) == r
p = Poly(1, x, domain='ZZ[a,b]')
q = Poly((x + a)*(x + b), x, domain='ZZ[a,b]')
r = 1/((a - b)*(b + x)) - 1/((a - b)*(a + x))
assert apart_undetermined_coeffs(p, q) == r
def test_apart_list():
from sympy.utilities.iterables import numbered_symbols
def dummy_eq(i, j):
if type(i) in (list, tuple):
return all(dummy_eq(i, j) for i, j in zip(i, j))
return i == j or i.dummy_eq(j)
w0, w1, w2 = Symbol("w0"), Symbol("w1"), Symbol("w2")
_a = Dummy("a")
f = (-2*x - 2*x**2) / (3*x**2 - 6*x)
got = apart_list(f, x, dummies=numbered_symbols("w"))
ans = (-1, Poly(Rational(2, 3), x, domain='QQ'),
[(Poly(w0 - 2, w0, domain='ZZ'), Lambda(_a, 2), Lambda(_a, -_a + x), 1)])
assert dummy_eq(got, ans)
got = apart_list(2/(x**2-2), x, dummies=numbered_symbols("w"))
ans = (1, Poly(0, x, domain='ZZ'), [(Poly(w0**2 - 2, w0, domain='ZZ'),
Lambda(_a, _a/2),
Lambda(_a, -_a + x), 1)])
assert dummy_eq(got, ans)
f = 36 / (x**5 - 2*x**4 - 2*x**3 + 4*x**2 + x - 2)
got = apart_list(f, x, dummies=numbered_symbols("w"))
ans = (1, Poly(0, x, domain='ZZ'),
[(Poly(w0 - 2, w0, domain='ZZ'), Lambda(_a, 4), Lambda(_a, -_a + x), 1),
(Poly(w1**2 - 1, w1, domain='ZZ'), Lambda(_a, -3*_a - 6), Lambda(_a, -_a + x), 2),
(Poly(w2 + 1, w2, domain='ZZ'), Lambda(_a, -4), Lambda(_a, -_a + x), 1)])
assert dummy_eq(got, ans)
def test_assemble_partfrac_list():
f = 36 / (x**5 - 2*x**4 - 2*x**3 + 4*x**2 + x - 2)
pfd = apart_list(f)
assert assemble_partfrac_list(pfd) == -4/(x + 1) - 3/(x + 1)**2 - 9/(x - 1)**2 + 4/(x - 2)
a = Dummy("a")
pfd = (1, Poly(0, x, domain='ZZ'), [([sqrt(2),-sqrt(2)], Lambda(a, a/2), Lambda(a, -a + x), 1)])
assert assemble_partfrac_list(pfd) == -1/(sqrt(2)*(x + sqrt(2))) + 1/(sqrt(2)*(x - sqrt(2)))
@XFAIL
def test_noncommutative_pseudomultivariate():
# apart doesn't go inside noncommutative expressions
class foo(Expr):
is_commutative=False
e = x/(x + x*y)
c = 1/(1 + y)
assert apart(e + foo(e)) == c + foo(c)
assert apart(e*foo(e)) == c*foo(c)
def test_noncommutative():
class foo(Expr):
is_commutative=False
e = x/(x + x*y)
c = 1/(1 + y)
assert apart(e + foo()) == c + foo()
def test_issue_5798():
assert apart(
2*x/(x**2 + 1) - (x - 1)/(2*(x**2 + 1)) + 1/(2*(x + 1)) - 2/x) == \
(3*x + 1)/(x**2 + 1)/2 + 1/(x + 1)/2 - 2/x
|
4f14166cd9fe036da7a71ccf9d165e11f7610f14517b2aef948c967f748962fb | """Tests for user-friendly public interface to polynomial functions. """
from sympy.polys.polytools import (
Poly, PurePoly, poly,
parallel_poly_from_expr,
degree, degree_list,
total_degree,
LC, LM, LT,
pdiv, prem, pquo, pexquo,
div, rem, quo, exquo,
half_gcdex, gcdex, invert,
subresultants,
resultant, discriminant,
terms_gcd, cofactors,
gcd, gcd_list,
lcm, lcm_list,
trunc,
monic, content, primitive,
compose, decompose,
sturm,
gff_list, gff,
sqf_norm, sqf_part, sqf_list, sqf,
factor_list, factor,
intervals, refine_root, count_roots,
real_roots, nroots, ground_roots,
nth_power_roots_poly,
cancel, reduced, groebner,
GroebnerBasis, is_zero_dimensional,
_torational_factor_list,
to_rational_coeffs)
from sympy.polys.polyerrors import (
MultivariatePolynomialError,
ExactQuotientFailed,
PolificationFailed,
ComputationFailed,
UnificationFailed,
RefinementFailed,
GeneratorsNeeded,
GeneratorsError,
PolynomialError,
CoercionFailed,
DomainError,
OptionError,
FlagError)
from sympy.polys.polyclasses import DMP
from sympy.polys.fields import field
from sympy.polys.domains import FF, ZZ, QQ, RR, EX
from sympy.polys.domains.realfield import RealField
from sympy.polys.orderings import lex, grlex, grevlex
from sympy import (
S, Integer, Rational, Float, Mul, Symbol, sqrt, Piecewise, Derivative,
exp, sin, tanh, expand, oo, I, pi, re, im, rootof, Eq, Tuple, Expr, diff)
from sympy.core.basic import _aresame
from sympy.core.compatibility import iterable
from sympy.core.mul import _keep_coeff
from sympy.testing.pytest import raises, warns_deprecated_sympy
from sympy.abc import a, b, c, d, p, q, t, w, x, y, z
from sympy import MatrixSymbol, Matrix
def _epsilon_eq(a, b):
for u, v in zip(a, b):
if abs(u - v) > 1e-10:
return False
return True
def _strict_eq(a, b):
if type(a) == type(b):
if iterable(a):
if len(a) == len(b):
return all(_strict_eq(c, d) for c, d in zip(a, b))
else:
return False
else:
return isinstance(a, Poly) and a.eq(b, strict=True)
else:
return False
def test_Poly_mixed_operations():
p = Poly(x, x)
with warns_deprecated_sympy():
p * exp(x)
with warns_deprecated_sympy():
p + exp(x)
with warns_deprecated_sympy():
p - exp(x)
def test_Poly_from_dict():
K = FF(3)
assert Poly.from_dict(
{0: 1, 1: 2}, gens=x, domain=K).rep == DMP([K(2), K(1)], K)
assert Poly.from_dict(
{0: 1, 1: 5}, gens=x, domain=K).rep == DMP([K(2), K(1)], K)
assert Poly.from_dict(
{(0,): 1, (1,): 2}, gens=x, domain=K).rep == DMP([K(2), K(1)], K)
assert Poly.from_dict(
{(0,): 1, (1,): 5}, gens=x, domain=K).rep == DMP([K(2), K(1)], K)
assert Poly.from_dict({(0, 0): 1, (1, 1): 2}, gens=(
x, y), domain=K).rep == DMP([[K(2), K(0)], [K(1)]], K)
assert Poly.from_dict({0: 1, 1: 2}, gens=x).rep == DMP([ZZ(2), ZZ(1)], ZZ)
assert Poly.from_dict(
{0: 1, 1: 2}, gens=x, field=True).rep == DMP([QQ(2), QQ(1)], QQ)
assert Poly.from_dict(
{0: 1, 1: 2}, gens=x, domain=ZZ).rep == DMP([ZZ(2), ZZ(1)], ZZ)
assert Poly.from_dict(
{0: 1, 1: 2}, gens=x, domain=QQ).rep == DMP([QQ(2), QQ(1)], QQ)
assert Poly.from_dict(
{(0,): 1, (1,): 2}, gens=x).rep == DMP([ZZ(2), ZZ(1)], ZZ)
assert Poly.from_dict(
{(0,): 1, (1,): 2}, gens=x, field=True).rep == DMP([QQ(2), QQ(1)], QQ)
assert Poly.from_dict(
{(0,): 1, (1,): 2}, gens=x, domain=ZZ).rep == DMP([ZZ(2), ZZ(1)], ZZ)
assert Poly.from_dict(
{(0,): 1, (1,): 2}, gens=x, domain=QQ).rep == DMP([QQ(2), QQ(1)], QQ)
assert Poly.from_dict({(1,): sin(y)}, gens=x, composite=False) == \
Poly(sin(y)*x, x, domain='EX')
assert Poly.from_dict({(1,): y}, gens=x, composite=False) == \
Poly(y*x, x, domain='EX')
assert Poly.from_dict({(1, 1): 1}, gens=(x, y), composite=False) == \
Poly(x*y, x, y, domain='ZZ')
assert Poly.from_dict({(1, 0): y}, gens=(x, z), composite=False) == \
Poly(y*x, x, z, domain='EX')
def test_Poly_from_list():
K = FF(3)
assert Poly.from_list([2, 1], gens=x, domain=K).rep == DMP([K(2), K(1)], K)
assert Poly.from_list([5, 1], gens=x, domain=K).rep == DMP([K(2), K(1)], K)
assert Poly.from_list([2, 1], gens=x).rep == DMP([ZZ(2), ZZ(1)], ZZ)
assert Poly.from_list([2, 1], gens=x, field=True).rep == DMP([QQ(2), QQ(1)], QQ)
assert Poly.from_list([2, 1], gens=x, domain=ZZ).rep == DMP([ZZ(2), ZZ(1)], ZZ)
assert Poly.from_list([2, 1], gens=x, domain=QQ).rep == DMP([QQ(2), QQ(1)], QQ)
assert Poly.from_list([0, 1.0], gens=x).rep == DMP([RR(1.0)], RR)
assert Poly.from_list([1.0, 0], gens=x).rep == DMP([RR(1.0), RR(0.0)], RR)
raises(MultivariatePolynomialError, lambda: Poly.from_list([[]], gens=(x, y)))
def test_Poly_from_poly():
f = Poly(x + 7, x, domain=ZZ)
g = Poly(x + 2, x, modulus=3)
h = Poly(x + y, x, y, domain=ZZ)
K = FF(3)
assert Poly.from_poly(f) == f
assert Poly.from_poly(f, domain=K).rep == DMP([K(1), K(1)], K)
assert Poly.from_poly(f, domain=ZZ).rep == DMP([1, 7], ZZ)
assert Poly.from_poly(f, domain=QQ).rep == DMP([1, 7], QQ)
assert Poly.from_poly(f, gens=x) == f
assert Poly.from_poly(f, gens=x, domain=K).rep == DMP([K(1), K(1)], K)
assert Poly.from_poly(f, gens=x, domain=ZZ).rep == DMP([1, 7], ZZ)
assert Poly.from_poly(f, gens=x, domain=QQ).rep == DMP([1, 7], QQ)
assert Poly.from_poly(f, gens=y) == Poly(x + 7, y, domain='ZZ[x]')
raises(CoercionFailed, lambda: Poly.from_poly(f, gens=y, domain=K))
raises(CoercionFailed, lambda: Poly.from_poly(f, gens=y, domain=ZZ))
raises(CoercionFailed, lambda: Poly.from_poly(f, gens=y, domain=QQ))
assert Poly.from_poly(f, gens=(x, y)) == Poly(x + 7, x, y, domain='ZZ')
assert Poly.from_poly(
f, gens=(x, y), domain=ZZ) == Poly(x + 7, x, y, domain='ZZ')
assert Poly.from_poly(
f, gens=(x, y), domain=QQ) == Poly(x + 7, x, y, domain='QQ')
assert Poly.from_poly(
f, gens=(x, y), modulus=3) == Poly(x + 7, x, y, domain='FF(3)')
K = FF(2)
assert Poly.from_poly(g) == g
assert Poly.from_poly(g, domain=ZZ).rep == DMP([1, -1], ZZ)
raises(CoercionFailed, lambda: Poly.from_poly(g, domain=QQ))
assert Poly.from_poly(g, domain=K).rep == DMP([K(1), K(0)], K)
assert Poly.from_poly(g, gens=x) == g
assert Poly.from_poly(g, gens=x, domain=ZZ).rep == DMP([1, -1], ZZ)
raises(CoercionFailed, lambda: Poly.from_poly(g, gens=x, domain=QQ))
assert Poly.from_poly(g, gens=x, domain=K).rep == DMP([K(1), K(0)], K)
K = FF(3)
assert Poly.from_poly(h) == h
assert Poly.from_poly(
h, domain=ZZ).rep == DMP([[ZZ(1)], [ZZ(1), ZZ(0)]], ZZ)
assert Poly.from_poly(
h, domain=QQ).rep == DMP([[QQ(1)], [QQ(1), QQ(0)]], QQ)
assert Poly.from_poly(h, domain=K).rep == DMP([[K(1)], [K(1), K(0)]], K)
assert Poly.from_poly(h, gens=x) == Poly(x + y, x, domain=ZZ[y])
raises(CoercionFailed, lambda: Poly.from_poly(h, gens=x, domain=ZZ))
assert Poly.from_poly(
h, gens=x, domain=ZZ[y]) == Poly(x + y, x, domain=ZZ[y])
raises(CoercionFailed, lambda: Poly.from_poly(h, gens=x, domain=QQ))
assert Poly.from_poly(
h, gens=x, domain=QQ[y]) == Poly(x + y, x, domain=QQ[y])
raises(CoercionFailed, lambda: Poly.from_poly(h, gens=x, modulus=3))
assert Poly.from_poly(h, gens=y) == Poly(x + y, y, domain=ZZ[x])
raises(CoercionFailed, lambda: Poly.from_poly(h, gens=y, domain=ZZ))
assert Poly.from_poly(
h, gens=y, domain=ZZ[x]) == Poly(x + y, y, domain=ZZ[x])
raises(CoercionFailed, lambda: Poly.from_poly(h, gens=y, domain=QQ))
assert Poly.from_poly(
h, gens=y, domain=QQ[x]) == Poly(x + y, y, domain=QQ[x])
raises(CoercionFailed, lambda: Poly.from_poly(h, gens=y, modulus=3))
assert Poly.from_poly(h, gens=(x, y)) == h
assert Poly.from_poly(
h, gens=(x, y), domain=ZZ).rep == DMP([[ZZ(1)], [ZZ(1), ZZ(0)]], ZZ)
assert Poly.from_poly(
h, gens=(x, y), domain=QQ).rep == DMP([[QQ(1)], [QQ(1), QQ(0)]], QQ)
assert Poly.from_poly(
h, gens=(x, y), domain=K).rep == DMP([[K(1)], [K(1), K(0)]], K)
assert Poly.from_poly(
h, gens=(y, x)).rep == DMP([[ZZ(1)], [ZZ(1), ZZ(0)]], ZZ)
assert Poly.from_poly(
h, gens=(y, x), domain=ZZ).rep == DMP([[ZZ(1)], [ZZ(1), ZZ(0)]], ZZ)
assert Poly.from_poly(
h, gens=(y, x), domain=QQ).rep == DMP([[QQ(1)], [QQ(1), QQ(0)]], QQ)
assert Poly.from_poly(
h, gens=(y, x), domain=K).rep == DMP([[K(1)], [K(1), K(0)]], K)
assert Poly.from_poly(
h, gens=(x, y), field=True).rep == DMP([[QQ(1)], [QQ(1), QQ(0)]], QQ)
assert Poly.from_poly(
h, gens=(x, y), field=True).rep == DMP([[QQ(1)], [QQ(1), QQ(0)]], QQ)
def test_Poly_from_expr():
raises(GeneratorsNeeded, lambda: Poly.from_expr(S.Zero))
raises(GeneratorsNeeded, lambda: Poly.from_expr(S(7)))
F3 = FF(3)
assert Poly.from_expr(x + 5, domain=F3).rep == DMP([F3(1), F3(2)], F3)
assert Poly.from_expr(y + 5, domain=F3).rep == DMP([F3(1), F3(2)], F3)
assert Poly.from_expr(x + 5, x, domain=F3).rep == DMP([F3(1), F3(2)], F3)
assert Poly.from_expr(y + 5, y, domain=F3).rep == DMP([F3(1), F3(2)], F3)
assert Poly.from_expr(x + y, domain=F3).rep == DMP([[F3(1)], [F3(1), F3(0)]], F3)
assert Poly.from_expr(x + y, x, y, domain=F3).rep == DMP([[F3(1)], [F3(1), F3(0)]], F3)
assert Poly.from_expr(x + 5).rep == DMP([1, 5], ZZ)
assert Poly.from_expr(y + 5).rep == DMP([1, 5], ZZ)
assert Poly.from_expr(x + 5, x).rep == DMP([1, 5], ZZ)
assert Poly.from_expr(y + 5, y).rep == DMP([1, 5], ZZ)
assert Poly.from_expr(x + 5, domain=ZZ).rep == DMP([1, 5], ZZ)
assert Poly.from_expr(y + 5, domain=ZZ).rep == DMP([1, 5], ZZ)
assert Poly.from_expr(x + 5, x, domain=ZZ).rep == DMP([1, 5], ZZ)
assert Poly.from_expr(y + 5, y, domain=ZZ).rep == DMP([1, 5], ZZ)
assert Poly.from_expr(x + 5, x, y, domain=ZZ).rep == DMP([[1], [5]], ZZ)
assert Poly.from_expr(y + 5, x, y, domain=ZZ).rep == DMP([[1, 5]], ZZ)
def test_Poly__new__():
raises(GeneratorsError, lambda: Poly(x + 1, x, x))
raises(GeneratorsError, lambda: Poly(x + y, x, y, domain=ZZ[x]))
raises(GeneratorsError, lambda: Poly(x + y, x, y, domain=ZZ[y]))
raises(OptionError, lambda: Poly(x, x, symmetric=True))
raises(OptionError, lambda: Poly(x + 2, x, modulus=3, domain=QQ))
raises(OptionError, lambda: Poly(x + 2, x, domain=ZZ, gaussian=True))
raises(OptionError, lambda: Poly(x + 2, x, modulus=3, gaussian=True))
raises(OptionError, lambda: Poly(x + 2, x, domain=ZZ, extension=[sqrt(3)]))
raises(OptionError, lambda: Poly(x + 2, x, modulus=3, extension=[sqrt(3)]))
raises(OptionError, lambda: Poly(x + 2, x, domain=ZZ, extension=True))
raises(OptionError, lambda: Poly(x + 2, x, modulus=3, extension=True))
raises(OptionError, lambda: Poly(x + 2, x, domain=ZZ, greedy=True))
raises(OptionError, lambda: Poly(x + 2, x, domain=QQ, field=True))
raises(OptionError, lambda: Poly(x + 2, x, domain=ZZ, greedy=False))
raises(OptionError, lambda: Poly(x + 2, x, domain=QQ, field=False))
raises(NotImplementedError, lambda: Poly(x + 1, x, modulus=3, order='grlex'))
raises(NotImplementedError, lambda: Poly(x + 1, x, order='grlex'))
raises(GeneratorsNeeded, lambda: Poly({1: 2, 0: 1}))
raises(GeneratorsNeeded, lambda: Poly([2, 1]))
raises(GeneratorsNeeded, lambda: Poly((2, 1)))
raises(GeneratorsNeeded, lambda: Poly(1))
f = a*x**2 + b*x + c
assert Poly({2: a, 1: b, 0: c}, x) == f
assert Poly(iter([a, b, c]), x) == f
assert Poly([a, b, c], x) == f
assert Poly((a, b, c), x) == f
f = Poly({}, x, y, z)
assert f.gens == (x, y, z) and f.as_expr() == 0
assert Poly(Poly(a*x + b*y, x, y), x) == Poly(a*x + b*y, x)
assert Poly(3*x**2 + 2*x + 1, domain='ZZ').all_coeffs() == [3, 2, 1]
assert Poly(3*x**2 + 2*x + 1, domain='QQ').all_coeffs() == [3, 2, 1]
assert Poly(3*x**2 + 2*x + 1, domain='RR').all_coeffs() == [3.0, 2.0, 1.0]
raises(CoercionFailed, lambda: Poly(3*x**2/5 + x*Rational(2, 5) + 1, domain='ZZ'))
assert Poly(
3*x**2/5 + x*Rational(2, 5) + 1, domain='QQ').all_coeffs() == [Rational(3, 5), Rational(2, 5), 1]
assert _epsilon_eq(
Poly(3*x**2/5 + x*Rational(2, 5) + 1, domain='RR').all_coeffs(), [0.6, 0.4, 1.0])
assert Poly(3.0*x**2 + 2.0*x + 1, domain='ZZ').all_coeffs() == [3, 2, 1]
assert Poly(3.0*x**2 + 2.0*x + 1, domain='QQ').all_coeffs() == [3, 2, 1]
assert Poly(
3.0*x**2 + 2.0*x + 1, domain='RR').all_coeffs() == [3.0, 2.0, 1.0]
raises(CoercionFailed, lambda: Poly(3.1*x**2 + 2.1*x + 1, domain='ZZ'))
assert Poly(3.1*x**2 + 2.1*x + 1, domain='QQ').all_coeffs() == [Rational(31, 10), Rational(21, 10), 1]
assert Poly(3.1*x**2 + 2.1*x + 1, domain='RR').all_coeffs() == [3.1, 2.1, 1.0]
assert Poly({(2, 1): 1, (1, 2): 2, (1, 1): 3}, x, y) == \
Poly(x**2*y + 2*x*y**2 + 3*x*y, x, y)
assert Poly(x**2 + 1, extension=I).get_domain() == QQ.algebraic_field(I)
f = 3*x**5 - x**4 + x**3 - x** 2 + 65538
assert Poly(f, x, modulus=65537, symmetric=True) == \
Poly(3*x**5 - x**4 + x**3 - x** 2 + 1, x, modulus=65537,
symmetric=True)
assert Poly(f, x, modulus=65537, symmetric=False) == \
Poly(3*x**5 + 65536*x**4 + x**3 + 65536*x** 2 + 1, x,
modulus=65537, symmetric=False)
assert isinstance(Poly(x**2 + x + 1.0).get_domain(), RealField)
def test_Poly__args():
assert Poly(x**2 + 1).args == (x**2 + 1, x)
def test_Poly__gens():
assert Poly((x - p)*(x - q), x).gens == (x,)
assert Poly((x - p)*(x - q), p).gens == (p,)
assert Poly((x - p)*(x - q), q).gens == (q,)
assert Poly((x - p)*(x - q), x, p).gens == (x, p)
assert Poly((x - p)*(x - q), x, q).gens == (x, q)
assert Poly((x - p)*(x - q), x, p, q).gens == (x, p, q)
assert Poly((x - p)*(x - q), p, x, q).gens == (p, x, q)
assert Poly((x - p)*(x - q), p, q, x).gens == (p, q, x)
assert Poly((x - p)*(x - q)).gens == (x, p, q)
assert Poly((x - p)*(x - q), sort='x > p > q').gens == (x, p, q)
assert Poly((x - p)*(x - q), sort='p > x > q').gens == (p, x, q)
assert Poly((x - p)*(x - q), sort='p > q > x').gens == (p, q, x)
assert Poly((x - p)*(x - q), x, p, q, sort='p > q > x').gens == (x, p, q)
assert Poly((x - p)*(x - q), wrt='x').gens == (x, p, q)
assert Poly((x - p)*(x - q), wrt='p').gens == (p, x, q)
assert Poly((x - p)*(x - q), wrt='q').gens == (q, x, p)
assert Poly((x - p)*(x - q), wrt=x).gens == (x, p, q)
assert Poly((x - p)*(x - q), wrt=p).gens == (p, x, q)
assert Poly((x - p)*(x - q), wrt=q).gens == (q, x, p)
assert Poly((x - p)*(x - q), x, p, q, wrt='p').gens == (x, p, q)
assert Poly((x - p)*(x - q), wrt='p', sort='q > x').gens == (p, q, x)
assert Poly((x - p)*(x - q), wrt='q', sort='p > x').gens == (q, p, x)
def test_Poly_zero():
assert Poly(x).zero == Poly(0, x, domain=ZZ)
assert Poly(x/2).zero == Poly(0, x, domain=QQ)
def test_Poly_one():
assert Poly(x).one == Poly(1, x, domain=ZZ)
assert Poly(x/2).one == Poly(1, x, domain=QQ)
def test_Poly__unify():
raises(UnificationFailed, lambda: Poly(x)._unify(y))
F3 = FF(3)
F5 = FF(5)
assert Poly(x, x, modulus=3)._unify(Poly(y, y, modulus=3))[2:] == (
DMP([[F3(1)], []], F3), DMP([[F3(1), F3(0)]], F3))
assert Poly(x, x, modulus=3)._unify(Poly(y, y, modulus=5))[2:] == (
DMP([[F5(1)], []], F5), DMP([[F5(1), F5(0)]], F5))
assert Poly(y, x, y)._unify(Poly(x, x, modulus=3))[2:] == (DMP([[F3(1), F3(0)]], F3), DMP([[F3(1)], []], F3))
assert Poly(x, x, modulus=3)._unify(Poly(y, x, y))[2:] == (DMP([[F3(1)], []], F3), DMP([[F3(1), F3(0)]], F3))
assert Poly(x + 1, x)._unify(Poly(x + 2, x))[2:] == (DMP([1, 1], ZZ), DMP([1, 2], ZZ))
assert Poly(x + 1, x, domain='QQ')._unify(Poly(x + 2, x))[2:] == (DMP([1, 1], QQ), DMP([1, 2], QQ))
assert Poly(x + 1, x)._unify(Poly(x + 2, x, domain='QQ'))[2:] == (DMP([1, 1], QQ), DMP([1, 2], QQ))
assert Poly(x + 1, x)._unify(Poly(x + 2, x, y))[2:] == (DMP([[1], [1]], ZZ), DMP([[1], [2]], ZZ))
assert Poly(x + 1, x, domain='QQ')._unify(Poly(x + 2, x, y))[2:] == (DMP([[1], [1]], QQ), DMP([[1], [2]], QQ))
assert Poly(x + 1, x)._unify(Poly(x + 2, x, y, domain='QQ'))[2:] == (DMP([[1], [1]], QQ), DMP([[1], [2]], QQ))
assert Poly(x + 1, x, y)._unify(Poly(x + 2, x))[2:] == (DMP([[1], [1]], ZZ), DMP([[1], [2]], ZZ))
assert Poly(x + 1, x, y, domain='QQ')._unify(Poly(x + 2, x))[2:] == (DMP([[1], [1]], QQ), DMP([[1], [2]], QQ))
assert Poly(x + 1, x, y)._unify(Poly(x + 2, x, domain='QQ'))[2:] == (DMP([[1], [1]], QQ), DMP([[1], [2]], QQ))
assert Poly(x + 1, x, y)._unify(Poly(x + 2, x, y))[2:] == (DMP([[1], [1]], ZZ), DMP([[1], [2]], ZZ))
assert Poly(x + 1, x, y, domain='QQ')._unify(Poly(x + 2, x, y))[2:] == (DMP([[1], [1]], QQ), DMP([[1], [2]], QQ))
assert Poly(x + 1, x, y)._unify(Poly(x + 2, x, y, domain='QQ'))[2:] == (DMP([[1], [1]], QQ), DMP([[1], [2]], QQ))
assert Poly(x + 1, x)._unify(Poly(x + 2, y, x))[2:] == (DMP([[1, 1]], ZZ), DMP([[1, 2]], ZZ))
assert Poly(x + 1, x, domain='QQ')._unify(Poly(x + 2, y, x))[2:] == (DMP([[1, 1]], QQ), DMP([[1, 2]], QQ))
assert Poly(x + 1, x)._unify(Poly(x + 2, y, x, domain='QQ'))[2:] == (DMP([[1, 1]], QQ), DMP([[1, 2]], QQ))
assert Poly(x + 1, y, x)._unify(Poly(x + 2, x))[2:] == (DMP([[1, 1]], ZZ), DMP([[1, 2]], ZZ))
assert Poly(x + 1, y, x, domain='QQ')._unify(Poly(x + 2, x))[2:] == (DMP([[1, 1]], QQ), DMP([[1, 2]], QQ))
assert Poly(x + 1, y, x)._unify(Poly(x + 2, x, domain='QQ'))[2:] == (DMP([[1, 1]], QQ), DMP([[1, 2]], QQ))
assert Poly(x + 1, x, y)._unify(Poly(x + 2, y, x))[2:] == (DMP([[1], [1]], ZZ), DMP([[1], [2]], ZZ))
assert Poly(x + 1, x, y, domain='QQ')._unify(Poly(x + 2, y, x))[2:] == (DMP([[1], [1]], QQ), DMP([[1], [2]], QQ))
assert Poly(x + 1, x, y)._unify(Poly(x + 2, y, x, domain='QQ'))[2:] == (DMP([[1], [1]], QQ), DMP([[1], [2]], QQ))
assert Poly(x + 1, y, x)._unify(Poly(x + 2, x, y))[2:] == (DMP([[1, 1]], ZZ), DMP([[1, 2]], ZZ))
assert Poly(x + 1, y, x, domain='QQ')._unify(Poly(x + 2, x, y))[2:] == (DMP([[1, 1]], QQ), DMP([[1, 2]], QQ))
assert Poly(x + 1, y, x)._unify(Poly(x + 2, x, y, domain='QQ'))[2:] == (DMP([[1, 1]], QQ), DMP([[1, 2]], QQ))
F, A, B = field("a,b", ZZ)
assert Poly(a*x, x, domain='ZZ[a]')._unify(Poly(a*b*x, x, domain='ZZ(a,b)'))[2:] == \
(DMP([A, F(0)], F.to_domain()), DMP([A*B, F(0)], F.to_domain()))
assert Poly(a*x, x, domain='ZZ(a)')._unify(Poly(a*b*x, x, domain='ZZ(a,b)'))[2:] == \
(DMP([A, F(0)], F.to_domain()), DMP([A*B, F(0)], F.to_domain()))
raises(CoercionFailed, lambda: Poly(Poly(x**2 + x**2*z, y, field=True), domain='ZZ(x)'))
f = Poly(t**2 + t/3 + x, t, domain='QQ(x)')
g = Poly(t**2 + t/3 + x, t, domain='QQ[x]')
assert f._unify(g)[2:] == (f.rep, f.rep)
def test_Poly_free_symbols():
assert Poly(x**2 + 1).free_symbols == {x}
assert Poly(x**2 + y*z).free_symbols == {x, y, z}
assert Poly(x**2 + y*z, x).free_symbols == {x, y, z}
assert Poly(x**2 + sin(y*z)).free_symbols == {x, y, z}
assert Poly(x**2 + sin(y*z), x).free_symbols == {x, y, z}
assert Poly(x**2 + sin(y*z), x, domain=EX).free_symbols == {x, y, z}
assert Poly(1 + x + x**2, x, y, z).free_symbols == {x}
assert Poly(x + sin(y), z).free_symbols == {x, y}
def test_PurePoly_free_symbols():
assert PurePoly(x**2 + 1).free_symbols == set([])
assert PurePoly(x**2 + y*z).free_symbols == set([])
assert PurePoly(x**2 + y*z, x).free_symbols == {y, z}
assert PurePoly(x**2 + sin(y*z)).free_symbols == set([])
assert PurePoly(x**2 + sin(y*z), x).free_symbols == {y, z}
assert PurePoly(x**2 + sin(y*z), x, domain=EX).free_symbols == {y, z}
def test_Poly__eq__():
assert (Poly(x, x) == Poly(x, x)) is True
assert (Poly(x, x, domain=QQ) == Poly(x, x)) is False
assert (Poly(x, x) == Poly(x, x, domain=QQ)) is False
assert (Poly(x, x, domain=ZZ[a]) == Poly(x, x)) is False
assert (Poly(x, x) == Poly(x, x, domain=ZZ[a])) is False
assert (Poly(x*y, x, y) == Poly(x, x)) is False
assert (Poly(x, x, y) == Poly(x, x)) is False
assert (Poly(x, x) == Poly(x, x, y)) is False
assert (Poly(x**2 + 1, x) == Poly(y**2 + 1, y)) is False
assert (Poly(y**2 + 1, y) == Poly(x**2 + 1, x)) is False
f = Poly(x, x, domain=ZZ)
g = Poly(x, x, domain=QQ)
assert f.eq(g) is False
assert f.ne(g) is True
assert f.eq(g, strict=True) is False
assert f.ne(g, strict=True) is True
t0 = Symbol('t0')
f = Poly((t0/2 + x**2)*t**2 - x**2*t, t, domain='QQ[x,t0]')
g = Poly((t0/2 + x**2)*t**2 - x**2*t, t, domain='ZZ(x,t0)')
assert (f == g) is False
def test_PurePoly__eq__():
assert (PurePoly(x, x) == PurePoly(x, x)) is True
assert (PurePoly(x, x, domain=QQ) == PurePoly(x, x)) is True
assert (PurePoly(x, x) == PurePoly(x, x, domain=QQ)) is True
assert (PurePoly(x, x, domain=ZZ[a]) == PurePoly(x, x)) is True
assert (PurePoly(x, x) == PurePoly(x, x, domain=ZZ[a])) is True
assert (PurePoly(x*y, x, y) == PurePoly(x, x)) is False
assert (PurePoly(x, x, y) == PurePoly(x, x)) is False
assert (PurePoly(x, x) == PurePoly(x, x, y)) is False
assert (PurePoly(x**2 + 1, x) == PurePoly(y**2 + 1, y)) is True
assert (PurePoly(y**2 + 1, y) == PurePoly(x**2 + 1, x)) is True
f = PurePoly(x, x, domain=ZZ)
g = PurePoly(x, x, domain=QQ)
assert f.eq(g) is True
assert f.ne(g) is False
assert f.eq(g, strict=True) is False
assert f.ne(g, strict=True) is True
f = PurePoly(x, x, domain=ZZ)
g = PurePoly(y, y, domain=QQ)
assert f.eq(g) is True
assert f.ne(g) is False
assert f.eq(g, strict=True) is False
assert f.ne(g, strict=True) is True
def test_PurePoly_Poly():
assert isinstance(PurePoly(Poly(x**2 + 1)), PurePoly) is True
assert isinstance(Poly(PurePoly(x**2 + 1)), Poly) is True
def test_Poly_get_domain():
assert Poly(2*x).get_domain() == ZZ
assert Poly(2*x, domain='ZZ').get_domain() == ZZ
assert Poly(2*x, domain='QQ').get_domain() == QQ
assert Poly(x/2).get_domain() == QQ
raises(CoercionFailed, lambda: Poly(x/2, domain='ZZ'))
assert Poly(x/2, domain='QQ').get_domain() == QQ
assert isinstance(Poly(0.2*x).get_domain(), RealField)
def test_Poly_set_domain():
assert Poly(2*x + 1).set_domain(ZZ) == Poly(2*x + 1)
assert Poly(2*x + 1).set_domain('ZZ') == Poly(2*x + 1)
assert Poly(2*x + 1).set_domain(QQ) == Poly(2*x + 1, domain='QQ')
assert Poly(2*x + 1).set_domain('QQ') == Poly(2*x + 1, domain='QQ')
assert Poly(Rational(2, 10)*x + Rational(1, 10)).set_domain('RR') == Poly(0.2*x + 0.1)
assert Poly(0.2*x + 0.1).set_domain('QQ') == Poly(Rational(2, 10)*x + Rational(1, 10))
raises(CoercionFailed, lambda: Poly(x/2 + 1).set_domain(ZZ))
raises(CoercionFailed, lambda: Poly(x + 1, modulus=2).set_domain(QQ))
raises(GeneratorsError, lambda: Poly(x*y, x, y).set_domain(ZZ[y]))
def test_Poly_get_modulus():
assert Poly(x**2 + 1, modulus=2).get_modulus() == 2
raises(PolynomialError, lambda: Poly(x**2 + 1).get_modulus())
def test_Poly_set_modulus():
assert Poly(
x**2 + 1, modulus=2).set_modulus(7) == Poly(x**2 + 1, modulus=7)
assert Poly(
x**2 + 5, modulus=7).set_modulus(2) == Poly(x**2 + 1, modulus=2)
assert Poly(x**2 + 1).set_modulus(2) == Poly(x**2 + 1, modulus=2)
raises(CoercionFailed, lambda: Poly(x/2 + 1).set_modulus(2))
def test_Poly_add_ground():
assert Poly(x + 1).add_ground(2) == Poly(x + 3)
def test_Poly_sub_ground():
assert Poly(x + 1).sub_ground(2) == Poly(x - 1)
def test_Poly_mul_ground():
assert Poly(x + 1).mul_ground(2) == Poly(2*x + 2)
def test_Poly_quo_ground():
assert Poly(2*x + 4).quo_ground(2) == Poly(x + 2)
assert Poly(2*x + 3).quo_ground(2) == Poly(x + 1)
def test_Poly_exquo_ground():
assert Poly(2*x + 4).exquo_ground(2) == Poly(x + 2)
raises(ExactQuotientFailed, lambda: Poly(2*x + 3).exquo_ground(2))
def test_Poly_abs():
assert Poly(-x + 1, x).abs() == abs(Poly(-x + 1, x)) == Poly(x + 1, x)
def test_Poly_neg():
assert Poly(-x + 1, x).neg() == -Poly(-x + 1, x) == Poly(x - 1, x)
def test_Poly_add():
assert Poly(0, x).add(Poly(0, x)) == Poly(0, x)
assert Poly(0, x) + Poly(0, x) == Poly(0, x)
assert Poly(1, x).add(Poly(0, x)) == Poly(1, x)
assert Poly(1, x, y) + Poly(0, x) == Poly(1, x, y)
assert Poly(0, x).add(Poly(1, x, y)) == Poly(1, x, y)
assert Poly(0, x, y) + Poly(1, x, y) == Poly(1, x, y)
assert Poly(1, x) + x == Poly(x + 1, x)
with warns_deprecated_sympy():
Poly(1, x) + sin(x)
assert Poly(x, x) + 1 == Poly(x + 1, x)
assert 1 + Poly(x, x) == Poly(x + 1, x)
def test_Poly_sub():
assert Poly(0, x).sub(Poly(0, x)) == Poly(0, x)
assert Poly(0, x) - Poly(0, x) == Poly(0, x)
assert Poly(1, x).sub(Poly(0, x)) == Poly(1, x)
assert Poly(1, x, y) - Poly(0, x) == Poly(1, x, y)
assert Poly(0, x).sub(Poly(1, x, y)) == Poly(-1, x, y)
assert Poly(0, x, y) - Poly(1, x, y) == Poly(-1, x, y)
assert Poly(1, x) - x == Poly(1 - x, x)
with warns_deprecated_sympy():
Poly(1, x) - sin(x)
assert Poly(x, x) - 1 == Poly(x - 1, x)
assert 1 - Poly(x, x) == Poly(1 - x, x)
def test_Poly_mul():
assert Poly(0, x).mul(Poly(0, x)) == Poly(0, x)
assert Poly(0, x) * Poly(0, x) == Poly(0, x)
assert Poly(2, x).mul(Poly(4, x)) == Poly(8, x)
assert Poly(2, x, y) * Poly(4, x) == Poly(8, x, y)
assert Poly(4, x).mul(Poly(2, x, y)) == Poly(8, x, y)
assert Poly(4, x, y) * Poly(2, x, y) == Poly(8, x, y)
assert Poly(1, x) * x == Poly(x, x)
with warns_deprecated_sympy():
Poly(1, x) * sin(x)
assert Poly(x, x) * 2 == Poly(2*x, x)
assert 2 * Poly(x, x) == Poly(2*x, x)
def test_issue_13079():
assert Poly(x)*x == Poly(x**2, x, domain='ZZ')
assert x*Poly(x) == Poly(x**2, x, domain='ZZ')
assert -2*Poly(x) == Poly(-2*x, x, domain='ZZ')
assert S(-2)*Poly(x) == Poly(-2*x, x, domain='ZZ')
assert Poly(x)*S(-2) == Poly(-2*x, x, domain='ZZ')
def test_Poly_sqr():
assert Poly(x*y, x, y).sqr() == Poly(x**2*y**2, x, y)
def test_Poly_pow():
assert Poly(x, x).pow(10) == Poly(x**10, x)
assert Poly(x, x).pow(Integer(10)) == Poly(x**10, x)
assert Poly(2*y, x, y).pow(4) == Poly(16*y**4, x, y)
assert Poly(2*y, x, y).pow(Integer(4)) == Poly(16*y**4, x, y)
assert Poly(7*x*y, x, y)**3 == Poly(343*x**3*y**3, x, y)
raises(TypeError, lambda: Poly(x*y + 1, x, y)**(-1))
raises(TypeError, lambda: Poly(x*y + 1, x, y)**x)
def test_Poly_divmod():
f, g = Poly(x**2), Poly(x)
q, r = g, Poly(0, x)
assert divmod(f, g) == (q, r)
assert f // g == q
assert f % g == r
assert divmod(f, x) == (q, r)
assert f // x == q
assert f % x == r
q, r = Poly(0, x), Poly(2, x)
assert divmod(2, g) == (q, r)
assert 2 // g == q
assert 2 % g == r
assert Poly(x)/Poly(x) == 1
assert Poly(x**2)/Poly(x) == x
assert Poly(x)/Poly(x**2) == 1/x
def test_Poly_eq_ne():
assert (Poly(x + y, x, y) == Poly(x + y, x, y)) is True
assert (Poly(x + y, x) == Poly(x + y, x, y)) is False
assert (Poly(x + y, x, y) == Poly(x + y, x)) is False
assert (Poly(x + y, x) == Poly(x + y, x)) is True
assert (Poly(x + y, y) == Poly(x + y, y)) is True
assert (Poly(x + y, x, y) == x + y) is True
assert (Poly(x + y, x) == x + y) is True
assert (Poly(x + y, x, y) == x + y) is True
assert (Poly(x + y, x) == x + y) is True
assert (Poly(x + y, y) == x + y) is True
assert (Poly(x + y, x, y) != Poly(x + y, x, y)) is False
assert (Poly(x + y, x) != Poly(x + y, x, y)) is True
assert (Poly(x + y, x, y) != Poly(x + y, x)) is True
assert (Poly(x + y, x) != Poly(x + y, x)) is False
assert (Poly(x + y, y) != Poly(x + y, y)) is False
assert (Poly(x + y, x, y) != x + y) is False
assert (Poly(x + y, x) != x + y) is False
assert (Poly(x + y, x, y) != x + y) is False
assert (Poly(x + y, x) != x + y) is False
assert (Poly(x + y, y) != x + y) is False
assert (Poly(x, x) == sin(x)) is False
assert (Poly(x, x) != sin(x)) is True
def test_Poly_nonzero():
assert not bool(Poly(0, x)) is True
assert not bool(Poly(1, x)) is False
def test_Poly_properties():
assert Poly(0, x).is_zero is True
assert Poly(1, x).is_zero is False
assert Poly(1, x).is_one is True
assert Poly(2, x).is_one is False
assert Poly(x - 1, x).is_sqf is True
assert Poly((x - 1)**2, x).is_sqf is False
assert Poly(x - 1, x).is_monic is True
assert Poly(2*x - 1, x).is_monic is False
assert Poly(3*x + 2, x).is_primitive is True
assert Poly(4*x + 2, x).is_primitive is False
assert Poly(1, x).is_ground is True
assert Poly(x, x).is_ground is False
assert Poly(x + y + z + 1).is_linear is True
assert Poly(x*y*z + 1).is_linear is False
assert Poly(x*y + z + 1).is_quadratic is True
assert Poly(x*y*z + 1).is_quadratic is False
assert Poly(x*y).is_monomial is True
assert Poly(x*y + 1).is_monomial is False
assert Poly(x**2 + x*y).is_homogeneous is True
assert Poly(x**3 + x*y).is_homogeneous is False
assert Poly(x).is_univariate is True
assert Poly(x*y).is_univariate is False
assert Poly(x*y).is_multivariate is True
assert Poly(x).is_multivariate is False
assert Poly(
x**16 + x**14 - x**10 + x**8 - x**6 + x**2 + 1).is_cyclotomic is False
assert Poly(
x**16 + x**14 - x**10 - x**8 - x**6 + x**2 + 1).is_cyclotomic is True
def test_Poly_is_irreducible():
assert Poly(x**2 + x + 1).is_irreducible is True
assert Poly(x**2 + 2*x + 1).is_irreducible is False
assert Poly(7*x + 3, modulus=11).is_irreducible is True
assert Poly(7*x**2 + 3*x + 1, modulus=11).is_irreducible is False
def test_Poly_subs():
assert Poly(x + 1).subs(x, 0) == 1
assert Poly(x + 1).subs(x, x) == Poly(x + 1)
assert Poly(x + 1).subs(x, y) == Poly(y + 1)
assert Poly(x*y, x).subs(y, x) == x**2
assert Poly(x*y, x).subs(x, y) == y**2
def test_Poly_replace():
assert Poly(x + 1).replace(x) == Poly(x + 1)
assert Poly(x + 1).replace(y) == Poly(y + 1)
raises(PolynomialError, lambda: Poly(x + y).replace(z))
assert Poly(x + 1).replace(x, x) == Poly(x + 1)
assert Poly(x + 1).replace(x, y) == Poly(y + 1)
assert Poly(x + y).replace(x, x) == Poly(x + y)
assert Poly(x + y).replace(x, z) == Poly(z + y, z, y)
assert Poly(x + y).replace(y, y) == Poly(x + y)
assert Poly(x + y).replace(y, z) == Poly(x + z, x, z)
assert Poly(x + y).replace(z, t) == Poly(x + y)
raises(PolynomialError, lambda: Poly(x + y).replace(x, y))
assert Poly(x + y, x).replace(x, z) == Poly(z + y, z)
assert Poly(x + y, y).replace(y, z) == Poly(x + z, z)
raises(PolynomialError, lambda: Poly(x + y, x).replace(x, y))
raises(PolynomialError, lambda: Poly(x + y, y).replace(y, x))
def test_Poly_reorder():
raises(PolynomialError, lambda: Poly(x + y).reorder(x, z))
assert Poly(x + y, x, y).reorder(x, y) == Poly(x + y, x, y)
assert Poly(x + y, x, y).reorder(y, x) == Poly(x + y, y, x)
assert Poly(x + y, y, x).reorder(x, y) == Poly(x + y, x, y)
assert Poly(x + y, y, x).reorder(y, x) == Poly(x + y, y, x)
assert Poly(x + y, x, y).reorder(wrt=x) == Poly(x + y, x, y)
assert Poly(x + y, x, y).reorder(wrt=y) == Poly(x + y, y, x)
def test_Poly_ltrim():
f = Poly(y**2 + y*z**2, x, y, z).ltrim(y)
assert f.as_expr() == y**2 + y*z**2 and f.gens == (y, z)
assert Poly(x*y - x, z, x, y).ltrim(1) == Poly(x*y - x, x, y)
raises(PolynomialError, lambda: Poly(x*y**2 + y**2, x, y).ltrim(y))
raises(PolynomialError, lambda: Poly(x*y - x, x, y).ltrim(-1))
def test_Poly_has_only_gens():
assert Poly(x*y + 1, x, y, z).has_only_gens(x, y) is True
assert Poly(x*y + z, x, y, z).has_only_gens(x, y) is False
raises(GeneratorsError, lambda: Poly(x*y**2 + y**2, x, y).has_only_gens(t))
def test_Poly_to_ring():
assert Poly(2*x + 1, domain='ZZ').to_ring() == Poly(2*x + 1, domain='ZZ')
assert Poly(2*x + 1, domain='QQ').to_ring() == Poly(2*x + 1, domain='ZZ')
raises(CoercionFailed, lambda: Poly(x/2 + 1).to_ring())
raises(DomainError, lambda: Poly(2*x + 1, modulus=3).to_ring())
def test_Poly_to_field():
assert Poly(2*x + 1, domain='ZZ').to_field() == Poly(2*x + 1, domain='QQ')
assert Poly(2*x + 1, domain='QQ').to_field() == Poly(2*x + 1, domain='QQ')
assert Poly(x/2 + 1, domain='QQ').to_field() == Poly(x/2 + 1, domain='QQ')
assert Poly(2*x + 1, modulus=3).to_field() == Poly(2*x + 1, modulus=3)
assert Poly(2.0*x + 1.0).to_field() == Poly(2.0*x + 1.0)
def test_Poly_to_exact():
assert Poly(2*x).to_exact() == Poly(2*x)
assert Poly(x/2).to_exact() == Poly(x/2)
assert Poly(0.1*x).to_exact() == Poly(x/10)
def test_Poly_retract():
f = Poly(x**2 + 1, x, domain=QQ[y])
assert f.retract() == Poly(x**2 + 1, x, domain='ZZ')
assert f.retract(field=True) == Poly(x**2 + 1, x, domain='QQ')
assert Poly(0, x, y).retract() == Poly(0, x, y)
def test_Poly_slice():
f = Poly(x**3 + 2*x**2 + 3*x + 4)
assert f.slice(0, 0) == Poly(0, x)
assert f.slice(0, 1) == Poly(4, x)
assert f.slice(0, 2) == Poly(3*x + 4, x)
assert f.slice(0, 3) == Poly(2*x**2 + 3*x + 4, x)
assert f.slice(0, 4) == Poly(x**3 + 2*x**2 + 3*x + 4, x)
assert f.slice(x, 0, 0) == Poly(0, x)
assert f.slice(x, 0, 1) == Poly(4, x)
assert f.slice(x, 0, 2) == Poly(3*x + 4, x)
assert f.slice(x, 0, 3) == Poly(2*x**2 + 3*x + 4, x)
assert f.slice(x, 0, 4) == Poly(x**3 + 2*x**2 + 3*x + 4, x)
def test_Poly_coeffs():
assert Poly(0, x).coeffs() == [0]
assert Poly(1, x).coeffs() == [1]
assert Poly(2*x + 1, x).coeffs() == [2, 1]
assert Poly(7*x**2 + 2*x + 1, x).coeffs() == [7, 2, 1]
assert Poly(7*x**4 + 2*x + 1, x).coeffs() == [7, 2, 1]
assert Poly(x*y**7 + 2*x**2*y**3).coeffs('lex') == [2, 1]
assert Poly(x*y**7 + 2*x**2*y**3).coeffs('grlex') == [1, 2]
def test_Poly_monoms():
assert Poly(0, x).monoms() == [(0,)]
assert Poly(1, x).monoms() == [(0,)]
assert Poly(2*x + 1, x).monoms() == [(1,), (0,)]
assert Poly(7*x**2 + 2*x + 1, x).monoms() == [(2,), (1,), (0,)]
assert Poly(7*x**4 + 2*x + 1, x).monoms() == [(4,), (1,), (0,)]
assert Poly(x*y**7 + 2*x**2*y**3).monoms('lex') == [(2, 3), (1, 7)]
assert Poly(x*y**7 + 2*x**2*y**3).monoms('grlex') == [(1, 7), (2, 3)]
def test_Poly_terms():
assert Poly(0, x).terms() == [((0,), 0)]
assert Poly(1, x).terms() == [((0,), 1)]
assert Poly(2*x + 1, x).terms() == [((1,), 2), ((0,), 1)]
assert Poly(7*x**2 + 2*x + 1, x).terms() == [((2,), 7), ((1,), 2), ((0,), 1)]
assert Poly(7*x**4 + 2*x + 1, x).terms() == [((4,), 7), ((1,), 2), ((0,), 1)]
assert Poly(
x*y**7 + 2*x**2*y**3).terms('lex') == [((2, 3), 2), ((1, 7), 1)]
assert Poly(
x*y**7 + 2*x**2*y**3).terms('grlex') == [((1, 7), 1), ((2, 3), 2)]
def test_Poly_all_coeffs():
assert Poly(0, x).all_coeffs() == [0]
assert Poly(1, x).all_coeffs() == [1]
assert Poly(2*x + 1, x).all_coeffs() == [2, 1]
assert Poly(7*x**2 + 2*x + 1, x).all_coeffs() == [7, 2, 1]
assert Poly(7*x**4 + 2*x + 1, x).all_coeffs() == [7, 0, 0, 2, 1]
def test_Poly_all_monoms():
assert Poly(0, x).all_monoms() == [(0,)]
assert Poly(1, x).all_monoms() == [(0,)]
assert Poly(2*x + 1, x).all_monoms() == [(1,), (0,)]
assert Poly(7*x**2 + 2*x + 1, x).all_monoms() == [(2,), (1,), (0,)]
assert Poly(7*x**4 + 2*x + 1, x).all_monoms() == [(4,), (3,), (2,), (1,), (0,)]
def test_Poly_all_terms():
assert Poly(0, x).all_terms() == [((0,), 0)]
assert Poly(1, x).all_terms() == [((0,), 1)]
assert Poly(2*x + 1, x).all_terms() == [((1,), 2), ((0,), 1)]
assert Poly(7*x**2 + 2*x + 1, x).all_terms() == \
[((2,), 7), ((1,), 2), ((0,), 1)]
assert Poly(7*x**4 + 2*x + 1, x).all_terms() == \
[((4,), 7), ((3,), 0), ((2,), 0), ((1,), 2), ((0,), 1)]
def test_Poly_termwise():
f = Poly(x**2 + 20*x + 400)
g = Poly(x**2 + 2*x + 4)
def func(monom, coeff):
(k,) = monom
return coeff//10**(2 - k)
assert f.termwise(func) == g
def func(monom, coeff):
(k,) = monom
return (k,), coeff//10**(2 - k)
assert f.termwise(func) == g
def test_Poly_length():
assert Poly(0, x).length() == 0
assert Poly(1, x).length() == 1
assert Poly(x, x).length() == 1
assert Poly(x + 1, x).length() == 2
assert Poly(x**2 + 1, x).length() == 2
assert Poly(x**2 + x + 1, x).length() == 3
def test_Poly_as_dict():
assert Poly(0, x).as_dict() == {}
assert Poly(0, x, y, z).as_dict() == {}
assert Poly(1, x).as_dict() == {(0,): 1}
assert Poly(1, x, y, z).as_dict() == {(0, 0, 0): 1}
assert Poly(x**2 + 3, x).as_dict() == {(2,): 1, (0,): 3}
assert Poly(x**2 + 3, x, y, z).as_dict() == {(2, 0, 0): 1, (0, 0, 0): 3}
assert Poly(3*x**2*y*z**3 + 4*x*y + 5*x*z).as_dict() == {(2, 1, 3): 3,
(1, 1, 0): 4, (1, 0, 1): 5}
def test_Poly_as_expr():
assert Poly(0, x).as_expr() == 0
assert Poly(0, x, y, z).as_expr() == 0
assert Poly(1, x).as_expr() == 1
assert Poly(1, x, y, z).as_expr() == 1
assert Poly(x**2 + 3, x).as_expr() == x**2 + 3
assert Poly(x**2 + 3, x, y, z).as_expr() == x**2 + 3
assert Poly(
3*x**2*y*z**3 + 4*x*y + 5*x*z).as_expr() == 3*x**2*y*z**3 + 4*x*y + 5*x*z
f = Poly(x**2 + 2*x*y**2 - y, x, y)
assert f.as_expr() == -y + x**2 + 2*x*y**2
assert f.as_expr({x: 5}) == 25 - y + 10*y**2
assert f.as_expr({y: 6}) == -6 + 72*x + x**2
assert f.as_expr({x: 5, y: 6}) == 379
assert f.as_expr(5, 6) == 379
raises(GeneratorsError, lambda: f.as_expr({z: 7}))
def test_Poly_lift():
assert Poly(x**4 - I*x + 17*I, x, gaussian=True).lift() == \
Poly(x**16 + 2*x**10 + 578*x**8 + x**4 - 578*x**2 + 83521,
x, domain='QQ')
def test_Poly_deflate():
assert Poly(0, x).deflate() == ((1,), Poly(0, x))
assert Poly(1, x).deflate() == ((1,), Poly(1, x))
assert Poly(x, x).deflate() == ((1,), Poly(x, x))
assert Poly(x**2, x).deflate() == ((2,), Poly(x, x))
assert Poly(x**17, x).deflate() == ((17,), Poly(x, x))
assert Poly(
x**2*y*z**11 + x**4*z**11).deflate() == ((2, 1, 11), Poly(x*y*z + x**2*z))
def test_Poly_inject():
f = Poly(x**2*y + x*y**3 + x*y + 1, x)
assert f.inject() == Poly(x**2*y + x*y**3 + x*y + 1, x, y)
assert f.inject(front=True) == Poly(y**3*x + y*x**2 + y*x + 1, y, x)
def test_Poly_eject():
f = Poly(x**2*y + x*y**3 + x*y + 1, x, y)
assert f.eject(x) == Poly(x*y**3 + (x**2 + x)*y + 1, y, domain='ZZ[x]')
assert f.eject(y) == Poly(y*x**2 + (y**3 + y)*x + 1, x, domain='ZZ[y]')
ex = x + y + z + t + w
g = Poly(ex, x, y, z, t, w)
assert g.eject(x) == Poly(ex, y, z, t, w, domain='ZZ[x]')
assert g.eject(x, y) == Poly(ex, z, t, w, domain='ZZ[x, y]')
assert g.eject(x, y, z) == Poly(ex, t, w, domain='ZZ[x, y, z]')
assert g.eject(w) == Poly(ex, x, y, z, t, domain='ZZ[w]')
assert g.eject(t, w) == Poly(ex, x, y, z, domain='ZZ[t, w]')
assert g.eject(z, t, w) == Poly(ex, x, y, domain='ZZ[z, t, w]')
raises(DomainError, lambda: Poly(x*y, x, y, domain=ZZ[z]).eject(y))
raises(NotImplementedError, lambda: Poly(x*y, x, y, z).eject(y))
def test_Poly_exclude():
assert Poly(x, x, y).exclude() == Poly(x, x)
assert Poly(x*y, x, y).exclude() == Poly(x*y, x, y)
assert Poly(1, x, y).exclude() == Poly(1, x, y)
def test_Poly__gen_to_level():
assert Poly(1, x, y)._gen_to_level(-2) == 0
assert Poly(1, x, y)._gen_to_level(-1) == 1
assert Poly(1, x, y)._gen_to_level( 0) == 0
assert Poly(1, x, y)._gen_to_level( 1) == 1
raises(PolynomialError, lambda: Poly(1, x, y)._gen_to_level(-3))
raises(PolynomialError, lambda: Poly(1, x, y)._gen_to_level( 2))
assert Poly(1, x, y)._gen_to_level(x) == 0
assert Poly(1, x, y)._gen_to_level(y) == 1
assert Poly(1, x, y)._gen_to_level('x') == 0
assert Poly(1, x, y)._gen_to_level('y') == 1
raises(PolynomialError, lambda: Poly(1, x, y)._gen_to_level(z))
raises(PolynomialError, lambda: Poly(1, x, y)._gen_to_level('z'))
def test_Poly_degree():
assert Poly(0, x).degree() is -oo
assert Poly(1, x).degree() == 0
assert Poly(x, x).degree() == 1
assert Poly(0, x).degree(gen=0) is -oo
assert Poly(1, x).degree(gen=0) == 0
assert Poly(x, x).degree(gen=0) == 1
assert Poly(0, x).degree(gen=x) is -oo
assert Poly(1, x).degree(gen=x) == 0
assert Poly(x, x).degree(gen=x) == 1
assert Poly(0, x).degree(gen='x') is -oo
assert Poly(1, x).degree(gen='x') == 0
assert Poly(x, x).degree(gen='x') == 1
raises(PolynomialError, lambda: Poly(1, x).degree(gen=1))
raises(PolynomialError, lambda: Poly(1, x).degree(gen=y))
raises(PolynomialError, lambda: Poly(1, x).degree(gen='y'))
assert Poly(1, x, y).degree() == 0
assert Poly(2*y, x, y).degree() == 0
assert Poly(x*y, x, y).degree() == 1
assert Poly(1, x, y).degree(gen=x) == 0
assert Poly(2*y, x, y).degree(gen=x) == 0
assert Poly(x*y, x, y).degree(gen=x) == 1
assert Poly(1, x, y).degree(gen=y) == 0
assert Poly(2*y, x, y).degree(gen=y) == 1
assert Poly(x*y, x, y).degree(gen=y) == 1
assert degree(0, x) is -oo
assert degree(1, x) == 0
assert degree(x, x) == 1
assert degree(x*y**2, x) == 1
assert degree(x*y**2, y) == 2
assert degree(x*y**2, z) == 0
assert degree(pi) == 1
raises(TypeError, lambda: degree(y**2 + x**3))
raises(TypeError, lambda: degree(y**2 + x**3, 1))
raises(PolynomialError, lambda: degree(x, 1.1))
raises(PolynomialError, lambda: degree(x**2/(x**3 + 1), x))
assert degree(Poly(0,x),z) is -oo
assert degree(Poly(1,x),z) == 0
assert degree(Poly(x**2+y**3,y)) == 3
assert degree(Poly(y**2 + x**3, y, x), 1) == 3
assert degree(Poly(y**2 + x**3, x), z) == 0
assert degree(Poly(y**2 + x**3 + z**4, x), z) == 4
def test_Poly_degree_list():
assert Poly(0, x).degree_list() == (-oo,)
assert Poly(0, x, y).degree_list() == (-oo, -oo)
assert Poly(0, x, y, z).degree_list() == (-oo, -oo, -oo)
assert Poly(1, x).degree_list() == (0,)
assert Poly(1, x, y).degree_list() == (0, 0)
assert Poly(1, x, y, z).degree_list() == (0, 0, 0)
assert Poly(x**2*y + x**3*z**2 + 1).degree_list() == (3, 1, 2)
assert degree_list(1, x) == (0,)
assert degree_list(x, x) == (1,)
assert degree_list(x*y**2) == (1, 2)
raises(ComputationFailed, lambda: degree_list(1))
def test_Poly_total_degree():
assert Poly(x**2*y + x**3*z**2 + 1).total_degree() == 5
assert Poly(x**2 + z**3).total_degree() == 3
assert Poly(x*y*z + z**4).total_degree() == 4
assert Poly(x**3 + x + 1).total_degree() == 3
assert total_degree(x*y + z**3) == 3
assert total_degree(x*y + z**3, x, y) == 2
assert total_degree(1) == 0
assert total_degree(Poly(y**2 + x**3 + z**4)) == 4
assert total_degree(Poly(y**2 + x**3 + z**4, x)) == 3
assert total_degree(Poly(y**2 + x**3 + z**4, x), z) == 4
assert total_degree(Poly(x**9 + x*z*y + x**3*z**2 + z**7,x), z) == 7
def test_Poly_homogenize():
assert Poly(x**2+y).homogenize(z) == Poly(x**2+y*z)
assert Poly(x+y).homogenize(z) == Poly(x+y, x, y, z)
assert Poly(x+y**2).homogenize(y) == Poly(x*y+y**2)
def test_Poly_homogeneous_order():
assert Poly(0, x, y).homogeneous_order() is -oo
assert Poly(1, x, y).homogeneous_order() == 0
assert Poly(x, x, y).homogeneous_order() == 1
assert Poly(x*y, x, y).homogeneous_order() == 2
assert Poly(x + 1, x, y).homogeneous_order() is None
assert Poly(x*y + x, x, y).homogeneous_order() is None
assert Poly(x**5 + 2*x**3*y**2 + 9*x*y**4).homogeneous_order() == 5
assert Poly(x**5 + 2*x**3*y**3 + 9*x*y**4).homogeneous_order() is None
def test_Poly_LC():
assert Poly(0, x).LC() == 0
assert Poly(1, x).LC() == 1
assert Poly(2*x**2 + x, x).LC() == 2
assert Poly(x*y**7 + 2*x**2*y**3).LC('lex') == 2
assert Poly(x*y**7 + 2*x**2*y**3).LC('grlex') == 1
assert LC(x*y**7 + 2*x**2*y**3, order='lex') == 2
assert LC(x*y**7 + 2*x**2*y**3, order='grlex') == 1
def test_Poly_TC():
assert Poly(0, x).TC() == 0
assert Poly(1, x).TC() == 1
assert Poly(2*x**2 + x, x).TC() == 0
def test_Poly_EC():
assert Poly(0, x).EC() == 0
assert Poly(1, x).EC() == 1
assert Poly(2*x**2 + x, x).EC() == 1
assert Poly(x*y**7 + 2*x**2*y**3).EC('lex') == 1
assert Poly(x*y**7 + 2*x**2*y**3).EC('grlex') == 2
def test_Poly_coeff():
assert Poly(0, x).coeff_monomial(1) == 0
assert Poly(0, x).coeff_monomial(x) == 0
assert Poly(1, x).coeff_monomial(1) == 1
assert Poly(1, x).coeff_monomial(x) == 0
assert Poly(x**8, x).coeff_monomial(1) == 0
assert Poly(x**8, x).coeff_monomial(x**7) == 0
assert Poly(x**8, x).coeff_monomial(x**8) == 1
assert Poly(x**8, x).coeff_monomial(x**9) == 0
assert Poly(3*x*y**2 + 1, x, y).coeff_monomial(1) == 1
assert Poly(3*x*y**2 + 1, x, y).coeff_monomial(x*y**2) == 3
p = Poly(24*x*y*exp(8) + 23*x, x, y)
assert p.coeff_monomial(x) == 23
assert p.coeff_monomial(y) == 0
assert p.coeff_monomial(x*y) == 24*exp(8)
assert p.as_expr().coeff(x) == 24*y*exp(8) + 23
raises(NotImplementedError, lambda: p.coeff(x))
raises(ValueError, lambda: Poly(x + 1).coeff_monomial(0))
raises(ValueError, lambda: Poly(x + 1).coeff_monomial(3*x))
raises(ValueError, lambda: Poly(x + 1).coeff_monomial(3*x*y))
def test_Poly_nth():
assert Poly(0, x).nth(0) == 0
assert Poly(0, x).nth(1) == 0
assert Poly(1, x).nth(0) == 1
assert Poly(1, x).nth(1) == 0
assert Poly(x**8, x).nth(0) == 0
assert Poly(x**8, x).nth(7) == 0
assert Poly(x**8, x).nth(8) == 1
assert Poly(x**8, x).nth(9) == 0
assert Poly(3*x*y**2 + 1, x, y).nth(0, 0) == 1
assert Poly(3*x*y**2 + 1, x, y).nth(1, 2) == 3
raises(ValueError, lambda: Poly(x*y + 1, x, y).nth(1))
def test_Poly_LM():
assert Poly(0, x).LM() == (0,)
assert Poly(1, x).LM() == (0,)
assert Poly(2*x**2 + x, x).LM() == (2,)
assert Poly(x*y**7 + 2*x**2*y**3).LM('lex') == (2, 3)
assert Poly(x*y**7 + 2*x**2*y**3).LM('grlex') == (1, 7)
assert LM(x*y**7 + 2*x**2*y**3, order='lex') == x**2*y**3
assert LM(x*y**7 + 2*x**2*y**3, order='grlex') == x*y**7
def test_Poly_LM_custom_order():
f = Poly(x**2*y**3*z + x**2*y*z**3 + x*y*z + 1)
rev_lex = lambda monom: tuple(reversed(monom))
assert f.LM(order='lex') == (2, 3, 1)
assert f.LM(order=rev_lex) == (2, 1, 3)
def test_Poly_EM():
assert Poly(0, x).EM() == (0,)
assert Poly(1, x).EM() == (0,)
assert Poly(2*x**2 + x, x).EM() == (1,)
assert Poly(x*y**7 + 2*x**2*y**3).EM('lex') == (1, 7)
assert Poly(x*y**7 + 2*x**2*y**3).EM('grlex') == (2, 3)
def test_Poly_LT():
assert Poly(0, x).LT() == ((0,), 0)
assert Poly(1, x).LT() == ((0,), 1)
assert Poly(2*x**2 + x, x).LT() == ((2,), 2)
assert Poly(x*y**7 + 2*x**2*y**3).LT('lex') == ((2, 3), 2)
assert Poly(x*y**7 + 2*x**2*y**3).LT('grlex') == ((1, 7), 1)
assert LT(x*y**7 + 2*x**2*y**3, order='lex') == 2*x**2*y**3
assert LT(x*y**7 + 2*x**2*y**3, order='grlex') == x*y**7
def test_Poly_ET():
assert Poly(0, x).ET() == ((0,), 0)
assert Poly(1, x).ET() == ((0,), 1)
assert Poly(2*x**2 + x, x).ET() == ((1,), 1)
assert Poly(x*y**7 + 2*x**2*y**3).ET('lex') == ((1, 7), 1)
assert Poly(x*y**7 + 2*x**2*y**3).ET('grlex') == ((2, 3), 2)
def test_Poly_max_norm():
assert Poly(-1, x).max_norm() == 1
assert Poly( 0, x).max_norm() == 0
assert Poly( 1, x).max_norm() == 1
def test_Poly_l1_norm():
assert Poly(-1, x).l1_norm() == 1
assert Poly( 0, x).l1_norm() == 0
assert Poly( 1, x).l1_norm() == 1
def test_Poly_clear_denoms():
coeff, poly = Poly(x + 2, x).clear_denoms()
assert coeff == 1 and poly == Poly(
x + 2, x, domain='ZZ') and poly.get_domain() == ZZ
coeff, poly = Poly(x/2 + 1, x).clear_denoms()
assert coeff == 2 and poly == Poly(
x + 2, x, domain='QQ') and poly.get_domain() == QQ
coeff, poly = Poly(x/2 + 1, x).clear_denoms(convert=True)
assert coeff == 2 and poly == Poly(
x + 2, x, domain='ZZ') and poly.get_domain() == ZZ
coeff, poly = Poly(x/y + 1, x).clear_denoms(convert=True)
assert coeff == y and poly == Poly(
x + y, x, domain='ZZ[y]') and poly.get_domain() == ZZ[y]
coeff, poly = Poly(x/3 + sqrt(2), x, domain='EX').clear_denoms()
assert coeff == 3 and poly == Poly(
x + 3*sqrt(2), x, domain='EX') and poly.get_domain() == EX
coeff, poly = Poly(
x/3 + sqrt(2), x, domain='EX').clear_denoms(convert=True)
assert coeff == 3 and poly == Poly(
x + 3*sqrt(2), x, domain='EX') and poly.get_domain() == EX
def test_Poly_rat_clear_denoms():
f = Poly(x**2/y + 1, x)
g = Poly(x**3 + y, x)
assert f.rat_clear_denoms(g) == \
(Poly(x**2 + y, x), Poly(y*x**3 + y**2, x))
f = f.set_domain(EX)
g = g.set_domain(EX)
assert f.rat_clear_denoms(g) == (f, g)
def test_Poly_integrate():
assert Poly(x + 1).integrate() == Poly(x**2/2 + x)
assert Poly(x + 1).integrate(x) == Poly(x**2/2 + x)
assert Poly(x + 1).integrate((x, 1)) == Poly(x**2/2 + x)
assert Poly(x*y + 1).integrate(x) == Poly(x**2*y/2 + x)
assert Poly(x*y + 1).integrate(y) == Poly(x*y**2/2 + y)
assert Poly(x*y + 1).integrate(x, x) == Poly(x**3*y/6 + x**2/2)
assert Poly(x*y + 1).integrate(y, y) == Poly(x*y**3/6 + y**2/2)
assert Poly(x*y + 1).integrate((x, 2)) == Poly(x**3*y/6 + x**2/2)
assert Poly(x*y + 1).integrate((y, 2)) == Poly(x*y**3/6 + y**2/2)
assert Poly(x*y + 1).integrate(x, y) == Poly(x**2*y**2/4 + x*y)
assert Poly(x*y + 1).integrate(y, x) == Poly(x**2*y**2/4 + x*y)
def test_Poly_diff():
assert Poly(x**2 + x).diff() == Poly(2*x + 1)
assert Poly(x**2 + x).diff(x) == Poly(2*x + 1)
assert Poly(x**2 + x).diff((x, 1)) == Poly(2*x + 1)
assert Poly(x**2*y**2 + x*y).diff(x) == Poly(2*x*y**2 + y)
assert Poly(x**2*y**2 + x*y).diff(y) == Poly(2*x**2*y + x)
assert Poly(x**2*y**2 + x*y).diff(x, x) == Poly(2*y**2, x, y)
assert Poly(x**2*y**2 + x*y).diff(y, y) == Poly(2*x**2, x, y)
assert Poly(x**2*y**2 + x*y).diff((x, 2)) == Poly(2*y**2, x, y)
assert Poly(x**2*y**2 + x*y).diff((y, 2)) == Poly(2*x**2, x, y)
assert Poly(x**2*y**2 + x*y).diff(x, y) == Poly(4*x*y + 1)
assert Poly(x**2*y**2 + x*y).diff(y, x) == Poly(4*x*y + 1)
def test_issue_9585():
assert diff(Poly(x**2 + x)) == Poly(2*x + 1)
assert diff(Poly(x**2 + x), x, evaluate=False) == \
Derivative(Poly(x**2 + x), x)
assert Derivative(Poly(x**2 + x), x).doit() == Poly(2*x + 1)
def test_Poly_eval():
assert Poly(0, x).eval(7) == 0
assert Poly(1, x).eval(7) == 1
assert Poly(x, x).eval(7) == 7
assert Poly(0, x).eval(0, 7) == 0
assert Poly(1, x).eval(0, 7) == 1
assert Poly(x, x).eval(0, 7) == 7
assert Poly(0, x).eval(x, 7) == 0
assert Poly(1, x).eval(x, 7) == 1
assert Poly(x, x).eval(x, 7) == 7
assert Poly(0, x).eval('x', 7) == 0
assert Poly(1, x).eval('x', 7) == 1
assert Poly(x, x).eval('x', 7) == 7
raises(PolynomialError, lambda: Poly(1, x).eval(1, 7))
raises(PolynomialError, lambda: Poly(1, x).eval(y, 7))
raises(PolynomialError, lambda: Poly(1, x).eval('y', 7))
assert Poly(123, x, y).eval(7) == Poly(123, y)
assert Poly(2*y, x, y).eval(7) == Poly(2*y, y)
assert Poly(x*y, x, y).eval(7) == Poly(7*y, y)
assert Poly(123, x, y).eval(x, 7) == Poly(123, y)
assert Poly(2*y, x, y).eval(x, 7) == Poly(2*y, y)
assert Poly(x*y, x, y).eval(x, 7) == Poly(7*y, y)
assert Poly(123, x, y).eval(y, 7) == Poly(123, x)
assert Poly(2*y, x, y).eval(y, 7) == Poly(14, x)
assert Poly(x*y, x, y).eval(y, 7) == Poly(7*x, x)
assert Poly(x*y + y, x, y).eval({x: 7}) == Poly(8*y, y)
assert Poly(x*y + y, x, y).eval({y: 7}) == Poly(7*x + 7, x)
assert Poly(x*y + y, x, y).eval({x: 6, y: 7}) == 49
assert Poly(x*y + y, x, y).eval({x: 7, y: 6}) == 48
assert Poly(x*y + y, x, y).eval((6, 7)) == 49
assert Poly(x*y + y, x, y).eval([6, 7]) == 49
assert Poly(x + 1, domain='ZZ').eval(S.Half) == Rational(3, 2)
assert Poly(x + 1, domain='ZZ').eval(sqrt(2)) == sqrt(2) + 1
raises(ValueError, lambda: Poly(x*y + y, x, y).eval((6, 7, 8)))
raises(DomainError, lambda: Poly(x + 1, domain='ZZ').eval(S.Half, auto=False))
# issue 6344
alpha = Symbol('alpha')
result = (2*alpha*z - 2*alpha + z**2 + 3)/(z**2 - 2*z + 1)
f = Poly(x**2 + (alpha - 1)*x - alpha + 1, x, domain='ZZ[alpha]')
assert f.eval((z + 1)/(z - 1)) == result
g = Poly(x**2 + (alpha - 1)*x - alpha + 1, x, y, domain='ZZ[alpha]')
assert g.eval((z + 1)/(z - 1)) == Poly(result, y, domain='ZZ(alpha,z)')
def test_Poly___call__():
f = Poly(2*x*y + 3*x + y + 2*z)
assert f(2) == Poly(5*y + 2*z + 6)
assert f(2, 5) == Poly(2*z + 31)
assert f(2, 5, 7) == 45
def test_parallel_poly_from_expr():
assert parallel_poly_from_expr(
[x - 1, x**2 - 1], x)[0] == [Poly(x - 1, x), Poly(x**2 - 1, x)]
assert parallel_poly_from_expr(
[Poly(x - 1, x), x**2 - 1], x)[0] == [Poly(x - 1, x), Poly(x**2 - 1, x)]
assert parallel_poly_from_expr(
[x - 1, Poly(x**2 - 1, x)], x)[0] == [Poly(x - 1, x), Poly(x**2 - 1, x)]
assert parallel_poly_from_expr([Poly(
x - 1, x), Poly(x**2 - 1, x)], x)[0] == [Poly(x - 1, x), Poly(x**2 - 1, x)]
assert parallel_poly_from_expr(
[x - 1, x**2 - 1], x, y)[0] == [Poly(x - 1, x, y), Poly(x**2 - 1, x, y)]
assert parallel_poly_from_expr([Poly(
x - 1, x), x**2 - 1], x, y)[0] == [Poly(x - 1, x, y), Poly(x**2 - 1, x, y)]
assert parallel_poly_from_expr([x - 1, Poly(
x**2 - 1, x)], x, y)[0] == [Poly(x - 1, x, y), Poly(x**2 - 1, x, y)]
assert parallel_poly_from_expr([Poly(x - 1, x), Poly(
x**2 - 1, x)], x, y)[0] == [Poly(x - 1, x, y), Poly(x**2 - 1, x, y)]
assert parallel_poly_from_expr(
[x - 1, x**2 - 1])[0] == [Poly(x - 1, x), Poly(x**2 - 1, x)]
assert parallel_poly_from_expr(
[Poly(x - 1, x), x**2 - 1])[0] == [Poly(x - 1, x), Poly(x**2 - 1, x)]
assert parallel_poly_from_expr(
[x - 1, Poly(x**2 - 1, x)])[0] == [Poly(x - 1, x), Poly(x**2 - 1, x)]
assert parallel_poly_from_expr(
[Poly(x - 1, x), Poly(x**2 - 1, x)])[0] == [Poly(x - 1, x), Poly(x**2 - 1, x)]
assert parallel_poly_from_expr(
[1, x**2 - 1])[0] == [Poly(1, x), Poly(x**2 - 1, x)]
assert parallel_poly_from_expr(
[1, x**2 - 1])[0] == [Poly(1, x), Poly(x**2 - 1, x)]
assert parallel_poly_from_expr(
[1, Poly(x**2 - 1, x)])[0] == [Poly(1, x), Poly(x**2 - 1, x)]
assert parallel_poly_from_expr(
[1, Poly(x**2 - 1, x)])[0] == [Poly(1, x), Poly(x**2 - 1, x)]
assert parallel_poly_from_expr(
[x**2 - 1, 1])[0] == [Poly(x**2 - 1, x), Poly(1, x)]
assert parallel_poly_from_expr(
[x**2 - 1, 1])[0] == [Poly(x**2 - 1, x), Poly(1, x)]
assert parallel_poly_from_expr(
[Poly(x**2 - 1, x), 1])[0] == [Poly(x**2 - 1, x), Poly(1, x)]
assert parallel_poly_from_expr(
[Poly(x**2 - 1, x), 1])[0] == [Poly(x**2 - 1, x), Poly(1, x)]
assert parallel_poly_from_expr([Poly(x, x, y), Poly(y, x, y)], x, y, order='lex')[0] == \
[Poly(x, x, y, domain='ZZ'), Poly(y, x, y, domain='ZZ')]
raises(PolificationFailed, lambda: parallel_poly_from_expr([0, 1]))
def test_pdiv():
f, g = x**2 - y**2, x - y
q, r = x + y, 0
F, G, Q, R = [ Poly(h, x, y) for h in (f, g, q, r) ]
assert F.pdiv(G) == (Q, R)
assert F.prem(G) == R
assert F.pquo(G) == Q
assert F.pexquo(G) == Q
assert pdiv(f, g) == (q, r)
assert prem(f, g) == r
assert pquo(f, g) == q
assert pexquo(f, g) == q
assert pdiv(f, g, x, y) == (q, r)
assert prem(f, g, x, y) == r
assert pquo(f, g, x, y) == q
assert pexquo(f, g, x, y) == q
assert pdiv(f, g, (x, y)) == (q, r)
assert prem(f, g, (x, y)) == r
assert pquo(f, g, (x, y)) == q
assert pexquo(f, g, (x, y)) == q
assert pdiv(F, G) == (Q, R)
assert prem(F, G) == R
assert pquo(F, G) == Q
assert pexquo(F, G) == Q
assert pdiv(f, g, polys=True) == (Q, R)
assert prem(f, g, polys=True) == R
assert pquo(f, g, polys=True) == Q
assert pexquo(f, g, polys=True) == Q
assert pdiv(F, G, polys=False) == (q, r)
assert prem(F, G, polys=False) == r
assert pquo(F, G, polys=False) == q
assert pexquo(F, G, polys=False) == q
raises(ComputationFailed, lambda: pdiv(4, 2))
raises(ComputationFailed, lambda: prem(4, 2))
raises(ComputationFailed, lambda: pquo(4, 2))
raises(ComputationFailed, lambda: pexquo(4, 2))
def test_div():
f, g = x**2 - y**2, x - y
q, r = x + y, 0
F, G, Q, R = [ Poly(h, x, y) for h in (f, g, q, r) ]
assert F.div(G) == (Q, R)
assert F.rem(G) == R
assert F.quo(G) == Q
assert F.exquo(G) == Q
assert div(f, g) == (q, r)
assert rem(f, g) == r
assert quo(f, g) == q
assert exquo(f, g) == q
assert div(f, g, x, y) == (q, r)
assert rem(f, g, x, y) == r
assert quo(f, g, x, y) == q
assert exquo(f, g, x, y) == q
assert div(f, g, (x, y)) == (q, r)
assert rem(f, g, (x, y)) == r
assert quo(f, g, (x, y)) == q
assert exquo(f, g, (x, y)) == q
assert div(F, G) == (Q, R)
assert rem(F, G) == R
assert quo(F, G) == Q
assert exquo(F, G) == Q
assert div(f, g, polys=True) == (Q, R)
assert rem(f, g, polys=True) == R
assert quo(f, g, polys=True) == Q
assert exquo(f, g, polys=True) == Q
assert div(F, G, polys=False) == (q, r)
assert rem(F, G, polys=False) == r
assert quo(F, G, polys=False) == q
assert exquo(F, G, polys=False) == q
raises(ComputationFailed, lambda: div(4, 2))
raises(ComputationFailed, lambda: rem(4, 2))
raises(ComputationFailed, lambda: quo(4, 2))
raises(ComputationFailed, lambda: exquo(4, 2))
f, g = x**2 + 1, 2*x - 4
qz, rz = 0, x**2 + 1
qq, rq = x/2 + 1, 5
assert div(f, g) == (qq, rq)
assert div(f, g, auto=True) == (qq, rq)
assert div(f, g, auto=False) == (qz, rz)
assert div(f, g, domain=ZZ) == (qz, rz)
assert div(f, g, domain=QQ) == (qq, rq)
assert div(f, g, domain=ZZ, auto=True) == (qq, rq)
assert div(f, g, domain=ZZ, auto=False) == (qz, rz)
assert div(f, g, domain=QQ, auto=True) == (qq, rq)
assert div(f, g, domain=QQ, auto=False) == (qq, rq)
assert rem(f, g) == rq
assert rem(f, g, auto=True) == rq
assert rem(f, g, auto=False) == rz
assert rem(f, g, domain=ZZ) == rz
assert rem(f, g, domain=QQ) == rq
assert rem(f, g, domain=ZZ, auto=True) == rq
assert rem(f, g, domain=ZZ, auto=False) == rz
assert rem(f, g, domain=QQ, auto=True) == rq
assert rem(f, g, domain=QQ, auto=False) == rq
assert quo(f, g) == qq
assert quo(f, g, auto=True) == qq
assert quo(f, g, auto=False) == qz
assert quo(f, g, domain=ZZ) == qz
assert quo(f, g, domain=QQ) == qq
assert quo(f, g, domain=ZZ, auto=True) == qq
assert quo(f, g, domain=ZZ, auto=False) == qz
assert quo(f, g, domain=QQ, auto=True) == qq
assert quo(f, g, domain=QQ, auto=False) == qq
f, g, q = x**2, 2*x, x/2
assert exquo(f, g) == q
assert exquo(f, g, auto=True) == q
raises(ExactQuotientFailed, lambda: exquo(f, g, auto=False))
raises(ExactQuotientFailed, lambda: exquo(f, g, domain=ZZ))
assert exquo(f, g, domain=QQ) == q
assert exquo(f, g, domain=ZZ, auto=True) == q
raises(ExactQuotientFailed, lambda: exquo(f, g, domain=ZZ, auto=False))
assert exquo(f, g, domain=QQ, auto=True) == q
assert exquo(f, g, domain=QQ, auto=False) == q
f, g = Poly(x**2), Poly(x)
q, r = f.div(g)
assert q.get_domain().is_ZZ and r.get_domain().is_ZZ
r = f.rem(g)
assert r.get_domain().is_ZZ
q = f.quo(g)
assert q.get_domain().is_ZZ
q = f.exquo(g)
assert q.get_domain().is_ZZ
f, g = Poly(x+y, x), Poly(2*x+y, x)
q, r = f.div(g)
assert q.get_domain().is_Frac and r.get_domain().is_Frac
def test_issue_7864():
q, r = div(a, .408248290463863*a)
assert abs(q - 2.44948974278318) < 1e-14
assert r == 0
def test_gcdex():
f, g = 2*x, x**2 - 16
s, t, h = x/32, Rational(-1, 16), 1
F, G, S, T, H = [ Poly(u, x, domain='QQ') for u in (f, g, s, t, h) ]
assert F.half_gcdex(G) == (S, H)
assert F.gcdex(G) == (S, T, H)
assert F.invert(G) == S
assert half_gcdex(f, g) == (s, h)
assert gcdex(f, g) == (s, t, h)
assert invert(f, g) == s
assert half_gcdex(f, g, x) == (s, h)
assert gcdex(f, g, x) == (s, t, h)
assert invert(f, g, x) == s
assert half_gcdex(f, g, (x,)) == (s, h)
assert gcdex(f, g, (x,)) == (s, t, h)
assert invert(f, g, (x,)) == s
assert half_gcdex(F, G) == (S, H)
assert gcdex(F, G) == (S, T, H)
assert invert(F, G) == S
assert half_gcdex(f, g, polys=True) == (S, H)
assert gcdex(f, g, polys=True) == (S, T, H)
assert invert(f, g, polys=True) == S
assert half_gcdex(F, G, polys=False) == (s, h)
assert gcdex(F, G, polys=False) == (s, t, h)
assert invert(F, G, polys=False) == s
assert half_gcdex(100, 2004) == (-20, 4)
assert gcdex(100, 2004) == (-20, 1, 4)
assert invert(3, 7) == 5
raises(DomainError, lambda: half_gcdex(x + 1, 2*x + 1, auto=False))
raises(DomainError, lambda: gcdex(x + 1, 2*x + 1, auto=False))
raises(DomainError, lambda: invert(x + 1, 2*x + 1, auto=False))
def test_revert():
f = Poly(1 - x**2/2 + x**4/24 - x**6/720)
g = Poly(61*x**6/720 + 5*x**4/24 + x**2/2 + 1)
assert f.revert(8) == g
def test_subresultants():
f, g, h = x**2 - 2*x + 1, x**2 - 1, 2*x - 2
F, G, H = Poly(f), Poly(g), Poly(h)
assert F.subresultants(G) == [F, G, H]
assert subresultants(f, g) == [f, g, h]
assert subresultants(f, g, x) == [f, g, h]
assert subresultants(f, g, (x,)) == [f, g, h]
assert subresultants(F, G) == [F, G, H]
assert subresultants(f, g, polys=True) == [F, G, H]
assert subresultants(F, G, polys=False) == [f, g, h]
raises(ComputationFailed, lambda: subresultants(4, 2))
def test_resultant():
f, g, h = x**2 - 2*x + 1, x**2 - 1, 0
F, G = Poly(f), Poly(g)
assert F.resultant(G) == h
assert resultant(f, g) == h
assert resultant(f, g, x) == h
assert resultant(f, g, (x,)) == h
assert resultant(F, G) == h
assert resultant(f, g, polys=True) == h
assert resultant(F, G, polys=False) == h
assert resultant(f, g, includePRS=True) == (h, [f, g, 2*x - 2])
f, g, h = x - a, x - b, a - b
F, G, H = Poly(f), Poly(g), Poly(h)
assert F.resultant(G) == H
assert resultant(f, g) == h
assert resultant(f, g, x) == h
assert resultant(f, g, (x,)) == h
assert resultant(F, G) == H
assert resultant(f, g, polys=True) == H
assert resultant(F, G, polys=False) == h
raises(ComputationFailed, lambda: resultant(4, 2))
def test_discriminant():
f, g = x**3 + 3*x**2 + 9*x - 13, -11664
F = Poly(f)
assert F.discriminant() == g
assert discriminant(f) == g
assert discriminant(f, x) == g
assert discriminant(f, (x,)) == g
assert discriminant(F) == g
assert discriminant(f, polys=True) == g
assert discriminant(F, polys=False) == g
f, g = a*x**2 + b*x + c, b**2 - 4*a*c
F, G = Poly(f), Poly(g)
assert F.discriminant() == G
assert discriminant(f) == g
assert discriminant(f, x, a, b, c) == g
assert discriminant(f, (x, a, b, c)) == g
assert discriminant(F) == G
assert discriminant(f, polys=True) == G
assert discriminant(F, polys=False) == g
raises(ComputationFailed, lambda: discriminant(4))
def test_dispersion():
# We test only the API here. For more mathematical
# tests see the dedicated test file.
fp = poly((x + 1)*(x + 2), x)
assert sorted(fp.dispersionset()) == [0, 1]
assert fp.dispersion() == 1
fp = poly(x**4 - 3*x**2 + 1, x)
gp = fp.shift(-3)
assert sorted(fp.dispersionset(gp)) == [2, 3, 4]
assert fp.dispersion(gp) == 4
def test_gcd_list():
F = [x**3 - 1, x**2 - 1, x**2 - 3*x + 2]
assert gcd_list(F) == x - 1
assert gcd_list(F, polys=True) == Poly(x - 1)
assert gcd_list([]) == 0
assert gcd_list([1, 2]) == 1
assert gcd_list([4, 6, 8]) == 2
assert gcd_list([x*(y + 42) - x*y - x*42]) == 0
gcd = gcd_list([], x)
assert gcd.is_Number and gcd is S.Zero
gcd = gcd_list([], x, polys=True)
assert gcd.is_Poly and gcd.is_zero
raises(ComputationFailed, lambda: gcd_list([], polys=True))
def test_lcm_list():
F = [x**3 - 1, x**2 - 1, x**2 - 3*x + 2]
assert lcm_list(F) == x**5 - x**4 - 2*x**3 - x**2 + x + 2
assert lcm_list(F, polys=True) == Poly(x**5 - x**4 - 2*x**3 - x**2 + x + 2)
assert lcm_list([]) == 1
assert lcm_list([1, 2]) == 2
assert lcm_list([4, 6, 8]) == 24
assert lcm_list([x*(y + 42) - x*y - x*42]) == 0
lcm = lcm_list([], x)
assert lcm.is_Number and lcm is S.One
lcm = lcm_list([], x, polys=True)
assert lcm.is_Poly and lcm.is_one
raises(ComputationFailed, lambda: lcm_list([], polys=True))
def test_gcd():
f, g = x**3 - 1, x**2 - 1
s, t = x**2 + x + 1, x + 1
h, r = x - 1, x**4 + x**3 - x - 1
F, G, S, T, H, R = [ Poly(u) for u in (f, g, s, t, h, r) ]
assert F.cofactors(G) == (H, S, T)
assert F.gcd(G) == H
assert F.lcm(G) == R
assert cofactors(f, g) == (h, s, t)
assert gcd(f, g) == h
assert lcm(f, g) == r
assert cofactors(f, g, x) == (h, s, t)
assert gcd(f, g, x) == h
assert lcm(f, g, x) == r
assert cofactors(f, g, (x,)) == (h, s, t)
assert gcd(f, g, (x,)) == h
assert lcm(f, g, (x,)) == r
assert cofactors(F, G) == (H, S, T)
assert gcd(F, G) == H
assert lcm(F, G) == R
assert cofactors(f, g, polys=True) == (H, S, T)
assert gcd(f, g, polys=True) == H
assert lcm(f, g, polys=True) == R
assert cofactors(F, G, polys=False) == (h, s, t)
assert gcd(F, G, polys=False) == h
assert lcm(F, G, polys=False) == r
f, g = 1.0*x**2 - 1.0, 1.0*x - 1.0
h, s, t = g, 1.0*x + 1.0, 1.0
assert cofactors(f, g) == (h, s, t)
assert gcd(f, g) == h
assert lcm(f, g) == f
f, g = 1.0*x**2 - 1.0, 1.0*x - 1.0
h, s, t = g, 1.0*x + 1.0, 1.0
assert cofactors(f, g) == (h, s, t)
assert gcd(f, g) == h
assert lcm(f, g) == f
assert cofactors(8, 6) == (2, 4, 3)
assert gcd(8, 6) == 2
assert lcm(8, 6) == 24
f, g = x**2 - 3*x - 4, x**3 - 4*x**2 + x - 4
l = x**4 - 3*x**3 - 3*x**2 - 3*x - 4
h, s, t = x - 4, x + 1, x**2 + 1
assert cofactors(f, g, modulus=11) == (h, s, t)
assert gcd(f, g, modulus=11) == h
assert lcm(f, g, modulus=11) == l
f, g = x**2 + 8*x + 7, x**3 + 7*x**2 + x + 7
l = x**4 + 8*x**3 + 8*x**2 + 8*x + 7
h, s, t = x + 7, x + 1, x**2 + 1
assert cofactors(f, g, modulus=11, symmetric=False) == (h, s, t)
assert gcd(f, g, modulus=11, symmetric=False) == h
assert lcm(f, g, modulus=11, symmetric=False) == l
raises(TypeError, lambda: gcd(x))
raises(TypeError, lambda: lcm(x))
def test_gcd_numbers_vs_polys():
assert isinstance(gcd(3, 9), Integer)
assert isinstance(gcd(3*x, 9), Integer)
assert gcd(3, 9) == 3
assert gcd(3*x, 9) == 3
assert isinstance(gcd(Rational(3, 2), Rational(9, 4)), Rational)
assert isinstance(gcd(Rational(3, 2)*x, Rational(9, 4)), Rational)
assert gcd(Rational(3, 2), Rational(9, 4)) == Rational(3, 4)
assert gcd(Rational(3, 2)*x, Rational(9, 4)) == 1
assert isinstance(gcd(3.0, 9.0), Float)
assert isinstance(gcd(3.0*x, 9.0), Float)
assert gcd(3.0, 9.0) == 1.0
assert gcd(3.0*x, 9.0) == 1.0
def test_terms_gcd():
assert terms_gcd(1) == 1
assert terms_gcd(1, x) == 1
assert terms_gcd(x - 1) == x - 1
assert terms_gcd(-x - 1) == -x - 1
assert terms_gcd(2*x + 3) == 2*x + 3
assert terms_gcd(6*x + 4) == Mul(2, 3*x + 2, evaluate=False)
assert terms_gcd(x**3*y + x*y**3) == x*y*(x**2 + y**2)
assert terms_gcd(2*x**3*y + 2*x*y**3) == 2*x*y*(x**2 + y**2)
assert terms_gcd(x**3*y/2 + x*y**3/2) == x*y/2*(x**2 + y**2)
assert terms_gcd(x**3*y + 2*x*y**3) == x*y*(x**2 + 2*y**2)
assert terms_gcd(2*x**3*y + 4*x*y**3) == 2*x*y*(x**2 + 2*y**2)
assert terms_gcd(2*x**3*y/3 + 4*x*y**3/5) == x*y*Rational(2, 15)*(5*x**2 + 6*y**2)
assert terms_gcd(2.0*x**3*y + 4.1*x*y**3) == x*y*(2.0*x**2 + 4.1*y**2)
assert _aresame(terms_gcd(2.0*x + 3), 2.0*x + 3)
assert terms_gcd((3 + 3*x)*(x + x*y), expand=False) == \
(3*x + 3)*(x*y + x)
assert terms_gcd((3 + 3*x)*(x + x*sin(3 + 3*y)), expand=False, deep=True) == \
3*x*(x + 1)*(sin(Mul(3, y + 1, evaluate=False)) + 1)
assert terms_gcd(sin(x + x*y), deep=True) == \
sin(x*(y + 1))
eq = Eq(2*x, 2*y + 2*z*y)
assert terms_gcd(eq) == Eq(2*x, 2*y*(z + 1))
assert terms_gcd(eq, deep=True) == Eq(2*x, 2*y*(z + 1))
raises(TypeError, lambda: terms_gcd(x < 2))
def test_trunc():
f, g = x**5 + 2*x**4 + 3*x**3 + 4*x**2 + 5*x + 6, x**5 - x**4 + x**2 - x
F, G = Poly(f), Poly(g)
assert F.trunc(3) == G
assert trunc(f, 3) == g
assert trunc(f, 3, x) == g
assert trunc(f, 3, (x,)) == g
assert trunc(F, 3) == G
assert trunc(f, 3, polys=True) == G
assert trunc(F, 3, polys=False) == g
f, g = 6*x**5 + 5*x**4 + 4*x**3 + 3*x**2 + 2*x + 1, -x**4 + x**3 - x + 1
F, G = Poly(f), Poly(g)
assert F.trunc(3) == G
assert trunc(f, 3) == g
assert trunc(f, 3, x) == g
assert trunc(f, 3, (x,)) == g
assert trunc(F, 3) == G
assert trunc(f, 3, polys=True) == G
assert trunc(F, 3, polys=False) == g
f = Poly(x**2 + 2*x + 3, modulus=5)
assert f.trunc(2) == Poly(x**2 + 1, modulus=5)
def test_monic():
f, g = 2*x - 1, x - S.Half
F, G = Poly(f, domain='QQ'), Poly(g)
assert F.monic() == G
assert monic(f) == g
assert monic(f, x) == g
assert monic(f, (x,)) == g
assert monic(F) == G
assert monic(f, polys=True) == G
assert monic(F, polys=False) == g
raises(ComputationFailed, lambda: monic(4))
assert monic(2*x**2 + 6*x + 4, auto=False) == x**2 + 3*x + 2
raises(ExactQuotientFailed, lambda: monic(2*x + 6*x + 1, auto=False))
assert monic(2.0*x**2 + 6.0*x + 4.0) == 1.0*x**2 + 3.0*x + 2.0
assert monic(2*x**2 + 3*x + 4, modulus=5) == x**2 - x + 2
def test_content():
f, F = 4*x + 2, Poly(4*x + 2)
assert F.content() == 2
assert content(f) == 2
raises(ComputationFailed, lambda: content(4))
f = Poly(2*x, modulus=3)
assert f.content() == 1
def test_primitive():
f, g = 4*x + 2, 2*x + 1
F, G = Poly(f), Poly(g)
assert F.primitive() == (2, G)
assert primitive(f) == (2, g)
assert primitive(f, x) == (2, g)
assert primitive(f, (x,)) == (2, g)
assert primitive(F) == (2, G)
assert primitive(f, polys=True) == (2, G)
assert primitive(F, polys=False) == (2, g)
raises(ComputationFailed, lambda: primitive(4))
f = Poly(2*x, modulus=3)
g = Poly(2.0*x, domain=RR)
assert f.primitive() == (1, f)
assert g.primitive() == (1.0, g)
assert primitive(S('-3*x/4 + y + 11/8')) == \
S('(1/8, -6*x + 8*y + 11)')
def test_compose():
f = x**12 + 20*x**10 + 150*x**8 + 500*x**6 + 625*x**4 - 2*x**3 - 10*x + 9
g = x**4 - 2*x + 9
h = x**3 + 5*x
F, G, H = map(Poly, (f, g, h))
assert G.compose(H) == F
assert compose(g, h) == f
assert compose(g, h, x) == f
assert compose(g, h, (x,)) == f
assert compose(G, H) == F
assert compose(g, h, polys=True) == F
assert compose(G, H, polys=False) == f
assert F.decompose() == [G, H]
assert decompose(f) == [g, h]
assert decompose(f, x) == [g, h]
assert decompose(f, (x,)) == [g, h]
assert decompose(F) == [G, H]
assert decompose(f, polys=True) == [G, H]
assert decompose(F, polys=False) == [g, h]
raises(ComputationFailed, lambda: compose(4, 2))
raises(ComputationFailed, lambda: decompose(4))
assert compose(x**2 - y**2, x - y, x, y) == x**2 - 2*x*y
assert compose(x**2 - y**2, x - y, y, x) == -y**2 + 2*x*y
def test_shift():
assert Poly(x**2 - 2*x + 1, x).shift(2) == Poly(x**2 + 2*x + 1, x)
def test_transform():
# Also test that 3-way unification is done correctly
assert Poly(x**2 - 2*x + 1, x).transform(Poly(x + 1), Poly(x - 1)) == \
Poly(4, x) == \
cancel((x - 1)**2*(x**2 - 2*x + 1).subs(x, (x + 1)/(x - 1)))
assert Poly(x**2 - x/2 + 1, x).transform(Poly(x + 1), Poly(x - 1)) == \
Poly(3*x**2/2 + Rational(5, 2), x) == \
cancel((x - 1)**2*(x**2 - x/2 + 1).subs(x, (x + 1)/(x - 1)))
assert Poly(x**2 - 2*x + 1, x).transform(Poly(x + S.Half), Poly(x - 1)) == \
Poly(Rational(9, 4), x) == \
cancel((x - 1)**2*(x**2 - 2*x + 1).subs(x, (x + S.Half)/(x - 1)))
assert Poly(x**2 - 2*x + 1, x).transform(Poly(x + 1), Poly(x - S.Half)) == \
Poly(Rational(9, 4), x) == \
cancel((x - S.Half)**2*(x**2 - 2*x + 1).subs(x, (x + 1)/(x - S.Half)))
# Unify ZZ, QQ, and RR
assert Poly(x**2 - 2*x + 1, x).transform(Poly(x + 1.0), Poly(x - S.Half)) == \
Poly(Rational(9, 4), x, domain='RR') == \
cancel((x - S.Half)**2*(x**2 - 2*x + 1).subs(x, (x + 1.0)/(x - S.Half)))
raises(ValueError, lambda: Poly(x*y).transform(Poly(x + 1), Poly(x - 1)))
raises(ValueError, lambda: Poly(x).transform(Poly(y + 1), Poly(x - 1)))
raises(ValueError, lambda: Poly(x).transform(Poly(x + 1), Poly(y - 1)))
raises(ValueError, lambda: Poly(x).transform(Poly(x*y + 1), Poly(x - 1)))
raises(ValueError, lambda: Poly(x).transform(Poly(x + 1), Poly(x*y - 1)))
def test_sturm():
f, F = x, Poly(x, domain='QQ')
g, G = 1, Poly(1, x, domain='QQ')
assert F.sturm() == [F, G]
assert sturm(f) == [f, g]
assert sturm(f, x) == [f, g]
assert sturm(f, (x,)) == [f, g]
assert sturm(F) == [F, G]
assert sturm(f, polys=True) == [F, G]
assert sturm(F, polys=False) == [f, g]
raises(ComputationFailed, lambda: sturm(4))
raises(DomainError, lambda: sturm(f, auto=False))
f = Poly(S(1024)/(15625*pi**8)*x**5
- S(4096)/(625*pi**8)*x**4
+ S(32)/(15625*pi**4)*x**3
- S(128)/(625*pi**4)*x**2
+ Rational(1, 62500)*x
- Rational(1, 625), x, domain='ZZ(pi)')
assert sturm(f) == \
[Poly(x**3 - 100*x**2 + pi**4/64*x - 25*pi**4/16, x, domain='ZZ(pi)'),
Poly(3*x**2 - 200*x + pi**4/64, x, domain='ZZ(pi)'),
Poly((Rational(20000, 9) - pi**4/96)*x + 25*pi**4/18, x, domain='ZZ(pi)'),
Poly((-3686400000000*pi**4 - 11520000*pi**8 - 9*pi**12)/(26214400000000 - 245760000*pi**4 + 576*pi**8), x, domain='ZZ(pi)')]
def test_gff():
f = x**5 + 2*x**4 - x**3 - 2*x**2
assert Poly(f).gff_list() == [(Poly(x), 1), (Poly(x + 2), 4)]
assert gff_list(f) == [(x, 1), (x + 2, 4)]
raises(NotImplementedError, lambda: gff(f))
f = x*(x - 1)**3*(x - 2)**2*(x - 4)**2*(x - 5)
assert Poly(f).gff_list() == [(
Poly(x**2 - 5*x + 4), 1), (Poly(x**2 - 5*x + 4), 2), (Poly(x), 3)]
assert gff_list(f) == [(x**2 - 5*x + 4, 1), (x**2 - 5*x + 4, 2), (x, 3)]
raises(NotImplementedError, lambda: gff(f))
def test_norm():
a, b = sqrt(2), sqrt(3)
f = Poly(a*x + b*y, x, y, extension=(a, b))
assert f.norm() == Poly(4*x**4 - 12*x**2*y**2 + 9*y**4, x, y, domain='QQ')
def test_sqf_norm():
assert sqf_norm(x**2 - 2, extension=sqrt(3)) == \
(1, x**2 - 2*sqrt(3)*x + 1, x**4 - 10*x**2 + 1)
assert sqf_norm(x**2 - 3, extension=sqrt(2)) == \
(1, x**2 - 2*sqrt(2)*x - 1, x**4 - 10*x**2 + 1)
assert Poly(x**2 - 2, extension=sqrt(3)).sqf_norm() == \
(1, Poly(x**2 - 2*sqrt(3)*x + 1, x, extension=sqrt(3)),
Poly(x**4 - 10*x**2 + 1, x, domain='QQ'))
assert Poly(x**2 - 3, extension=sqrt(2)).sqf_norm() == \
(1, Poly(x**2 - 2*sqrt(2)*x - 1, x, extension=sqrt(2)),
Poly(x**4 - 10*x**2 + 1, x, domain='QQ'))
def test_sqf():
f = x**5 - x**3 - x**2 + 1
g = x**3 + 2*x**2 + 2*x + 1
h = x - 1
p = x**4 + x**3 - x - 1
F, G, H, P = map(Poly, (f, g, h, p))
assert F.sqf_part() == P
assert sqf_part(f) == p
assert sqf_part(f, x) == p
assert sqf_part(f, (x,)) == p
assert sqf_part(F) == P
assert sqf_part(f, polys=True) == P
assert sqf_part(F, polys=False) == p
assert F.sqf_list() == (1, [(G, 1), (H, 2)])
assert sqf_list(f) == (1, [(g, 1), (h, 2)])
assert sqf_list(f, x) == (1, [(g, 1), (h, 2)])
assert sqf_list(f, (x,)) == (1, [(g, 1), (h, 2)])
assert sqf_list(F) == (1, [(G, 1), (H, 2)])
assert sqf_list(f, polys=True) == (1, [(G, 1), (H, 2)])
assert sqf_list(F, polys=False) == (1, [(g, 1), (h, 2)])
assert F.sqf_list_include() == [(G, 1), (H, 2)]
raises(ComputationFailed, lambda: sqf_part(4))
assert sqf(1) == 1
assert sqf_list(1) == (1, [])
assert sqf((2*x**2 + 2)**7) == 128*(x**2 + 1)**7
assert sqf(f) == g*h**2
assert sqf(f, x) == g*h**2
assert sqf(f, (x,)) == g*h**2
d = x**2 + y**2
assert sqf(f/d) == (g*h**2)/d
assert sqf(f/d, x) == (g*h**2)/d
assert sqf(f/d, (x,)) == (g*h**2)/d
assert sqf(x - 1) == x - 1
assert sqf(-x - 1) == -x - 1
assert sqf(x - 1) == x - 1
assert sqf(6*x - 10) == Mul(2, 3*x - 5, evaluate=False)
assert sqf((6*x - 10)/(3*x - 6)) == Rational(2, 3)*((3*x - 5)/(x - 2))
assert sqf(Poly(x**2 - 2*x + 1)) == (x - 1)**2
f = 3 + x - x*(1 + x) + x**2
assert sqf(f) == 3
f = (x**2 + 2*x + 1)**20000000000
assert sqf(f) == (x + 1)**40000000000
assert sqf_list(f) == (1, [(x + 1, 40000000000)])
def test_factor():
f = x**5 - x**3 - x**2 + 1
u = x + 1
v = x - 1
w = x**2 + x + 1
F, U, V, W = map(Poly, (f, u, v, w))
assert F.factor_list() == (1, [(U, 1), (V, 2), (W, 1)])
assert factor_list(f) == (1, [(u, 1), (v, 2), (w, 1)])
assert factor_list(f, x) == (1, [(u, 1), (v, 2), (w, 1)])
assert factor_list(f, (x,)) == (1, [(u, 1), (v, 2), (w, 1)])
assert factor_list(F) == (1, [(U, 1), (V, 2), (W, 1)])
assert factor_list(f, polys=True) == (1, [(U, 1), (V, 2), (W, 1)])
assert factor_list(F, polys=False) == (1, [(u, 1), (v, 2), (w, 1)])
assert F.factor_list_include() == [(U, 1), (V, 2), (W, 1)]
assert factor_list(1) == (1, [])
assert factor_list(6) == (6, [])
assert factor_list(sqrt(3), x) == (sqrt(3), [])
assert factor_list((-1)**x, x) == (1, [(-1, x)])
assert factor_list((2*x)**y, x) == (1, [(2, y), (x, y)])
assert factor_list(sqrt(x*y), x) == (1, [(x*y, S.Half)])
assert factor(6) == 6 and factor(6).is_Integer
assert factor_list(3*x) == (3, [(x, 1)])
assert factor_list(3*x**2) == (3, [(x, 2)])
assert factor(3*x) == 3*x
assert factor(3*x**2) == 3*x**2
assert factor((2*x**2 + 2)**7) == 128*(x**2 + 1)**7
assert factor(f) == u*v**2*w
assert factor(f, x) == u*v**2*w
assert factor(f, (x,)) == u*v**2*w
g, p, q, r = x**2 - y**2, x - y, x + y, x**2 + 1
assert factor(f/g) == (u*v**2*w)/(p*q)
assert factor(f/g, x) == (u*v**2*w)/(p*q)
assert factor(f/g, (x,)) == (u*v**2*w)/(p*q)
p = Symbol('p', positive=True)
i = Symbol('i', integer=True)
r = Symbol('r', real=True)
assert factor(sqrt(x*y)).is_Pow is True
assert factor(sqrt(3*x**2 - 3)) == sqrt(3)*sqrt((x - 1)*(x + 1))
assert factor(sqrt(3*x**2 + 3)) == sqrt(3)*sqrt(x**2 + 1)
assert factor((y*x**2 - y)**i) == y**i*(x - 1)**i*(x + 1)**i
assert factor((y*x**2 + y)**i) == y**i*(x**2 + 1)**i
assert factor((y*x**2 - y)**t) == (y*(x - 1)*(x + 1))**t
assert factor((y*x**2 + y)**t) == (y*(x**2 + 1))**t
f = sqrt(expand((r**2 + 1)*(p + 1)*(p - 1)*(p - 2)**3))
g = sqrt((p - 2)**3*(p - 1))*sqrt(p + 1)*sqrt(r**2 + 1)
assert factor(f) == g
assert factor(g) == g
g = (x - 1)**5*(r**2 + 1)
f = sqrt(expand(g))
assert factor(f) == sqrt(g)
f = Poly(sin(1)*x + 1, x, domain=EX)
assert f.factor_list() == (1, [(f, 1)])
f = x**4 + 1
assert factor(f) == f
assert factor(f, extension=I) == (x**2 - I)*(x**2 + I)
assert factor(f, gaussian=True) == (x**2 - I)*(x**2 + I)
assert factor(
f, extension=sqrt(2)) == (x**2 + sqrt(2)*x + 1)*(x**2 - sqrt(2)*x + 1)
f = x**2 + 2*sqrt(2)*x + 2
assert factor(f, extension=sqrt(2)) == (x + sqrt(2))**2
assert factor(f**3, extension=sqrt(2)) == (x + sqrt(2))**6
assert factor(x**2 - 2*y**2, extension=sqrt(2)) == \
(x + sqrt(2)*y)*(x - sqrt(2)*y)
assert factor(2*x**2 - 4*y**2, extension=sqrt(2)) == \
2*((x + sqrt(2)*y)*(x - sqrt(2)*y))
assert factor(x - 1) == x - 1
assert factor(-x - 1) == -x - 1
assert factor(x - 1) == x - 1
assert factor(6*x - 10) == Mul(2, 3*x - 5, evaluate=False)
assert factor(x**11 + x + 1, modulus=65537, symmetric=True) == \
(x**2 + x + 1)*(x**9 - x**8 + x**6 - x**5 + x**3 - x** 2 + 1)
assert factor(x**11 + x + 1, modulus=65537, symmetric=False) == \
(x**2 + x + 1)*(x**9 + 65536*x**8 + x**6 + 65536*x**5 +
x**3 + 65536*x** 2 + 1)
f = x/pi + x*sin(x)/pi
g = y/(pi**2 + 2*pi + 1) + y*sin(x)/(pi**2 + 2*pi + 1)
assert factor(f) == x*(sin(x) + 1)/pi
assert factor(g) == y*(sin(x) + 1)/(pi + 1)**2
assert factor(Eq(
x**2 + 2*x + 1, x**3 + 1)) == Eq((x + 1)**2, (x + 1)*(x**2 - x + 1))
f = (x**2 - 1)/(x**2 + 4*x + 4)
assert factor(f) == (x + 1)*(x - 1)/(x + 2)**2
assert factor(f, x) == (x + 1)*(x - 1)/(x + 2)**2
f = 3 + x - x*(1 + x) + x**2
assert factor(f) == 3
assert factor(f, x) == 3
assert factor(1/(x**2 + 2*x + 1/x) - 1) == -((1 - x + 2*x**2 +
x**3)/(1 + 2*x**2 + x**3))
assert factor(f, expand=False) == f
raises(PolynomialError, lambda: factor(f, x, expand=False))
raises(FlagError, lambda: factor(x**2 - 1, polys=True))
assert factor([x, Eq(x**2 - y**2, Tuple(x**2 - z**2, 1/x + 1/y))]) == \
[x, Eq((x - y)*(x + y), Tuple((x - z)*(x + z), (x + y)/x/y))]
assert not isinstance(
Poly(x**3 + x + 1).factor_list()[1][0][0], PurePoly) is True
assert isinstance(
PurePoly(x**3 + x + 1).factor_list()[1][0][0], PurePoly) is True
assert factor(sqrt(-x)) == sqrt(-x)
# issue 5917
e = (-2*x*(-x + 1)*(x - 1)*(-x*(-x + 1)*(x - 1) - x*(x - 1)**2)*(x**2*(x -
1) - x*(x - 1) - x) - (-2*x**2*(x - 1)**2 - x*(-x + 1)*(-x*(-x + 1) +
x*(x - 1)))*(x**2*(x - 1)**4 - x*(-x*(-x + 1)*(x - 1) - x*(x - 1)**2)))
assert factor(e) == 0
# deep option
assert factor(sin(x**2 + x) + x, deep=True) == sin(x*(x + 1)) + x
assert factor(sin(x**2 + x)*x, deep=True) == sin(x*(x + 1))*x
assert factor(sqrt(x**2)) == sqrt(x**2)
# issue 13149
assert factor(expand((0.5*x+1)*(0.5*y+1))) == Mul(1.0, 0.5*x + 1.0,
0.5*y + 1.0, evaluate = False)
assert factor(expand((0.5*x+0.5)**2)) == 0.25*(1.0*x + 1.0)**2
eq = x**2*y**2 + 11*x**2*y + 30*x**2 + 7*x*y**2 + 77*x*y + 210*x + 12*y**2 + 132*y + 360
assert factor(eq, x) == (x + 3)*(x + 4)*(y**2 + 11*y + 30)
assert factor(eq, x, deep=True) == (x + 3)*(x + 4)*(y**2 + 11*y + 30)
assert factor(eq, y, deep=True) == (y + 5)*(y + 6)*(x**2 + 7*x + 12)
# fraction option
f = 5*x + 3*exp(2 - 7*x)
assert factor(f, deep=True) == factor(f, deep=True, fraction=True)
assert factor(f, deep=True, fraction=False) == 5*x + 3*exp(2)*exp(-7*x)
def test_factor_large():
f = (x**2 + 4*x + 4)**10000000*(x**2 + 1)*(x**2 + 2*x + 1)**1234567
g = ((x**2 + 2*x + 1)**3000*y**2 + (x**2 + 2*x + 1)**3000*2*y + (
x**2 + 2*x + 1)**3000)
assert factor(f) == (x + 2)**20000000*(x**2 + 1)*(x + 1)**2469134
assert factor(g) == (x + 1)**6000*(y + 1)**2
assert factor_list(
f) == (1, [(x + 1, 2469134), (x + 2, 20000000), (x**2 + 1, 1)])
assert factor_list(g) == (1, [(y + 1, 2), (x + 1, 6000)])
f = (x**2 - y**2)**200000*(x**7 + 1)
g = (x**2 + y**2)**200000*(x**7 + 1)
assert factor(f) == \
(x + 1)*(x - y)**200000*(x + y)**200000*(x**6 - x**5 +
x**4 - x**3 + x**2 - x + 1)
assert factor(g, gaussian=True) == \
(x + 1)*(x - I*y)**200000*(x + I*y)**200000*(x**6 - x**5 +
x**4 - x**3 + x**2 - x + 1)
assert factor_list(f) == \
(1, [(x + 1, 1), (x - y, 200000), (x + y, 200000), (x**6 -
x**5 + x**4 - x**3 + x**2 - x + 1, 1)])
assert factor_list(g, gaussian=True) == \
(1, [(x + 1, 1), (x - I*y, 200000), (x + I*y, 200000), (
x**6 - x**5 + x**4 - x**3 + x**2 - x + 1, 1)])
def test_factor_noeval():
assert factor(6*x - 10) == Mul(2, 3*x - 5, evaluate=False)
assert factor((6*x - 10)/(3*x - 6)) == Mul(Rational(2, 3), 3*x - 5, 1/(x - 2))
def test_intervals():
assert intervals(0) == []
assert intervals(1) == []
assert intervals(x, sqf=True) == [(0, 0)]
assert intervals(x) == [((0, 0), 1)]
assert intervals(x**128) == [((0, 0), 128)]
assert intervals([x**2, x**4]) == [((0, 0), {0: 2, 1: 4})]
f = Poly((x*Rational(2, 5) - Rational(17, 3))*(4*x + Rational(1, 257)))
assert f.intervals(sqf=True) == [(-1, 0), (14, 15)]
assert f.intervals() == [((-1, 0), 1), ((14, 15), 1)]
assert f.intervals(fast=True, sqf=True) == [(-1, 0), (14, 15)]
assert f.intervals(fast=True) == [((-1, 0), 1), ((14, 15), 1)]
assert f.intervals(eps=Rational(1, 10)) == f.intervals(eps=0.1) == \
[((Rational(-1, 258), 0), 1), ((Rational(85, 6), Rational(85, 6)), 1)]
assert f.intervals(eps=Rational(1, 100)) == f.intervals(eps=0.01) == \
[((Rational(-1, 258), 0), 1), ((Rational(85, 6), Rational(85, 6)), 1)]
assert f.intervals(eps=Rational(1, 1000)) == f.intervals(eps=0.001) == \
[((Rational(-1, 1002), 0), 1), ((Rational(85, 6), Rational(85, 6)), 1)]
assert f.intervals(eps=Rational(1, 10000)) == f.intervals(eps=0.0001) == \
[((Rational(-1, 1028), Rational(-1, 1028)), 1), ((Rational(85, 6), Rational(85, 6)), 1)]
f = (x*Rational(2, 5) - Rational(17, 3))*(4*x + Rational(1, 257))
assert intervals(f, sqf=True) == [(-1, 0), (14, 15)]
assert intervals(f) == [((-1, 0), 1), ((14, 15), 1)]
assert intervals(f, eps=Rational(1, 10)) == intervals(f, eps=0.1) == \
[((Rational(-1, 258), 0), 1), ((Rational(85, 6), Rational(85, 6)), 1)]
assert intervals(f, eps=Rational(1, 100)) == intervals(f, eps=0.01) == \
[((Rational(-1, 258), 0), 1), ((Rational(85, 6), Rational(85, 6)), 1)]
assert intervals(f, eps=Rational(1, 1000)) == intervals(f, eps=0.001) == \
[((Rational(-1, 1002), 0), 1), ((Rational(85, 6), Rational(85, 6)), 1)]
assert intervals(f, eps=Rational(1, 10000)) == intervals(f, eps=0.0001) == \
[((Rational(-1, 1028), Rational(-1, 1028)), 1), ((Rational(85, 6), Rational(85, 6)), 1)]
f = Poly((x**2 - 2)*(x**2 - 3)**7*(x + 1)*(7*x + 3)**3)
assert f.intervals() == \
[((-2, Rational(-3, 2)), 7), ((Rational(-3, 2), -1), 1),
((-1, -1), 1), ((-1, 0), 3),
((1, Rational(3, 2)), 1), ((Rational(3, 2), 2), 7)]
assert intervals([x**5 - 200, x**5 - 201]) == \
[((Rational(75, 26), Rational(101, 35)), {0: 1}), ((Rational(309, 107), Rational(26, 9)), {1: 1})]
assert intervals([x**5 - 200, x**5 - 201], fast=True) == \
[((Rational(75, 26), Rational(101, 35)), {0: 1}), ((Rational(309, 107), Rational(26, 9)), {1: 1})]
assert intervals([x**2 - 200, x**2 - 201]) == \
[((Rational(-71, 5), Rational(-85, 6)), {1: 1}), ((Rational(-85, 6), -14), {0: 1}),
((14, Rational(85, 6)), {0: 1}), ((Rational(85, 6), Rational(71, 5)), {1: 1})]
assert intervals([x + 1, x + 2, x - 1, x + 1, 1, x - 1, x - 1, (x - 2)**2]) == \
[((-2, -2), {1: 1}), ((-1, -1), {0: 1, 3: 1}), ((1, 1), {2:
1, 5: 1, 6: 1}), ((2, 2), {7: 2})]
f, g, h = x**2 - 2, x**4 - 4*x**2 + 4, x - 1
assert intervals(f, inf=Rational(7, 4), sqf=True) == []
assert intervals(f, inf=Rational(7, 5), sqf=True) == [(Rational(7, 5), Rational(3, 2))]
assert intervals(f, sup=Rational(7, 4), sqf=True) == [(-2, -1), (1, Rational(3, 2))]
assert intervals(f, sup=Rational(7, 5), sqf=True) == [(-2, -1)]
assert intervals(g, inf=Rational(7, 4)) == []
assert intervals(g, inf=Rational(7, 5)) == [((Rational(7, 5), Rational(3, 2)), 2)]
assert intervals(g, sup=Rational(7, 4)) == [((-2, -1), 2), ((1, Rational(3, 2)), 2)]
assert intervals(g, sup=Rational(7, 5)) == [((-2, -1), 2)]
assert intervals([g, h], inf=Rational(7, 4)) == []
assert intervals([g, h], inf=Rational(7, 5)) == [((Rational(7, 5), Rational(3, 2)), {0: 2})]
assert intervals([g, h], sup=S(
7)/4) == [((-2, -1), {0: 2}), ((1, 1), {1: 1}), ((1, Rational(3, 2)), {0: 2})]
assert intervals(
[g, h], sup=Rational(7, 5)) == [((-2, -1), {0: 2}), ((1, 1), {1: 1})]
assert intervals([x + 2, x**2 - 2]) == \
[((-2, -2), {0: 1}), ((-2, -1), {1: 1}), ((1, 2), {1: 1})]
assert intervals([x + 2, x**2 - 2], strict=True) == \
[((-2, -2), {0: 1}), ((Rational(-3, 2), -1), {1: 1}), ((1, 2), {1: 1})]
f = 7*z**4 - 19*z**3 + 20*z**2 + 17*z + 20
assert intervals(f) == []
real_part, complex_part = intervals(f, all=True, sqf=True)
assert real_part == []
assert all(re(a) < re(r) < re(b) and im(
a) < im(r) < im(b) for (a, b), r in zip(complex_part, nroots(f)))
assert complex_part == [(Rational(-40, 7) - I*Rational(40, 7), 0),
(Rational(-40, 7), I*Rational(40, 7)),
(I*Rational(-40, 7), Rational(40, 7)),
(0, Rational(40, 7) + I*Rational(40, 7))]
real_part, complex_part = intervals(f, all=True, sqf=True, eps=Rational(1, 10))
assert real_part == []
assert all(re(a) < re(r) < re(b) and im(
a) < im(r) < im(b) for (a, b), r in zip(complex_part, nroots(f)))
raises(ValueError, lambda: intervals(x**2 - 2, eps=10**-100000))
raises(ValueError, lambda: Poly(x**2 - 2).intervals(eps=10**-100000))
raises(
ValueError, lambda: intervals([x**2 - 2, x**2 - 3], eps=10**-100000))
def test_refine_root():
f = Poly(x**2 - 2)
assert f.refine_root(1, 2, steps=0) == (1, 2)
assert f.refine_root(-2, -1, steps=0) == (-2, -1)
assert f.refine_root(1, 2, steps=None) == (1, Rational(3, 2))
assert f.refine_root(-2, -1, steps=None) == (Rational(-3, 2), -1)
assert f.refine_root(1, 2, steps=1) == (1, Rational(3, 2))
assert f.refine_root(-2, -1, steps=1) == (Rational(-3, 2), -1)
assert f.refine_root(1, 2, steps=1, fast=True) == (1, Rational(3, 2))
assert f.refine_root(-2, -1, steps=1, fast=True) == (Rational(-3, 2), -1)
assert f.refine_root(1, 2, eps=Rational(1, 100)) == (Rational(24, 17), Rational(17, 12))
assert f.refine_root(1, 2, eps=1e-2) == (Rational(24, 17), Rational(17, 12))
raises(PolynomialError, lambda: (f**2).refine_root(1, 2, check_sqf=True))
raises(RefinementFailed, lambda: (f**2).refine_root(1, 2))
raises(RefinementFailed, lambda: (f**2).refine_root(2, 3))
f = x**2 - 2
assert refine_root(f, 1, 2, steps=1) == (1, Rational(3, 2))
assert refine_root(f, -2, -1, steps=1) == (Rational(-3, 2), -1)
assert refine_root(f, 1, 2, steps=1, fast=True) == (1, Rational(3, 2))
assert refine_root(f, -2, -1, steps=1, fast=True) == (Rational(-3, 2), -1)
assert refine_root(f, 1, 2, eps=Rational(1, 100)) == (Rational(24, 17), Rational(17, 12))
assert refine_root(f, 1, 2, eps=1e-2) == (Rational(24, 17), Rational(17, 12))
raises(PolynomialError, lambda: refine_root(1, 7, 8, eps=Rational(1, 100)))
raises(ValueError, lambda: Poly(f).refine_root(1, 2, eps=10**-100000))
raises(ValueError, lambda: refine_root(f, 1, 2, eps=10**-100000))
def test_count_roots():
assert count_roots(x**2 - 2) == 2
assert count_roots(x**2 - 2, inf=-oo) == 2
assert count_roots(x**2 - 2, sup=+oo) == 2
assert count_roots(x**2 - 2, inf=-oo, sup=+oo) == 2
assert count_roots(x**2 - 2, inf=-2) == 2
assert count_roots(x**2 - 2, inf=-1) == 1
assert count_roots(x**2 - 2, sup=1) == 1
assert count_roots(x**2 - 2, sup=2) == 2
assert count_roots(x**2 - 2, inf=-1, sup=1) == 0
assert count_roots(x**2 - 2, inf=-2, sup=2) == 2
assert count_roots(x**2 - 2, inf=-1, sup=1) == 0
assert count_roots(x**2 - 2, inf=-2, sup=2) == 2
assert count_roots(x**2 + 2) == 0
assert count_roots(x**2 + 2, inf=-2*I) == 2
assert count_roots(x**2 + 2, sup=+2*I) == 2
assert count_roots(x**2 + 2, inf=-2*I, sup=+2*I) == 2
assert count_roots(x**2 + 2, inf=0) == 0
assert count_roots(x**2 + 2, sup=0) == 0
assert count_roots(x**2 + 2, inf=-I) == 1
assert count_roots(x**2 + 2, sup=+I) == 1
assert count_roots(x**2 + 2, inf=+I/2, sup=+I) == 0
assert count_roots(x**2 + 2, inf=-I, sup=-I/2) == 0
raises(PolynomialError, lambda: count_roots(1))
def test_Poly_root():
f = Poly(2*x**3 - 7*x**2 + 4*x + 4)
assert f.root(0) == Rational(-1, 2)
assert f.root(1) == 2
assert f.root(2) == 2
raises(IndexError, lambda: f.root(3))
assert Poly(x**5 + x + 1).root(0) == rootof(x**3 - x**2 + 1, 0)
def test_real_roots():
assert real_roots(x) == [0]
assert real_roots(x, multiple=False) == [(0, 1)]
assert real_roots(x**3) == [0, 0, 0]
assert real_roots(x**3, multiple=False) == [(0, 3)]
assert real_roots(x*(x**3 + x + 3)) == [rootof(x**3 + x + 3, 0), 0]
assert real_roots(x*(x**3 + x + 3), multiple=False) == [(rootof(
x**3 + x + 3, 0), 1), (0, 1)]
assert real_roots(
x**3*(x**3 + x + 3)) == [rootof(x**3 + x + 3, 0), 0, 0, 0]
assert real_roots(x**3*(x**3 + x + 3), multiple=False) == [(rootof(
x**3 + x + 3, 0), 1), (0, 3)]
f = 2*x**3 - 7*x**2 + 4*x + 4
g = x**3 + x + 1
assert Poly(f).real_roots() == [Rational(-1, 2), 2, 2]
assert Poly(g).real_roots() == [rootof(g, 0)]
def test_all_roots():
f = 2*x**3 - 7*x**2 + 4*x + 4
g = x**3 + x + 1
assert Poly(f).all_roots() == [Rational(-1, 2), 2, 2]
assert Poly(g).all_roots() == [rootof(g, 0), rootof(g, 1), rootof(g, 2)]
def test_nroots():
assert Poly(0, x).nroots() == []
assert Poly(1, x).nroots() == []
assert Poly(x**2 - 1, x).nroots() == [-1.0, 1.0]
assert Poly(x**2 + 1, x).nroots() == [-1.0*I, 1.0*I]
roots = Poly(x**2 - 1, x).nroots()
assert roots == [-1.0, 1.0]
roots = Poly(x**2 + 1, x).nroots()
assert roots == [-1.0*I, 1.0*I]
roots = Poly(x**2/3 - Rational(1, 3), x).nroots()
assert roots == [-1.0, 1.0]
roots = Poly(x**2/3 + Rational(1, 3), x).nroots()
assert roots == [-1.0*I, 1.0*I]
assert Poly(x**2 + 2*I, x).nroots() == [-1.0 + 1.0*I, 1.0 - 1.0*I]
assert Poly(
x**2 + 2*I, x, extension=I).nroots() == [-1.0 + 1.0*I, 1.0 - 1.0*I]
assert Poly(0.2*x + 0.1).nroots() == [-0.5]
roots = nroots(x**5 + x + 1, n=5)
eps = Float("1e-5")
assert re(roots[0]).epsilon_eq(-0.75487, eps) is S.true
assert im(roots[0]) == 0.0
assert re(roots[1]) == -0.5
assert im(roots[1]).epsilon_eq(-0.86602, eps) is S.true
assert re(roots[2]) == -0.5
assert im(roots[2]).epsilon_eq(+0.86602, eps) is S.true
assert re(roots[3]).epsilon_eq(+0.87743, eps) is S.true
assert im(roots[3]).epsilon_eq(-0.74486, eps) is S.true
assert re(roots[4]).epsilon_eq(+0.87743, eps) is S.true
assert im(roots[4]).epsilon_eq(+0.74486, eps) is S.true
eps = Float("1e-6")
assert re(roots[0]).epsilon_eq(-0.75487, eps) is S.false
assert im(roots[0]) == 0.0
assert re(roots[1]) == -0.5
assert im(roots[1]).epsilon_eq(-0.86602, eps) is S.false
assert re(roots[2]) == -0.5
assert im(roots[2]).epsilon_eq(+0.86602, eps) is S.false
assert re(roots[3]).epsilon_eq(+0.87743, eps) is S.false
assert im(roots[3]).epsilon_eq(-0.74486, eps) is S.false
assert re(roots[4]).epsilon_eq(+0.87743, eps) is S.false
assert im(roots[4]).epsilon_eq(+0.74486, eps) is S.false
raises(DomainError, lambda: Poly(x + y, x).nroots())
raises(MultivariatePolynomialError, lambda: Poly(x + y).nroots())
assert nroots(x**2 - 1) == [-1.0, 1.0]
roots = nroots(x**2 - 1)
assert roots == [-1.0, 1.0]
assert nroots(x + I) == [-1.0*I]
assert nroots(x + 2*I) == [-2.0*I]
raises(PolynomialError, lambda: nroots(0))
# issue 8296
f = Poly(x**4 - 1)
assert f.nroots(2) == [w.n(2) for w in f.all_roots()]
assert str(Poly(x**16 + 32*x**14 + 508*x**12 + 5440*x**10 +
39510*x**8 + 204320*x**6 + 755548*x**4 + 1434496*x**2 +
877969).nroots(2)) == ('[-1.7 - 1.9*I, -1.7 + 1.9*I, -1.7 '
'- 2.5*I, -1.7 + 2.5*I, -1.0*I, 1.0*I, -1.7*I, 1.7*I, -2.8*I, '
'2.8*I, -3.4*I, 3.4*I, 1.7 - 1.9*I, 1.7 + 1.9*I, 1.7 - 2.5*I, '
'1.7 + 2.5*I]')
def test_ground_roots():
f = x**6 - 4*x**4 + 4*x**3 - x**2
assert Poly(f).ground_roots() == {S.One: 2, S.Zero: 2}
assert ground_roots(f) == {S.One: 2, S.Zero: 2}
def test_nth_power_roots_poly():
f = x**4 - x**2 + 1
f_2 = (x**2 - x + 1)**2
f_3 = (x**2 + 1)**2
f_4 = (x**2 + x + 1)**2
f_12 = (x - 1)**4
assert nth_power_roots_poly(f, 1) == f
raises(ValueError, lambda: nth_power_roots_poly(f, 0))
raises(ValueError, lambda: nth_power_roots_poly(f, x))
assert factor(nth_power_roots_poly(f, 2)) == f_2
assert factor(nth_power_roots_poly(f, 3)) == f_3
assert factor(nth_power_roots_poly(f, 4)) == f_4
assert factor(nth_power_roots_poly(f, 12)) == f_12
raises(MultivariatePolynomialError, lambda: nth_power_roots_poly(
x + y, 2, x, y))
def test_torational_factor_list():
p = expand(((x**2-1)*(x-2)).subs({x:x*(1 + sqrt(2))}))
assert _torational_factor_list(p, x) == (-2, [
(-x*(1 + sqrt(2))/2 + 1, 1),
(-x*(1 + sqrt(2)) - 1, 1),
(-x*(1 + sqrt(2)) + 1, 1)])
p = expand(((x**2-1)*(x-2)).subs({x:x*(1 + 2**Rational(1, 4))}))
assert _torational_factor_list(p, x) is None
def test_cancel():
assert cancel(0) == 0
assert cancel(7) == 7
assert cancel(x) == x
assert cancel(oo) is oo
assert cancel((2, 3)) == (1, 2, 3)
assert cancel((1, 0), x) == (1, 1, 0)
assert cancel((0, 1), x) == (1, 0, 1)
f, g, p, q = 4*x**2 - 4, 2*x - 2, 2*x + 2, 1
F, G, P, Q = [ Poly(u, x) for u in (f, g, p, q) ]
assert F.cancel(G) == (1, P, Q)
assert cancel((f, g)) == (1, p, q)
assert cancel((f, g), x) == (1, p, q)
assert cancel((f, g), (x,)) == (1, p, q)
assert cancel((F, G)) == (1, P, Q)
assert cancel((f, g), polys=True) == (1, P, Q)
assert cancel((F, G), polys=False) == (1, p, q)
f = (x**2 - 2)/(x + sqrt(2))
assert cancel(f) == f
assert cancel(f, greedy=False) == x - sqrt(2)
f = (x**2 - 2)/(x - sqrt(2))
assert cancel(f) == f
assert cancel(f, greedy=False) == x + sqrt(2)
assert cancel((x**2/4 - 1, x/2 - 1)) == (S.Half, x + 2, 1)
assert cancel((x**2 - y)/(x - y)) == 1/(x - y)*(x**2 - y)
assert cancel((x**2 - y**2)/(x - y), x) == x + y
assert cancel((x**2 - y**2)/(x - y), y) == x + y
assert cancel((x**2 - y**2)/(x - y)) == x + y
assert cancel((x**3 - 1)/(x**2 - 1)) == (x**2 + x + 1)/(x + 1)
assert cancel((x**3/2 - S.Half)/(x**2 - 1)) == (x**2 + x + 1)/(2*x + 2)
assert cancel((exp(2*x) + 2*exp(x) + 1)/(exp(x) + 1)) == exp(x) + 1
f = Poly(x**2 - a**2, x)
g = Poly(x - a, x)
F = Poly(x + a, x, domain='ZZ[a]')
G = Poly(1, x, domain='ZZ[a]')
assert cancel((f, g)) == (1, F, G)
f = x**3 + (sqrt(2) - 2)*x**2 - (2*sqrt(2) + 3)*x - 3*sqrt(2)
g = x**2 - 2
assert cancel((f, g), extension=True) == (1, x**2 - 2*x - 3, x - sqrt(2))
f = Poly(-2*x + 3, x)
g = Poly(-x**9 + x**8 + x**6 - x**5 + 2*x**2 - 3*x + 1, x)
assert cancel((f, g)) == (1, -f, -g)
f = Poly(y, y, domain='ZZ(x)')
g = Poly(1, y, domain='ZZ[x]')
assert f.cancel(
g) == (1, Poly(y, y, domain='ZZ(x)'), Poly(1, y, domain='ZZ(x)'))
assert f.cancel(g, include=True) == (
Poly(y, y, domain='ZZ(x)'), Poly(1, y, domain='ZZ(x)'))
f = Poly(5*x*y + x, y, domain='ZZ(x)')
g = Poly(2*x**2*y, y, domain='ZZ(x)')
assert f.cancel(g, include=True) == (
Poly(5*y + 1, y, domain='ZZ(x)'), Poly(2*x*y, y, domain='ZZ(x)'))
f = -(-2*x - 4*y + 0.005*(z - y)**2)/((z - y)*(-z + y + 2))
assert cancel(f).is_Mul == True
P = tanh(x - 3.0)
Q = tanh(x + 3.0)
f = ((-2*P**2 + 2)*(-P**2 + 1)*Q**2/2 + (-2*P**2 + 2)*(-2*Q**2 + 2)*P*Q - (-2*P**2 + 2)*P**2*Q**2 + (-2*Q**2 + 2)*(-Q**2 + 1)*P**2/2 - (-2*Q**2 + 2)*P**2*Q**2)/(2*sqrt(P**2*Q**2 + 0.0001)) \
+ (-(-2*P**2 + 2)*P*Q**2/2 - (-2*Q**2 + 2)*P**2*Q/2)*((-2*P**2 + 2)*P*Q**2/2 + (-2*Q**2 + 2)*P**2*Q/2)/(2*(P**2*Q**2 + 0.0001)**Rational(3, 2))
assert cancel(f).is_Mul == True
# issue 7022
A = Symbol('A', commutative=False)
p1 = Piecewise((A*(x**2 - 1)/(x + 1), x > 1), ((x + 2)/(x**2 + 2*x), True))
p2 = Piecewise((A*(x - 1), x > 1), (1/x, True))
assert cancel(p1) == p2
assert cancel(2*p1) == 2*p2
assert cancel(1 + p1) == 1 + p2
assert cancel((x**2 - 1)/(x + 1)*p1) == (x - 1)*p2
assert cancel((x**2 - 1)/(x + 1) + p1) == (x - 1) + p2
p3 = Piecewise(((x**2 - 1)/(x + 1), x > 1), ((x + 2)/(x**2 + 2*x), True))
p4 = Piecewise(((x - 1), x > 1), (1/x, True))
assert cancel(p3) == p4
assert cancel(2*p3) == 2*p4
assert cancel(1 + p3) == 1 + p4
assert cancel((x**2 - 1)/(x + 1)*p3) == (x - 1)*p4
assert cancel((x**2 - 1)/(x + 1) + p3) == (x - 1) + p4
# issue 9363
M = MatrixSymbol('M', 5, 5)
assert cancel(M[0,0] + 7) == M[0,0] + 7
expr = sin(M[1, 4] + M[2, 1] * 5 * M[4, 0]) - 5 * M[1, 2] / z
assert cancel(expr) == (z*sin(M[1, 4] + M[2, 1] * 5 * M[4, 0]) - 5 * M[1, 2]) / z
def test_reduced():
f = 2*x**4 + y**2 - x**2 + y**3
G = [x**3 - x, y**3 - y]
Q = [2*x, 1]
r = x**2 + y**2 + y
assert reduced(f, G) == (Q, r)
assert reduced(f, G, x, y) == (Q, r)
H = groebner(G)
assert H.reduce(f) == (Q, r)
Q = [Poly(2*x, x, y), Poly(1, x, y)]
r = Poly(x**2 + y**2 + y, x, y)
assert _strict_eq(reduced(f, G, polys=True), (Q, r))
assert _strict_eq(reduced(f, G, x, y, polys=True), (Q, r))
H = groebner(G, polys=True)
assert _strict_eq(H.reduce(f), (Q, r))
f = 2*x**3 + y**3 + 3*y
G = groebner([x**2 + y**2 - 1, x*y - 2])
Q = [x**2 - x*y**3/2 + x*y/2 + y**6/4 - y**4/2 + y**2/4, -y**5/4 + y**3/2 + y*Rational(3, 4)]
r = 0
assert reduced(f, G) == (Q, r)
assert G.reduce(f) == (Q, r)
assert reduced(f, G, auto=False)[1] != 0
assert G.reduce(f, auto=False)[1] != 0
assert G.contains(f) is True
assert G.contains(f + 1) is False
assert reduced(1, [1], x) == ([1], 0)
raises(ComputationFailed, lambda: reduced(1, [1]))
def test_groebner():
assert groebner([], x, y, z) == []
assert groebner([x**2 + 1, y**4*x + x**3], x, y, order='lex') == [1 + x**2, -1 + y**4]
assert groebner([x**2 + 1, y**4*x + x**3, x*y*z**3], x, y, z, order='grevlex') == [-1 + y**4, z**3, 1 + x**2]
assert groebner([x**2 + 1, y**4*x + x**3], x, y, order='lex', polys=True) == \
[Poly(1 + x**2, x, y), Poly(-1 + y**4, x, y)]
assert groebner([x**2 + 1, y**4*x + x**3, x*y*z**3], x, y, z, order='grevlex', polys=True) == \
[Poly(-1 + y**4, x, y, z), Poly(z**3, x, y, z), Poly(1 + x**2, x, y, z)]
assert groebner([x**3 - 1, x**2 - 1]) == [x - 1]
assert groebner([Eq(x**3, 1), Eq(x**2, 1)]) == [x - 1]
F = [3*x**2 + y*z - 5*x - 1, 2*x + 3*x*y + y**2, x - 3*y + x*z - 2*z**2]
f = z**9 - x**2*y**3 - 3*x*y**2*z + 11*y*z**2 + x**2*z**2 - 5
G = groebner(F, x, y, z, modulus=7, symmetric=False)
assert G == [1 + x + y + 3*z + 2*z**2 + 2*z**3 + 6*z**4 + z**5,
1 + 3*y + y**2 + 6*z**2 + 3*z**3 + 3*z**4 + 3*z**5 + 4*z**6,
1 + 4*y + 4*z + y*z + 4*z**3 + z**4 + z**6,
6 + 6*z + z**2 + 4*z**3 + 3*z**4 + 6*z**5 + 3*z**6 + z**7]
Q, r = reduced(f, G, x, y, z, modulus=7, symmetric=False, polys=True)
assert sum([ q*g for q, g in zip(Q, G.polys)], r) == Poly(f, modulus=7)
F = [x*y - 2*y, 2*y**2 - x**2]
assert groebner(F, x, y, order='grevlex') == \
[y**3 - 2*y, x**2 - 2*y**2, x*y - 2*y]
assert groebner(F, y, x, order='grevlex') == \
[x**3 - 2*x**2, -x**2 + 2*y**2, x*y - 2*y]
assert groebner(F, order='grevlex', field=True) == \
[y**3 - 2*y, x**2 - 2*y**2, x*y - 2*y]
assert groebner([1], x) == [1]
assert groebner([x**2 + 2.0*y], x, y) == [1.0*x**2 + 2.0*y]
raises(ComputationFailed, lambda: groebner([1]))
assert groebner([x**2 - 1, x**3 + 1], method='buchberger') == [x + 1]
assert groebner([x**2 - 1, x**3 + 1], method='f5b') == [x + 1]
raises(ValueError, lambda: groebner([x, y], method='unknown'))
def test_fglm():
F = [a + b + c + d, a*b + a*d + b*c + b*d, a*b*c + a*b*d + a*c*d + b*c*d, a*b*c*d - 1]
G = groebner(F, a, b, c, d, order=grlex)
B = [
4*a + 3*d**9 - 4*d**5 - 3*d,
4*b + 4*c - 3*d**9 + 4*d**5 + 7*d,
4*c**2 + 3*d**10 - 4*d**6 - 3*d**2,
4*c*d**4 + 4*c - d**9 + 4*d**5 + 5*d,
d**12 - d**8 - d**4 + 1,
]
assert groebner(F, a, b, c, d, order=lex) == B
assert G.fglm(lex) == B
F = [9*x**8 + 36*x**7 - 32*x**6 - 252*x**5 - 78*x**4 + 468*x**3 + 288*x**2 - 108*x + 9,
-72*t*x**7 - 252*t*x**6 + 192*t*x**5 + 1260*t*x**4 + 312*t*x**3 - 404*t*x**2 - 576*t*x + \
108*t - 72*x**7 - 256*x**6 + 192*x**5 + 1280*x**4 + 312*x**3 - 576*x + 96]
G = groebner(F, t, x, order=grlex)
B = [
203577793572507451707*t + 627982239411707112*x**7 - 666924143779443762*x**6 - \
10874593056632447619*x**5 + 5119998792707079562*x**4 + 72917161949456066376*x**3 + \
20362663855832380362*x**2 - 142079311455258371571*x + 183756699868981873194,
9*x**8 + 36*x**7 - 32*x**6 - 252*x**5 - 78*x**4 + 468*x**3 + 288*x**2 - 108*x + 9,
]
assert groebner(F, t, x, order=lex) == B
assert G.fglm(lex) == B
F = [x**2 - x - 3*y + 1, -2*x + y**2 + y - 1]
G = groebner(F, x, y, order=lex)
B = [
x**2 - x - 3*y + 1,
y**2 - 2*x + y - 1,
]
assert groebner(F, x, y, order=grlex) == B
assert G.fglm(grlex) == B
def test_is_zero_dimensional():
assert is_zero_dimensional([x, y], x, y) is True
assert is_zero_dimensional([x**3 + y**2], x, y) is False
assert is_zero_dimensional([x, y, z], x, y, z) is True
assert is_zero_dimensional([x, y, z], x, y, z, t) is False
F = [x*y - z, y*z - x, x*y - y]
assert is_zero_dimensional(F, x, y, z) is True
F = [x**2 - 2*x*z + 5, x*y**2 + y*z**3, 3*y**2 - 8*z**2]
assert is_zero_dimensional(F, x, y, z) is True
def test_GroebnerBasis():
F = [x*y - 2*y, 2*y**2 - x**2]
G = groebner(F, x, y, order='grevlex')
H = [y**3 - 2*y, x**2 - 2*y**2, x*y - 2*y]
P = [ Poly(h, x, y) for h in H ]
assert groebner(F + [0], x, y, order='grevlex') == G
assert isinstance(G, GroebnerBasis) is True
assert len(G) == 3
assert G[0] == H[0] and not G[0].is_Poly
assert G[1] == H[1] and not G[1].is_Poly
assert G[2] == H[2] and not G[2].is_Poly
assert G[1:] == H[1:] and not any(g.is_Poly for g in G[1:])
assert G[:2] == H[:2] and not any(g.is_Poly for g in G[1:])
assert G.exprs == H
assert G.polys == P
assert G.gens == (x, y)
assert G.domain == ZZ
assert G.order == grevlex
assert G == H
assert G == tuple(H)
assert G == P
assert G == tuple(P)
assert G != []
G = groebner(F, x, y, order='grevlex', polys=True)
assert G[0] == P[0] and G[0].is_Poly
assert G[1] == P[1] and G[1].is_Poly
assert G[2] == P[2] and G[2].is_Poly
assert G[1:] == P[1:] and all(g.is_Poly for g in G[1:])
assert G[:2] == P[:2] and all(g.is_Poly for g in G[1:])
def test_poly():
assert poly(x) == Poly(x, x)
assert poly(y) == Poly(y, y)
assert poly(x + y) == Poly(x + y, x, y)
assert poly(x + sin(x)) == Poly(x + sin(x), x, sin(x))
assert poly(x + y, wrt=y) == Poly(x + y, y, x)
assert poly(x + sin(x), wrt=sin(x)) == Poly(x + sin(x), sin(x), x)
assert poly(x*y + 2*x*z**2 + 17) == Poly(x*y + 2*x*z**2 + 17, x, y, z)
assert poly(2*(y + z)**2 - 1) == Poly(2*y**2 + 4*y*z + 2*z**2 - 1, y, z)
assert poly(
x*(y + z)**2 - 1) == Poly(x*y**2 + 2*x*y*z + x*z**2 - 1, x, y, z)
assert poly(2*x*(
y + z)**2 - 1) == Poly(2*x*y**2 + 4*x*y*z + 2*x*z**2 - 1, x, y, z)
assert poly(2*(
y + z)**2 - x - 1) == Poly(2*y**2 + 4*y*z + 2*z**2 - x - 1, x, y, z)
assert poly(x*(
y + z)**2 - x - 1) == Poly(x*y**2 + 2*x*y*z + x*z**2 - x - 1, x, y, z)
assert poly(2*x*(y + z)**2 - x - 1) == Poly(2*x*y**2 + 4*x*y*z + 2*
x*z**2 - x - 1, x, y, z)
assert poly(x*y + (x + y)**2 + (x + z)**2) == \
Poly(2*x*z + 3*x*y + y**2 + z**2 + 2*x**2, x, y, z)
assert poly(x*y*(x + y)*(x + z)**2) == \
Poly(x**3*y**2 + x*y**2*z**2 + y*x**2*z**2 + 2*z*x**2*
y**2 + 2*y*z*x**3 + y*x**4, x, y, z)
assert poly(Poly(x + y + z, y, x, z)) == Poly(x + y + z, y, x, z)
assert poly((x + y)**2, x) == Poly(x**2 + 2*x*y + y**2, x, domain=ZZ[y])
assert poly((x + y)**2, y) == Poly(x**2 + 2*x*y + y**2, y, domain=ZZ[x])
assert poly(1, x) == Poly(1, x)
raises(GeneratorsNeeded, lambda: poly(1))
# issue 6184
assert poly(x + y, x, y) == Poly(x + y, x, y)
assert poly(x + y, y, x) == Poly(x + y, y, x)
def test_keep_coeff():
u = Mul(2, x + 1, evaluate=False)
assert _keep_coeff(S.One, x) == x
assert _keep_coeff(S.NegativeOne, x) == -x
assert _keep_coeff(S(1.0), x) == 1.0*x
assert _keep_coeff(S(-1.0), x) == -1.0*x
assert _keep_coeff(S.One, 2*x) == 2*x
assert _keep_coeff(S(2), x/2) == x
assert _keep_coeff(S(2), sin(x)) == 2*sin(x)
assert _keep_coeff(S(2), x + 1) == u
assert _keep_coeff(x, 1/x) == 1
assert _keep_coeff(x + 1, S(2)) == u
def test_poly_matching_consistency():
# Test for this issue:
# https://github.com/sympy/sympy/issues/5514
assert I * Poly(x, x) == Poly(I*x, x)
assert Poly(x, x) * I == Poly(I*x, x)
def test_issue_5786():
assert expand(factor(expand(
(x - I*y)*(z - I*t)), extension=[I])) == -I*t*x - t*y + x*z - I*y*z
def test_noncommutative():
class foo(Expr):
is_commutative=False
e = x/(x + x*y)
c = 1/( 1 + y)
assert cancel(foo(e)) == foo(c)
assert cancel(e + foo(e)) == c + foo(c)
assert cancel(e*foo(c)) == c*foo(c)
def test_to_rational_coeffs():
assert to_rational_coeffs(
Poly(x**3 + y*x**2 + sqrt(y), x, domain='EX')) is None
def test_factor_terms():
# issue 7067
assert factor_list(x*(x + y)) == (1, [(x, 1), (x + y, 1)])
assert sqf_list(x*(x + y)) == (1, [(x**2 + x*y, 1)])
def test_as_list():
# issue 14496
assert Poly(x**3 + 2, x, domain='ZZ').as_list() == [1, 0, 0, 2]
assert Poly(x**2 + y + 1, x, y, domain='ZZ').as_list() == [[1], [], [1, 1]]
assert Poly(x**2 + y + 1, x, y, z, domain='ZZ').as_list() == \
[[[1]], [[]], [[1], [1]]]
def test_issue_11198():
assert factor_list(sqrt(2)*x) == (sqrt(2), [(x, 1)])
assert factor_list(sqrt(2)*sin(x), sin(x)) == (sqrt(2), [(sin(x), 1)])
def test_Poly_precision():
# Make sure Poly doesn't lose precision
p = Poly(pi.evalf(100)*x)
assert p.as_expr() == pi.evalf(100)*x
def test_issue_12400():
# Correction of check for negative exponents
assert poly(1/(1+sqrt(2)), x) == \
Poly(1/(1+sqrt(2)), x , domain='EX')
def test_issue_14364():
assert gcd(S(6)*(1 + sqrt(3))/5, S(3)*(1 + sqrt(3))/10) == Rational(3, 10) * (1 + sqrt(3))
assert gcd(sqrt(5)*Rational(4, 7), sqrt(5)*Rational(2, 3)) == sqrt(5)*Rational(2, 21)
assert lcm(Rational(2, 3)*sqrt(3), Rational(5, 6)*sqrt(3)) == S(10)*sqrt(3)/3
assert lcm(3*sqrt(3), 4/sqrt(3)) == 12*sqrt(3)
assert lcm(S(5)*(1 + 2**Rational(1, 3))/6, S(3)*(1 + 2**Rational(1, 3))/8) == Rational(15, 2) * (1 + 2**Rational(1, 3))
assert gcd(Rational(2, 3)*sqrt(3), Rational(5, 6)/sqrt(3)) == sqrt(3)/18
assert gcd(S(4)*sqrt(13)/7, S(3)*sqrt(13)/14) == sqrt(13)/14
# gcd_list and lcm_list
assert gcd([S(2)*sqrt(47)/7, S(6)*sqrt(47)/5, S(8)*sqrt(47)/5]) == sqrt(47)*Rational(2, 35)
assert gcd([S(6)*(1 + sqrt(7))/5, S(2)*(1 + sqrt(7))/7, S(4)*(1 + sqrt(7))/13]) == (1 + sqrt(7))*Rational(2, 455)
assert lcm((Rational(7, 2)/sqrt(15), Rational(5, 6)/sqrt(15), Rational(5, 8)/sqrt(15))) == Rational(35, 2)/sqrt(15)
assert lcm([S(5)*(2 + 2**Rational(5, 7))/6, S(7)*(2 + 2**Rational(5, 7))/2, S(13)*(2 + 2**Rational(5, 7))/4]) == Rational(455, 2) * (2 + 2**Rational(5, 7))
def test_issue_15669():
x = Symbol("x", positive=True)
expr = (16*x**3/(-x**2 + sqrt(8*x**2 + (x**2 - 2)**2) + 2)**2 -
2*2**Rational(4, 5)*x*(-x**2 + sqrt(8*x**2 + (x**2 - 2)**2) + 2)**Rational(3, 5) + 10*x)
assert factor(expr, deep=True) == x*(x**2 + 2)
def test_issue_17988():
x = Symbol('x')
p = poly(x - 1)
M = Matrix([[poly(x + 1), poly(x + 1)]])
assert p * M == M * p == Matrix([[poly(x**2 - 1), poly(x**2 - 1)]])
def test_issue_18205():
assert cancel((2 + I)*(3 - I)) == 7 + I
assert cancel((2 + I)*(2 - I)) == 5
def test_issue_8695():
p = (x**2 + 1) * (x - 1)**2 * (x - 2)**3 * (x - 3)**3
result = (1, [(x**2 + 1, 1), (x - 1, 2), (x**2 - 5*x + 6, 3)])
assert sqf_list(p) == result
def test_issue_19113():
eq = sin(x)**3 - sin(x) + 1
raises(PolynomialError, lambda: refine_root(eq, 1, 2, 1e-2))
raises(PolynomialError, lambda: count_roots(eq, -1, 1))
raises(PolynomialError, lambda: real_roots(eq))
raises(PolynomialError, lambda: nroots(eq))
raises(PolynomialError, lambda: ground_roots(eq))
raises(PolynomialError, lambda: nth_power_roots_poly(eq, 2))
def test_issue_19360():
f = 2*x**2 - 2*sqrt(2)*x*y + y**2
assert factor(f, extension=sqrt(2)) == 2*(x - (sqrt(2)*y/2))**2
f = -I*t*x - t*y + x*z - I*y*z
assert factor(f, extension=I) == (x - I*y)*(-I*t + z)
|
ab2ee12462b99cda9a9777267fcc89799279a23e2ed33d54f2192df837e414a4 | """Tests for classes defining properties of ground domains, e.g. ZZ, QQ, ZZ[x] ... """
from sympy import S, sqrt, sin, oo, Poly, Float, Rational, pi
from sympy.abc import x, y, z
from sympy.polys.domains import ZZ, QQ, RR, CC, FF, GF, EX
from sympy.polys.domains.realfield import RealField
from sympy.polys.domains.gaussiandomains import ZZ_I, QQ_I
from sympy.polys.rings import ring
from sympy.polys.fields import field
from sympy.polys.polyerrors import (
UnificationFailed,
GeneratorsError,
CoercionFailed,
NotInvertible,
DomainError)
from sympy.polys.polyutils import illegal
from sympy.testing.pytest import raises
ALG = QQ.algebraic_field(sqrt(2), sqrt(3))
def unify(K0, K1):
return K0.unify(K1)
def test_Domain_unify():
F3 = GF(3)
assert unify(F3, F3) == F3
assert unify(F3, ZZ) == ZZ
assert unify(F3, QQ) == QQ
assert unify(F3, ALG) == ALG
assert unify(F3, RR) == RR
assert unify(F3, CC) == CC
assert unify(F3, ZZ[x]) == ZZ[x]
assert unify(F3, ZZ.frac_field(x)) == ZZ.frac_field(x)
assert unify(F3, EX) == EX
assert unify(ZZ, F3) == ZZ
assert unify(ZZ, ZZ) == ZZ
assert unify(ZZ, QQ) == QQ
assert unify(ZZ, ALG) == ALG
assert unify(ZZ, RR) == RR
assert unify(ZZ, CC) == CC
assert unify(ZZ, ZZ[x]) == ZZ[x]
assert unify(ZZ, ZZ.frac_field(x)) == ZZ.frac_field(x)
assert unify(ZZ, EX) == EX
assert unify(QQ, F3) == QQ
assert unify(QQ, ZZ) == QQ
assert unify(QQ, QQ) == QQ
assert unify(QQ, ALG) == ALG
assert unify(QQ, RR) == RR
assert unify(QQ, CC) == CC
assert unify(QQ, ZZ[x]) == QQ[x]
assert unify(QQ, ZZ.frac_field(x)) == QQ.frac_field(x)
assert unify(QQ, EX) == EX
assert unify(RR, F3) == RR
assert unify(RR, ZZ) == RR
assert unify(RR, QQ) == RR
assert unify(RR, ALG) == RR
assert unify(RR, RR) == RR
assert unify(RR, CC) == CC
assert unify(RR, ZZ[x]) == RR[x]
assert unify(RR, ZZ.frac_field(x)) == RR.frac_field(x)
assert unify(RR, EX) == EX
assert RR[x].unify(ZZ.frac_field(y)) == RR.frac_field(x, y)
assert unify(CC, F3) == CC
assert unify(CC, ZZ) == CC
assert unify(CC, QQ) == CC
assert unify(CC, ALG) == CC
assert unify(CC, RR) == CC
assert unify(CC, CC) == CC
assert unify(CC, ZZ[x]) == CC[x]
assert unify(CC, ZZ.frac_field(x)) == CC.frac_field(x)
assert unify(CC, EX) == EX
assert unify(ZZ[x], F3) == ZZ[x]
assert unify(ZZ[x], ZZ) == ZZ[x]
assert unify(ZZ[x], QQ) == QQ[x]
assert unify(ZZ[x], ALG) == ALG[x]
assert unify(ZZ[x], RR) == RR[x]
assert unify(ZZ[x], CC) == CC[x]
assert unify(ZZ[x], ZZ[x]) == ZZ[x]
assert unify(ZZ[x], ZZ.frac_field(x)) == ZZ.frac_field(x)
assert unify(ZZ[x], EX) == EX
assert unify(ZZ.frac_field(x), F3) == ZZ.frac_field(x)
assert unify(ZZ.frac_field(x), ZZ) == ZZ.frac_field(x)
assert unify(ZZ.frac_field(x), QQ) == QQ.frac_field(x)
assert unify(ZZ.frac_field(x), ALG) == ALG.frac_field(x)
assert unify(ZZ.frac_field(x), RR) == RR.frac_field(x)
assert unify(ZZ.frac_field(x), CC) == CC.frac_field(x)
assert unify(ZZ.frac_field(x), ZZ[x]) == ZZ.frac_field(x)
assert unify(ZZ.frac_field(x), ZZ.frac_field(x)) == ZZ.frac_field(x)
assert unify(ZZ.frac_field(x), EX) == EX
assert unify(EX, F3) == EX
assert unify(EX, ZZ) == EX
assert unify(EX, QQ) == EX
assert unify(EX, ALG) == EX
assert unify(EX, RR) == EX
assert unify(EX, CC) == EX
assert unify(EX, ZZ[x]) == EX
assert unify(EX, ZZ.frac_field(x)) == EX
assert unify(EX, EX) == EX
def test_Domain_unify_composite():
assert unify(ZZ.poly_ring(x), ZZ) == ZZ.poly_ring(x)
assert unify(ZZ.poly_ring(x), QQ) == QQ.poly_ring(x)
assert unify(QQ.poly_ring(x), ZZ) == QQ.poly_ring(x)
assert unify(QQ.poly_ring(x), QQ) == QQ.poly_ring(x)
assert unify(ZZ, ZZ.poly_ring(x)) == ZZ.poly_ring(x)
assert unify(QQ, ZZ.poly_ring(x)) == QQ.poly_ring(x)
assert unify(ZZ, QQ.poly_ring(x)) == QQ.poly_ring(x)
assert unify(QQ, QQ.poly_ring(x)) == QQ.poly_ring(x)
assert unify(ZZ.poly_ring(x, y), ZZ) == ZZ.poly_ring(x, y)
assert unify(ZZ.poly_ring(x, y), QQ) == QQ.poly_ring(x, y)
assert unify(QQ.poly_ring(x, y), ZZ) == QQ.poly_ring(x, y)
assert unify(QQ.poly_ring(x, y), QQ) == QQ.poly_ring(x, y)
assert unify(ZZ, ZZ.poly_ring(x, y)) == ZZ.poly_ring(x, y)
assert unify(QQ, ZZ.poly_ring(x, y)) == QQ.poly_ring(x, y)
assert unify(ZZ, QQ.poly_ring(x, y)) == QQ.poly_ring(x, y)
assert unify(QQ, QQ.poly_ring(x, y)) == QQ.poly_ring(x, y)
assert unify(ZZ.frac_field(x), ZZ) == ZZ.frac_field(x)
assert unify(ZZ.frac_field(x), QQ) == QQ.frac_field(x)
assert unify(QQ.frac_field(x), ZZ) == QQ.frac_field(x)
assert unify(QQ.frac_field(x), QQ) == QQ.frac_field(x)
assert unify(ZZ, ZZ.frac_field(x)) == ZZ.frac_field(x)
assert unify(QQ, ZZ.frac_field(x)) == QQ.frac_field(x)
assert unify(ZZ, QQ.frac_field(x)) == QQ.frac_field(x)
assert unify(QQ, QQ.frac_field(x)) == QQ.frac_field(x)
assert unify(ZZ.frac_field(x, y), ZZ) == ZZ.frac_field(x, y)
assert unify(ZZ.frac_field(x, y), QQ) == QQ.frac_field(x, y)
assert unify(QQ.frac_field(x, y), ZZ) == QQ.frac_field(x, y)
assert unify(QQ.frac_field(x, y), QQ) == QQ.frac_field(x, y)
assert unify(ZZ, ZZ.frac_field(x, y)) == ZZ.frac_field(x, y)
assert unify(QQ, ZZ.frac_field(x, y)) == QQ.frac_field(x, y)
assert unify(ZZ, QQ.frac_field(x, y)) == QQ.frac_field(x, y)
assert unify(QQ, QQ.frac_field(x, y)) == QQ.frac_field(x, y)
assert unify(ZZ.poly_ring(x), ZZ.poly_ring(x)) == ZZ.poly_ring(x)
assert unify(ZZ.poly_ring(x), QQ.poly_ring(x)) == QQ.poly_ring(x)
assert unify(QQ.poly_ring(x), ZZ.poly_ring(x)) == QQ.poly_ring(x)
assert unify(QQ.poly_ring(x), QQ.poly_ring(x)) == QQ.poly_ring(x)
assert unify(ZZ.poly_ring(x, y), ZZ.poly_ring(x)) == ZZ.poly_ring(x, y)
assert unify(ZZ.poly_ring(x, y), QQ.poly_ring(x)) == QQ.poly_ring(x, y)
assert unify(QQ.poly_ring(x, y), ZZ.poly_ring(x)) == QQ.poly_ring(x, y)
assert unify(QQ.poly_ring(x, y), QQ.poly_ring(x)) == QQ.poly_ring(x, y)
assert unify(ZZ.poly_ring(x), ZZ.poly_ring(x, y)) == ZZ.poly_ring(x, y)
assert unify(ZZ.poly_ring(x), QQ.poly_ring(x, y)) == QQ.poly_ring(x, y)
assert unify(QQ.poly_ring(x), ZZ.poly_ring(x, y)) == QQ.poly_ring(x, y)
assert unify(QQ.poly_ring(x), QQ.poly_ring(x, y)) == QQ.poly_ring(x, y)
assert unify(ZZ.poly_ring(x, y), ZZ.poly_ring(x, z)) == ZZ.poly_ring(x, y, z)
assert unify(ZZ.poly_ring(x, y), QQ.poly_ring(x, z)) == QQ.poly_ring(x, y, z)
assert unify(QQ.poly_ring(x, y), ZZ.poly_ring(x, z)) == QQ.poly_ring(x, y, z)
assert unify(QQ.poly_ring(x, y), QQ.poly_ring(x, z)) == QQ.poly_ring(x, y, z)
assert unify(ZZ.frac_field(x), ZZ.frac_field(x)) == ZZ.frac_field(x)
assert unify(ZZ.frac_field(x), QQ.frac_field(x)) == QQ.frac_field(x)
assert unify(QQ.frac_field(x), ZZ.frac_field(x)) == QQ.frac_field(x)
assert unify(QQ.frac_field(x), QQ.frac_field(x)) == QQ.frac_field(x)
assert unify(ZZ.frac_field(x, y), ZZ.frac_field(x)) == ZZ.frac_field(x, y)
assert unify(ZZ.frac_field(x, y), QQ.frac_field(x)) == QQ.frac_field(x, y)
assert unify(QQ.frac_field(x, y), ZZ.frac_field(x)) == QQ.frac_field(x, y)
assert unify(QQ.frac_field(x, y), QQ.frac_field(x)) == QQ.frac_field(x, y)
assert unify(ZZ.frac_field(x), ZZ.frac_field(x, y)) == ZZ.frac_field(x, y)
assert unify(ZZ.frac_field(x), QQ.frac_field(x, y)) == QQ.frac_field(x, y)
assert unify(QQ.frac_field(x), ZZ.frac_field(x, y)) == QQ.frac_field(x, y)
assert unify(QQ.frac_field(x), QQ.frac_field(x, y)) == QQ.frac_field(x, y)
assert unify(ZZ.frac_field(x, y), ZZ.frac_field(x, z)) == ZZ.frac_field(x, y, z)
assert unify(ZZ.frac_field(x, y), QQ.frac_field(x, z)) == QQ.frac_field(x, y, z)
assert unify(QQ.frac_field(x, y), ZZ.frac_field(x, z)) == QQ.frac_field(x, y, z)
assert unify(QQ.frac_field(x, y), QQ.frac_field(x, z)) == QQ.frac_field(x, y, z)
assert unify(ZZ.poly_ring(x), ZZ.frac_field(x)) == ZZ.frac_field(x)
assert unify(ZZ.poly_ring(x), QQ.frac_field(x)) == ZZ.frac_field(x)
assert unify(QQ.poly_ring(x), ZZ.frac_field(x)) == ZZ.frac_field(x)
assert unify(QQ.poly_ring(x), QQ.frac_field(x)) == QQ.frac_field(x)
assert unify(ZZ.poly_ring(x, y), ZZ.frac_field(x)) == ZZ.frac_field(x, y)
assert unify(ZZ.poly_ring(x, y), QQ.frac_field(x)) == ZZ.frac_field(x, y)
assert unify(QQ.poly_ring(x, y), ZZ.frac_field(x)) == ZZ.frac_field(x, y)
assert unify(QQ.poly_ring(x, y), QQ.frac_field(x)) == QQ.frac_field(x, y)
assert unify(ZZ.poly_ring(x), ZZ.frac_field(x, y)) == ZZ.frac_field(x, y)
assert unify(ZZ.poly_ring(x), QQ.frac_field(x, y)) == ZZ.frac_field(x, y)
assert unify(QQ.poly_ring(x), ZZ.frac_field(x, y)) == ZZ.frac_field(x, y)
assert unify(QQ.poly_ring(x), QQ.frac_field(x, y)) == QQ.frac_field(x, y)
assert unify(ZZ.poly_ring(x, y), ZZ.frac_field(x, z)) == ZZ.frac_field(x, y, z)
assert unify(ZZ.poly_ring(x, y), QQ.frac_field(x, z)) == ZZ.frac_field(x, y, z)
assert unify(QQ.poly_ring(x, y), ZZ.frac_field(x, z)) == ZZ.frac_field(x, y, z)
assert unify(QQ.poly_ring(x, y), QQ.frac_field(x, z)) == QQ.frac_field(x, y, z)
assert unify(ZZ.frac_field(x), ZZ.poly_ring(x)) == ZZ.frac_field(x)
assert unify(ZZ.frac_field(x), QQ.poly_ring(x)) == ZZ.frac_field(x)
assert unify(QQ.frac_field(x), ZZ.poly_ring(x)) == ZZ.frac_field(x)
assert unify(QQ.frac_field(x), QQ.poly_ring(x)) == QQ.frac_field(x)
assert unify(ZZ.frac_field(x, y), ZZ.poly_ring(x)) == ZZ.frac_field(x, y)
assert unify(ZZ.frac_field(x, y), QQ.poly_ring(x)) == ZZ.frac_field(x, y)
assert unify(QQ.frac_field(x, y), ZZ.poly_ring(x)) == ZZ.frac_field(x, y)
assert unify(QQ.frac_field(x, y), QQ.poly_ring(x)) == QQ.frac_field(x, y)
assert unify(ZZ.frac_field(x), ZZ.poly_ring(x, y)) == ZZ.frac_field(x, y)
assert unify(ZZ.frac_field(x), QQ.poly_ring(x, y)) == ZZ.frac_field(x, y)
assert unify(QQ.frac_field(x), ZZ.poly_ring(x, y)) == ZZ.frac_field(x, y)
assert unify(QQ.frac_field(x), QQ.poly_ring(x, y)) == QQ.frac_field(x, y)
assert unify(ZZ.frac_field(x, y), ZZ.poly_ring(x, z)) == ZZ.frac_field(x, y, z)
assert unify(ZZ.frac_field(x, y), QQ.poly_ring(x, z)) == ZZ.frac_field(x, y, z)
assert unify(QQ.frac_field(x, y), ZZ.poly_ring(x, z)) == ZZ.frac_field(x, y, z)
assert unify(QQ.frac_field(x, y), QQ.poly_ring(x, z)) == QQ.frac_field(x, y, z)
def test_Domain_unify_algebraic():
sqrt5 = QQ.algebraic_field(sqrt(5))
sqrt7 = QQ.algebraic_field(sqrt(7))
sqrt57 = QQ.algebraic_field(sqrt(5), sqrt(7))
assert sqrt5.unify(sqrt7) == sqrt57
assert sqrt5.unify(sqrt5[x, y]) == sqrt5[x, y]
assert sqrt5[x, y].unify(sqrt5) == sqrt5[x, y]
assert sqrt5.unify(sqrt5.frac_field(x, y)) == sqrt5.frac_field(x, y)
assert sqrt5.frac_field(x, y).unify(sqrt5) == sqrt5.frac_field(x, y)
assert sqrt5.unify(sqrt7[x, y]) == sqrt57[x, y]
assert sqrt5[x, y].unify(sqrt7) == sqrt57[x, y]
assert sqrt5.unify(sqrt7.frac_field(x, y)) == sqrt57.frac_field(x, y)
assert sqrt5.frac_field(x, y).unify(sqrt7) == sqrt57.frac_field(x, y)
def test_Domain_unify_with_symbols():
raises(UnificationFailed, lambda: ZZ[x, y].unify_with_symbols(ZZ, (y, z)))
raises(UnificationFailed, lambda: ZZ.unify_with_symbols(ZZ[x, y], (y, z)))
def test_Domain__contains__():
assert (0 in EX) is True
assert (0 in ZZ) is True
assert (0 in QQ) is True
assert (0 in RR) is True
assert (0 in CC) is True
assert (0 in ALG) is True
assert (0 in ZZ[x, y]) is True
assert (0 in QQ[x, y]) is True
assert (0 in RR[x, y]) is True
assert (-7 in EX) is True
assert (-7 in ZZ) is True
assert (-7 in QQ) is True
assert (-7 in RR) is True
assert (-7 in CC) is True
assert (-7 in ALG) is True
assert (-7 in ZZ[x, y]) is True
assert (-7 in QQ[x, y]) is True
assert (-7 in RR[x, y]) is True
assert (17 in EX) is True
assert (17 in ZZ) is True
assert (17 in QQ) is True
assert (17 in RR) is True
assert (17 in CC) is True
assert (17 in ALG) is True
assert (17 in ZZ[x, y]) is True
assert (17 in QQ[x, y]) is True
assert (17 in RR[x, y]) is True
assert (Rational(-1, 7) in EX) is True
assert (Rational(-1, 7) in ZZ) is False
assert (Rational(-1, 7) in QQ) is True
assert (Rational(-1, 7) in RR) is True
assert (Rational(-1, 7) in CC) is True
assert (Rational(-1, 7) in ALG) is True
assert (Rational(-1, 7) in ZZ[x, y]) is False
assert (Rational(-1, 7) in QQ[x, y]) is True
assert (Rational(-1, 7) in RR[x, y]) is True
assert (Rational(3, 5) in EX) is True
assert (Rational(3, 5) in ZZ) is False
assert (Rational(3, 5) in QQ) is True
assert (Rational(3, 5) in RR) is True
assert (Rational(3, 5) in CC) is True
assert (Rational(3, 5) in ALG) is True
assert (Rational(3, 5) in ZZ[x, y]) is False
assert (Rational(3, 5) in QQ[x, y]) is True
assert (Rational(3, 5) in RR[x, y]) is True
assert (3.0 in EX) is True
assert (3.0 in ZZ) is True
assert (3.0 in QQ) is True
assert (3.0 in RR) is True
assert (3.0 in CC) is True
assert (3.0 in ALG) is True
assert (3.0 in ZZ[x, y]) is True
assert (3.0 in QQ[x, y]) is True
assert (3.0 in RR[x, y]) is True
assert (3.14 in EX) is True
assert (3.14 in ZZ) is False
assert (3.14 in QQ) is True
assert (3.14 in RR) is True
assert (3.14 in CC) is True
assert (3.14 in ALG) is True
assert (3.14 in ZZ[x, y]) is False
assert (3.14 in QQ[x, y]) is True
assert (3.14 in RR[x, y]) is True
assert (oo in ALG) is False
assert (oo in ZZ[x, y]) is False
assert (oo in QQ[x, y]) is False
assert (-oo in ZZ) is False
assert (-oo in QQ) is False
assert (-oo in ALG) is False
assert (-oo in ZZ[x, y]) is False
assert (-oo in QQ[x, y]) is False
assert (sqrt(7) in EX) is True
assert (sqrt(7) in ZZ) is False
assert (sqrt(7) in QQ) is False
assert (sqrt(7) in RR) is True
assert (sqrt(7) in CC) is True
assert (sqrt(7) in ALG) is False
assert (sqrt(7) in ZZ[x, y]) is False
assert (sqrt(7) in QQ[x, y]) is False
assert (sqrt(7) in RR[x, y]) is True
assert (2*sqrt(3) + 1 in EX) is True
assert (2*sqrt(3) + 1 in ZZ) is False
assert (2*sqrt(3) + 1 in QQ) is False
assert (2*sqrt(3) + 1 in RR) is True
assert (2*sqrt(3) + 1 in CC) is True
assert (2*sqrt(3) + 1 in ALG) is True
assert (2*sqrt(3) + 1 in ZZ[x, y]) is False
assert (2*sqrt(3) + 1 in QQ[x, y]) is False
assert (2*sqrt(3) + 1 in RR[x, y]) is True
assert (sin(1) in EX) is True
assert (sin(1) in ZZ) is False
assert (sin(1) in QQ) is False
assert (sin(1) in RR) is True
assert (sin(1) in CC) is True
assert (sin(1) in ALG) is False
assert (sin(1) in ZZ[x, y]) is False
assert (sin(1) in QQ[x, y]) is False
assert (sin(1) in RR[x, y]) is True
assert (x**2 + 1 in EX) is True
assert (x**2 + 1 in ZZ) is False
assert (x**2 + 1 in QQ) is False
assert (x**2 + 1 in RR) is False
assert (x**2 + 1 in CC) is False
assert (x**2 + 1 in ALG) is False
assert (x**2 + 1 in ZZ[x]) is True
assert (x**2 + 1 in QQ[x]) is True
assert (x**2 + 1 in RR[x]) is True
assert (x**2 + 1 in ZZ[x, y]) is True
assert (x**2 + 1 in QQ[x, y]) is True
assert (x**2 + 1 in RR[x, y]) is True
assert (x**2 + y**2 in EX) is True
assert (x**2 + y**2 in ZZ) is False
assert (x**2 + y**2 in QQ) is False
assert (x**2 + y**2 in RR) is False
assert (x**2 + y**2 in CC) is False
assert (x**2 + y**2 in ALG) is False
assert (x**2 + y**2 in ZZ[x]) is False
assert (x**2 + y**2 in QQ[x]) is False
assert (x**2 + y**2 in RR[x]) is False
assert (x**2 + y**2 in ZZ[x, y]) is True
assert (x**2 + y**2 in QQ[x, y]) is True
assert (x**2 + y**2 in RR[x, y]) is True
assert (Rational(3, 2)*x/(y + 1) - z in QQ[x, y, z]) is False
def test_Domain_get_ring():
assert ZZ.has_assoc_Ring is True
assert QQ.has_assoc_Ring is True
assert ZZ[x].has_assoc_Ring is True
assert QQ[x].has_assoc_Ring is True
assert ZZ[x, y].has_assoc_Ring is True
assert QQ[x, y].has_assoc_Ring is True
assert ZZ.frac_field(x).has_assoc_Ring is True
assert QQ.frac_field(x).has_assoc_Ring is True
assert ZZ.frac_field(x, y).has_assoc_Ring is True
assert QQ.frac_field(x, y).has_assoc_Ring is True
assert EX.has_assoc_Ring is False
assert RR.has_assoc_Ring is False
assert ALG.has_assoc_Ring is False
assert ZZ.get_ring() == ZZ
assert QQ.get_ring() == ZZ
assert ZZ[x].get_ring() == ZZ[x]
assert QQ[x].get_ring() == QQ[x]
assert ZZ[x, y].get_ring() == ZZ[x, y]
assert QQ[x, y].get_ring() == QQ[x, y]
assert ZZ.frac_field(x).get_ring() == ZZ[x]
assert QQ.frac_field(x).get_ring() == QQ[x]
assert ZZ.frac_field(x, y).get_ring() == ZZ[x, y]
assert QQ.frac_field(x, y).get_ring() == QQ[x, y]
assert EX.get_ring() == EX
assert RR.get_ring() == RR
# XXX: This should also be like RR
raises(DomainError, lambda: ALG.get_ring())
def test_Domain_get_field():
assert EX.has_assoc_Field is True
assert ZZ.has_assoc_Field is True
assert QQ.has_assoc_Field is True
assert RR.has_assoc_Field is True
assert ALG.has_assoc_Field is True
assert ZZ[x].has_assoc_Field is True
assert QQ[x].has_assoc_Field is True
assert ZZ[x, y].has_assoc_Field is True
assert QQ[x, y].has_assoc_Field is True
assert EX.get_field() == EX
assert ZZ.get_field() == QQ
assert QQ.get_field() == QQ
assert RR.get_field() == RR
assert ALG.get_field() == ALG
assert ZZ[x].get_field() == ZZ.frac_field(x)
assert QQ[x].get_field() == QQ.frac_field(x)
assert ZZ[x, y].get_field() == ZZ.frac_field(x, y)
assert QQ[x, y].get_field() == QQ.frac_field(x, y)
def test_Domain_get_exact():
assert EX.get_exact() == EX
assert ZZ.get_exact() == ZZ
assert QQ.get_exact() == QQ
assert RR.get_exact() == QQ
assert ALG.get_exact() == ALG
assert ZZ[x].get_exact() == ZZ[x]
assert QQ[x].get_exact() == QQ[x]
assert ZZ[x, y].get_exact() == ZZ[x, y]
assert QQ[x, y].get_exact() == QQ[x, y]
assert ZZ.frac_field(x).get_exact() == ZZ.frac_field(x)
assert QQ.frac_field(x).get_exact() == QQ.frac_field(x)
assert ZZ.frac_field(x, y).get_exact() == ZZ.frac_field(x, y)
assert QQ.frac_field(x, y).get_exact() == QQ.frac_field(x, y)
def test_Domain_convert():
assert QQ.convert(10e-52) == QQ(1684996666696915, 1684996666696914987166688442938726917102321526408785780068975640576)
R, x = ring("x", ZZ)
assert ZZ.convert(x - x) == 0
assert ZZ.convert(x - x, R.to_domain()) == 0
def test_PolynomialRing__init():
R, = ring("", ZZ)
assert ZZ.poly_ring() == R.to_domain()
def test_FractionField__init():
F, = field("", ZZ)
assert ZZ.frac_field() == F.to_domain()
def test_inject():
assert ZZ.inject(x, y, z) == ZZ[x, y, z]
assert ZZ[x].inject(y, z) == ZZ[x, y, z]
assert ZZ.frac_field(x).inject(y, z) == ZZ.frac_field(x, y, z)
raises(GeneratorsError, lambda: ZZ[x].inject(x))
def test_Domain_map():
seq = ZZ.map([1, 2, 3, 4])
assert all(ZZ.of_type(elt) for elt in seq)
seq = ZZ.map([[1, 2, 3, 4]])
assert all(ZZ.of_type(elt) for elt in seq[0]) and len(seq) == 1
def test_Domain___eq__():
assert (ZZ[x, y] == ZZ[x, y]) is True
assert (QQ[x, y] == QQ[x, y]) is True
assert (ZZ[x, y] == QQ[x, y]) is False
assert (QQ[x, y] == ZZ[x, y]) is False
assert (ZZ.frac_field(x, y) == ZZ.frac_field(x, y)) is True
assert (QQ.frac_field(x, y) == QQ.frac_field(x, y)) is True
assert (ZZ.frac_field(x, y) == QQ.frac_field(x, y)) is False
assert (QQ.frac_field(x, y) == ZZ.frac_field(x, y)) is False
assert RealField()[x] == RR[x]
def test_Domain__algebraic_field():
alg = ZZ.algebraic_field(sqrt(2))
assert alg.ext.minpoly == Poly(x**2 - 2)
assert alg.dom == QQ
alg = QQ.algebraic_field(sqrt(2))
assert alg.ext.minpoly == Poly(x**2 - 2)
assert alg.dom == QQ
alg = alg.algebraic_field(sqrt(3))
assert alg.ext.minpoly == Poly(x**4 - 10*x**2 + 1)
assert alg.dom == QQ
def test_PolynomialRing_from_FractionField():
F, x,y = field("x,y", ZZ)
R, X,Y = ring("x,y", ZZ)
f = (x**2 + y**2)/(x + 1)
g = (x**2 + y**2)/4
h = x**2 + y**2
assert R.to_domain().from_FractionField(f, F.to_domain()) is None
assert R.to_domain().from_FractionField(g, F.to_domain()) == X**2/4 + Y**2/4
assert R.to_domain().from_FractionField(h, F.to_domain()) == X**2 + Y**2
F, x,y = field("x,y", QQ)
R, X,Y = ring("x,y", QQ)
f = (x**2 + y**2)/(x + 1)
g = (x**2 + y**2)/4
h = x**2 + y**2
assert R.to_domain().from_FractionField(f, F.to_domain()) is None
assert R.to_domain().from_FractionField(g, F.to_domain()) == X**2/4 + Y**2/4
assert R.to_domain().from_FractionField(h, F.to_domain()) == X**2 + Y**2
def test_FractionField_from_PolynomialRing():
R, x,y = ring("x,y", QQ)
F, X,Y = field("x,y", ZZ)
f = 3*x**2 + 5*y**2
g = x**2/3 + y**2/5
assert F.to_domain().from_PolynomialRing(f, R.to_domain()) == 3*X**2 + 5*Y**2
assert F.to_domain().from_PolynomialRing(g, R.to_domain()) == (5*X**2 + 3*Y**2)/15
def test_FF_of_type():
assert FF(3).of_type(FF(3)(1)) is True
assert FF(5).of_type(FF(5)(3)) is True
assert FF(5).of_type(FF(7)(3)) is False
def test___eq__():
assert not QQ[x] == ZZ[x]
assert not QQ.frac_field(x) == ZZ.frac_field(x)
def test_RealField_from_sympy():
assert RR.convert(S.Zero) == RR.dtype(0)
assert RR.convert(S(0.0)) == RR.dtype(0.0)
assert RR.convert(S.One) == RR.dtype(1)
assert RR.convert(S(1.0)) == RR.dtype(1.0)
assert RR.convert(sin(1)) == RR.dtype(sin(1).evalf())
def test_not_in_any_domain():
check = illegal + [x] + [
float(i) for i in illegal if i != S.ComplexInfinity]
for dom in (ZZ, QQ, RR, CC, EX):
for i in check:
if i == x and dom == EX:
continue
assert i not in dom, (i, dom)
raises(CoercionFailed, lambda: dom.convert(i))
def test_ModularInteger():
F3 = FF(3)
a = F3(0)
assert isinstance(a, F3.dtype) and a == 0
a = F3(1)
assert isinstance(a, F3.dtype) and a == 1
a = F3(2)
assert isinstance(a, F3.dtype) and a == 2
a = F3(3)
assert isinstance(a, F3.dtype) and a == 0
a = F3(4)
assert isinstance(a, F3.dtype) and a == 1
a = F3(F3(0))
assert isinstance(a, F3.dtype) and a == 0
a = F3(F3(1))
assert isinstance(a, F3.dtype) and a == 1
a = F3(F3(2))
assert isinstance(a, F3.dtype) and a == 2
a = F3(F3(3))
assert isinstance(a, F3.dtype) and a == 0
a = F3(F3(4))
assert isinstance(a, F3.dtype) and a == 1
a = -F3(1)
assert isinstance(a, F3.dtype) and a == 2
a = -F3(2)
assert isinstance(a, F3.dtype) and a == 1
a = 2 + F3(2)
assert isinstance(a, F3.dtype) and a == 1
a = F3(2) + 2
assert isinstance(a, F3.dtype) and a == 1
a = F3(2) + F3(2)
assert isinstance(a, F3.dtype) and a == 1
a = F3(2) + F3(2)
assert isinstance(a, F3.dtype) and a == 1
a = 3 - F3(2)
assert isinstance(a, F3.dtype) and a == 1
a = F3(3) - 2
assert isinstance(a, F3.dtype) and a == 1
a = F3(3) - F3(2)
assert isinstance(a, F3.dtype) and a == 1
a = F3(3) - F3(2)
assert isinstance(a, F3.dtype) and a == 1
a = 2*F3(2)
assert isinstance(a, F3.dtype) and a == 1
a = F3(2)*2
assert isinstance(a, F3.dtype) and a == 1
a = F3(2)*F3(2)
assert isinstance(a, F3.dtype) and a == 1
a = F3(2)*F3(2)
assert isinstance(a, F3.dtype) and a == 1
a = 2/F3(2)
assert isinstance(a, F3.dtype) and a == 1
a = F3(2)/2
assert isinstance(a, F3.dtype) and a == 1
a = F3(2)/F3(2)
assert isinstance(a, F3.dtype) and a == 1
a = F3(2)/F3(2)
assert isinstance(a, F3.dtype) and a == 1
a = 1 % F3(2)
assert isinstance(a, F3.dtype) and a == 1
a = F3(1) % 2
assert isinstance(a, F3.dtype) and a == 1
a = F3(1) % F3(2)
assert isinstance(a, F3.dtype) and a == 1
a = F3(1) % F3(2)
assert isinstance(a, F3.dtype) and a == 1
a = F3(2)**0
assert isinstance(a, F3.dtype) and a == 1
a = F3(2)**1
assert isinstance(a, F3.dtype) and a == 2
a = F3(2)**2
assert isinstance(a, F3.dtype) and a == 1
F7 = FF(7)
a = F7(3)**100000000000
assert isinstance(a, F7.dtype) and a == 4
a = F7(3)**-100000000000
assert isinstance(a, F7.dtype) and a == 2
a = F7(3)**S(2)
assert isinstance(a, F7.dtype) and a == 2
assert bool(F3(3)) is False
assert bool(F3(4)) is True
F5 = FF(5)
a = F5(1)**(-1)
assert isinstance(a, F5.dtype) and a == 1
a = F5(2)**(-1)
assert isinstance(a, F5.dtype) and a == 3
a = F5(3)**(-1)
assert isinstance(a, F5.dtype) and a == 2
a = F5(4)**(-1)
assert isinstance(a, F5.dtype) and a == 4
assert (F5(1) < F5(2)) is True
assert (F5(1) <= F5(2)) is True
assert (F5(1) > F5(2)) is False
assert (F5(1) >= F5(2)) is False
assert (F5(3) < F5(2)) is False
assert (F5(3) <= F5(2)) is False
assert (F5(3) > F5(2)) is True
assert (F5(3) >= F5(2)) is True
assert (F5(1) < F5(7)) is True
assert (F5(1) <= F5(7)) is True
assert (F5(1) > F5(7)) is False
assert (F5(1) >= F5(7)) is False
assert (F5(3) < F5(7)) is False
assert (F5(3) <= F5(7)) is False
assert (F5(3) > F5(7)) is True
assert (F5(3) >= F5(7)) is True
assert (F5(1) < 2) is True
assert (F5(1) <= 2) is True
assert (F5(1) > 2) is False
assert (F5(1) >= 2) is False
assert (F5(3) < 2) is False
assert (F5(3) <= 2) is False
assert (F5(3) > 2) is True
assert (F5(3) >= 2) is True
assert (F5(1) < 7) is True
assert (F5(1) <= 7) is True
assert (F5(1) > 7) is False
assert (F5(1) >= 7) is False
assert (F5(3) < 7) is False
assert (F5(3) <= 7) is False
assert (F5(3) > 7) is True
assert (F5(3) >= 7) is True
raises(NotInvertible, lambda: F5(0)**(-1))
raises(NotInvertible, lambda: F5(5)**(-1))
raises(ValueError, lambda: FF(0))
raises(ValueError, lambda: FF(2.1))
def test_QQ_int():
assert int(QQ(2**2000, 3**1250)) == 455431
assert int(QQ(2**100, 3)) == 422550200076076467165567735125
def test_RR_double():
assert RR(3.14) > 1e-50
assert RR(1e-13) > 1e-50
assert RR(1e-14) > 1e-50
assert RR(1e-15) > 1e-50
assert RR(1e-20) > 1e-50
assert RR(1e-40) > 1e-50
def test_RR_Float():
f1 = Float("1.01")
f2 = Float("1.0000000000000000000001")
assert f1._prec == 53
assert f2._prec == 80
assert RR(f1)-1 > 1e-50
assert RR(f2)-1 < 1e-50 # RR's precision is lower than f2's
RR2 = RealField(prec=f2._prec)
assert RR2(f1)-1 > 1e-50
assert RR2(f2)-1 > 1e-50 # RR's precision is equal to f2's
def test_CC_double():
assert CC(3.14).real > 1e-50
assert CC(1e-13).real > 1e-50
assert CC(1e-14).real > 1e-50
assert CC(1e-15).real > 1e-50
assert CC(1e-20).real > 1e-50
assert CC(1e-40).real > 1e-50
assert CC(3.14j).imag > 1e-50
assert CC(1e-13j).imag > 1e-50
assert CC(1e-14j).imag > 1e-50
assert CC(1e-15j).imag > 1e-50
assert CC(1e-20j).imag > 1e-50
assert CC(1e-40j).imag > 1e-50
def test_gaussian_domains():
I = S.ImaginaryUnit
a, b, c = [ZZ_I.convert(x) for x in (5, 2 + I, 3 - I)]
ZZ_I.gcd(a, b) == b
ZZ_I.gcd(a, c) == b
assert ZZ_I(3, 4) != QQ_I(3, 4) # XXX is this right or should QQ->ZZ if possible?
assert ZZ_I(3, 0) != 3 # and should this go to Integer?
assert QQ_I(S(3)/4, 0) != S(3)/4 # and this to Rational?
assert ZZ_I(0, 0).quadrant() == 0
assert ZZ_I(-1, 0).quadrant() == 2
for G in (QQ_I, ZZ_I):
q = G(3, 4)
assert q._get_xy(pi) == (None, None)
assert q._get_xy(2) == (2, 0)
assert q._get_xy(2*I) == (0, 2)
assert hash(q) == hash((3, 4))
assert q + q == G(6, 8)
assert q - q == G(0, 0)
assert 3 - q == -q + 3 == G(0, -4)
assert 3 + q == q + 3 == G(6, 4)
assert repr(q) in ('GaussianInteger(3, 4)', 'GaussianRational(3, 4)')
assert str(q) == '3 + 4*I'
assert q.parent() == G
assert q/3 == QQ_I(1, S(4)/3)
assert 3/q == QQ_I(S(9)/25, -S(12)/25)
i, r = divmod(q, 2)
assert 2*i + r == q
i, r = divmod(2, q)
assert G.from_sympy(S(2)) == G(2, 0)
raises(ZeroDivisionError, lambda: q % 0)
raises(ZeroDivisionError, lambda: q / 0)
raises(ZeroDivisionError, lambda: q // 0)
raises(ZeroDivisionError, lambda: divmod(q, 0))
raises(ZeroDivisionError, lambda: divmod(q, 0))
raises(CoercionFailed, lambda: G.from_sympy(pi))
if G == ZZ_I:
assert q//3 == G(1, 1)
assert 12//q == G(1, -2)
assert 12 % q == G(1, 2)
assert q % 2 == G(-1, 0)
assert i == G(0, 0)
assert r == G(2, 0)
assert G.get_ring() == G
assert G.get_field() == QQ_I
else:
assert G.get_ring() == ZZ_I
assert G.get_field() == G
assert q//3 == G(1, S(4)/3)
assert 12//q == G(S(36)/25, -S(48)/25)
assert 12 % q == G(0, 0)
assert q % 2 == G(0, 0)
assert i == G(S(6)/25, -S(8)/25), (G,i)
assert r == G(0, 0)
def test_issue_18278():
assert str(RR(2).parent()) == 'RR'
assert str(CC(2).parent()) == 'CC'
|
4f98aff5b15130d2bbbaabfafd8780b3b81275f0457abab997c99ccb163fb92a | from sympy.testing.pytest import raises, XFAIL
from sympy.external import import_module
from sympy import (
Symbol, Mul, Add, Eq, Abs, sin, asin, cos, Pow,
csc, sec, Limit, oo, Derivative, Integral, factorial,
sqrt, root, StrictLessThan, LessThan, StrictGreaterThan,
GreaterThan, Sum, Product, E, log, tan, Function, binomial, exp,
)
from sympy.abc import x, y, z, a, b, c, t, k, n
antlr4 = import_module("antlr4")
# disable tests if antlr4-python*-runtime is not present
if not antlr4:
disabled = True
theta = Symbol('theta')
f = Function('f')
# shorthand definitions
def _Add(a, b):
return Add(a, b, evaluate=False)
def _Mul(a, b):
return Mul(a, b, evaluate=False)
def _Pow(a, b):
return Pow(a, b, evaluate=False)
def _Abs(a):
return Abs(a, evaluate=False)
def _factorial(a):
return factorial(a, evaluate=False)
def _exp(a):
return exp(a, evaluate=False)
def _log(a, b):
return log(a, b, evaluate=False)
def _binomial(n, k):
return binomial(n, k, evaluate=False)
def test_import():
from sympy.parsing.latex._build_latex_antlr import (
build_parser,
check_antlr_version,
dir_latex_antlr
)
# XXX: It would be better to come up with a test for these...
del build_parser, check_antlr_version, dir_latex_antlr
# These LaTeX strings should parse to the corresponding SymPy expression
GOOD_PAIRS = [
("0", 0),
("1", 1),
("-3.14", _Mul(-1, 3.14)),
("(-7.13)(1.5)", _Mul(_Mul(-1, 7.13), 1.5)),
("x", x),
("2x", 2*x),
("x^2", x**2),
("x^{3 + 1}", x**_Add(3, 1)),
("-c", -c),
("a \\cdot b", a * b),
("a / b", a / b),
("a \\div b", a / b),
("a + b", a + b),
("a + b - a", _Add(a+b, -a)),
("a^2 + b^2 = c^2", Eq(a**2 + b**2, c**2)),
("1+1", Add(1, 1, evaluate=False)),
("0+1", Add(0, 1, evaluate=False)),
("1*2", Mul(1, 2, evaluate=False)),
("0*1", Mul(0, 1, evaluate=False)),
("\\sin \\theta", sin(theta)),
("\\sin(\\theta)", sin(theta)),
("\\sin^{-1} a", asin(a)),
("\\sin a \\cos b", _Mul(sin(a), cos(b))),
("\\sin \\cos \\theta", sin(cos(theta))),
("\\sin(\\cos \\theta)", sin(cos(theta))),
("\\frac{a}{b}", a / b),
("\\frac{a + b}{c}", _Mul(a + b, _Pow(c, -1))),
("\\frac{7}{3}", _Mul(7, _Pow(3, -1))),
("(\\csc x)(\\sec y)", csc(x)*sec(y)),
("\\lim_{x \\to 3} a", Limit(a, x, 3)),
("\\lim_{x \\rightarrow 3} a", Limit(a, x, 3)),
("\\lim_{x \\Rightarrow 3} a", Limit(a, x, 3)),
("\\lim_{x \\longrightarrow 3} a", Limit(a, x, 3)),
("\\lim_{x \\Longrightarrow 3} a", Limit(a, x, 3)),
("\\lim_{x \\to 3^{+}} a", Limit(a, x, 3, dir='+')),
("\\lim_{x \\to 3^{-}} a", Limit(a, x, 3, dir='-')),
("\\infty", oo),
("\\lim_{x \\to \\infty} \\frac{1}{x}", Limit(_Pow(x, -1), x, oo)),
("\\frac{d}{dx} x", Derivative(x, x)),
("\\frac{d}{dt} x", Derivative(x, t)),
("f(x)", f(x)),
("f(x, y)", f(x, y)),
("f(x, y, z)", f(x, y, z)),
("\\frac{d f(x)}{dx}", Derivative(f(x), x)),
("\\frac{d\\theta(x)}{dx}", Derivative(Function('theta')(x), x)),
("|x|", _Abs(x)),
("||x||", _Abs(Abs(x))),
("|x||y|", _Abs(x)*_Abs(y)),
("||x||y||", _Abs(_Abs(x)*_Abs(y))),
("\\pi^{|xy|}", Symbol('pi')**_Abs(x*y)),
("\\int x dx", Integral(x, x)),
("\\int x d\\theta", Integral(x, theta)),
("\\int (x^2 - y)dx", Integral(x**2 - y, x)),
("\\int x + a dx", Integral(_Add(x, a), x)),
("\\int da", Integral(1, a)),
("\\int_0^7 dx", Integral(1, (x, 0, 7))),
("\\int_a^b x dx", Integral(x, (x, a, b))),
("\\int^b_a x dx", Integral(x, (x, a, b))),
("\\int_{a}^b x dx", Integral(x, (x, a, b))),
("\\int^{b}_a x dx", Integral(x, (x, a, b))),
("\\int_{a}^{b} x dx", Integral(x, (x, a, b))),
("\\int^{b}_{a} x dx", Integral(x, (x, a, b))),
("\\int_{f(a)}^{f(b)} f(z) dz", Integral(f(z), (z, f(a), f(b)))),
("\\int (x+a)", Integral(_Add(x, a), x)),
("\\int a + b + c dx", Integral(_Add(_Add(a, b), c), x)),
("\\int \\frac{dz}{z}", Integral(Pow(z, -1), z)),
("\\int \\frac{3 dz}{z}", Integral(3*Pow(z, -1), z)),
("\\int \\frac{1}{x} dx", Integral(Pow(x, -1), x)),
("\\int \\frac{1}{a} + \\frac{1}{b} dx",
Integral(_Add(_Pow(a, -1), Pow(b, -1)), x)),
("\\int \\frac{3 \\cdot d\\theta}{\\theta}",
Integral(3*_Pow(theta, -1), theta)),
("\\int \\frac{1}{x} + 1 dx", Integral(_Add(_Pow(x, -1), 1), x)),
("x_0", Symbol('x_{0}')),
("x_{1}", Symbol('x_{1}')),
("x_a", Symbol('x_{a}')),
("x_{b}", Symbol('x_{b}')),
("h_\\theta", Symbol('h_{theta}')),
("h_{\\theta}", Symbol('h_{theta}')),
("h_{\\theta}(x_0, x_1)",
Function('h_{theta}')(Symbol('x_{0}'), Symbol('x_{1}'))),
("x!", _factorial(x)),
("100!", _factorial(100)),
("\\theta!", _factorial(theta)),
("(x + 1)!", _factorial(_Add(x, 1))),
("(x!)!", _factorial(_factorial(x))),
("x!!!", _factorial(_factorial(_factorial(x)))),
("5!7!", _Mul(_factorial(5), _factorial(7))),
("\\sqrt{x}", sqrt(x)),
("\\sqrt{x + b}", sqrt(_Add(x, b))),
("\\sqrt[3]{\\sin x}", root(sin(x), 3)),
("\\sqrt[y]{\\sin x}", root(sin(x), y)),
("\\sqrt[\\theta]{\\sin x}", root(sin(x), theta)),
("x < y", StrictLessThan(x, y)),
("x \\leq y", LessThan(x, y)),
("x > y", StrictGreaterThan(x, y)),
("x \\geq y", GreaterThan(x, y)),
("\\mathit{x}", Symbol('x')),
("\\mathit{test}", Symbol('test')),
("\\mathit{TEST}", Symbol('TEST')),
("\\mathit{HELLO world}", Symbol('HELLO world')),
("\\sum_{k = 1}^{3} c", Sum(c, (k, 1, 3))),
("\\sum_{k = 1}^3 c", Sum(c, (k, 1, 3))),
("\\sum^{3}_{k = 1} c", Sum(c, (k, 1, 3))),
("\\sum^3_{k = 1} c", Sum(c, (k, 1, 3))),
("\\sum_{k = 1}^{10} k^2", Sum(k**2, (k, 1, 10))),
("\\sum_{n = 0}^{\\infty} \\frac{1}{n!}",
Sum(_Pow(_factorial(n), -1), (n, 0, oo))),
("\\prod_{a = b}^{c} x", Product(x, (a, b, c))),
("\\prod_{a = b}^c x", Product(x, (a, b, c))),
("\\prod^{c}_{a = b} x", Product(x, (a, b, c))),
("\\prod^c_{a = b} x", Product(x, (a, b, c))),
("\\exp x", _exp(x)),
("\\exp(x)", _exp(x)),
("\\ln x", _log(x, E)),
("\\ln xy", _log(x*y, E)),
("\\log x", _log(x, 10)),
("\\log xy", _log(x*y, 10)),
("\\log_{2} x", _log(x, 2)),
("\\log_{a} x", _log(x, a)),
("\\log_{11} x", _log(x, 11)),
("\\log_{a^2} x", _log(x, _Pow(a, 2))),
("[x]", x),
("[a + b]", _Add(a, b)),
("\\frac{d}{dx} [ \\tan x ]", Derivative(tan(x), x)),
("\\binom{n}{k}", _binomial(n, k)),
("\\tbinom{n}{k}", _binomial(n, k)),
("\\dbinom{n}{k}", _binomial(n, k)),
("\\binom{n}{0}", _binomial(n, 0)),
("a \\, b", _Mul(a, b)),
("a \\thinspace b", _Mul(a, b)),
("a \\: b", _Mul(a, b)),
("a \\medspace b", _Mul(a, b)),
("a \\; b", _Mul(a, b)),
("a \\thickspace b", _Mul(a, b)),
("a \\quad b", _Mul(a, b)),
("a \\qquad b", _Mul(a, b)),
("a \\! b", _Mul(a, b)),
("a \\negthinspace b", _Mul(a, b)),
("a \\negmedspace b", _Mul(a, b)),
("a \\negthickspace b", _Mul(a, b)),
("\\int x \\, dx", Integral(x, x)),
]
def test_parseable():
from sympy.parsing.latex import parse_latex
for latex_str, sympy_expr in GOOD_PAIRS:
assert parse_latex(latex_str) == sympy_expr
# At time of migration from latex2sympy, should work but doesn't
FAILING_PAIRS = [
("\\log_2 x", _log(x, 2)),
("\\log_a x", _log(x, a)),
]
def test_failing_parseable():
from sympy.parsing.latex import parse_latex
for latex_str, sympy_expr in FAILING_PAIRS:
with raises(Exception):
assert parse_latex(latex_str) == sympy_expr
# These bad LaTeX strings should raise a LaTeXParsingError when parsed
BAD_STRINGS = [
"(",
")",
"\\frac{d}{dx}",
"(\\frac{d}{dx})"
"\\sqrt{}",
"\\sqrt",
"{",
"}",
"\\mathit{x + y}",
"\\mathit{21}",
"\\frac{2}{}",
"\\frac{}{2}",
"\\int",
"!",
"!0",
"_",
"^",
"|",
"||x|",
"()",
"((((((((((((((((()))))))))))))))))",
"-",
"\\frac{d}{dx} + \\frac{d}{dt}",
"f(x,,y)",
"f(x,y,",
"\\sin^x",
"\\cos^2",
"@",
"#",
"$",
"%",
"&",
"*",
"\\",
"~",
"\\frac{(2 + x}{1 - x)}"
]
def test_not_parseable():
from sympy.parsing.latex import parse_latex, LaTeXParsingError
for latex_str in BAD_STRINGS:
with raises(LaTeXParsingError):
parse_latex(latex_str)
# At time of migration from latex2sympy, should fail but doesn't
FAILING_BAD_STRINGS = [
"\\cos 1 \\cos",
"f(,",
"f()",
"a \\div \\div b",
"a \\cdot \\cdot b",
"a // b",
"a +",
"1.1.1",
"1 +",
"a / b /",
]
@XFAIL
def test_failing_not_parseable():
from sympy.parsing.latex import parse_latex, LaTeXParsingError
for latex_str in FAILING_BAD_STRINGS:
with raises(LaTeXParsingError):
parse_latex(latex_str)
|
b0413b94ecffeeccfda9de88c7443bdb4ff3e331fdec37bcd5128f49a87daf2e | # Ported from latex2sympy by @augustt198
# https://github.com/augustt198/latex2sympy
# See license in LICENSE.txt
import sympy
from sympy.external import import_module
from sympy.printing.str import StrPrinter
from .errors import LaTeXParsingError
LaTeXParser = LaTeXLexer = MathErrorListener = None
try:
LaTeXParser = import_module('sympy.parsing.latex._antlr.latexparser',
import_kwargs={'fromlist': ['LaTeXParser']}).LaTeXParser
LaTeXLexer = import_module('sympy.parsing.latex._antlr.latexlexer',
import_kwargs={'fromlist': ['LaTeXLexer']}).LaTeXLexer
except Exception:
pass
ErrorListener = import_module('antlr4.error.ErrorListener',
warn_not_installed=True,
import_kwargs={'fromlist': ['ErrorListener']}
)
if ErrorListener:
class MathErrorListener(ErrorListener.ErrorListener): # type: ignore
def __init__(self, src):
super(ErrorListener.ErrorListener, self).__init__()
self.src = src
def syntaxError(self, recog, symbol, line, col, msg, e):
fmt = "%s\n%s\n%s"
marker = "~" * col + "^"
if msg.startswith("missing"):
err = fmt % (msg, self.src, marker)
elif msg.startswith("no viable"):
err = fmt % ("I expected something else here", self.src, marker)
elif msg.startswith("mismatched"):
names = LaTeXParser.literalNames
expected = [
names[i] for i in e.getExpectedTokens() if i < len(names)
]
if len(expected) < 10:
expected = " ".join(expected)
err = (fmt % ("I expected one of these: " + expected, self.src,
marker))
else:
err = (fmt % ("I expected something else here", self.src,
marker))
else:
err = fmt % ("I don't understand this", self.src, marker)
raise LaTeXParsingError(err)
def parse_latex(sympy):
antlr4 = import_module('antlr4', warn_not_installed=True)
if None in [antlr4, MathErrorListener]:
raise ImportError("LaTeX parsing requires the antlr4 python package,"
" provided by pip (antlr4-python2-runtime or"
" antlr4-python3-runtime) or"
" conda (antlr-python-runtime)")
matherror = MathErrorListener(sympy)
stream = antlr4.InputStream(sympy)
lex = LaTeXLexer(stream)
lex.removeErrorListeners()
lex.addErrorListener(matherror)
tokens = antlr4.CommonTokenStream(lex)
parser = LaTeXParser(tokens)
# remove default console error listener
parser.removeErrorListeners()
parser.addErrorListener(matherror)
relation = parser.math().relation()
expr = convert_relation(relation)
return expr
def convert_relation(rel):
if rel.expr():
return convert_expr(rel.expr())
lh = convert_relation(rel.relation(0))
rh = convert_relation(rel.relation(1))
if rel.LT():
return sympy.StrictLessThan(lh, rh)
elif rel.LTE():
return sympy.LessThan(lh, rh)
elif rel.GT():
return sympy.StrictGreaterThan(lh, rh)
elif rel.GTE():
return sympy.GreaterThan(lh, rh)
elif rel.EQUAL():
return sympy.Eq(lh, rh)
def convert_expr(expr):
return convert_add(expr.additive())
def convert_add(add):
if add.ADD():
lh = convert_add(add.additive(0))
rh = convert_add(add.additive(1))
return sympy.Add(lh, rh, evaluate=False)
elif add.SUB():
lh = convert_add(add.additive(0))
rh = convert_add(add.additive(1))
return sympy.Add(lh, -1 * rh, evaluate=False)
else:
return convert_mp(add.mp())
def convert_mp(mp):
if hasattr(mp, 'mp'):
mp_left = mp.mp(0)
mp_right = mp.mp(1)
else:
mp_left = mp.mp_nofunc(0)
mp_right = mp.mp_nofunc(1)
if mp.MUL() or mp.CMD_TIMES() or mp.CMD_CDOT():
lh = convert_mp(mp_left)
rh = convert_mp(mp_right)
return sympy.Mul(lh, rh, evaluate=False)
elif mp.DIV() or mp.CMD_DIV() or mp.COLON():
lh = convert_mp(mp_left)
rh = convert_mp(mp_right)
return sympy.Mul(lh, sympy.Pow(rh, -1, evaluate=False), evaluate=False)
else:
if hasattr(mp, 'unary'):
return convert_unary(mp.unary())
else:
return convert_unary(mp.unary_nofunc())
def convert_unary(unary):
if hasattr(unary, 'unary'):
nested_unary = unary.unary()
else:
nested_unary = unary.unary_nofunc()
if hasattr(unary, 'postfix_nofunc'):
first = unary.postfix()
tail = unary.postfix_nofunc()
postfix = [first] + tail
else:
postfix = unary.postfix()
if unary.ADD():
return convert_unary(nested_unary)
elif unary.SUB():
numabs = convert_unary(nested_unary)
if numabs == 1:
# Use Integer(-1) instead of Mul(-1, 1)
return -numabs
else:
return sympy.Mul(-1, convert_unary(nested_unary), evaluate=False)
elif postfix:
return convert_postfix_list(postfix)
def convert_postfix_list(arr, i=0):
if i >= len(arr):
raise LaTeXParsingError("Index out of bounds")
res = convert_postfix(arr[i])
if isinstance(res, sympy.Expr):
if i == len(arr) - 1:
return res # nothing to multiply by
else:
if i > 0:
left = convert_postfix(arr[i - 1])
right = convert_postfix(arr[i + 1])
if isinstance(left, sympy.Expr) and isinstance(
right, sympy.Expr):
left_syms = convert_postfix(arr[i - 1]).atoms(sympy.Symbol)
right_syms = convert_postfix(arr[i + 1]).atoms(
sympy.Symbol)
# if the left and right sides contain no variables and the
# symbol in between is 'x', treat as multiplication.
if len(left_syms) == 0 and len(right_syms) == 0 and str(
res) == "x":
return convert_postfix_list(arr, i + 1)
# multiply by next
return sympy.Mul(
res, convert_postfix_list(arr, i + 1), evaluate=False)
else: # must be derivative
wrt = res[0]
if i == len(arr) - 1:
raise LaTeXParsingError("Expected expression for derivative")
else:
expr = convert_postfix_list(arr, i + 1)
return sympy.Derivative(expr, wrt)
def do_subs(expr, at):
if at.expr():
at_expr = convert_expr(at.expr())
syms = at_expr.atoms(sympy.Symbol)
if len(syms) == 0:
return expr
elif len(syms) > 0:
sym = next(iter(syms))
return expr.subs(sym, at_expr)
elif at.equality():
lh = convert_expr(at.equality().expr(0))
rh = convert_expr(at.equality().expr(1))
return expr.subs(lh, rh)
def convert_postfix(postfix):
if hasattr(postfix, 'exp'):
exp_nested = postfix.exp()
else:
exp_nested = postfix.exp_nofunc()
exp = convert_exp(exp_nested)
for op in postfix.postfix_op():
if op.BANG():
if isinstance(exp, list):
raise LaTeXParsingError("Cannot apply postfix to derivative")
exp = sympy.factorial(exp, evaluate=False)
elif op.eval_at():
ev = op.eval_at()
at_b = None
at_a = None
if ev.eval_at_sup():
at_b = do_subs(exp, ev.eval_at_sup())
if ev.eval_at_sub():
at_a = do_subs(exp, ev.eval_at_sub())
if at_b is not None and at_a is not None:
exp = sympy.Add(at_b, -1 * at_a, evaluate=False)
elif at_b is not None:
exp = at_b
elif at_a is not None:
exp = at_a
return exp
def convert_exp(exp):
if hasattr(exp, 'exp'):
exp_nested = exp.exp()
else:
exp_nested = exp.exp_nofunc()
if exp_nested:
base = convert_exp(exp_nested)
if isinstance(base, list):
raise LaTeXParsingError("Cannot raise derivative to power")
if exp.atom():
exponent = convert_atom(exp.atom())
elif exp.expr():
exponent = convert_expr(exp.expr())
return sympy.Pow(base, exponent, evaluate=False)
else:
if hasattr(exp, 'comp'):
return convert_comp(exp.comp())
else:
return convert_comp(exp.comp_nofunc())
def convert_comp(comp):
if comp.group():
return convert_expr(comp.group().expr())
elif comp.abs_group():
return sympy.Abs(convert_expr(comp.abs_group().expr()), evaluate=False)
elif comp.atom():
return convert_atom(comp.atom())
elif comp.frac():
return convert_frac(comp.frac())
elif comp.binom():
return convert_binom(comp.binom())
elif comp.func():
return convert_func(comp.func())
def convert_atom(atom):
if atom.LETTER():
subscriptName = ''
if atom.subexpr():
subscript = None
if atom.subexpr().expr(): # subscript is expr
subscript = convert_expr(atom.subexpr().expr())
else: # subscript is atom
subscript = convert_atom(atom.subexpr().atom())
subscriptName = '_{' + StrPrinter().doprint(subscript) + '}'
return sympy.Symbol(atom.LETTER().getText() + subscriptName)
elif atom.SYMBOL():
s = atom.SYMBOL().getText()[1:]
if s == "infty":
return sympy.oo
else:
if atom.subexpr():
subscript = None
if atom.subexpr().expr(): # subscript is expr
subscript = convert_expr(atom.subexpr().expr())
else: # subscript is atom
subscript = convert_atom(atom.subexpr().atom())
subscriptName = StrPrinter().doprint(subscript)
s += '_{' + subscriptName + '}'
return sympy.Symbol(s)
elif atom.NUMBER():
s = atom.NUMBER().getText().replace(",", "")
return sympy.Number(s)
elif atom.DIFFERENTIAL():
var = get_differential_var(atom.DIFFERENTIAL())
return sympy.Symbol('d' + var.name)
elif atom.mathit():
text = rule2text(atom.mathit().mathit_text())
return sympy.Symbol(text)
def rule2text(ctx):
stream = ctx.start.getInputStream()
# starting index of starting token
startIdx = ctx.start.start
# stopping index of stopping token
stopIdx = ctx.stop.stop
return stream.getText(startIdx, stopIdx)
def convert_frac(frac):
diff_op = False
partial_op = False
lower_itv = frac.lower.getSourceInterval()
lower_itv_len = lower_itv[1] - lower_itv[0] + 1
if (frac.lower.start == frac.lower.stop
and frac.lower.start.type == LaTeXLexer.DIFFERENTIAL):
wrt = get_differential_var_str(frac.lower.start.text)
diff_op = True
elif (lower_itv_len == 2 and frac.lower.start.type == LaTeXLexer.SYMBOL
and frac.lower.start.text == '\\partial'
and (frac.lower.stop.type == LaTeXLexer.LETTER
or frac.lower.stop.type == LaTeXLexer.SYMBOL)):
partial_op = True
wrt = frac.lower.stop.text
if frac.lower.stop.type == LaTeXLexer.SYMBOL:
wrt = wrt[1:]
if diff_op or partial_op:
wrt = sympy.Symbol(wrt)
if (diff_op and frac.upper.start == frac.upper.stop
and frac.upper.start.type == LaTeXLexer.LETTER
and frac.upper.start.text == 'd'):
return [wrt]
elif (partial_op and frac.upper.start == frac.upper.stop
and frac.upper.start.type == LaTeXLexer.SYMBOL
and frac.upper.start.text == '\\partial'):
return [wrt]
upper_text = rule2text(frac.upper)
expr_top = None
if diff_op and upper_text.startswith('d'):
expr_top = parse_latex(upper_text[1:])
elif partial_op and frac.upper.start.text == '\\partial':
expr_top = parse_latex(upper_text[len('\\partial'):])
if expr_top:
return sympy.Derivative(expr_top, wrt)
expr_top = convert_expr(frac.upper)
expr_bot = convert_expr(frac.lower)
inverse_denom = sympy.Pow(expr_bot, -1, evaluate=False)
if expr_top == 1:
return inverse_denom
else:
return sympy.Mul(expr_top, inverse_denom, evaluate=False)
def convert_binom(binom):
expr_n = convert_expr(binom.n)
expr_k = convert_expr(binom.k)
return sympy.binomial(expr_n, expr_k, evaluate=False)
def convert_func(func):
if func.func_normal():
if func.L_PAREN(): # function called with parenthesis
arg = convert_func_arg(func.func_arg())
else:
arg = convert_func_arg(func.func_arg_noparens())
name = func.func_normal().start.text[1:]
# change arc<trig> -> a<trig>
if name in [
"arcsin", "arccos", "arctan", "arccsc", "arcsec", "arccot"
]:
name = "a" + name[3:]
expr = getattr(sympy.functions, name)(arg, evaluate=False)
if name in ["arsinh", "arcosh", "artanh"]:
name = "a" + name[2:]
expr = getattr(sympy.functions, name)(arg, evaluate=False)
if name == "exp":
expr = sympy.exp(arg, evaluate=False)
if (name == "log" or name == "ln"):
if func.subexpr():
base = convert_expr(func.subexpr().expr())
elif name == "log":
base = 10
elif name == "ln":
base = sympy.E
expr = sympy.log(arg, base, evaluate=False)
func_pow = None
should_pow = True
if func.supexpr():
if func.supexpr().expr():
func_pow = convert_expr(func.supexpr().expr())
else:
func_pow = convert_atom(func.supexpr().atom())
if name in [
"sin", "cos", "tan", "csc", "sec", "cot", "sinh", "cosh",
"tanh"
]:
if func_pow == -1:
name = "a" + name
should_pow = False
expr = getattr(sympy.functions, name)(arg, evaluate=False)
if func_pow and should_pow:
expr = sympy.Pow(expr, func_pow, evaluate=False)
return expr
elif func.LETTER() or func.SYMBOL():
if func.LETTER():
fname = func.LETTER().getText()
elif func.SYMBOL():
fname = func.SYMBOL().getText()[1:]
fname = str(fname) # can't be unicode
if func.subexpr():
subscript = None
if func.subexpr().expr(): # subscript is expr
subscript = convert_expr(func.subexpr().expr())
else: # subscript is atom
subscript = convert_atom(func.subexpr().atom())
subscriptName = StrPrinter().doprint(subscript)
fname += '_{' + subscriptName + '}'
input_args = func.args()
output_args = []
while input_args.args(): # handle multiple arguments to function
output_args.append(convert_expr(input_args.expr()))
input_args = input_args.args()
output_args.append(convert_expr(input_args.expr()))
return sympy.Function(fname)(*output_args)
elif func.FUNC_INT():
return handle_integral(func)
elif func.FUNC_SQRT():
expr = convert_expr(func.base)
if func.root:
r = convert_expr(func.root)
return sympy.root(expr, r)
else:
return sympy.sqrt(expr)
elif func.FUNC_SUM():
return handle_sum_or_prod(func, "summation")
elif func.FUNC_PROD():
return handle_sum_or_prod(func, "product")
elif func.FUNC_LIM():
return handle_limit(func)
def convert_func_arg(arg):
if hasattr(arg, 'expr'):
return convert_expr(arg.expr())
else:
return convert_mp(arg.mp_nofunc())
def handle_integral(func):
if func.additive():
integrand = convert_add(func.additive())
elif func.frac():
integrand = convert_frac(func.frac())
else:
integrand = 1
int_var = None
if func.DIFFERENTIAL():
int_var = get_differential_var(func.DIFFERENTIAL())
else:
for sym in integrand.atoms(sympy.Symbol):
s = str(sym)
if len(s) > 1 and s[0] == 'd':
if s[1] == '\\':
int_var = sympy.Symbol(s[2:])
else:
int_var = sympy.Symbol(s[1:])
int_sym = sym
if int_var:
integrand = integrand.subs(int_sym, 1)
else:
# Assume dx by default
int_var = sympy.Symbol('x')
if func.subexpr():
if func.subexpr().atom():
lower = convert_atom(func.subexpr().atom())
else:
lower = convert_expr(func.subexpr().expr())
if func.supexpr().atom():
upper = convert_atom(func.supexpr().atom())
else:
upper = convert_expr(func.supexpr().expr())
return sympy.Integral(integrand, (int_var, lower, upper))
else:
return sympy.Integral(integrand, int_var)
def handle_sum_or_prod(func, name):
val = convert_mp(func.mp())
iter_var = convert_expr(func.subeq().equality().expr(0))
start = convert_expr(func.subeq().equality().expr(1))
if func.supexpr().expr(): # ^{expr}
end = convert_expr(func.supexpr().expr())
else: # ^atom
end = convert_atom(func.supexpr().atom())
if name == "summation":
return sympy.Sum(val, (iter_var, start, end))
elif name == "product":
return sympy.Product(val, (iter_var, start, end))
def handle_limit(func):
sub = func.limit_sub()
if sub.LETTER():
var = sympy.Symbol(sub.LETTER().getText())
elif sub.SYMBOL():
var = sympy.Symbol(sub.SYMBOL().getText()[1:])
else:
var = sympy.Symbol('x')
if sub.SUB():
direction = "-"
else:
direction = "+"
approaching = convert_expr(sub.expr())
content = convert_mp(func.mp())
return sympy.Limit(content, var, approaching, direction)
def get_differential_var(d):
text = get_differential_var_str(d.getText())
return sympy.Symbol(text)
def get_differential_var_str(text):
for i in range(1, len(text)):
c = text[i]
if not (c == " " or c == "\r" or c == "\n" or c == "\t"):
idx = i
break
text = text[idx:]
if text[0] == "\\":
text = text[1:]
return text
|
8a9bd4cced9e26c7ea7cab939987a41e0ef9a32dea0f6558eba6aacf0b887578 |
# encoding: utf-8
# *** GENERATED BY `setup.py antlr`, DO NOT EDIT BY HAND ***
#
# Generated from ../LaTeX.g4, derived from latex2sympy
# latex2sympy is licensed under the MIT license
# https://github.com/augustt198/latex2sympy/blob/master/LICENSE.txt
#
# Generated with antlr4
# antlr4 is licensed under the BSD-3-Clause License
# https://github.com/antlr/antlr4/blob/master/LICENSE.txt
from __future__ import print_function
from antlr4 import *
from io import StringIO
import sys
def serializedATN():
with StringIO() as buf:
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buf.write(u",\4-\t-\4.\t.\4/\t/\4\60\t\60\4\61\t\61\4\62\t\62\4\63")
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buf.write(u"\7`\2\2\u0238v\3\2\2\2\u0239\u023a\7<\2\2\u023ax\3\2")
buf.write(u"\2\2\u023b\u023c\t\2\2\2\u023cz\3\2\2\2\u023d\u0241\7")
buf.write(u"f\2\2\u023e\u0240\5y=\2\u023f\u023e\3\2\2\2\u0240\u0243")
buf.write(u"\3\2\2\2\u0241\u0242\3\2\2\2\u0241\u023f\3\2\2\2\u0242")
buf.write(u"\u024b\3\2\2\2\u0243\u0241\3\2\2\2\u0244\u024c\t\3\2")
buf.write(u"\2\u0245\u0247\7^\2\2\u0246\u0248\t\3\2\2\u0247\u0246")
buf.write(u"\3\2\2\2\u0248\u0249\3\2\2\2\u0249\u0247\3\2\2\2\u0249")
buf.write(u"\u024a\3\2\2\2\u024a\u024c\3\2\2\2\u024b\u0244\3\2\2")
buf.write(u"\2\u024b\u0245\3\2\2\2\u024c|\3\2\2\2\u024d\u024e\t\3")
buf.write(u"\2\2\u024e~\3\2\2\2\u024f\u0250\t\4\2\2\u0250\u0080\3")
buf.write(u"\2\2\2\u0251\u0253\5\177@\2\u0252\u0251\3\2\2\2\u0253")
buf.write(u"\u0254\3\2\2\2\u0254\u0252\3\2\2\2\u0254\u0255\3\2\2")
buf.write(u"\2\u0255\u025d\3\2\2\2\u0256\u0257\7.\2\2\u0257\u0258")
buf.write(u"\5\177@\2\u0258\u0259\5\177@\2\u0259\u025a\5\177@\2\u025a")
buf.write(u"\u025c\3\2\2\2\u025b\u0256\3\2\2\2\u025c\u025f\3\2\2")
buf.write(u"\2\u025d\u025b\3\2\2\2\u025d\u025e\3\2\2\2\u025e\u0277")
buf.write(u"\3\2\2\2\u025f\u025d\3\2\2\2\u0260\u0262\5\177@\2\u0261")
buf.write(u"\u0260\3\2\2\2\u0262\u0265\3\2\2\2\u0263\u0261\3\2\2")
buf.write(u"\2\u0263\u0264\3\2\2\2\u0264\u026d\3\2\2\2\u0265\u0263")
buf.write(u"\3\2\2\2\u0266\u0267\7.\2\2\u0267\u0268\5\177@\2\u0268")
buf.write(u"\u0269\5\177@\2\u0269\u026a\5\177@\2\u026a\u026c\3\2")
buf.write(u"\2\2\u026b\u0266\3\2\2\2\u026c\u026f\3\2\2\2\u026d\u026b")
buf.write(u"\3\2\2\2\u026d\u026e\3\2\2\2\u026e\u0270\3\2\2\2\u026f")
buf.write(u"\u026d\3\2\2\2\u0270\u0272\7\60\2\2\u0271\u0273\5\177")
buf.write(u"@\2\u0272\u0271\3\2\2\2\u0273\u0274\3\2\2\2\u0274\u0272")
buf.write(u"\3\2\2\2\u0274\u0275\3\2\2\2\u0275\u0277\3\2\2\2\u0276")
buf.write(u"\u0252\3\2\2\2\u0276\u0263\3\2\2\2\u0277\u0082\3\2\2")
buf.write(u"\2\u0278\u0279\7?\2\2\u0279\u0084\3\2\2\2\u027a\u027b")
buf.write(u"\7>\2\2\u027b\u0086\3\2\2\2\u027c\u027d\7^\2\2\u027d")
buf.write(u"\u027e\7n\2\2\u027e\u027f\7g\2\2\u027f\u0280\7s\2\2\u0280")
buf.write(u"\u0088\3\2\2\2\u0281\u0282\7@\2\2\u0282\u008a\3\2\2\2")
buf.write(u"\u0283\u0284\7^\2\2\u0284\u0285\7i\2\2\u0285\u0286\7")
buf.write(u"g\2\2\u0286\u0287\7s\2\2\u0287\u008c\3\2\2\2\u0288\u0289")
buf.write(u"\7#\2\2\u0289\u008e\3\2\2\2\u028a\u028c\7^\2\2\u028b")
buf.write(u"\u028d\t\3\2\2\u028c\u028b\3\2\2\2\u028d\u028e\3\2\2")
buf.write(u"\2\u028e\u028c\3\2\2\2\u028e\u028f\3\2\2\2\u028f\u0090")
buf.write(u"\3\2\2\2\23\2\u0096\u00a6\u00b5\u00c6\u00ea\u0160\u0241")
buf.write(u"\u0249\u024b\u0254\u025d\u0263\u026d\u0274\u0276\u028e")
buf.write(u"\3\b\2\2")
return buf.getvalue()
class LaTeXLexer(Lexer):
atn = ATNDeserializer().deserialize(serializedATN())
decisionsToDFA = [ DFA(ds, i) for i, ds in enumerate(atn.decisionToState) ]
T__0 = 1
WS = 2
THINSPACE = 3
MEDSPACE = 4
THICKSPACE = 5
QUAD = 6
QQUAD = 7
NEGTHINSPACE = 8
NEGMEDSPACE = 9
NEGTHICKSPACE = 10
ADD = 11
SUB = 12
MUL = 13
DIV = 14
L_PAREN = 15
R_PAREN = 16
L_BRACE = 17
R_BRACE = 18
L_BRACKET = 19
R_BRACKET = 20
BAR = 21
FUNC_LIM = 22
LIM_APPROACH_SYM = 23
FUNC_INT = 24
FUNC_SUM = 25
FUNC_PROD = 26
FUNC_EXP = 27
FUNC_LOG = 28
FUNC_LN = 29
FUNC_SIN = 30
FUNC_COS = 31
FUNC_TAN = 32
FUNC_CSC = 33
FUNC_SEC = 34
FUNC_COT = 35
FUNC_ARCSIN = 36
FUNC_ARCCOS = 37
FUNC_ARCTAN = 38
FUNC_ARCCSC = 39
FUNC_ARCSEC = 40
FUNC_ARCCOT = 41
FUNC_SINH = 42
FUNC_COSH = 43
FUNC_TANH = 44
FUNC_ARSINH = 45
FUNC_ARCOSH = 46
FUNC_ARTANH = 47
FUNC_SQRT = 48
CMD_TIMES = 49
CMD_CDOT = 50
CMD_DIV = 51
CMD_FRAC = 52
CMD_BINOM = 53
CMD_DBINOM = 54
CMD_TBINOM = 55
CMD_MATHIT = 56
UNDERSCORE = 57
CARET = 58
COLON = 59
DIFFERENTIAL = 60
LETTER = 61
NUMBER = 62
EQUAL = 63
LT = 64
LTE = 65
GT = 66
GTE = 67
BANG = 68
SYMBOL = 69
channelNames = [ u"DEFAULT_TOKEN_CHANNEL", u"HIDDEN" ]
modeNames = [ u"DEFAULT_MODE" ]
literalNames = [ u"<INVALID>",
u"','", u"'\\quad'", u"'\\qquad'", u"'\\negmedspace'", u"'\\negthickspace'",
u"'+'", u"'-'", u"'*'", u"'/'", u"'('", u"')'", u"'{'", u"'}'",
u"'['", u"']'", u"'|'", u"'\\lim'", u"'\\int'", u"'\\sum'",
u"'\\prod'", u"'\\exp'", u"'\\log'", u"'\\ln'", u"'\\sin'",
u"'\\cos'", u"'\\tan'", u"'\\csc'", u"'\\sec'", u"'\\cot'",
u"'\\arcsin'", u"'\\arccos'", u"'\\arctan'", u"'\\arccsc'",
u"'\\arcsec'", u"'\\arccot'", u"'\\sinh'", u"'\\cosh'", u"'\\tanh'",
u"'\\arsinh'", u"'\\arcosh'", u"'\\artanh'", u"'\\sqrt'", u"'\\times'",
u"'\\cdot'", u"'\\div'", u"'\\frac'", u"'\\binom'", u"'\\dbinom'",
u"'\\tbinom'", u"'\\mathit'", u"'_'", u"'^'", u"':'", u"'='",
u"'<'", u"'\\leq'", u"'>'", u"'\\geq'", u"'!'" ]
symbolicNames = [ u"<INVALID>",
u"WS", u"THINSPACE", u"MEDSPACE", u"THICKSPACE", u"QUAD", u"QQUAD",
u"NEGTHINSPACE", u"NEGMEDSPACE", u"NEGTHICKSPACE", u"ADD", u"SUB",
u"MUL", u"DIV", u"L_PAREN", u"R_PAREN", u"L_BRACE", u"R_BRACE",
u"L_BRACKET", u"R_BRACKET", u"BAR", u"FUNC_LIM", u"LIM_APPROACH_SYM",
u"FUNC_INT", u"FUNC_SUM", u"FUNC_PROD", u"FUNC_EXP", u"FUNC_LOG",
u"FUNC_LN", u"FUNC_SIN", u"FUNC_COS", u"FUNC_TAN", u"FUNC_CSC",
u"FUNC_SEC", u"FUNC_COT", u"FUNC_ARCSIN", u"FUNC_ARCCOS", u"FUNC_ARCTAN",
u"FUNC_ARCCSC", u"FUNC_ARCSEC", u"FUNC_ARCCOT", u"FUNC_SINH",
u"FUNC_COSH", u"FUNC_TANH", u"FUNC_ARSINH", u"FUNC_ARCOSH",
u"FUNC_ARTANH", u"FUNC_SQRT", u"CMD_TIMES", u"CMD_CDOT", u"CMD_DIV",
u"CMD_FRAC", u"CMD_BINOM", u"CMD_DBINOM", u"CMD_TBINOM", u"CMD_MATHIT",
u"UNDERSCORE", u"CARET", u"COLON", u"DIFFERENTIAL", u"LETTER",
u"NUMBER", u"EQUAL", u"LT", u"LTE", u"GT", u"GTE", u"BANG",
u"SYMBOL" ]
ruleNames = [ u"T__0", u"WS", u"THINSPACE", u"MEDSPACE", u"THICKSPACE",
u"QUAD", u"QQUAD", u"NEGTHINSPACE", u"NEGMEDSPACE", u"NEGTHICKSPACE",
u"ADD", u"SUB", u"MUL", u"DIV", u"L_PAREN", u"R_PAREN",
u"L_BRACE", u"R_BRACE", u"L_BRACKET", u"R_BRACKET", u"BAR",
u"FUNC_LIM", u"LIM_APPROACH_SYM", u"FUNC_INT", u"FUNC_SUM",
u"FUNC_PROD", u"FUNC_EXP", u"FUNC_LOG", u"FUNC_LN", u"FUNC_SIN",
u"FUNC_COS", u"FUNC_TAN", u"FUNC_CSC", u"FUNC_SEC", u"FUNC_COT",
u"FUNC_ARCSIN", u"FUNC_ARCCOS", u"FUNC_ARCTAN", u"FUNC_ARCCSC",
u"FUNC_ARCSEC", u"FUNC_ARCCOT", u"FUNC_SINH", u"FUNC_COSH",
u"FUNC_TANH", u"FUNC_ARSINH", u"FUNC_ARCOSH", u"FUNC_ARTANH",
u"FUNC_SQRT", u"CMD_TIMES", u"CMD_CDOT", u"CMD_DIV", u"CMD_FRAC",
u"CMD_BINOM", u"CMD_DBINOM", u"CMD_TBINOM", u"CMD_MATHIT",
u"UNDERSCORE", u"CARET", u"COLON", u"WS_CHAR", u"DIFFERENTIAL",
u"LETTER", u"DIGIT", u"NUMBER", u"EQUAL", u"LT", u"LTE",
u"GT", u"GTE", u"BANG", u"SYMBOL" ]
grammarFileName = u"LaTeX.g4"
def __init__(self, input=None, output=sys.stdout):
super(LaTeXLexer, self).__init__(input, output=output)
self.checkVersion("4.7.1")
self._interp = LexerATNSimulator(self, self.atn, self.decisionsToDFA, PredictionContextCache())
self._actions = None
self._predicates = None
|
2730760ebcf83a399b55f450991d96d537454af15e6e081d73dc28162d9fcd9e |
# encoding: utf-8
# *** GENERATED BY `setup.py antlr`, DO NOT EDIT BY HAND ***
#
# Generated from ../LaTeX.g4, derived from latex2sympy
# latex2sympy is licensed under the MIT license
# https://github.com/augustt198/latex2sympy/blob/master/LICENSE.txt
#
# Generated with antlr4
# antlr4 is licensed under the BSD-3-Clause License
# https://github.com/antlr/antlr4/blob/master/LICENSE.txt
from __future__ import print_function
from antlr4 import *
from io import StringIO
import sys
def serializedATN():
with StringIO() as buf:
buf.write(u"\3\u608b\ua72a\u8133\ub9ed\u417c\u3be7\u7786\u5964\3")
buf.write(u"G\u01ae\4\2\t\2\4\3\t\3\4\4\t\4\4\5\t\5\4\6\t\6\4\7\t")
buf.write(u"\7\4\b\t\b\4\t\t\t\4\n\t\n\4\13\t\13\4\f\t\f\4\r\t\r")
buf.write(u"\4\16\t\16\4\17\t\17\4\20\t\20\4\21\t\21\4\22\t\22\4")
buf.write(u"\23\t\23\4\24\t\24\4\25\t\25\4\26\t\26\4\27\t\27\4\30")
buf.write(u"\t\30\4\31\t\31\4\32\t\32\4\33\t\33\4\34\t\34\4\35\t")
buf.write(u"\35\4\36\t\36\4\37\t\37\4 \t \4!\t!\4\"\t\"\4#\t#\4$")
buf.write(u"\t$\4%\t%\3\2\3\2\3\3\3\3\3\3\3\3\3\3\3\3\7\3S\n\3\f")
buf.write(u"\3\16\3V\13\3\3\4\3\4\3\4\3\4\3\5\3\5\3\6\3\6\3\6\3\6")
buf.write(u"\3\6\3\6\7\6d\n\6\f\6\16\6g\13\6\3\7\3\7\3\7\3\7\3\7")
buf.write(u"\3\7\7\7o\n\7\f\7\16\7r\13\7\3\b\3\b\3\b\3\b\3\b\3\b")
buf.write(u"\7\bz\n\b\f\b\16\b}\13\b\3\t\3\t\3\t\6\t\u0082\n\t\r")
buf.write(u"\t\16\t\u0083\5\t\u0086\n\t\3\n\3\n\3\n\3\n\7\n\u008c")
buf.write(u"\n\n\f\n\16\n\u008f\13\n\5\n\u0091\n\n\3\13\3\13\7\13")
buf.write(u"\u0095\n\13\f\13\16\13\u0098\13\13\3\f\3\f\7\f\u009c")
buf.write(u"\n\f\f\f\16\f\u009f\13\f\3\r\3\r\5\r\u00a3\n\r\3\16\3")
buf.write(u"\16\3\16\3\16\3\16\3\16\5\16\u00ab\n\16\3\17\3\17\3\17")
buf.write(u"\3\17\5\17\u00b1\n\17\3\17\3\17\3\20\3\20\3\20\3\20\5")
buf.write(u"\20\u00b9\n\20\3\20\3\20\3\21\3\21\3\21\3\21\3\21\3\21")
buf.write(u"\3\21\3\21\3\21\3\21\5\21\u00c7\n\21\3\21\5\21\u00ca")
buf.write(u"\n\21\7\21\u00cc\n\21\f\21\16\21\u00cf\13\21\3\22\3\22")
buf.write(u"\3\22\3\22\3\22\3\22\3\22\3\22\3\22\3\22\5\22\u00db\n")
buf.write(u"\22\3\22\5\22\u00de\n\22\7\22\u00e0\n\22\f\22\16\22\u00e3")
buf.write(u"\13\22\3\23\3\23\3\23\3\23\3\23\3\23\5\23\u00eb\n\23")
buf.write(u"\3\24\3\24\3\24\3\24\3\24\5\24\u00f2\n\24\3\25\3\25\3")
buf.write(u"\25\3\25\3\25\3\25\3\25\3\25\3\25\3\25\3\25\3\25\5\25")
buf.write(u"\u0100\n\25\3\26\3\26\3\26\3\26\3\27\3\27\5\27\u0108")
buf.write(u"\n\27\3\27\3\27\3\27\5\27\u010d\n\27\3\30\3\30\3\30\3")
buf.write(u"\30\3\30\3\31\7\31\u0115\n\31\f\31\16\31\u0118\13\31")
buf.write(u"\3\32\3\32\3\32\3\32\3\32\3\32\3\32\3\32\3\33\3\33\3")
buf.write(u"\33\3\33\3\33\3\33\3\33\3\33\3\34\3\34\3\35\3\35\5\35")
buf.write(u"\u012e\n\35\3\35\5\35\u0131\n\35\3\35\5\35\u0134\n\35")
buf.write(u"\3\35\5\35\u0137\n\35\5\35\u0139\n\35\3\35\3\35\3\35")
buf.write(u"\3\35\3\35\5\35\u0140\n\35\3\35\3\35\5\35\u0144\n\35")
buf.write(u"\3\35\3\35\3\35\3\35\3\35\3\35\3\35\3\35\3\35\3\35\3")
buf.write(u"\35\5\35\u0151\n\35\3\35\5\35\u0154\n\35\3\35\3\35\3")
buf.write(u"\35\5\35\u0159\n\35\3\35\3\35\3\35\3\35\3\35\5\35\u0160")
buf.write(u"\n\35\3\35\3\35\3\35\3\35\3\35\3\35\3\35\3\35\3\35\3")
buf.write(u"\35\3\35\5\35\u016d\n\35\3\35\3\35\3\35\3\35\3\35\3\35")
buf.write(u"\5\35\u0175\n\35\3\36\3\36\3\36\3\36\3\36\5\36\u017c")
buf.write(u"\n\36\3\37\3\37\3\37\3\37\3\37\3\37\3\37\3\37\3\37\5")
buf.write(u"\37\u0187\n\37\3\37\3\37\3 \3 \3 \3 \3 \5 \u0190\n \3")
buf.write(u"!\3!\3\"\3\"\3\"\3\"\3\"\3\"\5\"\u019a\n\"\3#\3#\3#\3")
buf.write(u"#\3#\3#\5#\u01a2\n#\3$\3$\3$\3$\3$\3%\3%\3%\3%\3%\3%")
buf.write(u"\2\b\4\n\f\16 \"&\2\4\6\b\n\f\16\20\22\24\26\30\32\34")
buf.write(u"\36 \"$&(*,.\60\62\64\668:<>@BDFH\2\t\3\2AE\3\2\r\16")
buf.write(u"\5\2\17\20\63\65==\4\2??GG\3\2\679\3\2\35\61\3\2\33\34")
buf.write(u"\2\u01c6\2J\3\2\2\2\4L\3\2\2\2\6W\3\2\2\2\b[\3\2\2\2")
buf.write(u"\n]\3\2\2\2\fh\3\2\2\2\16s\3\2\2\2\20\u0085\3\2\2\2\22")
buf.write(u"\u0090\3\2\2\2\24\u0092\3\2\2\2\26\u0099\3\2\2\2\30\u00a2")
buf.write(u"\3\2\2\2\32\u00a4\3\2\2\2\34\u00ac\3\2\2\2\36\u00b4\3")
buf.write(u"\2\2\2 \u00bc\3\2\2\2\"\u00d0\3\2\2\2$\u00ea\3\2\2\2")
buf.write(u"&\u00f1\3\2\2\2(\u00ff\3\2\2\2*\u0101\3\2\2\2,\u010c")
buf.write(u"\3\2\2\2.\u010e\3\2\2\2\60\u0116\3\2\2\2\62\u0119\3\2")
buf.write(u"\2\2\64\u0121\3\2\2\2\66\u0129\3\2\2\28\u0174\3\2\2\2")
buf.write(u":\u017b\3\2\2\2<\u017d\3\2\2\2>\u018f\3\2\2\2@\u0191")
buf.write(u"\3\2\2\2B\u0193\3\2\2\2D\u019b\3\2\2\2F\u01a3\3\2\2\2")
buf.write(u"H\u01a8\3\2\2\2JK\5\4\3\2K\3\3\2\2\2LM\b\3\1\2MN\5\b")
buf.write(u"\5\2NT\3\2\2\2OP\f\4\2\2PQ\t\2\2\2QS\5\4\3\5RO\3\2\2")
buf.write(u"\2SV\3\2\2\2TR\3\2\2\2TU\3\2\2\2U\5\3\2\2\2VT\3\2\2\2")
buf.write(u"WX\5\b\5\2XY\7A\2\2YZ\5\b\5\2Z\7\3\2\2\2[\\\5\n\6\2\\")
buf.write(u"\t\3\2\2\2]^\b\6\1\2^_\5\f\7\2_e\3\2\2\2`a\f\4\2\2ab")
buf.write(u"\t\3\2\2bd\5\n\6\5c`\3\2\2\2dg\3\2\2\2ec\3\2\2\2ef\3")
buf.write(u"\2\2\2f\13\3\2\2\2ge\3\2\2\2hi\b\7\1\2ij\5\20\t\2jp\3")
buf.write(u"\2\2\2kl\f\4\2\2lm\t\4\2\2mo\5\f\7\5nk\3\2\2\2or\3\2")
buf.write(u"\2\2pn\3\2\2\2pq\3\2\2\2q\r\3\2\2\2rp\3\2\2\2st\b\b\1")
buf.write(u"\2tu\5\22\n\2u{\3\2\2\2vw\f\4\2\2wx\t\4\2\2xz\5\16\b")
buf.write(u"\5yv\3\2\2\2z}\3\2\2\2{y\3\2\2\2{|\3\2\2\2|\17\3\2\2")
buf.write(u"\2}{\3\2\2\2~\177\t\3\2\2\177\u0086\5\20\t\2\u0080\u0082")
buf.write(u"\5\24\13\2\u0081\u0080\3\2\2\2\u0082\u0083\3\2\2\2\u0083")
buf.write(u"\u0081\3\2\2\2\u0083\u0084\3\2\2\2\u0084\u0086\3\2\2")
buf.write(u"\2\u0085~\3\2\2\2\u0085\u0081\3\2\2\2\u0086\21\3\2\2")
buf.write(u"\2\u0087\u0088\t\3\2\2\u0088\u0091\5\22\n\2\u0089\u008d")
buf.write(u"\5\24\13\2\u008a\u008c\5\26\f\2\u008b\u008a\3\2\2\2\u008c")
buf.write(u"\u008f\3\2\2\2\u008d\u008b\3\2\2\2\u008d\u008e\3\2\2")
buf.write(u"\2\u008e\u0091\3\2\2\2\u008f\u008d\3\2\2\2\u0090\u0087")
buf.write(u"\3\2\2\2\u0090\u0089\3\2\2\2\u0091\23\3\2\2\2\u0092\u0096")
buf.write(u"\5 \21\2\u0093\u0095\5\30\r\2\u0094\u0093\3\2\2\2\u0095")
buf.write(u"\u0098\3\2\2\2\u0096\u0094\3\2\2\2\u0096\u0097\3\2\2")
buf.write(u"\2\u0097\25\3\2\2\2\u0098\u0096\3\2\2\2\u0099\u009d\5")
buf.write(u"\"\22\2\u009a\u009c\5\30\r\2\u009b\u009a\3\2\2\2\u009c")
buf.write(u"\u009f\3\2\2\2\u009d\u009b\3\2\2\2\u009d\u009e\3\2\2")
buf.write(u"\2\u009e\27\3\2\2\2\u009f\u009d\3\2\2\2\u00a0\u00a3\7")
buf.write(u"F\2\2\u00a1\u00a3\5\32\16\2\u00a2\u00a0\3\2\2\2\u00a2")
buf.write(u"\u00a1\3\2\2\2\u00a3\31\3\2\2\2\u00a4\u00aa\7\27\2\2")
buf.write(u"\u00a5\u00ab\5\36\20\2\u00a6\u00ab\5\34\17\2\u00a7\u00a8")
buf.write(u"\5\36\20\2\u00a8\u00a9\5\34\17\2\u00a9\u00ab\3\2\2\2")
buf.write(u"\u00aa\u00a5\3\2\2\2\u00aa\u00a6\3\2\2\2\u00aa\u00a7")
buf.write(u"\3\2\2\2\u00ab\33\3\2\2\2\u00ac\u00ad\7;\2\2\u00ad\u00b0")
buf.write(u"\7\23\2\2\u00ae\u00b1\5\b\5\2\u00af\u00b1\5\6\4\2\u00b0")
buf.write(u"\u00ae\3\2\2\2\u00b0\u00af\3\2\2\2\u00b1\u00b2\3\2\2")
buf.write(u"\2\u00b2\u00b3\7\24\2\2\u00b3\35\3\2\2\2\u00b4\u00b5")
buf.write(u"\7<\2\2\u00b5\u00b8\7\23\2\2\u00b6\u00b9\5\b\5\2\u00b7")
buf.write(u"\u00b9\5\6\4\2\u00b8\u00b6\3\2\2\2\u00b8\u00b7\3\2\2")
buf.write(u"\2\u00b9\u00ba\3\2\2\2\u00ba\u00bb\7\24\2\2\u00bb\37")
buf.write(u"\3\2\2\2\u00bc\u00bd\b\21\1\2\u00bd\u00be\5$\23\2\u00be")
buf.write(u"\u00cd\3\2\2\2\u00bf\u00c0\f\4\2\2\u00c0\u00c6\7<\2\2")
buf.write(u"\u00c1\u00c7\5,\27\2\u00c2\u00c3\7\23\2\2\u00c3\u00c4")
buf.write(u"\5\b\5\2\u00c4\u00c5\7\24\2\2\u00c5\u00c7\3\2\2\2\u00c6")
buf.write(u"\u00c1\3\2\2\2\u00c6\u00c2\3\2\2\2\u00c7\u00c9\3\2\2")
buf.write(u"\2\u00c8\u00ca\5B\"\2\u00c9\u00c8\3\2\2\2\u00c9\u00ca")
buf.write(u"\3\2\2\2\u00ca\u00cc\3\2\2\2\u00cb\u00bf\3\2\2\2\u00cc")
buf.write(u"\u00cf\3\2\2\2\u00cd\u00cb\3\2\2\2\u00cd\u00ce\3\2\2")
buf.write(u"\2\u00ce!\3\2\2\2\u00cf\u00cd\3\2\2\2\u00d0\u00d1\b\22")
buf.write(u"\1\2\u00d1\u00d2\5&\24\2\u00d2\u00e1\3\2\2\2\u00d3\u00d4")
buf.write(u"\f\4\2\2\u00d4\u00da\7<\2\2\u00d5\u00db\5,\27\2\u00d6")
buf.write(u"\u00d7\7\23\2\2\u00d7\u00d8\5\b\5\2\u00d8\u00d9\7\24")
buf.write(u"\2\2\u00d9\u00db\3\2\2\2\u00da\u00d5\3\2\2\2\u00da\u00d6")
buf.write(u"\3\2\2\2\u00db\u00dd\3\2\2\2\u00dc\u00de\5B\"\2\u00dd")
buf.write(u"\u00dc\3\2\2\2\u00dd\u00de\3\2\2\2\u00de\u00e0\3\2\2")
buf.write(u"\2\u00df\u00d3\3\2\2\2\u00e0\u00e3\3\2\2\2\u00e1\u00df")
buf.write(u"\3\2\2\2\u00e1\u00e2\3\2\2\2\u00e2#\3\2\2\2\u00e3\u00e1")
buf.write(u"\3\2\2\2\u00e4\u00eb\5(\25\2\u00e5\u00eb\5*\26\2\u00e6")
buf.write(u"\u00eb\58\35\2\u00e7\u00eb\5,\27\2\u00e8\u00eb\5\62\32")
buf.write(u"\2\u00e9\u00eb\5\64\33\2\u00ea\u00e4\3\2\2\2\u00ea\u00e5")
buf.write(u"\3\2\2\2\u00ea\u00e6\3\2\2\2\u00ea\u00e7\3\2\2\2\u00ea")
buf.write(u"\u00e8\3\2\2\2\u00ea\u00e9\3\2\2\2\u00eb%\3\2\2\2\u00ec")
buf.write(u"\u00f2\5(\25\2\u00ed\u00f2\5*\26\2\u00ee\u00f2\5,\27")
buf.write(u"\2\u00ef\u00f2\5\62\32\2\u00f0\u00f2\5\64\33\2\u00f1")
buf.write(u"\u00ec\3\2\2\2\u00f1\u00ed\3\2\2\2\u00f1\u00ee\3\2\2")
buf.write(u"\2\u00f1\u00ef\3\2\2\2\u00f1\u00f0\3\2\2\2\u00f2\'\3")
buf.write(u"\2\2\2\u00f3\u00f4\7\21\2\2\u00f4\u00f5\5\b\5\2\u00f5")
buf.write(u"\u00f6\7\22\2\2\u00f6\u0100\3\2\2\2\u00f7\u00f8\7\25")
buf.write(u"\2\2\u00f8\u00f9\5\b\5\2\u00f9\u00fa\7\26\2\2\u00fa\u0100")
buf.write(u"\3\2\2\2\u00fb\u00fc\7\23\2\2\u00fc\u00fd\5\b\5\2\u00fd")
buf.write(u"\u00fe\7\24\2\2\u00fe\u0100\3\2\2\2\u00ff\u00f3\3\2\2")
buf.write(u"\2\u00ff\u00f7\3\2\2\2\u00ff\u00fb\3\2\2\2\u0100)\3\2")
buf.write(u"\2\2\u0101\u0102\7\27\2\2\u0102\u0103\5\b\5\2\u0103\u0104")
buf.write(u"\7\27\2\2\u0104+\3\2\2\2\u0105\u0107\t\5\2\2\u0106\u0108")
buf.write(u"\5B\"\2\u0107\u0106\3\2\2\2\u0107\u0108\3\2\2\2\u0108")
buf.write(u"\u010d\3\2\2\2\u0109\u010d\7@\2\2\u010a\u010d\7>\2\2")
buf.write(u"\u010b\u010d\5.\30\2\u010c\u0105\3\2\2\2\u010c\u0109")
buf.write(u"\3\2\2\2\u010c\u010a\3\2\2\2\u010c\u010b\3\2\2\2\u010d")
buf.write(u"-\3\2\2\2\u010e\u010f\7:\2\2\u010f\u0110\7\23\2\2\u0110")
buf.write(u"\u0111\5\60\31\2\u0111\u0112\7\24\2\2\u0112/\3\2\2\2")
buf.write(u"\u0113\u0115\7?\2\2\u0114\u0113\3\2\2\2\u0115\u0118\3")
buf.write(u"\2\2\2\u0116\u0114\3\2\2\2\u0116\u0117\3\2\2\2\u0117")
buf.write(u"\61\3\2\2\2\u0118\u0116\3\2\2\2\u0119\u011a\7\66\2\2")
buf.write(u"\u011a\u011b\7\23\2\2\u011b\u011c\5\b\5\2\u011c\u011d")
buf.write(u"\7\24\2\2\u011d\u011e\7\23\2\2\u011e\u011f\5\b\5\2\u011f")
buf.write(u"\u0120\7\24\2\2\u0120\63\3\2\2\2\u0121\u0122\t\6\2\2")
buf.write(u"\u0122\u0123\7\23\2\2\u0123\u0124\5\b\5\2\u0124\u0125")
buf.write(u"\7\24\2\2\u0125\u0126\7\23\2\2\u0126\u0127\5\b\5\2\u0127")
buf.write(u"\u0128\7\24\2\2\u0128\65\3\2\2\2\u0129\u012a\t\7\2\2")
buf.write(u"\u012a\67\3\2\2\2\u012b\u0138\5\66\34\2\u012c\u012e\5")
buf.write(u"B\"\2\u012d\u012c\3\2\2\2\u012d\u012e\3\2\2\2\u012e\u0130")
buf.write(u"\3\2\2\2\u012f\u0131\5D#\2\u0130\u012f\3\2\2\2\u0130")
buf.write(u"\u0131\3\2\2\2\u0131\u0139\3\2\2\2\u0132\u0134\5D#\2")
buf.write(u"\u0133\u0132\3\2\2\2\u0133\u0134\3\2\2\2\u0134\u0136")
buf.write(u"\3\2\2\2\u0135\u0137\5B\"\2\u0136\u0135\3\2\2\2\u0136")
buf.write(u"\u0137\3\2\2\2\u0137\u0139\3\2\2\2\u0138\u012d\3\2\2")
buf.write(u"\2\u0138\u0133\3\2\2\2\u0139\u013f\3\2\2\2\u013a\u013b")
buf.write(u"\7\21\2\2\u013b\u013c\5> \2\u013c\u013d\7\22\2\2\u013d")
buf.write(u"\u0140\3\2\2\2\u013e\u0140\5@!\2\u013f\u013a\3\2\2\2")
buf.write(u"\u013f\u013e\3\2\2\2\u0140\u0175\3\2\2\2\u0141\u0143")
buf.write(u"\t\5\2\2\u0142\u0144\5B\"\2\u0143\u0142\3\2\2\2\u0143")
buf.write(u"\u0144\3\2\2\2\u0144\u0145\3\2\2\2\u0145\u0146\7\21\2")
buf.write(u"\2\u0146\u0147\5:\36\2\u0147\u0148\7\22\2\2\u0148\u0175")
buf.write(u"\3\2\2\2\u0149\u0150\7\32\2\2\u014a\u014b\5B\"\2\u014b")
buf.write(u"\u014c\5D#\2\u014c\u0151\3\2\2\2\u014d\u014e\5D#\2\u014e")
buf.write(u"\u014f\5B\"\2\u014f\u0151\3\2\2\2\u0150\u014a\3\2\2\2")
buf.write(u"\u0150\u014d\3\2\2\2\u0150\u0151\3\2\2\2\u0151\u0158")
buf.write(u"\3\2\2\2\u0152\u0154\5\n\6\2\u0153\u0152\3\2\2\2\u0153")
buf.write(u"\u0154\3\2\2\2\u0154\u0155\3\2\2\2\u0155\u0159\7>\2\2")
buf.write(u"\u0156\u0159\5\62\32\2\u0157\u0159\5\n\6\2\u0158\u0153")
buf.write(u"\3\2\2\2\u0158\u0156\3\2\2\2\u0158\u0157\3\2\2\2\u0159")
buf.write(u"\u0175\3\2\2\2\u015a\u015f\7\62\2\2\u015b\u015c\7\25")
buf.write(u"\2\2\u015c\u015d\5\b\5\2\u015d\u015e\7\26\2\2\u015e\u0160")
buf.write(u"\3\2\2\2\u015f\u015b\3\2\2\2\u015f\u0160\3\2\2\2\u0160")
buf.write(u"\u0161\3\2\2\2\u0161\u0162\7\23\2\2\u0162\u0163\5\b\5")
buf.write(u"\2\u0163\u0164\7\24\2\2\u0164\u0175\3\2\2\2\u0165\u016c")
buf.write(u"\t\b\2\2\u0166\u0167\5F$\2\u0167\u0168\5D#\2\u0168\u016d")
buf.write(u"\3\2\2\2\u0169\u016a\5D#\2\u016a\u016b\5F$\2\u016b\u016d")
buf.write(u"\3\2\2\2\u016c\u0166\3\2\2\2\u016c\u0169\3\2\2\2\u016d")
buf.write(u"\u016e\3\2\2\2\u016e\u016f\5\f\7\2\u016f\u0175\3\2\2")
buf.write(u"\2\u0170\u0171\7\30\2\2\u0171\u0172\5<\37\2\u0172\u0173")
buf.write(u"\5\f\7\2\u0173\u0175\3\2\2\2\u0174\u012b\3\2\2\2\u0174")
buf.write(u"\u0141\3\2\2\2\u0174\u0149\3\2\2\2\u0174\u015a\3\2\2")
buf.write(u"\2\u0174\u0165\3\2\2\2\u0174\u0170\3\2\2\2\u01759\3\2")
buf.write(u"\2\2\u0176\u0177\5\b\5\2\u0177\u0178\7\3\2\2\u0178\u0179")
buf.write(u"\5:\36\2\u0179\u017c\3\2\2\2\u017a\u017c\5\b\5\2\u017b")
buf.write(u"\u0176\3\2\2\2\u017b\u017a\3\2\2\2\u017c;\3\2\2\2\u017d")
buf.write(u"\u017e\7;\2\2\u017e\u017f\7\23\2\2\u017f\u0180\t\5\2")
buf.write(u"\2\u0180\u0181\7\31\2\2\u0181\u0186\5\b\5\2\u0182\u0183")
buf.write(u"\7<\2\2\u0183\u0184\7\23\2\2\u0184\u0185\t\3\2\2\u0185")
buf.write(u"\u0187\7\24\2\2\u0186\u0182\3\2\2\2\u0186\u0187\3\2\2")
buf.write(u"\2\u0187\u0188\3\2\2\2\u0188\u0189\7\24\2\2\u0189=\3")
buf.write(u"\2\2\2\u018a\u0190\5\b\5\2\u018b\u018c\5\b\5\2\u018c")
buf.write(u"\u018d\7\3\2\2\u018d\u018e\5> \2\u018e\u0190\3\2\2\2")
buf.write(u"\u018f\u018a\3\2\2\2\u018f\u018b\3\2\2\2\u0190?\3\2\2")
buf.write(u"\2\u0191\u0192\5\16\b\2\u0192A\3\2\2\2\u0193\u0199\7")
buf.write(u";\2\2\u0194\u019a\5,\27\2\u0195\u0196\7\23\2\2\u0196")
buf.write(u"\u0197\5\b\5\2\u0197\u0198\7\24\2\2\u0198\u019a\3\2\2")
buf.write(u"\2\u0199\u0194\3\2\2\2\u0199\u0195\3\2\2\2\u019aC\3\2")
buf.write(u"\2\2\u019b\u01a1\7<\2\2\u019c\u01a2\5,\27\2\u019d\u019e")
buf.write(u"\7\23\2\2\u019e\u019f\5\b\5\2\u019f\u01a0\7\24\2\2\u01a0")
buf.write(u"\u01a2\3\2\2\2\u01a1\u019c\3\2\2\2\u01a1\u019d\3\2\2")
buf.write(u"\2\u01a2E\3\2\2\2\u01a3\u01a4\7;\2\2\u01a4\u01a5\7\23")
buf.write(u"\2\2\u01a5\u01a6\5\6\4\2\u01a6\u01a7\7\24\2\2\u01a7G")
buf.write(u"\3\2\2\2\u01a8\u01a9\7;\2\2\u01a9\u01aa\7\23\2\2\u01aa")
buf.write(u"\u01ab\5\6\4\2\u01ab\u01ac\7\24\2\2\u01acI\3\2\2\2.T")
buf.write(u"ep{\u0083\u0085\u008d\u0090\u0096\u009d\u00a2\u00aa\u00b0")
buf.write(u"\u00b8\u00c6\u00c9\u00cd\u00da\u00dd\u00e1\u00ea\u00f1")
buf.write(u"\u00ff\u0107\u010c\u0116\u012d\u0130\u0133\u0136\u0138")
buf.write(u"\u013f\u0143\u0150\u0153\u0158\u015f\u016c\u0174\u017b")
buf.write(u"\u0186\u018f\u0199\u01a1")
return buf.getvalue()
class LaTeXParser ( Parser ):
grammarFileName = "LaTeX.g4"
atn = ATNDeserializer().deserialize(serializedATN())
decisionsToDFA = [ DFA(ds, i) for i, ds in enumerate(atn.decisionToState) ]
sharedContextCache = PredictionContextCache()
literalNames = [ u"<INVALID>", u"','", u"<INVALID>", u"<INVALID>", u"<INVALID>",
u"<INVALID>", u"'\\quad'", u"'\\qquad'", u"<INVALID>",
u"'\\negmedspace'", u"'\\negthickspace'", u"'+'", u"'-'",
u"'*'", u"'/'", u"'('", u"')'", u"'{'", u"'}'", u"'['",
u"']'", u"'|'", u"'\\lim'", u"<INVALID>", u"'\\int'",
u"'\\sum'", u"'\\prod'", u"'\\exp'", u"'\\log'", u"'\\ln'",
u"'\\sin'", u"'\\cos'", u"'\\tan'", u"'\\csc'", u"'\\sec'",
u"'\\cot'", u"'\\arcsin'", u"'\\arccos'", u"'\\arctan'",
u"'\\arccsc'", u"'\\arcsec'", u"'\\arccot'", u"'\\sinh'",
u"'\\cosh'", u"'\\tanh'", u"'\\arsinh'", u"'\\arcosh'",
u"'\\artanh'", u"'\\sqrt'", u"'\\times'", u"'\\cdot'",
u"'\\div'", u"'\\frac'", u"'\\binom'", u"'\\dbinom'",
u"'\\tbinom'", u"'\\mathit'", u"'_'", u"'^'", u"':'",
u"<INVALID>", u"<INVALID>", u"<INVALID>", u"'='", u"'<'",
u"'\\leq'", u"'>'", u"'\\geq'", u"'!'" ]
symbolicNames = [ u"<INVALID>", u"<INVALID>", u"WS", u"THINSPACE", u"MEDSPACE",
u"THICKSPACE", u"QUAD", u"QQUAD", u"NEGTHINSPACE",
u"NEGMEDSPACE", u"NEGTHICKSPACE", u"ADD", u"SUB",
u"MUL", u"DIV", u"L_PAREN", u"R_PAREN", u"L_BRACE",
u"R_BRACE", u"L_BRACKET", u"R_BRACKET", u"BAR", u"FUNC_LIM",
u"LIM_APPROACH_SYM", u"FUNC_INT", u"FUNC_SUM", u"FUNC_PROD",
u"FUNC_EXP", u"FUNC_LOG", u"FUNC_LN", u"FUNC_SIN",
u"FUNC_COS", u"FUNC_TAN", u"FUNC_CSC", u"FUNC_SEC",
u"FUNC_COT", u"FUNC_ARCSIN", u"FUNC_ARCCOS", u"FUNC_ARCTAN",
u"FUNC_ARCCSC", u"FUNC_ARCSEC", u"FUNC_ARCCOT", u"FUNC_SINH",
u"FUNC_COSH", u"FUNC_TANH", u"FUNC_ARSINH", u"FUNC_ARCOSH",
u"FUNC_ARTANH", u"FUNC_SQRT", u"CMD_TIMES", u"CMD_CDOT",
u"CMD_DIV", u"CMD_FRAC", u"CMD_BINOM", u"CMD_DBINOM",
u"CMD_TBINOM", u"CMD_MATHIT", u"UNDERSCORE", u"CARET",
u"COLON", u"DIFFERENTIAL", u"LETTER", u"NUMBER", u"EQUAL",
u"LT", u"LTE", u"GT", u"GTE", u"BANG", u"SYMBOL" ]
RULE_math = 0
RULE_relation = 1
RULE_equality = 2
RULE_expr = 3
RULE_additive = 4
RULE_mp = 5
RULE_mp_nofunc = 6
RULE_unary = 7
RULE_unary_nofunc = 8
RULE_postfix = 9
RULE_postfix_nofunc = 10
RULE_postfix_op = 11
RULE_eval_at = 12
RULE_eval_at_sub = 13
RULE_eval_at_sup = 14
RULE_exp = 15
RULE_exp_nofunc = 16
RULE_comp = 17
RULE_comp_nofunc = 18
RULE_group = 19
RULE_abs_group = 20
RULE_atom = 21
RULE_mathit = 22
RULE_mathit_text = 23
RULE_frac = 24
RULE_binom = 25
RULE_func_normal = 26
RULE_func = 27
RULE_args = 28
RULE_limit_sub = 29
RULE_func_arg = 30
RULE_func_arg_noparens = 31
RULE_subexpr = 32
RULE_supexpr = 33
RULE_subeq = 34
RULE_supeq = 35
ruleNames = [ u"math", u"relation", u"equality", u"expr", u"additive",
u"mp", u"mp_nofunc", u"unary", u"unary_nofunc", u"postfix",
u"postfix_nofunc", u"postfix_op", u"eval_at", u"eval_at_sub",
u"eval_at_sup", u"exp", u"exp_nofunc", u"comp", u"comp_nofunc",
u"group", u"abs_group", u"atom", u"mathit", u"mathit_text",
u"frac", u"binom", u"func_normal", u"func", u"args",
u"limit_sub", u"func_arg", u"func_arg_noparens", u"subexpr",
u"supexpr", u"subeq", u"supeq" ]
EOF = Token.EOF
T__0=1
WS=2
THINSPACE=3
MEDSPACE=4
THICKSPACE=5
QUAD=6
QQUAD=7
NEGTHINSPACE=8
NEGMEDSPACE=9
NEGTHICKSPACE=10
ADD=11
SUB=12
MUL=13
DIV=14
L_PAREN=15
R_PAREN=16
L_BRACE=17
R_BRACE=18
L_BRACKET=19
R_BRACKET=20
BAR=21
FUNC_LIM=22
LIM_APPROACH_SYM=23
FUNC_INT=24
FUNC_SUM=25
FUNC_PROD=26
FUNC_EXP=27
FUNC_LOG=28
FUNC_LN=29
FUNC_SIN=30
FUNC_COS=31
FUNC_TAN=32
FUNC_CSC=33
FUNC_SEC=34
FUNC_COT=35
FUNC_ARCSIN=36
FUNC_ARCCOS=37
FUNC_ARCTAN=38
FUNC_ARCCSC=39
FUNC_ARCSEC=40
FUNC_ARCCOT=41
FUNC_SINH=42
FUNC_COSH=43
FUNC_TANH=44
FUNC_ARSINH=45
FUNC_ARCOSH=46
FUNC_ARTANH=47
FUNC_SQRT=48
CMD_TIMES=49
CMD_CDOT=50
CMD_DIV=51
CMD_FRAC=52
CMD_BINOM=53
CMD_DBINOM=54
CMD_TBINOM=55
CMD_MATHIT=56
UNDERSCORE=57
CARET=58
COLON=59
DIFFERENTIAL=60
LETTER=61
NUMBER=62
EQUAL=63
LT=64
LTE=65
GT=66
GTE=67
BANG=68
SYMBOL=69
def __init__(self, input, output=sys.stdout):
super(LaTeXParser, self).__init__(input, output=output)
self.checkVersion("4.7.1")
self._interp = ParserATNSimulator(self, self.atn, self.decisionsToDFA, self.sharedContextCache)
self._predicates = None
class MathContext(ParserRuleContext):
def __init__(self, parser, parent=None, invokingState=-1):
super(LaTeXParser.MathContext, self).__init__(parent, invokingState)
self.parser = parser
def relation(self):
return self.getTypedRuleContext(LaTeXParser.RelationContext,0)
def getRuleIndex(self):
return LaTeXParser.RULE_math
def math(self):
localctx = LaTeXParser.MathContext(self, self._ctx, self.state)
self.enterRule(localctx, 0, self.RULE_math)
try:
self.enterOuterAlt(localctx, 1)
self.state = 72
self.relation(0)
except RecognitionException as re:
localctx.exception = re
self._errHandler.reportError(self, re)
self._errHandler.recover(self, re)
finally:
self.exitRule()
return localctx
class RelationContext(ParserRuleContext):
def __init__(self, parser, parent=None, invokingState=-1):
super(LaTeXParser.RelationContext, self).__init__(parent, invokingState)
self.parser = parser
def expr(self):
return self.getTypedRuleContext(LaTeXParser.ExprContext,0)
def relation(self, i=None):
if i is None:
return self.getTypedRuleContexts(LaTeXParser.RelationContext)
else:
return self.getTypedRuleContext(LaTeXParser.RelationContext,i)
def EQUAL(self):
return self.getToken(LaTeXParser.EQUAL, 0)
def LT(self):
return self.getToken(LaTeXParser.LT, 0)
def LTE(self):
return self.getToken(LaTeXParser.LTE, 0)
def GT(self):
return self.getToken(LaTeXParser.GT, 0)
def GTE(self):
return self.getToken(LaTeXParser.GTE, 0)
def getRuleIndex(self):
return LaTeXParser.RULE_relation
def relation(self, _p=0):
_parentctx = self._ctx
_parentState = self.state
localctx = LaTeXParser.RelationContext(self, self._ctx, _parentState)
_prevctx = localctx
_startState = 2
self.enterRecursionRule(localctx, 2, self.RULE_relation, _p)
self._la = 0 # Token type
try:
self.enterOuterAlt(localctx, 1)
self.state = 75
self.expr()
self._ctx.stop = self._input.LT(-1)
self.state = 82
self._errHandler.sync(self)
_alt = self._interp.adaptivePredict(self._input,0,self._ctx)
while _alt!=2 and _alt!=ATN.INVALID_ALT_NUMBER:
if _alt==1:
if self._parseListeners is not None:
self.triggerExitRuleEvent()
_prevctx = localctx
localctx = LaTeXParser.RelationContext(self, _parentctx, _parentState)
self.pushNewRecursionContext(localctx, _startState, self.RULE_relation)
self.state = 77
if not self.precpred(self._ctx, 2):
from antlr4.error.Errors import FailedPredicateException
raise FailedPredicateException(self, "self.precpred(self._ctx, 2)")
self.state = 78
_la = self._input.LA(1)
if not(((((_la - 63)) & ~0x3f) == 0 and ((1 << (_la - 63)) & ((1 << (LaTeXParser.EQUAL - 63)) | (1 << (LaTeXParser.LT - 63)) | (1 << (LaTeXParser.LTE - 63)) | (1 << (LaTeXParser.GT - 63)) | (1 << (LaTeXParser.GTE - 63)))) != 0)):
self._errHandler.recoverInline(self)
else:
self._errHandler.reportMatch(self)
self.consume()
self.state = 79
self.relation(3)
self.state = 84
self._errHandler.sync(self)
_alt = self._interp.adaptivePredict(self._input,0,self._ctx)
except RecognitionException as re:
localctx.exception = re
self._errHandler.reportError(self, re)
self._errHandler.recover(self, re)
finally:
self.unrollRecursionContexts(_parentctx)
return localctx
class EqualityContext(ParserRuleContext):
def __init__(self, parser, parent=None, invokingState=-1):
super(LaTeXParser.EqualityContext, self).__init__(parent, invokingState)
self.parser = parser
def expr(self, i=None):
if i is None:
return self.getTypedRuleContexts(LaTeXParser.ExprContext)
else:
return self.getTypedRuleContext(LaTeXParser.ExprContext,i)
def EQUAL(self):
return self.getToken(LaTeXParser.EQUAL, 0)
def getRuleIndex(self):
return LaTeXParser.RULE_equality
def equality(self):
localctx = LaTeXParser.EqualityContext(self, self._ctx, self.state)
self.enterRule(localctx, 4, self.RULE_equality)
try:
self.enterOuterAlt(localctx, 1)
self.state = 85
self.expr()
self.state = 86
self.match(LaTeXParser.EQUAL)
self.state = 87
self.expr()
except RecognitionException as re:
localctx.exception = re
self._errHandler.reportError(self, re)
self._errHandler.recover(self, re)
finally:
self.exitRule()
return localctx
class ExprContext(ParserRuleContext):
def __init__(self, parser, parent=None, invokingState=-1):
super(LaTeXParser.ExprContext, self).__init__(parent, invokingState)
self.parser = parser
def additive(self):
return self.getTypedRuleContext(LaTeXParser.AdditiveContext,0)
def getRuleIndex(self):
return LaTeXParser.RULE_expr
def expr(self):
localctx = LaTeXParser.ExprContext(self, self._ctx, self.state)
self.enterRule(localctx, 6, self.RULE_expr)
try:
self.enterOuterAlt(localctx, 1)
self.state = 89
self.additive(0)
except RecognitionException as re:
localctx.exception = re
self._errHandler.reportError(self, re)
self._errHandler.recover(self, re)
finally:
self.exitRule()
return localctx
class AdditiveContext(ParserRuleContext):
def __init__(self, parser, parent=None, invokingState=-1):
super(LaTeXParser.AdditiveContext, self).__init__(parent, invokingState)
self.parser = parser
def mp(self):
return self.getTypedRuleContext(LaTeXParser.MpContext,0)
def additive(self, i=None):
if i is None:
return self.getTypedRuleContexts(LaTeXParser.AdditiveContext)
else:
return self.getTypedRuleContext(LaTeXParser.AdditiveContext,i)
def ADD(self):
return self.getToken(LaTeXParser.ADD, 0)
def SUB(self):
return self.getToken(LaTeXParser.SUB, 0)
def getRuleIndex(self):
return LaTeXParser.RULE_additive
def additive(self, _p=0):
_parentctx = self._ctx
_parentState = self.state
localctx = LaTeXParser.AdditiveContext(self, self._ctx, _parentState)
_prevctx = localctx
_startState = 8
self.enterRecursionRule(localctx, 8, self.RULE_additive, _p)
self._la = 0 # Token type
try:
self.enterOuterAlt(localctx, 1)
self.state = 92
self.mp(0)
self._ctx.stop = self._input.LT(-1)
self.state = 99
self._errHandler.sync(self)
_alt = self._interp.adaptivePredict(self._input,1,self._ctx)
while _alt!=2 and _alt!=ATN.INVALID_ALT_NUMBER:
if _alt==1:
if self._parseListeners is not None:
self.triggerExitRuleEvent()
_prevctx = localctx
localctx = LaTeXParser.AdditiveContext(self, _parentctx, _parentState)
self.pushNewRecursionContext(localctx, _startState, self.RULE_additive)
self.state = 94
if not self.precpred(self._ctx, 2):
from antlr4.error.Errors import FailedPredicateException
raise FailedPredicateException(self, "self.precpred(self._ctx, 2)")
self.state = 95
_la = self._input.LA(1)
if not(_la==LaTeXParser.ADD or _la==LaTeXParser.SUB):
self._errHandler.recoverInline(self)
else:
self._errHandler.reportMatch(self)
self.consume()
self.state = 96
self.additive(3)
self.state = 101
self._errHandler.sync(self)
_alt = self._interp.adaptivePredict(self._input,1,self._ctx)
except RecognitionException as re:
localctx.exception = re
self._errHandler.reportError(self, re)
self._errHandler.recover(self, re)
finally:
self.unrollRecursionContexts(_parentctx)
return localctx
class MpContext(ParserRuleContext):
def __init__(self, parser, parent=None, invokingState=-1):
super(LaTeXParser.MpContext, self).__init__(parent, invokingState)
self.parser = parser
def unary(self):
return self.getTypedRuleContext(LaTeXParser.UnaryContext,0)
def mp(self, i=None):
if i is None:
return self.getTypedRuleContexts(LaTeXParser.MpContext)
else:
return self.getTypedRuleContext(LaTeXParser.MpContext,i)
def MUL(self):
return self.getToken(LaTeXParser.MUL, 0)
def CMD_TIMES(self):
return self.getToken(LaTeXParser.CMD_TIMES, 0)
def CMD_CDOT(self):
return self.getToken(LaTeXParser.CMD_CDOT, 0)
def DIV(self):
return self.getToken(LaTeXParser.DIV, 0)
def CMD_DIV(self):
return self.getToken(LaTeXParser.CMD_DIV, 0)
def COLON(self):
return self.getToken(LaTeXParser.COLON, 0)
def getRuleIndex(self):
return LaTeXParser.RULE_mp
def mp(self, _p=0):
_parentctx = self._ctx
_parentState = self.state
localctx = LaTeXParser.MpContext(self, self._ctx, _parentState)
_prevctx = localctx
_startState = 10
self.enterRecursionRule(localctx, 10, self.RULE_mp, _p)
self._la = 0 # Token type
try:
self.enterOuterAlt(localctx, 1)
self.state = 103
self.unary()
self._ctx.stop = self._input.LT(-1)
self.state = 110
self._errHandler.sync(self)
_alt = self._interp.adaptivePredict(self._input,2,self._ctx)
while _alt!=2 and _alt!=ATN.INVALID_ALT_NUMBER:
if _alt==1:
if self._parseListeners is not None:
self.triggerExitRuleEvent()
_prevctx = localctx
localctx = LaTeXParser.MpContext(self, _parentctx, _parentState)
self.pushNewRecursionContext(localctx, _startState, self.RULE_mp)
self.state = 105
if not self.precpred(self._ctx, 2):
from antlr4.error.Errors import FailedPredicateException
raise FailedPredicateException(self, "self.precpred(self._ctx, 2)")
self.state = 106
_la = self._input.LA(1)
if not((((_la) & ~0x3f) == 0 and ((1 << _la) & ((1 << LaTeXParser.MUL) | (1 << LaTeXParser.DIV) | (1 << LaTeXParser.CMD_TIMES) | (1 << LaTeXParser.CMD_CDOT) | (1 << LaTeXParser.CMD_DIV) | (1 << LaTeXParser.COLON))) != 0)):
self._errHandler.recoverInline(self)
else:
self._errHandler.reportMatch(self)
self.consume()
self.state = 107
self.mp(3)
self.state = 112
self._errHandler.sync(self)
_alt = self._interp.adaptivePredict(self._input,2,self._ctx)
except RecognitionException as re:
localctx.exception = re
self._errHandler.reportError(self, re)
self._errHandler.recover(self, re)
finally:
self.unrollRecursionContexts(_parentctx)
return localctx
class Mp_nofuncContext(ParserRuleContext):
def __init__(self, parser, parent=None, invokingState=-1):
super(LaTeXParser.Mp_nofuncContext, self).__init__(parent, invokingState)
self.parser = parser
def unary_nofunc(self):
return self.getTypedRuleContext(LaTeXParser.Unary_nofuncContext,0)
def mp_nofunc(self, i=None):
if i is None:
return self.getTypedRuleContexts(LaTeXParser.Mp_nofuncContext)
else:
return self.getTypedRuleContext(LaTeXParser.Mp_nofuncContext,i)
def MUL(self):
return self.getToken(LaTeXParser.MUL, 0)
def CMD_TIMES(self):
return self.getToken(LaTeXParser.CMD_TIMES, 0)
def CMD_CDOT(self):
return self.getToken(LaTeXParser.CMD_CDOT, 0)
def DIV(self):
return self.getToken(LaTeXParser.DIV, 0)
def CMD_DIV(self):
return self.getToken(LaTeXParser.CMD_DIV, 0)
def COLON(self):
return self.getToken(LaTeXParser.COLON, 0)
def getRuleIndex(self):
return LaTeXParser.RULE_mp_nofunc
def mp_nofunc(self, _p=0):
_parentctx = self._ctx
_parentState = self.state
localctx = LaTeXParser.Mp_nofuncContext(self, self._ctx, _parentState)
_prevctx = localctx
_startState = 12
self.enterRecursionRule(localctx, 12, self.RULE_mp_nofunc, _p)
self._la = 0 # Token type
try:
self.enterOuterAlt(localctx, 1)
self.state = 114
self.unary_nofunc()
self._ctx.stop = self._input.LT(-1)
self.state = 121
self._errHandler.sync(self)
_alt = self._interp.adaptivePredict(self._input,3,self._ctx)
while _alt!=2 and _alt!=ATN.INVALID_ALT_NUMBER:
if _alt==1:
if self._parseListeners is not None:
self.triggerExitRuleEvent()
_prevctx = localctx
localctx = LaTeXParser.Mp_nofuncContext(self, _parentctx, _parentState)
self.pushNewRecursionContext(localctx, _startState, self.RULE_mp_nofunc)
self.state = 116
if not self.precpred(self._ctx, 2):
from antlr4.error.Errors import FailedPredicateException
raise FailedPredicateException(self, "self.precpred(self._ctx, 2)")
self.state = 117
_la = self._input.LA(1)
if not((((_la) & ~0x3f) == 0 and ((1 << _la) & ((1 << LaTeXParser.MUL) | (1 << LaTeXParser.DIV) | (1 << LaTeXParser.CMD_TIMES) | (1 << LaTeXParser.CMD_CDOT) | (1 << LaTeXParser.CMD_DIV) | (1 << LaTeXParser.COLON))) != 0)):
self._errHandler.recoverInline(self)
else:
self._errHandler.reportMatch(self)
self.consume()
self.state = 118
self.mp_nofunc(3)
self.state = 123
self._errHandler.sync(self)
_alt = self._interp.adaptivePredict(self._input,3,self._ctx)
except RecognitionException as re:
localctx.exception = re
self._errHandler.reportError(self, re)
self._errHandler.recover(self, re)
finally:
self.unrollRecursionContexts(_parentctx)
return localctx
class UnaryContext(ParserRuleContext):
def __init__(self, parser, parent=None, invokingState=-1):
super(LaTeXParser.UnaryContext, self).__init__(parent, invokingState)
self.parser = parser
def unary(self):
return self.getTypedRuleContext(LaTeXParser.UnaryContext,0)
def ADD(self):
return self.getToken(LaTeXParser.ADD, 0)
def SUB(self):
return self.getToken(LaTeXParser.SUB, 0)
def postfix(self, i=None):
if i is None:
return self.getTypedRuleContexts(LaTeXParser.PostfixContext)
else:
return self.getTypedRuleContext(LaTeXParser.PostfixContext,i)
def getRuleIndex(self):
return LaTeXParser.RULE_unary
def unary(self):
localctx = LaTeXParser.UnaryContext(self, self._ctx, self.state)
self.enterRule(localctx, 14, self.RULE_unary)
self._la = 0 # Token type
try:
self.state = 131
self._errHandler.sync(self)
token = self._input.LA(1)
if token in [LaTeXParser.ADD, LaTeXParser.SUB]:
self.enterOuterAlt(localctx, 1)
self.state = 124
_la = self._input.LA(1)
if not(_la==LaTeXParser.ADD or _la==LaTeXParser.SUB):
self._errHandler.recoverInline(self)
else:
self._errHandler.reportMatch(self)
self.consume()
self.state = 125
self.unary()
pass
elif token in [LaTeXParser.L_PAREN, LaTeXParser.L_BRACE, LaTeXParser.L_BRACKET, LaTeXParser.BAR, LaTeXParser.FUNC_LIM, LaTeXParser.FUNC_INT, LaTeXParser.FUNC_SUM, LaTeXParser.FUNC_PROD, LaTeXParser.FUNC_EXP, LaTeXParser.FUNC_LOG, LaTeXParser.FUNC_LN, LaTeXParser.FUNC_SIN, LaTeXParser.FUNC_COS, LaTeXParser.FUNC_TAN, LaTeXParser.FUNC_CSC, LaTeXParser.FUNC_SEC, LaTeXParser.FUNC_COT, LaTeXParser.FUNC_ARCSIN, LaTeXParser.FUNC_ARCCOS, LaTeXParser.FUNC_ARCTAN, LaTeXParser.FUNC_ARCCSC, LaTeXParser.FUNC_ARCSEC, LaTeXParser.FUNC_ARCCOT, LaTeXParser.FUNC_SINH, LaTeXParser.FUNC_COSH, LaTeXParser.FUNC_TANH, LaTeXParser.FUNC_ARSINH, LaTeXParser.FUNC_ARCOSH, LaTeXParser.FUNC_ARTANH, LaTeXParser.FUNC_SQRT, LaTeXParser.CMD_FRAC, LaTeXParser.CMD_BINOM, LaTeXParser.CMD_DBINOM, LaTeXParser.CMD_TBINOM, LaTeXParser.CMD_MATHIT, LaTeXParser.DIFFERENTIAL, LaTeXParser.LETTER, LaTeXParser.NUMBER, LaTeXParser.SYMBOL]:
self.enterOuterAlt(localctx, 2)
self.state = 127
self._errHandler.sync(self)
_alt = 1
while _alt!=2 and _alt!=ATN.INVALID_ALT_NUMBER:
if _alt == 1:
self.state = 126
self.postfix()
else:
raise NoViableAltException(self)
self.state = 129
self._errHandler.sync(self)
_alt = self._interp.adaptivePredict(self._input,4,self._ctx)
pass
else:
raise NoViableAltException(self)
except RecognitionException as re:
localctx.exception = re
self._errHandler.reportError(self, re)
self._errHandler.recover(self, re)
finally:
self.exitRule()
return localctx
class Unary_nofuncContext(ParserRuleContext):
def __init__(self, parser, parent=None, invokingState=-1):
super(LaTeXParser.Unary_nofuncContext, self).__init__(parent, invokingState)
self.parser = parser
def unary_nofunc(self):
return self.getTypedRuleContext(LaTeXParser.Unary_nofuncContext,0)
def ADD(self):
return self.getToken(LaTeXParser.ADD, 0)
def SUB(self):
return self.getToken(LaTeXParser.SUB, 0)
def postfix(self):
return self.getTypedRuleContext(LaTeXParser.PostfixContext,0)
def postfix_nofunc(self, i=None):
if i is None:
return self.getTypedRuleContexts(LaTeXParser.Postfix_nofuncContext)
else:
return self.getTypedRuleContext(LaTeXParser.Postfix_nofuncContext,i)
def getRuleIndex(self):
return LaTeXParser.RULE_unary_nofunc
def unary_nofunc(self):
localctx = LaTeXParser.Unary_nofuncContext(self, self._ctx, self.state)
self.enterRule(localctx, 16, self.RULE_unary_nofunc)
self._la = 0 # Token type
try:
self.state = 142
self._errHandler.sync(self)
token = self._input.LA(1)
if token in [LaTeXParser.ADD, LaTeXParser.SUB]:
self.enterOuterAlt(localctx, 1)
self.state = 133
_la = self._input.LA(1)
if not(_la==LaTeXParser.ADD or _la==LaTeXParser.SUB):
self._errHandler.recoverInline(self)
else:
self._errHandler.reportMatch(self)
self.consume()
self.state = 134
self.unary_nofunc()
pass
elif token in [LaTeXParser.L_PAREN, LaTeXParser.L_BRACE, LaTeXParser.L_BRACKET, LaTeXParser.BAR, LaTeXParser.FUNC_LIM, LaTeXParser.FUNC_INT, LaTeXParser.FUNC_SUM, LaTeXParser.FUNC_PROD, LaTeXParser.FUNC_EXP, LaTeXParser.FUNC_LOG, LaTeXParser.FUNC_LN, LaTeXParser.FUNC_SIN, LaTeXParser.FUNC_COS, LaTeXParser.FUNC_TAN, LaTeXParser.FUNC_CSC, LaTeXParser.FUNC_SEC, LaTeXParser.FUNC_COT, LaTeXParser.FUNC_ARCSIN, LaTeXParser.FUNC_ARCCOS, LaTeXParser.FUNC_ARCTAN, LaTeXParser.FUNC_ARCCSC, LaTeXParser.FUNC_ARCSEC, LaTeXParser.FUNC_ARCCOT, LaTeXParser.FUNC_SINH, LaTeXParser.FUNC_COSH, LaTeXParser.FUNC_TANH, LaTeXParser.FUNC_ARSINH, LaTeXParser.FUNC_ARCOSH, LaTeXParser.FUNC_ARTANH, LaTeXParser.FUNC_SQRT, LaTeXParser.CMD_FRAC, LaTeXParser.CMD_BINOM, LaTeXParser.CMD_DBINOM, LaTeXParser.CMD_TBINOM, LaTeXParser.CMD_MATHIT, LaTeXParser.DIFFERENTIAL, LaTeXParser.LETTER, LaTeXParser.NUMBER, LaTeXParser.SYMBOL]:
self.enterOuterAlt(localctx, 2)
self.state = 135
self.postfix()
self.state = 139
self._errHandler.sync(self)
_alt = self._interp.adaptivePredict(self._input,6,self._ctx)
while _alt!=2 and _alt!=ATN.INVALID_ALT_NUMBER:
if _alt==1:
self.state = 136
self.postfix_nofunc()
self.state = 141
self._errHandler.sync(self)
_alt = self._interp.adaptivePredict(self._input,6,self._ctx)
pass
else:
raise NoViableAltException(self)
except RecognitionException as re:
localctx.exception = re
self._errHandler.reportError(self, re)
self._errHandler.recover(self, re)
finally:
self.exitRule()
return localctx
class PostfixContext(ParserRuleContext):
def __init__(self, parser, parent=None, invokingState=-1):
super(LaTeXParser.PostfixContext, self).__init__(parent, invokingState)
self.parser = parser
def exp(self):
return self.getTypedRuleContext(LaTeXParser.ExpContext,0)
def postfix_op(self, i=None):
if i is None:
return self.getTypedRuleContexts(LaTeXParser.Postfix_opContext)
else:
return self.getTypedRuleContext(LaTeXParser.Postfix_opContext,i)
def getRuleIndex(self):
return LaTeXParser.RULE_postfix
def postfix(self):
localctx = LaTeXParser.PostfixContext(self, self._ctx, self.state)
self.enterRule(localctx, 18, self.RULE_postfix)
try:
self.enterOuterAlt(localctx, 1)
self.state = 144
self.exp(0)
self.state = 148
self._errHandler.sync(self)
_alt = self._interp.adaptivePredict(self._input,8,self._ctx)
while _alt!=2 and _alt!=ATN.INVALID_ALT_NUMBER:
if _alt==1:
self.state = 145
self.postfix_op()
self.state = 150
self._errHandler.sync(self)
_alt = self._interp.adaptivePredict(self._input,8,self._ctx)
except RecognitionException as re:
localctx.exception = re
self._errHandler.reportError(self, re)
self._errHandler.recover(self, re)
finally:
self.exitRule()
return localctx
class Postfix_nofuncContext(ParserRuleContext):
def __init__(self, parser, parent=None, invokingState=-1):
super(LaTeXParser.Postfix_nofuncContext, self).__init__(parent, invokingState)
self.parser = parser
def exp_nofunc(self):
return self.getTypedRuleContext(LaTeXParser.Exp_nofuncContext,0)
def postfix_op(self, i=None):
if i is None:
return self.getTypedRuleContexts(LaTeXParser.Postfix_opContext)
else:
return self.getTypedRuleContext(LaTeXParser.Postfix_opContext,i)
def getRuleIndex(self):
return LaTeXParser.RULE_postfix_nofunc
def postfix_nofunc(self):
localctx = LaTeXParser.Postfix_nofuncContext(self, self._ctx, self.state)
self.enterRule(localctx, 20, self.RULE_postfix_nofunc)
try:
self.enterOuterAlt(localctx, 1)
self.state = 151
self.exp_nofunc(0)
self.state = 155
self._errHandler.sync(self)
_alt = self._interp.adaptivePredict(self._input,9,self._ctx)
while _alt!=2 and _alt!=ATN.INVALID_ALT_NUMBER:
if _alt==1:
self.state = 152
self.postfix_op()
self.state = 157
self._errHandler.sync(self)
_alt = self._interp.adaptivePredict(self._input,9,self._ctx)
except RecognitionException as re:
localctx.exception = re
self._errHandler.reportError(self, re)
self._errHandler.recover(self, re)
finally:
self.exitRule()
return localctx
class Postfix_opContext(ParserRuleContext):
def __init__(self, parser, parent=None, invokingState=-1):
super(LaTeXParser.Postfix_opContext, self).__init__(parent, invokingState)
self.parser = parser
def BANG(self):
return self.getToken(LaTeXParser.BANG, 0)
def eval_at(self):
return self.getTypedRuleContext(LaTeXParser.Eval_atContext,0)
def getRuleIndex(self):
return LaTeXParser.RULE_postfix_op
def postfix_op(self):
localctx = LaTeXParser.Postfix_opContext(self, self._ctx, self.state)
self.enterRule(localctx, 22, self.RULE_postfix_op)
try:
self.state = 160
self._errHandler.sync(self)
token = self._input.LA(1)
if token in [LaTeXParser.BANG]:
self.enterOuterAlt(localctx, 1)
self.state = 158
self.match(LaTeXParser.BANG)
pass
elif token in [LaTeXParser.BAR]:
self.enterOuterAlt(localctx, 2)
self.state = 159
self.eval_at()
pass
else:
raise NoViableAltException(self)
except RecognitionException as re:
localctx.exception = re
self._errHandler.reportError(self, re)
self._errHandler.recover(self, re)
finally:
self.exitRule()
return localctx
class Eval_atContext(ParserRuleContext):
def __init__(self, parser, parent=None, invokingState=-1):
super(LaTeXParser.Eval_atContext, self).__init__(parent, invokingState)
self.parser = parser
def BAR(self):
return self.getToken(LaTeXParser.BAR, 0)
def eval_at_sup(self):
return self.getTypedRuleContext(LaTeXParser.Eval_at_supContext,0)
def eval_at_sub(self):
return self.getTypedRuleContext(LaTeXParser.Eval_at_subContext,0)
def getRuleIndex(self):
return LaTeXParser.RULE_eval_at
def eval_at(self):
localctx = LaTeXParser.Eval_atContext(self, self._ctx, self.state)
self.enterRule(localctx, 24, self.RULE_eval_at)
try:
self.enterOuterAlt(localctx, 1)
self.state = 162
self.match(LaTeXParser.BAR)
self.state = 168
self._errHandler.sync(self)
la_ = self._interp.adaptivePredict(self._input,11,self._ctx)
if la_ == 1:
self.state = 163
self.eval_at_sup()
pass
elif la_ == 2:
self.state = 164
self.eval_at_sub()
pass
elif la_ == 3:
self.state = 165
self.eval_at_sup()
self.state = 166
self.eval_at_sub()
pass
except RecognitionException as re:
localctx.exception = re
self._errHandler.reportError(self, re)
self._errHandler.recover(self, re)
finally:
self.exitRule()
return localctx
class Eval_at_subContext(ParserRuleContext):
def __init__(self, parser, parent=None, invokingState=-1):
super(LaTeXParser.Eval_at_subContext, self).__init__(parent, invokingState)
self.parser = parser
def UNDERSCORE(self):
return self.getToken(LaTeXParser.UNDERSCORE, 0)
def L_BRACE(self):
return self.getToken(LaTeXParser.L_BRACE, 0)
def R_BRACE(self):
return self.getToken(LaTeXParser.R_BRACE, 0)
def expr(self):
return self.getTypedRuleContext(LaTeXParser.ExprContext,0)
def equality(self):
return self.getTypedRuleContext(LaTeXParser.EqualityContext,0)
def getRuleIndex(self):
return LaTeXParser.RULE_eval_at_sub
def eval_at_sub(self):
localctx = LaTeXParser.Eval_at_subContext(self, self._ctx, self.state)
self.enterRule(localctx, 26, self.RULE_eval_at_sub)
try:
self.enterOuterAlt(localctx, 1)
self.state = 170
self.match(LaTeXParser.UNDERSCORE)
self.state = 171
self.match(LaTeXParser.L_BRACE)
self.state = 174
self._errHandler.sync(self)
la_ = self._interp.adaptivePredict(self._input,12,self._ctx)
if la_ == 1:
self.state = 172
self.expr()
pass
elif la_ == 2:
self.state = 173
self.equality()
pass
self.state = 176
self.match(LaTeXParser.R_BRACE)
except RecognitionException as re:
localctx.exception = re
self._errHandler.reportError(self, re)
self._errHandler.recover(self, re)
finally:
self.exitRule()
return localctx
class Eval_at_supContext(ParserRuleContext):
def __init__(self, parser, parent=None, invokingState=-1):
super(LaTeXParser.Eval_at_supContext, self).__init__(parent, invokingState)
self.parser = parser
def CARET(self):
return self.getToken(LaTeXParser.CARET, 0)
def L_BRACE(self):
return self.getToken(LaTeXParser.L_BRACE, 0)
def R_BRACE(self):
return self.getToken(LaTeXParser.R_BRACE, 0)
def expr(self):
return self.getTypedRuleContext(LaTeXParser.ExprContext,0)
def equality(self):
return self.getTypedRuleContext(LaTeXParser.EqualityContext,0)
def getRuleIndex(self):
return LaTeXParser.RULE_eval_at_sup
def eval_at_sup(self):
localctx = LaTeXParser.Eval_at_supContext(self, self._ctx, self.state)
self.enterRule(localctx, 28, self.RULE_eval_at_sup)
try:
self.enterOuterAlt(localctx, 1)
self.state = 178
self.match(LaTeXParser.CARET)
self.state = 179
self.match(LaTeXParser.L_BRACE)
self.state = 182
self._errHandler.sync(self)
la_ = self._interp.adaptivePredict(self._input,13,self._ctx)
if la_ == 1:
self.state = 180
self.expr()
pass
elif la_ == 2:
self.state = 181
self.equality()
pass
self.state = 184
self.match(LaTeXParser.R_BRACE)
except RecognitionException as re:
localctx.exception = re
self._errHandler.reportError(self, re)
self._errHandler.recover(self, re)
finally:
self.exitRule()
return localctx
class ExpContext(ParserRuleContext):
def __init__(self, parser, parent=None, invokingState=-1):
super(LaTeXParser.ExpContext, self).__init__(parent, invokingState)
self.parser = parser
def comp(self):
return self.getTypedRuleContext(LaTeXParser.CompContext,0)
def exp(self):
return self.getTypedRuleContext(LaTeXParser.ExpContext,0)
def CARET(self):
return self.getToken(LaTeXParser.CARET, 0)
def atom(self):
return self.getTypedRuleContext(LaTeXParser.AtomContext,0)
def L_BRACE(self):
return self.getToken(LaTeXParser.L_BRACE, 0)
def expr(self):
return self.getTypedRuleContext(LaTeXParser.ExprContext,0)
def R_BRACE(self):
return self.getToken(LaTeXParser.R_BRACE, 0)
def subexpr(self):
return self.getTypedRuleContext(LaTeXParser.SubexprContext,0)
def getRuleIndex(self):
return LaTeXParser.RULE_exp
def exp(self, _p=0):
_parentctx = self._ctx
_parentState = self.state
localctx = LaTeXParser.ExpContext(self, self._ctx, _parentState)
_prevctx = localctx
_startState = 30
self.enterRecursionRule(localctx, 30, self.RULE_exp, _p)
try:
self.enterOuterAlt(localctx, 1)
self.state = 187
self.comp()
self._ctx.stop = self._input.LT(-1)
self.state = 203
self._errHandler.sync(self)
_alt = self._interp.adaptivePredict(self._input,16,self._ctx)
while _alt!=2 and _alt!=ATN.INVALID_ALT_NUMBER:
if _alt==1:
if self._parseListeners is not None:
self.triggerExitRuleEvent()
_prevctx = localctx
localctx = LaTeXParser.ExpContext(self, _parentctx, _parentState)
self.pushNewRecursionContext(localctx, _startState, self.RULE_exp)
self.state = 189
if not self.precpred(self._ctx, 2):
from antlr4.error.Errors import FailedPredicateException
raise FailedPredicateException(self, "self.precpred(self._ctx, 2)")
self.state = 190
self.match(LaTeXParser.CARET)
self.state = 196
self._errHandler.sync(self)
token = self._input.LA(1)
if token in [LaTeXParser.CMD_MATHIT, LaTeXParser.DIFFERENTIAL, LaTeXParser.LETTER, LaTeXParser.NUMBER, LaTeXParser.SYMBOL]:
self.state = 191
self.atom()
pass
elif token in [LaTeXParser.L_BRACE]:
self.state = 192
self.match(LaTeXParser.L_BRACE)
self.state = 193
self.expr()
self.state = 194
self.match(LaTeXParser.R_BRACE)
pass
else:
raise NoViableAltException(self)
self.state = 199
self._errHandler.sync(self)
la_ = self._interp.adaptivePredict(self._input,15,self._ctx)
if la_ == 1:
self.state = 198
self.subexpr()
self.state = 205
self._errHandler.sync(self)
_alt = self._interp.adaptivePredict(self._input,16,self._ctx)
except RecognitionException as re:
localctx.exception = re
self._errHandler.reportError(self, re)
self._errHandler.recover(self, re)
finally:
self.unrollRecursionContexts(_parentctx)
return localctx
class Exp_nofuncContext(ParserRuleContext):
def __init__(self, parser, parent=None, invokingState=-1):
super(LaTeXParser.Exp_nofuncContext, self).__init__(parent, invokingState)
self.parser = parser
def comp_nofunc(self):
return self.getTypedRuleContext(LaTeXParser.Comp_nofuncContext,0)
def exp_nofunc(self):
return self.getTypedRuleContext(LaTeXParser.Exp_nofuncContext,0)
def CARET(self):
return self.getToken(LaTeXParser.CARET, 0)
def atom(self):
return self.getTypedRuleContext(LaTeXParser.AtomContext,0)
def L_BRACE(self):
return self.getToken(LaTeXParser.L_BRACE, 0)
def expr(self):
return self.getTypedRuleContext(LaTeXParser.ExprContext,0)
def R_BRACE(self):
return self.getToken(LaTeXParser.R_BRACE, 0)
def subexpr(self):
return self.getTypedRuleContext(LaTeXParser.SubexprContext,0)
def getRuleIndex(self):
return LaTeXParser.RULE_exp_nofunc
def exp_nofunc(self, _p=0):
_parentctx = self._ctx
_parentState = self.state
localctx = LaTeXParser.Exp_nofuncContext(self, self._ctx, _parentState)
_prevctx = localctx
_startState = 32
self.enterRecursionRule(localctx, 32, self.RULE_exp_nofunc, _p)
try:
self.enterOuterAlt(localctx, 1)
self.state = 207
self.comp_nofunc()
self._ctx.stop = self._input.LT(-1)
self.state = 223
self._errHandler.sync(self)
_alt = self._interp.adaptivePredict(self._input,19,self._ctx)
while _alt!=2 and _alt!=ATN.INVALID_ALT_NUMBER:
if _alt==1:
if self._parseListeners is not None:
self.triggerExitRuleEvent()
_prevctx = localctx
localctx = LaTeXParser.Exp_nofuncContext(self, _parentctx, _parentState)
self.pushNewRecursionContext(localctx, _startState, self.RULE_exp_nofunc)
self.state = 209
if not self.precpred(self._ctx, 2):
from antlr4.error.Errors import FailedPredicateException
raise FailedPredicateException(self, "self.precpred(self._ctx, 2)")
self.state = 210
self.match(LaTeXParser.CARET)
self.state = 216
self._errHandler.sync(self)
token = self._input.LA(1)
if token in [LaTeXParser.CMD_MATHIT, LaTeXParser.DIFFERENTIAL, LaTeXParser.LETTER, LaTeXParser.NUMBER, LaTeXParser.SYMBOL]:
self.state = 211
self.atom()
pass
elif token in [LaTeXParser.L_BRACE]:
self.state = 212
self.match(LaTeXParser.L_BRACE)
self.state = 213
self.expr()
self.state = 214
self.match(LaTeXParser.R_BRACE)
pass
else:
raise NoViableAltException(self)
self.state = 219
self._errHandler.sync(self)
la_ = self._interp.adaptivePredict(self._input,18,self._ctx)
if la_ == 1:
self.state = 218
self.subexpr()
self.state = 225
self._errHandler.sync(self)
_alt = self._interp.adaptivePredict(self._input,19,self._ctx)
except RecognitionException as re:
localctx.exception = re
self._errHandler.reportError(self, re)
self._errHandler.recover(self, re)
finally:
self.unrollRecursionContexts(_parentctx)
return localctx
class CompContext(ParserRuleContext):
def __init__(self, parser, parent=None, invokingState=-1):
super(LaTeXParser.CompContext, self).__init__(parent, invokingState)
self.parser = parser
def group(self):
return self.getTypedRuleContext(LaTeXParser.GroupContext,0)
def abs_group(self):
return self.getTypedRuleContext(LaTeXParser.Abs_groupContext,0)
def func(self):
return self.getTypedRuleContext(LaTeXParser.FuncContext,0)
def atom(self):
return self.getTypedRuleContext(LaTeXParser.AtomContext,0)
def frac(self):
return self.getTypedRuleContext(LaTeXParser.FracContext,0)
def binom(self):
return self.getTypedRuleContext(LaTeXParser.BinomContext,0)
def getRuleIndex(self):
return LaTeXParser.RULE_comp
def comp(self):
localctx = LaTeXParser.CompContext(self, self._ctx, self.state)
self.enterRule(localctx, 34, self.RULE_comp)
try:
self.state = 232
self._errHandler.sync(self)
la_ = self._interp.adaptivePredict(self._input,20,self._ctx)
if la_ == 1:
self.enterOuterAlt(localctx, 1)
self.state = 226
self.group()
pass
elif la_ == 2:
self.enterOuterAlt(localctx, 2)
self.state = 227
self.abs_group()
pass
elif la_ == 3:
self.enterOuterAlt(localctx, 3)
self.state = 228
self.func()
pass
elif la_ == 4:
self.enterOuterAlt(localctx, 4)
self.state = 229
self.atom()
pass
elif la_ == 5:
self.enterOuterAlt(localctx, 5)
self.state = 230
self.frac()
pass
elif la_ == 6:
self.enterOuterAlt(localctx, 6)
self.state = 231
self.binom()
pass
except RecognitionException as re:
localctx.exception = re
self._errHandler.reportError(self, re)
self._errHandler.recover(self, re)
finally:
self.exitRule()
return localctx
class Comp_nofuncContext(ParserRuleContext):
def __init__(self, parser, parent=None, invokingState=-1):
super(LaTeXParser.Comp_nofuncContext, self).__init__(parent, invokingState)
self.parser = parser
def group(self):
return self.getTypedRuleContext(LaTeXParser.GroupContext,0)
def abs_group(self):
return self.getTypedRuleContext(LaTeXParser.Abs_groupContext,0)
def atom(self):
return self.getTypedRuleContext(LaTeXParser.AtomContext,0)
def frac(self):
return self.getTypedRuleContext(LaTeXParser.FracContext,0)
def binom(self):
return self.getTypedRuleContext(LaTeXParser.BinomContext,0)
def getRuleIndex(self):
return LaTeXParser.RULE_comp_nofunc
def comp_nofunc(self):
localctx = LaTeXParser.Comp_nofuncContext(self, self._ctx, self.state)
self.enterRule(localctx, 36, self.RULE_comp_nofunc)
try:
self.state = 239
self._errHandler.sync(self)
token = self._input.LA(1)
if token in [LaTeXParser.L_PAREN, LaTeXParser.L_BRACE, LaTeXParser.L_BRACKET]:
self.enterOuterAlt(localctx, 1)
self.state = 234
self.group()
pass
elif token in [LaTeXParser.BAR]:
self.enterOuterAlt(localctx, 2)
self.state = 235
self.abs_group()
pass
elif token in [LaTeXParser.CMD_MATHIT, LaTeXParser.DIFFERENTIAL, LaTeXParser.LETTER, LaTeXParser.NUMBER, LaTeXParser.SYMBOL]:
self.enterOuterAlt(localctx, 3)
self.state = 236
self.atom()
pass
elif token in [LaTeXParser.CMD_FRAC]:
self.enterOuterAlt(localctx, 4)
self.state = 237
self.frac()
pass
elif token in [LaTeXParser.CMD_BINOM, LaTeXParser.CMD_DBINOM, LaTeXParser.CMD_TBINOM]:
self.enterOuterAlt(localctx, 5)
self.state = 238
self.binom()
pass
else:
raise NoViableAltException(self)
except RecognitionException as re:
localctx.exception = re
self._errHandler.reportError(self, re)
self._errHandler.recover(self, re)
finally:
self.exitRule()
return localctx
class GroupContext(ParserRuleContext):
def __init__(self, parser, parent=None, invokingState=-1):
super(LaTeXParser.GroupContext, self).__init__(parent, invokingState)
self.parser = parser
def L_PAREN(self):
return self.getToken(LaTeXParser.L_PAREN, 0)
def expr(self):
return self.getTypedRuleContext(LaTeXParser.ExprContext,0)
def R_PAREN(self):
return self.getToken(LaTeXParser.R_PAREN, 0)
def L_BRACKET(self):
return self.getToken(LaTeXParser.L_BRACKET, 0)
def R_BRACKET(self):
return self.getToken(LaTeXParser.R_BRACKET, 0)
def L_BRACE(self):
return self.getToken(LaTeXParser.L_BRACE, 0)
def R_BRACE(self):
return self.getToken(LaTeXParser.R_BRACE, 0)
def getRuleIndex(self):
return LaTeXParser.RULE_group
def group(self):
localctx = LaTeXParser.GroupContext(self, self._ctx, self.state)
self.enterRule(localctx, 38, self.RULE_group)
try:
self.state = 253
self._errHandler.sync(self)
token = self._input.LA(1)
if token in [LaTeXParser.L_PAREN]:
self.enterOuterAlt(localctx, 1)
self.state = 241
self.match(LaTeXParser.L_PAREN)
self.state = 242
self.expr()
self.state = 243
self.match(LaTeXParser.R_PAREN)
pass
elif token in [LaTeXParser.L_BRACKET]:
self.enterOuterAlt(localctx, 2)
self.state = 245
self.match(LaTeXParser.L_BRACKET)
self.state = 246
self.expr()
self.state = 247
self.match(LaTeXParser.R_BRACKET)
pass
elif token in [LaTeXParser.L_BRACE]:
self.enterOuterAlt(localctx, 3)
self.state = 249
self.match(LaTeXParser.L_BRACE)
self.state = 250
self.expr()
self.state = 251
self.match(LaTeXParser.R_BRACE)
pass
else:
raise NoViableAltException(self)
except RecognitionException as re:
localctx.exception = re
self._errHandler.reportError(self, re)
self._errHandler.recover(self, re)
finally:
self.exitRule()
return localctx
class Abs_groupContext(ParserRuleContext):
def __init__(self, parser, parent=None, invokingState=-1):
super(LaTeXParser.Abs_groupContext, self).__init__(parent, invokingState)
self.parser = parser
def BAR(self, i=None):
if i is None:
return self.getTokens(LaTeXParser.BAR)
else:
return self.getToken(LaTeXParser.BAR, i)
def expr(self):
return self.getTypedRuleContext(LaTeXParser.ExprContext,0)
def getRuleIndex(self):
return LaTeXParser.RULE_abs_group
def abs_group(self):
localctx = LaTeXParser.Abs_groupContext(self, self._ctx, self.state)
self.enterRule(localctx, 40, self.RULE_abs_group)
try:
self.enterOuterAlt(localctx, 1)
self.state = 255
self.match(LaTeXParser.BAR)
self.state = 256
self.expr()
self.state = 257
self.match(LaTeXParser.BAR)
except RecognitionException as re:
localctx.exception = re
self._errHandler.reportError(self, re)
self._errHandler.recover(self, re)
finally:
self.exitRule()
return localctx
class AtomContext(ParserRuleContext):
def __init__(self, parser, parent=None, invokingState=-1):
super(LaTeXParser.AtomContext, self).__init__(parent, invokingState)
self.parser = parser
def LETTER(self):
return self.getToken(LaTeXParser.LETTER, 0)
def SYMBOL(self):
return self.getToken(LaTeXParser.SYMBOL, 0)
def subexpr(self):
return self.getTypedRuleContext(LaTeXParser.SubexprContext,0)
def NUMBER(self):
return self.getToken(LaTeXParser.NUMBER, 0)
def DIFFERENTIAL(self):
return self.getToken(LaTeXParser.DIFFERENTIAL, 0)
def mathit(self):
return self.getTypedRuleContext(LaTeXParser.MathitContext,0)
def getRuleIndex(self):
return LaTeXParser.RULE_atom
def atom(self):
localctx = LaTeXParser.AtomContext(self, self._ctx, self.state)
self.enterRule(localctx, 42, self.RULE_atom)
self._la = 0 # Token type
try:
self.state = 266
self._errHandler.sync(self)
token = self._input.LA(1)
if token in [LaTeXParser.LETTER, LaTeXParser.SYMBOL]:
self.enterOuterAlt(localctx, 1)
self.state = 259
_la = self._input.LA(1)
if not(_la==LaTeXParser.LETTER or _la==LaTeXParser.SYMBOL):
self._errHandler.recoverInline(self)
else:
self._errHandler.reportMatch(self)
self.consume()
self.state = 261
self._errHandler.sync(self)
la_ = self._interp.adaptivePredict(self._input,23,self._ctx)
if la_ == 1:
self.state = 260
self.subexpr()
pass
elif token in [LaTeXParser.NUMBER]:
self.enterOuterAlt(localctx, 2)
self.state = 263
self.match(LaTeXParser.NUMBER)
pass
elif token in [LaTeXParser.DIFFERENTIAL]:
self.enterOuterAlt(localctx, 3)
self.state = 264
self.match(LaTeXParser.DIFFERENTIAL)
pass
elif token in [LaTeXParser.CMD_MATHIT]:
self.enterOuterAlt(localctx, 4)
self.state = 265
self.mathit()
pass
else:
raise NoViableAltException(self)
except RecognitionException as re:
localctx.exception = re
self._errHandler.reportError(self, re)
self._errHandler.recover(self, re)
finally:
self.exitRule()
return localctx
class MathitContext(ParserRuleContext):
def __init__(self, parser, parent=None, invokingState=-1):
super(LaTeXParser.MathitContext, self).__init__(parent, invokingState)
self.parser = parser
def CMD_MATHIT(self):
return self.getToken(LaTeXParser.CMD_MATHIT, 0)
def L_BRACE(self):
return self.getToken(LaTeXParser.L_BRACE, 0)
def mathit_text(self):
return self.getTypedRuleContext(LaTeXParser.Mathit_textContext,0)
def R_BRACE(self):
return self.getToken(LaTeXParser.R_BRACE, 0)
def getRuleIndex(self):
return LaTeXParser.RULE_mathit
def mathit(self):
localctx = LaTeXParser.MathitContext(self, self._ctx, self.state)
self.enterRule(localctx, 44, self.RULE_mathit)
try:
self.enterOuterAlt(localctx, 1)
self.state = 268
self.match(LaTeXParser.CMD_MATHIT)
self.state = 269
self.match(LaTeXParser.L_BRACE)
self.state = 270
self.mathit_text()
self.state = 271
self.match(LaTeXParser.R_BRACE)
except RecognitionException as re:
localctx.exception = re
self._errHandler.reportError(self, re)
self._errHandler.recover(self, re)
finally:
self.exitRule()
return localctx
class Mathit_textContext(ParserRuleContext):
def __init__(self, parser, parent=None, invokingState=-1):
super(LaTeXParser.Mathit_textContext, self).__init__(parent, invokingState)
self.parser = parser
def LETTER(self, i=None):
if i is None:
return self.getTokens(LaTeXParser.LETTER)
else:
return self.getToken(LaTeXParser.LETTER, i)
def getRuleIndex(self):
return LaTeXParser.RULE_mathit_text
def mathit_text(self):
localctx = LaTeXParser.Mathit_textContext(self, self._ctx, self.state)
self.enterRule(localctx, 46, self.RULE_mathit_text)
self._la = 0 # Token type
try:
self.enterOuterAlt(localctx, 1)
self.state = 276
self._errHandler.sync(self)
_la = self._input.LA(1)
while _la==LaTeXParser.LETTER:
self.state = 273
self.match(LaTeXParser.LETTER)
self.state = 278
self._errHandler.sync(self)
_la = self._input.LA(1)
except RecognitionException as re:
localctx.exception = re
self._errHandler.reportError(self, re)
self._errHandler.recover(self, re)
finally:
self.exitRule()
return localctx
class FracContext(ParserRuleContext):
def __init__(self, parser, parent=None, invokingState=-1):
super(LaTeXParser.FracContext, self).__init__(parent, invokingState)
self.parser = parser
self.upper = None # ExprContext
self.lower = None # ExprContext
def CMD_FRAC(self):
return self.getToken(LaTeXParser.CMD_FRAC, 0)
def L_BRACE(self, i=None):
if i is None:
return self.getTokens(LaTeXParser.L_BRACE)
else:
return self.getToken(LaTeXParser.L_BRACE, i)
def R_BRACE(self, i=None):
if i is None:
return self.getTokens(LaTeXParser.R_BRACE)
else:
return self.getToken(LaTeXParser.R_BRACE, i)
def expr(self, i=None):
if i is None:
return self.getTypedRuleContexts(LaTeXParser.ExprContext)
else:
return self.getTypedRuleContext(LaTeXParser.ExprContext,i)
def getRuleIndex(self):
return LaTeXParser.RULE_frac
def frac(self):
localctx = LaTeXParser.FracContext(self, self._ctx, self.state)
self.enterRule(localctx, 48, self.RULE_frac)
try:
self.enterOuterAlt(localctx, 1)
self.state = 279
self.match(LaTeXParser.CMD_FRAC)
self.state = 280
self.match(LaTeXParser.L_BRACE)
self.state = 281
localctx.upper = self.expr()
self.state = 282
self.match(LaTeXParser.R_BRACE)
self.state = 283
self.match(LaTeXParser.L_BRACE)
self.state = 284
localctx.lower = self.expr()
self.state = 285
self.match(LaTeXParser.R_BRACE)
except RecognitionException as re:
localctx.exception = re
self._errHandler.reportError(self, re)
self._errHandler.recover(self, re)
finally:
self.exitRule()
return localctx
class BinomContext(ParserRuleContext):
def __init__(self, parser, parent=None, invokingState=-1):
super(LaTeXParser.BinomContext, self).__init__(parent, invokingState)
self.parser = parser
self.n = None # ExprContext
self.k = None # ExprContext
def L_BRACE(self, i=None):
if i is None:
return self.getTokens(LaTeXParser.L_BRACE)
else:
return self.getToken(LaTeXParser.L_BRACE, i)
def R_BRACE(self, i=None):
if i is None:
return self.getTokens(LaTeXParser.R_BRACE)
else:
return self.getToken(LaTeXParser.R_BRACE, i)
def CMD_BINOM(self):
return self.getToken(LaTeXParser.CMD_BINOM, 0)
def CMD_DBINOM(self):
return self.getToken(LaTeXParser.CMD_DBINOM, 0)
def CMD_TBINOM(self):
return self.getToken(LaTeXParser.CMD_TBINOM, 0)
def expr(self, i=None):
if i is None:
return self.getTypedRuleContexts(LaTeXParser.ExprContext)
else:
return self.getTypedRuleContext(LaTeXParser.ExprContext,i)
def getRuleIndex(self):
return LaTeXParser.RULE_binom
def binom(self):
localctx = LaTeXParser.BinomContext(self, self._ctx, self.state)
self.enterRule(localctx, 50, self.RULE_binom)
self._la = 0 # Token type
try:
self.enterOuterAlt(localctx, 1)
self.state = 287
_la = self._input.LA(1)
if not((((_la) & ~0x3f) == 0 and ((1 << _la) & ((1 << LaTeXParser.CMD_BINOM) | (1 << LaTeXParser.CMD_DBINOM) | (1 << LaTeXParser.CMD_TBINOM))) != 0)):
self._errHandler.recoverInline(self)
else:
self._errHandler.reportMatch(self)
self.consume()
self.state = 288
self.match(LaTeXParser.L_BRACE)
self.state = 289
localctx.n = self.expr()
self.state = 290
self.match(LaTeXParser.R_BRACE)
self.state = 291
self.match(LaTeXParser.L_BRACE)
self.state = 292
localctx.k = self.expr()
self.state = 293
self.match(LaTeXParser.R_BRACE)
except RecognitionException as re:
localctx.exception = re
self._errHandler.reportError(self, re)
self._errHandler.recover(self, re)
finally:
self.exitRule()
return localctx
class Func_normalContext(ParserRuleContext):
def __init__(self, parser, parent=None, invokingState=-1):
super(LaTeXParser.Func_normalContext, self).__init__(parent, invokingState)
self.parser = parser
def FUNC_EXP(self):
return self.getToken(LaTeXParser.FUNC_EXP, 0)
def FUNC_LOG(self):
return self.getToken(LaTeXParser.FUNC_LOG, 0)
def FUNC_LN(self):
return self.getToken(LaTeXParser.FUNC_LN, 0)
def FUNC_SIN(self):
return self.getToken(LaTeXParser.FUNC_SIN, 0)
def FUNC_COS(self):
return self.getToken(LaTeXParser.FUNC_COS, 0)
def FUNC_TAN(self):
return self.getToken(LaTeXParser.FUNC_TAN, 0)
def FUNC_CSC(self):
return self.getToken(LaTeXParser.FUNC_CSC, 0)
def FUNC_SEC(self):
return self.getToken(LaTeXParser.FUNC_SEC, 0)
def FUNC_COT(self):
return self.getToken(LaTeXParser.FUNC_COT, 0)
def FUNC_ARCSIN(self):
return self.getToken(LaTeXParser.FUNC_ARCSIN, 0)
def FUNC_ARCCOS(self):
return self.getToken(LaTeXParser.FUNC_ARCCOS, 0)
def FUNC_ARCTAN(self):
return self.getToken(LaTeXParser.FUNC_ARCTAN, 0)
def FUNC_ARCCSC(self):
return self.getToken(LaTeXParser.FUNC_ARCCSC, 0)
def FUNC_ARCSEC(self):
return self.getToken(LaTeXParser.FUNC_ARCSEC, 0)
def FUNC_ARCCOT(self):
return self.getToken(LaTeXParser.FUNC_ARCCOT, 0)
def FUNC_SINH(self):
return self.getToken(LaTeXParser.FUNC_SINH, 0)
def FUNC_COSH(self):
return self.getToken(LaTeXParser.FUNC_COSH, 0)
def FUNC_TANH(self):
return self.getToken(LaTeXParser.FUNC_TANH, 0)
def FUNC_ARSINH(self):
return self.getToken(LaTeXParser.FUNC_ARSINH, 0)
def FUNC_ARCOSH(self):
return self.getToken(LaTeXParser.FUNC_ARCOSH, 0)
def FUNC_ARTANH(self):
return self.getToken(LaTeXParser.FUNC_ARTANH, 0)
def getRuleIndex(self):
return LaTeXParser.RULE_func_normal
def func_normal(self):
localctx = LaTeXParser.Func_normalContext(self, self._ctx, self.state)
self.enterRule(localctx, 52, self.RULE_func_normal)
self._la = 0 # Token type
try:
self.enterOuterAlt(localctx, 1)
self.state = 295
_la = self._input.LA(1)
if not((((_la) & ~0x3f) == 0 and ((1 << _la) & ((1 << LaTeXParser.FUNC_EXP) | (1 << LaTeXParser.FUNC_LOG) | (1 << LaTeXParser.FUNC_LN) | (1 << LaTeXParser.FUNC_SIN) | (1 << LaTeXParser.FUNC_COS) | (1 << LaTeXParser.FUNC_TAN) | (1 << LaTeXParser.FUNC_CSC) | (1 << LaTeXParser.FUNC_SEC) | (1 << LaTeXParser.FUNC_COT) | (1 << LaTeXParser.FUNC_ARCSIN) | (1 << LaTeXParser.FUNC_ARCCOS) | (1 << LaTeXParser.FUNC_ARCTAN) | (1 << LaTeXParser.FUNC_ARCCSC) | (1 << LaTeXParser.FUNC_ARCSEC) | (1 << LaTeXParser.FUNC_ARCCOT) | (1 << LaTeXParser.FUNC_SINH) | (1 << LaTeXParser.FUNC_COSH) | (1 << LaTeXParser.FUNC_TANH) | (1 << LaTeXParser.FUNC_ARSINH) | (1 << LaTeXParser.FUNC_ARCOSH) | (1 << LaTeXParser.FUNC_ARTANH))) != 0)):
self._errHandler.recoverInline(self)
else:
self._errHandler.reportMatch(self)
self.consume()
except RecognitionException as re:
localctx.exception = re
self._errHandler.reportError(self, re)
self._errHandler.recover(self, re)
finally:
self.exitRule()
return localctx
class FuncContext(ParserRuleContext):
def __init__(self, parser, parent=None, invokingState=-1):
super(LaTeXParser.FuncContext, self).__init__(parent, invokingState)
self.parser = parser
self.root = None # ExprContext
self.base = None # ExprContext
def func_normal(self):
return self.getTypedRuleContext(LaTeXParser.Func_normalContext,0)
def L_PAREN(self):
return self.getToken(LaTeXParser.L_PAREN, 0)
def func_arg(self):
return self.getTypedRuleContext(LaTeXParser.Func_argContext,0)
def R_PAREN(self):
return self.getToken(LaTeXParser.R_PAREN, 0)
def func_arg_noparens(self):
return self.getTypedRuleContext(LaTeXParser.Func_arg_noparensContext,0)
def subexpr(self):
return self.getTypedRuleContext(LaTeXParser.SubexprContext,0)
def supexpr(self):
return self.getTypedRuleContext(LaTeXParser.SupexprContext,0)
def args(self):
return self.getTypedRuleContext(LaTeXParser.ArgsContext,0)
def LETTER(self):
return self.getToken(LaTeXParser.LETTER, 0)
def SYMBOL(self):
return self.getToken(LaTeXParser.SYMBOL, 0)
def FUNC_INT(self):
return self.getToken(LaTeXParser.FUNC_INT, 0)
def DIFFERENTIAL(self):
return self.getToken(LaTeXParser.DIFFERENTIAL, 0)
def frac(self):
return self.getTypedRuleContext(LaTeXParser.FracContext,0)
def additive(self):
return self.getTypedRuleContext(LaTeXParser.AdditiveContext,0)
def FUNC_SQRT(self):
return self.getToken(LaTeXParser.FUNC_SQRT, 0)
def L_BRACE(self):
return self.getToken(LaTeXParser.L_BRACE, 0)
def R_BRACE(self):
return self.getToken(LaTeXParser.R_BRACE, 0)
def expr(self, i=None):
if i is None:
return self.getTypedRuleContexts(LaTeXParser.ExprContext)
else:
return self.getTypedRuleContext(LaTeXParser.ExprContext,i)
def L_BRACKET(self):
return self.getToken(LaTeXParser.L_BRACKET, 0)
def R_BRACKET(self):
return self.getToken(LaTeXParser.R_BRACKET, 0)
def mp(self):
return self.getTypedRuleContext(LaTeXParser.MpContext,0)
def FUNC_SUM(self):
return self.getToken(LaTeXParser.FUNC_SUM, 0)
def FUNC_PROD(self):
return self.getToken(LaTeXParser.FUNC_PROD, 0)
def subeq(self):
return self.getTypedRuleContext(LaTeXParser.SubeqContext,0)
def FUNC_LIM(self):
return self.getToken(LaTeXParser.FUNC_LIM, 0)
def limit_sub(self):
return self.getTypedRuleContext(LaTeXParser.Limit_subContext,0)
def getRuleIndex(self):
return LaTeXParser.RULE_func
def func(self):
localctx = LaTeXParser.FuncContext(self, self._ctx, self.state)
self.enterRule(localctx, 54, self.RULE_func)
self._la = 0 # Token type
try:
self.state = 370
self._errHandler.sync(self)
token = self._input.LA(1)
if token in [LaTeXParser.FUNC_EXP, LaTeXParser.FUNC_LOG, LaTeXParser.FUNC_LN, LaTeXParser.FUNC_SIN, LaTeXParser.FUNC_COS, LaTeXParser.FUNC_TAN, LaTeXParser.FUNC_CSC, LaTeXParser.FUNC_SEC, LaTeXParser.FUNC_COT, LaTeXParser.FUNC_ARCSIN, LaTeXParser.FUNC_ARCCOS, LaTeXParser.FUNC_ARCTAN, LaTeXParser.FUNC_ARCCSC, LaTeXParser.FUNC_ARCSEC, LaTeXParser.FUNC_ARCCOT, LaTeXParser.FUNC_SINH, LaTeXParser.FUNC_COSH, LaTeXParser.FUNC_TANH, LaTeXParser.FUNC_ARSINH, LaTeXParser.FUNC_ARCOSH, LaTeXParser.FUNC_ARTANH]:
self.enterOuterAlt(localctx, 1)
self.state = 297
self.func_normal()
self.state = 310
self._errHandler.sync(self)
la_ = self._interp.adaptivePredict(self._input,30,self._ctx)
if la_ == 1:
self.state = 299
self._errHandler.sync(self)
_la = self._input.LA(1)
if _la==LaTeXParser.UNDERSCORE:
self.state = 298
self.subexpr()
self.state = 302
self._errHandler.sync(self)
_la = self._input.LA(1)
if _la==LaTeXParser.CARET:
self.state = 301
self.supexpr()
pass
elif la_ == 2:
self.state = 305
self._errHandler.sync(self)
_la = self._input.LA(1)
if _la==LaTeXParser.CARET:
self.state = 304
self.supexpr()
self.state = 308
self._errHandler.sync(self)
_la = self._input.LA(1)
if _la==LaTeXParser.UNDERSCORE:
self.state = 307
self.subexpr()
pass
self.state = 317
self._errHandler.sync(self)
la_ = self._interp.adaptivePredict(self._input,31,self._ctx)
if la_ == 1:
self.state = 312
self.match(LaTeXParser.L_PAREN)
self.state = 313
self.func_arg()
self.state = 314
self.match(LaTeXParser.R_PAREN)
pass
elif la_ == 2:
self.state = 316
self.func_arg_noparens()
pass
pass
elif token in [LaTeXParser.LETTER, LaTeXParser.SYMBOL]:
self.enterOuterAlt(localctx, 2)
self.state = 319
_la = self._input.LA(1)
if not(_la==LaTeXParser.LETTER or _la==LaTeXParser.SYMBOL):
self._errHandler.recoverInline(self)
else:
self._errHandler.reportMatch(self)
self.consume()
self.state = 321
self._errHandler.sync(self)
_la = self._input.LA(1)
if _la==LaTeXParser.UNDERSCORE:
self.state = 320
self.subexpr()
self.state = 323
self.match(LaTeXParser.L_PAREN)
self.state = 324
self.args()
self.state = 325
self.match(LaTeXParser.R_PAREN)
pass
elif token in [LaTeXParser.FUNC_INT]:
self.enterOuterAlt(localctx, 3)
self.state = 327
self.match(LaTeXParser.FUNC_INT)
self.state = 334
self._errHandler.sync(self)
token = self._input.LA(1)
if token in [LaTeXParser.UNDERSCORE]:
self.state = 328
self.subexpr()
self.state = 329
self.supexpr()
pass
elif token in [LaTeXParser.CARET]:
self.state = 331
self.supexpr()
self.state = 332
self.subexpr()
pass
elif token in [LaTeXParser.ADD, LaTeXParser.SUB, LaTeXParser.L_PAREN, LaTeXParser.L_BRACE, LaTeXParser.L_BRACKET, LaTeXParser.BAR, LaTeXParser.FUNC_LIM, LaTeXParser.FUNC_INT, LaTeXParser.FUNC_SUM, LaTeXParser.FUNC_PROD, LaTeXParser.FUNC_EXP, LaTeXParser.FUNC_LOG, LaTeXParser.FUNC_LN, LaTeXParser.FUNC_SIN, LaTeXParser.FUNC_COS, LaTeXParser.FUNC_TAN, LaTeXParser.FUNC_CSC, LaTeXParser.FUNC_SEC, LaTeXParser.FUNC_COT, LaTeXParser.FUNC_ARCSIN, LaTeXParser.FUNC_ARCCOS, LaTeXParser.FUNC_ARCTAN, LaTeXParser.FUNC_ARCCSC, LaTeXParser.FUNC_ARCSEC, LaTeXParser.FUNC_ARCCOT, LaTeXParser.FUNC_SINH, LaTeXParser.FUNC_COSH, LaTeXParser.FUNC_TANH, LaTeXParser.FUNC_ARSINH, LaTeXParser.FUNC_ARCOSH, LaTeXParser.FUNC_ARTANH, LaTeXParser.FUNC_SQRT, LaTeXParser.CMD_FRAC, LaTeXParser.CMD_BINOM, LaTeXParser.CMD_DBINOM, LaTeXParser.CMD_TBINOM, LaTeXParser.CMD_MATHIT, LaTeXParser.DIFFERENTIAL, LaTeXParser.LETTER, LaTeXParser.NUMBER, LaTeXParser.SYMBOL]:
pass
else:
pass
self.state = 342
self._errHandler.sync(self)
la_ = self._interp.adaptivePredict(self._input,35,self._ctx)
if la_ == 1:
self.state = 337
self._errHandler.sync(self)
la_ = self._interp.adaptivePredict(self._input,34,self._ctx)
if la_ == 1:
self.state = 336
self.additive(0)
self.state = 339
self.match(LaTeXParser.DIFFERENTIAL)
pass
elif la_ == 2:
self.state = 340
self.frac()
pass
elif la_ == 3:
self.state = 341
self.additive(0)
pass
pass
elif token in [LaTeXParser.FUNC_SQRT]:
self.enterOuterAlt(localctx, 4)
self.state = 344
self.match(LaTeXParser.FUNC_SQRT)
self.state = 349
self._errHandler.sync(self)
_la = self._input.LA(1)
if _la==LaTeXParser.L_BRACKET:
self.state = 345
self.match(LaTeXParser.L_BRACKET)
self.state = 346
localctx.root = self.expr()
self.state = 347
self.match(LaTeXParser.R_BRACKET)
self.state = 351
self.match(LaTeXParser.L_BRACE)
self.state = 352
localctx.base = self.expr()
self.state = 353
self.match(LaTeXParser.R_BRACE)
pass
elif token in [LaTeXParser.FUNC_SUM, LaTeXParser.FUNC_PROD]:
self.enterOuterAlt(localctx, 5)
self.state = 355
_la = self._input.LA(1)
if not(_la==LaTeXParser.FUNC_SUM or _la==LaTeXParser.FUNC_PROD):
self._errHandler.recoverInline(self)
else:
self._errHandler.reportMatch(self)
self.consume()
self.state = 362
self._errHandler.sync(self)
token = self._input.LA(1)
if token in [LaTeXParser.UNDERSCORE]:
self.state = 356
self.subeq()
self.state = 357
self.supexpr()
pass
elif token in [LaTeXParser.CARET]:
self.state = 359
self.supexpr()
self.state = 360
self.subeq()
pass
else:
raise NoViableAltException(self)
self.state = 364
self.mp(0)
pass
elif token in [LaTeXParser.FUNC_LIM]:
self.enterOuterAlt(localctx, 6)
self.state = 366
self.match(LaTeXParser.FUNC_LIM)
self.state = 367
self.limit_sub()
self.state = 368
self.mp(0)
pass
else:
raise NoViableAltException(self)
except RecognitionException as re:
localctx.exception = re
self._errHandler.reportError(self, re)
self._errHandler.recover(self, re)
finally:
self.exitRule()
return localctx
class ArgsContext(ParserRuleContext):
def __init__(self, parser, parent=None, invokingState=-1):
super(LaTeXParser.ArgsContext, self).__init__(parent, invokingState)
self.parser = parser
def expr(self):
return self.getTypedRuleContext(LaTeXParser.ExprContext,0)
def args(self):
return self.getTypedRuleContext(LaTeXParser.ArgsContext,0)
def getRuleIndex(self):
return LaTeXParser.RULE_args
def args(self):
localctx = LaTeXParser.ArgsContext(self, self._ctx, self.state)
self.enterRule(localctx, 56, self.RULE_args)
try:
self.state = 377
self._errHandler.sync(self)
la_ = self._interp.adaptivePredict(self._input,39,self._ctx)
if la_ == 1:
self.enterOuterAlt(localctx, 1)
self.state = 372
self.expr()
self.state = 373
self.match(LaTeXParser.T__0)
self.state = 374
self.args()
pass
elif la_ == 2:
self.enterOuterAlt(localctx, 2)
self.state = 376
self.expr()
pass
except RecognitionException as re:
localctx.exception = re
self._errHandler.reportError(self, re)
self._errHandler.recover(self, re)
finally:
self.exitRule()
return localctx
class Limit_subContext(ParserRuleContext):
def __init__(self, parser, parent=None, invokingState=-1):
super(LaTeXParser.Limit_subContext, self).__init__(parent, invokingState)
self.parser = parser
def UNDERSCORE(self):
return self.getToken(LaTeXParser.UNDERSCORE, 0)
def L_BRACE(self, i=None):
if i is None:
return self.getTokens(LaTeXParser.L_BRACE)
else:
return self.getToken(LaTeXParser.L_BRACE, i)
def LIM_APPROACH_SYM(self):
return self.getToken(LaTeXParser.LIM_APPROACH_SYM, 0)
def expr(self):
return self.getTypedRuleContext(LaTeXParser.ExprContext,0)
def R_BRACE(self, i=None):
if i is None:
return self.getTokens(LaTeXParser.R_BRACE)
else:
return self.getToken(LaTeXParser.R_BRACE, i)
def LETTER(self):
return self.getToken(LaTeXParser.LETTER, 0)
def SYMBOL(self):
return self.getToken(LaTeXParser.SYMBOL, 0)
def CARET(self):
return self.getToken(LaTeXParser.CARET, 0)
def ADD(self):
return self.getToken(LaTeXParser.ADD, 0)
def SUB(self):
return self.getToken(LaTeXParser.SUB, 0)
def getRuleIndex(self):
return LaTeXParser.RULE_limit_sub
def limit_sub(self):
localctx = LaTeXParser.Limit_subContext(self, self._ctx, self.state)
self.enterRule(localctx, 58, self.RULE_limit_sub)
self._la = 0 # Token type
try:
self.enterOuterAlt(localctx, 1)
self.state = 379
self.match(LaTeXParser.UNDERSCORE)
self.state = 380
self.match(LaTeXParser.L_BRACE)
self.state = 381
_la = self._input.LA(1)
if not(_la==LaTeXParser.LETTER or _la==LaTeXParser.SYMBOL):
self._errHandler.recoverInline(self)
else:
self._errHandler.reportMatch(self)
self.consume()
self.state = 382
self.match(LaTeXParser.LIM_APPROACH_SYM)
self.state = 383
self.expr()
self.state = 388
self._errHandler.sync(self)
_la = self._input.LA(1)
if _la==LaTeXParser.CARET:
self.state = 384
self.match(LaTeXParser.CARET)
self.state = 385
self.match(LaTeXParser.L_BRACE)
self.state = 386
_la = self._input.LA(1)
if not(_la==LaTeXParser.ADD or _la==LaTeXParser.SUB):
self._errHandler.recoverInline(self)
else:
self._errHandler.reportMatch(self)
self.consume()
self.state = 387
self.match(LaTeXParser.R_BRACE)
self.state = 390
self.match(LaTeXParser.R_BRACE)
except RecognitionException as re:
localctx.exception = re
self._errHandler.reportError(self, re)
self._errHandler.recover(self, re)
finally:
self.exitRule()
return localctx
class Func_argContext(ParserRuleContext):
def __init__(self, parser, parent=None, invokingState=-1):
super(LaTeXParser.Func_argContext, self).__init__(parent, invokingState)
self.parser = parser
def expr(self):
return self.getTypedRuleContext(LaTeXParser.ExprContext,0)
def func_arg(self):
return self.getTypedRuleContext(LaTeXParser.Func_argContext,0)
def getRuleIndex(self):
return LaTeXParser.RULE_func_arg
def func_arg(self):
localctx = LaTeXParser.Func_argContext(self, self._ctx, self.state)
self.enterRule(localctx, 60, self.RULE_func_arg)
try:
self.state = 397
self._errHandler.sync(self)
la_ = self._interp.adaptivePredict(self._input,41,self._ctx)
if la_ == 1:
self.enterOuterAlt(localctx, 1)
self.state = 392
self.expr()
pass
elif la_ == 2:
self.enterOuterAlt(localctx, 2)
self.state = 393
self.expr()
self.state = 394
self.match(LaTeXParser.T__0)
self.state = 395
self.func_arg()
pass
except RecognitionException as re:
localctx.exception = re
self._errHandler.reportError(self, re)
self._errHandler.recover(self, re)
finally:
self.exitRule()
return localctx
class Func_arg_noparensContext(ParserRuleContext):
def __init__(self, parser, parent=None, invokingState=-1):
super(LaTeXParser.Func_arg_noparensContext, self).__init__(parent, invokingState)
self.parser = parser
def mp_nofunc(self):
return self.getTypedRuleContext(LaTeXParser.Mp_nofuncContext,0)
def getRuleIndex(self):
return LaTeXParser.RULE_func_arg_noparens
def func_arg_noparens(self):
localctx = LaTeXParser.Func_arg_noparensContext(self, self._ctx, self.state)
self.enterRule(localctx, 62, self.RULE_func_arg_noparens)
try:
self.enterOuterAlt(localctx, 1)
self.state = 399
self.mp_nofunc(0)
except RecognitionException as re:
localctx.exception = re
self._errHandler.reportError(self, re)
self._errHandler.recover(self, re)
finally:
self.exitRule()
return localctx
class SubexprContext(ParserRuleContext):
def __init__(self, parser, parent=None, invokingState=-1):
super(LaTeXParser.SubexprContext, self).__init__(parent, invokingState)
self.parser = parser
def UNDERSCORE(self):
return self.getToken(LaTeXParser.UNDERSCORE, 0)
def atom(self):
return self.getTypedRuleContext(LaTeXParser.AtomContext,0)
def L_BRACE(self):
return self.getToken(LaTeXParser.L_BRACE, 0)
def expr(self):
return self.getTypedRuleContext(LaTeXParser.ExprContext,0)
def R_BRACE(self):
return self.getToken(LaTeXParser.R_BRACE, 0)
def getRuleIndex(self):
return LaTeXParser.RULE_subexpr
def subexpr(self):
localctx = LaTeXParser.SubexprContext(self, self._ctx, self.state)
self.enterRule(localctx, 64, self.RULE_subexpr)
try:
self.enterOuterAlt(localctx, 1)
self.state = 401
self.match(LaTeXParser.UNDERSCORE)
self.state = 407
self._errHandler.sync(self)
token = self._input.LA(1)
if token in [LaTeXParser.CMD_MATHIT, LaTeXParser.DIFFERENTIAL, LaTeXParser.LETTER, LaTeXParser.NUMBER, LaTeXParser.SYMBOL]:
self.state = 402
self.atom()
pass
elif token in [LaTeXParser.L_BRACE]:
self.state = 403
self.match(LaTeXParser.L_BRACE)
self.state = 404
self.expr()
self.state = 405
self.match(LaTeXParser.R_BRACE)
pass
else:
raise NoViableAltException(self)
except RecognitionException as re:
localctx.exception = re
self._errHandler.reportError(self, re)
self._errHandler.recover(self, re)
finally:
self.exitRule()
return localctx
class SupexprContext(ParserRuleContext):
def __init__(self, parser, parent=None, invokingState=-1):
super(LaTeXParser.SupexprContext, self).__init__(parent, invokingState)
self.parser = parser
def CARET(self):
return self.getToken(LaTeXParser.CARET, 0)
def atom(self):
return self.getTypedRuleContext(LaTeXParser.AtomContext,0)
def L_BRACE(self):
return self.getToken(LaTeXParser.L_BRACE, 0)
def expr(self):
return self.getTypedRuleContext(LaTeXParser.ExprContext,0)
def R_BRACE(self):
return self.getToken(LaTeXParser.R_BRACE, 0)
def getRuleIndex(self):
return LaTeXParser.RULE_supexpr
def supexpr(self):
localctx = LaTeXParser.SupexprContext(self, self._ctx, self.state)
self.enterRule(localctx, 66, self.RULE_supexpr)
try:
self.enterOuterAlt(localctx, 1)
self.state = 409
self.match(LaTeXParser.CARET)
self.state = 415
self._errHandler.sync(self)
token = self._input.LA(1)
if token in [LaTeXParser.CMD_MATHIT, LaTeXParser.DIFFERENTIAL, LaTeXParser.LETTER, LaTeXParser.NUMBER, LaTeXParser.SYMBOL]:
self.state = 410
self.atom()
pass
elif token in [LaTeXParser.L_BRACE]:
self.state = 411
self.match(LaTeXParser.L_BRACE)
self.state = 412
self.expr()
self.state = 413
self.match(LaTeXParser.R_BRACE)
pass
else:
raise NoViableAltException(self)
except RecognitionException as re:
localctx.exception = re
self._errHandler.reportError(self, re)
self._errHandler.recover(self, re)
finally:
self.exitRule()
return localctx
class SubeqContext(ParserRuleContext):
def __init__(self, parser, parent=None, invokingState=-1):
super(LaTeXParser.SubeqContext, self).__init__(parent, invokingState)
self.parser = parser
def UNDERSCORE(self):
return self.getToken(LaTeXParser.UNDERSCORE, 0)
def L_BRACE(self):
return self.getToken(LaTeXParser.L_BRACE, 0)
def equality(self):
return self.getTypedRuleContext(LaTeXParser.EqualityContext,0)
def R_BRACE(self):
return self.getToken(LaTeXParser.R_BRACE, 0)
def getRuleIndex(self):
return LaTeXParser.RULE_subeq
def subeq(self):
localctx = LaTeXParser.SubeqContext(self, self._ctx, self.state)
self.enterRule(localctx, 68, self.RULE_subeq)
try:
self.enterOuterAlt(localctx, 1)
self.state = 417
self.match(LaTeXParser.UNDERSCORE)
self.state = 418
self.match(LaTeXParser.L_BRACE)
self.state = 419
self.equality()
self.state = 420
self.match(LaTeXParser.R_BRACE)
except RecognitionException as re:
localctx.exception = re
self._errHandler.reportError(self, re)
self._errHandler.recover(self, re)
finally:
self.exitRule()
return localctx
class SupeqContext(ParserRuleContext):
def __init__(self, parser, parent=None, invokingState=-1):
super(LaTeXParser.SupeqContext, self).__init__(parent, invokingState)
self.parser = parser
def UNDERSCORE(self):
return self.getToken(LaTeXParser.UNDERSCORE, 0)
def L_BRACE(self):
return self.getToken(LaTeXParser.L_BRACE, 0)
def equality(self):
return self.getTypedRuleContext(LaTeXParser.EqualityContext,0)
def R_BRACE(self):
return self.getToken(LaTeXParser.R_BRACE, 0)
def getRuleIndex(self):
return LaTeXParser.RULE_supeq
def supeq(self):
localctx = LaTeXParser.SupeqContext(self, self._ctx, self.state)
self.enterRule(localctx, 70, self.RULE_supeq)
try:
self.enterOuterAlt(localctx, 1)
self.state = 422
self.match(LaTeXParser.UNDERSCORE)
self.state = 423
self.match(LaTeXParser.L_BRACE)
self.state = 424
self.equality()
self.state = 425
self.match(LaTeXParser.R_BRACE)
except RecognitionException as re:
localctx.exception = re
self._errHandler.reportError(self, re)
self._errHandler.recover(self, re)
finally:
self.exitRule()
return localctx
def sempred(self, localctx, ruleIndex, predIndex):
if self._predicates == None:
self._predicates = dict()
self._predicates[1] = self.relation_sempred
self._predicates[4] = self.additive_sempred
self._predicates[5] = self.mp_sempred
self._predicates[6] = self.mp_nofunc_sempred
self._predicates[15] = self.exp_sempred
self._predicates[16] = self.exp_nofunc_sempred
pred = self._predicates.get(ruleIndex, None)
if pred is None:
raise Exception("No predicate with index:" + str(ruleIndex))
else:
return pred(localctx, predIndex)
def relation_sempred(self, localctx, predIndex):
if predIndex == 0:
return self.precpred(self._ctx, 2)
def additive_sempred(self, localctx, predIndex):
if predIndex == 1:
return self.precpred(self._ctx, 2)
def mp_sempred(self, localctx, predIndex):
if predIndex == 2:
return self.precpred(self._ctx, 2)
def mp_nofunc_sempred(self, localctx, predIndex):
if predIndex == 3:
return self.precpred(self._ctx, 2)
def exp_sempred(self, localctx, predIndex):
if predIndex == 4:
return self.precpred(self._ctx, 2)
def exp_nofunc_sempred(self, localctx, predIndex):
if predIndex == 5:
return self.precpred(self._ctx, 2)
|
159d491a31059d71f325b8ec0325284b59ec0bca5d32ff9c41c1d7d1eaf78036 | from __future__ import print_function, division
from collections import deque
from random import randint
from sympy.external import import_module
from sympy import Mul, Basic, Number, Pow, Integer
from sympy.physics.quantum.represent import represent
from sympy.physics.quantum.dagger import Dagger
__all__ = [
# Public interfaces
'generate_gate_rules',
'generate_equivalent_ids',
'GateIdentity',
'bfs_identity_search',
'random_identity_search',
# "Private" functions
'is_scalar_sparse_matrix',
'is_scalar_nonsparse_matrix',
'is_degenerate',
'is_reducible',
]
np = import_module('numpy')
scipy = import_module('scipy', import_kwargs={'fromlist': ['sparse']})
def is_scalar_sparse_matrix(circuit, nqubits, identity_only, eps=1e-11):
"""Checks if a given scipy.sparse matrix is a scalar matrix.
A scalar matrix is such that B = bI, where B is the scalar
matrix, b is some scalar multiple, and I is the identity
matrix. A scalar matrix would have only the element b along
it's main diagonal and zeroes elsewhere.
Parameters
==========
circuit : Gate tuple
Sequence of quantum gates representing a quantum circuit
nqubits : int
Number of qubits in the circuit
identity_only : bool
Check for only identity matrices
eps : number
The tolerance value for zeroing out elements in the matrix.
Values in the range [-eps, +eps] will be changed to a zero.
"""
if not np or not scipy:
pass
matrix = represent(Mul(*circuit), nqubits=nqubits,
format='scipy.sparse')
# In some cases, represent returns a 1D scalar value in place
# of a multi-dimensional scalar matrix
if (isinstance(matrix, int)):
return matrix == 1 if identity_only else True
# If represent returns a matrix, check if the matrix is diagonal
# and if every item along the diagonal is the same
else:
# Due to floating pointing operations, must zero out
# elements that are "very" small in the dense matrix
# See parameter for default value.
# Get the ndarray version of the dense matrix
dense_matrix = matrix.todense().getA()
# Since complex values can't be compared, must split
# the matrix into real and imaginary components
# Find the real values in between -eps and eps
bool_real = np.logical_and(dense_matrix.real > -eps,
dense_matrix.real < eps)
# Find the imaginary values between -eps and eps
bool_imag = np.logical_and(dense_matrix.imag > -eps,
dense_matrix.imag < eps)
# Replaces values between -eps and eps with 0
corrected_real = np.where(bool_real, 0.0, dense_matrix.real)
corrected_imag = np.where(bool_imag, 0.0, dense_matrix.imag)
# Convert the matrix with real values into imaginary values
corrected_imag = corrected_imag * np.complex(1j)
# Recombine the real and imaginary components
corrected_dense = corrected_real + corrected_imag
# Check if it's diagonal
row_indices = corrected_dense.nonzero()[0]
col_indices = corrected_dense.nonzero()[1]
# Check if the rows indices and columns indices are the same
# If they match, then matrix only contains elements along diagonal
bool_indices = row_indices == col_indices
is_diagonal = bool_indices.all()
first_element = corrected_dense[0][0]
# If the first element is a zero, then can't rescale matrix
# and definitely not diagonal
if (first_element == 0.0 + 0.0j):
return False
# The dimensions of the dense matrix should still
# be 2^nqubits if there are elements all along the
# the main diagonal
trace_of_corrected = (corrected_dense/first_element).trace()
expected_trace = pow(2, nqubits)
has_correct_trace = trace_of_corrected == expected_trace
# If only looking for identity matrices
# first element must be a 1
real_is_one = abs(first_element.real - 1.0) < eps
imag_is_zero = abs(first_element.imag) < eps
is_one = real_is_one and imag_is_zero
is_identity = is_one if identity_only else True
return bool(is_diagonal and has_correct_trace and is_identity)
def is_scalar_nonsparse_matrix(circuit, nqubits, identity_only, eps=None):
"""Checks if a given circuit, in matrix form, is equivalent to
a scalar value.
Parameters
==========
circuit : Gate tuple
Sequence of quantum gates representing a quantum circuit
nqubits : int
Number of qubits in the circuit
identity_only : bool
Check for only identity matrices
eps : number
This argument is ignored. It is just for signature compatibility with
is_scalar_sparse_matrix.
Note: Used in situations when is_scalar_sparse_matrix has bugs
"""
matrix = represent(Mul(*circuit), nqubits=nqubits)
# In some cases, represent returns a 1D scalar value in place
# of a multi-dimensional scalar matrix
if (isinstance(matrix, Number)):
return matrix == 1 if identity_only else True
# If represent returns a matrix, check if the matrix is diagonal
# and if every item along the diagonal is the same
else:
# Added up the diagonal elements
matrix_trace = matrix.trace()
# Divide the trace by the first element in the matrix
# if matrix is not required to be the identity matrix
adjusted_matrix_trace = (matrix_trace/matrix[0]
if not identity_only
else matrix_trace)
is_identity = matrix[0] == 1.0 if identity_only else True
has_correct_trace = adjusted_matrix_trace == pow(2, nqubits)
# The matrix is scalar if it's diagonal and the adjusted trace
# value is equal to 2^nqubits
return bool(
matrix.is_diagonal() and has_correct_trace and is_identity)
if np and scipy:
is_scalar_matrix = is_scalar_sparse_matrix
else:
is_scalar_matrix = is_scalar_nonsparse_matrix
def _get_min_qubits(a_gate):
if isinstance(a_gate, Pow):
return a_gate.base.min_qubits
else:
return a_gate.min_qubits
def ll_op(left, right):
"""Perform a LL operation.
A LL operation multiplies both left and right circuits
with the dagger of the left circuit's leftmost gate, and
the dagger is multiplied on the left side of both circuits.
If a LL is possible, it returns the new gate rule as a
2-tuple (LHS, RHS), where LHS is the left circuit and
and RHS is the right circuit of the new rule.
If a LL is not possible, None is returned.
Parameters
==========
left : Gate tuple
The left circuit of a gate rule expression.
right : Gate tuple
The right circuit of a gate rule expression.
Examples
========
Generate a new gate rule using a LL operation:
>>> from sympy.physics.quantum.identitysearch import ll_op
>>> from sympy.physics.quantum.gate import X, Y, Z
>>> x = X(0); y = Y(0); z = Z(0)
>>> ll_op((x, y, z), ())
((Y(0), Z(0)), (X(0),))
>>> ll_op((y, z), (x,))
((Z(0),), (Y(0), X(0)))
"""
if (len(left) > 0):
ll_gate = left[0]
ll_gate_is_unitary = is_scalar_matrix(
(Dagger(ll_gate), ll_gate), _get_min_qubits(ll_gate), True)
if (len(left) > 0 and ll_gate_is_unitary):
# Get the new left side w/o the leftmost gate
new_left = left[1:len(left)]
# Add the leftmost gate to the left position on the right side
new_right = (Dagger(ll_gate),) + right
# Return the new gate rule
return (new_left, new_right)
return None
def lr_op(left, right):
"""Perform a LR operation.
A LR operation multiplies both left and right circuits
with the dagger of the left circuit's rightmost gate, and
the dagger is multiplied on the right side of both circuits.
If a LR is possible, it returns the new gate rule as a
2-tuple (LHS, RHS), where LHS is the left circuit and
and RHS is the right circuit of the new rule.
If a LR is not possible, None is returned.
Parameters
==========
left : Gate tuple
The left circuit of a gate rule expression.
right : Gate tuple
The right circuit of a gate rule expression.
Examples
========
Generate a new gate rule using a LR operation:
>>> from sympy.physics.quantum.identitysearch import lr_op
>>> from sympy.physics.quantum.gate import X, Y, Z
>>> x = X(0); y = Y(0); z = Z(0)
>>> lr_op((x, y, z), ())
((X(0), Y(0)), (Z(0),))
>>> lr_op((x, y), (z,))
((X(0),), (Z(0), Y(0)))
"""
if (len(left) > 0):
lr_gate = left[len(left) - 1]
lr_gate_is_unitary = is_scalar_matrix(
(Dagger(lr_gate), lr_gate), _get_min_qubits(lr_gate), True)
if (len(left) > 0 and lr_gate_is_unitary):
# Get the new left side w/o the rightmost gate
new_left = left[0:len(left) - 1]
# Add the rightmost gate to the right position on the right side
new_right = right + (Dagger(lr_gate),)
# Return the new gate rule
return (new_left, new_right)
return None
def rl_op(left, right):
"""Perform a RL operation.
A RL operation multiplies both left and right circuits
with the dagger of the right circuit's leftmost gate, and
the dagger is multiplied on the left side of both circuits.
If a RL is possible, it returns the new gate rule as a
2-tuple (LHS, RHS), where LHS is the left circuit and
and RHS is the right circuit of the new rule.
If a RL is not possible, None is returned.
Parameters
==========
left : Gate tuple
The left circuit of a gate rule expression.
right : Gate tuple
The right circuit of a gate rule expression.
Examples
========
Generate a new gate rule using a RL operation:
>>> from sympy.physics.quantum.identitysearch import rl_op
>>> from sympy.physics.quantum.gate import X, Y, Z
>>> x = X(0); y = Y(0); z = Z(0)
>>> rl_op((x,), (y, z))
((Y(0), X(0)), (Z(0),))
>>> rl_op((x, y), (z,))
((Z(0), X(0), Y(0)), ())
"""
if (len(right) > 0):
rl_gate = right[0]
rl_gate_is_unitary = is_scalar_matrix(
(Dagger(rl_gate), rl_gate), _get_min_qubits(rl_gate), True)
if (len(right) > 0 and rl_gate_is_unitary):
# Get the new right side w/o the leftmost gate
new_right = right[1:len(right)]
# Add the leftmost gate to the left position on the left side
new_left = (Dagger(rl_gate),) + left
# Return the new gate rule
return (new_left, new_right)
return None
def rr_op(left, right):
"""Perform a RR operation.
A RR operation multiplies both left and right circuits
with the dagger of the right circuit's rightmost gate, and
the dagger is multiplied on the right side of both circuits.
If a RR is possible, it returns the new gate rule as a
2-tuple (LHS, RHS), where LHS is the left circuit and
and RHS is the right circuit of the new rule.
If a RR is not possible, None is returned.
Parameters
==========
left : Gate tuple
The left circuit of a gate rule expression.
right : Gate tuple
The right circuit of a gate rule expression.
Examples
========
Generate a new gate rule using a RR operation:
>>> from sympy.physics.quantum.identitysearch import rr_op
>>> from sympy.physics.quantum.gate import X, Y, Z
>>> x = X(0); y = Y(0); z = Z(0)
>>> rr_op((x, y), (z,))
((X(0), Y(0), Z(0)), ())
>>> rr_op((x,), (y, z))
((X(0), Z(0)), (Y(0),))
"""
if (len(right) > 0):
rr_gate = right[len(right) - 1]
rr_gate_is_unitary = is_scalar_matrix(
(Dagger(rr_gate), rr_gate), _get_min_qubits(rr_gate), True)
if (len(right) > 0 and rr_gate_is_unitary):
# Get the new right side w/o the rightmost gate
new_right = right[0:len(right) - 1]
# Add the rightmost gate to the right position on the right side
new_left = left + (Dagger(rr_gate),)
# Return the new gate rule
return (new_left, new_right)
return None
def generate_gate_rules(gate_seq, return_as_muls=False):
"""Returns a set of gate rules. Each gate rules is represented
as a 2-tuple of tuples or Muls. An empty tuple represents an arbitrary
scalar value.
This function uses the four operations (LL, LR, RL, RR)
to generate the gate rules.
A gate rule is an expression such as ABC = D or AB = CD, where
A, B, C, and D are gates. Each value on either side of the
equal sign represents a circuit. The four operations allow
one to find a set of equivalent circuits from a gate identity.
The letters denoting the operation tell the user what
activities to perform on each expression. The first letter
indicates which side of the equal sign to focus on. The
second letter indicates which gate to focus on given the
side. Once this information is determined, the inverse
of the gate is multiplied on both circuits to create a new
gate rule.
For example, given the identity, ABCD = 1, a LL operation
means look at the left value and multiply both left sides by the
inverse of the leftmost gate A. If A is Hermitian, the inverse
of A is still A. The resulting new rule is BCD = A.
The following is a summary of the four operations. Assume
that in the examples, all gates are Hermitian.
LL : left circuit, left multiply
ABCD = E -> AABCD = AE -> BCD = AE
LR : left circuit, right multiply
ABCD = E -> ABCDD = ED -> ABC = ED
RL : right circuit, left multiply
ABC = ED -> EABC = EED -> EABC = D
RR : right circuit, right multiply
AB = CD -> ABD = CDD -> ABD = C
The number of gate rules generated is n*(n+1), where n
is the number of gates in the sequence (unproven).
Parameters
==========
gate_seq : Gate tuple, Mul, or Number
A variable length tuple or Mul of Gates whose product is equal to
a scalar matrix
return_as_muls : bool
True to return a set of Muls; False to return a set of tuples
Examples
========
Find the gate rules of the current circuit using tuples:
>>> from sympy.physics.quantum.identitysearch import generate_gate_rules
>>> from sympy.physics.quantum.gate import X, Y, Z
>>> x = X(0); y = Y(0); z = Z(0)
>>> generate_gate_rules((x, x))
{((X(0),), (X(0),)), ((X(0), X(0)), ())}
>>> generate_gate_rules((x, y, z))
{((), (X(0), Z(0), Y(0))), ((), (Y(0), X(0), Z(0))),
((), (Z(0), Y(0), X(0))), ((X(0),), (Z(0), Y(0))),
((Y(0),), (X(0), Z(0))), ((Z(0),), (Y(0), X(0))),
((X(0), Y(0)), (Z(0),)), ((Y(0), Z(0)), (X(0),)),
((Z(0), X(0)), (Y(0),)), ((X(0), Y(0), Z(0)), ()),
((Y(0), Z(0), X(0)), ()), ((Z(0), X(0), Y(0)), ())}
Find the gate rules of the current circuit using Muls:
>>> generate_gate_rules(x*x, return_as_muls=True)
{(1, 1)}
>>> generate_gate_rules(x*y*z, return_as_muls=True)
{(1, X(0)*Z(0)*Y(0)), (1, Y(0)*X(0)*Z(0)),
(1, Z(0)*Y(0)*X(0)), (X(0)*Y(0), Z(0)),
(Y(0)*Z(0), X(0)), (Z(0)*X(0), Y(0)),
(X(0)*Y(0)*Z(0), 1), (Y(0)*Z(0)*X(0), 1),
(Z(0)*X(0)*Y(0), 1), (X(0), Z(0)*Y(0)),
(Y(0), X(0)*Z(0)), (Z(0), Y(0)*X(0))}
"""
if isinstance(gate_seq, Number):
if return_as_muls:
return {(Integer(1), Integer(1))}
else:
return {((), ())}
elif isinstance(gate_seq, Mul):
gate_seq = gate_seq.args
# Each item in queue is a 3-tuple:
# i) first item is the left side of an equality
# ii) second item is the right side of an equality
# iii) third item is the number of operations performed
# The argument, gate_seq, will start on the left side, and
# the right side will be empty, implying the presence of an
# identity.
queue = deque()
# A set of gate rules
rules = set()
# Maximum number of operations to perform
max_ops = len(gate_seq)
def process_new_rule(new_rule, ops):
if new_rule is not None:
new_left, new_right = new_rule
if new_rule not in rules and (new_right, new_left) not in rules:
rules.add(new_rule)
# If haven't reached the max limit on operations
if ops + 1 < max_ops:
queue.append(new_rule + (ops + 1,))
queue.append((gate_seq, (), 0))
rules.add((gate_seq, ()))
while len(queue) > 0:
left, right, ops = queue.popleft()
# Do a LL
new_rule = ll_op(left, right)
process_new_rule(new_rule, ops)
# Do a LR
new_rule = lr_op(left, right)
process_new_rule(new_rule, ops)
# Do a RL
new_rule = rl_op(left, right)
process_new_rule(new_rule, ops)
# Do a RR
new_rule = rr_op(left, right)
process_new_rule(new_rule, ops)
if return_as_muls:
# Convert each rule as tuples into a rule as muls
mul_rules = set()
for rule in rules:
left, right = rule
mul_rules.add((Mul(*left), Mul(*right)))
rules = mul_rules
return rules
def generate_equivalent_ids(gate_seq, return_as_muls=False):
"""Returns a set of equivalent gate identities.
A gate identity is a quantum circuit such that the product
of the gates in the circuit is equal to a scalar value.
For example, XYZ = i, where X, Y, Z are the Pauli gates and
i is the imaginary value, is considered a gate identity.
This function uses the four operations (LL, LR, RL, RR)
to generate the gate rules and, subsequently, to locate equivalent
gate identities.
Note that all equivalent identities are reachable in n operations
from the starting gate identity, where n is the number of gates
in the sequence.
The max number of gate identities is 2n, where n is the number
of gates in the sequence (unproven).
Parameters
==========
gate_seq : Gate tuple, Mul, or Number
A variable length tuple or Mul of Gates whose product is equal to
a scalar matrix.
return_as_muls: bool
True to return as Muls; False to return as tuples
Examples
========
Find equivalent gate identities from the current circuit with tuples:
>>> from sympy.physics.quantum.identitysearch import generate_equivalent_ids
>>> from sympy.physics.quantum.gate import X, Y, Z
>>> x = X(0); y = Y(0); z = Z(0)
>>> generate_equivalent_ids((x, x))
{(X(0), X(0))}
>>> generate_equivalent_ids((x, y, z))
{(X(0), Y(0), Z(0)), (X(0), Z(0), Y(0)), (Y(0), X(0), Z(0)),
(Y(0), Z(0), X(0)), (Z(0), X(0), Y(0)), (Z(0), Y(0), X(0))}
Find equivalent gate identities from the current circuit with Muls:
>>> generate_equivalent_ids(x*x, return_as_muls=True)
{1}
>>> generate_equivalent_ids(x*y*z, return_as_muls=True)
{X(0)*Y(0)*Z(0), X(0)*Z(0)*Y(0), Y(0)*X(0)*Z(0),
Y(0)*Z(0)*X(0), Z(0)*X(0)*Y(0), Z(0)*Y(0)*X(0)}
"""
if isinstance(gate_seq, Number):
return {Integer(1)}
elif isinstance(gate_seq, Mul):
gate_seq = gate_seq.args
# Filter through the gate rules and keep the rules
# with an empty tuple either on the left or right side
# A set of equivalent gate identities
eq_ids = set()
gate_rules = generate_gate_rules(gate_seq)
for rule in gate_rules:
l, r = rule
if l == ():
eq_ids.add(r)
elif r == ():
eq_ids.add(l)
if return_as_muls:
convert_to_mul = lambda id_seq: Mul(*id_seq)
eq_ids = set(map(convert_to_mul, eq_ids))
return eq_ids
class GateIdentity(Basic):
"""Wrapper class for circuits that reduce to a scalar value.
A gate identity is a quantum circuit such that the product
of the gates in the circuit is equal to a scalar value.
For example, XYZ = i, where X, Y, Z are the Pauli gates and
i is the imaginary value, is considered a gate identity.
Parameters
==========
args : Gate tuple
A variable length tuple of Gates that form an identity.
Examples
========
Create a GateIdentity and look at its attributes:
>>> from sympy.physics.quantum.identitysearch import GateIdentity
>>> from sympy.physics.quantum.gate import X, Y, Z
>>> x = X(0); y = Y(0); z = Z(0)
>>> an_identity = GateIdentity(x, y, z)
>>> an_identity.circuit
X(0)*Y(0)*Z(0)
>>> an_identity.equivalent_ids
{(X(0), Y(0), Z(0)), (X(0), Z(0), Y(0)), (Y(0), X(0), Z(0)),
(Y(0), Z(0), X(0)), (Z(0), X(0), Y(0)), (Z(0), Y(0), X(0))}
"""
def __new__(cls, *args):
# args should be a tuple - a variable length argument list
obj = Basic.__new__(cls, *args)
obj._circuit = Mul(*args)
obj._rules = generate_gate_rules(args)
obj._eq_ids = generate_equivalent_ids(args)
return obj
@property
def circuit(self):
return self._circuit
@property
def gate_rules(self):
return self._rules
@property
def equivalent_ids(self):
return self._eq_ids
@property
def sequence(self):
return self.args
def __str__(self):
"""Returns the string of gates in a tuple."""
return str(self.circuit)
def is_degenerate(identity_set, gate_identity):
"""Checks if a gate identity is a permutation of another identity.
Parameters
==========
identity_set : set
A Python set with GateIdentity objects.
gate_identity : GateIdentity
The GateIdentity to check for existence in the set.
Examples
========
Check if the identity is a permutation of another identity:
>>> from sympy.physics.quantum.identitysearch import (
... GateIdentity, is_degenerate)
>>> from sympy.physics.quantum.gate import X, Y, Z
>>> x = X(0); y = Y(0); z = Z(0)
>>> an_identity = GateIdentity(x, y, z)
>>> id_set = {an_identity}
>>> another_id = (y, z, x)
>>> is_degenerate(id_set, another_id)
True
>>> another_id = (x, x)
>>> is_degenerate(id_set, another_id)
False
"""
# For now, just iteratively go through the set and check if the current
# gate_identity is a permutation of an identity in the set
for an_id in identity_set:
if (gate_identity in an_id.equivalent_ids):
return True
return False
def is_reducible(circuit, nqubits, begin, end):
"""Determines if a circuit is reducible by checking
if its subcircuits are scalar values.
Parameters
==========
circuit : Gate tuple
A tuple of Gates representing a circuit. The circuit to check
if a gate identity is contained in a subcircuit.
nqubits : int
The number of qubits the circuit operates on.
begin : int
The leftmost gate in the circuit to include in a subcircuit.
end : int
The rightmost gate in the circuit to include in a subcircuit.
Examples
========
Check if the circuit can be reduced:
>>> from sympy.physics.quantum.identitysearch import is_reducible
>>> from sympy.physics.quantum.gate import X, Y, Z
>>> x = X(0); y = Y(0); z = Z(0)
>>> is_reducible((x, y, z), 1, 0, 3)
True
Check if an interval in the circuit can be reduced:
>>> is_reducible((x, y, z), 1, 1, 3)
False
>>> is_reducible((x, y, y), 1, 1, 3)
True
"""
current_circuit = ()
# Start from the gate at "end" and go down to almost the gate at "begin"
for ndx in reversed(range(begin, end)):
next_gate = circuit[ndx]
current_circuit = (next_gate,) + current_circuit
# If a circuit as a matrix is equivalent to a scalar value
if (is_scalar_matrix(current_circuit, nqubits, False)):
return True
return False
def bfs_identity_search(gate_list, nqubits, max_depth=None,
identity_only=False):
"""Constructs a set of gate identities from the list of possible gates.
Performs a breadth first search over the space of gate identities.
This allows the finding of the shortest gate identities first.
Parameters
==========
gate_list : list, Gate
A list of Gates from which to search for gate identities.
nqubits : int
The number of qubits the quantum circuit operates on.
max_depth : int
The longest quantum circuit to construct from gate_list.
identity_only : bool
True to search for gate identities that reduce to identity;
False to search for gate identities that reduce to a scalar.
Examples
========
Find a list of gate identities:
>>> from sympy.physics.quantum.identitysearch import bfs_identity_search
>>> from sympy.physics.quantum.gate import X, Y, Z
>>> x = X(0); y = Y(0); z = Z(0)
>>> bfs_identity_search([x], 1, max_depth=2)
{GateIdentity(X(0), X(0))}
>>> bfs_identity_search([x, y, z], 1)
{GateIdentity(X(0), X(0)), GateIdentity(Y(0), Y(0)),
GateIdentity(Z(0), Z(0)), GateIdentity(X(0), Y(0), Z(0))}
Find a list of identities that only equal to 1:
>>> bfs_identity_search([x, y, z], 1, identity_only=True)
{GateIdentity(X(0), X(0)), GateIdentity(Y(0), Y(0)),
GateIdentity(Z(0), Z(0))}
"""
if max_depth is None or max_depth <= 0:
max_depth = len(gate_list)
id_only = identity_only
# Start with an empty sequence (implicitly contains an IdentityGate)
queue = deque([()])
# Create an empty set of gate identities
ids = set()
# Begin searching for gate identities in given space.
while (len(queue) > 0):
current_circuit = queue.popleft()
for next_gate in gate_list:
new_circuit = current_circuit + (next_gate,)
# Determines if a (strict) subcircuit is a scalar matrix
circuit_reducible = is_reducible(new_circuit, nqubits,
1, len(new_circuit))
# In many cases when the matrix is a scalar value,
# the evaluated matrix will actually be an integer
if (is_scalar_matrix(new_circuit, nqubits, id_only) and
not is_degenerate(ids, new_circuit) and
not circuit_reducible):
ids.add(GateIdentity(*new_circuit))
elif (len(new_circuit) < max_depth and
not circuit_reducible):
queue.append(new_circuit)
return ids
def random_identity_search(gate_list, numgates, nqubits):
"""Randomly selects numgates from gate_list and checks if it is
a gate identity.
If the circuit is a gate identity, the circuit is returned;
Otherwise, None is returned.
"""
gate_size = len(gate_list)
circuit = ()
for i in range(numgates):
next_gate = gate_list[randint(0, gate_size - 1)]
circuit = circuit + (next_gate,)
is_scalar = is_scalar_matrix(circuit, nqubits, False)
return circuit if is_scalar else None
|
ea364dbd7b6ff6e44181542b9c32069b2f0d8f483850f09380b51b3fb369284e | """Abstract tensor product."""
from __future__ import print_function, division
from sympy import Expr, Add, Mul, Matrix, Pow, sympify
from sympy.core.trace import Tr
from sympy.printing.pretty.stringpict import prettyForm
from sympy.physics.quantum.qexpr import QuantumError
from sympy.physics.quantum.dagger import Dagger
from sympy.physics.quantum.commutator import Commutator
from sympy.physics.quantum.anticommutator import AntiCommutator
from sympy.physics.quantum.state import Ket, Bra
from sympy.physics.quantum.matrixutils import (
numpy_ndarray,
scipy_sparse_matrix,
matrix_tensor_product
)
__all__ = [
'TensorProduct',
'tensor_product_simp'
]
#-----------------------------------------------------------------------------
# Tensor product
#-----------------------------------------------------------------------------
_combined_printing = False
def combined_tensor_printing(combined):
"""Set flag controlling whether tensor products of states should be
printed as a combined bra/ket or as an explicit tensor product of different
bra/kets. This is a global setting for all TensorProduct class instances.
Parameters
----------
combine : bool
When true, tensor product states are combined into one ket/bra, and
when false explicit tensor product notation is used between each
ket/bra.
"""
global _combined_printing
_combined_printing = combined
class TensorProduct(Expr):
"""The tensor product of two or more arguments.
For matrices, this uses ``matrix_tensor_product`` to compute the Kronecker
or tensor product matrix. For other objects a symbolic ``TensorProduct``
instance is returned. The tensor product is a non-commutative
multiplication that is used primarily with operators and states in quantum
mechanics.
Currently, the tensor product distinguishes between commutative and
non-commutative arguments. Commutative arguments are assumed to be scalars
and are pulled out in front of the ``TensorProduct``. Non-commutative
arguments remain in the resulting ``TensorProduct``.
Parameters
==========
args : tuple
A sequence of the objects to take the tensor product of.
Examples
========
Start with a simple tensor product of sympy matrices::
>>> from sympy import Matrix
>>> from sympy.physics.quantum import TensorProduct
>>> m1 = Matrix([[1,2],[3,4]])
>>> m2 = Matrix([[1,0],[0,1]])
>>> TensorProduct(m1, m2)
Matrix([
[1, 0, 2, 0],
[0, 1, 0, 2],
[3, 0, 4, 0],
[0, 3, 0, 4]])
>>> TensorProduct(m2, m1)
Matrix([
[1, 2, 0, 0],
[3, 4, 0, 0],
[0, 0, 1, 2],
[0, 0, 3, 4]])
We can also construct tensor products of non-commutative symbols:
>>> from sympy import Symbol
>>> A = Symbol('A',commutative=False)
>>> B = Symbol('B',commutative=False)
>>> tp = TensorProduct(A, B)
>>> tp
AxB
We can take the dagger of a tensor product (note the order does NOT reverse
like the dagger of a normal product):
>>> from sympy.physics.quantum import Dagger
>>> Dagger(tp)
Dagger(A)xDagger(B)
Expand can be used to distribute a tensor product across addition:
>>> C = Symbol('C',commutative=False)
>>> tp = TensorProduct(A+B,C)
>>> tp
(A + B)xC
>>> tp.expand(tensorproduct=True)
AxC + BxC
"""
is_commutative = False
def __new__(cls, *args):
if isinstance(args[0], (Matrix, numpy_ndarray, scipy_sparse_matrix)):
return matrix_tensor_product(*args)
c_part, new_args = cls.flatten(sympify(args))
c_part = Mul(*c_part)
if len(new_args) == 0:
return c_part
elif len(new_args) == 1:
return c_part * new_args[0]
else:
tp = Expr.__new__(cls, *new_args)
return c_part * tp
@classmethod
def flatten(cls, args):
# TODO: disallow nested TensorProducts.
c_part = []
nc_parts = []
for arg in args:
cp, ncp = arg.args_cnc()
c_part.extend(list(cp))
nc_parts.append(Mul._from_args(ncp))
return c_part, nc_parts
def _eval_adjoint(self):
return TensorProduct(*[Dagger(i) for i in self.args])
def _eval_rewrite(self, pattern, rule, **hints):
sargs = self.args
terms = [t._eval_rewrite(pattern, rule, **hints) for t in sargs]
return TensorProduct(*terms).expand(tensorproduct=True)
def _sympystr(self, printer, *args):
length = len(self.args)
s = ''
for i in range(length):
if isinstance(self.args[i], (Add, Pow, Mul)):
s = s + '('
s = s + printer._print(self.args[i])
if isinstance(self.args[i], (Add, Pow, Mul)):
s = s + ')'
if i != length - 1:
s = s + 'x'
return s
def _pretty(self, printer, *args):
if (_combined_printing and
(all([isinstance(arg, Ket) for arg in self.args]) or
all([isinstance(arg, Bra) for arg in self.args]))):
length = len(self.args)
pform = printer._print('', *args)
for i in range(length):
next_pform = printer._print('', *args)
length_i = len(self.args[i].args)
for j in range(length_i):
part_pform = printer._print(self.args[i].args[j], *args)
next_pform = prettyForm(*next_pform.right(part_pform))
if j != length_i - 1:
next_pform = prettyForm(*next_pform.right(', '))
if len(self.args[i].args) > 1:
next_pform = prettyForm(
*next_pform.parens(left='{', right='}'))
pform = prettyForm(*pform.right(next_pform))
if i != length - 1:
pform = prettyForm(*pform.right(',' + ' '))
pform = prettyForm(*pform.left(self.args[0].lbracket))
pform = prettyForm(*pform.right(self.args[0].rbracket))
return pform
length = len(self.args)
pform = printer._print('', *args)
for i in range(length):
next_pform = printer._print(self.args[i], *args)
if isinstance(self.args[i], (Add, Mul)):
next_pform = prettyForm(
*next_pform.parens(left='(', right=')')
)
pform = prettyForm(*pform.right(next_pform))
if i != length - 1:
if printer._use_unicode:
pform = prettyForm(*pform.right(u'\N{N-ARY CIRCLED TIMES OPERATOR}' + u' '))
else:
pform = prettyForm(*pform.right('x' + ' '))
return pform
def _latex(self, printer, *args):
if (_combined_printing and
(all([isinstance(arg, Ket) for arg in self.args]) or
all([isinstance(arg, Bra) for arg in self.args]))):
def _label_wrap(label, nlabels):
return label if nlabels == 1 else r"\left\{%s\right\}" % label
s = r", ".join([_label_wrap(arg._print_label_latex(printer, *args),
len(arg.args)) for arg in self.args])
return r"{%s%s%s}" % (self.args[0].lbracket_latex, s,
self.args[0].rbracket_latex)
length = len(self.args)
s = ''
for i in range(length):
if isinstance(self.args[i], (Add, Mul)):
s = s + '\\left('
# The extra {} brackets are needed to get matplotlib's latex
# rendered to render this properly.
s = s + '{' + printer._print(self.args[i], *args) + '}'
if isinstance(self.args[i], (Add, Mul)):
s = s + '\\right)'
if i != length - 1:
s = s + '\\otimes '
return s
def doit(self, **hints):
return TensorProduct(*[item.doit(**hints) for item in self.args])
def _eval_expand_tensorproduct(self, **hints):
"""Distribute TensorProducts across addition."""
args = self.args
add_args = []
for i in range(len(args)):
if isinstance(args[i], Add):
for aa in args[i].args:
tp = TensorProduct(*args[:i] + (aa,) + args[i + 1:])
if isinstance(tp, TensorProduct):
tp = tp._eval_expand_tensorproduct()
add_args.append(tp)
break
if add_args:
return Add(*add_args)
else:
return self
def _eval_trace(self, **kwargs):
indices = kwargs.get('indices', None)
exp = tensor_product_simp(self)
if indices is None or len(indices) == 0:
return Mul(*[Tr(arg).doit() for arg in exp.args])
else:
return Mul(*[Tr(value).doit() if idx in indices else value
for idx, value in enumerate(exp.args)])
def tensor_product_simp_Mul(e):
"""Simplify a Mul with TensorProducts.
Current the main use of this is to simplify a ``Mul`` of ``TensorProduct``s
to a ``TensorProduct`` of ``Muls``. It currently only works for relatively
simple cases where the initial ``Mul`` only has scalars and raw
``TensorProduct``s, not ``Add``, ``Pow``, ``Commutator``s of
``TensorProduct``s.
Parameters
==========
e : Expr
A ``Mul`` of ``TensorProduct``s to be simplified.
Returns
=======
e : Expr
A ``TensorProduct`` of ``Mul``s.
Examples
========
This is an example of the type of simplification that this function
performs::
>>> from sympy.physics.quantum.tensorproduct import \
tensor_product_simp_Mul, TensorProduct
>>> from sympy import Symbol
>>> A = Symbol('A',commutative=False)
>>> B = Symbol('B',commutative=False)
>>> C = Symbol('C',commutative=False)
>>> D = Symbol('D',commutative=False)
>>> e = TensorProduct(A,B)*TensorProduct(C,D)
>>> e
AxB*CxD
>>> tensor_product_simp_Mul(e)
(A*C)x(B*D)
"""
# TODO: This won't work with Muls that have other composites of
# TensorProducts, like an Add, Commutator, etc.
# TODO: This only works for the equivalent of single Qbit gates.
if not isinstance(e, Mul):
return e
c_part, nc_part = e.args_cnc()
n_nc = len(nc_part)
if n_nc == 0:
return e
elif n_nc == 1:
if isinstance(nc_part[0], Pow):
return Mul(*c_part) * tensor_product_simp_Pow(nc_part[0])
return e
elif e.has(TensorProduct):
current = nc_part[0]
if not isinstance(current, TensorProduct):
if isinstance(current, Pow):
if isinstance(current.base, TensorProduct):
current = tensor_product_simp_Pow(current)
else:
raise TypeError('TensorProduct expected, got: %r' % current)
n_terms = len(current.args)
new_args = list(current.args)
for next in nc_part[1:]:
# TODO: check the hilbert spaces of next and current here.
if isinstance(next, TensorProduct):
if n_terms != len(next.args):
raise QuantumError(
'TensorProducts of different lengths: %r and %r' %
(current, next)
)
for i in range(len(new_args)):
new_args[i] = new_args[i] * next.args[i]
else:
if isinstance(next, Pow):
if isinstance(next.base, TensorProduct):
new_tp = tensor_product_simp_Pow(next)
for i in range(len(new_args)):
new_args[i] = new_args[i] * new_tp.args[i]
else:
raise TypeError('TensorProduct expected, got: %r' % next)
else:
raise TypeError('TensorProduct expected, got: %r' % next)
current = next
return Mul(*c_part) * TensorProduct(*new_args)
elif e.has(Pow):
new_args = [ tensor_product_simp_Pow(nc) for nc in nc_part ]
return tensor_product_simp_Mul(Mul(*c_part) * TensorProduct(*new_args))
else:
return e
def tensor_product_simp_Pow(e):
"""Evaluates ``Pow`` expressions whose base is ``TensorProduct``"""
if not isinstance(e, Pow):
return e
if isinstance(e.base, TensorProduct):
return TensorProduct(*[ b**e.exp for b in e.base.args])
else:
return e
def tensor_product_simp(e, **hints):
"""Try to simplify and combine TensorProducts.
In general this will try to pull expressions inside of ``TensorProducts``.
It currently only works for relatively simple cases where the products have
only scalars, raw ``TensorProducts``, not ``Add``, ``Pow``, ``Commutators``
of ``TensorProducts``. It is best to see what it does by showing examples.
Examples
========
>>> from sympy.physics.quantum import tensor_product_simp
>>> from sympy.physics.quantum import TensorProduct
>>> from sympy import Symbol
>>> A = Symbol('A',commutative=False)
>>> B = Symbol('B',commutative=False)
>>> C = Symbol('C',commutative=False)
>>> D = Symbol('D',commutative=False)
First see what happens to products of tensor products:
>>> e = TensorProduct(A,B)*TensorProduct(C,D)
>>> e
AxB*CxD
>>> tensor_product_simp(e)
(A*C)x(B*D)
This is the core logic of this function, and it works inside, powers, sums,
commutators and anticommutators as well:
>>> tensor_product_simp(e**2)
(A*C)x(B*D)**2
"""
if isinstance(e, Add):
return Add(*[tensor_product_simp(arg) for arg in e.args])
elif isinstance(e, Pow):
if isinstance(e.base, TensorProduct):
return tensor_product_simp_Pow(e)
else:
return tensor_product_simp(e.base) ** e.exp
elif isinstance(e, Mul):
return tensor_product_simp_Mul(e)
elif isinstance(e, Commutator):
return Commutator(*[tensor_product_simp(arg) for arg in e.args])
elif isinstance(e, AntiCommutator):
return AntiCommutator(*[tensor_product_simp(arg) for arg in e.args])
else:
return e
|
6416fa0d5d77c5ee9141e3e6a3196d1f7594b40e4ee1fb5d404269442e734a40 | """The anti-commutator: ``{A,B} = A*B + B*A``."""
from __future__ import print_function, division
from sympy import S, Expr, Mul, Integer
from sympy.printing.pretty.stringpict import prettyForm
from sympy.physics.quantum.operator import Operator
from sympy.physics.quantum.dagger import Dagger
__all__ = [
'AntiCommutator'
]
#-----------------------------------------------------------------------------
# Anti-commutator
#-----------------------------------------------------------------------------
class AntiCommutator(Expr):
"""The standard anticommutator, in an unevaluated state.
Evaluating an anticommutator is defined [1]_ as: ``{A, B} = A*B + B*A``.
This class returns the anticommutator in an unevaluated form. To evaluate
the anticommutator, use the ``.doit()`` method.
Canonical ordering of an anticommutator is ``{A, B}`` for ``A < B``. The
arguments of the anticommutator are put into canonical order using
``__cmp__``. If ``B < A``, then ``{A, B}`` is returned as ``{B, A}``.
Parameters
==========
A : Expr
The first argument of the anticommutator {A,B}.
B : Expr
The second argument of the anticommutator {A,B}.
Examples
========
>>> from sympy import symbols
>>> from sympy.physics.quantum import AntiCommutator
>>> from sympy.physics.quantum import Operator, Dagger
>>> x, y = symbols('x,y')
>>> A = Operator('A')
>>> B = Operator('B')
Create an anticommutator and use ``doit()`` to multiply them out.
>>> ac = AntiCommutator(A,B); ac
{A,B}
>>> ac.doit()
A*B + B*A
The commutator orders it arguments in canonical order:
>>> ac = AntiCommutator(B,A); ac
{A,B}
Commutative constants are factored out:
>>> AntiCommutator(3*x*A,x*y*B)
3*x**2*y*{A,B}
Adjoint operations applied to the anticommutator are properly applied to
the arguments:
>>> Dagger(AntiCommutator(A,B))
{Dagger(A),Dagger(B)}
References
==========
.. [1] https://en.wikipedia.org/wiki/Commutator
"""
is_commutative = False
def __new__(cls, A, B):
r = cls.eval(A, B)
if r is not None:
return r
obj = Expr.__new__(cls, A, B)
return obj
@classmethod
def eval(cls, a, b):
if not (a and b):
return S.Zero
if a == b:
return Integer(2)*a**2
if a.is_commutative or b.is_commutative:
return Integer(2)*a*b
# [xA,yB] -> xy*[A,B]
ca, nca = a.args_cnc()
cb, ncb = b.args_cnc()
c_part = ca + cb
if c_part:
return Mul(Mul(*c_part), cls(Mul._from_args(nca), Mul._from_args(ncb)))
# Canonical ordering of arguments
#The Commutator [A,B] is on canonical form if A < B.
if a.compare(b) == 1:
return cls(b, a)
def doit(self, **hints):
""" Evaluate anticommutator """
A = self.args[0]
B = self.args[1]
if isinstance(A, Operator) and isinstance(B, Operator):
try:
comm = A._eval_anticommutator(B, **hints)
except NotImplementedError:
try:
comm = B._eval_anticommutator(A, **hints)
except NotImplementedError:
comm = None
if comm is not None:
return comm.doit(**hints)
return (A*B + B*A).doit(**hints)
def _eval_adjoint(self):
return AntiCommutator(Dagger(self.args[0]), Dagger(self.args[1]))
def _sympyrepr(self, printer, *args):
return "%s(%s,%s)" % (
self.__class__.__name__, printer._print(
self.args[0]), printer._print(self.args[1])
)
def _sympystr(self, printer, *args):
return "{%s,%s}" % (
printer._print(self.args[0]), printer._print(self.args[1]))
def _pretty(self, printer, *args):
pform = printer._print(self.args[0], *args)
pform = prettyForm(*pform.right((prettyForm(u','))))
pform = prettyForm(*pform.right((printer._print(self.args[1], *args))))
pform = prettyForm(*pform.parens(left='{', right='}'))
return pform
def _latex(self, printer, *args):
return "\\left\\{%s,%s\\right\\}" % tuple([
printer._print(arg, *args) for arg in self.args])
|
41c1f7c304159fb38a21807f38694b1184b5e7122421772ded9b416f367645a0 | """Dirac notation for states."""
from __future__ import print_function, division
from sympy import (cacheit, conjugate, Expr, Function, integrate, oo, sqrt,
Tuple)
from sympy.printing.pretty.stringpict import stringPict
from sympy.physics.quantum.qexpr import QExpr, dispatch_method
__all__ = [
'KetBase',
'BraBase',
'StateBase',
'State',
'Ket',
'Bra',
'TimeDepState',
'TimeDepBra',
'TimeDepKet',
'OrthogonalKet',
'OrthogonalBra',
'OrthogonalState',
'Wavefunction'
]
#-----------------------------------------------------------------------------
# States, bras and kets.
#-----------------------------------------------------------------------------
# ASCII brackets
_lbracket = "<"
_rbracket = ">"
_straight_bracket = "|"
# Unicode brackets
# MATHEMATICAL ANGLE BRACKETS
_lbracket_ucode = u"\N{MATHEMATICAL LEFT ANGLE BRACKET}"
_rbracket_ucode = u"\N{MATHEMATICAL RIGHT ANGLE BRACKET}"
# LIGHT VERTICAL BAR
_straight_bracket_ucode = u"\N{LIGHT VERTICAL BAR}"
# Other options for unicode printing of <, > and | for Dirac notation.
# LEFT-POINTING ANGLE BRACKET
# _lbracket = u"\u2329"
# _rbracket = u"\u232A"
# LEFT ANGLE BRACKET
# _lbracket = u"\u3008"
# _rbracket = u"\u3009"
# VERTICAL LINE
# _straight_bracket = u"\u007C"
class StateBase(QExpr):
"""Abstract base class for general abstract states in quantum mechanics.
All other state classes defined will need to inherit from this class. It
carries the basic structure for all other states such as dual, _eval_adjoint
and label.
This is an abstract base class and you should not instantiate it directly,
instead use State.
"""
@classmethod
def _operators_to_state(self, ops, **options):
""" Returns the eigenstate instance for the passed operators.
This method should be overridden in subclasses. It will handle being
passed either an Operator instance or set of Operator instances. It
should return the corresponding state INSTANCE or simply raise a
NotImplementedError. See cartesian.py for an example.
"""
raise NotImplementedError("Cannot map operators to states in this class. Method not implemented!")
def _state_to_operators(self, op_classes, **options):
""" Returns the operators which this state instance is an eigenstate
of.
This method should be overridden in subclasses. It will be called on
state instances and be passed the operator classes that we wish to make
into instances. The state instance will then transform the classes
appropriately, or raise a NotImplementedError if it cannot return
operator instances. See cartesian.py for examples,
"""
raise NotImplementedError(
"Cannot map this state to operators. Method not implemented!")
@property
def operators(self):
"""Return the operator(s) that this state is an eigenstate of"""
from .operatorset import state_to_operators # import internally to avoid circular import errors
return state_to_operators(self)
def _enumerate_state(self, num_states, **options):
raise NotImplementedError("Cannot enumerate this state!")
def _represent_default_basis(self, **options):
return self._represent(basis=self.operators)
#-------------------------------------------------------------------------
# Dagger/dual
#-------------------------------------------------------------------------
@property
def dual(self):
"""Return the dual state of this one."""
return self.dual_class()._new_rawargs(self.hilbert_space, *self.args)
@classmethod
def dual_class(self):
"""Return the class used to construct the dual."""
raise NotImplementedError(
'dual_class must be implemented in a subclass'
)
def _eval_adjoint(self):
"""Compute the dagger of this state using the dual."""
return self.dual
#-------------------------------------------------------------------------
# Printing
#-------------------------------------------------------------------------
def _pretty_brackets(self, height, use_unicode=True):
# Return pretty printed brackets for the state
# Ideally, this could be done by pform.parens but it does not support the angled < and >
# Setup for unicode vs ascii
if use_unicode:
lbracket, rbracket = self.lbracket_ucode, self.rbracket_ucode
slash, bslash, vert = u'\N{BOX DRAWINGS LIGHT DIAGONAL UPPER RIGHT TO LOWER LEFT}', \
u'\N{BOX DRAWINGS LIGHT DIAGONAL UPPER LEFT TO LOWER RIGHT}', \
u'\N{BOX DRAWINGS LIGHT VERTICAL}'
else:
lbracket, rbracket = self.lbracket, self.rbracket
slash, bslash, vert = '/', '\\', '|'
# If height is 1, just return brackets
if height == 1:
return stringPict(lbracket), stringPict(rbracket)
# Make height even
height += (height % 2)
brackets = []
for bracket in lbracket, rbracket:
# Create left bracket
if bracket in {_lbracket, _lbracket_ucode}:
bracket_args = [ ' ' * (height//2 - i - 1) +
slash for i in range(height // 2)]
bracket_args.extend(
[ ' ' * i + bslash for i in range(height // 2)])
# Create right bracket
elif bracket in {_rbracket, _rbracket_ucode}:
bracket_args = [ ' ' * i + bslash for i in range(height // 2)]
bracket_args.extend([ ' ' * (
height//2 - i - 1) + slash for i in range(height // 2)])
# Create straight bracket
elif bracket in {_straight_bracket, _straight_bracket_ucode}:
bracket_args = [vert for i in range(height)]
else:
raise ValueError(bracket)
brackets.append(
stringPict('\n'.join(bracket_args), baseline=height//2))
return brackets
def _sympystr(self, printer, *args):
contents = self._print_contents(printer, *args)
return '%s%s%s' % (self.lbracket, contents, self.rbracket)
def _pretty(self, printer, *args):
from sympy.printing.pretty.stringpict import prettyForm
# Get brackets
pform = self._print_contents_pretty(printer, *args)
lbracket, rbracket = self._pretty_brackets(
pform.height(), printer._use_unicode)
# Put together state
pform = prettyForm(*pform.left(lbracket))
pform = prettyForm(*pform.right(rbracket))
return pform
def _latex(self, printer, *args):
contents = self._print_contents_latex(printer, *args)
# The extra {} brackets are needed to get matplotlib's latex
# rendered to render this properly.
return '{%s%s%s}' % (self.lbracket_latex, contents, self.rbracket_latex)
class KetBase(StateBase):
"""Base class for Kets.
This class defines the dual property and the brackets for printing. This is
an abstract base class and you should not instantiate it directly, instead
use Ket.
"""
lbracket = _straight_bracket
rbracket = _rbracket
lbracket_ucode = _straight_bracket_ucode
rbracket_ucode = _rbracket_ucode
lbracket_latex = r'\left|'
rbracket_latex = r'\right\rangle '
@classmethod
def default_args(self):
return ("psi",)
@classmethod
def dual_class(self):
return BraBase
def __mul__(self, other):
"""KetBase*other"""
from sympy.physics.quantum.operator import OuterProduct
if isinstance(other, BraBase):
return OuterProduct(self, other)
else:
return Expr.__mul__(self, other)
def __rmul__(self, other):
"""other*KetBase"""
from sympy.physics.quantum.innerproduct import InnerProduct
if isinstance(other, BraBase):
return InnerProduct(other, self)
else:
return Expr.__rmul__(self, other)
#-------------------------------------------------------------------------
# _eval_* methods
#-------------------------------------------------------------------------
def _eval_innerproduct(self, bra, **hints):
"""Evaluate the inner product between this ket and a bra.
This is called to compute <bra|ket>, where the ket is ``self``.
This method will dispatch to sub-methods having the format::
``def _eval_innerproduct_BraClass(self, **hints):``
Subclasses should define these methods (one for each BraClass) to
teach the ket how to take inner products with bras.
"""
return dispatch_method(self, '_eval_innerproduct', bra, **hints)
def _apply_operator(self, op, **options):
"""Apply an Operator to this Ket.
This method will dispatch to methods having the format::
``def _apply_operator_OperatorName(op, **options):``
Subclasses should define these methods (one for each OperatorName) to
teach the Ket how operators act on it.
Parameters
==========
op : Operator
The Operator that is acting on the Ket.
options : dict
A dict of key/value pairs that control how the operator is applied
to the Ket.
"""
return dispatch_method(self, '_apply_operator', op, **options)
class BraBase(StateBase):
"""Base class for Bras.
This class defines the dual property and the brackets for printing. This
is an abstract base class and you should not instantiate it directly,
instead use Bra.
"""
lbracket = _lbracket
rbracket = _straight_bracket
lbracket_ucode = _lbracket_ucode
rbracket_ucode = _straight_bracket_ucode
lbracket_latex = r'\left\langle '
rbracket_latex = r'\right|'
@classmethod
def _operators_to_state(self, ops, **options):
state = self.dual_class()._operators_to_state(ops, **options)
return state.dual
def _state_to_operators(self, op_classes, **options):
return self.dual._state_to_operators(op_classes, **options)
def _enumerate_state(self, num_states, **options):
dual_states = self.dual._enumerate_state(num_states, **options)
return [x.dual for x in dual_states]
@classmethod
def default_args(self):
return self.dual_class().default_args()
@classmethod
def dual_class(self):
return KetBase
def __mul__(self, other):
"""BraBase*other"""
from sympy.physics.quantum.innerproduct import InnerProduct
if isinstance(other, KetBase):
return InnerProduct(self, other)
else:
return Expr.__mul__(self, other)
def __rmul__(self, other):
"""other*BraBase"""
from sympy.physics.quantum.operator import OuterProduct
if isinstance(other, KetBase):
return OuterProduct(other, self)
else:
return Expr.__rmul__(self, other)
def _represent(self, **options):
"""A default represent that uses the Ket's version."""
from sympy.physics.quantum.dagger import Dagger
return Dagger(self.dual._represent(**options))
class State(StateBase):
"""General abstract quantum state used as a base class for Ket and Bra."""
pass
class Ket(State, KetBase):
"""A general time-independent Ket in quantum mechanics.
Inherits from State and KetBase. This class should be used as the base
class for all physical, time-independent Kets in a system. This class
and its subclasses will be the main classes that users will use for
expressing Kets in Dirac notation [1]_.
Parameters
==========
args : tuple
The list of numbers or parameters that uniquely specify the
ket. This will usually be its symbol or its quantum numbers. For
time-dependent state, this will include the time.
Examples
========
Create a simple Ket and looking at its properties::
>>> from sympy.physics.quantum import Ket
>>> from sympy import symbols, I
>>> k = Ket('psi')
>>> k
|psi>
>>> k.hilbert_space
H
>>> k.is_commutative
False
>>> k.label
(psi,)
Ket's know about their associated bra::
>>> k.dual
<psi|
>>> k.dual_class()
<class 'sympy.physics.quantum.state.Bra'>
Take a linear combination of two kets::
>>> k0 = Ket(0)
>>> k1 = Ket(1)
>>> 2*I*k0 - 4*k1
2*I*|0> - 4*|1>
Compound labels are passed as tuples::
>>> n, m = symbols('n,m')
>>> k = Ket(n,m)
>>> k
|nm>
References
==========
.. [1] https://en.wikipedia.org/wiki/Bra-ket_notation
"""
@classmethod
def dual_class(self):
return Bra
class Bra(State, BraBase):
"""A general time-independent Bra in quantum mechanics.
Inherits from State and BraBase. A Bra is the dual of a Ket [1]_. This
class and its subclasses will be the main classes that users will use for
expressing Bras in Dirac notation.
Parameters
==========
args : tuple
The list of numbers or parameters that uniquely specify the
ket. This will usually be its symbol or its quantum numbers. For
time-dependent state, this will include the time.
Examples
========
Create a simple Bra and look at its properties::
>>> from sympy.physics.quantum import Bra
>>> from sympy import symbols, I
>>> b = Bra('psi')
>>> b
<psi|
>>> b.hilbert_space
H
>>> b.is_commutative
False
Bra's know about their dual Ket's::
>>> b.dual
|psi>
>>> b.dual_class()
<class 'sympy.physics.quantum.state.Ket'>
Like Kets, Bras can have compound labels and be manipulated in a similar
manner::
>>> n, m = symbols('n,m')
>>> b = Bra(n,m) - I*Bra(m,n)
>>> b
-I*<mn| + <nm|
Symbols in a Bra can be substituted using ``.subs``::
>>> b.subs(n,m)
<mm| - I*<mm|
References
==========
.. [1] https://en.wikipedia.org/wiki/Bra-ket_notation
"""
@classmethod
def dual_class(self):
return Ket
#-----------------------------------------------------------------------------
# Time dependent states, bras and kets.
#-----------------------------------------------------------------------------
class TimeDepState(StateBase):
"""Base class for a general time-dependent quantum state.
This class is used as a base class for any time-dependent state. The main
difference between this class and the time-independent state is that this
class takes a second argument that is the time in addition to the usual
label argument.
Parameters
==========
args : tuple
The list of numbers or parameters that uniquely specify the ket. This
will usually be its symbol or its quantum numbers. For time-dependent
state, this will include the time as the final argument.
"""
#-------------------------------------------------------------------------
# Initialization
#-------------------------------------------------------------------------
@classmethod
def default_args(self):
return ("psi", "t")
#-------------------------------------------------------------------------
# Properties
#-------------------------------------------------------------------------
@property
def label(self):
"""The label of the state."""
return self.args[:-1]
@property
def time(self):
"""The time of the state."""
return self.args[-1]
#-------------------------------------------------------------------------
# Printing
#-------------------------------------------------------------------------
def _print_time(self, printer, *args):
return printer._print(self.time, *args)
_print_time_repr = _print_time
_print_time_latex = _print_time
def _print_time_pretty(self, printer, *args):
pform = printer._print(self.time, *args)
return pform
def _print_contents(self, printer, *args):
label = self._print_label(printer, *args)
time = self._print_time(printer, *args)
return '%s;%s' % (label, time)
def _print_label_repr(self, printer, *args):
label = self._print_sequence(self.label, ',', printer, *args)
time = self._print_time_repr(printer, *args)
return '%s,%s' % (label, time)
def _print_contents_pretty(self, printer, *args):
label = self._print_label_pretty(printer, *args)
time = self._print_time_pretty(printer, *args)
return printer._print_seq((label, time), delimiter=';')
def _print_contents_latex(self, printer, *args):
label = self._print_sequence(
self.label, self._label_separator, printer, *args)
time = self._print_time_latex(printer, *args)
return '%s;%s' % (label, time)
class TimeDepKet(TimeDepState, KetBase):
"""General time-dependent Ket in quantum mechanics.
This inherits from ``TimeDepState`` and ``KetBase`` and is the main class
that should be used for Kets that vary with time. Its dual is a
``TimeDepBra``.
Parameters
==========
args : tuple
The list of numbers or parameters that uniquely specify the ket. This
will usually be its symbol or its quantum numbers. For time-dependent
state, this will include the time as the final argument.
Examples
========
Create a TimeDepKet and look at its attributes::
>>> from sympy.physics.quantum import TimeDepKet
>>> k = TimeDepKet('psi', 't')
>>> k
|psi;t>
>>> k.time
t
>>> k.label
(psi,)
>>> k.hilbert_space
H
TimeDepKets know about their dual bra::
>>> k.dual
<psi;t|
>>> k.dual_class()
<class 'sympy.physics.quantum.state.TimeDepBra'>
"""
@classmethod
def dual_class(self):
return TimeDepBra
class TimeDepBra(TimeDepState, BraBase):
"""General time-dependent Bra in quantum mechanics.
This inherits from TimeDepState and BraBase and is the main class that
should be used for Bras that vary with time. Its dual is a TimeDepBra.
Parameters
==========
args : tuple
The list of numbers or parameters that uniquely specify the ket. This
will usually be its symbol or its quantum numbers. For time-dependent
state, this will include the time as the final argument.
Examples
========
>>> from sympy.physics.quantum import TimeDepBra
>>> b = TimeDepBra('psi', 't')
>>> b
<psi;t|
>>> b.time
t
>>> b.label
(psi,)
>>> b.hilbert_space
H
>>> b.dual
|psi;t>
"""
@classmethod
def dual_class(self):
return TimeDepKet
class OrthogonalState(State, StateBase):
"""General abstract quantum state used as a base class for Ket and Bra."""
pass
class OrthogonalKet(OrthogonalState, KetBase):
"""Orthogonal Ket in quantum mechanics.
The inner product of two states with different labels will give zero,
states with the same label will give one.
>>> from sympy.physics.quantum import OrthogonalBra, OrthogonalKet
>>> from sympy.abc import m, n
>>> (OrthogonalBra(n)*OrthogonalKet(n)).doit()
1
>>> (OrthogonalBra(n)*OrthogonalKet(n+1)).doit()
0
>>> (OrthogonalBra(n)*OrthogonalKet(m)).doit()
<n|m>
"""
@classmethod
def dual_class(self):
return OrthogonalBra
def _eval_innerproduct(self, bra, **hints):
if len(self.args) != len(bra.args):
raise ValueError('Cannot multiply a ket that has a different number of labels.')
for i in range(len(self.args)):
diff = self.args[i] - bra.args[i]
diff = diff.expand()
if diff.is_zero is False:
return 0
if diff.is_zero is None:
return None
return 1
class OrthogonalBra(OrthogonalState, BraBase):
"""Orthogonal Bra in quantum mechanics.
"""
@classmethod
def dual_class(self):
return OrthogonalKet
class Wavefunction(Function):
"""Class for representations in continuous bases
This class takes an expression and coordinates in its constructor. It can
be used to easily calculate normalizations and probabilities.
Parameters
==========
expr : Expr
The expression representing the functional form of the w.f.
coords : Symbol or tuple
The coordinates to be integrated over, and their bounds
Examples
========
Particle in a box, specifying bounds in the more primitive way of using
Piecewise:
>>> from sympy import Symbol, Piecewise, pi, N
>>> from sympy.functions import sqrt, sin
>>> from sympy.physics.quantum.state import Wavefunction
>>> x = Symbol('x', real=True)
>>> n = 1
>>> L = 1
>>> g = Piecewise((0, x < 0), (0, x > L), (sqrt(2//L)*sin(n*pi*x/L), True))
>>> f = Wavefunction(g, x)
>>> f.norm
1
>>> f.is_normalized
True
>>> p = f.prob()
>>> p(0)
0
>>> p(L)
0
>>> p(0.5)
2
>>> p(0.85*L)
2*sin(0.85*pi)**2
>>> N(p(0.85*L))
0.412214747707527
Additionally, you can specify the bounds of the function and the indices in
a more compact way:
>>> from sympy import symbols, pi, diff
>>> from sympy.functions import sqrt, sin
>>> from sympy.physics.quantum.state import Wavefunction
>>> x, L = symbols('x,L', positive=True)
>>> n = symbols('n', integer=True, positive=True)
>>> g = sqrt(2/L)*sin(n*pi*x/L)
>>> f = Wavefunction(g, (x, 0, L))
>>> f.norm
1
>>> f(L+1)
0
>>> f(L-1)
sqrt(2)*sin(pi*n*(L - 1)/L)/sqrt(L)
>>> f(-1)
0
>>> f(0.85)
sqrt(2)*sin(0.85*pi*n/L)/sqrt(L)
>>> f(0.85, n=1, L=1)
sqrt(2)*sin(0.85*pi)
>>> f.is_commutative
False
All arguments are automatically sympified, so you can define the variables
as strings rather than symbols:
>>> expr = x**2
>>> f = Wavefunction(expr, 'x')
>>> type(f.variables[0])
<class 'sympy.core.symbol.Symbol'>
Derivatives of Wavefunctions will return Wavefunctions:
>>> diff(f, x)
Wavefunction(2*x, x)
"""
#Any passed tuples for coordinates and their bounds need to be
#converted to Tuples before Function's constructor is called, to
#avoid errors from calling is_Float in the constructor
def __new__(cls, *args, **options):
new_args = [None for i in args]
ct = 0
for arg in args:
if isinstance(arg, tuple):
new_args[ct] = Tuple(*arg)
else:
new_args[ct] = arg
ct += 1
return super(Wavefunction, cls).__new__(cls, *new_args, **options)
def __call__(self, *args, **options):
var = self.variables
if len(args) != len(var):
raise NotImplementedError(
"Incorrect number of arguments to function!")
ct = 0
#If the passed value is outside the specified bounds, return 0
for v in var:
lower, upper = self.limits[v]
#Do the comparison to limits only if the passed symbol is actually
#a symbol present in the limits;
#Had problems with a comparison of x > L
if isinstance(args[ct], Expr) and \
not (lower in args[ct].free_symbols
or upper in args[ct].free_symbols):
continue
if (args[ct] < lower) == True or (args[ct] > upper) == True:
return 0
ct += 1
expr = self.expr
#Allows user to make a call like f(2, 4, m=1, n=1)
for symbol in list(expr.free_symbols):
if str(symbol) in options.keys():
val = options[str(symbol)]
expr = expr.subs(symbol, val)
return expr.subs(zip(var, args))
def _eval_derivative(self, symbol):
expr = self.expr
deriv = expr._eval_derivative(symbol)
return Wavefunction(deriv, *self.args[1:])
def _eval_conjugate(self):
return Wavefunction(conjugate(self.expr), *self.args[1:])
def _eval_transpose(self):
return self
@property
def free_symbols(self):
return self.expr.free_symbols
@property
def is_commutative(self):
"""
Override Function's is_commutative so that order is preserved in
represented expressions
"""
return False
@classmethod
def eval(self, *args):
return None
@property
def variables(self):
"""
Return the coordinates which the wavefunction depends on
Examples
========
>>> from sympy.physics.quantum.state import Wavefunction
>>> from sympy import symbols
>>> x,y = symbols('x,y')
>>> f = Wavefunction(x*y, x, y)
>>> f.variables
(x, y)
>>> g = Wavefunction(x*y, x)
>>> g.variables
(x,)
"""
var = [g[0] if isinstance(g, Tuple) else g for g in self._args[1:]]
return tuple(var)
@property
def limits(self):
"""
Return the limits of the coordinates which the w.f. depends on If no
limits are specified, defaults to ``(-oo, oo)``.
Examples
========
>>> from sympy.physics.quantum.state import Wavefunction
>>> from sympy import symbols
>>> x, y = symbols('x, y')
>>> f = Wavefunction(x**2, (x, 0, 1))
>>> f.limits
{x: (0, 1)}
>>> f = Wavefunction(x**2, x)
>>> f.limits
{x: (-oo, oo)}
>>> f = Wavefunction(x**2 + y**2, x, (y, -1, 2))
>>> f.limits
{x: (-oo, oo), y: (-1, 2)}
"""
limits = [(g[1], g[2]) if isinstance(g, Tuple) else (-oo, oo)
for g in self._args[1:]]
return dict(zip(self.variables, tuple(limits)))
@property
def expr(self):
"""
Return the expression which is the functional form of the Wavefunction
Examples
========
>>> from sympy.physics.quantum.state import Wavefunction
>>> from sympy import symbols
>>> x, y = symbols('x, y')
>>> f = Wavefunction(x**2, x)
>>> f.expr
x**2
"""
return self._args[0]
@property
def is_normalized(self):
"""
Returns true if the Wavefunction is properly normalized
Examples
========
>>> from sympy import symbols, pi
>>> from sympy.functions import sqrt, sin
>>> from sympy.physics.quantum.state import Wavefunction
>>> x, L = symbols('x,L', positive=True)
>>> n = symbols('n', integer=True, positive=True)
>>> g = sqrt(2/L)*sin(n*pi*x/L)
>>> f = Wavefunction(g, (x, 0, L))
>>> f.is_normalized
True
"""
return (self.norm == 1.0)
@property # type: ignore
@cacheit
def norm(self):
"""
Return the normalization of the specified functional form.
This function integrates over the coordinates of the Wavefunction, with
the bounds specified.
Examples
========
>>> from sympy import symbols, pi
>>> from sympy.functions import sqrt, sin
>>> from sympy.physics.quantum.state import Wavefunction
>>> x, L = symbols('x,L', positive=True)
>>> n = symbols('n', integer=True, positive=True)
>>> g = sqrt(2/L)*sin(n*pi*x/L)
>>> f = Wavefunction(g, (x, 0, L))
>>> f.norm
1
>>> g = sin(n*pi*x/L)
>>> f = Wavefunction(g, (x, 0, L))
>>> f.norm
sqrt(2)*sqrt(L)/2
"""
exp = self.expr*conjugate(self.expr)
var = self.variables
limits = self.limits
for v in var:
curr_limits = limits[v]
exp = integrate(exp, (v, curr_limits[0], curr_limits[1]))
return sqrt(exp)
def normalize(self):
"""
Return a normalized version of the Wavefunction
Examples
========
>>> from sympy import symbols, pi
>>> from sympy.functions import sin
>>> from sympy.physics.quantum.state import Wavefunction
>>> x = symbols('x', real=True)
>>> L = symbols('L', positive=True)
>>> n = symbols('n', integer=True, positive=True)
>>> g = sin(n*pi*x/L)
>>> f = Wavefunction(g, (x, 0, L))
>>> f.normalize()
Wavefunction(sqrt(2)*sin(pi*n*x/L)/sqrt(L), (x, 0, L))
"""
const = self.norm
if const is oo:
raise NotImplementedError("The function is not normalizable!")
else:
return Wavefunction((const)**(-1)*self.expr, *self.args[1:])
def prob(self):
r"""
Return the absolute magnitude of the w.f., `|\psi(x)|^2`
Examples
========
>>> from sympy import symbols, pi
>>> from sympy.functions import sin
>>> from sympy.physics.quantum.state import Wavefunction
>>> x, L = symbols('x,L', real=True)
>>> n = symbols('n', integer=True)
>>> g = sin(n*pi*x/L)
>>> f = Wavefunction(g, (x, 0, L))
>>> f.prob()
Wavefunction(sin(pi*n*x/L)**2, x)
"""
return Wavefunction(self.expr*conjugate(self.expr), *self.variables)
|
449df2a047594fd3cc7d554a4bd9fbc99961899c2a15d93d17918842d2f58547 | """Logic for representing operators in state in various bases.
TODO:
* Get represent working with continuous hilbert spaces.
* Document default basis functionality.
"""
from __future__ import print_function, division
from sympy import Add, Expr, I, integrate, Mul, Pow
from sympy.physics.quantum.dagger import Dagger
from sympy.physics.quantum.commutator import Commutator
from sympy.physics.quantum.anticommutator import AntiCommutator
from sympy.physics.quantum.innerproduct import InnerProduct
from sympy.physics.quantum.qexpr import QExpr
from sympy.physics.quantum.tensorproduct import TensorProduct
from sympy.physics.quantum.matrixutils import flatten_scalar
from sympy.physics.quantum.state import KetBase, BraBase, StateBase
from sympy.physics.quantum.operator import Operator, OuterProduct
from sympy.physics.quantum.qapply import qapply
from sympy.physics.quantum.operatorset import operators_to_state, state_to_operators
__all__ = [
'represent',
'rep_innerproduct',
'rep_expectation',
'integrate_result',
'get_basis',
'enumerate_states'
]
#-----------------------------------------------------------------------------
# Represent
#-----------------------------------------------------------------------------
def _sympy_to_scalar(e):
"""Convert from a sympy scalar to a Python scalar."""
if isinstance(e, Expr):
if e.is_Integer:
return int(e)
elif e.is_Float:
return float(e)
elif e.is_Rational:
return float(e)
elif e.is_Number or e.is_NumberSymbol or e == I:
return complex(e)
raise TypeError('Expected number, got: %r' % e)
def represent(expr, **options):
"""Represent the quantum expression in the given basis.
In quantum mechanics abstract states and operators can be represented in
various basis sets. Under this operation the follow transforms happen:
* Ket -> column vector or function
* Bra -> row vector of function
* Operator -> matrix or differential operator
This function is the top-level interface for this action.
This function walks the sympy expression tree looking for ``QExpr``
instances that have a ``_represent`` method. This method is then called
and the object is replaced by the representation returned by this method.
By default, the ``_represent`` method will dispatch to other methods
that handle the representation logic for a particular basis set. The
naming convention for these methods is the following::
def _represent_FooBasis(self, e, basis, **options)
This function will have the logic for representing instances of its class
in the basis set having a class named ``FooBasis``.
Parameters
==========
expr : Expr
The expression to represent.
basis : Operator, basis set
An object that contains the information about the basis set. If an
operator is used, the basis is assumed to be the orthonormal
eigenvectors of that operator. In general though, the basis argument
can be any object that contains the basis set information.
options : dict
Key/value pairs of options that are passed to the underlying method
that finds the representation. These options can be used to
control how the representation is done. For example, this is where
the size of the basis set would be set.
Returns
=======
e : Expr
The SymPy expression of the represented quantum expression.
Examples
========
Here we subclass ``Operator`` and ``Ket`` to create the z-spin operator
and its spin 1/2 up eigenstate. By defining the ``_represent_SzOp``
method, the ket can be represented in the z-spin basis.
>>> from sympy.physics.quantum import Operator, represent, Ket
>>> from sympy import Matrix
>>> class SzUpKet(Ket):
... def _represent_SzOp(self, basis, **options):
... return Matrix([1,0])
...
>>> class SzOp(Operator):
... pass
...
>>> sz = SzOp('Sz')
>>> up = SzUpKet('up')
>>> represent(up, basis=sz)
Matrix([
[1],
[0]])
Here we see an example of representations in a continuous
basis. We see that the result of representing various combinations
of cartesian position operators and kets give us continuous
expressions involving DiracDelta functions.
>>> from sympy.physics.quantum.cartesian import XOp, XKet, XBra
>>> X = XOp()
>>> x = XKet()
>>> y = XBra('y')
>>> represent(X*x)
x*DiracDelta(x - x_2)
>>> represent(X*x*y)
x*DiracDelta(x - x_3)*DiracDelta(x_1 - y)
"""
format = options.get('format', 'sympy')
if isinstance(expr, QExpr) and not isinstance(expr, OuterProduct):
options['replace_none'] = False
temp_basis = get_basis(expr, **options)
if temp_basis is not None:
options['basis'] = temp_basis
try:
return expr._represent(**options)
except NotImplementedError as strerr:
#If no _represent_FOO method exists, map to the
#appropriate basis state and try
#the other methods of representation
options['replace_none'] = True
if isinstance(expr, (KetBase, BraBase)):
try:
return rep_innerproduct(expr, **options)
except NotImplementedError:
raise NotImplementedError(strerr)
elif isinstance(expr, Operator):
try:
return rep_expectation(expr, **options)
except NotImplementedError:
raise NotImplementedError(strerr)
else:
raise NotImplementedError(strerr)
elif isinstance(expr, Add):
result = represent(expr.args[0], **options)
for args in expr.args[1:]:
# scipy.sparse doesn't support += so we use plain = here.
result = result + represent(args, **options)
return result
elif isinstance(expr, Pow):
base, exp = expr.as_base_exp()
if format == 'numpy' or format == 'scipy.sparse':
exp = _sympy_to_scalar(exp)
base = represent(base, **options)
# scipy.sparse doesn't support negative exponents
# and warns when inverting a matrix in csr format.
if format == 'scipy.sparse' and exp < 0:
from scipy.sparse.linalg import inv
exp = - exp
base = inv(base.tocsc()).tocsr()
return base ** exp
elif isinstance(expr, TensorProduct):
new_args = [represent(arg, **options) for arg in expr.args]
return TensorProduct(*new_args)
elif isinstance(expr, Dagger):
return Dagger(represent(expr.args[0], **options))
elif isinstance(expr, Commutator):
A = represent(expr.args[0], **options)
B = represent(expr.args[1], **options)
return A*B - B*A
elif isinstance(expr, AntiCommutator):
A = represent(expr.args[0], **options)
B = represent(expr.args[1], **options)
return A*B + B*A
elif isinstance(expr, InnerProduct):
return represent(Mul(expr.bra, expr.ket), **options)
elif not (isinstance(expr, Mul) or isinstance(expr, OuterProduct)):
# For numpy and scipy.sparse, we can only handle numerical prefactors.
if format == 'numpy' or format == 'scipy.sparse':
return _sympy_to_scalar(expr)
return expr
if not (isinstance(expr, Mul) or isinstance(expr, OuterProduct)):
raise TypeError('Mul expected, got: %r' % expr)
if "index" in options:
options["index"] += 1
else:
options["index"] = 1
if not "unities" in options:
options["unities"] = []
result = represent(expr.args[-1], **options)
last_arg = expr.args[-1]
for arg in reversed(expr.args[:-1]):
if isinstance(last_arg, Operator):
options["index"] += 1
options["unities"].append(options["index"])
elif isinstance(last_arg, BraBase) and isinstance(arg, KetBase):
options["index"] += 1
elif isinstance(last_arg, KetBase) and isinstance(arg, Operator):
options["unities"].append(options["index"])
elif isinstance(last_arg, KetBase) and isinstance(arg, BraBase):
options["unities"].append(options["index"])
result = represent(arg, **options)*result
last_arg = arg
# All three matrix formats create 1 by 1 matrices when inner products of
# vectors are taken. In these cases, we simply return a scalar.
result = flatten_scalar(result)
result = integrate_result(expr, result, **options)
return result
def rep_innerproduct(expr, **options):
"""
Returns an innerproduct like representation (e.g. ``<x'|x>``) for the
given state.
Attempts to calculate inner product with a bra from the specified
basis. Should only be passed an instance of KetBase or BraBase
Parameters
==========
expr : KetBase or BraBase
The expression to be represented
Examples
========
>>> from sympy.physics.quantum.represent import rep_innerproduct
>>> from sympy.physics.quantum.cartesian import XOp, XKet, PxOp, PxKet
>>> rep_innerproduct(XKet())
DiracDelta(x - x_1)
>>> rep_innerproduct(XKet(), basis=PxOp())
sqrt(2)*exp(-I*px_1*x/hbar)/(2*sqrt(hbar)*sqrt(pi))
>>> rep_innerproduct(PxKet(), basis=XOp())
sqrt(2)*exp(I*px*x_1/hbar)/(2*sqrt(hbar)*sqrt(pi))
"""
if not isinstance(expr, (KetBase, BraBase)):
raise TypeError("expr passed is not a Bra or Ket")
basis = get_basis(expr, **options)
if not isinstance(basis, StateBase):
raise NotImplementedError("Can't form this representation!")
if not "index" in options:
options["index"] = 1
basis_kets = enumerate_states(basis, options["index"], 2)
if isinstance(expr, BraBase):
bra = expr
ket = (basis_kets[1] if basis_kets[0].dual == expr else basis_kets[0])
else:
bra = (basis_kets[1].dual if basis_kets[0]
== expr else basis_kets[0].dual)
ket = expr
prod = InnerProduct(bra, ket)
result = prod.doit()
format = options.get('format', 'sympy')
return expr._format_represent(result, format)
def rep_expectation(expr, **options):
"""
Returns an ``<x'|A|x>`` type representation for the given operator.
Parameters
==========
expr : Operator
Operator to be represented in the specified basis
Examples
========
>>> from sympy.physics.quantum.cartesian import XOp, PxOp, PxKet
>>> from sympy.physics.quantum.represent import rep_expectation
>>> rep_expectation(XOp())
x_1*DiracDelta(x_1 - x_2)
>>> rep_expectation(XOp(), basis=PxOp())
<px_2|*X*|px_1>
>>> rep_expectation(XOp(), basis=PxKet())
<px_2|*X*|px_1>
"""
if not "index" in options:
options["index"] = 1
if not isinstance(expr, Operator):
raise TypeError("The passed expression is not an operator")
basis_state = get_basis(expr, **options)
if basis_state is None or not isinstance(basis_state, StateBase):
raise NotImplementedError("Could not get basis kets for this operator")
basis_kets = enumerate_states(basis_state, options["index"], 2)
bra = basis_kets[1].dual
ket = basis_kets[0]
return qapply(bra*expr*ket)
def integrate_result(orig_expr, result, **options):
"""
Returns the result of integrating over any unities ``(|x><x|)`` in
the given expression. Intended for integrating over the result of
representations in continuous bases.
This function integrates over any unities that may have been
inserted into the quantum expression and returns the result.
It uses the interval of the Hilbert space of the basis state
passed to it in order to figure out the limits of integration.
The unities option must be
specified for this to work.
Note: This is mostly used internally by represent(). Examples are
given merely to show the use cases.
Parameters
==========
orig_expr : quantum expression
The original expression which was to be represented
result: Expr
The resulting representation that we wish to integrate over
Examples
========
>>> from sympy import symbols, DiracDelta
>>> from sympy.physics.quantum.represent import integrate_result
>>> from sympy.physics.quantum.cartesian import XOp, XKet
>>> x_ket = XKet()
>>> X_op = XOp()
>>> x, x_1, x_2 = symbols('x, x_1, x_2')
>>> integrate_result(X_op*x_ket, x*DiracDelta(x-x_1)*DiracDelta(x_1-x_2))
x*DiracDelta(x - x_1)*DiracDelta(x_1 - x_2)
>>> integrate_result(X_op*x_ket, x*DiracDelta(x-x_1)*DiracDelta(x_1-x_2),
... unities=[1])
x*DiracDelta(x - x_2)
"""
if not isinstance(result, Expr):
return result
options['replace_none'] = True
if not "basis" in options:
arg = orig_expr.args[-1]
options["basis"] = get_basis(arg, **options)
elif not isinstance(options["basis"], StateBase):
options["basis"] = get_basis(orig_expr, **options)
basis = options.pop("basis", None)
if basis is None:
return result
unities = options.pop("unities", [])
if len(unities) == 0:
return result
kets = enumerate_states(basis, unities)
coords = [k.label[0] for k in kets]
for coord in coords:
if coord in result.free_symbols:
#TODO: Add support for sets of operators
basis_op = state_to_operators(basis)
start = basis_op.hilbert_space.interval.start
end = basis_op.hilbert_space.interval.end
result = integrate(result, (coord, start, end))
return result
def get_basis(expr, **options):
"""
Returns a basis state instance corresponding to the basis specified in
options=s. If no basis is specified, the function tries to form a default
basis state of the given expression.
There are three behaviors:
1. The basis specified in options is already an instance of StateBase. If
this is the case, it is simply returned. If the class is specified but
not an instance, a default instance is returned.
2. The basis specified is an operator or set of operators. If this
is the case, the operator_to_state mapping method is used.
3. No basis is specified. If expr is a state, then a default instance of
its class is returned. If expr is an operator, then it is mapped to the
corresponding state. If it is neither, then we cannot obtain the basis
state.
If the basis cannot be mapped, then it is not changed.
This will be called from within represent, and represent will
only pass QExpr's.
TODO (?): Support for Muls and other types of expressions?
Parameters
==========
expr : Operator or StateBase
Expression whose basis is sought
Examples
========
>>> from sympy.physics.quantum.represent import get_basis
>>> from sympy.physics.quantum.cartesian import XOp, XKet, PxOp, PxKet
>>> x = XKet()
>>> X = XOp()
>>> get_basis(x)
|x>
>>> get_basis(X)
|x>
>>> get_basis(x, basis=PxOp())
|px>
>>> get_basis(x, basis=PxKet)
|px>
"""
basis = options.pop("basis", None)
replace_none = options.pop("replace_none", True)
if basis is None and not replace_none:
return None
if basis is None:
if isinstance(expr, KetBase):
return _make_default(expr.__class__)
elif isinstance(expr, BraBase):
return _make_default((expr.dual_class()))
elif isinstance(expr, Operator):
state_inst = operators_to_state(expr)
return (state_inst if state_inst is not None else None)
else:
return None
elif (isinstance(basis, Operator) or
(not isinstance(basis, StateBase) and issubclass(basis, Operator))):
state = operators_to_state(basis)
if state is None:
return None
elif isinstance(state, StateBase):
return state
else:
return _make_default(state)
elif isinstance(basis, StateBase):
return basis
elif issubclass(basis, StateBase):
return _make_default(basis)
else:
return None
def _make_default(expr):
# XXX: Catching TypeError like this is a bad way of distinguishing
# instances from classes. The logic using this function should be
# rewritten somehow.
try:
expr = expr()
except TypeError:
return expr
return expr
def enumerate_states(*args, **options):
"""
Returns instances of the given state with dummy indices appended
Operates in two different modes:
1. Two arguments are passed to it. The first is the base state which is to
be indexed, and the second argument is a list of indices to append.
2. Three arguments are passed. The first is again the base state to be
indexed. The second is the start index for counting. The final argument
is the number of kets you wish to receive.
Tries to call state._enumerate_state. If this fails, returns an empty list
Parameters
==========
args : list
See list of operation modes above for explanation
Examples
========
>>> from sympy.physics.quantum.cartesian import XBra, XKet
>>> from sympy.physics.quantum.represent import enumerate_states
>>> test = XKet('foo')
>>> enumerate_states(test, 1, 3)
[|foo_1>, |foo_2>, |foo_3>]
>>> test2 = XBra('bar')
>>> enumerate_states(test2, [4, 5, 10])
[<bar_4|, <bar_5|, <bar_10|]
"""
state = args[0]
if not (len(args) == 2 or len(args) == 3):
raise NotImplementedError("Wrong number of arguments!")
if not isinstance(state, StateBase):
raise TypeError("First argument is not a state!")
if len(args) == 3:
num_states = args[2]
options['start_index'] = args[1]
else:
num_states = len(args[1])
options['index_list'] = args[1]
try:
ret = state._enumerate_state(num_states, **options)
except NotImplementedError:
ret = []
return ret
|
8c2a8c1ab3b4184a33fbd104bfb6a0beab40eaeafd2f956f324d635628eae2ea | """Quantum mechanical operators.
TODO:
* Fix early 0 in apply_operators.
* Debug and test apply_operators.
* Get cse working with classes in this file.
* Doctests and documentation of special methods for InnerProduct, Commutator,
AntiCommutator, represent, apply_operators.
"""
from __future__ import print_function, division
from sympy import Derivative, Expr, Integer, oo, Mul, expand, Add
from sympy.printing.pretty.stringpict import prettyForm
from sympy.physics.quantum.dagger import Dagger
from sympy.physics.quantum.qexpr import QExpr, dispatch_method
from sympy.matrices import eye
__all__ = [
'Operator',
'HermitianOperator',
'UnitaryOperator',
'IdentityOperator',
'OuterProduct',
'DifferentialOperator'
]
#-----------------------------------------------------------------------------
# Operators and outer products
#-----------------------------------------------------------------------------
class Operator(QExpr):
"""Base class for non-commuting quantum operators.
An operator maps between quantum states [1]_. In quantum mechanics,
observables (including, but not limited to, measured physical values) are
represented as Hermitian operators [2]_.
Parameters
==========
args : tuple
The list of numbers or parameters that uniquely specify the
operator. For time-dependent operators, this will include the time.
Examples
========
Create an operator and examine its attributes::
>>> from sympy.physics.quantum import Operator
>>> from sympy import I
>>> A = Operator('A')
>>> A
A
>>> A.hilbert_space
H
>>> A.label
(A,)
>>> A.is_commutative
False
Create another operator and do some arithmetic operations::
>>> B = Operator('B')
>>> C = 2*A*A + I*B
>>> C
2*A**2 + I*B
Operators don't commute::
>>> A.is_commutative
False
>>> B.is_commutative
False
>>> A*B == B*A
False
Polymonials of operators respect the commutation properties::
>>> e = (A+B)**3
>>> e.expand()
A*B*A + A*B**2 + A**2*B + A**3 + B*A*B + B*A**2 + B**2*A + B**3
Operator inverses are handle symbolically::
>>> A.inv()
A**(-1)
>>> A*A.inv()
1
References
==========
.. [1] https://en.wikipedia.org/wiki/Operator_%28physics%29
.. [2] https://en.wikipedia.org/wiki/Observable
"""
@classmethod
def default_args(self):
return ("O",)
#-------------------------------------------------------------------------
# Printing
#-------------------------------------------------------------------------
_label_separator = ','
def _print_operator_name(self, printer, *args):
return printer._print(self.__class__.__name__, *args)
_print_operator_name_latex = _print_operator_name
def _print_operator_name_pretty(self, printer, *args):
return prettyForm(self.__class__.__name__)
def _print_contents(self, printer, *args):
if len(self.label) == 1:
return self._print_label(printer, *args)
else:
return '%s(%s)' % (
self._print_operator_name(printer, *args),
self._print_label(printer, *args)
)
def _print_contents_pretty(self, printer, *args):
if len(self.label) == 1:
return self._print_label_pretty(printer, *args)
else:
pform = self._print_operator_name_pretty(printer, *args)
label_pform = self._print_label_pretty(printer, *args)
label_pform = prettyForm(
*label_pform.parens(left='(', right=')')
)
pform = prettyForm(*pform.right((label_pform)))
return pform
def _print_contents_latex(self, printer, *args):
if len(self.label) == 1:
return self._print_label_latex(printer, *args)
else:
return r'%s\left(%s\right)' % (
self._print_operator_name_latex(printer, *args),
self._print_label_latex(printer, *args)
)
#-------------------------------------------------------------------------
# _eval_* methods
#-------------------------------------------------------------------------
def _eval_commutator(self, other, **options):
"""Evaluate [self, other] if known, return None if not known."""
return dispatch_method(self, '_eval_commutator', other, **options)
def _eval_anticommutator(self, other, **options):
"""Evaluate [self, other] if known."""
return dispatch_method(self, '_eval_anticommutator', other, **options)
#-------------------------------------------------------------------------
# Operator application
#-------------------------------------------------------------------------
def _apply_operator(self, ket, **options):
return dispatch_method(self, '_apply_operator', ket, **options)
def matrix_element(self, *args):
raise NotImplementedError('matrix_elements is not defined')
def inverse(self):
return self._eval_inverse()
inv = inverse
def _eval_inverse(self):
return self**(-1)
def __mul__(self, other):
if isinstance(other, IdentityOperator):
return self
return Mul(self, other)
class HermitianOperator(Operator):
"""A Hermitian operator that satisfies H == Dagger(H).
Parameters
==========
args : tuple
The list of numbers or parameters that uniquely specify the
operator. For time-dependent operators, this will include the time.
Examples
========
>>> from sympy.physics.quantum import Dagger, HermitianOperator
>>> H = HermitianOperator('H')
>>> Dagger(H)
H
"""
is_hermitian = True
def _eval_inverse(self):
if isinstance(self, UnitaryOperator):
return self
else:
return Operator._eval_inverse(self)
def _eval_power(self, exp):
if isinstance(self, UnitaryOperator):
if exp == -1:
return Operator._eval_power(self, exp)
elif abs(exp) % 2 == 0:
return self*(Operator._eval_inverse(self))
else:
return self
else:
return Operator._eval_power(self, exp)
class UnitaryOperator(Operator):
"""A unitary operator that satisfies U*Dagger(U) == 1.
Parameters
==========
args : tuple
The list of numbers or parameters that uniquely specify the
operator. For time-dependent operators, this will include the time.
Examples
========
>>> from sympy.physics.quantum import Dagger, UnitaryOperator
>>> U = UnitaryOperator('U')
>>> U*Dagger(U)
1
"""
def _eval_adjoint(self):
return self._eval_inverse()
class IdentityOperator(Operator):
"""An identity operator I that satisfies op * I == I * op == op for any
operator op.
Parameters
==========
N : Integer
Optional parameter that specifies the dimension of the Hilbert space
of operator. This is used when generating a matrix representation.
Examples
========
>>> from sympy.physics.quantum import IdentityOperator
>>> IdentityOperator()
I
"""
@property
def dimension(self):
return self.N
@classmethod
def default_args(self):
return (oo,)
def __init__(self, *args, **hints):
if not len(args) in [0, 1]:
raise ValueError('0 or 1 parameters expected, got %s' % args)
self.N = args[0] if (len(args) == 1 and args[0]) else oo
def _eval_commutator(self, other, **hints):
return Integer(0)
def _eval_anticommutator(self, other, **hints):
return 2 * other
def _eval_inverse(self):
return self
def _eval_adjoint(self):
return self
def _apply_operator(self, ket, **options):
return ket
def _eval_power(self, exp):
return self
def _print_contents(self, printer, *args):
return 'I'
def _print_contents_pretty(self, printer, *args):
return prettyForm('I')
def _print_contents_latex(self, printer, *args):
return r'{\mathcal{I}}'
def __mul__(self, other):
if isinstance(other, Operator):
return other
return Mul(self, other)
def _represent_default_basis(self, **options):
if not self.N or self.N == oo:
raise NotImplementedError('Cannot represent infinite dimensional' +
' identity operator as a matrix')
format = options.get('format', 'sympy')
if format != 'sympy':
raise NotImplementedError('Representation in format ' +
'%s not implemented.' % format)
return eye(self.N)
class OuterProduct(Operator):
"""An unevaluated outer product between a ket and bra.
This constructs an outer product between any subclass of ``KetBase`` and
``BraBase`` as ``|a><b|``. An ``OuterProduct`` inherits from Operator as they act as
operators in quantum expressions. For reference see [1]_.
Parameters
==========
ket : KetBase
The ket on the left side of the outer product.
bar : BraBase
The bra on the right side of the outer product.
Examples
========
Create a simple outer product by hand and take its dagger::
>>> from sympy.physics.quantum import Ket, Bra, OuterProduct, Dagger
>>> from sympy.physics.quantum import Operator
>>> k = Ket('k')
>>> b = Bra('b')
>>> op = OuterProduct(k, b)
>>> op
|k><b|
>>> op.hilbert_space
H
>>> op.ket
|k>
>>> op.bra
<b|
>>> Dagger(op)
|b><k|
In simple products of kets and bras outer products will be automatically
identified and created::
>>> k*b
|k><b|
But in more complex expressions, outer products are not automatically
created::
>>> A = Operator('A')
>>> A*k*b
A*|k>*<b|
A user can force the creation of an outer product in a complex expression
by using parentheses to group the ket and bra::
>>> A*(k*b)
A*|k><b|
References
==========
.. [1] https://en.wikipedia.org/wiki/Outer_product
"""
is_commutative = False
def __new__(cls, *args, **old_assumptions):
from sympy.physics.quantum.state import KetBase, BraBase
if len(args) != 2:
raise ValueError('2 parameters expected, got %d' % len(args))
ket_expr = expand(args[0])
bra_expr = expand(args[1])
if (isinstance(ket_expr, (KetBase, Mul)) and
isinstance(bra_expr, (BraBase, Mul))):
ket_c, kets = ket_expr.args_cnc()
bra_c, bras = bra_expr.args_cnc()
if len(kets) != 1 or not isinstance(kets[0], KetBase):
raise TypeError('KetBase subclass expected'
', got: %r' % Mul(*kets))
if len(bras) != 1 or not isinstance(bras[0], BraBase):
raise TypeError('BraBase subclass expected'
', got: %r' % Mul(*bras))
if not kets[0].dual_class() == bras[0].__class__:
raise TypeError(
'ket and bra are not dual classes: %r, %r' %
(kets[0].__class__, bras[0].__class__)
)
# TODO: make sure the hilbert spaces of the bra and ket are
# compatible
obj = Expr.__new__(cls, *(kets[0], bras[0]), **old_assumptions)
obj.hilbert_space = kets[0].hilbert_space
return Mul(*(ket_c + bra_c)) * obj
op_terms = []
if isinstance(ket_expr, Add) and isinstance(bra_expr, Add):
for ket_term in ket_expr.args:
for bra_term in bra_expr.args:
op_terms.append(OuterProduct(ket_term, bra_term,
**old_assumptions))
elif isinstance(ket_expr, Add):
for ket_term in ket_expr.args:
op_terms.append(OuterProduct(ket_term, bra_expr,
**old_assumptions))
elif isinstance(bra_expr, Add):
for bra_term in bra_expr.args:
op_terms.append(OuterProduct(ket_expr, bra_term,
**old_assumptions))
else:
raise TypeError(
'Expected ket and bra expression, got: %r, %r' %
(ket_expr, bra_expr)
)
return Add(*op_terms)
@property
def ket(self):
"""Return the ket on the left side of the outer product."""
return self.args[0]
@property
def bra(self):
"""Return the bra on the right side of the outer product."""
return self.args[1]
def _eval_adjoint(self):
return OuterProduct(Dagger(self.bra), Dagger(self.ket))
def _sympystr(self, printer, *args):
return printer._print(self.ket) + printer._print(self.bra)
def _sympyrepr(self, printer, *args):
return '%s(%s,%s)' % (self.__class__.__name__,
printer._print(self.ket, *args), printer._print(self.bra, *args))
def _pretty(self, printer, *args):
pform = self.ket._pretty(printer, *args)
return prettyForm(*pform.right(self.bra._pretty(printer, *args)))
def _latex(self, printer, *args):
k = printer._print(self.ket, *args)
b = printer._print(self.bra, *args)
return k + b
def _represent(self, **options):
k = self.ket._represent(**options)
b = self.bra._represent(**options)
return k*b
def _eval_trace(self, **kwargs):
# TODO if operands are tensorproducts this may be will be handled
# differently.
return self.ket._eval_trace(self.bra, **kwargs)
class DifferentialOperator(Operator):
"""An operator for representing the differential operator, i.e. d/dx
It is initialized by passing two arguments. The first is an arbitrary
expression that involves a function, such as ``Derivative(f(x), x)``. The
second is the function (e.g. ``f(x)``) which we are to replace with the
``Wavefunction`` that this ``DifferentialOperator`` is applied to.
Parameters
==========
expr : Expr
The arbitrary expression which the appropriate Wavefunction is to be
substituted into
func : Expr
A function (e.g. f(x)) which is to be replaced with the appropriate
Wavefunction when this DifferentialOperator is applied
Examples
========
You can define a completely arbitrary expression and specify where the
Wavefunction is to be substituted
>>> from sympy import Derivative, Function, Symbol
>>> from sympy.physics.quantum.operator import DifferentialOperator
>>> from sympy.physics.quantum.state import Wavefunction
>>> from sympy.physics.quantum.qapply import qapply
>>> f = Function('f')
>>> x = Symbol('x')
>>> d = DifferentialOperator(1/x*Derivative(f(x), x), f(x))
>>> w = Wavefunction(x**2, x)
>>> d.function
f(x)
>>> d.variables
(x,)
>>> qapply(d*w)
Wavefunction(2, x)
"""
@property
def variables(self):
"""
Returns the variables with which the function in the specified
arbitrary expression is evaluated
Examples
========
>>> from sympy.physics.quantum.operator import DifferentialOperator
>>> from sympy import Symbol, Function, Derivative
>>> x = Symbol('x')
>>> f = Function('f')
>>> d = DifferentialOperator(1/x*Derivative(f(x), x), f(x))
>>> d.variables
(x,)
>>> y = Symbol('y')
>>> d = DifferentialOperator(Derivative(f(x, y), x) +
... Derivative(f(x, y), y), f(x, y))
>>> d.variables
(x, y)
"""
return self.args[-1].args
@property
def function(self):
"""
Returns the function which is to be replaced with the Wavefunction
Examples
========
>>> from sympy.physics.quantum.operator import DifferentialOperator
>>> from sympy import Function, Symbol, Derivative
>>> x = Symbol('x')
>>> f = Function('f')
>>> d = DifferentialOperator(Derivative(f(x), x), f(x))
>>> d.function
f(x)
>>> y = Symbol('y')
>>> d = DifferentialOperator(Derivative(f(x, y), x) +
... Derivative(f(x, y), y), f(x, y))
>>> d.function
f(x, y)
"""
return self.args[-1]
@property
def expr(self):
"""
Returns the arbitrary expression which is to have the Wavefunction
substituted into it
Examples
========
>>> from sympy.physics.quantum.operator import DifferentialOperator
>>> from sympy import Function, Symbol, Derivative
>>> x = Symbol('x')
>>> f = Function('f')
>>> d = DifferentialOperator(Derivative(f(x), x), f(x))
>>> d.expr
Derivative(f(x), x)
>>> y = Symbol('y')
>>> d = DifferentialOperator(Derivative(f(x, y), x) +
... Derivative(f(x, y), y), f(x, y))
>>> d.expr
Derivative(f(x, y), x) + Derivative(f(x, y), y)
"""
return self.args[0]
@property
def free_symbols(self):
"""
Return the free symbols of the expression.
"""
return self.expr.free_symbols
def _apply_operator_Wavefunction(self, func):
from sympy.physics.quantum.state import Wavefunction
var = self.variables
wf_vars = func.args[1:]
f = self.function
new_expr = self.expr.subs(f, func(*var))
new_expr = new_expr.doit()
return Wavefunction(new_expr, *wf_vars)
def _eval_derivative(self, symbol):
new_expr = Derivative(self.expr, symbol)
return DifferentialOperator(new_expr, self.args[-1])
#-------------------------------------------------------------------------
# Printing
#-------------------------------------------------------------------------
def _print(self, printer, *args):
return '%s(%s)' % (
self._print_operator_name(printer, *args),
self._print_label(printer, *args)
)
def _print_pretty(self, printer, *args):
pform = self._print_operator_name_pretty(printer, *args)
label_pform = self._print_label_pretty(printer, *args)
label_pform = prettyForm(
*label_pform.parens(left='(', right=')')
)
pform = prettyForm(*pform.right((label_pform)))
return pform
|
e97f127696adfdc678c456ceb84880ae6100fce35d415b641904eb34cc04b772 | """Qubits for quantum computing.
Todo:
* Finish implementing measurement logic. This should include POVM.
* Update docstrings.
* Update tests.
"""
from __future__ import print_function, division
import math
from sympy import Integer, log, Mul, Add, Pow, conjugate
from sympy.core.basic import sympify
from sympy.core.compatibility import SYMPY_INTS
from sympy.matrices import Matrix, zeros
from sympy.printing.pretty.stringpict import prettyForm
from sympy.physics.quantum.hilbert import ComplexSpace
from sympy.physics.quantum.state import Ket, Bra, State
from sympy.physics.quantum.qexpr import QuantumError
from sympy.physics.quantum.represent import represent
from sympy.physics.quantum.matrixutils import (
numpy_ndarray, scipy_sparse_matrix
)
from mpmath.libmp.libintmath import bitcount
__all__ = [
'Qubit',
'QubitBra',
'IntQubit',
'IntQubitBra',
'qubit_to_matrix',
'matrix_to_qubit',
'matrix_to_density',
'measure_all',
'measure_partial',
'measure_partial_oneshot',
'measure_all_oneshot'
]
#-----------------------------------------------------------------------------
# Qubit Classes
#-----------------------------------------------------------------------------
class QubitState(State):
"""Base class for Qubit and QubitBra."""
#-------------------------------------------------------------------------
# Initialization/creation
#-------------------------------------------------------------------------
@classmethod
def _eval_args(cls, args):
# If we are passed a QubitState or subclass, we just take its qubit
# values directly.
if len(args) == 1 and isinstance(args[0], QubitState):
return args[0].qubit_values
# Turn strings into tuple of strings
if len(args) == 1 and isinstance(args[0], str):
args = tuple(args[0])
args = sympify(args)
# Validate input (must have 0 or 1 input)
for element in args:
if not (element == 1 or element == 0):
raise ValueError(
"Qubit values must be 0 or 1, got: %r" % element)
return args
@classmethod
def _eval_hilbert_space(cls, args):
return ComplexSpace(2)**len(args)
#-------------------------------------------------------------------------
# Properties
#-------------------------------------------------------------------------
@property
def dimension(self):
"""The number of Qubits in the state."""
return len(self.qubit_values)
@property
def nqubits(self):
return self.dimension
@property
def qubit_values(self):
"""Returns the values of the qubits as a tuple."""
return self.label
#-------------------------------------------------------------------------
# Special methods
#-------------------------------------------------------------------------
def __len__(self):
return self.dimension
def __getitem__(self, bit):
return self.qubit_values[int(self.dimension - bit - 1)]
#-------------------------------------------------------------------------
# Utility methods
#-------------------------------------------------------------------------
def flip(self, *bits):
"""Flip the bit(s) given."""
newargs = list(self.qubit_values)
for i in bits:
bit = int(self.dimension - i - 1)
if newargs[bit] == 1:
newargs[bit] = 0
else:
newargs[bit] = 1
return self.__class__(*tuple(newargs))
class Qubit(QubitState, Ket):
"""A multi-qubit ket in the computational (z) basis.
We use the normal convention that the least significant qubit is on the
right, so ``|00001>`` has a 1 in the least significant qubit.
Parameters
==========
values : list, str
The qubit values as a list of ints ([0,0,0,1,1,]) or a string ('011').
Examples
========
Create a qubit in a couple of different ways and look at their attributes:
>>> from sympy.physics.quantum.qubit import Qubit
>>> Qubit(0,0,0)
|000>
>>> q = Qubit('0101')
>>> q
|0101>
>>> q.nqubits
4
>>> len(q)
4
>>> q.dimension
4
>>> q.qubit_values
(0, 1, 0, 1)
We can flip the value of an individual qubit:
>>> q.flip(1)
|0111>
We can take the dagger of a Qubit to get a bra:
>>> from sympy.physics.quantum.dagger import Dagger
>>> Dagger(q)
<0101|
>>> type(Dagger(q))
<class 'sympy.physics.quantum.qubit.QubitBra'>
Inner products work as expected:
>>> ip = Dagger(q)*q
>>> ip
<0101|0101>
>>> ip.doit()
1
"""
@classmethod
def dual_class(self):
return QubitBra
def _eval_innerproduct_QubitBra(self, bra, **hints):
if self.label == bra.label:
return Integer(1)
else:
return Integer(0)
def _represent_default_basis(self, **options):
return self._represent_ZGate(None, **options)
def _represent_ZGate(self, basis, **options):
"""Represent this qubits in the computational basis (ZGate).
"""
_format = options.get('format', 'sympy')
n = 1
definite_state = 0
for it in reversed(self.qubit_values):
definite_state += n*it
n = n*2
result = [0]*(2**self.dimension)
result[int(definite_state)] = 1
if _format == 'sympy':
return Matrix(result)
elif _format == 'numpy':
import numpy as np
return np.matrix(result, dtype='complex').transpose()
elif _format == 'scipy.sparse':
from scipy import sparse
return sparse.csr_matrix(result, dtype='complex').transpose()
def _eval_trace(self, bra, **kwargs):
indices = kwargs.get('indices', [])
#sort index list to begin trace from most-significant
#qubit
sorted_idx = list(indices)
if len(sorted_idx) == 0:
sorted_idx = list(range(0, self.nqubits))
sorted_idx.sort()
#trace out for each of index
new_mat = self*bra
for i in range(len(sorted_idx) - 1, -1, -1):
# start from tracing out from leftmost qubit
new_mat = self._reduced_density(new_mat, int(sorted_idx[i]))
if (len(sorted_idx) == self.nqubits):
#in case full trace was requested
return new_mat[0]
else:
return matrix_to_density(new_mat)
def _reduced_density(self, matrix, qubit, **options):
"""Compute the reduced density matrix by tracing out one qubit.
The qubit argument should be of type python int, since it is used
in bit operations
"""
def find_index_that_is_projected(j, k, qubit):
bit_mask = 2**qubit - 1
return ((j >> qubit) << (1 + qubit)) + (j & bit_mask) + (k << qubit)
old_matrix = represent(matrix, **options)
old_size = old_matrix.cols
#we expect the old_size to be even
new_size = old_size//2
new_matrix = Matrix().zeros(new_size)
for i in range(new_size):
for j in range(new_size):
for k in range(2):
col = find_index_that_is_projected(j, k, qubit)
row = find_index_that_is_projected(i, k, qubit)
new_matrix[i, j] += old_matrix[row, col]
return new_matrix
class QubitBra(QubitState, Bra):
"""A multi-qubit bra in the computational (z) basis.
We use the normal convention that the least significant qubit is on the
right, so ``|00001>`` has a 1 in the least significant qubit.
Parameters
==========
values : list, str
The qubit values as a list of ints ([0,0,0,1,1,]) or a string ('011').
See also
========
Qubit: Examples using qubits
"""
@classmethod
def dual_class(self):
return Qubit
class IntQubitState(QubitState):
"""A base class for qubits that work with binary representations."""
@classmethod
def _eval_args(cls, args, nqubits=None):
# The case of a QubitState instance
if len(args) == 1 and isinstance(args[0], QubitState):
return QubitState._eval_args(args)
# otherwise, args should be integer
elif not all((isinstance(a, (int, Integer)) for a in args)):
raise ValueError('values must be integers, got (%s)' % (tuple(type(a) for a in args),))
# use nqubits if specified
if nqubits is not None:
if not isinstance(nqubits, (int, Integer)):
raise ValueError('nqubits must be an integer, got (%s)' % type(nqubits))
if len(args) != 1:
raise ValueError(
'too many positional arguments (%s). should be (number, nqubits=n)' % (args,))
return cls._eval_args_with_nqubits(args[0], nqubits)
# For a single argument, we construct the binary representation of
# that integer with the minimal number of bits.
if len(args) == 1 and args[0] > 1:
#rvalues is the minimum number of bits needed to express the number
rvalues = reversed(range(bitcount(abs(args[0]))))
qubit_values = [(args[0] >> i) & 1 for i in rvalues]
return QubitState._eval_args(qubit_values)
# For two numbers, the second number is the number of bits
# on which it is expressed, so IntQubit(0,5) == |00000>.
elif len(args) == 2 and args[1] > 1:
return cls._eval_args_with_nqubits(args[0], args[1])
else:
return QubitState._eval_args(args)
@classmethod
def _eval_args_with_nqubits(cls, number, nqubits):
need = bitcount(abs(number))
if nqubits < need:
raise ValueError(
'cannot represent %s with %s bits' % (number, nqubits))
qubit_values = [(number >> i) & 1 for i in reversed(range(nqubits))]
return QubitState._eval_args(qubit_values)
def as_int(self):
"""Return the numerical value of the qubit."""
number = 0
n = 1
for i in reversed(self.qubit_values):
number += n*i
n = n << 1
return number
def _print_label(self, printer, *args):
return str(self.as_int())
def _print_label_pretty(self, printer, *args):
label = self._print_label(printer, *args)
return prettyForm(label)
_print_label_repr = _print_label
_print_label_latex = _print_label
class IntQubit(IntQubitState, Qubit):
"""A qubit ket that store integers as binary numbers in qubit values.
The differences between this class and ``Qubit`` are:
* The form of the constructor.
* The qubit values are printed as their corresponding integer, rather
than the raw qubit values. The internal storage format of the qubit
values in the same as ``Qubit``.
Parameters
==========
values : int, tuple
If a single argument, the integer we want to represent in the qubit
values. This integer will be represented using the fewest possible
number of qubits.
If a pair of integers and the second value is more than one, the first
integer gives the integer to represent in binary form and the second
integer gives the number of qubits to use.
List of zeros and ones is also accepted to generate qubit by bit pattern.
nqubits : int
The integer that represents the number of qubits.
This number should be passed with keyword ``nqubits=N``.
You can use this in order to avoid ambiguity of Qubit-style tuple of bits.
Please see the example below for more details.
Examples
========
Create a qubit for the integer 5:
>>> from sympy.physics.quantum.qubit import IntQubit
>>> from sympy.physics.quantum.qubit import Qubit
>>> q = IntQubit(5)
>>> q
|5>
We can also create an ``IntQubit`` by passing a ``Qubit`` instance.
>>> q = IntQubit(Qubit('101'))
>>> q
|5>
>>> q.as_int()
5
>>> q.nqubits
3
>>> q.qubit_values
(1, 0, 1)
We can go back to the regular qubit form.
>>> Qubit(q)
|101>
Please note that ``IntQubit`` also accepts a ``Qubit``-style list of bits.
So, the code below yields qubits 3, not a single bit ``1``.
>>> IntQubit(1, 1)
|3>
To avoid ambiguity, use ``nqubits`` parameter.
Use of this keyword is recommended especially when you provide the values by variables.
>>> IntQubit(1, nqubits=1)
|1>
>>> a = 1
>>> IntQubit(a, nqubits=1)
|1>
"""
@classmethod
def dual_class(self):
return IntQubitBra
def _eval_innerproduct_IntQubitBra(self, bra, **hints):
return Qubit._eval_innerproduct_QubitBra(self, bra)
class IntQubitBra(IntQubitState, QubitBra):
"""A qubit bra that store integers as binary numbers in qubit values."""
@classmethod
def dual_class(self):
return IntQubit
#-----------------------------------------------------------------------------
# Qubit <---> Matrix conversion functions
#-----------------------------------------------------------------------------
def matrix_to_qubit(matrix):
"""Convert from the matrix repr. to a sum of Qubit objects.
Parameters
----------
matrix : Matrix, numpy.matrix, scipy.sparse
The matrix to build the Qubit representation of. This works with
sympy matrices, numpy matrices and scipy.sparse sparse matrices.
Examples
========
Represent a state and then go back to its qubit form:
>>> from sympy.physics.quantum.qubit import matrix_to_qubit, Qubit
>>> from sympy.physics.quantum.represent import represent
>>> q = Qubit('01')
>>> matrix_to_qubit(represent(q))
|01>
"""
# Determine the format based on the type of the input matrix
format = 'sympy'
if isinstance(matrix, numpy_ndarray):
format = 'numpy'
if isinstance(matrix, scipy_sparse_matrix):
format = 'scipy.sparse'
# Make sure it is of correct dimensions for a Qubit-matrix representation.
# This logic should work with sympy, numpy or scipy.sparse matrices.
if matrix.shape[0] == 1:
mlistlen = matrix.shape[1]
nqubits = log(mlistlen, 2)
ket = False
cls = QubitBra
elif matrix.shape[1] == 1:
mlistlen = matrix.shape[0]
nqubits = log(mlistlen, 2)
ket = True
cls = Qubit
else:
raise QuantumError(
'Matrix must be a row/column vector, got %r' % matrix
)
if not isinstance(nqubits, Integer):
raise QuantumError('Matrix must be a row/column vector of size '
'2**nqubits, got: %r' % matrix)
# Go through each item in matrix, if element is non-zero, make it into a
# Qubit item times the element.
result = 0
for i in range(mlistlen):
if ket:
element = matrix[i, 0]
else:
element = matrix[0, i]
if format == 'numpy' or format == 'scipy.sparse':
element = complex(element)
if element != 0.0:
# Form Qubit array; 0 in bit-locations where i is 0, 1 in
# bit-locations where i is 1
qubit_array = [int(i & (1 << x) != 0) for x in range(nqubits)]
qubit_array.reverse()
result = result + element*cls(*qubit_array)
# If sympy simplified by pulling out a constant coefficient, undo that.
if isinstance(result, (Mul, Add, Pow)):
result = result.expand()
return result
def matrix_to_density(mat):
"""
Works by finding the eigenvectors and eigenvalues of the matrix.
We know we can decompose rho by doing:
sum(EigenVal*|Eigenvect><Eigenvect|)
"""
from sympy.physics.quantum.density import Density
eigen = mat.eigenvects()
args = [[matrix_to_qubit(Matrix(
[vector, ])), x[0]] for x in eigen for vector in x[2] if x[0] != 0]
if (len(args) == 0):
return 0
else:
return Density(*args)
def qubit_to_matrix(qubit, format='sympy'):
"""Converts an Add/Mul of Qubit objects into it's matrix representation
This function is the inverse of ``matrix_to_qubit`` and is a shorthand
for ``represent(qubit)``.
"""
return represent(qubit, format=format)
#-----------------------------------------------------------------------------
# Measurement
#-----------------------------------------------------------------------------
def measure_all(qubit, format='sympy', normalize=True):
"""Perform an ensemble measurement of all qubits.
Parameters
==========
qubit : Qubit, Add
The qubit to measure. This can be any Qubit or a linear combination
of them.
format : str
The format of the intermediate matrices to use. Possible values are
('sympy','numpy','scipy.sparse'). Currently only 'sympy' is
implemented.
Returns
=======
result : list
A list that consists of primitive states and their probabilities.
Examples
========
>>> from sympy.physics.quantum.qubit import Qubit, measure_all
>>> from sympy.physics.quantum.gate import H
>>> from sympy.physics.quantum.qapply import qapply
>>> c = H(0)*H(1)*Qubit('00')
>>> c
H(0)*H(1)*|00>
>>> q = qapply(c)
>>> measure_all(q)
[(|00>, 1/4), (|01>, 1/4), (|10>, 1/4), (|11>, 1/4)]
"""
m = qubit_to_matrix(qubit, format)
if format == 'sympy':
results = []
if normalize:
m = m.normalized()
size = max(m.shape) # Max of shape to account for bra or ket
nqubits = int(math.log(size)/math.log(2))
for i in range(size):
if m[i] != 0.0:
results.append(
(Qubit(IntQubit(i, nqubits=nqubits)), m[i]*conjugate(m[i]))
)
return results
else:
raise NotImplementedError(
"This function can't handle non-sympy matrix formats yet"
)
def measure_partial(qubit, bits, format='sympy', normalize=True):
"""Perform a partial ensemble measure on the specified qubits.
Parameters
==========
qubits : Qubit
The qubit to measure. This can be any Qubit or a linear combination
of them.
bits : tuple
The qubits to measure.
format : str
The format of the intermediate matrices to use. Possible values are
('sympy','numpy','scipy.sparse'). Currently only 'sympy' is
implemented.
Returns
=======
result : list
A list that consists of primitive states and their probabilities.
Examples
========
>>> from sympy.physics.quantum.qubit import Qubit, measure_partial
>>> from sympy.physics.quantum.gate import H
>>> from sympy.physics.quantum.qapply import qapply
>>> c = H(0)*H(1)*Qubit('00')
>>> c
H(0)*H(1)*|00>
>>> q = qapply(c)
>>> measure_partial(q, (0,))
[(sqrt(2)*|00>/2 + sqrt(2)*|10>/2, 1/2), (sqrt(2)*|01>/2 + sqrt(2)*|11>/2, 1/2)]
"""
m = qubit_to_matrix(qubit, format)
if isinstance(bits, (SYMPY_INTS, Integer)):
bits = (int(bits),)
if format == 'sympy':
if normalize:
m = m.normalized()
possible_outcomes = _get_possible_outcomes(m, bits)
# Form output from function.
output = []
for outcome in possible_outcomes:
# Calculate probability of finding the specified bits with
# given values.
prob_of_outcome = 0
prob_of_outcome += (outcome.H*outcome)[0]
# If the output has a chance, append it to output with found
# probability.
if prob_of_outcome != 0:
if normalize:
next_matrix = matrix_to_qubit(outcome.normalized())
else:
next_matrix = matrix_to_qubit(outcome)
output.append((
next_matrix,
prob_of_outcome
))
return output
else:
raise NotImplementedError(
"This function can't handle non-sympy matrix formats yet"
)
def measure_partial_oneshot(qubit, bits, format='sympy'):
"""Perform a partial oneshot measurement on the specified qubits.
A oneshot measurement is equivalent to performing a measurement on a
quantum system. This type of measurement does not return the probabilities
like an ensemble measurement does, but rather returns *one* of the
possible resulting states. The exact state that is returned is determined
by picking a state randomly according to the ensemble probabilities.
Parameters
----------
qubits : Qubit
The qubit to measure. This can be any Qubit or a linear combination
of them.
bits : tuple
The qubits to measure.
format : str
The format of the intermediate matrices to use. Possible values are
('sympy','numpy','scipy.sparse'). Currently only 'sympy' is
implemented.
Returns
-------
result : Qubit
The qubit that the system collapsed to upon measurement.
"""
import random
m = qubit_to_matrix(qubit, format)
if format == 'sympy':
m = m.normalized()
possible_outcomes = _get_possible_outcomes(m, bits)
# Form output from function
random_number = random.random()
total_prob = 0
for outcome in possible_outcomes:
# Calculate probability of finding the specified bits
# with given values
total_prob += (outcome.H*outcome)[0]
if total_prob >= random_number:
return matrix_to_qubit(outcome.normalized())
else:
raise NotImplementedError(
"This function can't handle non-sympy matrix formats yet"
)
def _get_possible_outcomes(m, bits):
"""Get the possible states that can be produced in a measurement.
Parameters
----------
m : Matrix
The matrix representing the state of the system.
bits : tuple, list
Which bits will be measured.
Returns
-------
result : list
The list of possible states which can occur given this measurement.
These are un-normalized so we can derive the probability of finding
this state by taking the inner product with itself
"""
# This is filled with loads of dirty binary tricks...You have been warned
size = max(m.shape) # Max of shape to account for bra or ket
nqubits = int(math.log(size, 2) + .1) # Number of qubits possible
# Make the output states and put in output_matrices, nothing in them now.
# Each state will represent a possible outcome of the measurement
# Thus, output_matrices[0] is the matrix which we get when all measured
# bits return 0. and output_matrices[1] is the matrix for only the 0th
# bit being true
output_matrices = []
for i in range(1 << len(bits)):
output_matrices.append(zeros(2**nqubits, 1))
# Bitmasks will help sort how to determine possible outcomes.
# When the bit mask is and-ed with a matrix-index,
# it will determine which state that index belongs to
bit_masks = []
for bit in bits:
bit_masks.append(1 << bit)
# Make possible outcome states
for i in range(2**nqubits):
trueness = 0 # This tells us to which output_matrix this value belongs
# Find trueness
for j in range(len(bit_masks)):
if i & bit_masks[j]:
trueness += j + 1
# Put the value in the correct output matrix
output_matrices[trueness][i] = m[i]
return output_matrices
def measure_all_oneshot(qubit, format='sympy'):
"""Perform a oneshot ensemble measurement on all qubits.
A oneshot measurement is equivalent to performing a measurement on a
quantum system. This type of measurement does not return the probabilities
like an ensemble measurement does, but rather returns *one* of the
possible resulting states. The exact state that is returned is determined
by picking a state randomly according to the ensemble probabilities.
Parameters
----------
qubits : Qubit
The qubit to measure. This can be any Qubit or a linear combination
of them.
format : str
The format of the intermediate matrices to use. Possible values are
('sympy','numpy','scipy.sparse'). Currently only 'sympy' is
implemented.
Returns
-------
result : Qubit
The qubit that the system collapsed to upon measurement.
"""
import random
m = qubit_to_matrix(qubit)
if format == 'sympy':
m = m.normalized()
random_number = random.random()
total = 0
result = 0
for i in m:
total += i*i.conjugate()
if total > random_number:
break
result += 1
return Qubit(IntQubit(result, int(math.log(max(m.shape), 2) + .1)))
else:
raise NotImplementedError(
"This function can't handle non-sympy matrix formats yet"
)
|
f0a671b76147cd2bf97f6daf9373eeb7610d2985cd1c2caa62058aaede5be097 | """Symbolic inner product."""
from __future__ import print_function, division
from sympy import Expr, conjugate
from sympy.printing.pretty.stringpict import prettyForm
from sympy.physics.quantum.dagger import Dagger
from sympy.physics.quantum.state import KetBase, BraBase
__all__ = [
'InnerProduct'
]
# InnerProduct is not an QExpr because it is really just a regular commutative
# number. We have gone back and forth about this, but we gain a lot by having
# it subclass Expr. The main challenges were getting Dagger to work
# (we use _eval_conjugate) and represent (we can use atoms and subs). Having
# it be an Expr, mean that there are no commutative QExpr subclasses,
# which simplifies the design of everything.
class InnerProduct(Expr):
"""An unevaluated inner product between a Bra and a Ket [1].
Parameters
==========
bra : BraBase or subclass
The bra on the left side of the inner product.
ket : KetBase or subclass
The ket on the right side of the inner product.
Examples
========
Create an InnerProduct and check its properties:
>>> from sympy.physics.quantum import Bra, Ket
>>> b = Bra('b')
>>> k = Ket('k')
>>> ip = b*k
>>> ip
<b|k>
>>> ip.bra
<b|
>>> ip.ket
|k>
In simple products of kets and bras inner products will be automatically
identified and created::
>>> b*k
<b|k>
But in more complex expressions, there is ambiguity in whether inner or
outer products should be created::
>>> k*b*k*b
|k><b|*|k>*<b|
A user can force the creation of a inner products in a complex expression
by using parentheses to group the bra and ket::
>>> k*(b*k)*b
<b|k>*|k>*<b|
Notice how the inner product <b|k> moved to the left of the expression
because inner products are commutative complex numbers.
References
==========
.. [1] https://en.wikipedia.org/wiki/Inner_product
"""
is_complex = True
def __new__(cls, bra, ket):
if not isinstance(ket, KetBase):
raise TypeError('KetBase subclass expected, got: %r' % ket)
if not isinstance(bra, BraBase):
raise TypeError('BraBase subclass expected, got: %r' % ket)
obj = Expr.__new__(cls, bra, ket)
return obj
@property
def bra(self):
return self.args[0]
@property
def ket(self):
return self.args[1]
def _eval_conjugate(self):
return InnerProduct(Dagger(self.ket), Dagger(self.bra))
def _sympyrepr(self, printer, *args):
return '%s(%s,%s)' % (self.__class__.__name__,
printer._print(self.bra, *args), printer._print(self.ket, *args))
def _sympystr(self, printer, *args):
sbra = printer._print(self.bra)
sket = printer._print(self.ket)
return '%s|%s' % (sbra[:-1], sket[1:])
def _pretty(self, printer, *args):
# Print state contents
bra = self.bra._print_contents_pretty(printer, *args)
ket = self.ket._print_contents_pretty(printer, *args)
# Print brackets
height = max(bra.height(), ket.height())
use_unicode = printer._use_unicode
lbracket, _ = self.bra._pretty_brackets(height, use_unicode)
cbracket, rbracket = self.ket._pretty_brackets(height, use_unicode)
# Build innerproduct
pform = prettyForm(*bra.left(lbracket))
pform = prettyForm(*pform.right(cbracket))
pform = prettyForm(*pform.right(ket))
pform = prettyForm(*pform.right(rbracket))
return pform
def _latex(self, printer, *args):
bra_label = self.bra._print_contents_latex(printer, *args)
ket = printer._print(self.ket, *args)
return r'\left\langle %s \right. %s' % (bra_label, ket)
def doit(self, **hints):
try:
r = self.ket._eval_innerproduct(self.bra, **hints)
except NotImplementedError:
try:
r = conjugate(
self.bra.dual._eval_innerproduct(self.ket.dual, **hints)
)
except NotImplementedError:
r = None
if r is not None:
return r
return self
|
c7210da9a8334c7bedd114b80918811f42ce1537340afcc5f6c3e44c8027c268 | from __future__ import print_function, division
from itertools import product
from sympy import Tuple, Add, Mul, Matrix, log, expand, S
from sympy.core.trace import Tr
from sympy.printing.pretty.stringpict import prettyForm
from sympy.physics.quantum.dagger import Dagger
from sympy.physics.quantum.operator import HermitianOperator
from sympy.physics.quantum.represent import represent
from sympy.physics.quantum.matrixutils import numpy_ndarray, scipy_sparse_matrix, to_numpy
from sympy.physics.quantum.tensorproduct import TensorProduct, tensor_product_simp
class Density(HermitianOperator):
"""Density operator for representing mixed states.
TODO: Density operator support for Qubits
Parameters
==========
values : tuples/lists
Each tuple/list should be of form (state, prob) or [state,prob]
Examples
========
Create a density operator with 2 states represented by Kets.
>>> from sympy.physics.quantum.state import Ket
>>> from sympy.physics.quantum.density import Density
>>> d = Density([Ket(0), 0.5], [Ket(1),0.5])
>>> d
'Density'((|0>, 0.5),(|1>, 0.5))
"""
@classmethod
def _eval_args(cls, args):
# call this to qsympify the args
args = super(Density, cls)._eval_args(args)
for arg in args:
# Check if arg is a tuple
if not (isinstance(arg, Tuple) and len(arg) == 2):
raise ValueError("Each argument should be of form [state,prob]"
" or ( state, prob )")
return args
def states(self):
"""Return list of all states.
Examples
========
>>> from sympy.physics.quantum.state import Ket
>>> from sympy.physics.quantum.density import Density
>>> d = Density([Ket(0), 0.5], [Ket(1),0.5])
>>> d.states()
(|0>, |1>)
"""
return Tuple(*[arg[0] for arg in self.args])
def probs(self):
"""Return list of all probabilities.
Examples
========
>>> from sympy.physics.quantum.state import Ket
>>> from sympy.physics.quantum.density import Density
>>> d = Density([Ket(0), 0.5], [Ket(1),0.5])
>>> d.probs()
(0.5, 0.5)
"""
return Tuple(*[arg[1] for arg in self.args])
def get_state(self, index):
"""Return specific state by index.
Parameters
==========
index : index of state to be returned
Examples
========
>>> from sympy.physics.quantum.state import Ket
>>> from sympy.physics.quantum.density import Density
>>> d = Density([Ket(0), 0.5], [Ket(1),0.5])
>>> d.states()[1]
|1>
"""
state = self.args[index][0]
return state
def get_prob(self, index):
"""Return probability of specific state by index.
Parameters
===========
index : index of states whose probability is returned.
Examples
========
>>> from sympy.physics.quantum.state import Ket
>>> from sympy.physics.quantum.density import Density
>>> d = Density([Ket(0), 0.5], [Ket(1),0.5])
>>> d.probs()[1]
0.500000000000000
"""
prob = self.args[index][1]
return prob
def apply_op(self, op):
"""op will operate on each individual state.
Parameters
==========
op : Operator
Examples
========
>>> from sympy.physics.quantum.state import Ket
>>> from sympy.physics.quantum.density import Density
>>> from sympy.physics.quantum.operator import Operator
>>> A = Operator('A')
>>> d = Density([Ket(0), 0.5], [Ket(1),0.5])
>>> d.apply_op(A)
'Density'((A*|0>, 0.5),(A*|1>, 0.5))
"""
new_args = [(op*state, prob) for (state, prob) in self.args]
return Density(*new_args)
def doit(self, **hints):
"""Expand the density operator into an outer product format.
Examples
========
>>> from sympy.physics.quantum.state import Ket
>>> from sympy.physics.quantum.density import Density
>>> from sympy.physics.quantum.operator import Operator
>>> A = Operator('A')
>>> d = Density([Ket(0), 0.5], [Ket(1),0.5])
>>> d.doit()
0.5*|0><0| + 0.5*|1><1|
"""
terms = []
for (state, prob) in self.args:
state = state.expand() # needed to break up (a+b)*c
if (isinstance(state, Add)):
for arg in product(state.args, repeat=2):
terms.append(prob*self._generate_outer_prod(arg[0],
arg[1]))
else:
terms.append(prob*self._generate_outer_prod(state, state))
return Add(*terms)
def _generate_outer_prod(self, arg1, arg2):
c_part1, nc_part1 = arg1.args_cnc()
c_part2, nc_part2 = arg2.args_cnc()
if (len(nc_part1) == 0 or len(nc_part2) == 0):
raise ValueError('Atleast one-pair of'
' Non-commutative instance required'
' for outer product.')
# Muls of Tensor Products should be expanded
# before this function is called
if (isinstance(nc_part1[0], TensorProduct) and len(nc_part1) == 1
and len(nc_part2) == 1):
op = tensor_product_simp(nc_part1[0]*Dagger(nc_part2[0]))
else:
op = Mul(*nc_part1)*Dagger(Mul(*nc_part2))
return Mul(*c_part1)*Mul(*c_part2) * op
def _represent(self, **options):
return represent(self.doit(), **options)
def _print_operator_name_latex(self, printer, *args):
return printer._print(r'\rho', *args)
def _print_operator_name_pretty(self, printer, *args):
return prettyForm('\N{GREEK SMALL LETTER RHO}')
def _eval_trace(self, **kwargs):
indices = kwargs.get('indices', [])
return Tr(self.doit(), indices).doit()
def entropy(self):
""" Compute the entropy of a density matrix.
Refer to density.entropy() method for examples.
"""
return entropy(self)
def entropy(density):
"""Compute the entropy of a matrix/density object.
This computes -Tr(density*ln(density)) using the eigenvalue decomposition
of density, which is given as either a Density instance or a matrix
(numpy.ndarray, sympy.Matrix or scipy.sparse).
Parameters
==========
density : density matrix of type Density, sympy matrix,
scipy.sparse or numpy.ndarray
Examples
========
>>> from sympy.physics.quantum.density import Density, entropy
>>> from sympy.physics.quantum.spin import JzKet
>>> from sympy import S
>>> up = JzKet(S(1)/2,S(1)/2)
>>> down = JzKet(S(1)/2,-S(1)/2)
>>> d = Density((up,S(1)/2),(down,S(1)/2))
>>> entropy(d)
log(2)/2
"""
if isinstance(density, Density):
density = represent(density) # represent in Matrix
if isinstance(density, scipy_sparse_matrix):
density = to_numpy(density)
if isinstance(density, Matrix):
eigvals = density.eigenvals().keys()
return expand(-sum(e*log(e) for e in eigvals))
elif isinstance(density, numpy_ndarray):
import numpy as np
eigvals = np.linalg.eigvals(density)
return -np.sum(eigvals*np.log(eigvals))
else:
raise ValueError(
"numpy.ndarray, scipy.sparse or sympy matrix expected")
def fidelity(state1, state2):
""" Computes the fidelity [1]_ between two quantum states
The arguments provided to this function should be a square matrix or a
Density object. If it is a square matrix, it is assumed to be diagonalizable.
Parameters
==========
state1, state2 : a density matrix or Matrix
Examples
========
>>> from sympy import S, sqrt
>>> from sympy.physics.quantum.dagger import Dagger
>>> from sympy.physics.quantum.spin import JzKet
>>> from sympy.physics.quantum.density import fidelity
>>> from sympy.physics.quantum.represent import represent
>>>
>>> up = JzKet(S(1)/2,S(1)/2)
>>> down = JzKet(S(1)/2,-S(1)/2)
>>> amp = 1/sqrt(2)
>>> updown = (amp*up) + (amp*down)
>>>
>>> # represent turns Kets into matrices
>>> up_dm = represent(up*Dagger(up))
>>> down_dm = represent(down*Dagger(down))
>>> updown_dm = represent(updown*Dagger(updown))
>>>
>>> fidelity(up_dm, up_dm)
1
>>> fidelity(up_dm, down_dm) #orthogonal states
0
>>> fidelity(up_dm, updown_dm).evalf().round(3)
0.707
References
==========
.. [1] https://en.wikipedia.org/wiki/Fidelity_of_quantum_states
"""
state1 = represent(state1) if isinstance(state1, Density) else state1
state2 = represent(state2) if isinstance(state2, Density) else state2
if not isinstance(state1, Matrix) or not isinstance(state2, Matrix):
raise ValueError("state1 and state2 must be of type Density or Matrix "
"received type=%s for state1 and type=%s for state2" %
(type(state1), type(state2)))
if state1.shape != state2.shape and state1.is_square:
raise ValueError("The dimensions of both args should be equal and the "
"matrix obtained should be a square matrix")
sqrt_state1 = state1**S.Half
return Tr((sqrt_state1*state2*sqrt_state1)**S.Half).doit()
|
ba630478598e6b06fa530a17e8cee0e02a5c53b004a70840c22b2473df650c53 | """The commutator: [A,B] = A*B - B*A."""
from __future__ import print_function, division
from sympy import S, Expr, Mul, Add, Pow
from sympy.printing.pretty.stringpict import prettyForm
from sympy.physics.quantum.dagger import Dagger
from sympy.physics.quantum.operator import Operator
__all__ = [
'Commutator'
]
#-----------------------------------------------------------------------------
# Commutator
#-----------------------------------------------------------------------------
class Commutator(Expr):
"""The standard commutator, in an unevaluated state.
Evaluating a commutator is defined [1]_ as: ``[A, B] = A*B - B*A``. This
class returns the commutator in an unevaluated form. To evaluate the
commutator, use the ``.doit()`` method.
Canonical ordering of a commutator is ``[A, B]`` for ``A < B``. The
arguments of the commutator are put into canonical order using ``__cmp__``.
If ``B < A``, then ``[B, A]`` is returned as ``-[A, B]``.
Parameters
==========
A : Expr
The first argument of the commutator [A,B].
B : Expr
The second argument of the commutator [A,B].
Examples
========
>>> from sympy.physics.quantum import Commutator, Dagger, Operator
>>> from sympy.abc import x, y
>>> A = Operator('A')
>>> B = Operator('B')
>>> C = Operator('C')
Create a commutator and use ``.doit()`` to evaluate it:
>>> comm = Commutator(A, B)
>>> comm
[A,B]
>>> comm.doit()
A*B - B*A
The commutator orders it arguments in canonical order:
>>> comm = Commutator(B, A); comm
-[A,B]
Commutative constants are factored out:
>>> Commutator(3*x*A, x*y*B)
3*x**2*y*[A,B]
Using ``.expand(commutator=True)``, the standard commutator expansion rules
can be applied:
>>> Commutator(A+B, C).expand(commutator=True)
[A,C] + [B,C]
>>> Commutator(A, B+C).expand(commutator=True)
[A,B] + [A,C]
>>> Commutator(A*B, C).expand(commutator=True)
[A,C]*B + A*[B,C]
>>> Commutator(A, B*C).expand(commutator=True)
[A,B]*C + B*[A,C]
Adjoint operations applied to the commutator are properly applied to the
arguments:
>>> Dagger(Commutator(A, B))
-[Dagger(A),Dagger(B)]
References
==========
.. [1] https://en.wikipedia.org/wiki/Commutator
"""
is_commutative = False
def __new__(cls, A, B):
r = cls.eval(A, B)
if r is not None:
return r
obj = Expr.__new__(cls, A, B)
return obj
@classmethod
def eval(cls, a, b):
if not (a and b):
return S.Zero
if a == b:
return S.Zero
if a.is_commutative or b.is_commutative:
return S.Zero
# [xA,yB] -> xy*[A,B]
ca, nca = a.args_cnc()
cb, ncb = b.args_cnc()
c_part = ca + cb
if c_part:
return Mul(Mul(*c_part), cls(Mul._from_args(nca), Mul._from_args(ncb)))
# Canonical ordering of arguments
# The Commutator [A, B] is in canonical form if A < B.
if a.compare(b) == 1:
return S.NegativeOne*cls(b, a)
def _expand_pow(self, A, B, sign):
exp = A.exp
if not exp.is_integer or not exp.is_constant() or abs(exp) <= 1:
# nothing to do
return self
base = A.base
if exp.is_negative:
base = A.base**-1
exp = -exp
comm = Commutator(base, B).expand(commutator=True)
result = base**(exp - 1) * comm
for i in range(1, exp):
result += base**(exp - 1 - i) * comm * base**i
return sign*result.expand()
def _eval_expand_commutator(self, **hints):
A = self.args[0]
B = self.args[1]
if isinstance(A, Add):
# [A + B, C] -> [A, C] + [B, C]
sargs = []
for term in A.args:
comm = Commutator(term, B)
if isinstance(comm, Commutator):
comm = comm._eval_expand_commutator()
sargs.append(comm)
return Add(*sargs)
elif isinstance(B, Add):
# [A, B + C] -> [A, B] + [A, C]
sargs = []
for term in B.args:
comm = Commutator(A, term)
if isinstance(comm, Commutator):
comm = comm._eval_expand_commutator()
sargs.append(comm)
return Add(*sargs)
elif isinstance(A, Mul):
# [A*B, C] -> A*[B, C] + [A, C]*B
a = A.args[0]
b = Mul(*A.args[1:])
c = B
comm1 = Commutator(b, c)
comm2 = Commutator(a, c)
if isinstance(comm1, Commutator):
comm1 = comm1._eval_expand_commutator()
if isinstance(comm2, Commutator):
comm2 = comm2._eval_expand_commutator()
first = Mul(a, comm1)
second = Mul(comm2, b)
return Add(first, second)
elif isinstance(B, Mul):
# [A, B*C] -> [A, B]*C + B*[A, C]
a = A
b = B.args[0]
c = Mul(*B.args[1:])
comm1 = Commutator(a, b)
comm2 = Commutator(a, c)
if isinstance(comm1, Commutator):
comm1 = comm1._eval_expand_commutator()
if isinstance(comm2, Commutator):
comm2 = comm2._eval_expand_commutator()
first = Mul(comm1, c)
second = Mul(b, comm2)
return Add(first, second)
elif isinstance(A, Pow):
# [A**n, C] -> A**(n - 1)*[A, C] + A**(n - 2)*[A, C]*A + ... + [A, C]*A**(n-1)
return self._expand_pow(A, B, 1)
elif isinstance(B, Pow):
# [A, C**n] -> C**(n - 1)*[C, A] + C**(n - 2)*[C, A]*C + ... + [C, A]*C**(n-1)
return self._expand_pow(B, A, -1)
# No changes, so return self
return self
def doit(self, **hints):
""" Evaluate commutator """
A = self.args[0]
B = self.args[1]
if isinstance(A, Operator) and isinstance(B, Operator):
try:
comm = A._eval_commutator(B, **hints)
except NotImplementedError:
try:
comm = -1*B._eval_commutator(A, **hints)
except NotImplementedError:
comm = None
if comm is not None:
return comm.doit(**hints)
return (A*B - B*A).doit(**hints)
def _eval_adjoint(self):
return Commutator(Dagger(self.args[1]), Dagger(self.args[0]))
def _sympyrepr(self, printer, *args):
return "%s(%s,%s)" % (
self.__class__.__name__, printer._print(
self.args[0]), printer._print(self.args[1])
)
def _sympystr(self, printer, *args):
return "[%s,%s]" % (
printer._print(self.args[0]), printer._print(self.args[1]))
def _pretty(self, printer, *args):
pform = printer._print(self.args[0], *args)
pform = prettyForm(*pform.right((prettyForm(u','))))
pform = prettyForm(*pform.right((printer._print(self.args[1], *args))))
pform = prettyForm(*pform.parens(left='[', right=']'))
return pform
def _latex(self, printer, *args):
return "\\left[%s,%s\\right]" % tuple([
printer._print(arg, *args) for arg in self.args])
|
91bfd75f89507429bc9e32bfd93165ee267a088db2fcddc5a553333cbb689633 | #TODO:
# -Implement Clebsch-Gordan symmetries
# -Improve simplification method
# -Implement new simpifications
"""Clebsch-Gordon Coefficients."""
from __future__ import print_function, division
from sympy import (Add, expand, Eq, Expr, Mul, Piecewise, Pow, sqrt, Sum,
symbols, sympify, Wild)
from sympy.printing.pretty.stringpict import prettyForm, stringPict
from sympy.functions.special.tensor_functions import KroneckerDelta
from sympy.physics.wigner import clebsch_gordan, wigner_3j, wigner_6j, wigner_9j
__all__ = [
'CG',
'Wigner3j',
'Wigner6j',
'Wigner9j',
'cg_simp'
]
#-----------------------------------------------------------------------------
# CG Coefficients
#-----------------------------------------------------------------------------
class Wigner3j(Expr):
"""Class for the Wigner-3j symbols
Wigner 3j-symbols are coefficients determined by the coupling of
two angular momenta. When created, they are expressed as symbolic
quantities that, for numerical parameters, can be evaluated using the
``.doit()`` method [1]_.
Parameters
==========
j1, m1, j2, m2, j3, m3 : Number, Symbol
Terms determining the angular momentum of coupled angular momentum
systems.
Examples
========
Declare a Wigner-3j coefficient and calculate its value
>>> from sympy.physics.quantum.cg import Wigner3j
>>> w3j = Wigner3j(6,0,4,0,2,0)
>>> w3j
Wigner3j(6, 0, 4, 0, 2, 0)
>>> w3j.doit()
sqrt(715)/143
See Also
========
CG: Clebsch-Gordan coefficients
References
==========
.. [1] Varshalovich, D A, Quantum Theory of Angular Momentum. 1988.
"""
is_commutative = True
def __new__(cls, j1, m1, j2, m2, j3, m3):
args = map(sympify, (j1, m1, j2, m2, j3, m3))
return Expr.__new__(cls, *args)
@property
def j1(self):
return self.args[0]
@property
def m1(self):
return self.args[1]
@property
def j2(self):
return self.args[2]
@property
def m2(self):
return self.args[3]
@property
def j3(self):
return self.args[4]
@property
def m3(self):
return self.args[5]
@property
def is_symbolic(self):
return not all([arg.is_number for arg in self.args])
# This is modified from the _print_Matrix method
def _pretty(self, printer, *args):
m = ((printer._print(self.j1), printer._print(self.m1)),
(printer._print(self.j2), printer._print(self.m2)),
(printer._print(self.j3), printer._print(self.m3)))
hsep = 2
vsep = 1
maxw = [-1]*3
for j in range(3):
maxw[j] = max([ m[j][i].width() for i in range(2) ])
D = None
for i in range(2):
D_row = None
for j in range(3):
s = m[j][i]
wdelta = maxw[j] - s.width()
wleft = wdelta //2
wright = wdelta - wleft
s = prettyForm(*s.right(' '*wright))
s = prettyForm(*s.left(' '*wleft))
if D_row is None:
D_row = s
continue
D_row = prettyForm(*D_row.right(' '*hsep))
D_row = prettyForm(*D_row.right(s))
if D is None:
D = D_row
continue
for _ in range(vsep):
D = prettyForm(*D.below(' '))
D = prettyForm(*D.below(D_row))
D = prettyForm(*D.parens())
return D
def _latex(self, printer, *args):
label = map(printer._print, (self.j1, self.j2, self.j3,
self.m1, self.m2, self.m3))
return r'\left(\begin{array}{ccc} %s & %s & %s \\ %s & %s & %s \end{array}\right)' % \
tuple(label)
def doit(self, **hints):
if self.is_symbolic:
raise ValueError("Coefficients must be numerical")
return wigner_3j(self.j1, self.j2, self.j3, self.m1, self.m2, self.m3)
class CG(Wigner3j):
r"""Class for Clebsch-Gordan coefficient
Clebsch-Gordan coefficients describe the angular momentum coupling between
two systems. The coefficients give the expansion of a coupled total angular
momentum state and an uncoupled tensor product state. The Clebsch-Gordan
coefficients are defined as [1]_:
.. math ::
C^{j_1,m_1}_{j_2,m_2,j_3,m_3} = \left\langle j_1,m_1;j_2,m_2 | j_3,m_3\right\rangle
Parameters
==========
j1, m1, j2, m2, j3, m3 : Number, Symbol
Terms determining the angular momentum of coupled angular momentum
systems.
Examples
========
Define a Clebsch-Gordan coefficient and evaluate its value
>>> from sympy.physics.quantum.cg import CG
>>> from sympy import S
>>> cg = CG(S(3)/2, S(3)/2, S(1)/2, -S(1)/2, 1, 1)
>>> cg
CG(3/2, 3/2, 1/2, -1/2, 1, 1)
>>> cg.doit()
sqrt(3)/2
See Also
========
Wigner3j: Wigner-3j symbols
References
==========
.. [1] Varshalovich, D A, Quantum Theory of Angular Momentum. 1988.
"""
def doit(self, **hints):
if self.is_symbolic:
raise ValueError("Coefficients must be numerical")
return clebsch_gordan(self.j1, self.j2, self.j3, self.m1, self.m2, self.m3)
def _pretty(self, printer, *args):
bot = printer._print_seq(
(self.j1, self.m1, self.j2, self.m2), delimiter=',')
top = printer._print_seq((self.j3, self.m3), delimiter=',')
pad = max(top.width(), bot.width())
bot = prettyForm(*bot.left(' '))
top = prettyForm(*top.left(' '))
if not pad == bot.width():
bot = prettyForm(*bot.right(' '*(pad - bot.width())))
if not pad == top.width():
top = prettyForm(*top.right(' '*(pad - top.width())))
s = stringPict('C' + ' '*pad)
s = prettyForm(*s.below(bot))
s = prettyForm(*s.above(top))
return s
def _latex(self, printer, *args):
label = map(printer._print, (self.j3, self.m3, self.j1,
self.m1, self.j2, self.m2))
return r'C^{%s,%s}_{%s,%s,%s,%s}' % tuple(label)
class Wigner6j(Expr):
"""Class for the Wigner-6j symbols
See Also
========
Wigner3j: Wigner-3j symbols
"""
def __new__(cls, j1, j2, j12, j3, j, j23):
args = map(sympify, (j1, j2, j12, j3, j, j23))
return Expr.__new__(cls, *args)
@property
def j1(self):
return self.args[0]
@property
def j2(self):
return self.args[1]
@property
def j12(self):
return self.args[2]
@property
def j3(self):
return self.args[3]
@property
def j(self):
return self.args[4]
@property
def j23(self):
return self.args[5]
@property
def is_symbolic(self):
return not all([arg.is_number for arg in self.args])
# This is modified from the _print_Matrix method
def _pretty(self, printer, *args):
m = ((printer._print(self.j1), printer._print(self.j3)),
(printer._print(self.j2), printer._print(self.j)),
(printer._print(self.j12), printer._print(self.j23)))
hsep = 2
vsep = 1
maxw = [-1]*3
for j in range(3):
maxw[j] = max([ m[j][i].width() for i in range(2) ])
D = None
for i in range(2):
D_row = None
for j in range(3):
s = m[j][i]
wdelta = maxw[j] - s.width()
wleft = wdelta //2
wright = wdelta - wleft
s = prettyForm(*s.right(' '*wright))
s = prettyForm(*s.left(' '*wleft))
if D_row is None:
D_row = s
continue
D_row = prettyForm(*D_row.right(' '*hsep))
D_row = prettyForm(*D_row.right(s))
if D is None:
D = D_row
continue
for _ in range(vsep):
D = prettyForm(*D.below(' '))
D = prettyForm(*D.below(D_row))
D = prettyForm(*D.parens(left='{', right='}'))
return D
def _latex(self, printer, *args):
label = map(printer._print, (self.j1, self.j2, self.j12,
self.j3, self.j, self.j23))
return r'\left\{\begin{array}{ccc} %s & %s & %s \\ %s & %s & %s \end{array}\right\}' % \
tuple(label)
def doit(self, **hints):
if self.is_symbolic:
raise ValueError("Coefficients must be numerical")
return wigner_6j(self.j1, self.j2, self.j12, self.j3, self.j, self.j23)
class Wigner9j(Expr):
"""Class for the Wigner-9j symbols
See Also
========
Wigner3j: Wigner-3j symbols
"""
def __new__(cls, j1, j2, j12, j3, j4, j34, j13, j24, j):
args = map(sympify, (j1, j2, j12, j3, j4, j34, j13, j24, j))
return Expr.__new__(cls, *args)
@property
def j1(self):
return self.args[0]
@property
def j2(self):
return self.args[1]
@property
def j12(self):
return self.args[2]
@property
def j3(self):
return self.args[3]
@property
def j4(self):
return self.args[4]
@property
def j34(self):
return self.args[5]
@property
def j13(self):
return self.args[6]
@property
def j24(self):
return self.args[7]
@property
def j(self):
return self.args[8]
@property
def is_symbolic(self):
return not all([arg.is_number for arg in self.args])
# This is modified from the _print_Matrix method
def _pretty(self, printer, *args):
m = (
(printer._print(
self.j1), printer._print(self.j3), printer._print(self.j13)),
(printer._print(
self.j2), printer._print(self.j4), printer._print(self.j24)),
(printer._print(self.j12), printer._print(self.j34), printer._print(self.j)))
hsep = 2
vsep = 1
maxw = [-1]*3
for j in range(3):
maxw[j] = max([ m[j][i].width() for i in range(3) ])
D = None
for i in range(3):
D_row = None
for j in range(3):
s = m[j][i]
wdelta = maxw[j] - s.width()
wleft = wdelta //2
wright = wdelta - wleft
s = prettyForm(*s.right(' '*wright))
s = prettyForm(*s.left(' '*wleft))
if D_row is None:
D_row = s
continue
D_row = prettyForm(*D_row.right(' '*hsep))
D_row = prettyForm(*D_row.right(s))
if D is None:
D = D_row
continue
for _ in range(vsep):
D = prettyForm(*D.below(' '))
D = prettyForm(*D.below(D_row))
D = prettyForm(*D.parens(left='{', right='}'))
return D
def _latex(self, printer, *args):
label = map(printer._print, (self.j1, self.j2, self.j12, self.j3,
self.j4, self.j34, self.j13, self.j24, self.j))
return r'\left\{\begin{array}{ccc} %s & %s & %s \\ %s & %s & %s \\ %s & %s & %s \end{array}\right\}' % \
tuple(label)
def doit(self, **hints):
if self.is_symbolic:
raise ValueError("Coefficients must be numerical")
return wigner_9j(self.j1, self.j2, self.j12, self.j3, self.j4, self.j34, self.j13, self.j24, self.j)
def cg_simp(e):
"""Simplify and combine CG coefficients
This function uses various symmetry and properties of sums and
products of Clebsch-Gordan coefficients to simplify statements
involving these terms [1]_.
Examples
========
Simplify the sum over CG(a,alpha,0,0,a,alpha) for all alpha to
2*a+1
>>> from sympy.physics.quantum.cg import CG, cg_simp
>>> a = CG(1,1,0,0,1,1)
>>> b = CG(1,0,0,0,1,0)
>>> c = CG(1,-1,0,0,1,-1)
>>> cg_simp(a+b+c)
3
See Also
========
CG: Clebsh-Gordan coefficients
References
==========
.. [1] Varshalovich, D A, Quantum Theory of Angular Momentum. 1988.
"""
if isinstance(e, Add):
return _cg_simp_add(e)
elif isinstance(e, Sum):
return _cg_simp_sum(e)
elif isinstance(e, Mul):
return Mul(*[cg_simp(arg) for arg in e.args])
elif isinstance(e, Pow):
return Pow(cg_simp(e.base), e.exp)
else:
return e
def _cg_simp_add(e):
#TODO: Improve simplification method
"""Takes a sum of terms involving Clebsch-Gordan coefficients and
simplifies the terms.
First, we create two lists, cg_part, which is all the terms involving CG
coefficients, and other_part, which is all other terms. The cg_part list
is then passed to the simplification methods, which return the new cg_part
and any additional terms that are added to other_part
"""
cg_part = []
other_part = []
e = expand(e)
for arg in e.args:
if arg.has(CG):
if isinstance(arg, Sum):
other_part.append(_cg_simp_sum(arg))
elif isinstance(arg, Mul):
terms = 1
for term in arg.args:
if isinstance(term, Sum):
terms *= _cg_simp_sum(term)
else:
terms *= term
if terms.has(CG):
cg_part.append(terms)
else:
other_part.append(terms)
else:
cg_part.append(arg)
else:
other_part.append(arg)
cg_part, other = _check_varsh_871_1(cg_part)
other_part.append(other)
cg_part, other = _check_varsh_871_2(cg_part)
other_part.append(other)
cg_part, other = _check_varsh_872_9(cg_part)
other_part.append(other)
return Add(*cg_part) + Add(*other_part)
def _check_varsh_871_1(term_list):
# Sum( CG(a,alpha,b,0,a,alpha), (alpha, -a, a)) == KroneckerDelta(b,0)
a, alpha, b, lt = map(Wild, ('a', 'alpha', 'b', 'lt'))
expr = lt*CG(a, alpha, b, 0, a, alpha)
simp = (2*a + 1)*KroneckerDelta(b, 0)
sign = lt/abs(lt)
build_expr = 2*a + 1
index_expr = a + alpha
return _check_cg_simp(expr, simp, sign, lt, term_list, (a, alpha, b, lt), (a, b), build_expr, index_expr)
def _check_varsh_871_2(term_list):
# Sum((-1)**(a-alpha)*CG(a,alpha,a,-alpha,c,0),(alpha,-a,a))
a, alpha, c, lt = map(Wild, ('a', 'alpha', 'c', 'lt'))
expr = lt*CG(a, alpha, a, -alpha, c, 0)
simp = sqrt(2*a + 1)*KroneckerDelta(c, 0)
sign = (-1)**(a - alpha)*lt/abs(lt)
build_expr = 2*a + 1
index_expr = a + alpha
return _check_cg_simp(expr, simp, sign, lt, term_list, (a, alpha, c, lt), (a, c), build_expr, index_expr)
def _check_varsh_872_9(term_list):
# Sum( CG(a,alpha,b,beta,c,gamma)*CG(a,alpha',b,beta',c,gamma), (gamma, -c, c), (c, abs(a-b), a+b))
a, alpha, alphap, b, beta, betap, c, gamma, lt = map(Wild, (
'a', 'alpha', 'alphap', 'b', 'beta', 'betap', 'c', 'gamma', 'lt'))
# Case alpha==alphap, beta==betap
# For numerical alpha,beta
expr = lt*CG(a, alpha, b, beta, c, gamma)**2
simp = 1
sign = lt/abs(lt)
x = abs(a - b)
y = abs(alpha + beta)
build_expr = a + b + 1 - Piecewise((x, x > y), (0, Eq(x, y)), (y, y > x))
index_expr = a + b - c
term_list, other1 = _check_cg_simp(expr, simp, sign, lt, term_list, (a, alpha, b, beta, c, gamma, lt), (a, alpha, b, beta), build_expr, index_expr)
# For symbolic alpha,beta
x = abs(a - b)
y = a + b
build_expr = (y + 1 - x)*(x + y + 1)
index_expr = (c - x)*(x + c) + c + gamma
term_list, other2 = _check_cg_simp(expr, simp, sign, lt, term_list, (a, alpha, b, beta, c, gamma, lt), (a, alpha, b, beta), build_expr, index_expr)
# Case alpha!=alphap or beta!=betap
# Note: this only works with leading term of 1, pattern matching is unable to match when there is a Wild leading term
# For numerical alpha,alphap,beta,betap
expr = CG(a, alpha, b, beta, c, gamma)*CG(a, alphap, b, betap, c, gamma)
simp = KroneckerDelta(alpha, alphap)*KroneckerDelta(beta, betap)
sign = sympify(1)
x = abs(a - b)
y = abs(alpha + beta)
build_expr = a + b + 1 - Piecewise((x, x > y), (0, Eq(x, y)), (y, y > x))
index_expr = a + b - c
term_list, other3 = _check_cg_simp(expr, simp, sign, sympify(1), term_list, (a, alpha, alphap, b, beta, betap, c, gamma), (a, alpha, alphap, b, beta, betap), build_expr, index_expr)
# For symbolic alpha,alphap,beta,betap
x = abs(a - b)
y = a + b
build_expr = (y + 1 - x)*(x + y + 1)
index_expr = (c - x)*(x + c) + c + gamma
term_list, other4 = _check_cg_simp(expr, simp, sign, sympify(1), term_list, (a, alpha, alphap, b, beta, betap, c, gamma), (a, alpha, alphap, b, beta, betap), build_expr, index_expr)
return term_list, other1 + other2 + other4
def _check_cg_simp(expr, simp, sign, lt, term_list, variables, dep_variables, build_index_expr, index_expr):
""" Checks for simplifications that can be made, returning a tuple of the
simplified list of terms and any terms generated by simplification.
Parameters
==========
expr: expression
The expression with Wild terms that will be matched to the terms in
the sum
simp: expression
The expression with Wild terms that is substituted in place of the CG
terms in the case of simplification
sign: expression
The expression with Wild terms denoting the sign that is on expr that
must match
lt: expression
The expression with Wild terms that gives the leading term of the
matched expr
term_list: list
A list of all of the terms is the sum to be simplified
variables: list
A list of all the variables that appears in expr
dep_variables: list
A list of the variables that must match for all the terms in the sum,
i.e. the dependent variables
build_index_expr: expression
Expression with Wild terms giving the number of elements in cg_index
index_expr: expression
Expression with Wild terms giving the index terms have when storing
them to cg_index
"""
other_part = 0
i = 0
while i < len(term_list):
sub_1 = _check_cg(term_list[i], expr, len(variables))
if sub_1 is None:
i += 1
continue
if not sympify(build_index_expr.subs(sub_1)).is_number:
i += 1
continue
sub_dep = [(x, sub_1[x]) for x in dep_variables]
cg_index = [None]*build_index_expr.subs(sub_1)
for j in range(i, len(term_list)):
sub_2 = _check_cg(term_list[j], expr.subs(sub_dep), len(variables) - len(dep_variables), sign=(sign.subs(sub_1), sign.subs(sub_dep)))
if sub_2 is None:
continue
if not sympify(index_expr.subs(sub_dep).subs(sub_2)).is_number:
continue
cg_index[index_expr.subs(sub_dep).subs(sub_2)] = j, expr.subs(lt, 1).subs(sub_dep).subs(sub_2), lt.subs(sub_2), sign.subs(sub_dep).subs(sub_2)
if all(i is not None for i in cg_index):
min_lt = min(*[ abs(term[2]) for term in cg_index ])
indices = [ term[0] for term in cg_index]
indices.sort()
indices.reverse()
[ term_list.pop(j) for j in indices ]
for term in cg_index:
if abs(term[2]) > min_lt:
term_list.append( (term[2] - min_lt*term[3])*term[1] )
other_part += min_lt*(sign*simp).subs(sub_1)
else:
i += 1
return term_list, other_part
def _check_cg(cg_term, expr, length, sign=None):
"""Checks whether a term matches the given expression"""
# TODO: Check for symmetries
matches = cg_term.match(expr)
if matches is None:
return
if sign is not None:
if not isinstance(sign, tuple):
raise TypeError('sign must be a tuple')
if not sign[0] == (sign[1]).subs(matches):
return
if len(matches) == length:
return matches
def _cg_simp_sum(e):
e = _check_varsh_sum_871_1(e)
e = _check_varsh_sum_871_2(e)
e = _check_varsh_sum_872_4(e)
return e
def _check_varsh_sum_871_1(e):
a = Wild('a')
alpha = symbols('alpha')
b = Wild('b')
match = e.match(Sum(CG(a, alpha, b, 0, a, alpha), (alpha, -a, a)))
if match is not None and len(match) == 2:
return ((2*a + 1)*KroneckerDelta(b, 0)).subs(match)
return e
def _check_varsh_sum_871_2(e):
a = Wild('a')
alpha = symbols('alpha')
c = Wild('c')
match = e.match(
Sum((-1)**(a - alpha)*CG(a, alpha, a, -alpha, c, 0), (alpha, -a, a)))
if match is not None and len(match) == 2:
return (sqrt(2*a + 1)*KroneckerDelta(c, 0)).subs(match)
return e
def _check_varsh_sum_872_4(e):
alpha = symbols('alpha')
beta = symbols('beta')
a = Wild('a')
b = Wild('b')
c = Wild('c')
cp = Wild('cp')
gamma = Wild('gamma')
gammap = Wild('gammap')
cg1 = CG(a, alpha, b, beta, c, gamma)
cg2 = CG(a, alpha, b, beta, cp, gammap)
match1 = e.match(Sum(cg1*cg2, (alpha, -a, a), (beta, -b, b)))
if match1 is not None and len(match1) == 6:
return (KroneckerDelta(c, cp)*KroneckerDelta(gamma, gammap)).subs(match1)
match2 = e.match(Sum(cg1**2, (alpha, -a, a), (beta, -b, b)))
if match2 is not None and len(match2) == 4:
return 1
return e
def _cg_list(term):
if isinstance(term, CG):
return (term,), 1, 1
cg = []
coeff = 1
if not (isinstance(term, Mul) or isinstance(term, Pow)):
raise NotImplementedError('term must be CG, Add, Mul or Pow')
if isinstance(term, Pow) and sympify(term.exp).is_number:
if sympify(term.exp).is_number:
[ cg.append(term.base) for _ in range(term.exp) ]
else:
return (term,), 1, 1
if isinstance(term, Mul):
for arg in term.args:
if isinstance(arg, CG):
cg.append(arg)
else:
coeff *= arg
return cg, coeff, coeff/abs(coeff)
|
620fb6942cf019bac6fec9636faccc5cab00c175073eebdbea2cb3078ada42c3 | """Primitive circuit operations on quantum circuits."""
from __future__ import print_function, division
from sympy import Symbol, Tuple, Mul, sympify, default_sort_key
from sympy.utilities import numbered_symbols
from sympy.core.compatibility import reduce
from sympy.physics.quantum.gate import Gate
__all__ = [
'kmp_table',
'find_subcircuit',
'replace_subcircuit',
'convert_to_symbolic_indices',
'convert_to_real_indices',
'random_reduce',
'random_insert'
]
def kmp_table(word):
"""Build the 'partial match' table of the Knuth-Morris-Pratt algorithm.
Note: This is applicable to strings or
quantum circuits represented as tuples.
"""
# Current position in subcircuit
pos = 2
# Beginning position of candidate substring that
# may reappear later in word
cnd = 0
# The 'partial match' table that helps one determine
# the next location to start substring search
table = list()
table.append(-1)
table.append(0)
while pos < len(word):
if word[pos - 1] == word[cnd]:
cnd = cnd + 1
table.append(cnd)
pos = pos + 1
elif cnd > 0:
cnd = table[cnd]
else:
table.append(0)
pos = pos + 1
return table
def find_subcircuit(circuit, subcircuit, start=0, end=0):
"""Finds the subcircuit in circuit, if it exists.
If the subcircuit exists, the index of the start of
the subcircuit in circuit is returned; otherwise,
-1 is returned. The algorithm that is implemented
is the Knuth-Morris-Pratt algorithm.
Parameters
==========
circuit : tuple, Gate or Mul
A tuple of Gates or Mul representing a quantum circuit
subcircuit : tuple, Gate or Mul
A tuple of Gates or Mul to find in circuit
start : int
The location to start looking for subcircuit.
If start is the same or past end, -1 is returned.
end : int
The last place to look for a subcircuit. If end
is less than 1 (one), then the length of circuit
is taken to be end.
Examples
========
Find the first instance of a subcircuit:
>>> from sympy.physics.quantum.circuitutils import find_subcircuit
>>> from sympy.physics.quantum.gate import X, Y, Z, H
>>> circuit = X(0)*Z(0)*Y(0)*H(0)
>>> subcircuit = Z(0)*Y(0)
>>> find_subcircuit(circuit, subcircuit)
1
Find the first instance starting at a specific position:
>>> find_subcircuit(circuit, subcircuit, start=1)
1
>>> find_subcircuit(circuit, subcircuit, start=2)
-1
>>> circuit = circuit*subcircuit
>>> find_subcircuit(circuit, subcircuit, start=2)
4
Find the subcircuit within some interval:
>>> find_subcircuit(circuit, subcircuit, start=2, end=2)
-1
"""
if isinstance(circuit, Mul):
circuit = circuit.args
if isinstance(subcircuit, Mul):
subcircuit = subcircuit.args
if len(subcircuit) == 0 or len(subcircuit) > len(circuit):
return -1
if end < 1:
end = len(circuit)
# Location in circuit
pos = start
# Location in the subcircuit
index = 0
# 'Partial match' table
table = kmp_table(subcircuit)
while (pos + index) < end:
if subcircuit[index] == circuit[pos + index]:
index = index + 1
else:
pos = pos + index - table[index]
index = table[index] if table[index] > -1 else 0
if index == len(subcircuit):
return pos
return -1
def replace_subcircuit(circuit, subcircuit, replace=None, pos=0):
"""Replaces a subcircuit with another subcircuit in circuit,
if it exists.
If multiple instances of subcircuit exists, the first instance is
replaced. The position to being searching from (if different from
0) may be optionally given. If subcircuit can't be found, circuit
is returned.
Parameters
==========
circuit : tuple, Gate or Mul
A quantum circuit
subcircuit : tuple, Gate or Mul
The circuit to be replaced
replace : tuple, Gate or Mul
The replacement circuit
pos : int
The location to start search and replace
subcircuit, if it exists. This may be used
if it is known beforehand that multiple
instances exist, and it is desirable to
replace a specific instance. If a negative number
is given, pos will be defaulted to 0.
Examples
========
Find and remove the subcircuit:
>>> from sympy.physics.quantum.circuitutils import replace_subcircuit
>>> from sympy.physics.quantum.gate import X, Y, Z, H
>>> circuit = X(0)*Z(0)*Y(0)*H(0)*X(0)*H(0)*Y(0)
>>> subcircuit = Z(0)*Y(0)
>>> replace_subcircuit(circuit, subcircuit)
(X(0), H(0), X(0), H(0), Y(0))
Remove the subcircuit given a starting search point:
>>> replace_subcircuit(circuit, subcircuit, pos=1)
(X(0), H(0), X(0), H(0), Y(0))
>>> replace_subcircuit(circuit, subcircuit, pos=2)
(X(0), Z(0), Y(0), H(0), X(0), H(0), Y(0))
Replace the subcircuit:
>>> replacement = H(0)*Z(0)
>>> replace_subcircuit(circuit, subcircuit, replace=replacement)
(X(0), H(0), Z(0), H(0), X(0), H(0), Y(0))
"""
if pos < 0:
pos = 0
if isinstance(circuit, Mul):
circuit = circuit.args
if isinstance(subcircuit, Mul):
subcircuit = subcircuit.args
if isinstance(replace, Mul):
replace = replace.args
elif replace is None:
replace = ()
# Look for the subcircuit starting at pos
loc = find_subcircuit(circuit, subcircuit, start=pos)
# If subcircuit was found
if loc > -1:
# Get the gates to the left of subcircuit
left = circuit[0:loc]
# Get the gates to the right of subcircuit
right = circuit[loc + len(subcircuit):len(circuit)]
# Recombine the left and right side gates into a circuit
circuit = left + replace + right
return circuit
def _sympify_qubit_map(mapping):
new_map = {}
for key in mapping:
new_map[key] = sympify(mapping[key])
return new_map
def convert_to_symbolic_indices(seq, start=None, gen=None, qubit_map=None):
"""Returns the circuit with symbolic indices and the
dictionary mapping symbolic indices to real indices.
The mapping is 1 to 1 and onto (bijective).
Parameters
==========
seq : tuple, Gate/Integer/tuple or Mul
A tuple of Gate, Integer, or tuple objects, or a Mul
start : Symbol
An optional starting symbolic index
gen : object
An optional numbered symbol generator
qubit_map : dict
An existing mapping of symbolic indices to real indices
All symbolic indices have the format 'i#', where # is
some number >= 0.
"""
if isinstance(seq, Mul):
seq = seq.args
# A numbered symbol generator
index_gen = numbered_symbols(prefix='i', start=-1)
cur_ndx = next(index_gen)
# keys are symbolic indices; values are real indices
ndx_map = {}
def create_inverse_map(symb_to_real_map):
rev_items = lambda item: tuple([item[1], item[0]])
return dict(map(rev_items, symb_to_real_map.items()))
if start is not None:
if not isinstance(start, Symbol):
msg = 'Expected Symbol for starting index, got %r.' % start
raise TypeError(msg)
cur_ndx = start
if gen is not None:
if not isinstance(gen, numbered_symbols().__class__):
msg = 'Expected a generator, got %r.' % gen
raise TypeError(msg)
index_gen = gen
if qubit_map is not None:
if not isinstance(qubit_map, dict):
msg = ('Expected dict for existing map, got ' +
'%r.' % qubit_map)
raise TypeError(msg)
ndx_map = qubit_map
ndx_map = _sympify_qubit_map(ndx_map)
# keys are real indices; keys are symbolic indices
inv_map = create_inverse_map(ndx_map)
sym_seq = ()
for item in seq:
# Nested items, so recurse
if isinstance(item, Gate):
result = convert_to_symbolic_indices(item.args,
qubit_map=ndx_map,
start=cur_ndx,
gen=index_gen)
sym_item, new_map, cur_ndx, index_gen = result
ndx_map.update(new_map)
inv_map = create_inverse_map(ndx_map)
elif isinstance(item, tuple) or isinstance(item, Tuple):
result = convert_to_symbolic_indices(item,
qubit_map=ndx_map,
start=cur_ndx,
gen=index_gen)
sym_item, new_map, cur_ndx, index_gen = result
ndx_map.update(new_map)
inv_map = create_inverse_map(ndx_map)
elif item in inv_map:
sym_item = inv_map[item]
else:
cur_ndx = next(gen)
ndx_map[cur_ndx] = item
inv_map[item] = cur_ndx
sym_item = cur_ndx
if isinstance(item, Gate):
sym_item = item.__class__(*sym_item)
sym_seq = sym_seq + (sym_item,)
return sym_seq, ndx_map, cur_ndx, index_gen
def convert_to_real_indices(seq, qubit_map):
"""Returns the circuit with real indices.
Parameters
==========
seq : tuple, Gate/Integer/tuple or Mul
A tuple of Gate, Integer, or tuple objects or a Mul
qubit_map : dict
A dictionary mapping symbolic indices to real indices.
Examples
========
Change the symbolic indices to real integers:
>>> from sympy import symbols
>>> from sympy.physics.quantum.circuitutils import convert_to_real_indices
>>> from sympy.physics.quantum.gate import X, Y, H
>>> i0, i1 = symbols('i:2')
>>> index_map = {i0 : 0, i1 : 1}
>>> convert_to_real_indices(X(i0)*Y(i1)*H(i0)*X(i1), index_map)
(X(0), Y(1), H(0), X(1))
"""
if isinstance(seq, Mul):
seq = seq.args
if not isinstance(qubit_map, dict):
msg = 'Expected dict for qubit_map, got %r.' % qubit_map
raise TypeError(msg)
qubit_map = _sympify_qubit_map(qubit_map)
real_seq = ()
for item in seq:
# Nested items, so recurse
if isinstance(item, Gate):
real_item = convert_to_real_indices(item.args, qubit_map)
elif isinstance(item, tuple) or isinstance(item, Tuple):
real_item = convert_to_real_indices(item, qubit_map)
else:
real_item = qubit_map[item]
if isinstance(item, Gate):
real_item = item.__class__(*real_item)
real_seq = real_seq + (real_item,)
return real_seq
def random_reduce(circuit, gate_ids, seed=None):
"""Shorten the length of a quantum circuit.
random_reduce looks for circuit identities in circuit, randomly chooses
one to remove, and returns a shorter yet equivalent circuit. If no
identities are found, the same circuit is returned.
Parameters
==========
circuit : Gate tuple of Mul
A tuple of Gates representing a quantum circuit
gate_ids : list, GateIdentity
List of gate identities to find in circuit
seed : int or list
seed used for _randrange; to override the random selection, provide a
list of integers: the elements of gate_ids will be tested in the order
given by the list
"""
from sympy.testing.randtest import _randrange
if not gate_ids:
return circuit
if isinstance(circuit, Mul):
circuit = circuit.args
ids = flatten_ids(gate_ids)
# Create the random integer generator with the seed
randrange = _randrange(seed)
# Look for an identity in the circuit
while ids:
i = randrange(len(ids))
id = ids.pop(i)
if find_subcircuit(circuit, id) != -1:
break
else:
# no identity was found
return circuit
# return circuit with the identity removed
return replace_subcircuit(circuit, id)
def random_insert(circuit, choices, seed=None):
"""Insert a circuit into another quantum circuit.
random_insert randomly chooses a location in the circuit to insert
a randomly selected circuit from amongst the given choices.
Parameters
==========
circuit : Gate tuple or Mul
A tuple or Mul of Gates representing a quantum circuit
choices : list
Set of circuit choices
seed : int or list
seed used for _randrange; to override the random selections, give
a list two integers, [i, j] where i is the circuit location where
choice[j] will be inserted.
Notes
=====
Indices for insertion should be [0, n] if n is the length of the
circuit.
"""
from sympy.testing.randtest import _randrange
if not choices:
return circuit
if isinstance(circuit, Mul):
circuit = circuit.args
# get the location in the circuit and the element to insert from choices
randrange = _randrange(seed)
loc = randrange(len(circuit) + 1)
choice = choices[randrange(len(choices))]
circuit = list(circuit)
circuit[loc: loc] = choice
return tuple(circuit)
# Flatten the GateIdentity objects (with gate rules) into one single list
def flatten_ids(ids):
collapse = lambda acc, an_id: acc + sorted(an_id.equivalent_ids,
key=default_sort_key)
ids = reduce(collapse, ids, [])
ids.sort(key=default_sort_key)
return ids
|
bbc065e626ae41a8c2573bcac1415a2f85f0312337f6ea3e91418ac64cc5cc6e | """Hilbert spaces for quantum mechanics.
Authors:
* Brian Granger
* Matt Curry
"""
from __future__ import print_function, division
from sympy import Basic, Interval, oo, sympify
from sympy.printing.pretty.stringpict import prettyForm
from sympy.physics.quantum.qexpr import QuantumError
from sympy.core.compatibility import reduce
__all__ = [
'HilbertSpaceError',
'HilbertSpace',
'TensorProductHilbertSpace',
'TensorPowerHilbertSpace',
'DirectSumHilbertSpace',
'ComplexSpace',
'L2',
'FockSpace'
]
#-----------------------------------------------------------------------------
# Main objects
#-----------------------------------------------------------------------------
class HilbertSpaceError(QuantumError):
pass
#-----------------------------------------------------------------------------
# Main objects
#-----------------------------------------------------------------------------
class HilbertSpace(Basic):
"""An abstract Hilbert space for quantum mechanics.
In short, a Hilbert space is an abstract vector space that is complete
with inner products defined [1]_.
Examples
========
>>> from sympy.physics.quantum.hilbert import HilbertSpace
>>> hs = HilbertSpace()
>>> hs
H
References
==========
.. [1] https://en.wikipedia.org/wiki/Hilbert_space
"""
def __new__(cls):
obj = Basic.__new__(cls)
return obj
@property
def dimension(self):
"""Return the Hilbert dimension of the space."""
raise NotImplementedError('This Hilbert space has no dimension.')
def __add__(self, other):
return DirectSumHilbertSpace(self, other)
def __radd__(self, other):
return DirectSumHilbertSpace(other, self)
def __mul__(self, other):
return TensorProductHilbertSpace(self, other)
def __rmul__(self, other):
return TensorProductHilbertSpace(other, self)
def __pow__(self, other, mod=None):
if mod is not None:
raise ValueError('The third argument to __pow__ is not supported \
for Hilbert spaces.')
return TensorPowerHilbertSpace(self, other)
def __contains__(self, other):
"""Is the operator or state in this Hilbert space.
This is checked by comparing the classes of the Hilbert spaces, not
the instances. This is to allow Hilbert Spaces with symbolic
dimensions.
"""
if other.hilbert_space.__class__ == self.__class__:
return True
else:
return False
def _sympystr(self, printer, *args):
return u'H'
def _pretty(self, printer, *args):
ustr = u'\N{LATIN CAPITAL LETTER H}'
return prettyForm(ustr)
def _latex(self, printer, *args):
return r'\mathcal{H}'
class ComplexSpace(HilbertSpace):
"""Finite dimensional Hilbert space of complex vectors.
The elements of this Hilbert space are n-dimensional complex valued
vectors with the usual inner product that takes the complex conjugate
of the vector on the right.
A classic example of this type of Hilbert space is spin-1/2, which is
``ComplexSpace(2)``. Generalizing to spin-s, the space is
``ComplexSpace(2*s+1)``. Quantum computing with N qubits is done with the
direct product space ``ComplexSpace(2)**N``.
Examples
========
>>> from sympy import symbols
>>> from sympy.physics.quantum.hilbert import ComplexSpace
>>> c1 = ComplexSpace(2)
>>> c1
C(2)
>>> c1.dimension
2
>>> n = symbols('n')
>>> c2 = ComplexSpace(n)
>>> c2
C(n)
>>> c2.dimension
n
"""
def __new__(cls, dimension):
dimension = sympify(dimension)
r = cls.eval(dimension)
if isinstance(r, Basic):
return r
obj = Basic.__new__(cls, dimension)
return obj
@classmethod
def eval(cls, dimension):
if len(dimension.atoms()) == 1:
if not (dimension.is_Integer and dimension > 0 or dimension is oo
or dimension.is_Symbol):
raise TypeError('The dimension of a ComplexSpace can only'
'be a positive integer, oo, or a Symbol: %r'
% dimension)
else:
for dim in dimension.atoms():
if not (dim.is_Integer or dim is oo or dim.is_Symbol):
raise TypeError('The dimension of a ComplexSpace can only'
' contain integers, oo, or a Symbol: %r'
% dim)
@property
def dimension(self):
return self.args[0]
def _sympyrepr(self, printer, *args):
return "%s(%s)" % (self.__class__.__name__,
printer._print(self.dimension, *args))
def _sympystr(self, printer, *args):
return "C(%s)" % printer._print(self.dimension, *args)
def _pretty(self, printer, *args):
ustr = u'\N{LATIN CAPITAL LETTER C}'
pform_exp = printer._print(self.dimension, *args)
pform_base = prettyForm(ustr)
return pform_base**pform_exp
def _latex(self, printer, *args):
return r'\mathcal{C}^{%s}' % printer._print(self.dimension, *args)
class L2(HilbertSpace):
"""The Hilbert space of square integrable functions on an interval.
An L2 object takes in a single sympy Interval argument which represents
the interval its functions (vectors) are defined on.
Examples
========
>>> from sympy import Interval, oo
>>> from sympy.physics.quantum.hilbert import L2
>>> hs = L2(Interval(0,oo))
>>> hs
L2(Interval(0, oo))
>>> hs.dimension
oo
>>> hs.interval
Interval(0, oo)
"""
def __new__(cls, interval):
if not isinstance(interval, Interval):
raise TypeError('L2 interval must be an Interval instance: %r'
% interval)
obj = Basic.__new__(cls, interval)
return obj
@property
def dimension(self):
return oo
@property
def interval(self):
return self.args[0]
def _sympyrepr(self, printer, *args):
return "L2(%s)" % printer._print(self.interval, *args)
def _sympystr(self, printer, *args):
return "L2(%s)" % printer._print(self.interval, *args)
def _pretty(self, printer, *args):
pform_exp = prettyForm(u'2')
pform_base = prettyForm(u'L')
return pform_base**pform_exp
def _latex(self, printer, *args):
interval = printer._print(self.interval, *args)
return r'{\mathcal{L}^2}\left( %s \right)' % interval
class FockSpace(HilbertSpace):
"""The Hilbert space for second quantization.
Technically, this Hilbert space is a infinite direct sum of direct
products of single particle Hilbert spaces [1]_. This is a mess, so we have
a class to represent it directly.
Examples
========
>>> from sympy.physics.quantum.hilbert import FockSpace
>>> hs = FockSpace()
>>> hs
F
>>> hs.dimension
oo
References
==========
.. [1] https://en.wikipedia.org/wiki/Fock_space
"""
def __new__(cls):
obj = Basic.__new__(cls)
return obj
@property
def dimension(self):
return oo
def _sympyrepr(self, printer, *args):
return "FockSpace()"
def _sympystr(self, printer, *args):
return "F"
def _pretty(self, printer, *args):
ustr = u'\N{LATIN CAPITAL LETTER F}'
return prettyForm(ustr)
def _latex(self, printer, *args):
return r'\mathcal{F}'
class TensorProductHilbertSpace(HilbertSpace):
"""A tensor product of Hilbert spaces [1]_.
The tensor product between Hilbert spaces is represented by the
operator ``*`` Products of the same Hilbert space will be combined into
tensor powers.
A ``TensorProductHilbertSpace`` object takes in an arbitrary number of
``HilbertSpace`` objects as its arguments. In addition, multiplication of
``HilbertSpace`` objects will automatically return this tensor product
object.
Examples
========
>>> from sympy.physics.quantum.hilbert import ComplexSpace, FockSpace
>>> from sympy import symbols
>>> c = ComplexSpace(2)
>>> f = FockSpace()
>>> hs = c*f
>>> hs
C(2)*F
>>> hs.dimension
oo
>>> hs.spaces
(C(2), F)
>>> c1 = ComplexSpace(2)
>>> n = symbols('n')
>>> c2 = ComplexSpace(n)
>>> hs = c1*c2
>>> hs
C(2)*C(n)
>>> hs.dimension
2*n
References
==========
.. [1] https://en.wikipedia.org/wiki/Hilbert_space#Tensor_products
"""
def __new__(cls, *args):
r = cls.eval(args)
if isinstance(r, Basic):
return r
obj = Basic.__new__(cls, *args)
return obj
@classmethod
def eval(cls, args):
"""Evaluates the direct product."""
new_args = []
recall = False
#flatten arguments
for arg in args:
if isinstance(arg, TensorProductHilbertSpace):
new_args.extend(arg.args)
recall = True
elif isinstance(arg, (HilbertSpace, TensorPowerHilbertSpace)):
new_args.append(arg)
else:
raise TypeError('Hilbert spaces can only be multiplied by \
other Hilbert spaces: %r' % arg)
#combine like arguments into direct powers
comb_args = []
prev_arg = None
for new_arg in new_args:
if prev_arg is not None:
if isinstance(new_arg, TensorPowerHilbertSpace) and \
isinstance(prev_arg, TensorPowerHilbertSpace) and \
new_arg.base == prev_arg.base:
prev_arg = new_arg.base**(new_arg.exp + prev_arg.exp)
elif isinstance(new_arg, TensorPowerHilbertSpace) and \
new_arg.base == prev_arg:
prev_arg = prev_arg**(new_arg.exp + 1)
elif isinstance(prev_arg, TensorPowerHilbertSpace) and \
new_arg == prev_arg.base:
prev_arg = new_arg**(prev_arg.exp + 1)
elif new_arg == prev_arg:
prev_arg = new_arg**2
else:
comb_args.append(prev_arg)
prev_arg = new_arg
elif prev_arg is None:
prev_arg = new_arg
comb_args.append(prev_arg)
if recall:
return TensorProductHilbertSpace(*comb_args)
elif len(comb_args) == 1:
return TensorPowerHilbertSpace(comb_args[0].base, comb_args[0].exp)
else:
return None
@property
def dimension(self):
arg_list = [arg.dimension for arg in self.args]
if oo in arg_list:
return oo
else:
return reduce(lambda x, y: x*y, arg_list)
@property
def spaces(self):
"""A tuple of the Hilbert spaces in this tensor product."""
return self.args
def _spaces_printer(self, printer, *args):
spaces_strs = []
for arg in self.args:
s = printer._print(arg, *args)
if isinstance(arg, DirectSumHilbertSpace):
s = '(%s)' % s
spaces_strs.append(s)
return spaces_strs
def _sympyrepr(self, printer, *args):
spaces_reprs = self._spaces_printer(printer, *args)
return "TensorProductHilbertSpace(%s)" % ','.join(spaces_reprs)
def _sympystr(self, printer, *args):
spaces_strs = self._spaces_printer(printer, *args)
return '*'.join(spaces_strs)
def _pretty(self, printer, *args):
length = len(self.args)
pform = printer._print('', *args)
for i in range(length):
next_pform = printer._print(self.args[i], *args)
if isinstance(self.args[i], (DirectSumHilbertSpace,
TensorProductHilbertSpace)):
next_pform = prettyForm(
*next_pform.parens(left='(', right=')')
)
pform = prettyForm(*pform.right(next_pform))
if i != length - 1:
if printer._use_unicode:
pform = prettyForm(*pform.right(u' ' + u'\N{N-ARY CIRCLED TIMES OPERATOR}' + u' '))
else:
pform = prettyForm(*pform.right(' x '))
return pform
def _latex(self, printer, *args):
length = len(self.args)
s = ''
for i in range(length):
arg_s = printer._print(self.args[i], *args)
if isinstance(self.args[i], (DirectSumHilbertSpace,
TensorProductHilbertSpace)):
arg_s = r'\left(%s\right)' % arg_s
s = s + arg_s
if i != length - 1:
s = s + r'\otimes '
return s
class DirectSumHilbertSpace(HilbertSpace):
"""A direct sum of Hilbert spaces [1]_.
This class uses the ``+`` operator to represent direct sums between
different Hilbert spaces.
A ``DirectSumHilbertSpace`` object takes in an arbitrary number of
``HilbertSpace`` objects as its arguments. Also, addition of
``HilbertSpace`` objects will automatically return a direct sum object.
Examples
========
>>> from sympy.physics.quantum.hilbert import ComplexSpace, FockSpace
>>> c = ComplexSpace(2)
>>> f = FockSpace()
>>> hs = c+f
>>> hs
C(2)+F
>>> hs.dimension
oo
>>> list(hs.spaces)
[C(2), F]
References
==========
.. [1] https://en.wikipedia.org/wiki/Hilbert_space#Direct_sums
"""
def __new__(cls, *args):
r = cls.eval(args)
if isinstance(r, Basic):
return r
obj = Basic.__new__(cls, *args)
return obj
@classmethod
def eval(cls, args):
"""Evaluates the direct product."""
new_args = []
recall = False
#flatten arguments
for arg in args:
if isinstance(arg, DirectSumHilbertSpace):
new_args.extend(arg.args)
recall = True
elif isinstance(arg, HilbertSpace):
new_args.append(arg)
else:
raise TypeError('Hilbert spaces can only be summed with other \
Hilbert spaces: %r' % arg)
if recall:
return DirectSumHilbertSpace(*new_args)
else:
return None
@property
def dimension(self):
arg_list = [arg.dimension for arg in self.args]
if oo in arg_list:
return oo
else:
return reduce(lambda x, y: x + y, arg_list)
@property
def spaces(self):
"""A tuple of the Hilbert spaces in this direct sum."""
return self.args
def _sympyrepr(self, printer, *args):
spaces_reprs = [printer._print(arg, *args) for arg in self.args]
return "DirectSumHilbertSpace(%s)" % ','.join(spaces_reprs)
def _sympystr(self, printer, *args):
spaces_strs = [printer._print(arg, *args) for arg in self.args]
return '+'.join(spaces_strs)
def _pretty(self, printer, *args):
length = len(self.args)
pform = printer._print('', *args)
for i in range(length):
next_pform = printer._print(self.args[i], *args)
if isinstance(self.args[i], (DirectSumHilbertSpace,
TensorProductHilbertSpace)):
next_pform = prettyForm(
*next_pform.parens(left='(', right=')')
)
pform = prettyForm(*pform.right(next_pform))
if i != length - 1:
if printer._use_unicode:
pform = prettyForm(*pform.right(u' \N{CIRCLED PLUS} '))
else:
pform = prettyForm(*pform.right(' + '))
return pform
def _latex(self, printer, *args):
length = len(self.args)
s = ''
for i in range(length):
arg_s = printer._print(self.args[i], *args)
if isinstance(self.args[i], (DirectSumHilbertSpace,
TensorProductHilbertSpace)):
arg_s = r'\left(%s\right)' % arg_s
s = s + arg_s
if i != length - 1:
s = s + r'\oplus '
return s
class TensorPowerHilbertSpace(HilbertSpace):
"""An exponentiated Hilbert space [1]_.
Tensor powers (repeated tensor products) are represented by the
operator ``**`` Identical Hilbert spaces that are multiplied together
will be automatically combined into a single tensor power object.
Any Hilbert space, product, or sum may be raised to a tensor power. The
``TensorPowerHilbertSpace`` takes two arguments: the Hilbert space; and the
tensor power (number).
Examples
========
>>> from sympy.physics.quantum.hilbert import ComplexSpace, FockSpace
>>> from sympy import symbols
>>> n = symbols('n')
>>> c = ComplexSpace(2)
>>> hs = c**n
>>> hs
C(2)**n
>>> hs.dimension
2**n
>>> c = ComplexSpace(2)
>>> c*c
C(2)**2
>>> f = FockSpace()
>>> c*f*f
C(2)*F**2
References
==========
.. [1] https://en.wikipedia.org/wiki/Hilbert_space#Tensor_products
"""
def __new__(cls, *args):
r = cls.eval(args)
if isinstance(r, Basic):
return r
return Basic.__new__(cls, *r)
@classmethod
def eval(cls, args):
new_args = args[0], sympify(args[1])
exp = new_args[1]
#simplify hs**1 -> hs
if exp == 1:
return args[0]
#simplify hs**0 -> 1
if exp == 0:
return sympify(1)
#check (and allow) for hs**(x+42+y...) case
if len(exp.atoms()) == 1:
if not (exp.is_Integer and exp >= 0 or exp.is_Symbol):
raise ValueError('Hilbert spaces can only be raised to \
positive integers or Symbols: %r' % exp)
else:
for power in exp.atoms():
if not (power.is_Integer or power.is_Symbol):
raise ValueError('Tensor powers can only contain integers \
or Symbols: %r' % power)
return new_args
@property
def base(self):
return self.args[0]
@property
def exp(self):
return self.args[1]
@property
def dimension(self):
if self.base.dimension is oo:
return oo
else:
return self.base.dimension**self.exp
def _sympyrepr(self, printer, *args):
return "TensorPowerHilbertSpace(%s,%s)" % (printer._print(self.base,
*args), printer._print(self.exp, *args))
def _sympystr(self, printer, *args):
return "%s**%s" % (printer._print(self.base, *args),
printer._print(self.exp, *args))
def _pretty(self, printer, *args):
pform_exp = printer._print(self.exp, *args)
if printer._use_unicode:
pform_exp = prettyForm(*pform_exp.left(prettyForm(u'\N{N-ARY CIRCLED TIMES OPERATOR}')))
else:
pform_exp = prettyForm(*pform_exp.left(prettyForm('x')))
pform_base = printer._print(self.base, *args)
return pform_base**pform_exp
def _latex(self, printer, *args):
base = printer._print(self.base, *args)
exp = printer._print(self.exp, *args)
return r'{%s}^{\otimes %s}' % (base, exp)
|
6e94f66df4d52bff22dd137787b233188a863f7bddbfd55737a3d2470bfef459 | from __future__ import print_function, division
from sympy.core.backend import sympify
from sympy.physics.vector import Point, ReferenceFrame, Dyadic
from sympy.utilities.exceptions import SymPyDeprecationWarning
__all__ = ['RigidBody']
class RigidBody(object):
"""An idealized rigid body.
This is essentially a container which holds the various components which
describe a rigid body: a name, mass, center of mass, reference frame, and
inertia.
All of these need to be supplied on creation, but can be changed
afterwards.
Attributes
==========
name : string
The body's name.
masscenter : Point
The point which represents the center of mass of the rigid body.
frame : ReferenceFrame
The ReferenceFrame which the rigid body is fixed in.
mass : Sympifyable
The body's mass.
inertia : (Dyadic, Point)
The body's inertia about a point; stored in a tuple as shown above.
Examples
========
>>> from sympy import Symbol
>>> from sympy.physics.mechanics import ReferenceFrame, Point, RigidBody
>>> from sympy.physics.mechanics import outer
>>> m = Symbol('m')
>>> A = ReferenceFrame('A')
>>> P = Point('P')
>>> I = outer (A.x, A.x)
>>> inertia_tuple = (I, P)
>>> B = RigidBody('B', P, A, m, inertia_tuple)
>>> # Or you could change them afterwards
>>> m2 = Symbol('m2')
>>> B.mass = m2
"""
def __init__(self, name, masscenter, frame, mass, inertia):
if not isinstance(name, str):
raise TypeError('Supply a valid name.')
self._name = name
self.masscenter = masscenter
self.mass = mass
self.frame = frame
self.inertia = inertia
self.potential_energy = 0
def __str__(self):
return self._name
def __repr__(self):
return self.__str__()
@property
def frame(self):
return self._frame
@frame.setter
def frame(self, F):
if not isinstance(F, ReferenceFrame):
raise TypeError("RigdBody frame must be a ReferenceFrame object.")
self._frame = F
@property
def masscenter(self):
return self._masscenter
@masscenter.setter
def masscenter(self, p):
if not isinstance(p, Point):
raise TypeError("RigidBody center of mass must be a Point object.")
self._masscenter = p
@property
def mass(self):
return self._mass
@mass.setter
def mass(self, m):
self._mass = sympify(m)
@property
def inertia(self):
return (self._inertia, self._inertia_point)
@inertia.setter
def inertia(self, I):
if not isinstance(I[0], Dyadic):
raise TypeError("RigidBody inertia must be a Dyadic object.")
if not isinstance(I[1], Point):
raise TypeError("RigidBody inertia must be about a Point.")
self._inertia = I[0]
self._inertia_point = I[1]
# have I S/O, want I S/S*
# I S/O = I S/S* + I S*/O; I S/S* = I S/O - I S*/O
# I_S/S* = I_S/O - I_S*/O
from sympy.physics.mechanics.functions import inertia_of_point_mass
I_Ss_O = inertia_of_point_mass(self.mass,
self.masscenter.pos_from(I[1]),
self.frame)
self._central_inertia = I[0] - I_Ss_O
@property
def central_inertia(self):
"""The body's central inertia dyadic."""
return self._central_inertia
def linear_momentum(self, frame):
""" Linear momentum of the rigid body.
The linear momentum L, of a rigid body B, with respect to frame N is
given by
L = M * v*
where M is the mass of the rigid body and v* is the velocity of
the mass center of B in the frame, N.
Parameters
==========
frame : ReferenceFrame
The frame in which linear momentum is desired.
Examples
========
>>> from sympy.physics.mechanics import Point, ReferenceFrame, outer
>>> from sympy.physics.mechanics import RigidBody, dynamicsymbols
>>> M, v = dynamicsymbols('M v')
>>> N = ReferenceFrame('N')
>>> P = Point('P')
>>> P.set_vel(N, v * N.x)
>>> I = outer (N.x, N.x)
>>> Inertia_tuple = (I, P)
>>> B = RigidBody('B', P, N, M, Inertia_tuple)
>>> B.linear_momentum(N)
M*v*N.x
"""
return self.mass * self.masscenter.vel(frame)
def angular_momentum(self, point, frame):
"""Returns the angular momentum of the rigid body about a point in the
given frame.
The angular momentum H of a rigid body B about some point O in a frame
N is given by:
H = I . w + r x Mv
where I is the central inertia dyadic of B, w is the angular velocity
of body B in the frame, N, r is the position vector from point O to the
mass center of B, and v is the velocity of the mass center in the
frame, N.
Parameters
==========
point : Point
The point about which angular momentum is desired.
frame : ReferenceFrame
The frame in which angular momentum is desired.
Examples
========
>>> from sympy.physics.mechanics import Point, ReferenceFrame, outer
>>> from sympy.physics.mechanics import RigidBody, dynamicsymbols
>>> M, v, r, omega = dynamicsymbols('M v r omega')
>>> N = ReferenceFrame('N')
>>> b = ReferenceFrame('b')
>>> b.set_ang_vel(N, omega * b.x)
>>> P = Point('P')
>>> P.set_vel(N, 1 * N.x)
>>> I = outer(b.x, b.x)
>>> B = RigidBody('B', P, b, M, (I, P))
>>> B.angular_momentum(P, N)
omega*b.x
"""
I = self.central_inertia
w = self.frame.ang_vel_in(frame)
m = self.mass
r = self.masscenter.pos_from(point)
v = self.masscenter.vel(frame)
return I.dot(w) + r.cross(m * v)
def kinetic_energy(self, frame):
"""Kinetic energy of the rigid body
The kinetic energy, T, of a rigid body, B, is given by
'T = 1/2 (I omega^2 + m v^2)'
where I and m are the central inertia dyadic and mass of rigid body B,
respectively, omega is the body's angular velocity and v is the
velocity of the body's mass center in the supplied ReferenceFrame.
Parameters
==========
frame : ReferenceFrame
The RigidBody's angular velocity and the velocity of it's mass
center are typically defined with respect to an inertial frame but
any relevant frame in which the velocities are known can be supplied.
Examples
========
>>> from sympy.physics.mechanics import Point, ReferenceFrame, outer
>>> from sympy.physics.mechanics import RigidBody
>>> from sympy import symbols
>>> M, v, r, omega = symbols('M v r omega')
>>> N = ReferenceFrame('N')
>>> b = ReferenceFrame('b')
>>> b.set_ang_vel(N, omega * b.x)
>>> P = Point('P')
>>> P.set_vel(N, v * N.x)
>>> I = outer (b.x, b.x)
>>> inertia_tuple = (I, P)
>>> B = RigidBody('B', P, b, M, inertia_tuple)
>>> B.kinetic_energy(N)
M*v**2/2 + omega**2/2
"""
rotational_KE = (self.frame.ang_vel_in(frame) & (self.central_inertia &
self.frame.ang_vel_in(frame)) / sympify(2))
translational_KE = (self.mass * (self.masscenter.vel(frame) &
self.masscenter.vel(frame)) / sympify(2))
return rotational_KE + translational_KE
@property
def potential_energy(self):
"""The potential energy of the RigidBody.
Examples
========
>>> from sympy.physics.mechanics import RigidBody, Point, outer, ReferenceFrame
>>> from sympy import symbols
>>> M, g, h = symbols('M g h')
>>> b = ReferenceFrame('b')
>>> P = Point('P')
>>> I = outer (b.x, b.x)
>>> Inertia_tuple = (I, P)
>>> B = RigidBody('B', P, b, M, Inertia_tuple)
>>> B.potential_energy = M * g * h
>>> B.potential_energy
M*g*h
"""
return self._pe
@potential_energy.setter
def potential_energy(self, scalar):
"""Used to set the potential energy of this RigidBody.
Parameters
==========
scalar: Sympifyable
The potential energy (a scalar) of the RigidBody.
Examples
========
>>> from sympy.physics.mechanics import Point, outer
>>> from sympy.physics.mechanics import RigidBody, ReferenceFrame
>>> from sympy import symbols
>>> b = ReferenceFrame('b')
>>> M, g, h = symbols('M g h')
>>> P = Point('P')
>>> I = outer (b.x, b.x)
>>> Inertia_tuple = (I, P)
>>> B = RigidBody('B', P, b, M, Inertia_tuple)
>>> B.potential_energy = M * g * h
"""
self._pe = sympify(scalar)
def set_potential_energy(self, scalar):
SymPyDeprecationWarning(
feature="Method sympy.physics.mechanics." +
"RigidBody.set_potential_energy(self, scalar)",
useinstead="property sympy.physics.mechanics." +
"RigidBody.potential_energy",
deprecated_since_version="1.5", issue=9800).warn()
self.potential_energy = scalar
# XXX: To be consistent with the parallel_axis method in Particle this
# should have a frame argument...
def parallel_axis(self, point):
"""Returns the inertia dyadic of the body with respect to another
point.
Parameters
==========
point : sympy.physics.vector.Point
The point to express the inertia dyadic about.
Returns
=======
inertia : sympy.physics.vector.Dyadic
The inertia dyadic of the rigid body expressed about the provided
point.
"""
# circular import issue
from sympy.physics.mechanics.functions import inertia
a, b, c = self.masscenter.pos_from(point).to_matrix(self.frame)
I = self.mass * inertia(self.frame, b**2 + c**2, c**2 + a**2, a**2 +
b**2, -a * b, -b * c, -a * c)
return self.central_inertia + I
|
53ea695ce5621f4d60d0b70566d8019643521d88935ae1301cc5b88cdffd00d1 | from __future__ import print_function, division
from sympy.core.backend import diff, zeros, Matrix, eye, sympify
from sympy.physics.vector import dynamicsymbols, ReferenceFrame
from sympy.physics.mechanics.functions import (find_dynamicsymbols, msubs,
_f_list_parser)
from sympy.physics.mechanics.linearize import Linearizer
from sympy.utilities import default_sort_key
from sympy.utilities.iterables import iterable
__all__ = ['LagrangesMethod']
class LagrangesMethod(object):
"""Lagrange's method object.
This object generates the equations of motion in a two step procedure. The
first step involves the initialization of LagrangesMethod by supplying the
Lagrangian and the generalized coordinates, at the bare minimum. If there
are any constraint equations, they can be supplied as keyword arguments.
The Lagrange multipliers are automatically generated and are equal in
number to the constraint equations. Similarly any non-conservative forces
can be supplied in an iterable (as described below and also shown in the
example) along with a ReferenceFrame. This is also discussed further in the
__init__ method.
Attributes
==========
q, u : Matrix
Matrices of the generalized coordinates and speeds
forcelist : iterable
Iterable of (Point, vector) or (ReferenceFrame, vector) tuples
describing the forces on the system.
bodies : iterable
Iterable containing the rigid bodies and particles of the system.
mass_matrix : Matrix
The system's mass matrix
forcing : Matrix
The system's forcing vector
mass_matrix_full : Matrix
The "mass matrix" for the qdot's, qdoubledot's, and the
lagrange multipliers (lam)
forcing_full : Matrix
The forcing vector for the qdot's, qdoubledot's and
lagrange multipliers (lam)
Examples
========
This is a simple example for a one degree of freedom translational
spring-mass-damper.
In this example, we first need to do the kinematics.
This involves creating generalized coordinates and their derivatives.
Then we create a point and set its velocity in a frame.
>>> from sympy.physics.mechanics import LagrangesMethod, Lagrangian
>>> from sympy.physics.mechanics import ReferenceFrame, Particle, Point
>>> from sympy.physics.mechanics import dynamicsymbols
>>> from sympy import symbols
>>> q = dynamicsymbols('q')
>>> qd = dynamicsymbols('q', 1)
>>> m, k, b = symbols('m k b')
>>> N = ReferenceFrame('N')
>>> P = Point('P')
>>> P.set_vel(N, qd * N.x)
We need to then prepare the information as required by LagrangesMethod to
generate equations of motion.
First we create the Particle, which has a point attached to it.
Following this the lagrangian is created from the kinetic and potential
energies.
Then, an iterable of nonconservative forces/torques must be constructed,
where each item is a (Point, Vector) or (ReferenceFrame, Vector) tuple,
with the Vectors representing the nonconservative forces or torques.
>>> Pa = Particle('Pa', P, m)
>>> Pa.potential_energy = k * q**2 / 2.0
>>> L = Lagrangian(N, Pa)
>>> fl = [(P, -b * qd * N.x)]
Finally we can generate the equations of motion.
First we create the LagrangesMethod object. To do this one must supply
the Lagrangian, and the generalized coordinates. The constraint equations,
the forcelist, and the inertial frame may also be provided, if relevant.
Next we generate Lagrange's equations of motion, such that:
Lagrange's equations of motion = 0.
We have the equations of motion at this point.
>>> l = LagrangesMethod(L, [q], forcelist = fl, frame = N)
>>> print(l.form_lagranges_equations())
Matrix([[b*Derivative(q(t), t) + 1.0*k*q(t) + m*Derivative(q(t), (t, 2))]])
We can also solve for the states using the 'rhs' method.
>>> print(l.rhs())
Matrix([[Derivative(q(t), t)], [(-b*Derivative(q(t), t) - 1.0*k*q(t))/m]])
Please refer to the docstrings on each method for more details.
"""
def __init__(self, Lagrangian, qs, forcelist=None, bodies=None, frame=None,
hol_coneqs=None, nonhol_coneqs=None):
"""Supply the following for the initialization of LagrangesMethod
Lagrangian : Sympifyable
qs : array_like
The generalized coordinates
hol_coneqs : array_like, optional
The holonomic constraint equations
nonhol_coneqs : array_like, optional
The nonholonomic constraint equations
forcelist : iterable, optional
Takes an iterable of (Point, Vector) or (ReferenceFrame, Vector)
tuples which represent the force at a point or torque on a frame.
This feature is primarily to account for the nonconservative forces
and/or moments.
bodies : iterable, optional
Takes an iterable containing the rigid bodies and particles of the
system.
frame : ReferenceFrame, optional
Supply the inertial frame. This is used to determine the
generalized forces due to non-conservative forces.
"""
self._L = Matrix([sympify(Lagrangian)])
self.eom = None
self._m_cd = Matrix() # Mass Matrix of differentiated coneqs
self._m_d = Matrix() # Mass Matrix of dynamic equations
self._f_cd = Matrix() # Forcing part of the diff coneqs
self._f_d = Matrix() # Forcing part of the dynamic equations
self.lam_coeffs = Matrix() # The coeffecients of the multipliers
forcelist = forcelist if forcelist else []
if not iterable(forcelist):
raise TypeError('Force pairs must be supplied in an iterable.')
self._forcelist = forcelist
if frame and not isinstance(frame, ReferenceFrame):
raise TypeError('frame must be a valid ReferenceFrame')
self._bodies = bodies
self.inertial = frame
self.lam_vec = Matrix()
self._term1 = Matrix()
self._term2 = Matrix()
self._term3 = Matrix()
self._term4 = Matrix()
# Creating the qs, qdots and qdoubledots
if not iterable(qs):
raise TypeError('Generalized coordinates must be an iterable')
self._q = Matrix(qs)
self._qdots = self.q.diff(dynamicsymbols._t)
self._qdoubledots = self._qdots.diff(dynamicsymbols._t)
mat_build = lambda x: Matrix(x) if x else Matrix()
hol_coneqs = mat_build(hol_coneqs)
nonhol_coneqs = mat_build(nonhol_coneqs)
self.coneqs = Matrix([hol_coneqs.diff(dynamicsymbols._t),
nonhol_coneqs])
self._hol_coneqs = hol_coneqs
def form_lagranges_equations(self):
"""Method to form Lagrange's equations of motion.
Returns a vector of equations of motion using Lagrange's equations of
the second kind.
"""
qds = self._qdots
qdd_zero = dict((i, 0) for i in self._qdoubledots)
n = len(self.q)
# Internally we represent the EOM as four terms:
# EOM = term1 - term2 - term3 - term4 = 0
# First term
self._term1 = self._L.jacobian(qds)
self._term1 = self._term1.diff(dynamicsymbols._t).T
# Second term
self._term2 = self._L.jacobian(self.q).T
# Third term
if self.coneqs:
coneqs = self.coneqs
m = len(coneqs)
# Creating the multipliers
self.lam_vec = Matrix(dynamicsymbols('lam1:' + str(m + 1)))
self.lam_coeffs = -coneqs.jacobian(qds)
self._term3 = self.lam_coeffs.T * self.lam_vec
# Extracting the coeffecients of the qdds from the diff coneqs
diffconeqs = coneqs.diff(dynamicsymbols._t)
self._m_cd = diffconeqs.jacobian(self._qdoubledots)
# The remaining terms i.e. the 'forcing' terms in diff coneqs
self._f_cd = -diffconeqs.subs(qdd_zero)
else:
self._term3 = zeros(n, 1)
# Fourth term
if self.forcelist:
N = self.inertial
self._term4 = zeros(n, 1)
for i, qd in enumerate(qds):
flist = zip(*_f_list_parser(self.forcelist, N))
self._term4[i] = sum(v.diff(qd, N) & f for (v, f) in flist)
else:
self._term4 = zeros(n, 1)
# Form the dynamic mass and forcing matrices
without_lam = self._term1 - self._term2 - self._term4
self._m_d = without_lam.jacobian(self._qdoubledots)
self._f_d = -without_lam.subs(qdd_zero)
# Form the EOM
self.eom = without_lam - self._term3
return self.eom
@property
def mass_matrix(self):
"""Returns the mass matrix, which is augmented by the Lagrange
multipliers, if necessary.
If the system is described by 'n' generalized coordinates and there are
no constraint equations then an n X n matrix is returned.
If there are 'n' generalized coordinates and 'm' constraint equations
have been supplied during initialization then an n X (n+m) matrix is
returned. The (n + m - 1)th and (n + m)th columns contain the
coefficients of the Lagrange multipliers.
"""
if self.eom is None:
raise ValueError('Need to compute the equations of motion first')
if self.coneqs:
return (self._m_d).row_join(self.lam_coeffs.T)
else:
return self._m_d
@property
def mass_matrix_full(self):
"""Augments the coefficients of qdots to the mass_matrix."""
if self.eom is None:
raise ValueError('Need to compute the equations of motion first')
n = len(self.q)
m = len(self.coneqs)
row1 = eye(n).row_join(zeros(n, n + m))
row2 = zeros(n, n).row_join(self.mass_matrix)
if self.coneqs:
row3 = zeros(m, n).row_join(self._m_cd).row_join(zeros(m, m))
return row1.col_join(row2).col_join(row3)
else:
return row1.col_join(row2)
@property
def forcing(self):
"""Returns the forcing vector from 'lagranges_equations' method."""
if self.eom is None:
raise ValueError('Need to compute the equations of motion first')
return self._f_d
@property
def forcing_full(self):
"""Augments qdots to the forcing vector above."""
if self.eom is None:
raise ValueError('Need to compute the equations of motion first')
if self.coneqs:
return self._qdots.col_join(self.forcing).col_join(self._f_cd)
else:
return self._qdots.col_join(self.forcing)
def to_linearizer(self, q_ind=None, qd_ind=None, q_dep=None, qd_dep=None):
"""Returns an instance of the Linearizer class, initiated from the
data in the LagrangesMethod class. This may be more desirable than using
the linearize class method, as the Linearizer object will allow more
efficient recalculation (i.e. about varying operating points).
Parameters
==========
q_ind, qd_ind : array_like, optional
The independent generalized coordinates and speeds.
q_dep, qd_dep : array_like, optional
The dependent generalized coordinates and speeds.
"""
# Compose vectors
t = dynamicsymbols._t
q = self.q
u = self._qdots
ud = u.diff(t)
# Get vector of lagrange multipliers
lams = self.lam_vec
mat_build = lambda x: Matrix(x) if x else Matrix()
q_i = mat_build(q_ind)
q_d = mat_build(q_dep)
u_i = mat_build(qd_ind)
u_d = mat_build(qd_dep)
# Compose general form equations
f_c = self._hol_coneqs
f_v = self.coneqs
f_a = f_v.diff(t)
f_0 = u
f_1 = -u
f_2 = self._term1
f_3 = -(self._term2 + self._term4)
f_4 = -self._term3
# Check that there are an appropriate number of independent and
# dependent coordinates
if len(q_d) != len(f_c) or len(u_d) != len(f_v):
raise ValueError(("Must supply {:} dependent coordinates, and " +
"{:} dependent speeds").format(len(f_c), len(f_v)))
if set(Matrix([q_i, q_d])) != set(q):
raise ValueError("Must partition q into q_ind and q_dep, with " +
"no extra or missing symbols.")
if set(Matrix([u_i, u_d])) != set(u):
raise ValueError("Must partition qd into qd_ind and qd_dep, " +
"with no extra or missing symbols.")
# Find all other dynamic symbols, forming the forcing vector r.
# Sort r to make it canonical.
insyms = set(Matrix([q, u, ud, lams]))
r = list(find_dynamicsymbols(f_3, insyms))
r.sort(key=default_sort_key)
# Check for any derivatives of variables in r that are also found in r.
for i in r:
if diff(i, dynamicsymbols._t) in r:
raise ValueError('Cannot have derivatives of specified \
quantities when linearizing forcing terms.')
return Linearizer(f_0, f_1, f_2, f_3, f_4, f_c, f_v, f_a, q, u, q_i,
q_d, u_i, u_d, r, lams)
def linearize(self, q_ind=None, qd_ind=None, q_dep=None, qd_dep=None,
**kwargs):
"""Linearize the equations of motion about a symbolic operating point.
If kwarg A_and_B is False (default), returns M, A, B, r for the
linearized form, M*[q', u']^T = A*[q_ind, u_ind]^T + B*r.
If kwarg A_and_B is True, returns A, B, r for the linearized form
dx = A*x + B*r, where x = [q_ind, u_ind]^T. Note that this is
computationally intensive if there are many symbolic parameters. For
this reason, it may be more desirable to use the default A_and_B=False,
returning M, A, and B. Values may then be substituted in to these
matrices, and the state space form found as
A = P.T*M.inv()*A, B = P.T*M.inv()*B, where P = Linearizer.perm_mat.
In both cases, r is found as all dynamicsymbols in the equations of
motion that are not part of q, u, q', or u'. They are sorted in
canonical form.
The operating points may be also entered using the ``op_point`` kwarg.
This takes a dictionary of {symbol: value}, or a an iterable of such
dictionaries. The values may be numeric or symbolic. The more values
you can specify beforehand, the faster this computation will run.
For more documentation, please see the ``Linearizer`` class."""
linearizer = self.to_linearizer(q_ind, qd_ind, q_dep, qd_dep)
result = linearizer.linearize(**kwargs)
return result + (linearizer.r,)
def solve_multipliers(self, op_point=None, sol_type='dict'):
"""Solves for the values of the lagrange multipliers symbolically at
the specified operating point
Parameters
==========
op_point : dict or iterable of dicts, optional
Point at which to solve at. The operating point is specified as
a dictionary or iterable of dictionaries of {symbol: value}. The
value may be numeric or symbolic itself.
sol_type : str, optional
Solution return type. Valid options are:
- 'dict': A dict of {symbol : value} (default)
- 'Matrix': An ordered column matrix of the solution
"""
# Determine number of multipliers
k = len(self.lam_vec)
if k == 0:
raise ValueError("System has no lagrange multipliers to solve for.")
# Compose dict of operating conditions
if isinstance(op_point, dict):
op_point_dict = op_point
elif iterable(op_point):
op_point_dict = {}
for op in op_point:
op_point_dict.update(op)
elif op_point is None:
op_point_dict = {}
else:
raise TypeError("op_point must be either a dictionary or an "
"iterable of dictionaries.")
# Compose the system to be solved
mass_matrix = self.mass_matrix.col_join((-self.lam_coeffs.row_join(
zeros(k, k))))
force_matrix = self.forcing.col_join(self._f_cd)
# Sub in the operating point
mass_matrix = msubs(mass_matrix, op_point_dict)
force_matrix = msubs(force_matrix, op_point_dict)
# Solve for the multipliers
sol_list = mass_matrix.LUsolve(-force_matrix)[-k:]
if sol_type == 'dict':
return dict(zip(self.lam_vec, sol_list))
elif sol_type == 'Matrix':
return Matrix(sol_list)
else:
raise ValueError("Unknown sol_type {:}.".format(sol_type))
def rhs(self, inv_method=None, **kwargs):
"""Returns equations that can be solved numerically
Parameters
==========
inv_method : str
The specific sympy inverse matrix calculation method to use. For a
list of valid methods, see
:meth:`~sympy.matrices.matrices.MatrixBase.inv`
"""
if inv_method is None:
self._rhs = self.mass_matrix_full.LUsolve(self.forcing_full)
else:
self._rhs = (self.mass_matrix_full.inv(inv_method,
try_block_diag=True) * self.forcing_full)
return self._rhs
@property
def q(self):
return self._q
@property
def u(self):
return self._qdots
@property
def bodies(self):
return self._bodies
@property
def forcelist(self):
return self._forcelist
|
f5df2d45ad681f1b4951351535849a53b25e51b8c90a6d109f6332a6090184a6 | """
Module defining unit prefixe class and some constants.
Constant dict for SI and binary prefixes are defined as PREFIXES and
BIN_PREFIXES.
"""
from sympy import Expr, sympify
class Prefix(Expr):
"""
This class represent prefixes, with their name, symbol and factor.
Prefixes are used to create derived units from a given unit. They should
always be encapsulated into units.
The factor is constructed from a base (default is 10) to some power, and
it gives the total multiple or fraction. For example the kilometer km
is constructed from the meter (factor 1) and the kilo (10 to the power 3,
i.e. 1000). The base can be changed to allow e.g. binary prefixes.
A prefix multiplied by something will always return the product of this
other object times the factor, except if the other object:
- is a prefix and they can be combined into a new prefix;
- defines multiplication with prefixes (which is the case for the Unit
class).
"""
_op_priority = 13.0
is_commutative = True
def __new__(cls, name, abbrev, exponent, base=sympify(10)):
name = sympify(name)
abbrev = sympify(abbrev)
exponent = sympify(exponent)
base = sympify(base)
obj = Expr.__new__(cls, name, abbrev, exponent, base)
obj._name = name
obj._abbrev = abbrev
obj._scale_factor = base**exponent
obj._exponent = exponent
obj._base = base
return obj
@property
def name(self):
return self._name
@property
def abbrev(self):
return self._abbrev
@property
def scale_factor(self):
return self._scale_factor
@property
def base(self):
return self._base
def __str__(self):
# TODO: add proper printers and tests:
if self.base == 10:
return "Prefix(%r, %r, %r)" % (
str(self.name), str(self.abbrev), self._exponent)
else:
return "Prefix(%r, %r, %r, %r)" % (
str(self.name), str(self.abbrev), self._exponent, self.base)
__repr__ = __str__
def __mul__(self, other):
from sympy.physics.units import Quantity
if not isinstance(other, (Quantity, Prefix)):
return super(Prefix, self).__mul__(other)
fact = self.scale_factor * other.scale_factor
if fact == 1:
return 1
elif isinstance(other, Prefix):
# simplify prefix
for p in PREFIXES:
if PREFIXES[p].scale_factor == fact:
return PREFIXES[p]
return fact
return self.scale_factor * other
def __div__(self, other):
if not hasattr(other, "scale_factor"):
return super(Prefix, self).__div__(other)
fact = self.scale_factor / other.scale_factor
if fact == 1:
return 1
elif isinstance(other, Prefix):
for p in PREFIXES:
if PREFIXES[p].scale_factor == fact:
return PREFIXES[p]
return fact
return self.scale_factor / other
__truediv__ = __div__
def __rdiv__(self, other):
if other == 1:
for p in PREFIXES:
if PREFIXES[p].scale_factor == 1 / self.scale_factor:
return PREFIXES[p]
return other / self.scale_factor
__rtruediv__ = __rdiv__
def prefix_unit(unit, prefixes):
"""
Return a list of all units formed by unit and the given prefixes.
You can use the predefined PREFIXES or BIN_PREFIXES, but you can also
pass as argument a subdict of them if you don't want all prefixed units.
>>> from sympy.physics.units.prefixes import (PREFIXES,
... prefix_unit)
>>> from sympy.physics.units import m
>>> pref = {"m": PREFIXES["m"], "c": PREFIXES["c"], "d": PREFIXES["d"]}
>>> prefix_unit(m, pref) # doctest: +SKIP
[millimeter, centimeter, decimeter]
"""
from sympy.physics.units.quantities import Quantity
from sympy.physics.units import UnitSystem
prefixed_units = []
for prefix_abbr, prefix in prefixes.items():
quantity = Quantity(
"%s%s" % (prefix.name, unit.name),
abbrev=("%s%s" % (prefix.abbrev, unit.abbrev))
)
UnitSystem._quantity_dimensional_equivalence_map_global[quantity] = unit
UnitSystem._quantity_scale_factors_global[quantity] = (prefix.scale_factor, unit)
prefixed_units.append(quantity)
return prefixed_units
yotta = Prefix('yotta', 'Y', 24)
zetta = Prefix('zetta', 'Z', 21)
exa = Prefix('exa', 'E', 18)
peta = Prefix('peta', 'P', 15)
tera = Prefix('tera', 'T', 12)
giga = Prefix('giga', 'G', 9)
mega = Prefix('mega', 'M', 6)
kilo = Prefix('kilo', 'k', 3)
hecto = Prefix('hecto', 'h', 2)
deca = Prefix('deca', 'da', 1)
deci = Prefix('deci', 'd', -1)
centi = Prefix('centi', 'c', -2)
milli = Prefix('milli', 'm', -3)
micro = Prefix('micro', 'mu', -6)
nano = Prefix('nano', 'n', -9)
pico = Prefix('pico', 'p', -12)
femto = Prefix('femto', 'f', -15)
atto = Prefix('atto', 'a', -18)
zepto = Prefix('zepto', 'z', -21)
yocto = Prefix('yocto', 'y', -24)
# http://physics.nist.gov/cuu/Units/prefixes.html
PREFIXES = {
'Y': yotta,
'Z': zetta,
'E': exa,
'P': peta,
'T': tera,
'G': giga,
'M': mega,
'k': kilo,
'h': hecto,
'da': deca,
'd': deci,
'c': centi,
'm': milli,
'mu': micro,
'n': nano,
'p': pico,
'f': femto,
'a': atto,
'z': zepto,
'y': yocto,
}
kibi = Prefix('kibi', 'Y', 10, 2)
mebi = Prefix('mebi', 'Y', 20, 2)
gibi = Prefix('gibi', 'Y', 30, 2)
tebi = Prefix('tebi', 'Y', 40, 2)
pebi = Prefix('pebi', 'Y', 50, 2)
exbi = Prefix('exbi', 'Y', 60, 2)
# http://physics.nist.gov/cuu/Units/binary.html
BIN_PREFIXES = {
'Ki': kibi,
'Mi': mebi,
'Gi': gibi,
'Ti': tebi,
'Pi': pebi,
'Ei': exbi,
}
|
23dfe680711ae7c884997dac0c1b7c13bd238c6d0a77eee55996c9c8b26dce9f | """
Several methods to simplify expressions involving unit objects.
"""
from __future__ import division
from sympy import Add, Mul, Pow, Tuple, sympify
from sympy.core.compatibility import reduce, Iterable, ordered
from sympy.physics.units.dimensions import Dimension
from sympy.physics.units.prefixes import Prefix
from sympy.physics.units.quantities import Quantity
from sympy.utilities.iterables import sift
def _get_conversion_matrix_for_expr(expr, target_units, unit_system):
from sympy import Matrix
dimension_system = unit_system.get_dimension_system()
expr_dim = Dimension(unit_system.get_dimensional_expr(expr))
dim_dependencies = dimension_system.get_dimensional_dependencies(expr_dim, mark_dimensionless=True)
target_dims = [Dimension(unit_system.get_dimensional_expr(x)) for x in target_units]
canon_dim_units = [i for x in target_dims for i in dimension_system.get_dimensional_dependencies(x, mark_dimensionless=True)]
canon_expr_units = {i for i in dim_dependencies}
if not canon_expr_units.issubset(set(canon_dim_units)):
return None
seen = set([])
canon_dim_units = [i for i in canon_dim_units if not (i in seen or seen.add(i))]
camat = Matrix([[dimension_system.get_dimensional_dependencies(i, mark_dimensionless=True).get(j, 0) for i in target_dims] for j in canon_dim_units])
exprmat = Matrix([dim_dependencies.get(k, 0) for k in canon_dim_units])
res_exponents = camat.solve_least_squares(exprmat, method=None)
return res_exponents
def convert_to(expr, target_units, unit_system="SI"):
"""
Convert ``expr`` to the same expression with all of its units and quantities
represented as factors of ``target_units``, whenever the dimension is compatible.
``target_units`` may be a single unit/quantity, or a collection of
units/quantities.
Examples
========
>>> from sympy.physics.units import speed_of_light, meter, gram, second, day
>>> from sympy.physics.units import mile, newton, kilogram, atomic_mass_constant
>>> from sympy.physics.units import kilometer, centimeter
>>> from sympy.physics.units import gravitational_constant, hbar
>>> from sympy.physics.units import convert_to
>>> convert_to(mile, kilometer)
25146*kilometer/15625
>>> convert_to(mile, kilometer).n()
1.609344*kilometer
>>> convert_to(speed_of_light, meter/second)
299792458*meter/second
>>> convert_to(day, second)
86400*second
>>> 3*newton
3*newton
>>> convert_to(3*newton, kilogram*meter/second**2)
3*kilogram*meter/second**2
>>> convert_to(atomic_mass_constant, gram)
1.660539060e-24*gram
Conversion to multiple units:
>>> convert_to(speed_of_light, [meter, second])
299792458*meter/second
>>> convert_to(3*newton, [centimeter, gram, second])
300000*centimeter*gram/second**2
Conversion to Planck units:
>>> convert_to(atomic_mass_constant, [gravitational_constant, speed_of_light, hbar]).n()
7.62963085040767e-20*gravitational_constant**(-0.5)*hbar**0.5*speed_of_light**0.5
"""
from sympy.physics.units import UnitSystem
unit_system = UnitSystem.get_unit_system(unit_system)
if not isinstance(target_units, (Iterable, Tuple)):
target_units = [target_units]
if isinstance(expr, Add):
return Add.fromiter(convert_to(i, target_units, unit_system) for i in expr.args)
expr = sympify(expr)
if not isinstance(expr, Quantity) and expr.has(Quantity):
expr = expr.replace(lambda x: isinstance(x, Quantity), lambda x: x.convert_to(target_units, unit_system))
def get_total_scale_factor(expr):
if isinstance(expr, Mul):
return reduce(lambda x, y: x * y, [get_total_scale_factor(i) for i in expr.args])
elif isinstance(expr, Pow):
return get_total_scale_factor(expr.base) ** expr.exp
elif isinstance(expr, Quantity):
return unit_system.get_quantity_scale_factor(expr)
return expr
depmat = _get_conversion_matrix_for_expr(expr, target_units, unit_system)
if depmat is None:
return expr
expr_scale_factor = get_total_scale_factor(expr)
return expr_scale_factor * Mul.fromiter((1/get_total_scale_factor(u) * u) ** p for u, p in zip(target_units, depmat))
def quantity_simplify(expr):
"""Return an equivalent expression in which prefixes are replaced
with numerical values and all units of a given dimension are the
unified in a canonical manner.
Examples
========
>>> from sympy.physics.units.util import quantity_simplify
>>> from sympy.physics.units.prefixes import kilo
>>> from sympy.physics.units import foot, inch
>>> quantity_simplify(kilo*foot*inch)
250*foot**2/3
>>> quantity_simplify(foot - 6*inch)
foot/2
"""
if expr.is_Atom or not expr.has(Prefix, Quantity):
return expr
# replace all prefixes with numerical values
p = expr.atoms(Prefix)
expr = expr.xreplace({p: p.scale_factor for p in p})
# replace all quantities of given dimension with a canonical
# quantity, chosen from those in the expression
d = sift(expr.atoms(Quantity), lambda i: i.dimension)
for k in d:
if len(d[k]) == 1:
continue
v = list(ordered(d[k]))
ref = v[0]/v[0].scale_factor
expr = expr.xreplace({vi: ref*vi.scale_factor for vi in v[1:]})
return expr
def check_dimensions(expr, unit_system="SI"):
"""Return expr if there are not unitless values added to
dimensional quantities, else raise a ValueError."""
# the case of adding a number to a dimensional quantity
# is ignored for the sake of SymPy core routines, so this
# function will raise an error now if such an addend is
# found.
# Also, when doing substitutions, multiplicative constants
# might be introduced, so remove those now
from sympy.physics.units import UnitSystem
unit_system = UnitSystem.get_unit_system(unit_system)
adds = expr.atoms(Add)
DIM_OF = unit_system.get_dimension_system().get_dimensional_dependencies
for a in adds:
deset = set()
for ai in a.args:
if ai.is_number:
deset.add(())
continue
dims = []
skip = False
for i in Mul.make_args(ai):
if i.has(Quantity):
i = Dimension(unit_system.get_dimensional_expr(i))
if i.has(Dimension):
dims.extend(DIM_OF(i).items())
elif i.free_symbols:
skip = True
break
if not skip:
deset.add(tuple(sorted(dims)))
if len(deset) > 1:
raise ValueError(
"addends have incompatible dimensions")
# clear multiplicative constants on Dimensions which may be
# left after substitution
reps = {}
for m in expr.atoms(Mul):
if any(isinstance(i, Dimension) for i in m.args):
reps[m] = m.func(*[
i for i in m.args if not i.is_number])
return expr.xreplace(reps)
|
084e5a8a5cc509e19cc1c36efd3fde4b2453d33ab4c6a4bfb262d00727252e65 | from sympy.core.backend import (S, sympify, expand, sqrt, Add, zeros,
ImmutableMatrix as Matrix)
from sympy import trigsimp
from sympy.core.compatibility import unicode
from sympy.utilities.misc import filldedent
__all__ = ['Vector']
class Vector(object):
"""The class used to define vectors.
It along with ReferenceFrame are the building blocks of describing a
classical mechanics system in PyDy and sympy.physics.vector.
Attributes
==========
simp : Boolean
Let certain methods use trigsimp on their outputs
"""
simp = False
def __init__(self, inlist):
"""This is the constructor for the Vector class. You shouldn't be
calling this, it should only be used by other functions. You should be
treating Vectors like you would with if you were doing the math by
hand, and getting the first 3 from the standard basis vectors from a
ReferenceFrame.
The only exception is to create a zero vector:
zv = Vector(0)
"""
self.args = []
if inlist == 0:
inlist = []
if isinstance(inlist, dict):
d = inlist
else:
d = {}
for inp in inlist:
if inp[1] in d:
d[inp[1]] += inp[0]
else:
d[inp[1]] = inp[0]
for k, v in d.items():
if v != Matrix([0, 0, 0]):
self.args.append((v, k))
def __hash__(self):
return hash(tuple(self.args))
def __add__(self, other):
"""The add operator for Vector. """
if other == 0:
return self
other = _check_vector(other)
return Vector(self.args + other.args)
def __and__(self, other):
"""Dot product of two vectors.
Returns a scalar, the dot product of the two Vectors
Parameters
==========
other : Vector
The Vector which we are dotting with
Examples
========
>>> from sympy.physics.vector import ReferenceFrame, dot
>>> from sympy import symbols
>>> q1 = symbols('q1')
>>> N = ReferenceFrame('N')
>>> dot(N.x, N.x)
1
>>> dot(N.x, N.y)
0
>>> A = N.orientnew('A', 'Axis', [q1, N.x])
>>> dot(N.y, A.y)
cos(q1)
"""
from sympy.physics.vector.dyadic import Dyadic
if isinstance(other, Dyadic):
return NotImplemented
other = _check_vector(other)
out = S.Zero
for i, v1 in enumerate(self.args):
for j, v2 in enumerate(other.args):
out += ((v2[0].T)
* (v2[1].dcm(v1[1]))
* (v1[0]))[0]
if Vector.simp:
return trigsimp(sympify(out), recursive=True)
else:
return sympify(out)
def __div__(self, other):
"""This uses mul and inputs self and 1 divided by other. """
return self.__mul__(sympify(1) / other)
__truediv__ = __div__
def __eq__(self, other):
"""Tests for equality.
It is very import to note that this is only as good as the SymPy
equality test; False does not always mean they are not equivalent
Vectors.
If other is 0, and self is empty, returns True.
If other is 0 and self is not empty, returns False.
If none of the above, only accepts other as a Vector.
"""
if other == 0:
other = Vector(0)
try:
other = _check_vector(other)
except TypeError:
return False
if (self.args == []) and (other.args == []):
return True
elif (self.args == []) or (other.args == []):
return False
frame = self.args[0][1]
for v in frame:
if expand((self - other) & v) != 0:
return False
return True
def __mul__(self, other):
"""Multiplies the Vector by a sympifyable expression.
Parameters
==========
other : Sympifyable
The scalar to multiply this Vector with
Examples
========
>>> from sympy.physics.vector import ReferenceFrame
>>> from sympy import Symbol
>>> N = ReferenceFrame('N')
>>> b = Symbol('b')
>>> V = 10 * b * N.x
>>> print(V)
10*b*N.x
"""
newlist = [v for v in self.args]
for i, v in enumerate(newlist):
newlist[i] = (sympify(other) * newlist[i][0], newlist[i][1])
return Vector(newlist)
def __ne__(self, other):
return not self == other
def __neg__(self):
return self * -1
def __or__(self, other):
"""Outer product between two Vectors.
A rank increasing operation, which returns a Dyadic from two Vectors
Parameters
==========
other : Vector
The Vector to take the outer product with
Examples
========
>>> from sympy.physics.vector import ReferenceFrame, outer
>>> N = ReferenceFrame('N')
>>> outer(N.x, N.x)
(N.x|N.x)
"""
from sympy.physics.vector.dyadic import Dyadic
other = _check_vector(other)
ol = Dyadic(0)
for i, v in enumerate(self.args):
for i2, v2 in enumerate(other.args):
# it looks this way because if we are in the same frame and
# use the enumerate function on the same frame in a nested
# fashion, then bad things happen
ol += Dyadic([(v[0][0] * v2[0][0], v[1].x, v2[1].x)])
ol += Dyadic([(v[0][0] * v2[0][1], v[1].x, v2[1].y)])
ol += Dyadic([(v[0][0] * v2[0][2], v[1].x, v2[1].z)])
ol += Dyadic([(v[0][1] * v2[0][0], v[1].y, v2[1].x)])
ol += Dyadic([(v[0][1] * v2[0][1], v[1].y, v2[1].y)])
ol += Dyadic([(v[0][1] * v2[0][2], v[1].y, v2[1].z)])
ol += Dyadic([(v[0][2] * v2[0][0], v[1].z, v2[1].x)])
ol += Dyadic([(v[0][2] * v2[0][1], v[1].z, v2[1].y)])
ol += Dyadic([(v[0][2] * v2[0][2], v[1].z, v2[1].z)])
return ol
def _latex(self, printer=None):
"""Latex Printing method. """
from sympy.physics.vector.printing import VectorLatexPrinter
ar = self.args # just to shorten things
if len(ar) == 0:
return str(0)
ol = [] # output list, to be concatenated to a string
for i, v in enumerate(ar):
for j in 0, 1, 2:
# if the coef of the basis vector is 1, we skip the 1
if ar[i][0][j] == 1:
ol.append(' + ' + ar[i][1].latex_vecs[j])
# if the coef of the basis vector is -1, we skip the 1
elif ar[i][0][j] == -1:
ol.append(' - ' + ar[i][1].latex_vecs[j])
elif ar[i][0][j] != 0:
# If the coefficient of the basis vector is not 1 or -1;
# also, we might wrap it in parentheses, for readability.
arg_str = VectorLatexPrinter().doprint(ar[i][0][j])
if isinstance(ar[i][0][j], Add):
arg_str = "(%s)" % arg_str
if arg_str[0] == '-':
arg_str = arg_str[1:]
str_start = ' - '
else:
str_start = ' + '
ol.append(str_start + arg_str + ar[i][1].latex_vecs[j])
outstr = ''.join(ol)
if outstr.startswith(' + '):
outstr = outstr[3:]
elif outstr.startswith(' '):
outstr = outstr[1:]
return outstr
def _pretty(self, printer=None):
"""Pretty Printing method. """
from sympy.physics.vector.printing import VectorPrettyPrinter
from sympy.printing.pretty.stringpict import prettyForm
e = self
class Fake(object):
def render(self, *args, **kwargs):
ar = e.args # just to shorten things
if len(ar) == 0:
return unicode(0)
settings = printer._settings if printer else {}
vp = printer if printer else VectorPrettyPrinter(settings)
pforms = [] # output list, to be concatenated to a string
for i, v in enumerate(ar):
for j in 0, 1, 2:
# if the coef of the basis vector is 1, we skip the 1
if ar[i][0][j] == 1:
pform = vp._print(ar[i][1].pretty_vecs[j])
# if the coef of the basis vector is -1, we skip the 1
elif ar[i][0][j] == -1:
pform = vp._print(ar[i][1].pretty_vecs[j])
pform = prettyForm(*pform.left(" - "))
bin = prettyForm.NEG
pform = prettyForm(binding=bin, *pform)
elif ar[i][0][j] != 0:
# If the basis vector coeff is not 1 or -1,
# we might wrap it in parentheses, for readability.
pform = vp._print(ar[i][0][j])
if isinstance(ar[i][0][j], Add):
tmp = pform.parens()
pform = prettyForm(tmp[0], tmp[1])
pform = prettyForm(*pform.right(" ",
ar[i][1].pretty_vecs[j]))
else:
continue
pforms.append(pform)
pform = prettyForm.__add__(*pforms)
kwargs["wrap_line"] = kwargs.get("wrap_line")
kwargs["num_columns"] = kwargs.get("num_columns")
out_str = pform.render(*args, **kwargs)
mlines = [line.rstrip() for line in out_str.split("\n")]
return "\n".join(mlines)
return Fake()
def __ror__(self, other):
"""Outer product between two Vectors.
A rank increasing operation, which returns a Dyadic from two Vectors
Parameters
==========
other : Vector
The Vector to take the outer product with
Examples
========
>>> from sympy.physics.vector import ReferenceFrame, outer
>>> N = ReferenceFrame('N')
>>> outer(N.x, N.x)
(N.x|N.x)
"""
from sympy.physics.vector.dyadic import Dyadic
other = _check_vector(other)
ol = Dyadic(0)
for i, v in enumerate(other.args):
for i2, v2 in enumerate(self.args):
# it looks this way because if we are in the same frame and
# use the enumerate function on the same frame in a nested
# fashion, then bad things happen
ol += Dyadic([(v[0][0] * v2[0][0], v[1].x, v2[1].x)])
ol += Dyadic([(v[0][0] * v2[0][1], v[1].x, v2[1].y)])
ol += Dyadic([(v[0][0] * v2[0][2], v[1].x, v2[1].z)])
ol += Dyadic([(v[0][1] * v2[0][0], v[1].y, v2[1].x)])
ol += Dyadic([(v[0][1] * v2[0][1], v[1].y, v2[1].y)])
ol += Dyadic([(v[0][1] * v2[0][2], v[1].y, v2[1].z)])
ol += Dyadic([(v[0][2] * v2[0][0], v[1].z, v2[1].x)])
ol += Dyadic([(v[0][2] * v2[0][1], v[1].z, v2[1].y)])
ol += Dyadic([(v[0][2] * v2[0][2], v[1].z, v2[1].z)])
return ol
def __rsub__(self, other):
return (-1 * self) + other
def __str__(self, printer=None, order=True):
"""Printing method. """
from sympy.physics.vector.printing import VectorStrPrinter
if not order or len(self.args) == 1:
ar = list(self.args)
elif len(self.args) == 0:
return str(0)
else:
d = {v[1]: v[0] for v in self.args}
keys = sorted(d.keys(), key=lambda x: x.index)
ar = []
for key in keys:
ar.append((d[key], key))
ol = [] # output list, to be concatenated to a string
for i, v in enumerate(ar):
for j in 0, 1, 2:
# if the coef of the basis vector is 1, we skip the 1
if ar[i][0][j] == 1:
ol.append(' + ' + ar[i][1].str_vecs[j])
# if the coef of the basis vector is -1, we skip the 1
elif ar[i][0][j] == -1:
ol.append(' - ' + ar[i][1].str_vecs[j])
elif ar[i][0][j] != 0:
# If the coefficient of the basis vector is not 1 or -1;
# also, we might wrap it in parentheses, for readability.
arg_str = VectorStrPrinter().doprint(ar[i][0][j])
if isinstance(ar[i][0][j], Add):
arg_str = "(%s)" % arg_str
if arg_str[0] == '-':
arg_str = arg_str[1:]
str_start = ' - '
else:
str_start = ' + '
ol.append(str_start + arg_str + '*' + ar[i][1].str_vecs[j])
outstr = ''.join(ol)
if outstr.startswith(' + '):
outstr = outstr[3:]
elif outstr.startswith(' '):
outstr = outstr[1:]
return outstr
def __sub__(self, other):
"""The subtraction operator. """
return self.__add__(other * -1)
def __xor__(self, other):
"""The cross product operator for two Vectors.
Returns a Vector, expressed in the same ReferenceFrames as self.
Parameters
==========
other : Vector
The Vector which we are crossing with
Examples
========
>>> from sympy.physics.vector import ReferenceFrame
>>> from sympy import symbols
>>> q1 = symbols('q1')
>>> N = ReferenceFrame('N')
>>> N.x ^ N.y
N.z
>>> A = N.orientnew('A', 'Axis', [q1, N.x])
>>> A.x ^ N.y
N.z
>>> N.y ^ A.x
- sin(q1)*A.y - cos(q1)*A.z
"""
from sympy.physics.vector.dyadic import Dyadic
if isinstance(other, Dyadic):
return NotImplemented
other = _check_vector(other)
if other.args == []:
return Vector(0)
def _det(mat):
"""This is needed as a little method for to find the determinant
of a list in python; needs to work for a 3x3 list.
SymPy's Matrix won't take in Vector, so need a custom function.
You shouldn't be calling this.
"""
return (mat[0][0] * (mat[1][1] * mat[2][2] - mat[1][2] * mat[2][1])
+ mat[0][1] * (mat[1][2] * mat[2][0] - mat[1][0] *
mat[2][2]) + mat[0][2] * (mat[1][0] * mat[2][1] -
mat[1][1] * mat[2][0]))
outlist = []
ar = other.args # For brevity
for i, v in enumerate(ar):
tempx = v[1].x
tempy = v[1].y
tempz = v[1].z
tempm = ([[tempx, tempy, tempz], [self & tempx, self & tempy,
self & tempz], [Vector([ar[i]]) & tempx,
Vector([ar[i]]) & tempy, Vector([ar[i]]) & tempz]])
outlist += _det(tempm).args
return Vector(outlist)
# We don't define _repr_png_ here because it would add a large amount of
# data to any notebook containing SymPy expressions, without adding
# anything useful to the notebook. It can still enabled manually, e.g.,
# for the qtconsole, with init_printing().
def _repr_latex_(self):
"""
IPython/Jupyter LaTeX printing
To change the behavior of this (e.g., pass in some settings to LaTeX),
use init_printing(). init_printing() will also enable LaTeX printing
for built in numeric types like ints and container types that contain
SymPy objects, like lists and dictionaries of expressions.
"""
from sympy.printing.latex import latex
s = latex(self, mode='plain')
return "$\\displaystyle %s$" % s
_repr_latex_orig = _repr_latex_
_sympystr = __str__
_sympyrepr = _sympystr
__repr__ = __str__
__radd__ = __add__
__rand__ = __and__
__rmul__ = __mul__
def separate(self):
"""
The constituents of this vector in different reference frames,
as per its definition.
Returns a dict mapping each ReferenceFrame to the corresponding
constituent Vector.
Examples
========
>>> from sympy.physics.vector import ReferenceFrame
>>> R1 = ReferenceFrame('R1')
>>> R2 = ReferenceFrame('R2')
>>> v = R1.x + R2.x
>>> v.separate() == {R1: R1.x, R2: R2.x}
True
"""
components = {}
for x in self.args:
components[x[1]] = Vector([x])
return components
def dot(self, other):
return self & other
dot.__doc__ = __and__.__doc__
def cross(self, other):
return self ^ other
cross.__doc__ = __xor__.__doc__
def outer(self, other):
return self | other
outer.__doc__ = __or__.__doc__
def diff(self, var, frame, var_in_dcm=True):
"""Returns the partial derivative of the vector with respect to a
variable in the provided reference frame.
Parameters
==========
var : Symbol
What the partial derivative is taken with respect to.
frame : ReferenceFrame
The reference frame that the partial derivative is taken in.
var_in_dcm : boolean
If true, the differentiation algorithm assumes that the variable
may be present in any of the direction cosine matrices that relate
the frame to the frames of any component of the vector. But if it
is known that the variable is not present in the direction cosine
matrices, false can be set to skip full reexpression in the desired
frame.
Examples
========
>>> from sympy import Symbol
>>> from sympy.physics.vector import dynamicsymbols, ReferenceFrame
>>> from sympy.physics.vector import Vector
>>> Vector.simp = True
>>> t = Symbol('t')
>>> q1 = dynamicsymbols('q1')
>>> N = ReferenceFrame('N')
>>> A = N.orientnew('A', 'Axis', [q1, N.y])
>>> A.x.diff(t, N)
- q1'*A.z
>>> B = ReferenceFrame('B')
>>> u1, u2 = dynamicsymbols('u1, u2')
>>> v = u1 * A.x + u2 * B.y
>>> v.diff(u2, N, var_in_dcm=False)
B.y
"""
from sympy.physics.vector.frame import _check_frame
var = sympify(var)
_check_frame(frame)
inlist = []
for vector_component in self.args:
measure_number = vector_component[0]
component_frame = vector_component[1]
if component_frame == frame:
inlist += [(measure_number.diff(var), frame)]
else:
# If the direction cosine matrix relating the component frame
# with the derivative frame does not contain the variable.
if not var_in_dcm or (frame.dcm(component_frame).diff(var) ==
zeros(3, 3)):
inlist += [(measure_number.diff(var),
component_frame)]
else: # else express in the frame
reexp_vec_comp = Vector([vector_component]).express(frame)
deriv = reexp_vec_comp.args[0][0].diff(var)
inlist += Vector([(deriv, frame)]).express(component_frame).args
return Vector(inlist)
def express(self, otherframe, variables=False):
"""
Returns a Vector equivalent to this one, expressed in otherframe.
Uses the global express method.
Parameters
==========
otherframe : ReferenceFrame
The frame for this Vector to be described in
variables : boolean
If True, the coordinate symbols(if present) in this Vector
are re-expressed in terms otherframe
Examples
========
>>> from sympy.physics.vector import ReferenceFrame, dynamicsymbols
>>> q1 = dynamicsymbols('q1')
>>> N = ReferenceFrame('N')
>>> A = N.orientnew('A', 'Axis', [q1, N.y])
>>> A.x.express(N)
cos(q1)*N.x - sin(q1)*N.z
"""
from sympy.physics.vector import express
return express(self, otherframe, variables=variables)
def to_matrix(self, reference_frame):
"""Returns the matrix form of the vector with respect to the given
frame.
Parameters
----------
reference_frame : ReferenceFrame
The reference frame that the rows of the matrix correspond to.
Returns
-------
matrix : ImmutableMatrix, shape(3,1)
The matrix that gives the 1D vector.
Examples
========
>>> from sympy import symbols
>>> from sympy.physics.vector import ReferenceFrame
>>> a, b, c = symbols('a, b, c')
>>> N = ReferenceFrame('N')
>>> vector = a * N.x + b * N.y + c * N.z
>>> vector.to_matrix(N)
Matrix([
[a],
[b],
[c]])
>>> beta = symbols('beta')
>>> A = N.orientnew('A', 'Axis', (beta, N.x))
>>> vector.to_matrix(A)
Matrix([
[ a],
[ b*cos(beta) + c*sin(beta)],
[-b*sin(beta) + c*cos(beta)]])
"""
return Matrix([self.dot(unit_vec) for unit_vec in
reference_frame]).reshape(3, 1)
def doit(self, **hints):
"""Calls .doit() on each term in the Vector"""
d = {}
for v in self.args:
d[v[1]] = v[0].applyfunc(lambda x: x.doit(**hints))
return Vector(d)
def dt(self, otherframe):
"""
Returns a Vector which is the time derivative of
the self Vector, taken in frame otherframe.
Calls the global time_derivative method
Parameters
==========
otherframe : ReferenceFrame
The frame to calculate the time derivative in
"""
from sympy.physics.vector import time_derivative
return time_derivative(self, otherframe)
def simplify(self):
"""Returns a simplified Vector."""
d = {}
for v in self.args:
d[v[1]] = v[0].simplify()
return Vector(d)
def subs(self, *args, **kwargs):
"""Substitution on the Vector.
Examples
========
>>> from sympy.physics.vector import ReferenceFrame
>>> from sympy import Symbol
>>> N = ReferenceFrame('N')
>>> s = Symbol('s')
>>> a = N.x * s
>>> a.subs({s: 2})
2*N.x
"""
d = {}
for v in self.args:
d[v[1]] = v[0].subs(*args, **kwargs)
return Vector(d)
def magnitude(self):
"""Returns the magnitude (Euclidean norm) of self."""
return sqrt(self & self)
def normalize(self):
"""Returns a Vector of magnitude 1, codirectional with self."""
return Vector(self.args + []) / self.magnitude()
def applyfunc(self, f):
"""Apply a function to each component of a vector."""
if not callable(f):
raise TypeError("`f` must be callable.")
d = {}
for v in self.args:
d[v[1]] = v[0].applyfunc(f)
return Vector(d)
def free_symbols(self, reference_frame):
"""
Returns the free symbols in the measure numbers of the vector
expressed in the given reference frame.
Parameter
=========
reference_frame : ReferenceFrame
The frame with respect to which the free symbols of the
given vector is to be determined.
"""
return self.to_matrix(reference_frame).free_symbols
class VectorTypeError(TypeError):
def __init__(self, other, want):
msg = filldedent("Expected an instance of %s, but received object "
"'%s' of %s." % (type(want), other, type(other)))
super(VectorTypeError, self).__init__(msg)
def _check_vector(other):
if not isinstance(other, Vector):
raise TypeError('A Vector must be supplied')
return other
|
f1400a214170f4e8e2f8fe5d0267f90ddfacbd750ccf982adf236c77dd01e85d | from __future__ import print_function, division
from .vector import Vector, _check_vector
from .frame import _check_frame
__all__ = ['Point']
class Point(object):
"""This object represents a point in a dynamic system.
It stores the: position, velocity, and acceleration of a point.
The position is a vector defined as the vector distance from a parent
point to this point.
Parameters
==========
name : string
The display name of the Point
Examples
========
>>> from sympy.physics.vector import Point, ReferenceFrame, dynamicsymbols
>>> N = ReferenceFrame('N')
>>> O = Point('O')
>>> P = Point('P')
>>> u1, u2, u3 = dynamicsymbols('u1 u2 u3')
>>> O.set_vel(N, u1 * N.x + u2 * N.y + u3 * N.z)
>>> O.acc(N)
u1'*N.x + u2'*N.y + u3'*N.z
symbols() can be used to create multiple Points in a single step, for example:
>>> from sympy.physics.vector import Point, ReferenceFrame, dynamicsymbols
>>> from sympy import symbols
>>> N = ReferenceFrame('N')
>>> u1, u2 = dynamicsymbols('u1 u2')
>>> A, B = symbols('A B', cls=Point)
>>> type(A)
<class 'sympy.physics.vector.point.Point'>
>>> A.set_vel(N, u1 * N.x + u2 * N.y)
>>> B.set_vel(N, u2 * N.x + u1 * N.y)
>>> A.acc(N) - B.acc(N)
(u1' - u2')*N.x + (-u1' + u2')*N.y
"""
def __init__(self, name):
"""Initialization of a Point object. """
self.name = name
self._pos_dict = {}
self._vel_dict = {}
self._acc_dict = {}
self._pdlist = [self._pos_dict, self._vel_dict, self._acc_dict]
def __str__(self):
return self.name
__repr__ = __str__
def _check_point(self, other):
if not isinstance(other, Point):
raise TypeError('A Point must be supplied')
def _pdict_list(self, other, num):
"""Creates a list from self to other using _dcm_dict. """
outlist = [[self]]
oldlist = [[]]
while outlist != oldlist:
oldlist = outlist[:]
for i, v in enumerate(outlist):
templist = v[-1]._pdlist[num].keys()
for i2, v2 in enumerate(templist):
if not v.__contains__(v2):
littletemplist = v + [v2]
if not outlist.__contains__(littletemplist):
outlist.append(littletemplist)
for i, v in enumerate(oldlist):
if v[-1] != other:
outlist.remove(v)
outlist.sort(key=len)
if len(outlist) != 0:
return outlist[0]
raise ValueError('No Connecting Path found between ' + other.name +
' and ' + self.name)
def a1pt_theory(self, otherpoint, outframe, interframe):
"""Sets the acceleration of this point with the 1-point theory.
The 1-point theory for point acceleration looks like this:
^N a^P = ^B a^P + ^N a^O + ^N alpha^B x r^OP + ^N omega^B x (^N omega^B
x r^OP) + 2 ^N omega^B x ^B v^P
where O is a point fixed in B, P is a point moving in B, and B is
rotating in frame N.
Parameters
==========
otherpoint : Point
The first point of the 1-point theory (O)
outframe : ReferenceFrame
The frame we want this point's acceleration defined in (N)
fixedframe : ReferenceFrame
The intermediate frame in this calculation (B)
Examples
========
>>> from sympy.physics.vector import Point, ReferenceFrame
>>> from sympy.physics.vector import dynamicsymbols
>>> q = dynamicsymbols('q')
>>> q2 = dynamicsymbols('q2')
>>> qd = dynamicsymbols('q', 1)
>>> q2d = dynamicsymbols('q2', 1)
>>> N = ReferenceFrame('N')
>>> B = ReferenceFrame('B')
>>> B.set_ang_vel(N, 5 * B.y)
>>> O = Point('O')
>>> P = O.locatenew('P', q * B.x)
>>> P.set_vel(B, qd * B.x + q2d * B.y)
>>> O.set_vel(N, 0)
>>> P.a1pt_theory(O, N, B)
(-25*q + q'')*B.x + q2''*B.y - 10*q'*B.z
"""
_check_frame(outframe)
_check_frame(interframe)
self._check_point(otherpoint)
dist = self.pos_from(otherpoint)
v = self.vel(interframe)
a1 = otherpoint.acc(outframe)
a2 = self.acc(interframe)
omega = interframe.ang_vel_in(outframe)
alpha = interframe.ang_acc_in(outframe)
self.set_acc(outframe, a2 + 2 * (omega ^ v) + a1 + (alpha ^ dist) +
(omega ^ (omega ^ dist)))
return self.acc(outframe)
def a2pt_theory(self, otherpoint, outframe, fixedframe):
"""Sets the acceleration of this point with the 2-point theory.
The 2-point theory for point acceleration looks like this:
^N a^P = ^N a^O + ^N alpha^B x r^OP + ^N omega^B x (^N omega^B x r^OP)
where O and P are both points fixed in frame B, which is rotating in
frame N.
Parameters
==========
otherpoint : Point
The first point of the 2-point theory (O)
outframe : ReferenceFrame
The frame we want this point's acceleration defined in (N)
fixedframe : ReferenceFrame
The frame in which both points are fixed (B)
Examples
========
>>> from sympy.physics.vector import Point, ReferenceFrame, dynamicsymbols
>>> q = dynamicsymbols('q')
>>> qd = dynamicsymbols('q', 1)
>>> N = ReferenceFrame('N')
>>> B = N.orientnew('B', 'Axis', [q, N.z])
>>> O = Point('O')
>>> P = O.locatenew('P', 10 * B.x)
>>> O.set_vel(N, 5 * N.x)
>>> P.a2pt_theory(O, N, B)
- 10*q'**2*B.x + 10*q''*B.y
"""
_check_frame(outframe)
_check_frame(fixedframe)
self._check_point(otherpoint)
dist = self.pos_from(otherpoint)
a = otherpoint.acc(outframe)
omega = fixedframe.ang_vel_in(outframe)
alpha = fixedframe.ang_acc_in(outframe)
self.set_acc(outframe, a + (alpha ^ dist) + (omega ^ (omega ^ dist)))
return self.acc(outframe)
def acc(self, frame):
"""The acceleration Vector of this Point in a ReferenceFrame.
Parameters
==========
frame : ReferenceFrame
The frame in which the returned acceleration vector will be defined in
Examples
========
>>> from sympy.physics.vector import Point, ReferenceFrame
>>> N = ReferenceFrame('N')
>>> p1 = Point('p1')
>>> p1.set_acc(N, 10 * N.x)
>>> p1.acc(N)
10*N.x
"""
_check_frame(frame)
if not (frame in self._acc_dict):
if self._vel_dict[frame] != 0:
return (self._vel_dict[frame]).dt(frame)
else:
return Vector(0)
return self._acc_dict[frame]
def locatenew(self, name, value):
"""Creates a new point with a position defined from this point.
Parameters
==========
name : str
The name for the new point
value : Vector
The position of the new point relative to this point
Examples
========
>>> from sympy.physics.vector import ReferenceFrame, Point
>>> N = ReferenceFrame('N')
>>> P1 = Point('P1')
>>> P2 = P1.locatenew('P2', 10 * N.x)
"""
if not isinstance(name, str):
raise TypeError('Must supply a valid name')
if value == 0:
value = Vector(0)
value = _check_vector(value)
p = Point(name)
p.set_pos(self, value)
self.set_pos(p, -value)
return p
def pos_from(self, otherpoint):
"""Returns a Vector distance between this Point and the other Point.
Parameters
==========
otherpoint : Point
The otherpoint we are locating this one relative to
Examples
========
>>> from sympy.physics.vector import Point, ReferenceFrame
>>> N = ReferenceFrame('N')
>>> p1 = Point('p1')
>>> p2 = Point('p2')
>>> p1.set_pos(p2, 10 * N.x)
>>> p1.pos_from(p2)
10*N.x
"""
outvec = Vector(0)
plist = self._pdict_list(otherpoint, 0)
for i in range(len(plist) - 1):
outvec += plist[i]._pos_dict[plist[i + 1]]
return outvec
def set_acc(self, frame, value):
"""Used to set the acceleration of this Point in a ReferenceFrame.
Parameters
==========
frame : ReferenceFrame
The frame in which this point's acceleration is defined
value : Vector
The vector value of this point's acceleration in the frame
Examples
========
>>> from sympy.physics.vector import Point, ReferenceFrame
>>> N = ReferenceFrame('N')
>>> p1 = Point('p1')
>>> p1.set_acc(N, 10 * N.x)
>>> p1.acc(N)
10*N.x
"""
if value == 0:
value = Vector(0)
value = _check_vector(value)
_check_frame(frame)
self._acc_dict.update({frame: value})
def set_pos(self, otherpoint, value):
"""Used to set the position of this point w.r.t. another point.
Parameters
==========
otherpoint : Point
The other point which this point's location is defined relative to
value : Vector
The vector which defines the location of this point
Examples
========
>>> from sympy.physics.vector import Point, ReferenceFrame
>>> N = ReferenceFrame('N')
>>> p1 = Point('p1')
>>> p2 = Point('p2')
>>> p1.set_pos(p2, 10 * N.x)
>>> p1.pos_from(p2)
10*N.x
"""
if value == 0:
value = Vector(0)
value = _check_vector(value)
self._check_point(otherpoint)
self._pos_dict.update({otherpoint: value})
otherpoint._pos_dict.update({self: -value})
def set_vel(self, frame, value):
"""Sets the velocity Vector of this Point in a ReferenceFrame.
Parameters
==========
frame : ReferenceFrame
The frame in which this point's velocity is defined
value : Vector
The vector value of this point's velocity in the frame
Examples
========
>>> from sympy.physics.vector import Point, ReferenceFrame
>>> N = ReferenceFrame('N')
>>> p1 = Point('p1')
>>> p1.set_vel(N, 10 * N.x)
>>> p1.vel(N)
10*N.x
"""
if value == 0:
value = Vector(0)
value = _check_vector(value)
_check_frame(frame)
self._vel_dict.update({frame: value})
def v1pt_theory(self, otherpoint, outframe, interframe):
"""Sets the velocity of this point with the 1-point theory.
The 1-point theory for point velocity looks like this:
^N v^P = ^B v^P + ^N v^O + ^N omega^B x r^OP
where O is a point fixed in B, P is a point moving in B, and B is
rotating in frame N.
Parameters
==========
otherpoint : Point
The first point of the 2-point theory (O)
outframe : ReferenceFrame
The frame we want this point's velocity defined in (N)
interframe : ReferenceFrame
The intermediate frame in this calculation (B)
Examples
========
>>> from sympy.physics.vector import Point, ReferenceFrame
>>> from sympy.physics.vector import dynamicsymbols
>>> q = dynamicsymbols('q')
>>> q2 = dynamicsymbols('q2')
>>> qd = dynamicsymbols('q', 1)
>>> q2d = dynamicsymbols('q2', 1)
>>> N = ReferenceFrame('N')
>>> B = ReferenceFrame('B')
>>> B.set_ang_vel(N, 5 * B.y)
>>> O = Point('O')
>>> P = O.locatenew('P', q * B.x)
>>> P.set_vel(B, qd * B.x + q2d * B.y)
>>> O.set_vel(N, 0)
>>> P.v1pt_theory(O, N, B)
q'*B.x + q2'*B.y - 5*q*B.z
"""
_check_frame(outframe)
_check_frame(interframe)
self._check_point(otherpoint)
dist = self.pos_from(otherpoint)
v1 = self.vel(interframe)
v2 = otherpoint.vel(outframe)
omega = interframe.ang_vel_in(outframe)
self.set_vel(outframe, v1 + v2 + (omega ^ dist))
return self.vel(outframe)
def v2pt_theory(self, otherpoint, outframe, fixedframe):
"""Sets the velocity of this point with the 2-point theory.
The 2-point theory for point velocity looks like this:
^N v^P = ^N v^O + ^N omega^B x r^OP
where O and P are both points fixed in frame B, which is rotating in
frame N.
Parameters
==========
otherpoint : Point
The first point of the 2-point theory (O)
outframe : ReferenceFrame
The frame we want this point's velocity defined in (N)
fixedframe : ReferenceFrame
The frame in which both points are fixed (B)
Examples
========
>>> from sympy.physics.vector import Point, ReferenceFrame, dynamicsymbols
>>> q = dynamicsymbols('q')
>>> qd = dynamicsymbols('q', 1)
>>> N = ReferenceFrame('N')
>>> B = N.orientnew('B', 'Axis', [q, N.z])
>>> O = Point('O')
>>> P = O.locatenew('P', 10 * B.x)
>>> O.set_vel(N, 5 * N.x)
>>> P.v2pt_theory(O, N, B)
5*N.x + 10*q'*B.y
"""
_check_frame(outframe)
_check_frame(fixedframe)
self._check_point(otherpoint)
dist = self.pos_from(otherpoint)
v = otherpoint.vel(outframe)
omega = fixedframe.ang_vel_in(outframe)
self.set_vel(outframe, v + (omega ^ dist))
return self.vel(outframe)
def vel(self, frame):
"""The velocity Vector of this Point in the ReferenceFrame.
Parameters
==========
frame : ReferenceFrame
The frame in which the returned velocity vector will be defined in
Examples
========
>>> from sympy.physics.vector import Point, ReferenceFrame
>>> N = ReferenceFrame('N')
>>> p1 = Point('p1')
>>> p1.set_vel(N, 10 * N.x)
>>> p1.vel(N)
10*N.x
"""
_check_frame(frame)
if not (frame in self._vel_dict):
raise ValueError('Velocity of point ' + self.name + ' has not been'
' defined in ReferenceFrame ' + frame.name)
return self._vel_dict[frame]
def partial_velocity(self, frame, *gen_speeds):
"""Returns the partial velocities of the linear velocity vector of this
point in the given frame with respect to one or more provided
generalized speeds.
Parameters
==========
frame : ReferenceFrame
The frame with which the velocity is defined in.
gen_speeds : functions of time
The generalized speeds.
Returns
=======
partial_velocities : tuple of Vector
The partial velocity vectors corresponding to the provided
generalized speeds.
Examples
========
>>> from sympy.physics.vector import ReferenceFrame, Point
>>> from sympy.physics.vector import dynamicsymbols
>>> N = ReferenceFrame('N')
>>> A = ReferenceFrame('A')
>>> p = Point('p')
>>> u1, u2 = dynamicsymbols('u1, u2')
>>> p.set_vel(N, u1 * N.x + u2 * A.y)
>>> p.partial_velocity(N, u1)
N.x
>>> p.partial_velocity(N, u1, u2)
(N.x, A.y)
"""
partials = [self.vel(frame).diff(speed, frame, var_in_dcm=False) for
speed in gen_speeds]
if len(partials) == 1:
return partials[0]
else:
return tuple(partials)
|
1aa3a406e00d86885acadd9a5468e2eb996681a84c72e0b38117d7db0df5433e | from sympy.core.backend import (diff, expand, sin, cos, sympify,
eye, symbols, ImmutableMatrix as Matrix, MatrixBase)
from sympy import (trigsimp, solve, Symbol, Dummy)
from sympy.physics.vector.vector import Vector, _check_vector
from sympy.utilities.misc import translate
__all__ = ['CoordinateSym', 'ReferenceFrame']
class CoordinateSym(Symbol):
"""
A coordinate symbol/base scalar associated wrt a Reference Frame.
Ideally, users should not instantiate this class. Instances of
this class must only be accessed through the corresponding frame
as 'frame[index]'.
CoordinateSyms having the same frame and index parameters are equal
(even though they may be instantiated separately).
Parameters
==========
name : string
The display name of the CoordinateSym
frame : ReferenceFrame
The reference frame this base scalar belongs to
index : 0, 1 or 2
The index of the dimension denoted by this coordinate variable
Examples
========
>>> from sympy.physics.vector import ReferenceFrame, CoordinateSym
>>> A = ReferenceFrame('A')
>>> A[1]
A_y
>>> type(A[0])
<class 'sympy.physics.vector.frame.CoordinateSym'>
>>> a_y = CoordinateSym('a_y', A, 1)
>>> a_y == A[1]
True
"""
def __new__(cls, name, frame, index):
# We can't use the cached Symbol.__new__ because this class depends on
# frame and index, which are not passed to Symbol.__xnew__.
assumptions = {}
super(CoordinateSym, cls)._sanitize(assumptions, cls)
obj = super(CoordinateSym, cls).__xnew__(cls, name, **assumptions)
_check_frame(frame)
if index not in range(0, 3):
raise ValueError("Invalid index specified")
obj._id = (frame, index)
return obj
@property
def frame(self):
return self._id[0]
def __eq__(self, other):
#Check if the other object is a CoordinateSym of the same frame
#and same index
if isinstance(other, CoordinateSym):
if other._id == self._id:
return True
return False
def __ne__(self, other):
return not self == other
def __hash__(self):
return tuple((self._id[0].__hash__(), self._id[1])).__hash__()
class ReferenceFrame(object):
"""A reference frame in classical mechanics.
ReferenceFrame is a class used to represent a reference frame in classical
mechanics. It has a standard basis of three unit vectors in the frame's
x, y, and z directions.
It also can have a rotation relative to a parent frame; this rotation is
defined by a direction cosine matrix relating this frame's basis vectors to
the parent frame's basis vectors. It can also have an angular velocity
vector, defined in another frame.
"""
_count = 0
def __init__(self, name, indices=None, latexs=None, variables=None):
"""ReferenceFrame initialization method.
A ReferenceFrame has a set of orthonormal basis vectors, along with
orientations relative to other ReferenceFrames and angular velocities
relative to other ReferenceFrames.
Parameters
==========
indices : tuple of str
Enables the reference frame's basis unit vectors to be accessed by
Python's square bracket indexing notation using the provided three
indice strings and alters the printing of the unit vectors to
reflect this choice.
latexs : tuple of str
Alters the LaTeX printing of the reference frame's basis unit
vectors to the provided three valid LaTeX strings.
Examples
========
>>> from sympy.physics.vector import ReferenceFrame, vlatex
>>> N = ReferenceFrame('N')
>>> N.x
N.x
>>> O = ReferenceFrame('O', indices=('1', '2', '3'))
>>> O.x
O['1']
>>> O['1']
O['1']
>>> P = ReferenceFrame('P', latexs=('A1', 'A2', 'A3'))
>>> vlatex(P.x)
'A1'
symbols() can be used to create multiple Reference Frames in one step, for example:
>>> from sympy.physics.vector import ReferenceFrame
>>> from sympy import symbols
>>> A, B, C = symbols('A B C', cls=ReferenceFrame)
>>> D, E = symbols('D E', cls=ReferenceFrame, indices=('1', '2', '3'))
>>> A[0]
A_x
>>> D.x
D['1']
>>> E.y
E['2']
>>> type(A) == type(D)
True
"""
if not isinstance(name, str):
raise TypeError('Need to supply a valid name')
# The if statements below are for custom printing of basis-vectors for
# each frame.
# First case, when custom indices are supplied
if indices is not None:
if not isinstance(indices, (tuple, list)):
raise TypeError('Supply the indices as a list')
if len(indices) != 3:
raise ValueError('Supply 3 indices')
for i in indices:
if not isinstance(i, str):
raise TypeError('Indices must be strings')
self.str_vecs = [(name + '[\'' + indices[0] + '\']'),
(name + '[\'' + indices[1] + '\']'),
(name + '[\'' + indices[2] + '\']')]
self.pretty_vecs = [(name.lower() + u"_" + indices[0]),
(name.lower() + u"_" + indices[1]),
(name.lower() + u"_" + indices[2])]
self.latex_vecs = [(r"\mathbf{\hat{%s}_{%s}}" % (name.lower(),
indices[0])), (r"\mathbf{\hat{%s}_{%s}}" %
(name.lower(), indices[1])),
(r"\mathbf{\hat{%s}_{%s}}" % (name.lower(),
indices[2]))]
self.indices = indices
# Second case, when no custom indices are supplied
else:
self.str_vecs = [(name + '.x'), (name + '.y'), (name + '.z')]
self.pretty_vecs = [name.lower() + u"_x",
name.lower() + u"_y",
name.lower() + u"_z"]
self.latex_vecs = [(r"\mathbf{\hat{%s}_x}" % name.lower()),
(r"\mathbf{\hat{%s}_y}" % name.lower()),
(r"\mathbf{\hat{%s}_z}" % name.lower())]
self.indices = ['x', 'y', 'z']
# Different step, for custom latex basis vectors
if latexs is not None:
if not isinstance(latexs, (tuple, list)):
raise TypeError('Supply the indices as a list')
if len(latexs) != 3:
raise ValueError('Supply 3 indices')
for i in latexs:
if not isinstance(i, str):
raise TypeError('Latex entries must be strings')
self.latex_vecs = latexs
self.name = name
self._var_dict = {}
#The _dcm_dict dictionary will only store the dcms of parent-child
#relationships. The _dcm_cache dictionary will work as the dcm
#cache.
self._dcm_dict = {}
self._dcm_cache = {}
self._ang_vel_dict = {}
self._ang_acc_dict = {}
self._dlist = [self._dcm_dict, self._ang_vel_dict, self._ang_acc_dict]
self._cur = 0
self._x = Vector([(Matrix([1, 0, 0]), self)])
self._y = Vector([(Matrix([0, 1, 0]), self)])
self._z = Vector([(Matrix([0, 0, 1]), self)])
#Associate coordinate symbols wrt this frame
if variables is not None:
if not isinstance(variables, (tuple, list)):
raise TypeError('Supply the variable names as a list/tuple')
if len(variables) != 3:
raise ValueError('Supply 3 variable names')
for i in variables:
if not isinstance(i, str):
raise TypeError('Variable names must be strings')
else:
variables = [name + '_x', name + '_y', name + '_z']
self.varlist = (CoordinateSym(variables[0], self, 0), \
CoordinateSym(variables[1], self, 1), \
CoordinateSym(variables[2], self, 2))
ReferenceFrame._count += 1
self.index = ReferenceFrame._count
def __getitem__(self, ind):
"""
Returns basis vector for the provided index, if the index is a string.
If the index is a number, returns the coordinate variable correspon-
-ding to that index.
"""
if not isinstance(ind, str):
if ind < 3:
return self.varlist[ind]
else:
raise ValueError("Invalid index provided")
if self.indices[0] == ind:
return self.x
if self.indices[1] == ind:
return self.y
if self.indices[2] == ind:
return self.z
else:
raise ValueError('Not a defined index')
def __iter__(self):
return iter([self.x, self.y, self.z])
def __str__(self):
"""Returns the name of the frame. """
return self.name
__repr__ = __str__
def _dict_list(self, other, num):
"""Creates a list from self to other using _dcm_dict. """
outlist = [[self]]
oldlist = [[]]
while outlist != oldlist:
oldlist = outlist[:]
for i, v in enumerate(outlist):
templist = v[-1]._dlist[num].keys()
for i2, v2 in enumerate(templist):
if not v.__contains__(v2):
littletemplist = v + [v2]
if not outlist.__contains__(littletemplist):
outlist.append(littletemplist)
for i, v in enumerate(oldlist):
if v[-1] != other:
outlist.remove(v)
outlist.sort(key=len)
if len(outlist) != 0:
return outlist[0]
raise ValueError('No Connecting Path found between ' + self.name +
' and ' + other.name)
def _w_diff_dcm(self, otherframe):
"""Angular velocity from time differentiating the DCM. """
from sympy.physics.vector.functions import dynamicsymbols
dcm2diff = otherframe.dcm(self)
diffed = dcm2diff.diff(dynamicsymbols._t)
angvelmat = diffed * dcm2diff.T
w1 = trigsimp(expand(angvelmat[7]), recursive=True)
w2 = trigsimp(expand(angvelmat[2]), recursive=True)
w3 = trigsimp(expand(angvelmat[3]), recursive=True)
return Vector([(Matrix([w1, w2, w3]), otherframe)])
def variable_map(self, otherframe):
"""
Returns a dictionary which expresses the coordinate variables
of this frame in terms of the variables of otherframe.
If Vector.simp is True, returns a simplified version of the mapped
values. Else, returns them without simplification.
Simplification of the expressions may take time.
Parameters
==========
otherframe : ReferenceFrame
The other frame to map the variables to
Examples
========
>>> from sympy.physics.vector import ReferenceFrame, dynamicsymbols
>>> A = ReferenceFrame('A')
>>> q = dynamicsymbols('q')
>>> B = A.orientnew('B', 'Axis', [q, A.z])
>>> A.variable_map(B)
{A_x: B_x*cos(q(t)) - B_y*sin(q(t)), A_y: B_x*sin(q(t)) + B_y*cos(q(t)), A_z: B_z}
"""
_check_frame(otherframe)
if (otherframe, Vector.simp) in self._var_dict:
return self._var_dict[(otherframe, Vector.simp)]
else:
vars_matrix = self.dcm(otherframe) * Matrix(otherframe.varlist)
mapping = {}
for i, x in enumerate(self):
if Vector.simp:
mapping[self.varlist[i]] = trigsimp(vars_matrix[i], method='fu')
else:
mapping[self.varlist[i]] = vars_matrix[i]
self._var_dict[(otherframe, Vector.simp)] = mapping
return mapping
def ang_acc_in(self, otherframe):
"""Returns the angular acceleration Vector of the ReferenceFrame.
Effectively returns the Vector:
^N alpha ^B
which represent the angular acceleration of B in N, where B is self, and
N is otherframe.
Parameters
==========
otherframe : ReferenceFrame
The ReferenceFrame which the angular acceleration is returned in.
Examples
========
>>> from sympy.physics.vector import ReferenceFrame
>>> N = ReferenceFrame('N')
>>> A = ReferenceFrame('A')
>>> V = 10 * N.x
>>> A.set_ang_acc(N, V)
>>> A.ang_acc_in(N)
10*N.x
"""
_check_frame(otherframe)
if otherframe in self._ang_acc_dict:
return self._ang_acc_dict[otherframe]
else:
return self.ang_vel_in(otherframe).dt(otherframe)
def ang_vel_in(self, otherframe):
"""Returns the angular velocity Vector of the ReferenceFrame.
Effectively returns the Vector:
^N omega ^B
which represent the angular velocity of B in N, where B is self, and
N is otherframe.
Parameters
==========
otherframe : ReferenceFrame
The ReferenceFrame which the angular velocity is returned in.
Examples
========
>>> from sympy.physics.vector import ReferenceFrame
>>> N = ReferenceFrame('N')
>>> A = ReferenceFrame('A')
>>> V = 10 * N.x
>>> A.set_ang_vel(N, V)
>>> A.ang_vel_in(N)
10*N.x
"""
_check_frame(otherframe)
flist = self._dict_list(otherframe, 1)
outvec = Vector(0)
for i in range(len(flist) - 1):
outvec += flist[i]._ang_vel_dict[flist[i + 1]]
return outvec
def dcm(self, otherframe):
r"""Returns the direction cosine matrix relative to the provided
reference frame.
The returned matrix can be used to express the orthogonal unit vectors
of this frame in terms of the orthogonal unit vectors of
``otherframe``.
Parameters
==========
otherframe : ReferenceFrame
The reference frame which the direction cosine matrix of this frame
is formed relative to.
Examples
========
The following example rotates the reference frame A relative to N by a
simple rotation and then calculates the direction cosine matrix of N
relative to A.
>>> from sympy import symbols, sin, cos
>>> from sympy.physics.vector import ReferenceFrame
>>> q1 = symbols('q1')
>>> N = ReferenceFrame('N')
>>> A = N.orientnew('A', 'Axis', (q1, N.x))
>>> N.dcm(A)
Matrix([
[1, 0, 0],
[0, cos(q1), -sin(q1)],
[0, sin(q1), cos(q1)]])
The second row of the above direction cosine matrix represents the
``N.y`` unit vector in N expressed in A. Like so:
>>> Ny = 0*A.x + cos(q1)*A.y - sin(q1)*A.z
Thus, expressing ``N.y`` in A should return the same result:
>>> N.y.express(A)
cos(q1)*A.y - sin(q1)*A.z
Notes
=====
It is import to know what form of the direction cosine matrix is
returned. If ``B.dcm(A)`` is called, it means the "direction cosine
matrix of B relative to A". This is the matrix :math:`{}^A\mathbf{R}^B`
shown in the following relationship:
.. math::
\begin{bmatrix}
\hat{\mathbf{b}}_1 \\
\hat{\mathbf{b}}_2 \\
\hat{\mathbf{b}}_3
\end{bmatrix}
=
{}^A\mathbf{R}^B
\begin{bmatrix}
\hat{\mathbf{a}}_1 \\
\hat{\mathbf{a}}_2 \\
\hat{\mathbf{a}}_3
\end{bmatrix}.
:math:`^{}A\mathbf{R}^B` is the matrix that expresses the B unit
vectors in terms of the A unit vectors.
"""
_check_frame(otherframe)
# Check if the dcm wrt that frame has already been calculated
if otherframe in self._dcm_cache:
return self._dcm_cache[otherframe]
flist = self._dict_list(otherframe, 0)
outdcm = eye(3)
for i in range(len(flist) - 1):
outdcm = outdcm * flist[i]._dcm_dict[flist[i + 1]]
# After calculation, store the dcm in dcm cache for faster future
# retrieval
self._dcm_cache[otherframe] = outdcm
otherframe._dcm_cache[self] = outdcm.T
return outdcm
def orient(self, parent, rot_type, amounts, rot_order=''):
"""Sets the orientation of this reference frame relative to another
(parent) reference frame.
Parameters
==========
parent : ReferenceFrame
Reference frame that this reference frame will be rotated relative
to.
rot_type : str
The method used to generate the direction cosine matrix. Supported
methods are:
- ``'Axis'``: simple rotations about a single common axis
- ``'DCM'``: for setting the direction cosine matrix directly
- ``'Body'``: three successive rotations about new intermediate
axes, also called "Euler and Tait-Bryan angles"
- ``'Space'``: three successive rotations about the parent
frames' unit vectors
- ``'Quaternion'``: rotations defined by four parameters which
result in a singularity free direction cosine matrix
amounts :
Expressions defining the rotation angles or direction cosine
matrix. These must match the ``rot_type``. See examples below for
details. The input types are:
- ``'Axis'``: 2-tuple (expr/sym/func, Vector)
- ``'DCM'``: Matrix, shape(3,3)
- ``'Body'``: 3-tuple of expressions, symbols, or functions
- ``'Space'``: 3-tuple of expressions, symbols, or functions
- ``'Quaternion'``: 4-tuple of expressions, symbols, or
functions
rot_order : str or int, optional
If applicable, the order of the successive of rotations. The string
``'123'`` and integer ``123`` are equivalent, for example. Required
for ``'Body'`` and ``'Space'``.
Examples
========
Setup variables for the examples:
>>> from sympy import symbols
>>> from sympy.physics.vector import ReferenceFrame
>>> q0, q1, q2, q3 = symbols('q0 q1 q2 q3')
>>> N = ReferenceFrame('N')
>>> B = ReferenceFrame('B')
>>> B1 = ReferenceFrame('B')
>>> B2 = ReferenceFrame('B2')
Axis
----
``rot_type='Axis'`` creates a direction cosine matrix defined by a
simple rotation about a single axis fixed in both reference frames.
This is a rotation about an arbitrary, non-time-varying
axis by some angle. The axis is supplied as a Vector. This is how
simple rotations are defined.
>>> B.orient(N, 'Axis', (q1, N.x))
The ``orient()`` method generates a direction cosine matrix and its
transpose which defines the orientation of B relative to N and vice
versa. Once orient is called, ``dcm()`` outputs the appropriate
direction cosine matrix.
>>> B.dcm(N)
Matrix([
[1, 0, 0],
[0, cos(q1), sin(q1)],
[0, -sin(q1), cos(q1)]])
The following two lines show how the sense of the rotation can be
defined. Both lines produce the same result.
>>> B.orient(N, 'Axis', (q1, -N.x))
>>> B.orient(N, 'Axis', (-q1, N.x))
The axis does not have to be defined by a unit vector, it can be any
vector in the parent frame.
>>> B.orient(N, 'Axis', (q1, N.x + 2 * N.y))
DCM
---
The direction cosine matrix can be set directly. The orientation of a
frame A can be set to be the same as the frame B above like so:
>>> B.orient(N, 'Axis', (q1, N.x))
>>> A = ReferenceFrame('A')
>>> A.orient(N, 'DCM', N.dcm(B))
>>> A.dcm(N)
Matrix([
[1, 0, 0],
[0, cos(q1), sin(q1)],
[0, -sin(q1), cos(q1)]])
**Note carefully that** ``N.dcm(B)`` **was passed into** ``orient()``
**for** ``A.dcm(N)`` **to match** ``B.dcm(N)``.
Body
----
``rot_type='Body'`` rotates this reference frame relative to the
provided reference frame by rotating through three successive simple
rotations. Each subsequent axis of rotation is about the "body fixed"
unit vectors of the new intermediate reference frame. This type of
rotation is also referred to rotating through the `Euler and Tait-Bryan
Angles <https://en.wikipedia.org/wiki/Euler_angles>`_.
For example, the classic Euler Angle rotation can be done by:
>>> B.orient(N, 'Body', (q1, q2, q3), 'XYX')
>>> B.dcm(N)
Matrix([
[ cos(q2), sin(q1)*sin(q2), -sin(q2)*cos(q1)],
[sin(q2)*sin(q3), -sin(q1)*sin(q3)*cos(q2) + cos(q1)*cos(q3), sin(q1)*cos(q3) + sin(q3)*cos(q1)*cos(q2)],
[sin(q2)*cos(q3), -sin(q1)*cos(q2)*cos(q3) - sin(q3)*cos(q1), -sin(q1)*sin(q3) + cos(q1)*cos(q2)*cos(q3)]])
This rotates B relative to N through ``q1`` about ``N.x``, then rotates
B again through q2 about B.y, and finally through q3 about B.x. It is
equivalent to:
>>> B1.orient(N, 'Axis', (q1, N.x))
>>> B2.orient(B1, 'Axis', (q2, B1.y))
>>> B.orient(B2, 'Axis', (q3, B2.x))
>>> B.dcm(N)
Matrix([
[ cos(q2), sin(q1)*sin(q2), -sin(q2)*cos(q1)],
[sin(q2)*sin(q3), -sin(q1)*sin(q3)*cos(q2) + cos(q1)*cos(q3), sin(q1)*cos(q3) + sin(q3)*cos(q1)*cos(q2)],
[sin(q2)*cos(q3), -sin(q1)*cos(q2)*cos(q3) - sin(q3)*cos(q1), -sin(q1)*sin(q3) + cos(q1)*cos(q2)*cos(q3)]])
Acceptable rotation orders are of length 3, expressed in as a string
``'XYZ'`` or ``'123'`` or integer ``123``. Rotations about an axis
twice in a row are prohibited.
>>> B.orient(N, 'Body', (q1, q2, 0), 'ZXZ')
>>> B.orient(N, 'Body', (q1, q2, 0), '121')
>>> B.orient(N, 'Body', (q1, q2, q3), 123)
Space
-----
``rot_type='Space'`` also rotates the reference frame in three
successive simple rotations but the axes of rotation are the
"Space-fixed" axes. For example:
>>> B.orient(N, 'Space', (q1, q2, q3), '312')
>>> B.dcm(N)
Matrix([
[ sin(q1)*sin(q2)*sin(q3) + cos(q1)*cos(q3), sin(q1)*cos(q2), sin(q1)*sin(q2)*cos(q3) - sin(q3)*cos(q1)],
[-sin(q1)*cos(q3) + sin(q2)*sin(q3)*cos(q1), cos(q1)*cos(q2), sin(q1)*sin(q3) + sin(q2)*cos(q1)*cos(q3)],
[ sin(q3)*cos(q2), -sin(q2), cos(q2)*cos(q3)]])
is equivalent to:
>>> B1.orient(N, 'Axis', (q1, N.z))
>>> B2.orient(B1, 'Axis', (q2, N.x))
>>> B.orient(B2, 'Axis', (q3, N.y))
>>> B.dcm(N).simplify() # doctest: +SKIP
Matrix([
[ sin(q1)*sin(q2)*sin(q3) + cos(q1)*cos(q3), sin(q1)*cos(q2), sin(q1)*sin(q2)*cos(q3) - sin(q3)*cos(q1)],
[-sin(q1)*cos(q3) + sin(q2)*sin(q3)*cos(q1), cos(q1)*cos(q2), sin(q1)*sin(q3) + sin(q2)*cos(q1)*cos(q3)],
[ sin(q3)*cos(q2), -sin(q2), cos(q2)*cos(q3)]])
It is worth noting that space-fixed and body-fixed rotations are
related by the order of the rotations, i.e. the reverse order of body
fixed will give space fixed and vice versa.
>>> B.orient(N, 'Space', (q1, q2, q3), '231')
>>> B.dcm(N)
Matrix([
[cos(q1)*cos(q2), sin(q1)*sin(q3) + sin(q2)*cos(q1)*cos(q3), -sin(q1)*cos(q3) + sin(q2)*sin(q3)*cos(q1)],
[ -sin(q2), cos(q2)*cos(q3), sin(q3)*cos(q2)],
[sin(q1)*cos(q2), sin(q1)*sin(q2)*cos(q3) - sin(q3)*cos(q1), sin(q1)*sin(q2)*sin(q3) + cos(q1)*cos(q3)]])
>>> B.orient(N, 'Body', (q3, q2, q1), '132')
>>> B.dcm(N)
Matrix([
[cos(q1)*cos(q2), sin(q1)*sin(q3) + sin(q2)*cos(q1)*cos(q3), -sin(q1)*cos(q3) + sin(q2)*sin(q3)*cos(q1)],
[ -sin(q2), cos(q2)*cos(q3), sin(q3)*cos(q2)],
[sin(q1)*cos(q2), sin(q1)*sin(q2)*cos(q3) - sin(q3)*cos(q1), sin(q1)*sin(q2)*sin(q3) + cos(q1)*cos(q3)]])
Quaternion
----------
``rot_type='Quaternion'`` orients the reference frame using
quaternions. Quaternion rotation is defined as a finite rotation about
lambda, a unit vector, by an amount theta. This orientation is
described by four parameters:
- ``q0 = cos(theta/2)``
- ``q1 = lambda_x sin(theta/2)``
- ``q2 = lambda_y sin(theta/2)``
- ``q3 = lambda_z sin(theta/2)``
This type does not need a ``rot_order``.
>>> B.orient(N, 'Quaternion', (q0, q1, q2, q3))
>>> B.dcm(N)
Matrix([
[q0**2 + q1**2 - q2**2 - q3**2, 2*q0*q3 + 2*q1*q2, -2*q0*q2 + 2*q1*q3],
[ -2*q0*q3 + 2*q1*q2, q0**2 - q1**2 + q2**2 - q3**2, 2*q0*q1 + 2*q2*q3],
[ 2*q0*q2 + 2*q1*q3, -2*q0*q1 + 2*q2*q3, q0**2 - q1**2 - q2**2 + q3**2]])
"""
from sympy.physics.vector.functions import dynamicsymbols
_check_frame(parent)
# Allow passing a rotation matrix manually.
if rot_type == 'DCM':
# When rot_type == 'DCM', then amounts must be a Matrix type object
# (e.g. sympy.matrices.dense.MutableDenseMatrix).
if not isinstance(amounts, MatrixBase):
raise TypeError("Amounts must be a sympy Matrix type object.")
else:
amounts = list(amounts)
for i, v in enumerate(amounts):
if not isinstance(v, Vector):
amounts[i] = sympify(v)
def _rot(axis, angle):
"""DCM for simple axis 1,2,or 3 rotations. """
if axis == 1:
return Matrix([[1, 0, 0],
[0, cos(angle), -sin(angle)],
[0, sin(angle), cos(angle)]])
elif axis == 2:
return Matrix([[cos(angle), 0, sin(angle)],
[0, 1, 0],
[-sin(angle), 0, cos(angle)]])
elif axis == 3:
return Matrix([[cos(angle), -sin(angle), 0],
[sin(angle), cos(angle), 0],
[0, 0, 1]])
approved_orders = ('123', '231', '312', '132', '213', '321', '121',
'131', '212', '232', '313', '323', '')
# make sure XYZ => 123 and rot_type is in upper case
rot_order = translate(str(rot_order), 'XYZxyz', '123123')
rot_type = rot_type.upper()
if rot_order not in approved_orders:
raise TypeError('The supplied order is not an approved type')
parent_orient = []
if rot_type == 'AXIS':
if not rot_order == '':
raise TypeError('Axis orientation takes no rotation order')
if not (isinstance(amounts, (list, tuple)) & (len(amounts) == 2)):
raise TypeError('Amounts are a list or tuple of length 2')
theta = amounts[0]
axis = amounts[1]
axis = _check_vector(axis)
if not axis.dt(parent) == 0:
raise ValueError('Axis cannot be time-varying')
axis = axis.express(parent).normalize()
axis = axis.args[0][0]
parent_orient = ((eye(3) - axis * axis.T) * cos(theta) +
Matrix([[0, -axis[2], axis[1]],
[axis[2], 0, -axis[0]],
[-axis[1], axis[0], 0]]) *
sin(theta) + axis * axis.T)
elif rot_type == 'QUATERNION':
if not rot_order == '':
raise TypeError(
'Quaternion orientation takes no rotation order')
if not (isinstance(amounts, (list, tuple)) & (len(amounts) == 4)):
raise TypeError('Amounts are a list or tuple of length 4')
q0, q1, q2, q3 = amounts
parent_orient = (Matrix([[q0**2 + q1**2 - q2**2 - q3**2,
2 * (q1 * q2 - q0 * q3),
2 * (q0 * q2 + q1 * q3)],
[2 * (q1 * q2 + q0 * q3),
q0**2 - q1**2 + q2**2 - q3**2,
2 * (q2 * q3 - q0 * q1)],
[2 * (q1 * q3 - q0 * q2),
2 * (q0 * q1 + q2 * q3),
q0**2 - q1**2 - q2**2 + q3**2]]))
elif rot_type == 'BODY':
if not (len(amounts) == 3 & len(rot_order) == 3):
raise TypeError('Body orientation takes 3 values & 3 orders')
a1 = int(rot_order[0])
a2 = int(rot_order[1])
a3 = int(rot_order[2])
parent_orient = (_rot(a1, amounts[0]) * _rot(a2, amounts[1]) *
_rot(a3, amounts[2]))
elif rot_type == 'SPACE':
if not (len(amounts) == 3 & len(rot_order) == 3):
raise TypeError('Space orientation takes 3 values & 3 orders')
a1 = int(rot_order[0])
a2 = int(rot_order[1])
a3 = int(rot_order[2])
parent_orient = (_rot(a3, amounts[2]) * _rot(a2, amounts[1]) *
_rot(a1, amounts[0]))
elif rot_type == 'DCM':
parent_orient = amounts
else:
raise NotImplementedError('That is not an implemented rotation')
# Reset the _dcm_cache of this frame, and remove it from the
# _dcm_caches of the frames it is linked to. Also remove it from the
# _dcm_dict of its parent
frames = self._dcm_cache.keys()
dcm_dict_del = []
dcm_cache_del = []
for frame in frames:
if frame in self._dcm_dict:
dcm_dict_del += [frame]
dcm_cache_del += [frame]
for frame in dcm_dict_del:
del frame._dcm_dict[self]
for frame in dcm_cache_del:
del frame._dcm_cache[self]
# Add the dcm relationship to _dcm_dict
self._dcm_dict = self._dlist[0] = {}
self._dcm_dict.update({parent: parent_orient.T})
parent._dcm_dict.update({self: parent_orient})
# Also update the dcm cache after resetting it
self._dcm_cache = {}
self._dcm_cache.update({parent: parent_orient.T})
parent._dcm_cache.update({self: parent_orient})
if rot_type == 'QUATERNION':
t = dynamicsymbols._t
q0, q1, q2, q3 = amounts
q0d = diff(q0, t)
q1d = diff(q1, t)
q2d = diff(q2, t)
q3d = diff(q3, t)
w1 = 2 * (q1d * q0 + q2d * q3 - q3d * q2 - q0d * q1)
w2 = 2 * (q2d * q0 + q3d * q1 - q1d * q3 - q0d * q2)
w3 = 2 * (q3d * q0 + q1d * q2 - q2d * q1 - q0d * q3)
wvec = Vector([(Matrix([w1, w2, w3]), self)])
elif rot_type == 'AXIS':
thetad = (amounts[0]).diff(dynamicsymbols._t)
wvec = thetad * amounts[1].express(parent).normalize()
elif rot_type == 'DCM':
wvec = self._w_diff_dcm(parent)
else:
try:
from sympy.polys.polyerrors import CoercionFailed
from sympy.physics.vector.functions import kinematic_equations
q1, q2, q3 = amounts
u1, u2, u3 = symbols('u1, u2, u3', cls=Dummy)
templist = kinematic_equations([u1, u2, u3], [q1, q2, q3],
rot_type, rot_order)
templist = [expand(i) for i in templist]
td = solve(templist, [u1, u2, u3])
u1 = expand(td[u1])
u2 = expand(td[u2])
u3 = expand(td[u3])
wvec = u1 * self.x + u2 * self.y + u3 * self.z
except (CoercionFailed, AssertionError):
wvec = self._w_diff_dcm(parent)
self._ang_vel_dict.update({parent: wvec})
parent._ang_vel_dict.update({self: -wvec})
self._var_dict = {}
def orientnew(self, newname, rot_type, amounts, rot_order='',
variables=None, indices=None, latexs=None):
r"""Returns a new reference frame oriented with respect to this
reference frame.
See ``ReferenceFrame.orient()`` for detailed examples of how to orient
reference frames.
Parameters
==========
newname : str
Name for the new reference frame.
rot_type : str
The method used to generate the direction cosine matrix. Supported
methods are:
- ``'Axis'``: simple rotations about a single common axis
- ``'DCM'``: for setting the direction cosine matrix directly
- ``'Body'``: three successive rotations about new intermediate
axes, also called "Euler and Tait-Bryan angles"
- ``'Space'``: three successive rotations about the parent
frames' unit vectors
- ``'Quaternion'``: rotations defined by four parameters which
result in a singularity free direction cosine matrix
amounts :
Expressions defining the rotation angles or direction cosine
matrix. These must match the ``rot_type``. See examples below for
details. The input types are:
- ``'Axis'``: 2-tuple (expr/sym/func, Vector)
- ``'DCM'``: Matrix, shape(3,3)
- ``'Body'``: 3-tuple of expressions, symbols, or functions
- ``'Space'``: 3-tuple of expressions, symbols, or functions
- ``'Quaternion'``: 4-tuple of expressions, symbols, or
functions
rot_order : str or int, optional
If applicable, the order of the successive of rotations. The string
``'123'`` and integer ``123`` are equivalent, for example. Required
for ``'Body'`` and ``'Space'``.
indices : tuple of str
Enables the reference frame's basis unit vectors to be accessed by
Python's square bracket indexing notation using the provided three
indice strings and alters the printing of the unit vectors to
reflect this choice.
latexs : tuple of str
Alters the LaTeX printing of the reference frame's basis unit
vectors to the provided three valid LaTeX strings.
Examples
========
>>> from sympy import symbols
>>> from sympy.physics.vector import ReferenceFrame, vlatex
>>> q0, q1, q2, q3 = symbols('q0 q1 q2 q3')
>>> N = ReferenceFrame('N')
Create a new reference frame A rotated relative to N through a simple
rotation.
>>> A = N.orientnew('A', 'Axis', (q0, N.x))
Create a new reference frame B rotated relative to N through body-fixed
rotations.
>>> B = N.orientnew('B', 'Body', (q1, q2, q3), '123')
Create a new reference frame C rotated relative to N through a simple
rotation with unique indices and LaTeX printing.
>>> C = N.orientnew('C', 'Axis', (q0, N.x), indices=('1', '2', '3'),
... latexs=(r'\hat{\mathbf{c}}_1',r'\hat{\mathbf{c}}_2',
... r'\hat{\mathbf{c}}_3'))
>>> C['1']
C['1']
>>> print(vlatex(C['1']))
\hat{\mathbf{c}}_1
"""
newframe = self.__class__(newname, variables=variables,
indices=indices, latexs=latexs)
newframe.orient(self, rot_type, amounts, rot_order)
return newframe
def set_ang_acc(self, otherframe, value):
"""Define the angular acceleration Vector in a ReferenceFrame.
Defines the angular acceleration of this ReferenceFrame, in another.
Angular acceleration can be defined with respect to multiple different
ReferenceFrames. Care must be taken to not create loops which are
inconsistent.
Parameters
==========
otherframe : ReferenceFrame
A ReferenceFrame to define the angular acceleration in
value : Vector
The Vector representing angular acceleration
Examples
========
>>> from sympy.physics.vector import ReferenceFrame
>>> N = ReferenceFrame('N')
>>> A = ReferenceFrame('A')
>>> V = 10 * N.x
>>> A.set_ang_acc(N, V)
>>> A.ang_acc_in(N)
10*N.x
"""
if value == 0:
value = Vector(0)
value = _check_vector(value)
_check_frame(otherframe)
self._ang_acc_dict.update({otherframe: value})
otherframe._ang_acc_dict.update({self: -value})
def set_ang_vel(self, otherframe, value):
"""Define the angular velocity vector in a ReferenceFrame.
Defines the angular velocity of this ReferenceFrame, in another.
Angular velocity can be defined with respect to multiple different
ReferenceFrames. Care must be taken to not create loops which are
inconsistent.
Parameters
==========
otherframe : ReferenceFrame
A ReferenceFrame to define the angular velocity in
value : Vector
The Vector representing angular velocity
Examples
========
>>> from sympy.physics.vector import ReferenceFrame
>>> N = ReferenceFrame('N')
>>> A = ReferenceFrame('A')
>>> V = 10 * N.x
>>> A.set_ang_vel(N, V)
>>> A.ang_vel_in(N)
10*N.x
"""
if value == 0:
value = Vector(0)
value = _check_vector(value)
_check_frame(otherframe)
self._ang_vel_dict.update({otherframe: value})
otherframe._ang_vel_dict.update({self: -value})
@property
def x(self):
"""The basis Vector for the ReferenceFrame, in the x direction. """
return self._x
@property
def y(self):
"""The basis Vector for the ReferenceFrame, in the y direction. """
return self._y
@property
def z(self):
"""The basis Vector for the ReferenceFrame, in the z direction. """
return self._z
def partial_velocity(self, frame, *gen_speeds):
"""Returns the partial angular velocities of this frame in the given
frame with respect to one or more provided generalized speeds.
Parameters
==========
frame : ReferenceFrame
The frame with which the angular velocity is defined in.
gen_speeds : functions of time
The generalized speeds.
Returns
=======
partial_velocities : tuple of Vector
The partial angular velocity vectors corresponding to the provided
generalized speeds.
Examples
========
>>> from sympy.physics.vector import ReferenceFrame, dynamicsymbols
>>> N = ReferenceFrame('N')
>>> A = ReferenceFrame('A')
>>> u1, u2 = dynamicsymbols('u1, u2')
>>> A.set_ang_vel(N, u1 * A.x + u2 * N.y)
>>> A.partial_velocity(N, u1)
A.x
>>> A.partial_velocity(N, u1, u2)
(A.x, N.y)
"""
partials = [self.ang_vel_in(frame).diff(speed, frame, var_in_dcm=False)
for speed in gen_speeds]
if len(partials) == 1:
return partials[0]
else:
return tuple(partials)
def _check_frame(other):
from .vector import VectorTypeError
if not isinstance(other, ReferenceFrame):
raise VectorTypeError(other, ReferenceFrame('A'))
|
cd5ab3364c9a4445cae97404fbb19136e25425d45210845f29b986bea1fe9535 | """
This module can be used to solve 2D beam bending problems with
singularity functions in mechanics.
"""
from __future__ import print_function, division
from sympy.core import S, Symbol, diff, symbols
from sympy.solvers import linsolve
from sympy.printing import sstr
from sympy.functions import SingularityFunction, Piecewise, factorial
from sympy.core import sympify
from sympy.integrals import integrate
from sympy.series import limit
from sympy.plotting import plot, PlotGrid
from sympy.geometry.entity import GeometryEntity
from sympy.external import import_module
from sympy import lambdify, Add
from sympy.core.compatibility import iterable
from sympy.utilities.decorator import doctest_depends_on
numpy = import_module('numpy', import_kwargs={'fromlist':['arange']})
class Beam(object):
"""
A Beam is a structural element that is capable of withstanding load
primarily by resisting against bending. Beams are characterized by
their cross sectional profile(Second moment of area), their length
and their material.
.. note::
While solving a beam bending problem, a user should choose its
own sign convention and should stick to it. The results will
automatically follow the chosen sign convention.
Examples
========
There is a beam of length 4 meters. A constant distributed load of 6 N/m
is applied from half of the beam till the end. There are two simple supports
below the beam, one at the starting point and another at the ending point
of the beam. The deflection of the beam at the end is restricted.
Using the sign convention of downwards forces being positive.
>>> from sympy.physics.continuum_mechanics.beam import Beam
>>> from sympy import symbols, Piecewise
>>> E, I = symbols('E, I')
>>> R1, R2 = symbols('R1, R2')
>>> b = Beam(4, E, I)
>>> b.apply_load(R1, 0, -1)
>>> b.apply_load(6, 2, 0)
>>> b.apply_load(R2, 4, -1)
>>> b.bc_deflection = [(0, 0), (4, 0)]
>>> b.boundary_conditions
{'deflection': [(0, 0), (4, 0)], 'slope': []}
>>> b.load
R1*SingularityFunction(x, 0, -1) + R2*SingularityFunction(x, 4, -1) + 6*SingularityFunction(x, 2, 0)
>>> b.solve_for_reaction_loads(R1, R2)
>>> b.load
-3*SingularityFunction(x, 0, -1) + 6*SingularityFunction(x, 2, 0) - 9*SingularityFunction(x, 4, -1)
>>> b.shear_force()
-3*SingularityFunction(x, 0, 0) + 6*SingularityFunction(x, 2, 1) - 9*SingularityFunction(x, 4, 0)
>>> b.bending_moment()
-3*SingularityFunction(x, 0, 1) + 3*SingularityFunction(x, 2, 2) - 9*SingularityFunction(x, 4, 1)
>>> b.slope()
(-3*SingularityFunction(x, 0, 2)/2 + SingularityFunction(x, 2, 3) - 9*SingularityFunction(x, 4, 2)/2 + 7)/(E*I)
>>> b.deflection()
(7*x - SingularityFunction(x, 0, 3)/2 + SingularityFunction(x, 2, 4)/4 - 3*SingularityFunction(x, 4, 3)/2)/(E*I)
>>> b.deflection().rewrite(Piecewise)
(7*x - Piecewise((x**3, x > 0), (0, True))/2
- 3*Piecewise(((x - 4)**3, x - 4 > 0), (0, True))/2
+ Piecewise(((x - 2)**4, x - 2 > 0), (0, True))/4)/(E*I)
"""
def __init__(self, length, elastic_modulus, second_moment, area=Symbol('A'), variable=Symbol('x'), base_char='C'):
"""Initializes the class.
Parameters
==========
length : Sympifyable
A Symbol or value representing the Beam's length.
elastic_modulus : Sympifyable
A SymPy expression representing the Beam's Modulus of Elasticity.
It is a measure of the stiffness of the Beam material. It can
also be a continuous function of position along the beam.
second_moment : Sympifyable or Geometry object
Describes the cross-section of the beam via a SymPy expression
representing the Beam's second moment of area. It is a geometrical
property of an area which reflects how its points are distributed
with respect to its neutral axis. It can also be a continuous
function of position along the beam. Alternatively ``second_moment``
can be a shape object such as a ``Polygon`` from the geometry module
representing the shape of the cross-section of the beam. In such cases,
it is assumed that the x-axis of the shape object is aligned with the
bending axis of the beam. The second moment of area will be computed
from the shape object internally.
area : Symbol/float
Represents the cross-section area of beam
variable : Symbol, optional
A Symbol object that will be used as the variable along the beam
while representing the load, shear, moment, slope and deflection
curve. By default, it is set to ``Symbol('x')``.
base_char : String, optional
A String that will be used as base character to generate sequential
symbols for integration constants in cases where boundary conditions
are not sufficient to solve them.
"""
self.length = length
self.elastic_modulus = elastic_modulus
if isinstance(second_moment, GeometryEntity):
self.cross_section = second_moment
else:
self.cross_section = None
self.second_moment = second_moment
self.variable = variable
self._base_char = base_char
self._boundary_conditions = {'deflection': [], 'slope': []}
self._load = 0
self._area = area
self._applied_supports = []
self._support_as_loads = []
self._applied_loads = []
self._reaction_loads = {}
self._composite_type = None
self._hinge_position = None
def __str__(self):
shape_description = self._cross_section if self._cross_section else self._second_moment
str_sol = 'Beam({}, {}, {})'.format(sstr(self._length), sstr(self._elastic_modulus), sstr(shape_description))
return str_sol
@property
def reaction_loads(self):
""" Returns the reaction forces in a dictionary."""
return self._reaction_loads
@property
def length(self):
"""Length of the Beam."""
return self._length
@length.setter
def length(self, l):
self._length = sympify(l)
@property
def area(self):
"""Cross-sectional area of the Beam. """
return self._area
@area.setter
def area(self, a):
self._area = sympify(a)
@property
def variable(self):
"""
A symbol that can be used as a variable along the length of the beam
while representing load distribution, shear force curve, bending
moment, slope curve and the deflection curve. By default, it is set
to ``Symbol('x')``, but this property is mutable.
Examples
========
>>> from sympy.physics.continuum_mechanics.beam import Beam
>>> from sympy import symbols
>>> E, I, A = symbols('E, I, A')
>>> x, y, z = symbols('x, y, z')
>>> b = Beam(4, E, I)
>>> b.variable
x
>>> b.variable = y
>>> b.variable
y
>>> b = Beam(4, E, I, A, z)
>>> b.variable
z
"""
return self._variable
@variable.setter
def variable(self, v):
if isinstance(v, Symbol):
self._variable = v
else:
raise TypeError("""The variable should be a Symbol object.""")
@property
def elastic_modulus(self):
"""Young's Modulus of the Beam. """
return self._elastic_modulus
@elastic_modulus.setter
def elastic_modulus(self, e):
self._elastic_modulus = sympify(e)
@property
def second_moment(self):
"""Second moment of area of the Beam. """
return self._second_moment
@second_moment.setter
def second_moment(self, i):
self._cross_section = None
if isinstance(i, GeometryEntity):
raise ValueError("To update cross-section geometry use `cross_section` attribute")
else:
self._second_moment = sympify(i)
@property
def cross_section(self):
"""Cross-section of the beam"""
return self._cross_section
@cross_section.setter
def cross_section(self, s):
if s:
self._second_moment = s.second_moment_of_area()[0]
self._cross_section = s
@property
def boundary_conditions(self):
"""
Returns a dictionary of boundary conditions applied on the beam.
The dictionary has three keywords namely moment, slope and deflection.
The value of each keyword is a list of tuple, where each tuple
contains location and value of a boundary condition in the format
(location, value).
Examples
========
There is a beam of length 4 meters. The bending moment at 0 should be 4
and at 4 it should be 0. The slope of the beam should be 1 at 0. The
deflection should be 2 at 0.
>>> from sympy.physics.continuum_mechanics.beam import Beam
>>> from sympy import symbols
>>> E, I = symbols('E, I')
>>> b = Beam(4, E, I)
>>> b.bc_deflection = [(0, 2)]
>>> b.bc_slope = [(0, 1)]
>>> b.boundary_conditions
{'deflection': [(0, 2)], 'slope': [(0, 1)]}
Here the deflection of the beam should be ``2`` at ``0``.
Similarly, the slope of the beam should be ``1`` at ``0``.
"""
return self._boundary_conditions
@property
def bc_slope(self):
return self._boundary_conditions['slope']
@bc_slope.setter
def bc_slope(self, s_bcs):
self._boundary_conditions['slope'] = s_bcs
@property
def bc_deflection(self):
return self._boundary_conditions['deflection']
@bc_deflection.setter
def bc_deflection(self, d_bcs):
self._boundary_conditions['deflection'] = d_bcs
def join(self, beam, via="fixed"):
"""
This method joins two beams to make a new composite beam system.
Passed Beam class instance is attached to the right end of calling
object. This method can be used to form beams having Discontinuous
values of Elastic modulus or Second moment.
Parameters
==========
beam : Beam class object
The Beam object which would be connected to the right of calling
object.
via : String
States the way two Beam object would get connected
- For axially fixed Beams, via="fixed"
- For Beams connected via hinge, via="hinge"
Examples
========
There is a cantilever beam of length 4 meters. For first 2 meters
its moment of inertia is `1.5*I` and `I` for the other end.
A pointload of magnitude 4 N is applied from the top at its free end.
>>> from sympy.physics.continuum_mechanics.beam import Beam
>>> from sympy import symbols
>>> E, I = symbols('E, I')
>>> R1, R2 = symbols('R1, R2')
>>> b1 = Beam(2, E, 1.5*I)
>>> b2 = Beam(2, E, I)
>>> b = b1.join(b2, "fixed")
>>> b.apply_load(20, 4, -1)
>>> b.apply_load(R1, 0, -1)
>>> b.apply_load(R2, 0, -2)
>>> b.bc_slope = [(0, 0)]
>>> b.bc_deflection = [(0, 0)]
>>> b.solve_for_reaction_loads(R1, R2)
>>> b.load
80*SingularityFunction(x, 0, -2) - 20*SingularityFunction(x, 0, -1) + 20*SingularityFunction(x, 4, -1)
>>> b.slope()
(((80*SingularityFunction(x, 0, 1) - 10*SingularityFunction(x, 0, 2) + 10*SingularityFunction(x, 4, 2))/I - 120/I)/E + 80.0/(E*I))*SingularityFunction(x, 2, 0)
+ 0.666666666666667*(80*SingularityFunction(x, 0, 1) - 10*SingularityFunction(x, 0, 2) + 10*SingularityFunction(x, 4, 2))*SingularityFunction(x, 0, 0)/(E*I)
- 0.666666666666667*(80*SingularityFunction(x, 0, 1) - 10*SingularityFunction(x, 0, 2) + 10*SingularityFunction(x, 4, 2))*SingularityFunction(x, 2, 0)/(E*I)
"""
x = self.variable
E = self.elastic_modulus
new_length = self.length + beam.length
if self.second_moment != beam.second_moment:
new_second_moment = Piecewise((self.second_moment, x<=self.length),
(beam.second_moment, x<=new_length))
else:
new_second_moment = self.second_moment
if via == "fixed":
new_beam = Beam(new_length, E, new_second_moment, x)
new_beam._composite_type = "fixed"
return new_beam
if via == "hinge":
new_beam = Beam(new_length, E, new_second_moment, x)
new_beam._composite_type = "hinge"
new_beam._hinge_position = self.length
return new_beam
def apply_support(self, loc, type="fixed"):
"""
This method applies support to a particular beam object.
Parameters
==========
loc : Sympifyable
Location of point at which support is applied.
type : String
Determines type of Beam support applied. To apply support structure
with
- zero degree of freedom, type = "fixed"
- one degree of freedom, type = "pin"
- two degrees of freedom, type = "roller"
Examples
========
There is a beam of length 30 meters. A moment of magnitude 120 Nm is
applied in the clockwise direction at the end of the beam. A pointload
of magnitude 8 N is applied from the top of the beam at the starting
point. There are two simple supports below the beam. One at the end
and another one at a distance of 10 meters from the start. The
deflection is restricted at both the supports.
Using the sign convention of upward forces and clockwise moment
being positive.
>>> from sympy.physics.continuum_mechanics.beam import Beam
>>> from sympy import symbols
>>> E, I = symbols('E, I')
>>> b = Beam(30, E, I)
>>> b.apply_support(10, 'roller')
>>> b.apply_support(30, 'roller')
>>> b.apply_load(-8, 0, -1)
>>> b.apply_load(120, 30, -2)
>>> R_10, R_30 = symbols('R_10, R_30')
>>> b.solve_for_reaction_loads(R_10, R_30)
>>> b.load
-8*SingularityFunction(x, 0, -1) + 6*SingularityFunction(x, 10, -1)
+ 120*SingularityFunction(x, 30, -2) + 2*SingularityFunction(x, 30, -1)
>>> b.slope()
(-4*SingularityFunction(x, 0, 2) + 3*SingularityFunction(x, 10, 2)
+ 120*SingularityFunction(x, 30, 1) + SingularityFunction(x, 30, 2) + 4000/3)/(E*I)
"""
loc = sympify(loc)
self._applied_supports.append((loc, type))
if type == "pin" or type == "roller":
reaction_load = Symbol('R_'+str(loc))
self.apply_load(reaction_load, loc, -1)
self.bc_deflection.append((loc, 0))
else:
reaction_load = Symbol('R_'+str(loc))
reaction_moment = Symbol('M_'+str(loc))
self.apply_load(reaction_load, loc, -1)
self.apply_load(reaction_moment, loc, -2)
self.bc_deflection.append((loc, 0))
self.bc_slope.append((loc, 0))
self._support_as_loads.append((reaction_moment, loc, -2, None))
self._support_as_loads.append((reaction_load, loc, -1, None))
def apply_load(self, value, start, order, end=None):
"""
This method adds up the loads given to a particular beam object.
Parameters
==========
value : Sympifyable
The value inserted should have the units [Force/(Distance**(n+1)]
where n is the order of applied load.
Units for applied loads:
- For moments, unit = kN*m
- For point loads, unit = kN
- For constant distributed load, unit = kN/m
- For ramp loads, unit = kN/m/m
- For parabolic ramp loads, unit = kN/m/m/m
- ... so on.
start : Sympifyable
The starting point of the applied load. For point moments and
point forces this is the location of application.
order : Integer
The order of the applied load.
- For moments, order = -2
- For point loads, order =-1
- For constant distributed load, order = 0
- For ramp loads, order = 1
- For parabolic ramp loads, order = 2
- ... so on.
end : Sympifyable, optional
An optional argument that can be used if the load has an end point
within the length of the beam.
Examples
========
There is a beam of length 4 meters. A moment of magnitude 3 Nm is
applied in the clockwise direction at the starting point of the beam.
A point load of magnitude 4 N is applied from the top of the beam at
2 meters from the starting point and a parabolic ramp load of magnitude
2 N/m is applied below the beam starting from 2 meters to 3 meters
away from the starting point of the beam.
>>> from sympy.physics.continuum_mechanics.beam import Beam
>>> from sympy import symbols
>>> E, I = symbols('E, I')
>>> b = Beam(4, E, I)
>>> b.apply_load(-3, 0, -2)
>>> b.apply_load(4, 2, -1)
>>> b.apply_load(-2, 2, 2, end=3)
>>> b.load
-3*SingularityFunction(x, 0, -2) + 4*SingularityFunction(x, 2, -1) - 2*SingularityFunction(x, 2, 2) + 2*SingularityFunction(x, 3, 0) + 4*SingularityFunction(x, 3, 1) + 2*SingularityFunction(x, 3, 2)
"""
x = self.variable
value = sympify(value)
start = sympify(start)
order = sympify(order)
self._applied_loads.append((value, start, order, end))
self._load += value*SingularityFunction(x, start, order)
if end:
if order.is_negative:
msg = ("If 'end' is provided the 'order' of the load cannot "
"be negative, i.e. 'end' is only valid for distributed "
"loads.")
raise ValueError(msg)
# NOTE : A Taylor series can be used to define the summation of
# singularity functions that subtract from the load past the end
# point such that it evaluates to zero past 'end'.
f = value*x**order
for i in range(0, order + 1):
self._load -= (f.diff(x, i).subs(x, end - start) *
SingularityFunction(x, end, i)/factorial(i))
def remove_load(self, value, start, order, end=None):
"""
This method removes a particular load present on the beam object.
Returns a ValueError if the load passed as an argument is not
present on the beam.
Parameters
==========
value : Sympifyable
The magnitude of an applied load.
start : Sympifyable
The starting point of the applied load. For point moments and
point forces this is the location of application.
order : Integer
The order of the applied load.
- For moments, order= -2
- For point loads, order=-1
- For constant distributed load, order=0
- For ramp loads, order=1
- For parabolic ramp loads, order=2
- ... so on.
end : Sympifyable, optional
An optional argument that can be used if the load has an end point
within the length of the beam.
Examples
========
There is a beam of length 4 meters. A moment of magnitude 3 Nm is
applied in the clockwise direction at the starting point of the beam.
A pointload of magnitude 4 N is applied from the top of the beam at
2 meters from the starting point and a parabolic ramp load of magnitude
2 N/m is applied below the beam starting from 2 meters to 3 meters
away from the starting point of the beam.
>>> from sympy.physics.continuum_mechanics.beam import Beam
>>> from sympy import symbols
>>> E, I = symbols('E, I')
>>> b = Beam(4, E, I)
>>> b.apply_load(-3, 0, -2)
>>> b.apply_load(4, 2, -1)
>>> b.apply_load(-2, 2, 2, end=3)
>>> b.load
-3*SingularityFunction(x, 0, -2) + 4*SingularityFunction(x, 2, -1) - 2*SingularityFunction(x, 2, 2) + 2*SingularityFunction(x, 3, 0) + 4*SingularityFunction(x, 3, 1) + 2*SingularityFunction(x, 3, 2)
>>> b.remove_load(-2, 2, 2, end = 3)
>>> b.load
-3*SingularityFunction(x, 0, -2) + 4*SingularityFunction(x, 2, -1)
"""
x = self.variable
value = sympify(value)
start = sympify(start)
order = sympify(order)
if (value, start, order, end) in self._applied_loads:
self._load -= value*SingularityFunction(x, start, order)
self._applied_loads.remove((value, start, order, end))
else:
msg = "No such load distribution exists on the beam object."
raise ValueError(msg)
if end:
# TODO : This is essentially duplicate code wrt to apply_load,
# would be better to move it to one location and both methods use
# it.
if order.is_negative:
msg = ("If 'end' is provided the 'order' of the load cannot "
"be negative, i.e. 'end' is only valid for distributed "
"loads.")
raise ValueError(msg)
# NOTE : A Taylor series can be used to define the summation of
# singularity functions that subtract from the load past the end
# point such that it evaluates to zero past 'end'.
f = value*x**order
for i in range(0, order + 1):
self._load += (f.diff(x, i).subs(x, end - start) *
SingularityFunction(x, end, i)/factorial(i))
@property
def load(self):
"""
Returns a Singularity Function expression which represents
the load distribution curve of the Beam object.
Examples
========
There is a beam of length 4 meters. A moment of magnitude 3 Nm is
applied in the clockwise direction at the starting point of the beam.
A point load of magnitude 4 N is applied from the top of the beam at
2 meters from the starting point and a parabolic ramp load of magnitude
2 N/m is applied below the beam starting from 3 meters away from the
starting point of the beam.
>>> from sympy.physics.continuum_mechanics.beam import Beam
>>> from sympy import symbols
>>> E, I = symbols('E, I')
>>> b = Beam(4, E, I)
>>> b.apply_load(-3, 0, -2)
>>> b.apply_load(4, 2, -1)
>>> b.apply_load(-2, 3, 2)
>>> b.load
-3*SingularityFunction(x, 0, -2) + 4*SingularityFunction(x, 2, -1) - 2*SingularityFunction(x, 3, 2)
"""
return self._load
@property
def applied_loads(self):
"""
Returns a list of all loads applied on the beam object.
Each load in the list is a tuple of form (value, start, order, end).
Examples
========
There is a beam of length 4 meters. A moment of magnitude 3 Nm is
applied in the clockwise direction at the starting point of the beam.
A pointload of magnitude 4 N is applied from the top of the beam at
2 meters from the starting point. Another pointload of magnitude 5 N
is applied at same position.
>>> from sympy.physics.continuum_mechanics.beam import Beam
>>> from sympy import symbols
>>> E, I = symbols('E, I')
>>> b = Beam(4, E, I)
>>> b.apply_load(-3, 0, -2)
>>> b.apply_load(4, 2, -1)
>>> b.apply_load(5, 2, -1)
>>> b.load
-3*SingularityFunction(x, 0, -2) + 9*SingularityFunction(x, 2, -1)
>>> b.applied_loads
[(-3, 0, -2, None), (4, 2, -1, None), (5, 2, -1, None)]
"""
return self._applied_loads
def _solve_hinge_beams(self, *reactions):
"""Method to find integration constants and reactional variables in a
composite beam connected via hinge.
This method resolves the composite Beam into its sub-beams and then
equations of shear force, bending moment, slope and deflection are
evaluated for both of them separately. These equations are then solved
for unknown reactions and integration constants using the boundary
conditions applied on the Beam. Equal deflection of both sub-beams
at the hinge joint gives us another equation to solve the system.
Examples
========
A combined beam, with constant fkexural rigidity E*I, is formed by joining
a Beam of length 2*l to the right of another Beam of length l. The whole beam
is fixed at both of its both end. A point load of magnitude P is also applied
from the top at a distance of 2*l from starting point.
>>> from sympy.physics.continuum_mechanics.beam import Beam
>>> from sympy import symbols
>>> E, I = symbols('E, I')
>>> l=symbols('l', positive=True)
>>> b1=Beam(l ,E,I)
>>> b2=Beam(2*l ,E,I)
>>> b=b1.join(b2,"hinge")
>>> M1, A1, M2, A2, P = symbols('M1 A1 M2 A2 P')
>>> b.apply_load(A1,0,-1)
>>> b.apply_load(M1,0,-2)
>>> b.apply_load(P,2*l,-1)
>>> b.apply_load(A2,3*l,-1)
>>> b.apply_load(M2,3*l,-2)
>>> b.bc_slope=[(0,0), (3*l, 0)]
>>> b.bc_deflection=[(0,0), (3*l, 0)]
>>> b.solve_for_reaction_loads(M1, A1, M2, A2)
>>> b.reaction_loads
{A1: -5*P/18, A2: -13*P/18, M1: 5*P*l/18, M2: -4*P*l/9}
>>> b.slope()
(5*P*l*SingularityFunction(x, 0, 1)/18 - 5*P*SingularityFunction(x, 0, 2)/36 + 5*P*SingularityFunction(x, l, 2)/36)*SingularityFunction(x, 0, 0)/(E*I)
- (5*P*l*SingularityFunction(x, 0, 1)/18 - 5*P*SingularityFunction(x, 0, 2)/36 + 5*P*SingularityFunction(x, l, 2)/36)*SingularityFunction(x, l, 0)/(E*I)
+ (P*l**2/18 - 4*P*l*SingularityFunction(-l + x, 2*l, 1)/9 - 5*P*SingularityFunction(-l + x, 0, 2)/36 + P*SingularityFunction(-l + x, l, 2)/2
- 13*P*SingularityFunction(-l + x, 2*l, 2)/36)*SingularityFunction(x, l, 0)/(E*I)
>>> b.deflection()
(5*P*l*SingularityFunction(x, 0, 2)/36 - 5*P*SingularityFunction(x, 0, 3)/108 + 5*P*SingularityFunction(x, l, 3)/108)*SingularityFunction(x, 0, 0)/(E*I)
- (5*P*l*SingularityFunction(x, 0, 2)/36 - 5*P*SingularityFunction(x, 0, 3)/108 + 5*P*SingularityFunction(x, l, 3)/108)*SingularityFunction(x, l, 0)/(E*I)
+ (5*P*l**3/54 + P*l**2*(-l + x)/18 - 2*P*l*SingularityFunction(-l + x, 2*l, 2)/9 - 5*P*SingularityFunction(-l + x, 0, 3)/108 + P*SingularityFunction(-l + x, l, 3)/6
- 13*P*SingularityFunction(-l + x, 2*l, 3)/108)*SingularityFunction(x, l, 0)/(E*I)
"""
x = self.variable
l = self._hinge_position
E = self._elastic_modulus
I = self._second_moment
if isinstance(I, Piecewise):
I1 = I.args[0][0]
I2 = I.args[1][0]
else:
I1 = I2 = I
load_1 = 0 # Load equation on first segment of composite beam
load_2 = 0 # Load equation on second segment of composite beam
# Distributing load on both segments
for load in self.applied_loads:
if load[1] < l:
load_1 += load[0]*SingularityFunction(x, load[1], load[2])
if load[2] == 0:
load_1 -= load[0]*SingularityFunction(x, load[3], load[2])
elif load[2] > 0:
load_1 -= load[0]*SingularityFunction(x, load[3], load[2]) + load[0]*SingularityFunction(x, load[3], 0)
elif load[1] == l:
load_1 += load[0]*SingularityFunction(x, load[1], load[2])
load_2 += load[0]*SingularityFunction(x, load[1] - l, load[2])
elif load[1] > l:
load_2 += load[0]*SingularityFunction(x, load[1] - l, load[2])
if load[2] == 0:
load_2 -= load[0]*SingularityFunction(x, load[3] - l, load[2])
elif load[2] > 0:
load_2 -= load[0]*SingularityFunction(x, load[3] - l, load[2]) + load[0]*SingularityFunction(x, load[3] - l, 0)
h = Symbol('h') # Force due to hinge
load_1 += h*SingularityFunction(x, l, -1)
load_2 -= h*SingularityFunction(x, 0, -1)
eq = []
shear_1 = integrate(load_1, x)
shear_curve_1 = limit(shear_1, x, l)
eq.append(shear_curve_1)
bending_1 = integrate(shear_1, x)
moment_curve_1 = limit(bending_1, x, l)
eq.append(moment_curve_1)
shear_2 = integrate(load_2, x)
shear_curve_2 = limit(shear_2, x, self.length - l)
eq.append(shear_curve_2)
bending_2 = integrate(shear_2, x)
moment_curve_2 = limit(bending_2, x, self.length - l)
eq.append(moment_curve_2)
C1 = Symbol('C1')
C2 = Symbol('C2')
C3 = Symbol('C3')
C4 = Symbol('C4')
slope_1 = S.One/(E*I1)*(integrate(bending_1, x) + C1)
def_1 = S.One/(E*I1)*(integrate((E*I)*slope_1, x) + C1*x + C2)
slope_2 = S.One/(E*I2)*(integrate(integrate(integrate(load_2, x), x), x) + C3)
def_2 = S.One/(E*I2)*(integrate((E*I)*slope_2, x) + C4)
for position, value in self.bc_slope:
if position<l:
eq.append(slope_1.subs(x, position) - value)
else:
eq.append(slope_2.subs(x, position - l) - value)
for position, value in self.bc_deflection:
if position<l:
eq.append(def_1.subs(x, position) - value)
else:
eq.append(def_2.subs(x, position - l) - value)
eq.append(def_1.subs(x, l) - def_2.subs(x, 0)) # Deflection of both the segments at hinge would be equal
constants = list(linsolve(eq, C1, C2, C3, C4, h, *reactions))
reaction_values = list(constants[0])[5:]
self._reaction_loads = dict(zip(reactions, reaction_values))
self._load = self._load.subs(self._reaction_loads)
# Substituting constants and reactional load and moments with their corresponding values
slope_1 = slope_1.subs({C1: constants[0][0], h:constants[0][4]}).subs(self._reaction_loads)
def_1 = def_1.subs({C1: constants[0][0], C2: constants[0][1], h:constants[0][4]}).subs(self._reaction_loads)
slope_2 = slope_2.subs({x: x-l, C3: constants[0][2], h:constants[0][4]}).subs(self._reaction_loads)
def_2 = def_2.subs({x: x-l,C3: constants[0][2], C4: constants[0][3], h:constants[0][4]}).subs(self._reaction_loads)
self._hinge_beam_slope = slope_1*SingularityFunction(x, 0, 0) - slope_1*SingularityFunction(x, l, 0) + slope_2*SingularityFunction(x, l, 0)
self._hinge_beam_deflection = def_1*SingularityFunction(x, 0, 0) - def_1*SingularityFunction(x, l, 0) + def_2*SingularityFunction(x, l, 0)
def solve_for_reaction_loads(self, *reactions):
"""
Solves for the reaction forces.
Examples
========
There is a beam of length 30 meters. A moment of magnitude 120 Nm is
applied in the clockwise direction at the end of the beam. A pointload
of magnitude 8 N is applied from the top of the beam at the starting
point. There are two simple supports below the beam. One at the end
and another one at a distance of 10 meters from the start. The
deflection is restricted at both the supports.
Using the sign convention of upward forces and clockwise moment
being positive.
>>> from sympy.physics.continuum_mechanics.beam import Beam
>>> from sympy import symbols
>>> E, I = symbols('E, I')
>>> R1, R2 = symbols('R1, R2')
>>> b = Beam(30, E, I)
>>> b.apply_load(-8, 0, -1)
>>> b.apply_load(R1, 10, -1) # Reaction force at x = 10
>>> b.apply_load(R2, 30, -1) # Reaction force at x = 30
>>> b.apply_load(120, 30, -2)
>>> b.bc_deflection = [(10, 0), (30, 0)]
>>> b.load
R1*SingularityFunction(x, 10, -1) + R2*SingularityFunction(x, 30, -1)
- 8*SingularityFunction(x, 0, -1) + 120*SingularityFunction(x, 30, -2)
>>> b.solve_for_reaction_loads(R1, R2)
>>> b.reaction_loads
{R1: 6, R2: 2}
>>> b.load
-8*SingularityFunction(x, 0, -1) + 6*SingularityFunction(x, 10, -1)
+ 120*SingularityFunction(x, 30, -2) + 2*SingularityFunction(x, 30, -1)
"""
if self._composite_type == "hinge":
return self._solve_hinge_beams(*reactions)
x = self.variable
l = self.length
C3 = Symbol('C3')
C4 = Symbol('C4')
shear_curve = limit(self.shear_force(), x, l)
moment_curve = limit(self.bending_moment(), x, l)
slope_eqs = []
deflection_eqs = []
slope_curve = integrate(self.bending_moment(), x) + C3
for position, value in self._boundary_conditions['slope']:
eqs = slope_curve.subs(x, position) - value
slope_eqs.append(eqs)
deflection_curve = integrate(slope_curve, x) + C4
for position, value in self._boundary_conditions['deflection']:
eqs = deflection_curve.subs(x, position) - value
deflection_eqs.append(eqs)
solution = list((linsolve([shear_curve, moment_curve] + slope_eqs
+ deflection_eqs, (C3, C4) + reactions).args)[0])
solution = solution[2:]
self._reaction_loads = dict(zip(reactions, solution))
self._load = self._load.subs(self._reaction_loads)
def shear_force(self):
"""
Returns a Singularity Function expression which represents
the shear force curve of the Beam object.
Examples
========
There is a beam of length 30 meters. A moment of magnitude 120 Nm is
applied in the clockwise direction at the end of the beam. A pointload
of magnitude 8 N is applied from the top of the beam at the starting
point. There are two simple supports below the beam. One at the end
and another one at a distance of 10 meters from the start. The
deflection is restricted at both the supports.
Using the sign convention of upward forces and clockwise moment
being positive.
>>> from sympy.physics.continuum_mechanics.beam import Beam
>>> from sympy import symbols
>>> E, I = symbols('E, I')
>>> R1, R2 = symbols('R1, R2')
>>> b = Beam(30, E, I)
>>> b.apply_load(-8, 0, -1)
>>> b.apply_load(R1, 10, -1)
>>> b.apply_load(R2, 30, -1)
>>> b.apply_load(120, 30, -2)
>>> b.bc_deflection = [(10, 0), (30, 0)]
>>> b.solve_for_reaction_loads(R1, R2)
>>> b.shear_force()
-8*SingularityFunction(x, 0, 0) + 6*SingularityFunction(x, 10, 0) + 120*SingularityFunction(x, 30, -1) + 2*SingularityFunction(x, 30, 0)
"""
x = self.variable
return integrate(self.load, x)
def max_shear_force(self):
"""Returns maximum Shear force and its coordinate
in the Beam object."""
from sympy import solve, Mul, Interval
shear_curve = self.shear_force()
x = self.variable
terms = shear_curve.args
singularity = [] # Points at which shear function changes
for term in terms:
if isinstance(term, Mul):
term = term.args[-1] # SingularityFunction in the term
singularity.append(term.args[1])
singularity.sort()
singularity = list(set(singularity))
intervals = [] # List of Intervals with discrete value of shear force
shear_values = [] # List of values of shear force in each interval
for i, s in enumerate(singularity):
if s == 0:
continue
try:
shear_slope = Piecewise((float("nan"), x<=singularity[i-1]),(self._load.rewrite(Piecewise), x<s), (float("nan"), True))
points = solve(shear_slope, x)
val = []
for point in points:
val.append(shear_curve.subs(x, point))
points.extend([singularity[i-1], s])
val.extend([limit(shear_curve, x, singularity[i-1], '+'), limit(shear_curve, x, s, '-')])
val = list(map(abs, val))
max_shear = max(val)
shear_values.append(max_shear)
intervals.append(points[val.index(max_shear)])
# If shear force in a particular Interval has zero or constant
# slope, then above block gives NotImplementedError as
# solve can't represent Interval solutions.
except NotImplementedError:
initial_shear = limit(shear_curve, x, singularity[i-1], '+')
final_shear = limit(shear_curve, x, s, '-')
# If shear_curve has a constant slope(it is a line).
if shear_curve.subs(x, (singularity[i-1] + s)/2) == (initial_shear + final_shear)/2 and initial_shear != final_shear:
shear_values.extend([initial_shear, final_shear])
intervals.extend([singularity[i-1], s])
else: # shear_curve has same value in whole Interval
shear_values.append(final_shear)
intervals.append(Interval(singularity[i-1], s))
shear_values = list(map(abs, shear_values))
maximum_shear = max(shear_values)
point = intervals[shear_values.index(maximum_shear)]
return (point, maximum_shear)
def bending_moment(self):
"""
Returns a Singularity Function expression which represents
the bending moment curve of the Beam object.
Examples
========
There is a beam of length 30 meters. A moment of magnitude 120 Nm is
applied in the clockwise direction at the end of the beam. A pointload
of magnitude 8 N is applied from the top of the beam at the starting
point. There are two simple supports below the beam. One at the end
and another one at a distance of 10 meters from the start. The
deflection is restricted at both the supports.
Using the sign convention of upward forces and clockwise moment
being positive.
>>> from sympy.physics.continuum_mechanics.beam import Beam
>>> from sympy import symbols
>>> E, I = symbols('E, I')
>>> R1, R2 = symbols('R1, R2')
>>> b = Beam(30, E, I)
>>> b.apply_load(-8, 0, -1)
>>> b.apply_load(R1, 10, -1)
>>> b.apply_load(R2, 30, -1)
>>> b.apply_load(120, 30, -2)
>>> b.bc_deflection = [(10, 0), (30, 0)]
>>> b.solve_for_reaction_loads(R1, R2)
>>> b.bending_moment()
-8*SingularityFunction(x, 0, 1) + 6*SingularityFunction(x, 10, 1) + 120*SingularityFunction(x, 30, 0) + 2*SingularityFunction(x, 30, 1)
"""
x = self.variable
return integrate(self.shear_force(), x)
def max_bmoment(self):
"""Returns maximum Shear force and its coordinate
in the Beam object."""
from sympy import solve, Mul, Interval
bending_curve = self.bending_moment()
x = self.variable
terms = bending_curve.args
singularity = [] # Points at which bending moment changes
for term in terms:
if isinstance(term, Mul):
term = term.args[-1] # SingularityFunction in the term
singularity.append(term.args[1])
singularity.sort()
singularity = list(set(singularity))
intervals = [] # List of Intervals with discrete value of bending moment
moment_values = [] # List of values of bending moment in each interval
for i, s in enumerate(singularity):
if s == 0:
continue
try:
moment_slope = Piecewise((float("nan"), x<=singularity[i-1]),(self.shear_force().rewrite(Piecewise), x<s), (float("nan"), True))
points = solve(moment_slope, x)
val = []
for point in points:
val.append(bending_curve.subs(x, point))
points.extend([singularity[i-1], s])
val.extend([limit(bending_curve, x, singularity[i-1], '+'), limit(bending_curve, x, s, '-')])
val = list(map(abs, val))
max_moment = max(val)
moment_values.append(max_moment)
intervals.append(points[val.index(max_moment)])
# If bending moment in a particular Interval has zero or constant
# slope, then above block gives NotImplementedError as solve
# can't represent Interval solutions.
except NotImplementedError:
initial_moment = limit(bending_curve, x, singularity[i-1], '+')
final_moment = limit(bending_curve, x, s, '-')
# If bending_curve has a constant slope(it is a line).
if bending_curve.subs(x, (singularity[i-1] + s)/2) == (initial_moment + final_moment)/2 and initial_moment != final_moment:
moment_values.extend([initial_moment, final_moment])
intervals.extend([singularity[i-1], s])
else: # bending_curve has same value in whole Interval
moment_values.append(final_moment)
intervals.append(Interval(singularity[i-1], s))
moment_values = list(map(abs, moment_values))
maximum_moment = max(moment_values)
point = intervals[moment_values.index(maximum_moment)]
return (point, maximum_moment)
def point_cflexure(self):
"""
Returns a Set of point(s) with zero bending moment and
where bending moment curve of the beam object changes
its sign from negative to positive or vice versa.
Examples
========
There is is 10 meter long overhanging beam. There are
two simple supports below the beam. One at the start
and another one at a distance of 6 meters from the start.
Point loads of magnitude 10KN and 20KN are applied at
2 meters and 4 meters from start respectively. A Uniformly
distribute load of magnitude of magnitude 3KN/m is also
applied on top starting from 6 meters away from starting
point till end.
Using the sign convention of upward forces and clockwise moment
being positive.
>>> from sympy.physics.continuum_mechanics.beam import Beam
>>> from sympy import symbols
>>> E, I = symbols('E, I')
>>> b = Beam(10, E, I)
>>> b.apply_load(-4, 0, -1)
>>> b.apply_load(-46, 6, -1)
>>> b.apply_load(10, 2, -1)
>>> b.apply_load(20, 4, -1)
>>> b.apply_load(3, 6, 0)
>>> b.point_cflexure()
[10/3]
"""
from sympy import solve, Piecewise
# To restrict the range within length of the Beam
moment_curve = Piecewise((float("nan"), self.variable<=0),
(self.bending_moment(), self.variable<self.length),
(float("nan"), True))
points = solve(moment_curve.rewrite(Piecewise), self.variable,
domain=S.Reals)
return points
def slope(self):
"""
Returns a Singularity Function expression which represents
the slope the elastic curve of the Beam object.
Examples
========
There is a beam of length 30 meters. A moment of magnitude 120 Nm is
applied in the clockwise direction at the end of the beam. A pointload
of magnitude 8 N is applied from the top of the beam at the starting
point. There are two simple supports below the beam. One at the end
and another one at a distance of 10 meters from the start. The
deflection is restricted at both the supports.
Using the sign convention of upward forces and clockwise moment
being positive.
>>> from sympy.physics.continuum_mechanics.beam import Beam
>>> from sympy import symbols
>>> E, I = symbols('E, I')
>>> R1, R2 = symbols('R1, R2')
>>> b = Beam(30, E, I)
>>> b.apply_load(-8, 0, -1)
>>> b.apply_load(R1, 10, -1)
>>> b.apply_load(R2, 30, -1)
>>> b.apply_load(120, 30, -2)
>>> b.bc_deflection = [(10, 0), (30, 0)]
>>> b.solve_for_reaction_loads(R1, R2)
>>> b.slope()
(-4*SingularityFunction(x, 0, 2) + 3*SingularityFunction(x, 10, 2)
+ 120*SingularityFunction(x, 30, 1) + SingularityFunction(x, 30, 2) + 4000/3)/(E*I)
"""
x = self.variable
E = self.elastic_modulus
I = self.second_moment
if self._composite_type == "hinge":
return self._hinge_beam_slope
if not self._boundary_conditions['slope']:
return diff(self.deflection(), x)
if isinstance(I, Piecewise) and self._composite_type == "fixed":
args = I.args
slope = 0
prev_slope = 0
prev_end = 0
for i in range(len(args)):
if i != 0:
prev_end = args[i-1][1].args[1]
slope_value = S.One/E*integrate(self.bending_moment()/args[i][0], (x, prev_end, x))
if i != len(args) - 1:
slope += (prev_slope + slope_value)*SingularityFunction(x, prev_end, 0) - \
(prev_slope + slope_value)*SingularityFunction(x, args[i][1].args[1], 0)
else:
slope += (prev_slope + slope_value)*SingularityFunction(x, prev_end, 0)
prev_slope = slope_value.subs(x, args[i][1].args[1])
return slope
C3 = Symbol('C3')
slope_curve = integrate(S.One/(E*I)*self.bending_moment(), x) + C3
bc_eqs = []
for position, value in self._boundary_conditions['slope']:
eqs = slope_curve.subs(x, position) - value
bc_eqs.append(eqs)
constants = list(linsolve(bc_eqs, C3))
slope_curve = slope_curve.subs({C3: constants[0][0]})
return slope_curve
def deflection(self):
"""
Returns a Singularity Function expression which represents
the elastic curve or deflection of the Beam object.
Examples
========
There is a beam of length 30 meters. A moment of magnitude 120 Nm is
applied in the clockwise direction at the end of the beam. A pointload
of magnitude 8 N is applied from the top of the beam at the starting
point. There are two simple supports below the beam. One at the end
and another one at a distance of 10 meters from the start. The
deflection is restricted at both the supports.
Using the sign convention of upward forces and clockwise moment
being positive.
>>> from sympy.physics.continuum_mechanics.beam import Beam
>>> from sympy import symbols
>>> E, I = symbols('E, I')
>>> R1, R2 = symbols('R1, R2')
>>> b = Beam(30, E, I)
>>> b.apply_load(-8, 0, -1)
>>> b.apply_load(R1, 10, -1)
>>> b.apply_load(R2, 30, -1)
>>> b.apply_load(120, 30, -2)
>>> b.bc_deflection = [(10, 0), (30, 0)]
>>> b.solve_for_reaction_loads(R1, R2)
>>> b.deflection()
(4000*x/3 - 4*SingularityFunction(x, 0, 3)/3 + SingularityFunction(x, 10, 3)
+ 60*SingularityFunction(x, 30, 2) + SingularityFunction(x, 30, 3)/3 - 12000)/(E*I)
"""
x = self.variable
E = self.elastic_modulus
I = self.second_moment
if self._composite_type == "hinge":
return self._hinge_beam_deflection
if not self._boundary_conditions['deflection'] and not self._boundary_conditions['slope']:
if isinstance(I, Piecewise) and self._composite_type == "fixed":
args = I.args
prev_slope = 0
prev_def = 0
prev_end = 0
deflection = 0
for i in range(len(args)):
if i != 0:
prev_end = args[i-1][1].args[1]
slope_value = S.One/E*integrate(self.bending_moment()/args[i][0], (x, prev_end, x))
recent_segment_slope = prev_slope + slope_value
deflection_value = integrate(recent_segment_slope, (x, prev_end, x))
if i != len(args) - 1:
deflection += (prev_def + deflection_value)*SingularityFunction(x, prev_end, 0) \
- (prev_def + deflection_value)*SingularityFunction(x, args[i][1].args[1], 0)
else:
deflection += (prev_def + deflection_value)*SingularityFunction(x, prev_end, 0)
prev_slope = slope_value.subs(x, args[i][1].args[1])
prev_def = deflection_value.subs(x, args[i][1].args[1])
return deflection
base_char = self._base_char
constants = symbols(base_char + '3:5')
return S.One/(E*I)*integrate(integrate(self.bending_moment(), x), x) + constants[0]*x + constants[1]
elif not self._boundary_conditions['deflection']:
base_char = self._base_char
constant = symbols(base_char + '4')
return integrate(self.slope(), x) + constant
elif not self._boundary_conditions['slope'] and self._boundary_conditions['deflection']:
if isinstance(I, Piecewise) and self._composite_type == "fixed":
args = I.args
prev_slope = 0
prev_def = 0
prev_end = 0
deflection = 0
for i in range(len(args)):
if i != 0:
prev_end = args[i-1][1].args[1]
slope_value = S.One/E*integrate(self.bending_moment()/args[i][0], (x, prev_end, x))
recent_segment_slope = prev_slope + slope_value
deflection_value = integrate(recent_segment_slope, (x, prev_end, x))
if i != len(args) - 1:
deflection += (prev_def + deflection_value)*SingularityFunction(x, prev_end, 0) \
- (prev_def + deflection_value)*SingularityFunction(x, args[i][1].args[1], 0)
else:
deflection += (prev_def + deflection_value)*SingularityFunction(x, prev_end, 0)
prev_slope = slope_value.subs(x, args[i][1].args[1])
prev_def = deflection_value.subs(x, args[i][1].args[1])
return deflection
base_char = self._base_char
C3, C4 = symbols(base_char + '3:5') # Integration constants
slope_curve = integrate(self.bending_moment(), x) + C3
deflection_curve = integrate(slope_curve, x) + C4
bc_eqs = []
for position, value in self._boundary_conditions['deflection']:
eqs = deflection_curve.subs(x, position) - value
bc_eqs.append(eqs)
constants = list(linsolve(bc_eqs, (C3, C4)))
deflection_curve = deflection_curve.subs({C3: constants[0][0], C4: constants[0][1]})
return S.One/(E*I)*deflection_curve
if isinstance(I, Piecewise) and self._composite_type == "fixed":
args = I.args
prev_slope = 0
prev_def = 0
prev_end = 0
deflection = 0
for i in range(len(args)):
if i != 0:
prev_end = args[i-1][1].args[1]
slope_value = S.One/E*integrate(self.bending_moment()/args[i][0], (x, prev_end, x))
recent_segment_slope = prev_slope + slope_value
deflection_value = integrate(recent_segment_slope, (x, prev_end, x))
if i != len(args) - 1:
deflection += (prev_def + deflection_value)*SingularityFunction(x, prev_end, 0) \
- (prev_def + deflection_value)*SingularityFunction(x, args[i][1].args[1], 0)
else:
deflection += (prev_def + deflection_value)*SingularityFunction(x, prev_end, 0)
prev_slope = slope_value.subs(x, args[i][1].args[1])
prev_def = deflection_value.subs(x, args[i][1].args[1])
return deflection
C4 = Symbol('C4')
deflection_curve = integrate(self.slope(), x) + C4
bc_eqs = []
for position, value in self._boundary_conditions['deflection']:
eqs = deflection_curve.subs(x, position) - value
bc_eqs.append(eqs)
constants = list(linsolve(bc_eqs, C4))
deflection_curve = deflection_curve.subs({C4: constants[0][0]})
return deflection_curve
def max_deflection(self):
"""
Returns point of max deflection and its corresponding deflection value
in a Beam object.
"""
from sympy import solve, Piecewise
# To restrict the range within length of the Beam
slope_curve = Piecewise((float("nan"), self.variable<=0),
(self.slope(), self.variable<self.length),
(float("nan"), True))
points = solve(slope_curve.rewrite(Piecewise), self.variable,
domain=S.Reals)
deflection_curve = self.deflection()
deflections = [deflection_curve.subs(self.variable, x) for x in points]
deflections = list(map(abs, deflections))
if len(deflections) != 0:
max_def = max(deflections)
return (points[deflections.index(max_def)], max_def)
else:
return None
def shear_stress(self):
"""
Returns an expression representing the Shear Stress
curve of the Beam object.
"""
return self.shear_force()/self._area
def plot_shear_force(self, subs=None):
"""
Returns a plot for Shear force present in the Beam object.
Parameters
==========
subs : dictionary
Python dictionary containing Symbols as key and their
corresponding values.
Examples
========
There is a beam of length 8 meters. A constant distributed load of 10 KN/m
is applied from half of the beam till the end. There are two simple supports
below the beam, one at the starting point and another at the ending point
of the beam. A pointload of magnitude 5 KN is also applied from top of the
beam, at a distance of 4 meters from the starting point.
Take E = 200 GPa and I = 400*(10**-6) meter**4.
Using the sign convention of downwards forces being positive.
.. plot::
:context: close-figs
:format: doctest
:include-source: True
>>> from sympy.physics.continuum_mechanics.beam import Beam
>>> from sympy import symbols
>>> R1, R2 = symbols('R1, R2')
>>> b = Beam(8, 200*(10**9), 400*(10**-6))
>>> b.apply_load(5000, 2, -1)
>>> b.apply_load(R1, 0, -1)
>>> b.apply_load(R2, 8, -1)
>>> b.apply_load(10000, 4, 0, end=8)
>>> b.bc_deflection = [(0, 0), (8, 0)]
>>> b.solve_for_reaction_loads(R1, R2)
>>> b.plot_shear_force()
Plot object containing:
[0]: cartesian line: -13750*SingularityFunction(x, 0, 0) + 5000*SingularityFunction(x, 2, 0)
+ 10000*SingularityFunction(x, 4, 1) - 31250*SingularityFunction(x, 8, 0)
- 10000*SingularityFunction(x, 8, 1) for x over (0.0, 8.0)
"""
shear_force = self.shear_force()
if subs is None:
subs = {}
for sym in shear_force.atoms(Symbol):
if sym == self.variable:
continue
if sym not in subs:
raise ValueError('Value of %s was not passed.' %sym)
if self.length in subs:
length = subs[self.length]
else:
length = self.length
return plot(shear_force.subs(subs), (self.variable, 0, length), title='Shear Force',
xlabel=r'$\mathrm{x}$', ylabel=r'$\mathrm{V}$', line_color='g')
def plot_bending_moment(self, subs=None):
"""
Returns a plot for Bending moment present in the Beam object.
Parameters
==========
subs : dictionary
Python dictionary containing Symbols as key and their
corresponding values.
Examples
========
There is a beam of length 8 meters. A constant distributed load of 10 KN/m
is applied from half of the beam till the end. There are two simple supports
below the beam, one at the starting point and another at the ending point
of the beam. A pointload of magnitude 5 KN is also applied from top of the
beam, at a distance of 4 meters from the starting point.
Take E = 200 GPa and I = 400*(10**-6) meter**4.
Using the sign convention of downwards forces being positive.
.. plot::
:context: close-figs
:format: doctest
:include-source: True
>>> from sympy.physics.continuum_mechanics.beam import Beam
>>> from sympy import symbols
>>> R1, R2 = symbols('R1, R2')
>>> b = Beam(8, 200*(10**9), 400*(10**-6))
>>> b.apply_load(5000, 2, -1)
>>> b.apply_load(R1, 0, -1)
>>> b.apply_load(R2, 8, -1)
>>> b.apply_load(10000, 4, 0, end=8)
>>> b.bc_deflection = [(0, 0), (8, 0)]
>>> b.solve_for_reaction_loads(R1, R2)
>>> b.plot_bending_moment()
Plot object containing:
[0]: cartesian line: -13750*SingularityFunction(x, 0, 1) + 5000*SingularityFunction(x, 2, 1)
+ 5000*SingularityFunction(x, 4, 2) - 31250*SingularityFunction(x, 8, 1)
- 5000*SingularityFunction(x, 8, 2) for x over (0.0, 8.0)
"""
bending_moment = self.bending_moment()
if subs is None:
subs = {}
for sym in bending_moment.atoms(Symbol):
if sym == self.variable:
continue
if sym not in subs:
raise ValueError('Value of %s was not passed.' %sym)
if self.length in subs:
length = subs[self.length]
else:
length = self.length
return plot(bending_moment.subs(subs), (self.variable, 0, length), title='Bending Moment',
xlabel=r'$\mathrm{x}$', ylabel=r'$\mathrm{M}$', line_color='b')
def plot_slope(self, subs=None):
"""
Returns a plot for slope of deflection curve of the Beam object.
Parameters
==========
subs : dictionary
Python dictionary containing Symbols as key and their
corresponding values.
Examples
========
There is a beam of length 8 meters. A constant distributed load of 10 KN/m
is applied from half of the beam till the end. There are two simple supports
below the beam, one at the starting point and another at the ending point
of the beam. A pointload of magnitude 5 KN is also applied from top of the
beam, at a distance of 4 meters from the starting point.
Take E = 200 GPa and I = 400*(10**-6) meter**4.
Using the sign convention of downwards forces being positive.
.. plot::
:context: close-figs
:format: doctest
:include-source: True
>>> from sympy.physics.continuum_mechanics.beam import Beam
>>> from sympy import symbols
>>> R1, R2 = symbols('R1, R2')
>>> b = Beam(8, 200*(10**9), 400*(10**-6))
>>> b.apply_load(5000, 2, -1)
>>> b.apply_load(R1, 0, -1)
>>> b.apply_load(R2, 8, -1)
>>> b.apply_load(10000, 4, 0, end=8)
>>> b.bc_deflection = [(0, 0), (8, 0)]
>>> b.solve_for_reaction_loads(R1, R2)
>>> b.plot_slope()
Plot object containing:
[0]: cartesian line: -8.59375e-5*SingularityFunction(x, 0, 2) + 3.125e-5*SingularityFunction(x, 2, 2)
+ 2.08333333333333e-5*SingularityFunction(x, 4, 3) - 0.0001953125*SingularityFunction(x, 8, 2)
- 2.08333333333333e-5*SingularityFunction(x, 8, 3) + 0.00138541666666667 for x over (0.0, 8.0)
"""
slope = self.slope()
if subs is None:
subs = {}
for sym in slope.atoms(Symbol):
if sym == self.variable:
continue
if sym not in subs:
raise ValueError('Value of %s was not passed.' %sym)
if self.length in subs:
length = subs[self.length]
else:
length = self.length
return plot(slope.subs(subs), (self.variable, 0, length), title='Slope',
xlabel=r'$\mathrm{x}$', ylabel=r'$\theta$', line_color='m')
def plot_deflection(self, subs=None):
"""
Returns a plot for deflection curve of the Beam object.
Parameters
==========
subs : dictionary
Python dictionary containing Symbols as key and their
corresponding values.
Examples
========
There is a beam of length 8 meters. A constant distributed load of 10 KN/m
is applied from half of the beam till the end. There are two simple supports
below the beam, one at the starting point and another at the ending point
of the beam. A pointload of magnitude 5 KN is also applied from top of the
beam, at a distance of 4 meters from the starting point.
Take E = 200 GPa and I = 400*(10**-6) meter**4.
Using the sign convention of downwards forces being positive.
.. plot::
:context: close-figs
:format: doctest
:include-source: True
>>> from sympy.physics.continuum_mechanics.beam import Beam
>>> from sympy import symbols
>>> R1, R2 = symbols('R1, R2')
>>> b = Beam(8, 200*(10**9), 400*(10**-6))
>>> b.apply_load(5000, 2, -1)
>>> b.apply_load(R1, 0, -1)
>>> b.apply_load(R2, 8, -1)
>>> b.apply_load(10000, 4, 0, end=8)
>>> b.bc_deflection = [(0, 0), (8, 0)]
>>> b.solve_for_reaction_loads(R1, R2)
>>> b.plot_deflection()
Plot object containing:
[0]: cartesian line: 0.00138541666666667*x - 2.86458333333333e-5*SingularityFunction(x, 0, 3)
+ 1.04166666666667e-5*SingularityFunction(x, 2, 3) + 5.20833333333333e-6*SingularityFunction(x, 4, 4)
- 6.51041666666667e-5*SingularityFunction(x, 8, 3) - 5.20833333333333e-6*SingularityFunction(x, 8, 4)
for x over (0.0, 8.0)
"""
deflection = self.deflection()
if subs is None:
subs = {}
for sym in deflection.atoms(Symbol):
if sym == self.variable:
continue
if sym not in subs:
raise ValueError('Value of %s was not passed.' %sym)
if self.length in subs:
length = subs[self.length]
else:
length = self.length
return plot(deflection.subs(subs), (self.variable, 0, length),
title='Deflection', xlabel=r'$\mathrm{x}$', ylabel=r'$\delta$',
line_color='r')
def plot_loading_results(self, subs=None):
"""
Returns a subplot of Shear Force, Bending Moment,
Slope and Deflection of the Beam object.
Parameters
==========
subs : dictionary
Python dictionary containing Symbols as key and their
corresponding values.
Examples
========
There is a beam of length 8 meters. A constant distributed load of 10 KN/m
is applied from half of the beam till the end. There are two simple supports
below the beam, one at the starting point and another at the ending point
of the beam. A pointload of magnitude 5 KN is also applied from top of the
beam, at a distance of 4 meters from the starting point.
Take E = 200 GPa and I = 400*(10**-6) meter**4.
Using the sign convention of downwards forces being positive.
.. plot::
:context: close-figs
:format: doctest
:include-source: True
>>> from sympy.physics.continuum_mechanics.beam import Beam
>>> from sympy import symbols
>>> R1, R2 = symbols('R1, R2')
>>> b = Beam(8, 200*(10**9), 400*(10**-6))
>>> b.apply_load(5000, 2, -1)
>>> b.apply_load(R1, 0, -1)
>>> b.apply_load(R2, 8, -1)
>>> b.apply_load(10000, 4, 0, end=8)
>>> b.bc_deflection = [(0, 0), (8, 0)]
>>> b.solve_for_reaction_loads(R1, R2)
>>> axes = b.plot_loading_results()
"""
length = self.length
variable = self.variable
if subs is None:
subs = {}
for sym in self.deflection().atoms(Symbol):
if sym == self.variable:
continue
if sym not in subs:
raise ValueError('Value of %s was not passed.' %sym)
if self.length in subs:
length = subs[self.length]
else:
length = self.length
ax1 = plot(self.shear_force().subs(subs), (variable, 0, length),
title="Shear Force", xlabel=r'$\mathrm{x}$', ylabel=r'$\mathrm{V}$',
line_color='g', show=False)
ax2 = plot(self.bending_moment().subs(subs), (variable, 0, length),
title="Bending Moment", xlabel=r'$\mathrm{x}$', ylabel=r'$\mathrm{M}$',
line_color='b', show=False)
ax3 = plot(self.slope().subs(subs), (variable, 0, length),
title="Slope", xlabel=r'$\mathrm{x}$', ylabel=r'$\theta$',
line_color='m', show=False)
ax4 = plot(self.deflection().subs(subs), (variable, 0, length),
title="Deflection", xlabel=r'$\mathrm{x}$', ylabel=r'$\delta$',
line_color='r', show=False)
return PlotGrid(4, 1, ax1, ax2, ax3, ax4)
@doctest_depends_on(modules=('numpy',))
def draw(self, pictorial=True):
"""
Returns a plot object representing the beam diagram of the beam.
.. note::
The user must be careful while entering load values.
The draw function assumes a sign convention which is used
for plotting loads.
Given a right handed coordinate system with XYZ coordinates,
the beam's length is assumed to be along the positive X axis.
The draw function recognizes positve loads(with n>-2) as loads
acting along negative Y direction and positve moments acting
along positive Z direction.
Parameters
==========
pictorial: Boolean (default=True)
Setting ``pictorial=True`` would simply create a pictorial (scaled) view
of the beam diagram not with the exact dimensions.
Although setting ``pictorial=False`` would create a beam diagram with
the exact dimensions on the plot
Examples
========
.. plot::
:context: close-figs
:format: doctest
:include-source: True
>>> from sympy.physics.continuum_mechanics.beam import Beam
>>> from sympy import symbols
>>> R1, R2 = symbols('R1, R2')
>>> E, I = symbols('E, I')
>>> b = Beam(50, 20, 30)
>>> b.apply_load(10, 2, -1)
>>> b.apply_load(R1, 10, -1)
>>> b.apply_load(R2, 30, -1)
>>> b.apply_load(90, 5, 0, 23)
>>> b.apply_load(10, 30, 1, 50)
>>> b.apply_support(50, "pin")
>>> b.apply_support(0, "fixed")
>>> b.apply_support(20, "roller")
>>> b.draw()
Plot object containing:
[0]: cartesian line: 25*SingularityFunction(x, 5, 0) - 25*SingularityFunction(x, 23, 0)
+ SingularityFunction(x, 30, 1) - 20*SingularityFunction(x, 50, 0)
- SingularityFunction(x, 50, 1) + 5 for x over (0.0, 50.0)
[1]: cartesian line: 5 for x over (0.0, 50.0)
"""
if not numpy:
raise ImportError("To use this function numpy module is required")
x = self.variable
# checking whether length is an expression in terms of any Symbol.
from sympy import Expr
if isinstance(self.length, Expr):
l = list(self.length.atoms(Symbol))
# assigning every Symbol a default value of 10
l = {i:10 for i in l}
length = self.length.subs(l)
else:
l = {}
length = self.length
height = length/10
rectangles = []
rectangles.append({'xy':(0, 0), 'width':length, 'height': height, 'facecolor':"brown"})
annotations, markers, load_eq,load_eq1, fill = self._draw_load(pictorial, length, l)
support_markers, support_rectangles = self._draw_supports(length, l)
rectangles += support_rectangles
markers += support_markers
sing_plot = plot(height + load_eq, height + load_eq1, (x, 0, length),
xlim=(-height, length + height), ylim=(-length, 1.25*length), annotations=annotations,
markers=markers, rectangles=rectangles, line_color='brown', fill=fill, axis=False, show=False)
return sing_plot
def _draw_load(self, pictorial, length, l):
loads = list(set(self.applied_loads) - set(self._support_as_loads))
height = length/10
x = self.variable
annotations = []
markers = []
load_args = []
scaled_load = 0
load_args1 = []
scaled_load1 = 0
load_eq = 0 # For positive valued higher order loads
load_eq1 = 0 # For negative valued higher order loads
fill = None
plus = 0 # For positive valued higher order loads
minus = 0 # For negative valued higher order loads
for load in loads:
# check if the position of load is in terms of the beam length.
if l:
pos = load[1].subs(l)
else:
pos = load[1]
# point loads
if load[2] == -1:
if isinstance(load[0], Symbol) or load[0].is_negative:
annotations.append({'s':'', 'xy':(pos, 0), 'xytext':(pos, height - 4*height), 'arrowprops':dict(width= 1.5, headlength=5, headwidth=5, facecolor='black')})
else:
annotations.append({'s':'', 'xy':(pos, height), 'xytext':(pos, height*4), 'arrowprops':dict(width= 1.5, headlength=4, headwidth=4, facecolor='black')})
# moment loads
elif load[2] == -2:
if load[0].is_negative:
markers.append({'args':[[pos], [height/2]], 'marker': r'$\circlearrowright$', 'markersize':15})
else:
markers.append({'args':[[pos], [height/2]], 'marker': r'$\circlearrowleft$', 'markersize':15})
# higher order loads
elif load[2] >= 0:
# `fill` will be assigned only when higher order loads are present
value, start, order, end = load
# Positive loads have their seperate equations
if(value>0):
plus = 1
# if pictorial is True we remake the load equation again with
# some constant magnitude values.
if pictorial:
value = 10**(1-order) if order > 0 else length/2
scaled_load += value*SingularityFunction(x, start, order)
if end:
f2 = 10**(1-order)*x**order if order > 0 else length/2*x**order
for i in range(0, order + 1):
scaled_load -= (f2.diff(x, i).subs(x, end - start)*
SingularityFunction(x, end, i)/factorial(i))
if pictorial:
if isinstance(scaled_load, Add):
load_args = scaled_load.args
else:
# when the load equation consists of only a single term
load_args = (scaled_load,)
load_eq = [i.subs(l) for i in load_args]
else:
if isinstance(self.load, Add):
load_args = self.load.args
else:
load_args = (self.load,)
load_eq = [i.subs(l) for i in load_args if list(i.atoms(SingularityFunction))[0].args[2] >= 0]
load_eq = Add(*load_eq)
# filling higher order loads with colour
expr = height + load_eq.rewrite(Piecewise)
y1 = lambdify(x, expr, 'numpy')
# For loads with negative value
else:
minus = 1
# if pictorial is True we remake the load equation again with
# some constant magnitude values.
if pictorial:
value = 10**(1-order) if order > 0 else length/2
scaled_load1 += value*SingularityFunction(x, start, order)
if end:
f2 = 10**(1-order)*x**order if order > 0 else length/2*x**order
for i in range(0, order + 1):
scaled_load1 -= (f2.diff(x, i).subs(x, end - start)*
SingularityFunction(x, end, i)/factorial(i))
if pictorial:
if isinstance(scaled_load1, Add):
load_args1 = scaled_load1.args
else:
# when the load equation consists of only a single term
load_args1 = (scaled_load1,)
load_eq1 = [i.subs(l) for i in load_args1]
else:
if isinstance(self.load, Add):
load_args1 = self.load.args1
else:
load_args1 = (self.load,)
load_eq1 = [i.subs(l) for i in load_args if list(i.atoms(SingularityFunction))[0].args[2] >= 0]
load_eq1 = -Add(*load_eq1)-height
# filling higher order loads with colour
expr = height + load_eq1.rewrite(Piecewise)
y1_ = lambdify(x, expr, 'numpy')
y = numpy.arange(0, float(length), 0.001)
y2 = float(height)
if(plus == 1 and minus == 1):
fill = {'x': y, 'y1': y1(y), 'y2': y1_(y), 'color':'darkkhaki'}
elif(plus == 1):
fill = {'x': y, 'y1': y1(y), 'y2': y2, 'color':'darkkhaki'}
else:
fill = {'x': y, 'y1': y1_(y), 'y2': y2 , 'color':'darkkhaki'}
return annotations, markers, load_eq, load_eq1, fill
def _draw_supports(self, length, l):
height = float(length/10)
support_markers = []
support_rectangles = []
for support in self._applied_supports:
if l:
pos = support[0].subs(l)
else:
pos = support[0]
if support[1] == "pin":
support_markers.append({'args':[pos, [0]], 'marker':6, 'markersize':13, 'color':"black"})
elif support[1] == "roller":
support_markers.append({'args':[pos, [-height/2.5]], 'marker':'o', 'markersize':11, 'color':"black"})
elif support[1] == "fixed":
if pos == 0:
support_rectangles.append({'xy':(0, -3*height), 'width':-length/20, 'height':6*height + height, 'fill':False, 'hatch':'/////'})
else:
support_rectangles.append({'xy':(length, -3*height), 'width':length/20, 'height': 6*height + height, 'fill':False, 'hatch':'/////'})
return support_markers, support_rectangles
class Beam3D(Beam):
"""
This class handles loads applied in any direction of a 3D space along
with unequal values of Second moment along different axes.
.. note::
While solving a beam bending problem, a user should choose its
own sign convention and should stick to it. The results will
automatically follow the chosen sign convention.
This class assumes that any kind of distributed load/moment is
applied through out the span of a beam.
Examples
========
There is a beam of l meters long. A constant distributed load of magnitude q
is applied along y-axis from start till the end of beam. A constant distributed
moment of magnitude m is also applied along z-axis from start till the end of beam.
Beam is fixed at both of its end. So, deflection of the beam at the both ends
is restricted.
>>> from sympy.physics.continuum_mechanics.beam import Beam3D
>>> from sympy import symbols, simplify, collect
>>> l, E, G, I, A = symbols('l, E, G, I, A')
>>> b = Beam3D(l, E, G, I, A)
>>> x, q, m = symbols('x, q, m')
>>> b.apply_load(q, 0, 0, dir="y")
>>> b.apply_moment_load(m, 0, -1, dir="z")
>>> b.shear_force()
[0, -q*x, 0]
>>> b.bending_moment()
[0, 0, -m*x + q*x**2/2]
>>> b.bc_slope = [(0, [0, 0, 0]), (l, [0, 0, 0])]
>>> b.bc_deflection = [(0, [0, 0, 0]), (l, [0, 0, 0])]
>>> b.solve_slope_deflection()
>>> b.slope()
[0, 0, x*(l*(-l*q + 3*l*(A*G*l**2*q - 2*A*G*l*m + 12*E*I*q)/(2*(A*G*l**2 + 12*E*I)) + 3*m)/6
+ q*x**2/6 + x*(-l*(A*G*l**2*q - 2*A*G*l*m + 12*E*I*q)/(2*(A*G*l**2 + 12*E*I)) - m)/2)/(E*I)]
>>> dx, dy, dz = b.deflection()
>>> dy = collect(simplify(dy), x)
>>> dx == dz == 0
True
>>> dy == (x*(12*A*E*G*I*l**3*q - 24*A*E*G*I*l**2*m + 144*E**2*I**2*l*q +
... x**3*(A**2*G**2*l**2*q + 12*A*E*G*I*q) +
... x**2*(-2*A**2*G**2*l**3*q - 24*A*E*G*I*l*q - 48*A*E*G*I*m) +
... x*(A**2*G**2*l**4*q + 72*A*E*G*I*l*m - 144*E**2*I**2*q)
... )/(24*A*E*G*I*(A*G*l**2 + 12*E*I)))
True
References
==========
.. [1] http://homes.civil.aau.dk/jc/FemteSemester/Beams3D.pdf
"""
def __init__(self, length, elastic_modulus, shear_modulus , second_moment, area, variable=Symbol('x')):
"""Initializes the class.
Parameters
==========
length : Sympifyable
A Symbol or value representing the Beam's length.
elastic_modulus : Sympifyable
A SymPy expression representing the Beam's Modulus of Elasticity.
It is a measure of the stiffness of the Beam material.
shear_modulus : Sympifyable
A SymPy expression representing the Beam's Modulus of rigidity.
It is a measure of rigidity of the Beam material.
second_moment : Sympifyable or list
A list of two elements having SymPy expression representing the
Beam's Second moment of area. First value represent Second moment
across y-axis and second across z-axis.
Single SymPy expression can be passed if both values are same
area : Sympifyable
A SymPy expression representing the Beam's cross-sectional area
in a plane prependicular to length of the Beam.
variable : Symbol, optional
A Symbol object that will be used as the variable along the beam
while representing the load, shear, moment, slope and deflection
curve. By default, it is set to ``Symbol('x')``.
"""
super(Beam3D, self).__init__(length, elastic_modulus, second_moment, variable)
self.shear_modulus = shear_modulus
self._area = area
self._load_vector = [0, 0, 0]
self._moment_load_vector = [0, 0, 0]
self._load_Singularity = [0, 0, 0]
self._slope = [0, 0, 0]
self._deflection = [0, 0, 0]
@property
def shear_modulus(self):
"""Young's Modulus of the Beam. """
return self._shear_modulus
@shear_modulus.setter
def shear_modulus(self, e):
self._shear_modulus = sympify(e)
@property
def second_moment(self):
"""Second moment of area of the Beam. """
return self._second_moment
@second_moment.setter
def second_moment(self, i):
if isinstance(i, list):
i = [sympify(x) for x in i]
self._second_moment = i
else:
self._second_moment = sympify(i)
@property
def area(self):
"""Cross-sectional area of the Beam. """
return self._area
@area.setter
def area(self, a):
self._area = sympify(a)
@property
def load_vector(self):
"""
Returns a three element list representing the load vector.
"""
return self._load_vector
@property
def moment_load_vector(self):
"""
Returns a three element list representing moment loads on Beam.
"""
return self._moment_load_vector
@property
def boundary_conditions(self):
"""
Returns a dictionary of boundary conditions applied on the beam.
The dictionary has two keywords namely slope and deflection.
The value of each keyword is a list of tuple, where each tuple
contains location and value of a boundary condition in the format
(location, value). Further each value is a list corresponding to
slope or deflection(s) values along three axes at that location.
Examples
========
There is a beam of length 4 meters. The slope at 0 should be 4 along
the x-axis and 0 along others. At the other end of beam, deflection
along all the three axes should be zero.
>>> from sympy.physics.continuum_mechanics.beam import Beam3D
>>> from sympy import symbols
>>> l, E, G, I, A, x = symbols('l, E, G, I, A, x')
>>> b = Beam3D(30, E, G, I, A, x)
>>> b.bc_slope = [(0, (4, 0, 0))]
>>> b.bc_deflection = [(4, [0, 0, 0])]
>>> b.boundary_conditions
{'deflection': [(4, [0, 0, 0])], 'slope': [(0, (4, 0, 0))]}
Here the deflection of the beam should be ``0`` along all the three axes at ``4``.
Similarly, the slope of the beam should be ``4`` along x-axis and ``0``
along y and z axis at ``0``.
"""
return self._boundary_conditions
def polar_moment(self):
"""
Returns the polar moment of area of the beam
about the X axis with respect to the centroid.
Examples
========
>>> from sympy.physics.continuum_mechanics.beam import Beam3D
>>> from sympy import symbols
>>> l, E, G, I, A = symbols('l, E, G, I, A')
>>> b = Beam3D(l, E, G, I, A)
>>> b.polar_moment()
2*I
>>> I1 = [9, 15]
>>> b = Beam3D(l, E, G, I1, A)
>>> b.polar_moment()
24
"""
if not iterable(self.second_moment):
return 2*self.second_moment
return sum(self.second_moment)
def apply_load(self, value, start, order, dir="y"):
"""
This method adds up the force load to a particular beam object.
Parameters
==========
value : Sympifyable
The magnitude of an applied load.
dir : String
Axis along which load is applied.
order : Integer
The order of the applied load.
- For point loads, order=-1
- For constant distributed load, order=0
- For ramp loads, order=1
- For parabolic ramp loads, order=2
- ... so on.
"""
x = self.variable
value = sympify(value)
start = sympify(start)
order = sympify(order)
if dir == "x":
if not order == -1:
self._load_vector[0] += value
self._load_Singularity[0] += value*SingularityFunction(x, start, order)
elif dir == "y":
if not order == -1:
self._load_vector[1] += value
self._load_Singularity[1] += value*SingularityFunction(x, start, order)
else:
if not order == -1:
self._load_vector[2] += value
self._load_Singularity[2] += value*SingularityFunction(x, start, order)
def apply_moment_load(self, value, start, order, dir="y"):
"""
This method adds up the moment loads to a particular beam object.
Parameters
==========
value : Sympifyable
The magnitude of an applied moment.
dir : String
Axis along which moment is applied.
order : Integer
The order of the applied load.
- For point moments, order=-2
- For constant distributed moment, order=-1
- For ramp moments, order=0
- For parabolic ramp moments, order=1
- ... so on.
"""
x = self.variable
value = sympify(value)
start = sympify(start)
order = sympify(order)
if dir == "x":
if not order == -2:
self._moment_load_vector[0] += value
self._load_Singularity[0] += value*SingularityFunction(x, start, order)
elif dir == "y":
if not order == -2:
self._moment_load_vector[1] += value
self._load_Singularity[0] += value*SingularityFunction(x, start, order)
else:
if not order == -2:
self._moment_load_vector[2] += value
self._load_Singularity[0] += value*SingularityFunction(x, start, order)
def apply_support(self, loc, type="fixed"):
if type == "pin" or type == "roller":
reaction_load = Symbol('R_'+str(loc))
self._reaction_loads[reaction_load] = reaction_load
self.bc_deflection.append((loc, [0, 0, 0]))
else:
reaction_load = Symbol('R_'+str(loc))
reaction_moment = Symbol('M_'+str(loc))
self._reaction_loads[reaction_load] = [reaction_load, reaction_moment]
self.bc_deflection.append((loc, [0, 0, 0]))
self.bc_slope.append((loc, [0, 0, 0]))
def solve_for_reaction_loads(self, *reaction):
"""
Solves for the reaction forces.
Examples
========
There is a beam of length 30 meters. It it supported by rollers at
of its end. A constant distributed load of magnitude 8 N is applied
from start till its end along y-axis. Another linear load having
slope equal to 9 is applied along z-axis.
>>> from sympy.physics.continuum_mechanics.beam import Beam3D
>>> from sympy import symbols
>>> l, E, G, I, A, x = symbols('l, E, G, I, A, x')
>>> b = Beam3D(30, E, G, I, A, x)
>>> b.apply_load(8, start=0, order=0, dir="y")
>>> b.apply_load(9*x, start=0, order=0, dir="z")
>>> b.bc_deflection = [(0, [0, 0, 0]), (30, [0, 0, 0])]
>>> R1, R2, R3, R4 = symbols('R1, R2, R3, R4')
>>> b.apply_load(R1, start=0, order=-1, dir="y")
>>> b.apply_load(R2, start=30, order=-1, dir="y")
>>> b.apply_load(R3, start=0, order=-1, dir="z")
>>> b.apply_load(R4, start=30, order=-1, dir="z")
>>> b.solve_for_reaction_loads(R1, R2, R3, R4)
>>> b.reaction_loads
{R1: -120, R2: -120, R3: -1350, R4: -2700}
"""
x = self.variable
l = self.length
q = self._load_Singularity
shear_curves = [integrate(load, x) for load in q]
moment_curves = [integrate(shear, x) for shear in shear_curves]
for i in range(3):
react = [r for r in reaction if (shear_curves[i].has(r) or moment_curves[i].has(r))]
if len(react) == 0:
continue
shear_curve = limit(shear_curves[i], x, l)
moment_curve = limit(moment_curves[i], x, l)
sol = list((linsolve([shear_curve, moment_curve], react).args)[0])
sol_dict = dict(zip(react, sol))
reaction_loads = self._reaction_loads
# Check if any of the evaluated rection exists in another direction
# and if it exists then it should have same value.
for key in sol_dict:
if key in reaction_loads and sol_dict[key] != reaction_loads[key]:
raise ValueError("Ambiguous solution for %s in different directions." % key)
self._reaction_loads.update(sol_dict)
def shear_force(self):
"""
Returns a list of three expressions which represents the shear force
curve of the Beam object along all three axes.
"""
x = self.variable
q = self._load_vector
return [integrate(-q[0], x), integrate(-q[1], x), integrate(-q[2], x)]
def axial_force(self):
"""
Returns expression of Axial shear force present inside the Beam object.
"""
return self.shear_force()[0]
def shear_stress(self):
"""
Returns a list of three expressions which represents the shear stress
curve of the Beam object along all three axes.
"""
return [self.shear_force()[0]/self._area, self.shear_force()[1]/self._area, self.shear_force()[2]/self._area]
def axial_stress(self):
"""
Returns expression of Axial stress present inside the Beam object.
"""
return self.axial_force()/self._area
def bending_moment(self):
"""
Returns a list of three expressions which represents the bending moment
curve of the Beam object along all three axes.
"""
x = self.variable
m = self._moment_load_vector
shear = self.shear_force()
return [integrate(-m[0], x), integrate(-m[1] + shear[2], x),
integrate(-m[2] - shear[1], x) ]
def torsional_moment(self):
"""
Returns expression of Torsional moment present inside the Beam object.
"""
return self.bending_moment()[0]
def solve_slope_deflection(self):
from sympy import dsolve, Function, Derivative, Eq
x = self.variable
l = self.length
E = self.elastic_modulus
G = self.shear_modulus
I = self.second_moment
if isinstance(I, list):
I_y, I_z = I[0], I[1]
else:
I_y = I_z = I
A = self._area
load = self._load_vector
moment = self._moment_load_vector
defl = Function('defl')
theta = Function('theta')
# Finding deflection along x-axis(and corresponding slope value by differentiating it)
# Equation used: Derivative(E*A*Derivative(def_x(x), x), x) + load_x = 0
eq = Derivative(E*A*Derivative(defl(x), x), x) + load[0]
def_x = dsolve(Eq(eq, 0), defl(x)).args[1]
# Solving constants originated from dsolve
C1 = Symbol('C1')
C2 = Symbol('C2')
constants = list((linsolve([def_x.subs(x, 0), def_x.subs(x, l)], C1, C2).args)[0])
def_x = def_x.subs({C1:constants[0], C2:constants[1]})
slope_x = def_x.diff(x)
self._deflection[0] = def_x
self._slope[0] = slope_x
# Finding deflection along y-axis and slope across z-axis. System of equation involved:
# 1: Derivative(E*I_z*Derivative(theta_z(x), x), x) + G*A*(Derivative(defl_y(x), x) - theta_z(x)) + moment_z = 0
# 2: Derivative(G*A*(Derivative(defl_y(x), x) - theta_z(x)), x) + load_y = 0
C_i = Symbol('C_i')
# Substitute value of `G*A*(Derivative(defl_y(x), x) - theta_z(x))` from (2) in (1)
eq1 = Derivative(E*I_z*Derivative(theta(x), x), x) + (integrate(-load[1], x) + C_i) + moment[2]
slope_z = dsolve(Eq(eq1, 0)).args[1]
# Solve for constants originated from using dsolve on eq1
constants = list((linsolve([slope_z.subs(x, 0), slope_z.subs(x, l)], C1, C2).args)[0])
slope_z = slope_z.subs({C1:constants[0], C2:constants[1]})
# Put value of slope obtained back in (2) to solve for `C_i` and find deflection across y-axis
eq2 = G*A*(Derivative(defl(x), x)) + load[1]*x - C_i - G*A*slope_z
def_y = dsolve(Eq(eq2, 0), defl(x)).args[1]
# Solve for constants originated from using dsolve on eq2
constants = list((linsolve([def_y.subs(x, 0), def_y.subs(x, l)], C1, C_i).args)[0])
self._deflection[1] = def_y.subs({C1:constants[0], C_i:constants[1]})
self._slope[2] = slope_z.subs(C_i, constants[1])
# Finding deflection along z-axis and slope across y-axis. System of equation involved:
# 1: Derivative(E*I_y*Derivative(theta_y(x), x), x) - G*A*(Derivative(defl_z(x), x) + theta_y(x)) + moment_y = 0
# 2: Derivative(G*A*(Derivative(defl_z(x), x) + theta_y(x)), x) + load_z = 0
# Substitute value of `G*A*(Derivative(defl_y(x), x) + theta_z(x))` from (2) in (1)
eq1 = Derivative(E*I_y*Derivative(theta(x), x), x) + (integrate(load[2], x) - C_i) + moment[1]
slope_y = dsolve(Eq(eq1, 0)).args[1]
# Solve for constants originated from using dsolve on eq1
constants = list((linsolve([slope_y.subs(x, 0), slope_y.subs(x, l)], C1, C2).args)[0])
slope_y = slope_y.subs({C1:constants[0], C2:constants[1]})
# Put value of slope obtained back in (2) to solve for `C_i` and find deflection across z-axis
eq2 = G*A*(Derivative(defl(x), x)) + load[2]*x - C_i + G*A*slope_y
def_z = dsolve(Eq(eq2,0)).args[1]
# Solve for constants originated from using dsolve on eq2
constants = list((linsolve([def_z.subs(x, 0), def_z.subs(x, l)], C1, C_i).args)[0])
self._deflection[2] = def_z.subs({C1:constants[0], C_i:constants[1]})
self._slope[1] = slope_y.subs(C_i, constants[1])
def slope(self):
"""
Returns a three element list representing slope of deflection curve
along all the three axes.
"""
return self._slope
def deflection(self):
"""
Returns a three element list representing deflection curve along all
the three axes.
"""
return self._deflection
|
41ec6dbaefc956c47ed55f638e7f3e6f354c23c5f4419679f9dc0ab5e5a65f0a | #!/usr/bin/env python
# -*- coding: utf-8 -*-
u"""
The module implements routines to model the polarization of optical fields
and can be used to calculate the effects of polarization optical elements on
the fields.
- Jones vectors.
- Stokes vectors.
- Jones matrices.
- Mueller matrices.
Examples
--------
We calculate a generic Jones vector:
>>> from sympy import symbols, pprint, zeros, simplify
>>> from sympy.physics.optics.polarization import (jones_vector, stokes_vector,
... half_wave_retarder, polarizing_beam_splitter, jones_2_stokes)
>>> psi, chi, p, I0 = symbols("psi, chi, p, I0", real=True)
>>> x0 = jones_vector(psi, chi)
>>> pprint(x0, use_unicode=True)
⎡-ⅈ⋅sin(χ)⋅sin(ψ) + cos(χ)⋅cos(ψ)⎤
⎢ ⎥
⎣ⅈ⋅sin(χ)⋅cos(ψ) + sin(ψ)⋅cos(χ) ⎦
And the more general Stokes vector:
>>> s0 = stokes_vector(psi, chi, p, I0)
>>> pprint(s0, use_unicode=True)
⎡ I₀ ⎤
⎢ ⎥
⎢I₀⋅p⋅cos(2⋅χ)⋅cos(2⋅ψ)⎥
⎢ ⎥
⎢I₀⋅p⋅sin(2⋅ψ)⋅cos(2⋅χ)⎥
⎢ ⎥
⎣ I₀⋅p⋅sin(2⋅χ) ⎦
We calculate how the Jones vector is modified by a half-wave plate:
>>> alpha = symbols("alpha", real=True)
>>> HWP = half_wave_retarder(alpha)
>>> x1 = simplify(HWP*x0)
We calculate the very common operation of passing a beam through a half-wave
plate and then through a polarizing beam-splitter. We do this by putting this
Jones vector as the first entry of a two-Jones-vector state that is transformed
by a 4x4 Jones matrix modelling the polarizing beam-splitter to get the
transmitted and reflected Jones vectors:
>>> PBS = polarizing_beam_splitter()
>>> X1 = zeros(4, 1)
>>> X1[:2, :] = x1
>>> X2 = PBS*X1
>>> transmitted_port = X2[:2, :]
>>> reflected_port = X2[2:, :]
This allows us to calculate how the power in both ports depends on the initial
polarization:
>>> transmitted_power = jones_2_stokes(transmitted_port)[0]
>>> reflected_power = jones_2_stokes(reflected_port)[0]
>>> print(transmitted_power)
cos(-2*alpha + chi + psi)**2/2 + cos(2*alpha + chi - psi)**2/2
>>> print(reflected_power)
sin(-2*alpha + chi + psi)**2/2 + sin(2*alpha + chi - psi)**2/2
Please see the description of the individual functions for further
details and examples.
References
==========
.. [1] https://en.wikipedia.org/wiki/Jones_calculus
.. [2] https://en.wikipedia.org/wiki/Mueller_calculus
.. [3] https://en.wikipedia.org/wiki/Stokes_parameters
"""
from sympy import sin, cos, exp, I, pi, sqrt, Matrix, Abs, re, im, simplify
from sympy.physics.quantum import TensorProduct
def jones_vector(psi, chi):
u"""A Jones vector corresponding to a polarization ellipse with `psi` tilt,
and `chi` circularity.
Parameters
----------
``psi`` : numeric type or sympy Symbol
The tilt of the polarization relative to the `x` axis.
``chi`` : numeric type or sympy Symbol
The angle adjacent to the mayor axis of the polarization ellipse.
Returns
-------
Matrix
A Jones vector.
Examples
--------
The axes on the Poincaré sphere.
>>> from sympy import pprint, symbols, pi
>>> from sympy.physics.optics.polarization import jones_vector
>>> psi, chi = symbols("psi, chi", real=True)
A general Jones vector.
>>> pprint(jones_vector(psi, chi), use_unicode=True)
⎡-ⅈ⋅sin(χ)⋅sin(ψ) + cos(χ)⋅cos(ψ)⎤
⎢ ⎥
⎣ⅈ⋅sin(χ)⋅cos(ψ) + sin(ψ)⋅cos(χ) ⎦
Horizontal polarization.
>>> pprint(jones_vector(0, 0), use_unicode=True)
⎡1⎤
⎢ ⎥
⎣0⎦
Vertical polarization.
>>> pprint(jones_vector(pi/2, 0), use_unicode=True)
⎡0⎤
⎢ ⎥
⎣1⎦
Diagonal polarization.
>>> pprint(jones_vector(pi/4, 0), use_unicode=True)
⎡√2⎤
⎢──⎥
⎢2 ⎥
⎢ ⎥
⎢√2⎥
⎢──⎥
⎣2 ⎦
Anti-diagonal polarization.
>>> pprint(jones_vector(-pi/4, 0), use_unicode=True)
⎡ √2 ⎤
⎢ ── ⎥
⎢ 2 ⎥
⎢ ⎥
⎢-√2 ⎥
⎢────⎥
⎣ 2 ⎦
Right-hand circular polarization.
>>> pprint(jones_vector(0, pi/4), use_unicode=True)
⎡ √2 ⎤
⎢ ── ⎥
⎢ 2 ⎥
⎢ ⎥
⎢√2⋅ⅈ⎥
⎢────⎥
⎣ 2 ⎦
Left-hand circular polarization.
>>> pprint(jones_vector(0, -pi/4), use_unicode=True)
⎡ √2 ⎤
⎢ ── ⎥
⎢ 2 ⎥
⎢ ⎥
⎢-√2⋅ⅈ ⎥
⎢──────⎥
⎣ 2 ⎦
"""
return Matrix([-I*sin(chi)*sin(psi) + cos(chi)*cos(psi),
I*sin(chi)*cos(psi) + sin(psi)*cos(chi)])
def stokes_vector(psi, chi, p=1, I=1):
u"""A Stokes vector corresponding to a polarization ellipse with `psi`
tilt, and `chi` circularity.
Parameters
----------
``psi`` : numeric type or sympy Symbol
The tilt of the polarization relative to the `x` axis.
``chi`` : numeric type or sympy Symbol
The angle adjacent to the mayor axis of the polarization ellipse.
``p`` : numeric type or sympy Symbol
The degree of polarization.
``I`` : numeric type or sympy Symbol
The intensity of the field.
Returns
-------
Matrix
A Stokes vector.
Examples
--------
The axes on the Poincaré sphere.
>>> from sympy import pprint, symbols, pi
>>> from sympy.physics.optics.polarization import stokes_vector
>>> psi, chi, p, I = symbols("psi, chi, p, I", real=True)
>>> pprint(stokes_vector(psi, chi, p, I), use_unicode=True)
⎡ I ⎤
⎢ ⎥
⎢I⋅p⋅cos(2⋅χ)⋅cos(2⋅ψ)⎥
⎢ ⎥
⎢I⋅p⋅sin(2⋅ψ)⋅cos(2⋅χ)⎥
⎢ ⎥
⎣ I⋅p⋅sin(2⋅χ) ⎦
Horizontal polarization
>>> pprint(stokes_vector(0, 0), use_unicode=True)
⎡1⎤
⎢ ⎥
⎢1⎥
⎢ ⎥
⎢0⎥
⎢ ⎥
⎣0⎦
Vertical polarization
>>> pprint(stokes_vector(pi/2, 0), use_unicode=True)
⎡1 ⎤
⎢ ⎥
⎢-1⎥
⎢ ⎥
⎢0 ⎥
⎢ ⎥
⎣0 ⎦
Diagonal polarization
>>> pprint(stokes_vector(pi/4, 0), use_unicode=True)
⎡1⎤
⎢ ⎥
⎢0⎥
⎢ ⎥
⎢1⎥
⎢ ⎥
⎣0⎦
Anti-diagonal polarization
>>> pprint(stokes_vector(-pi/4, 0), use_unicode=True)
⎡1 ⎤
⎢ ⎥
⎢0 ⎥
⎢ ⎥
⎢-1⎥
⎢ ⎥
⎣0 ⎦
Right-hand circular polarization
>>> pprint(stokes_vector(0, pi/4), use_unicode=True)
⎡1⎤
⎢ ⎥
⎢0⎥
⎢ ⎥
⎢0⎥
⎢ ⎥
⎣1⎦
Left-hand circular polarization
>>> pprint(stokes_vector(0, -pi/4), use_unicode=True)
⎡1 ⎤
⎢ ⎥
⎢0 ⎥
⎢ ⎥
⎢0 ⎥
⎢ ⎥
⎣-1⎦
Unpolarized light
>>> pprint(stokes_vector(0, 0, 0), use_unicode=True)
⎡1⎤
⎢ ⎥
⎢0⎥
⎢ ⎥
⎢0⎥
⎢ ⎥
⎣0⎦
"""
S0 = I
S1 = I*p*cos(2*psi)*cos(2*chi)
S2 = I*p*sin(2*psi)*cos(2*chi)
S3 = I*p*sin(2*chi)
return Matrix([S0, S1, S2, S3])
def jones_2_stokes(e):
u"""Return the Stokes vector for a Jones vector `e`.
Parameters
----------
``e`` : sympy Matrix
A Jones vector.
Returns
-------
sympy Matrix
A Jones vector.
Examples
--------
The axes on the Poincaré sphere.
>>> from sympy import pprint, pi
>>> from sympy.physics.optics.polarization import jones_vector
>>> from sympy.physics.optics.polarization import jones_2_stokes
>>> H = jones_vector(0, 0)
>>> V = jones_vector(pi/2, 0)
>>> D = jones_vector(pi/4, 0)
>>> A = jones_vector(-pi/4, 0)
>>> R = jones_vector(0, pi/4)
>>> L = jones_vector(0, -pi/4)
>>> pprint([jones_2_stokes(e) for e in [H, V, D, A, R, L]],
... use_unicode=True)
⎡⎡1⎤ ⎡1 ⎤ ⎡1⎤ ⎡1 ⎤ ⎡1⎤ ⎡1 ⎤⎤
⎢⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎥
⎢⎢1⎥ ⎢-1⎥ ⎢0⎥ ⎢0 ⎥ ⎢0⎥ ⎢0 ⎥⎥
⎢⎢ ⎥, ⎢ ⎥, ⎢ ⎥, ⎢ ⎥, ⎢ ⎥, ⎢ ⎥⎥
⎢⎢0⎥ ⎢0 ⎥ ⎢1⎥ ⎢-1⎥ ⎢0⎥ ⎢0 ⎥⎥
⎢⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎥
⎣⎣0⎦ ⎣0 ⎦ ⎣0⎦ ⎣0 ⎦ ⎣1⎦ ⎣-1⎦⎦
"""
ex, ey = e
return Matrix([Abs(ex)**2 + Abs(ey)**2,
Abs(ex)**2 - Abs(ey)**2,
2*re(ex*ey.conjugate()),
-2*im(ex*ey.conjugate())])
def linear_polarizer(theta=0):
u"""A linear polarizer Jones matrix with transmission axis at
an angle `theta`.
Parameters
----------
``theta`` : numeric type or sympy Symbol
The angle of the transmission axis relative to the horizontal plane.
Returns
-------
sympy Matrix
A Jones matrix representing the polarizer.
Examples
--------
A generic polarizer.
>>> from sympy import pprint, symbols
>>> from sympy.physics.optics.polarization import linear_polarizer
>>> theta = symbols("theta", real=True)
>>> J = linear_polarizer(theta)
>>> pprint(J, use_unicode=True)
⎡ 2 ⎤
⎢ cos (θ) sin(θ)⋅cos(θ)⎥
⎢ ⎥
⎢ 2 ⎥
⎣sin(θ)⋅cos(θ) sin (θ) ⎦
"""
M = Matrix([[cos(theta)**2, sin(theta)*cos(theta)],
[sin(theta)*cos(theta), sin(theta)**2]])
return M
def phase_retarder(theta=0, delta=0):
u"""A phase retarder Jones matrix with retardance `delta` at angle `theta`.
Parameters
----------
``theta`` : numeric type or sympy Symbol
The angle of the fast axis relative to the horizontal plane.
``delta`` : numeric type or sympy Symbol
The phase difference between the fast and slow axes of the
transmitted light.
Returns
-------
sympy Matrix
A Jones matrix representing the retarder.
Examples
--------
A generic retarder.
>>> from sympy import pprint, symbols
>>> from sympy.physics.optics.polarization import phase_retarder
>>> theta, delta = symbols("theta, delta", real=True)
>>> R = phase_retarder(theta, delta)
>>> pprint(R, use_unicode=True)
⎡ -ⅈ⋅δ -ⅈ⋅δ ⎤
⎢ ───── ───── ⎥
⎢⎛ ⅈ⋅δ 2 2 ⎞ 2 ⎛ ⅈ⋅δ⎞ 2 ⎥
⎢⎝ℯ ⋅sin (θ) + cos (θ)⎠⋅ℯ ⎝1 - ℯ ⎠⋅ℯ ⋅sin(θ)⋅cos(θ)⎥
⎢ ⎥
⎢ -ⅈ⋅δ -ⅈ⋅δ ⎥
⎢ ───── ─────⎥
⎢⎛ ⅈ⋅δ⎞ 2 ⎛ ⅈ⋅δ 2 2 ⎞ 2 ⎥
⎣⎝1 - ℯ ⎠⋅ℯ ⋅sin(θ)⋅cos(θ) ⎝ℯ ⋅cos (θ) + sin (θ)⎠⋅ℯ ⎦
"""
R = Matrix([[cos(theta)**2 + exp(I*delta)*sin(theta)**2,
(1-exp(I*delta))*cos(theta)*sin(theta)],
[(1-exp(I*delta))*cos(theta)*sin(theta),
sin(theta)**2 + exp(I*delta)*cos(theta)**2]])
return R*exp(-I*delta/2)
def half_wave_retarder(theta):
u"""A half-wave retarder Jones matrix at angle `theta`.
Parameters
----------
``theta`` : numeric type or sympy Symbol
The angle of the fast axis relative to the horizontal plane.
Returns
-------
sympy Matrix
A Jones matrix representing the retarder.
Examples
--------
A generic half-wave plate.
>>> from sympy import pprint, symbols
>>> from sympy.physics.optics.polarization import half_wave_retarder
>>> theta= symbols("theta", real=True)
>>> HWP = half_wave_retarder(theta)
>>> pprint(HWP, use_unicode=True)
⎡ ⎛ 2 2 ⎞ ⎤
⎢-ⅈ⋅⎝- sin (θ) + cos (θ)⎠ -2⋅ⅈ⋅sin(θ)⋅cos(θ) ⎥
⎢ ⎥
⎢ ⎛ 2 2 ⎞⎥
⎣ -2⋅ⅈ⋅sin(θ)⋅cos(θ) -ⅈ⋅⎝sin (θ) - cos (θ)⎠⎦
"""
return phase_retarder(theta, pi)
def quarter_wave_retarder(theta):
u"""A quarter-wave retarder Jones matrix at angle `theta`.
Parameters
----------
``theta`` : numeric type or sympy Symbol
The angle of the fast axis relative to the horizontal plane.
Returns
-------
sympy Matrix
A Jones matrix representing the retarder.
Examples
--------
A generic quarter-wave plate.
>>> from sympy import pprint, symbols
>>> from sympy.physics.optics.polarization import quarter_wave_retarder
>>> theta= symbols("theta", real=True)
>>> QWP = quarter_wave_retarder(theta)
>>> pprint(QWP, use_unicode=True)
⎡ -ⅈ⋅π -ⅈ⋅π ⎤
⎢ ───── ───── ⎥
⎢⎛ 2 2 ⎞ 4 4 ⎥
⎢⎝ⅈ⋅sin (θ) + cos (θ)⎠⋅ℯ (1 - ⅈ)⋅ℯ ⋅sin(θ)⋅cos(θ)⎥
⎢ ⎥
⎢ -ⅈ⋅π -ⅈ⋅π ⎥
⎢ ───── ─────⎥
⎢ 4 ⎛ 2 2 ⎞ 4 ⎥
⎣(1 - ⅈ)⋅ℯ ⋅sin(θ)⋅cos(θ) ⎝sin (θ) + ⅈ⋅cos (θ)⎠⋅ℯ ⎦
"""
return phase_retarder(theta, pi/2)
def transmissive_filter(T):
u"""An attenuator Jones matrix with transmittance `T`.
Parameters
----------
``T`` : numeric type or sympy Symbol
The transmittance of the attenuator.
Returns
-------
sympy Matrix
A Jones matrix representing the filter.
Examples
--------
A generic filter.
>>> from sympy import pprint, symbols
>>> from sympy.physics.optics.polarization import transmissive_filter
>>> T = symbols("T", real=True)
>>> NDF = transmissive_filter(T)
>>> pprint(NDF, use_unicode=True)
⎡√T 0 ⎤
⎢ ⎥
⎣0 √T⎦
"""
return Matrix([[sqrt(T), 0], [0, sqrt(T)]])
def reflective_filter(R):
u"""A reflective filter Jones matrix with reflectance `R`.
Parameters
----------
``R`` : numeric type or sympy Symbol
The reflectance of the filter.
Returns
-------
sympy Matrix
A Jones matrix representing the filter.
Examples
--------
A generic filter.
>>> from sympy import pprint, symbols
>>> from sympy.physics.optics.polarization import reflective_filter
>>> R = symbols("R", real=True)
>>> pprint(reflective_filter(R), use_unicode=True)
⎡√R 0 ⎤
⎢ ⎥
⎣0 -√R⎦
"""
return Matrix([[sqrt(R), 0], [0, -sqrt(R)]])
def mueller_matrix(J):
u"""The Mueller matrix corresponding to Jones matrix `J`.
Parameters
----------
``J`` : sympy Matrix
A Jones matrix.
Returns
-------
sympy Matrix
The corresponding Mueller matrix.
Examples
--------
Generic optical components.
>>> from sympy import pprint, symbols
>>> from sympy.physics.optics.polarization import (mueller_matrix,
... linear_polarizer, half_wave_retarder, quarter_wave_retarder)
>>> theta = symbols("theta", real=True)
A linear_polarizer
>>> pprint(mueller_matrix(linear_polarizer(theta)), use_unicode=True)
⎡ cos(2⋅θ) sin(2⋅θ) ⎤
⎢ 1/2 ──────── ──────── 0⎥
⎢ 2 2 ⎥
⎢ ⎥
⎢cos(2⋅θ) cos(4⋅θ) 1 sin(4⋅θ) ⎥
⎢──────── ──────── + ─ ──────── 0⎥
⎢ 2 4 4 4 ⎥
⎢ ⎥
⎢sin(2⋅θ) sin(4⋅θ) 1 cos(4⋅θ) ⎥
⎢──────── ──────── ─ - ──────── 0⎥
⎢ 2 4 4 4 ⎥
⎢ ⎥
⎣ 0 0 0 0⎦
A half-wave plate
>>> pprint(mueller_matrix(half_wave_retarder(theta)), use_unicode=True)
⎡1 0 0 0 ⎤
⎢ ⎥
⎢ 4 2 ⎥
⎢0 8⋅sin (θ) - 8⋅sin (θ) + 1 sin(4⋅θ) 0 ⎥
⎢ ⎥
⎢ 4 2 ⎥
⎢0 sin(4⋅θ) - 8⋅sin (θ) + 8⋅sin (θ) - 1 0 ⎥
⎢ ⎥
⎣0 0 0 -1⎦
A quarter-wave plate
>>> pprint(mueller_matrix(quarter_wave_retarder(theta)), use_unicode=True)
⎡1 0 0 0 ⎤
⎢ ⎥
⎢ cos(4⋅θ) 1 sin(4⋅θ) ⎥
⎢0 ──────── + ─ ──────── -sin(2⋅θ)⎥
⎢ 2 2 2 ⎥
⎢ ⎥
⎢ sin(4⋅θ) 1 cos(4⋅θ) ⎥
⎢0 ──────── ─ - ──────── cos(2⋅θ) ⎥
⎢ 2 2 2 ⎥
⎢ ⎥
⎣0 sin(2⋅θ) -cos(2⋅θ) 0 ⎦
"""
A = Matrix([[1, 0, 0, 1],
[1, 0, 0, -1],
[0, 1, 1, 0],
[0, -I, I, 0]])
return simplify(A*TensorProduct(J, J.conjugate())*A.inv())
def polarizing_beam_splitter(Tp=1, Rs=1, Ts=0, Rp=0, phia=0, phib=0):
r"""A polarizing beam splitter Jones matrix at angle `theta`.
Parameters
----------
``J`` : sympy Matrix
A Jones matrix.
``Tp`` : numeric type or sympy Symbol
The transmissivity of the P-polarized component.
``Rs`` : numeric type or sympy Symbol
The reflectivity of the S-polarized component.
``Ts`` : numeric type or sympy Symbol
The transmissivity of the S-polarized component.
``Rp`` : numeric type or sympy Symbol
The reflectivity of the P-polarized component.
``phia`` : numeric type or sympy Symbol
The phase difference between transmitted and reflected component for
output mode a.
``phib`` : numeric type or sympy Symbol
The phase difference between transmitted and reflected component for
output mode b.
Returns
-------
sympy Matrix
A 4x4 matrix representing the PBS. This matrix acts on a 4x1 vector
whose first two entries are the Jones vector on one of the PBS ports,
and the last two entries the Jones vector on the other port.
Examples
--------
Generic polarizing beam-splitter.
>>> from sympy import pprint, symbols
>>> from sympy.physics.optics.polarization import polarizing_beam_splitter
>>> Ts, Rs, Tp, Rp = symbols(r"Ts, Rs, Tp, Rp", positive=True)
>>> phia, phib = symbols("phi_a, phi_b", real=True)
>>> PBS = polarizing_beam_splitter(Tp, Rs, Ts, Rp, phia, phib)
>>> pprint(PBS, use_unicode=False)
[ ____ ____ ]
[ \/ Tp 0 I*\/ Rp 0 ]
[ ]
[ ____ ____ I*phi_a]
[ 0 \/ Ts 0 -I*\/ Rs *e ]
[ ]
[ ____ ____ ]
[I*\/ Rp 0 \/ Tp 0 ]
[ ]
[ ____ I*phi_b ____ ]
[ 0 -I*\/ Rs *e 0 \/ Ts ]
"""
PBS = Matrix([[sqrt(Tp), 0, I*sqrt(Rp), 0],
[0, sqrt(Ts), 0, -I*sqrt(Rs)*exp(I*phia)],
[I*sqrt(Rp), 0, sqrt(Tp), 0],
[0, -I*sqrt(Rs)*exp(I*phib), 0, sqrt(Ts)]])
return PBS
|
5c1b89ca90cb33c0dc1b93ec653a17d9892e4edd9f6e607ccfbce3aa4934bbb1 | """
**Contains**
* refraction_angle
* fresnel_coefficients
* deviation
* brewster_angle
* critical_angle
* lens_makers_formula
* mirror_formula
* lens_formula
* hyperfocal_distance
* transverse_magnification
"""
from __future__ import division
__all__ = ['refraction_angle',
'deviation',
'fresnel_coefficients',
'brewster_angle',
'critical_angle',
'lens_makers_formula',
'mirror_formula',
'lens_formula',
'hyperfocal_distance',
'transverse_magnification'
]
from sympy import Symbol, sympify, sqrt, Matrix, acos, oo, Limit, atan2, asin,\
cos, sin, tan, I, cancel, pi, Float
from sympy.core.compatibility import is_sequence
from sympy.geometry.line import Ray3D
from sympy.geometry.util import intersection
from sympy.geometry.plane import Plane
from .medium import Medium
def refractive_index_of_medium(medium):
"""
Helper function that returns refractive index, given a medium
"""
if isinstance(medium, Medium):
n = medium.refractive_index
else:
n = sympify(medium)
return n
def refraction_angle(incident, medium1, medium2, normal=None, plane=None):
"""
This function calculates transmitted vector after refraction at planar
surface. `medium1` and `medium2` can be `Medium` or any sympifiable object.
If `incident` is a number then treated as angle of incidence (in radians)
in which case refraction angle is returned.
If `incident` is an object of `Ray3D`, `normal` also has to be an instance
of `Ray3D` in order to get the output as a `Ray3D`. Please note that if
plane of separation is not provided and normal is an instance of `Ray3D`,
normal will be assumed to be intersecting incident ray at the plane of
separation. This will not be the case when `normal` is a `Matrix` or
any other sequence.
If `incident` is an instance of `Ray3D` and `plane` has not been provided
and `normal` is not `Ray3D`, output will be a `Matrix`.
Parameters
==========
incident : Matrix, Ray3D, sequence or a number
Incident vector or angle of incidence
medium1 : sympy.physics.optics.medium.Medium or sympifiable
Medium 1 or its refractive index
medium2 : sympy.physics.optics.medium.Medium or sympifiable
Medium 2 or its refractive index
normal : Matrix, Ray3D, or sequence
Normal vector
plane : Plane
Plane of separation of the two media.
Returns an angle of refraction or a refracted ray depending on inputs.
Examples
========
>>> from sympy.physics.optics import refraction_angle
>>> from sympy.geometry import Point3D, Ray3D, Plane
>>> from sympy.matrices import Matrix
>>> from sympy import symbols, pi
>>> n = Matrix([0, 0, 1])
>>> P = Plane(Point3D(0, 0, 0), normal_vector=[0, 0, 1])
>>> r1 = Ray3D(Point3D(-1, -1, 1), Point3D(0, 0, 0))
>>> refraction_angle(r1, 1, 1, n)
Matrix([
[ 1],
[ 1],
[-1]])
>>> refraction_angle(r1, 1, 1, plane=P)
Ray3D(Point3D(0, 0, 0), Point3D(1, 1, -1))
With different index of refraction of the two media
>>> n1, n2 = symbols('n1, n2')
>>> refraction_angle(r1, n1, n2, n)
Matrix([
[ n1/n2],
[ n1/n2],
[-sqrt(3)*sqrt(-2*n1**2/(3*n2**2) + 1)]])
>>> refraction_angle(r1, n1, n2, plane=P)
Ray3D(Point3D(0, 0, 0), Point3D(n1/n2, n1/n2, -sqrt(3)*sqrt(-2*n1**2/(3*n2**2) + 1)))
>>> round(refraction_angle(pi/6, 1.2, 1.5), 5)
0.41152
"""
n1 = refractive_index_of_medium(medium1)
n2 = refractive_index_of_medium(medium2)
# check if an incidence angle was supplied instead of a ray
try:
angle_of_incidence = float(incident)
except TypeError:
angle_of_incidence = None
try:
critical_angle_ = critical_angle(medium1, medium2)
except (ValueError, TypeError):
critical_angle_ = None
if angle_of_incidence is not None:
if normal is not None or plane is not None:
raise ValueError('Normal/plane not allowed if incident is an angle')
if not 0.0 <= angle_of_incidence < pi*0.5:
raise ValueError('Angle of incidence not in range [0:pi/2)')
if critical_angle_ and angle_of_incidence > critical_angle_:
raise ValueError('Ray undergoes total internal reflection')
return asin(n1*sin(angle_of_incidence)/n2)
if angle_of_incidence and not 0 <= angle_of_incidence < pi*0.5:
raise ValueError
# Treat the incident as ray below
# A flag to check whether to return Ray3D or not
return_ray = False
if plane is not None and normal is not None:
raise ValueError("Either plane or normal is acceptable.")
if not isinstance(incident, Matrix):
if is_sequence(incident):
_incident = Matrix(incident)
elif isinstance(incident, Ray3D):
_incident = Matrix(incident.direction_ratio)
else:
raise TypeError(
"incident should be a Matrix, Ray3D, or sequence")
else:
_incident = incident
# If plane is provided, get direction ratios of the normal
# to the plane from the plane else go with `normal` param.
if plane is not None:
if not isinstance(plane, Plane):
raise TypeError("plane should be an instance of geometry.plane.Plane")
# If we have the plane, we can get the intersection
# point of incident ray and the plane and thus return
# an instance of Ray3D.
if isinstance(incident, Ray3D):
return_ray = True
intersection_pt = plane.intersection(incident)[0]
_normal = Matrix(plane.normal_vector)
else:
if not isinstance(normal, Matrix):
if is_sequence(normal):
_normal = Matrix(normal)
elif isinstance(normal, Ray3D):
_normal = Matrix(normal.direction_ratio)
if isinstance(incident, Ray3D):
intersection_pt = intersection(incident, normal)
if len(intersection_pt) == 0:
raise ValueError(
"Normal isn't concurrent with the incident ray.")
else:
return_ray = True
intersection_pt = intersection_pt[0]
else:
raise TypeError(
"Normal should be a Matrix, Ray3D, or sequence")
else:
_normal = normal
eta = n1/n2 # Relative index of refraction
# Calculating magnitude of the vectors
mag_incident = sqrt(sum([i**2 for i in _incident]))
mag_normal = sqrt(sum([i**2 for i in _normal]))
# Converting vectors to unit vectors by dividing
# them with their magnitudes
_incident /= mag_incident
_normal /= mag_normal
c1 = -_incident.dot(_normal) # cos(angle_of_incidence)
cs2 = 1 - eta**2*(1 - c1**2) # cos(angle_of_refraction)**2
if cs2.is_negative: # This is the case of total internal reflection(TIR).
return 0
drs = eta*_incident + (eta*c1 - sqrt(cs2))*_normal
# Multiplying unit vector by its magnitude
drs = drs*mag_incident
if not return_ray:
return drs
else:
return Ray3D(intersection_pt, direction_ratio=drs)
def fresnel_coefficients(angle_of_incidence, medium1, medium2):
"""
This function uses Fresnel equations to calculate reflection and
transmission coefficients. Those are obtained for both polarisations
when the electric field vector is in the plane of incidence (labelled 'p')
and when the electric field vector is perpendicular to the plane of
incidence (labelled 's'). There are four real coefficients unless the
incident ray reflects in total internal in which case there are two complex
ones. Angle of incidence is the angle between the incident ray and the
surface normal. ``medium1`` and ``medium2`` can be ``Medium`` or any
sympifiable object.
Parameters
==========
angle_of_incidence : sympifiable
medium1 : Medium or sympifiable
Medium 1 or its refractive index
medium2 : Medium or sympifiable
Medium 2 or its refractive index
Returns a list with four real Fresnel coefficients:
[reflection p (TM), reflection s (TE),
transmission p (TM), transmission s (TE)]
If the ray is undergoes total internal reflection then returns a
list of two complex Fresnel coefficients:
[reflection p (TM), reflection s (TE)]
Examples
========
>>> from sympy.physics.optics import fresnel_coefficients
>>> fresnel_coefficients(0.3, 1, 2)
[0.317843553417859, -0.348645229818821,
0.658921776708929, 0.651354770181179]
>>> fresnel_coefficients(0.6, 2, 1)
[-0.235625382192159 - 0.971843958291041*I,
0.816477005968898 - 0.577377951366403*I]
References
==========
https://en.wikipedia.org/wiki/Fresnel_equations
"""
if not 0 <= 2*angle_of_incidence < pi:
raise ValueError('Angle of incidence not in range [0:pi/2)')
n1 = refractive_index_of_medium(medium1)
n2 = refractive_index_of_medium(medium2)
angle_of_refraction = asin(n1*sin(angle_of_incidence)/n2)
try:
angle_of_total_internal_reflection_onset = critical_angle(n1, n2)
except ValueError:
angle_of_total_internal_reflection_onset = None
if angle_of_total_internal_reflection_onset == None or\
angle_of_total_internal_reflection_onset > angle_of_incidence:
R_s = -sin(angle_of_incidence - angle_of_refraction)\
/sin(angle_of_incidence + angle_of_refraction)
R_p = tan(angle_of_incidence - angle_of_refraction)\
/tan(angle_of_incidence + angle_of_refraction)
T_s = 2*sin(angle_of_refraction)*cos(angle_of_incidence)\
/sin(angle_of_incidence + angle_of_refraction)
T_p = 2*sin(angle_of_refraction)*cos(angle_of_incidence)\
/(sin(angle_of_incidence + angle_of_refraction)\
*cos(angle_of_incidence - angle_of_refraction))
return [R_p, R_s, T_p, T_s]
else:
n = n2/n1
R_s = cancel((cos(angle_of_incidence)-\
I*sqrt(sin(angle_of_incidence)**2 - n**2))\
/(cos(angle_of_incidence)+\
I*sqrt(sin(angle_of_incidence)**2 - n**2)))
R_p = cancel((n**2*cos(angle_of_incidence)-\
I*sqrt(sin(angle_of_incidence)**2 - n**2))\
/(n**2*cos(angle_of_incidence)+\
I*sqrt(sin(angle_of_incidence)**2 - n**2)))
return [R_p, R_s]
def deviation(incident, medium1, medium2, normal=None, plane=None):
"""
This function calculates the angle of deviation of a ray
due to refraction at planar surface.
Parameters
==========
incident : Matrix, Ray3D, sequence or float
Incident vector or angle of incidence
medium1 : sympy.physics.optics.medium.Medium or sympifiable
Medium 1 or its refractive index
medium2 : sympy.physics.optics.medium.Medium or sympifiable
Medium 2 or its refractive index
normal : Matrix, Ray3D, or sequence
Normal vector
plane : Plane
Plane of separation of the two media.
Returns angular deviation between incident and refracted rays
Examples
========
>>> from sympy.physics.optics import deviation
>>> from sympy.geometry import Point3D, Ray3D, Plane
>>> from sympy.matrices import Matrix
>>> from sympy import symbols
>>> n1, n2 = symbols('n1, n2')
>>> n = Matrix([0, 0, 1])
>>> P = Plane(Point3D(0, 0, 0), normal_vector=[0, 0, 1])
>>> r1 = Ray3D(Point3D(-1, -1, 1), Point3D(0, 0, 0))
>>> deviation(r1, 1, 1, n)
0
>>> deviation(r1, n1, n2, plane=P)
-acos(-sqrt(-2*n1**2/(3*n2**2) + 1)) + acos(-sqrt(3)/3)
>>> round(deviation(0.1, 1.2, 1.5), 5)
-0.02005
"""
refracted = refraction_angle(incident,
medium1,
medium2,
normal=normal,
plane=plane)
try:
angle_of_incidence = Float(incident)
except TypeError:
angle_of_incidence = None
if angle_of_incidence is not None:
return float(refracted) - angle_of_incidence
if refracted != 0:
if isinstance(refracted, Ray3D):
refracted = Matrix(refracted.direction_ratio)
if not isinstance(incident, Matrix):
if is_sequence(incident):
_incident = Matrix(incident)
elif isinstance(incident, Ray3D):
_incident = Matrix(incident.direction_ratio)
else:
raise TypeError(
"incident should be a Matrix, Ray3D, or sequence")
else:
_incident = incident
if plane is None:
if not isinstance(normal, Matrix):
if is_sequence(normal):
_normal = Matrix(normal)
elif isinstance(normal, Ray3D):
_normal = Matrix(normal.direction_ratio)
else:
raise TypeError(
"normal should be a Matrix, Ray3D, or sequence")
else:
_normal = normal
else:
_normal = Matrix(plane.normal_vector)
mag_incident = sqrt(sum([i**2 for i in _incident]))
mag_normal = sqrt(sum([i**2 for i in _normal]))
mag_refracted = sqrt(sum([i**2 for i in refracted]))
_incident /= mag_incident
_normal /= mag_normal
refracted /= mag_refracted
i = acos(_incident.dot(_normal))
r = acos(refracted.dot(_normal))
return i - r
def brewster_angle(medium1, medium2):
"""
This function calculates the Brewster's angle of incidence to Medium 2 from
Medium 1 in radians.
Parameters
==========
medium 1 : Medium or sympifiable
Refractive index of Medium 1
medium 2 : Medium or sympifiable
Refractive index of Medium 1
Examples
========
>>> from sympy.physics.optics import brewster_angle
>>> brewster_angle(1, 1.33)
0.926093295503462
"""
n1 = refractive_index_of_medium(medium1)
n2 = refractive_index_of_medium(medium2)
return atan2(n2, n1)
def critical_angle(medium1, medium2):
"""
This function calculates the critical angle of incidence (marking the onset
of total internal) to Medium 2 from Medium 1 in radians.
Parameters
==========
medium 1 : Medium or sympifiable
Refractive index of Medium 1
medium 2 : Medium or sympifiable
Refractive index of Medium 1
Examples
========
>>> from sympy.physics.optics import critical_angle
>>> critical_angle(1.33, 1)
0.850908514477849
"""
n1 = refractive_index_of_medium(medium1)
n2 = refractive_index_of_medium(medium2)
if n2 > n1:
raise ValueError('Total internal reflection impossible for n1 < n2')
else:
return asin(n2/n1)
def lens_makers_formula(n_lens, n_surr, r1, r2):
"""
This function calculates focal length of a thin lens.
It follows cartesian sign convention.
Parameters
==========
n_lens : Medium or sympifiable
Index of refraction of lens.
n_surr : Medium or sympifiable
Index of reflection of surrounding.
r1 : sympifiable
Radius of curvature of first surface.
r2 : sympifiable
Radius of curvature of second surface.
Examples
========
>>> from sympy.physics.optics import lens_makers_formula
>>> lens_makers_formula(1.33, 1, 10, -10)
15.1515151515151
"""
if isinstance(n_lens, Medium):
n_lens = n_lens.refractive_index
else:
n_lens = sympify(n_lens)
if isinstance(n_surr, Medium):
n_surr = n_surr.refractive_index
else:
n_surr = sympify(n_surr)
r1 = sympify(r1)
r2 = sympify(r2)
return 1/((n_lens - n_surr)/n_surr*(1/r1 - 1/r2))
def mirror_formula(focal_length=None, u=None, v=None):
"""
This function provides one of the three parameters
when two of them are supplied.
This is valid only for paraxial rays.
Parameters
==========
focal_length : sympifiable
Focal length of the mirror.
u : sympifiable
Distance of object from the pole on
the principal axis.
v : sympifiable
Distance of the image from the pole
on the principal axis.
Examples
========
>>> from sympy.physics.optics import mirror_formula
>>> from sympy.abc import f, u, v
>>> mirror_formula(focal_length=f, u=u)
f*u/(-f + u)
>>> mirror_formula(focal_length=f, v=v)
f*v/(-f + v)
>>> mirror_formula(u=u, v=v)
u*v/(u + v)
"""
if focal_length and u and v:
raise ValueError("Please provide only two parameters")
focal_length = sympify(focal_length)
u = sympify(u)
v = sympify(v)
if u is oo:
_u = Symbol('u')
if v is oo:
_v = Symbol('v')
if focal_length is oo:
_f = Symbol('f')
if focal_length is None:
if u is oo and v is oo:
return Limit(Limit(_v*_u/(_v + _u), _u, oo), _v, oo).doit()
if u is oo:
return Limit(v*_u/(v + _u), _u, oo).doit()
if v is oo:
return Limit(_v*u/(_v + u), _v, oo).doit()
return v*u/(v + u)
if u is None:
if v is oo and focal_length is oo:
return Limit(Limit(_v*_f/(_v - _f), _v, oo), _f, oo).doit()
if v is oo:
return Limit(_v*focal_length/(_v - focal_length), _v, oo).doit()
if focal_length is oo:
return Limit(v*_f/(v - _f), _f, oo).doit()
return v*focal_length/(v - focal_length)
if v is None:
if u is oo and focal_length is oo:
return Limit(Limit(_u*_f/(_u - _f), _u, oo), _f, oo).doit()
if u is oo:
return Limit(_u*focal_length/(_u - focal_length), _u, oo).doit()
if focal_length is oo:
return Limit(u*_f/(u - _f), _f, oo).doit()
return u*focal_length/(u - focal_length)
def lens_formula(focal_length=None, u=None, v=None):
"""
This function provides one of the three parameters
when two of them are supplied.
This is valid only for paraxial rays.
Parameters
==========
focal_length : sympifiable
Focal length of the mirror.
u : sympifiable
Distance of object from the optical center on
the principal axis.
v : sympifiable
Distance of the image from the optical center
on the principal axis.
Examples
========
>>> from sympy.physics.optics import lens_formula
>>> from sympy.abc import f, u, v
>>> lens_formula(focal_length=f, u=u)
f*u/(f + u)
>>> lens_formula(focal_length=f, v=v)
f*v/(f - v)
>>> lens_formula(u=u, v=v)
u*v/(u - v)
"""
if focal_length and u and v:
raise ValueError("Please provide only two parameters")
focal_length = sympify(focal_length)
u = sympify(u)
v = sympify(v)
if u is oo:
_u = Symbol('u')
if v is oo:
_v = Symbol('v')
if focal_length is oo:
_f = Symbol('f')
if focal_length is None:
if u is oo and v is oo:
return Limit(Limit(_v*_u/(_u - _v), _u, oo), _v, oo).doit()
if u is oo:
return Limit(v*_u/(_u - v), _u, oo).doit()
if v is oo:
return Limit(_v*u/(u - _v), _v, oo).doit()
return v*u/(u - v)
if u is None:
if v is oo and focal_length is oo:
return Limit(Limit(_v*_f/(_f - _v), _v, oo), _f, oo).doit()
if v is oo:
return Limit(_v*focal_length/(focal_length - _v), _v, oo).doit()
if focal_length is oo:
return Limit(v*_f/(_f - v), _f, oo).doit()
return v*focal_length/(focal_length - v)
if v is None:
if u is oo and focal_length is oo:
return Limit(Limit(_u*_f/(_u + _f), _u, oo), _f, oo).doit()
if u is oo:
return Limit(_u*focal_length/(_u + focal_length), _u, oo).doit()
if focal_length is oo:
return Limit(u*_f/(u + _f), _f, oo).doit()
return u*focal_length/(u + focal_length)
def hyperfocal_distance(f, N, c):
"""
Parameters
==========
f: sympifiable
Focal length of a given lens
N: sympifiable
F-number of a given lens
c: sympifiable
Circle of Confusion (CoC) of a given image format
Example
=======
>>> from sympy.physics.optics import hyperfocal_distance
>>> round(hyperfocal_distance(f = 0.5, N = 8, c = 0.0033), 2)
9.47
"""
f = sympify(f)
N = sympify(N)
c = sympify(c)
return (1/(N * c))*(f**2)
def transverse_magnification(si, so):
"""
Calculates the transverse magnification, which is the ratio of the
image size to the object size.
Parameters
==========
so: sympifiable
Lens-object distance
si: sympifiable
Lens-image distance
Example
=======
>>> from sympy.physics.optics import transverse_magnification
>>> transverse_magnification(30, 15)
-2
"""
si = sympify(si)
so = sympify(so)
return (-(si/so))
|
a596c7cc6e474fca6ebd01ca4c0ba0ed56548c9003cc373e22a5c37b7dfda4d6 | import sympy
from sympy.testing.pytest import warns_deprecated_sympy
def test_deprecated_imports():
# https://github.com/sympy/sympy/pull/18245
# Before 1.6 these names were importable with e.g.
# from sympy import *
# from sympy import add
# Now sympy/__init__.py uses __all__ so these names are no longer
# accidentally imported. However many of the names now give a warning and
# this test checks that they are importable but a warning is given
from sympy import add
with warns_deprecated_sympy():
add.Add
modnames = type(add)._DEPRECATED_IMPORTS
assert len(modnames) == 80
for modname in modnames:
name = modname.split('.')[-1]
mod = getattr(sympy, name)
attr = dir(mod.mod)[0]
with warns_deprecated_sympy():
getattr(mod, attr)
|
c081c616b18fb047695c1c9141c3fc74de2e5c89434e158410eb9f7b5dec73cc | """Implementation of DPLL algorithm
Further improvements: eliminate calls to pl_true, implement branching rules,
efficient unit propagation.
References:
- https://en.wikipedia.org/wiki/DPLL_algorithm
- https://www.researchgate.net/publication/242384772_Implementations_of_the_DPLL_Algorithm
"""
from sympy import default_sort_key
from sympy.logic.boolalg import Or, Not, conjuncts, disjuncts, to_cnf, \
to_int_repr, _find_predicates
from sympy.assumptions.cnf import CNF
from sympy.logic.inference import pl_true, literal_symbol
def dpll_satisfiable(expr):
"""
Check satisfiability of a propositional sentence.
It returns a model rather than True when it succeeds
>>> from sympy.abc import A, B
>>> from sympy.logic.algorithms.dpll import dpll_satisfiable
>>> dpll_satisfiable(A & ~B)
{A: True, B: False}
>>> dpll_satisfiable(A & ~A)
False
"""
if not isinstance(expr, CNF):
clauses = conjuncts(to_cnf(expr))
else:
clauses = expr.clauses
if False in clauses:
return False
symbols = sorted(_find_predicates(expr), key=default_sort_key)
symbols_int_repr = set(range(1, len(symbols) + 1))
clauses_int_repr = to_int_repr(clauses, symbols)
result = dpll_int_repr(clauses_int_repr, symbols_int_repr, {})
if not result:
return result
output = {}
for key in result:
output.update({symbols[key - 1]: result[key]})
return output
def dpll(clauses, symbols, model):
"""
Compute satisfiability in a partial model.
Clauses is an array of conjuncts.
>>> from sympy.abc import A, B, D
>>> from sympy.logic.algorithms.dpll import dpll
>>> dpll([A, B, D], [A, B], {D: False})
False
"""
# compute DP kernel
P, value = find_unit_clause(clauses, model)
while P:
model.update({P: value})
symbols.remove(P)
if not value:
P = ~P
clauses = unit_propagate(clauses, P)
P, value = find_unit_clause(clauses, model)
P, value = find_pure_symbol(symbols, clauses)
while P:
model.update({P: value})
symbols.remove(P)
if not value:
P = ~P
clauses = unit_propagate(clauses, P)
P, value = find_pure_symbol(symbols, clauses)
# end DP kernel
unknown_clauses = []
for c in clauses:
val = pl_true(c, model)
if val is False:
return False
if val is not True:
unknown_clauses.append(c)
if not unknown_clauses:
return model
if not clauses:
return model
P = symbols.pop()
model_copy = model.copy()
model.update({P: True})
model_copy.update({P: False})
symbols_copy = symbols[:]
return (dpll(unit_propagate(unknown_clauses, P), symbols, model) or
dpll(unit_propagate(unknown_clauses, Not(P)), symbols_copy, model_copy))
def dpll_int_repr(clauses, symbols, model):
"""
Compute satisfiability in a partial model.
Arguments are expected to be in integer representation
>>> from sympy.logic.algorithms.dpll import dpll_int_repr
>>> dpll_int_repr([{1}, {2}, {3}], {1, 2}, {3: False})
False
"""
# compute DP kernel
P, value = find_unit_clause_int_repr(clauses, model)
while P:
model.update({P: value})
symbols.remove(P)
if not value:
P = -P
clauses = unit_propagate_int_repr(clauses, P)
P, value = find_unit_clause_int_repr(clauses, model)
P, value = find_pure_symbol_int_repr(symbols, clauses)
while P:
model.update({P: value})
symbols.remove(P)
if not value:
P = -P
clauses = unit_propagate_int_repr(clauses, P)
P, value = find_pure_symbol_int_repr(symbols, clauses)
# end DP kernel
unknown_clauses = []
for c in clauses:
val = pl_true_int_repr(c, model)
if val is False:
return False
if val is not True:
unknown_clauses.append(c)
if not unknown_clauses:
return model
P = symbols.pop()
model_copy = model.copy()
model.update({P: True})
model_copy.update({P: False})
symbols_copy = symbols.copy()
return (dpll_int_repr(unit_propagate_int_repr(unknown_clauses, P), symbols, model) or
dpll_int_repr(unit_propagate_int_repr(unknown_clauses, -P), symbols_copy, model_copy))
### helper methods for DPLL
def pl_true_int_repr(clause, model={}):
"""
Lightweight version of pl_true.
Argument clause represents the set of args of an Or clause. This is used
inside dpll_int_repr, it is not meant to be used directly.
>>> from sympy.logic.algorithms.dpll import pl_true_int_repr
>>> pl_true_int_repr({1, 2}, {1: False})
>>> pl_true_int_repr({1, 2}, {1: False, 2: False})
False
"""
result = False
for lit in clause:
if lit < 0:
p = model.get(-lit)
if p is not None:
p = not p
else:
p = model.get(lit)
if p is True:
return True
elif p is None:
result = None
return result
def unit_propagate(clauses, symbol):
"""
Returns an equivalent set of clauses
If a set of clauses contains the unit clause l, the other clauses are
simplified by the application of the two following rules:
1. every clause containing l is removed
2. in every clause that contains ~l this literal is deleted
Arguments are expected to be in CNF.
>>> from sympy.abc import A, B, D
>>> from sympy.logic.algorithms.dpll import unit_propagate
>>> unit_propagate([A | B, D | ~B, B], B)
[D, B]
"""
output = []
for c in clauses:
if c.func != Or:
output.append(c)
continue
for arg in c.args:
if arg == ~symbol:
output.append(Or(*[x for x in c.args if x != ~symbol]))
break
if arg == symbol:
break
else:
output.append(c)
return output
def unit_propagate_int_repr(clauses, s):
"""
Same as unit_propagate, but arguments are expected to be in integer
representation
>>> from sympy.logic.algorithms.dpll import unit_propagate_int_repr
>>> unit_propagate_int_repr([{1, 2}, {3, -2}, {2}], 2)
[{3}]
"""
negated = {-s}
return [clause - negated for clause in clauses if s not in clause]
def find_pure_symbol(symbols, unknown_clauses):
"""
Find a symbol and its value if it appears only as a positive literal
(or only as a negative) in clauses.
>>> from sympy.abc import A, B, D
>>> from sympy.logic.algorithms.dpll import find_pure_symbol
>>> find_pure_symbol([A, B, D], [A|~B,~B|~D,D|A])
(A, True)
"""
for sym in symbols:
found_pos, found_neg = False, False
for c in unknown_clauses:
if not found_pos and sym in disjuncts(c):
found_pos = True
if not found_neg and Not(sym) in disjuncts(c):
found_neg = True
if found_pos != found_neg:
return sym, found_pos
return None, None
def find_pure_symbol_int_repr(symbols, unknown_clauses):
"""
Same as find_pure_symbol, but arguments are expected
to be in integer representation
>>> from sympy.logic.algorithms.dpll import find_pure_symbol_int_repr
>>> find_pure_symbol_int_repr({1,2,3},
... [{1, -2}, {-2, -3}, {3, 1}])
(1, True)
"""
all_symbols = set().union(*unknown_clauses)
found_pos = all_symbols.intersection(symbols)
found_neg = all_symbols.intersection([-s for s in symbols])
for p in found_pos:
if -p not in found_neg:
return p, True
for p in found_neg:
if -p not in found_pos:
return -p, False
return None, None
def find_unit_clause(clauses, model):
"""
A unit clause has only 1 variable that is not bound in the model.
>>> from sympy.abc import A, B, D
>>> from sympy.logic.algorithms.dpll import find_unit_clause
>>> find_unit_clause([A | B | D, B | ~D, A | ~B], {A:True})
(B, False)
"""
for clause in clauses:
num_not_in_model = 0
for literal in disjuncts(clause):
sym = literal_symbol(literal)
if sym not in model:
num_not_in_model += 1
P, value = sym, not isinstance(literal, Not)
if num_not_in_model == 1:
return P, value
return None, None
def find_unit_clause_int_repr(clauses, model):
"""
Same as find_unit_clause, but arguments are expected to be in
integer representation.
>>> from sympy.logic.algorithms.dpll import find_unit_clause_int_repr
>>> find_unit_clause_int_repr([{1, 2, 3},
... {2, -3}, {1, -2}], {1: True})
(2, False)
"""
bound = set(model) | {-sym for sym in model}
for clause in clauses:
unbound = clause - bound
if len(unbound) == 1:
p = unbound.pop()
if p < 0:
return -p, False
else:
return p, True
return None, None
|
25313b97df2e9a82fed202d4919656b7ae82fd249b227a98b32fc1f1b113056c | from sympy import (
Rational, Symbol, N, I, Abs, sqrt, exp, Float, sin,
cos, symbols)
from sympy.matrices import eye, Matrix
from sympy.core.singleton import S
from sympy.testing.pytest import raises, XFAIL
from sympy.matrices.matrices import NonSquareMatrixError, MatrixError
from sympy.simplify.simplify import simplify
from sympy.matrices.immutable import ImmutableMatrix
from sympy.testing.pytest import slow
from sympy.testing.matrices import allclose
def test_eigen():
R = Rational
M = Matrix.eye(3)
assert M.eigenvals(multiple=False) == {S.One: 3}
assert M.eigenvals(multiple=True) == [1, 1, 1]
assert M.eigenvects() == (
[(1, 3, [Matrix([1, 0, 0]),
Matrix([0, 1, 0]),
Matrix([0, 0, 1])])])
assert M.left_eigenvects() == (
[(1, 3, [Matrix([[1, 0, 0]]),
Matrix([[0, 1, 0]]),
Matrix([[0, 0, 1]])])])
M = Matrix([[0, 1, 1],
[1, 0, 0],
[1, 1, 1]])
assert M.eigenvals() == {2*S.One: 1, -S.One: 1, S.Zero: 1}
assert M.eigenvects() == (
[
(-1, 1, [Matrix([-1, 1, 0])]),
( 0, 1, [Matrix([0, -1, 1])]),
( 2, 1, [Matrix([R(2, 3), R(1, 3), 1])])
])
assert M.left_eigenvects() == (
[
(-1, 1, [Matrix([[-2, 1, 1]])]),
(0, 1, [Matrix([[-1, -1, 1]])]),
(2, 1, [Matrix([[1, 1, 1]])])
])
a = Symbol('a')
M = Matrix([[a, 0],
[0, 1]])
assert M.eigenvals() == {a: 1, S.One: 1}
M = Matrix([[1, -1],
[1, 3]])
assert M.eigenvects() == ([(2, 2, [Matrix(2, 1, [-1, 1])])])
assert M.left_eigenvects() == ([(2, 2, [Matrix([[1, 1]])])])
M = Matrix([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
a = R(15, 2)
b = 3*33**R(1, 2)
c = R(13, 2)
d = (R(33, 8) + 3*b/8)
e = (R(33, 8) - 3*b/8)
def NS(e, n):
return str(N(e, n))
r = [
(a - b/2, 1, [Matrix([(12 + 24/(c - b/2))/((c - b/2)*e) + 3/(c - b/2),
(6 + 12/(c - b/2))/e, 1])]),
( 0, 1, [Matrix([1, -2, 1])]),
(a + b/2, 1, [Matrix([(12 + 24/(c + b/2))/((c + b/2)*d) + 3/(c + b/2),
(6 + 12/(c + b/2))/d, 1])]),
]
r1 = [(NS(r[i][0], 2), NS(r[i][1], 2),
[NS(j, 2) for j in r[i][2][0]]) for i in range(len(r))]
r = M.eigenvects()
r2 = [(NS(r[i][0], 2), NS(r[i][1], 2),
[NS(j, 2) for j in r[i][2][0]]) for i in range(len(r))]
assert sorted(r1) == sorted(r2)
eps = Symbol('eps', real=True)
M = Matrix([[abs(eps), I*eps ],
[-I*eps, abs(eps) ]])
assert M.eigenvects() == (
[
( 0, 1, [Matrix([[-I*eps/abs(eps)], [1]])]),
( 2*abs(eps), 1, [ Matrix([[I*eps/abs(eps)], [1]]) ] ),
])
assert M.left_eigenvects() == (
[
(0, 1, [Matrix([[I*eps/Abs(eps), 1]])]),
(2*Abs(eps), 1, [Matrix([[-I*eps/Abs(eps), 1]])])
])
M = Matrix(3, 3, [1, 2, 0, 0, 3, 0, 2, -4, 2])
M._eigenvects = M.eigenvects(simplify=False)
assert max(i.q for i in M._eigenvects[0][2][0]) > 1
M._eigenvects = M.eigenvects(simplify=True)
assert max(i.q for i in M._eigenvects[0][2][0]) == 1
M = Matrix([[Rational(1, 4), 1], [1, 1]])
assert M.eigenvects(simplify=True) == [
(Rational(5, 8) - sqrt(73)/8, 1, [Matrix([[-sqrt(73)/8 - Rational(3, 8)], [1]])]),
(Rational(5, 8) + sqrt(73)/8, 1, [Matrix([[Rational(-3, 8) + sqrt(73)/8], [1]])])]
assert M.eigenvects(simplify=False) == [
(Rational(5, 8) - sqrt(73)/8, 1, [Matrix([[-1/(-Rational(3, 8) + sqrt(73)/8)], [1]])]),
(Rational(5, 8) + sqrt(73)/8, 1, [Matrix([[8/(3 + sqrt(73))], [1]])])]
# issue 10719
assert Matrix([]).eigenvals() == {}
assert Matrix([]).eigenvals(multiple=True) == []
assert Matrix([]).eigenvects() == []
# issue 15119
raises(NonSquareMatrixError, lambda : Matrix([[1, 2], [0, 4], [0, 0]]).eigenvals())
raises(NonSquareMatrixError, lambda : Matrix([[1, 0], [3, 4], [5, 6]]).eigenvals())
raises(NonSquareMatrixError, lambda : Matrix([[1, 2, 3], [0, 5, 6]]).eigenvals())
raises(NonSquareMatrixError, lambda : Matrix([[1, 0, 0], [4, 5, 0]]).eigenvals())
raises(NonSquareMatrixError, lambda : Matrix([[1, 2, 3], [0, 5, 6]]).eigenvals(error_when_incomplete = False))
raises(NonSquareMatrixError, lambda : Matrix([[1, 0, 0], [4, 5, 0]]).eigenvals(error_when_incomplete = False))
# issue 15125
from sympy.core.function import count_ops
q = Symbol("q", positive = True)
m = Matrix([[-2, exp(-q), 1], [exp(q), -2, 1], [1, 1, -2]])
assert count_ops(m.eigenvals(simplify=False)) > count_ops(m.eigenvals(simplify=True))
assert count_ops(m.eigenvals(simplify=lambda x: x)) > count_ops(m.eigenvals(simplify=True))
assert isinstance(m.eigenvals(simplify=True, multiple=False), dict)
assert isinstance(m.eigenvals(simplify=True, multiple=True), list)
assert isinstance(m.eigenvals(simplify=lambda x: x, multiple=False), dict)
assert isinstance(m.eigenvals(simplify=lambda x: x, multiple=True), list)
def test_float_eigenvals():
m = Matrix([[1, .6, .6], [.6, .9, .9], [.9, .6, .6]])
evals = [
Rational(5, 4) - sqrt(385)/20,
sqrt(385)/20 + Rational(5, 4),
S.Zero]
n_evals = m.eigenvals(rational=True, multiple=True)
n_evals = sorted(n_evals)
s_evals = [x.evalf() for x in evals]
s_evals = sorted(s_evals)
for x, y in zip(n_evals, s_evals):
assert abs(x-y) < 10**-9
@XFAIL
def test_eigen_vects():
m = Matrix(2, 2, [1, 0, 0, I])
raises(NotImplementedError, lambda: m.is_diagonalizable(True))
# !!! bug because of eigenvects() or roots(x**2 + (-1 - I)*x + I, x)
# see issue 5292
assert not m.is_diagonalizable(True)
raises(MatrixError, lambda: m.diagonalize(True))
(P, D) = m.diagonalize(True)
def test_issue_8240():
# Eigenvalues of large triangular matrices
x, y = symbols('x y')
n = 200
diagonal_variables = [Symbol('x%s' % i) for i in range(n)]
M = [[0 for i in range(n)] for j in range(n)]
for i in range(n):
M[i][i] = diagonal_variables[i]
M = Matrix(M)
eigenvals = M.eigenvals()
assert len(eigenvals) == n
for i in range(n):
assert eigenvals[diagonal_variables[i]] == 1
eigenvals = M.eigenvals(multiple=True)
assert set(eigenvals) == set(diagonal_variables)
# with multiplicity
M = Matrix([[x, 0, 0], [1, y, 0], [2, 3, x]])
eigenvals = M.eigenvals()
assert eigenvals == {x: 2, y: 1}
eigenvals = M.eigenvals(multiple=True)
assert len(eigenvals) == 3
assert eigenvals.count(x) == 2
assert eigenvals.count(y) == 1
def test_eigenvals():
M = Matrix([[0, 1, 1],
[1, 0, 0],
[1, 1, 1]])
assert M.eigenvals() == {2*S.One: 1, -S.One: 1, S.Zero: 1}
# if we cannot factor the char poly, we raise an error
m = Matrix([
[3, 0, 0, 0, -3],
[0, -3, -3, 0, 3],
[0, 3, 0, 3, 0],
[0, 0, 3, 0, 3],
[3, 0, 0, 3, 0]])
raises(MatrixError, lambda: m.eigenvals())
def test_eigenvects():
M = Matrix([[0, 1, 1],
[1, 0, 0],
[1, 1, 1]])
vecs = M.eigenvects()
for val, mult, vec_list in vecs:
assert len(vec_list) == 1
assert M*vec_list[0] == val*vec_list[0]
def test_left_eigenvects():
M = Matrix([[0, 1, 1],
[1, 0, 0],
[1, 1, 1]])
vecs = M.left_eigenvects()
for val, mult, vec_list in vecs:
assert len(vec_list) == 1
assert vec_list[0]*M == val*vec_list[0]
@slow
def test_bidiagonalize():
M = Matrix([[1, 0, 0],
[0, 1, 0],
[0, 0, 1]])
assert M.bidiagonalize() == M
assert M.bidiagonalize(upper=False) == M
assert M.bidiagonalize() == M
assert M.bidiagonal_decomposition() == (M, M, M)
assert M.bidiagonal_decomposition(upper=False) == (M, M, M)
assert M.bidiagonalize() == M
import random
#Real Tests
for real_test in range(2):
test_values = []
row = 2
col = 2
for _ in range(row * col):
value = random.randint(-1000000000, 1000000000)
test_values = test_values + [value]
# L -> Lower Bidiagonalization
# M -> Mutable Matrix
# N -> Immutable Matrix
# 0 -> Bidiagonalized form
# 1,2,3 -> Bidiagonal_decomposition matrices
# 4 -> Product of 1 2 3
M = Matrix(row, col, test_values)
N = ImmutableMatrix(M)
N1, N2, N3 = N.bidiagonal_decomposition()
M1, M2, M3 = M.bidiagonal_decomposition()
M0 = M.bidiagonalize()
N0 = N.bidiagonalize()
N4 = N1 * N2 * N3
M4 = M1 * M2 * M3
N2.simplify()
N4.simplify()
N0.simplify()
M0.simplify()
M2.simplify()
M4.simplify()
LM0 = M.bidiagonalize(upper=False)
LM1, LM2, LM3 = M.bidiagonal_decomposition(upper=False)
LN0 = N.bidiagonalize(upper=False)
LN1, LN2, LN3 = N.bidiagonal_decomposition(upper=False)
LN4 = LN1 * LN2 * LN3
LM4 = LM1 * LM2 * LM3
LN2.simplify()
LN4.simplify()
LN0.simplify()
LM0.simplify()
LM2.simplify()
LM4.simplify()
assert M == M4
assert M2 == M0
assert N == N4
assert N2 == N0
assert M == LM4
assert LM2 == LM0
assert N == LN4
assert LN2 == LN0
#Complex Tests
for complex_test in range(2):
test_values = []
size = 2
for _ in range(size * size):
real = random.randint(-1000000000, 1000000000)
comp = random.randint(-1000000000, 1000000000)
value = real + comp * I
test_values = test_values + [value]
M = Matrix(size, size, test_values)
N = ImmutableMatrix(M)
# L -> Lower Bidiagonalization
# M -> Mutable Matrix
# N -> Immutable Matrix
# 0 -> Bidiagonalized form
# 1,2,3 -> Bidiagonal_decomposition matrices
# 4 -> Product of 1 2 3
N1, N2, N3 = N.bidiagonal_decomposition()
M1, M2, M3 = M.bidiagonal_decomposition()
M0 = M.bidiagonalize()
N0 = N.bidiagonalize()
N4 = N1 * N2 * N3
M4 = M1 * M2 * M3
N2.simplify()
N4.simplify()
N0.simplify()
M0.simplify()
M2.simplify()
M4.simplify()
LM0 = M.bidiagonalize(upper=False)
LM1, LM2, LM3 = M.bidiagonal_decomposition(upper=False)
LN0 = N.bidiagonalize(upper=False)
LN1, LN2, LN3 = N.bidiagonal_decomposition(upper=False)
LN4 = LN1 * LN2 * LN3
LM4 = LM1 * LM2 * LM3
LN2.simplify()
LN4.simplify()
LN0.simplify()
LM0.simplify()
LM2.simplify()
LM4.simplify()
assert M == M4
assert M2 == M0
assert N == N4
assert N2 == N0
assert M == LM4
assert LM2 == LM0
assert N == LN4
assert LN2 == LN0
M = Matrix(18, 8, range(1, 145))
M = M.applyfunc(lambda i: Float(i))
assert M.bidiagonal_decomposition()[1] == M.bidiagonalize()
assert M.bidiagonal_decomposition(upper=False)[1] == M.bidiagonalize(upper=False)
a, b, c = M.bidiagonal_decomposition()
diff = a * b * c - M
assert abs(max(diff)) < 10**-12
def test_diagonalize():
m = Matrix(2, 2, [0, -1, 1, 0])
raises(MatrixError, lambda: m.diagonalize(reals_only=True))
P, D = m.diagonalize()
assert D.is_diagonal()
assert D == Matrix([
[-I, 0],
[ 0, I]])
# make sure we use floats out if floats are passed in
m = Matrix(2, 2, [0, .5, .5, 0])
P, D = m.diagonalize()
assert all(isinstance(e, Float) for e in D.values())
assert all(isinstance(e, Float) for e in P.values())
_, D2 = m.diagonalize(reals_only=True)
assert D == D2
m = Matrix(
[[0, 1, 0, 0], [1, 0, 0, 0.002], [0.002, 0, 0, 1], [0, 0, 1, 0]])
P, D = m.diagonalize()
assert allclose(P*D, m*P)
def test_is_diagonalizable():
a, b, c = symbols('a b c')
m = Matrix(2, 2, [a, c, c, b])
assert m.is_symmetric()
assert m.is_diagonalizable()
assert not Matrix(2, 2, [1, 1, 0, 1]).is_diagonalizable()
m = Matrix(2, 2, [0, -1, 1, 0])
assert m.is_diagonalizable()
assert not m.is_diagonalizable(reals_only=True)
def test_jordan_form():
m = Matrix(3, 2, [-3, 1, -3, 20, 3, 10])
raises(NonSquareMatrixError, lambda: m.jordan_form())
# the next two tests test the cases where the old
# algorithm failed due to the fact that the block structure can
# *NOT* be determined from algebraic and geometric multiplicity alone
# This can be seen most easily when one lets compute the J.c.f. of a matrix that
# is in J.c.f already.
m = Matrix(4, 4, [2, 1, 0, 0,
0, 2, 1, 0,
0, 0, 2, 0,
0, 0, 0, 2
])
P, J = m.jordan_form()
assert m == J
m = Matrix(4, 4, [2, 1, 0, 0,
0, 2, 0, 0,
0, 0, 2, 1,
0, 0, 0, 2
])
P, J = m.jordan_form()
assert m == J
A = Matrix([[ 2, 4, 1, 0],
[-4, 2, 0, 1],
[ 0, 0, 2, 4],
[ 0, 0, -4, 2]])
P, J = A.jordan_form()
assert simplify(P*J*P.inv()) == A
assert Matrix(1, 1, [1]).jordan_form() == (Matrix([1]), Matrix([1]))
assert Matrix(1, 1, [1]).jordan_form(calc_transform=False) == Matrix([1])
# make sure if we cannot factor the characteristic polynomial, we raise an error
m = Matrix([[3, 0, 0, 0, -3], [0, -3, -3, 0, 3], [0, 3, 0, 3, 0], [0, 0, 3, 0, 3], [3, 0, 0, 3, 0]])
raises(MatrixError, lambda: m.jordan_form())
# make sure that if the input has floats, the output does too
m = Matrix([
[ 0.6875, 0.125 + 0.1875*sqrt(3)],
[0.125 + 0.1875*sqrt(3), 0.3125]])
P, J = m.jordan_form()
assert all(isinstance(x, Float) or x == 0 for x in P)
assert all(isinstance(x, Float) or x == 0 for x in J)
def test_singular_values():
x = Symbol('x', real=True)
A = Matrix([[0, 1*I], [2, 0]])
# if singular values can be sorted, they should be in decreasing order
assert A.singular_values() == [2, 1]
A = eye(3)
A[1, 1] = x
A[2, 2] = 5
vals = A.singular_values()
# since Abs(x) cannot be sorted, test set equality
assert set(vals) == {5, 1, Abs(x)}
A = Matrix([[sin(x), cos(x)], [-cos(x), sin(x)]])
vals = [sv.trigsimp() for sv in A.singular_values()]
assert vals == [S.One, S.One]
A = Matrix([
[2, 4],
[1, 3],
[0, 0],
[0, 0]
])
assert A.singular_values() == \
[sqrt(sqrt(221) + 15), sqrt(15 - sqrt(221))]
assert A.T.singular_values() == \
[sqrt(sqrt(221) + 15), sqrt(15 - sqrt(221)), 0, 0]
def test___eq__():
assert (Matrix(
[[0, 1, 1],
[1, 0, 0],
[1, 1, 1]]) == {}) is False
def test_definite():
# Examples from Gilbert Strang, "Introduction to Linear Algebra"
# Positive definite matrices
m = Matrix([[2, -1, 0], [-1, 2, -1], [0, -1, 2]])
assert m.is_positive_definite == True
assert m.is_positive_semidefinite == True
assert m.is_negative_definite == False
assert m.is_negative_semidefinite == False
assert m.is_indefinite == False
m = Matrix([[5, 4], [4, 5]])
assert m.is_positive_definite == True
assert m.is_positive_semidefinite == True
assert m.is_negative_definite == False
assert m.is_negative_semidefinite == False
assert m.is_indefinite == False
# Positive semidefinite matrices
m = Matrix([[2, -1, -1], [-1, 2, -1], [-1, -1, 2]])
assert m.is_positive_definite == False
assert m.is_positive_semidefinite == True
assert m.is_negative_definite == False
assert m.is_negative_semidefinite == False
assert m.is_indefinite == False
m = Matrix([[1, 2], [2, 4]])
assert m.is_positive_definite == False
assert m.is_positive_semidefinite == True
assert m.is_negative_definite == False
assert m.is_negative_semidefinite == False
assert m.is_indefinite == False
# Examples from Mathematica documentation
# Non-hermitian positive definite matrices
m = Matrix([[2, 3], [4, 8]])
assert m.is_positive_definite == True
assert m.is_positive_semidefinite == True
assert m.is_negative_definite == False
assert m.is_negative_semidefinite == False
assert m.is_indefinite == False
m = Matrix([[1, 2*I], [-I, 4]])
assert m.is_positive_definite == True
assert m.is_positive_semidefinite == True
assert m.is_negative_definite == False
assert m.is_negative_semidefinite == False
assert m.is_indefinite == False
# Symbolic matrices examples
a = Symbol('a', positive=True)
b = Symbol('b', negative=True)
m = Matrix([[a, 0, 0], [0, a, 0], [0, 0, a]])
assert m.is_positive_definite == True
assert m.is_positive_semidefinite == True
assert m.is_negative_definite == False
assert m.is_negative_semidefinite == False
assert m.is_indefinite == False
m = Matrix([[b, 0, 0], [0, b, 0], [0, 0, b]])
assert m.is_positive_definite == False
assert m.is_positive_semidefinite == False
assert m.is_negative_definite == True
assert m.is_negative_semidefinite == True
assert m.is_indefinite == False
m = Matrix([[a, 0], [0, b]])
assert m.is_positive_definite == False
assert m.is_positive_semidefinite == False
assert m.is_negative_definite == False
assert m.is_negative_semidefinite == False
assert m.is_indefinite == True
m = Matrix([
[0.0228202735623867, 0.00518748979085398,
-0.0743036351048907, -0.00709135324903921],
[0.00518748979085398, 0.0349045359786350,
0.0830317991056637, 0.00233147902806909],
[-0.0743036351048907, 0.0830317991056637,
1.15859676366277, 0.340359081555988],
[-0.00709135324903921, 0.00233147902806909,
0.340359081555988, 0.928147644848199]
])
assert m.is_positive_definite == True
assert m.is_positive_semidefinite == True
assert m.is_indefinite == False
|
4bf4ecb19100533ac1d8501507ec50791b127048a64d6800b2559763f40fd3cb | from sympy.assumptions import Q
from sympy.core.add import Add
from sympy.core.function import Function
from sympy.core.numbers import I, Integer, oo, pi, Rational
from sympy.core.singleton import S
from sympy.core.symbol import Symbol, symbols
from sympy.functions.elementary.complexes import Abs
from sympy.functions.elementary.exponential import exp
from sympy.functions.elementary.miscellaneous import sqrt
from sympy.functions.elementary.trigonometric import cos, sin
from sympy.matrices.common import (ShapeError, NonSquareMatrixError,
_MinimalMatrix, _CastableMatrix, MatrixShaping, MatrixProperties,
MatrixOperations, MatrixArithmetic, MatrixSpecial)
from sympy.matrices.matrices import MatrixCalculus
from sympy.matrices import (Matrix, diag, eye,
matrix_multiply_elementwise, ones, zeros, SparseMatrix, banded,
MutableDenseMatrix, MutableSparseMatrix, ImmutableDenseMatrix,
ImmutableSparseMatrix)
from sympy.polys.polytools import Poly
from sympy.utilities.iterables import flatten
from sympy.testing.pytest import raises, XFAIL, warns_deprecated_sympy
from sympy.abc import x, y, z
# classes to test the basic matrix classes
class ShapingOnlyMatrix(_MinimalMatrix, _CastableMatrix, MatrixShaping):
pass
def eye_Shaping(n):
return ShapingOnlyMatrix(n, n, lambda i, j: int(i == j))
def zeros_Shaping(n):
return ShapingOnlyMatrix(n, n, lambda i, j: 0)
class PropertiesOnlyMatrix(_MinimalMatrix, _CastableMatrix, MatrixProperties):
pass
def eye_Properties(n):
return PropertiesOnlyMatrix(n, n, lambda i, j: int(i == j))
def zeros_Properties(n):
return PropertiesOnlyMatrix(n, n, lambda i, j: 0)
class OperationsOnlyMatrix(_MinimalMatrix, _CastableMatrix, MatrixOperations):
pass
def eye_Operations(n):
return OperationsOnlyMatrix(n, n, lambda i, j: int(i == j))
def zeros_Operations(n):
return OperationsOnlyMatrix(n, n, lambda i, j: 0)
class ArithmeticOnlyMatrix(_MinimalMatrix, _CastableMatrix, MatrixArithmetic):
pass
def eye_Arithmetic(n):
return ArithmeticOnlyMatrix(n, n, lambda i, j: int(i == j))
def zeros_Arithmetic(n):
return ArithmeticOnlyMatrix(n, n, lambda i, j: 0)
class SpecialOnlyMatrix(_MinimalMatrix, _CastableMatrix, MatrixSpecial):
pass
class CalculusOnlyMatrix(_MinimalMatrix, _CastableMatrix, MatrixCalculus):
pass
def test__MinimalMatrix():
x = _MinimalMatrix(2, 3, [1, 2, 3, 4, 5, 6])
assert x.rows == 2
assert x.cols == 3
assert x[2] == 3
assert x[1, 1] == 5
assert list(x) == [1, 2, 3, 4, 5, 6]
assert list(x[1, :]) == [4, 5, 6]
assert list(x[:, 1]) == [2, 5]
assert list(x[:, :]) == list(x)
assert x[:, :] == x
assert _MinimalMatrix(x) == x
assert _MinimalMatrix([[1, 2, 3], [4, 5, 6]]) == x
assert _MinimalMatrix(([1, 2, 3], [4, 5, 6])) == x
assert _MinimalMatrix([(1, 2, 3), (4, 5, 6)]) == x
assert _MinimalMatrix(((1, 2, 3), (4, 5, 6))) == x
assert not (_MinimalMatrix([[1, 2], [3, 4], [5, 6]]) == x)
# ShapingOnlyMatrix tests
def test_vec():
m = ShapingOnlyMatrix(2, 2, [1, 3, 2, 4])
m_vec = m.vec()
assert m_vec.cols == 1
for i in range(4):
assert m_vec[i] == i + 1
def test_todok():
a, b, c, d = symbols('a:d')
m1 = MutableDenseMatrix([[a, b], [c, d]])
m2 = ImmutableDenseMatrix([[a, b], [c, d]])
m3 = MutableSparseMatrix([[a, b], [c, d]])
m4 = ImmutableSparseMatrix([[a, b], [c, d]])
assert m1.todok() == m2.todok() == m3.todok() == m4.todok() == \
{(0, 0): a, (0, 1): b, (1, 0): c, (1, 1): d}
def test_tolist():
lst = [[S.One, S.Half, x*y, S.Zero], [x, y, z, x**2], [y, -S.One, z*x, 3]]
flat_lst = [S.One, S.Half, x*y, S.Zero, x, y, z, x**2, y, -S.One, z*x, 3]
m = ShapingOnlyMatrix(3, 4, flat_lst)
assert m.tolist() == lst
def test_row_col_del():
e = ShapingOnlyMatrix(3, 3, [1, 2, 3, 4, 5, 6, 7, 8, 9])
raises(ValueError, lambda: e.row_del(5))
raises(ValueError, lambda: e.row_del(-5))
raises(ValueError, lambda: e.col_del(5))
raises(ValueError, lambda: e.col_del(-5))
assert e.row_del(2) == e.row_del(-1) == Matrix([[1, 2, 3], [4, 5, 6]])
assert e.col_del(2) == e.col_del(-1) == Matrix([[1, 2], [4, 5], [7, 8]])
assert e.row_del(1) == e.row_del(-2) == Matrix([[1, 2, 3], [7, 8, 9]])
assert e.col_del(1) == e.col_del(-2) == Matrix([[1, 3], [4, 6], [7, 9]])
def test_get_diag_blocks1():
a = Matrix([[1, 2], [2, 3]])
b = Matrix([[3, x], [y, 3]])
c = Matrix([[3, x, 3], [y, 3, z], [x, y, z]])
assert a.get_diag_blocks() == [a]
assert b.get_diag_blocks() == [b]
assert c.get_diag_blocks() == [c]
def test_get_diag_blocks2():
a = Matrix([[1, 2], [2, 3]])
b = Matrix([[3, x], [y, 3]])
c = Matrix([[3, x, 3], [y, 3, z], [x, y, z]])
A, B, C, D = diag(a, b, b), diag(a, b, c), diag(a, c, b), diag(c, c, b)
A = ShapingOnlyMatrix(A.rows, A.cols, A)
B = ShapingOnlyMatrix(B.rows, B.cols, B)
C = ShapingOnlyMatrix(C.rows, C.cols, C)
D = ShapingOnlyMatrix(D.rows, D.cols, D)
assert A.get_diag_blocks() == [a, b, b]
assert B.get_diag_blocks() == [a, b, c]
assert C.get_diag_blocks() == [a, c, b]
assert D.get_diag_blocks() == [c, c, b]
def test_shape():
m = ShapingOnlyMatrix(1, 2, [0, 0])
m.shape == (1, 2)
def test_reshape():
m0 = eye_Shaping(3)
assert m0.reshape(1, 9) == Matrix(1, 9, (1, 0, 0, 0, 1, 0, 0, 0, 1))
m1 = ShapingOnlyMatrix(3, 4, lambda i, j: i + j)
assert m1.reshape(
4, 3) == Matrix(((0, 1, 2), (3, 1, 2), (3, 4, 2), (3, 4, 5)))
assert m1.reshape(2, 6) == Matrix(((0, 1, 2, 3, 1, 2), (3, 4, 2, 3, 4, 5)))
def test_row_col():
m = ShapingOnlyMatrix(3, 3, [1, 2, 3, 4, 5, 6, 7, 8, 9])
assert m.row(0) == Matrix(1, 3, [1, 2, 3])
assert m.col(0) == Matrix(3, 1, [1, 4, 7])
def test_row_join():
assert eye_Shaping(3).row_join(Matrix([7, 7, 7])) == \
Matrix([[1, 0, 0, 7],
[0, 1, 0, 7],
[0, 0, 1, 7]])
def test_col_join():
assert eye_Shaping(3).col_join(Matrix([[7, 7, 7]])) == \
Matrix([[1, 0, 0],
[0, 1, 0],
[0, 0, 1],
[7, 7, 7]])
def test_row_insert():
r4 = Matrix([[4, 4, 4]])
for i in range(-4, 5):
l = [1, 0, 0]
l.insert(i, 4)
assert flatten(eye_Shaping(3).row_insert(i, r4).col(0).tolist()) == l
def test_col_insert():
c4 = Matrix([4, 4, 4])
for i in range(-4, 5):
l = [0, 0, 0]
l.insert(i, 4)
assert flatten(zeros_Shaping(3).col_insert(i, c4).row(0).tolist()) == l
# issue 13643
assert eye_Shaping(6).col_insert(3, Matrix([[2, 2], [2, 2], [2, 2], [2, 2], [2, 2], [2, 2]])) == \
Matrix([[1, 0, 0, 2, 2, 0, 0, 0],
[0, 1, 0, 2, 2, 0, 0, 0],
[0, 0, 1, 2, 2, 0, 0, 0],
[0, 0, 0, 2, 2, 1, 0, 0],
[0, 0, 0, 2, 2, 0, 1, 0],
[0, 0, 0, 2, 2, 0, 0, 1]])
def test_extract():
m = ShapingOnlyMatrix(4, 3, lambda i, j: i*3 + j)
assert m.extract([0, 1, 3], [0, 1]) == Matrix(3, 2, [0, 1, 3, 4, 9, 10])
assert m.extract([0, 3], [0, 0, 2]) == Matrix(2, 3, [0, 0, 2, 9, 9, 11])
assert m.extract(range(4), range(3)) == m
raises(IndexError, lambda: m.extract([4], [0]))
raises(IndexError, lambda: m.extract([0], [3]))
def test_hstack():
m = ShapingOnlyMatrix(4, 3, lambda i, j: i*3 + j)
m2 = ShapingOnlyMatrix(3, 4, lambda i, j: i*3 + j)
assert m == m.hstack(m)
assert m.hstack(m, m, m) == ShapingOnlyMatrix.hstack(m, m, m) == Matrix([
[0, 1, 2, 0, 1, 2, 0, 1, 2],
[3, 4, 5, 3, 4, 5, 3, 4, 5],
[6, 7, 8, 6, 7, 8, 6, 7, 8],
[9, 10, 11, 9, 10, 11, 9, 10, 11]])
raises(ShapeError, lambda: m.hstack(m, m2))
assert Matrix.hstack() == Matrix()
# test regression #12938
M1 = Matrix.zeros(0, 0)
M2 = Matrix.zeros(0, 1)
M3 = Matrix.zeros(0, 2)
M4 = Matrix.zeros(0, 3)
m = ShapingOnlyMatrix.hstack(M1, M2, M3, M4)
assert m.rows == 0 and m.cols == 6
def test_vstack():
m = ShapingOnlyMatrix(4, 3, lambda i, j: i*3 + j)
m2 = ShapingOnlyMatrix(3, 4, lambda i, j: i*3 + j)
assert m == m.vstack(m)
assert m.vstack(m, m, m) == ShapingOnlyMatrix.vstack(m, m, m) == Matrix([
[0, 1, 2],
[3, 4, 5],
[6, 7, 8],
[9, 10, 11],
[0, 1, 2],
[3, 4, 5],
[6, 7, 8],
[9, 10, 11],
[0, 1, 2],
[3, 4, 5],
[6, 7, 8],
[9, 10, 11]])
raises(ShapeError, lambda: m.vstack(m, m2))
assert Matrix.vstack() == Matrix()
# PropertiesOnlyMatrix tests
def test_atoms():
m = PropertiesOnlyMatrix(2, 2, [1, 2, x, 1 - 1/x])
assert m.atoms() == {S.One, S(2), S.NegativeOne, x}
assert m.atoms(Symbol) == {x}
def test_free_symbols():
assert PropertiesOnlyMatrix([[x], [0]]).free_symbols == {x}
def test_has():
A = PropertiesOnlyMatrix(((x, y), (2, 3)))
assert A.has(x)
assert not A.has(z)
assert A.has(Symbol)
A = PropertiesOnlyMatrix(((2, y), (2, 3)))
assert not A.has(x)
def test_is_anti_symmetric():
x = symbols('x')
assert PropertiesOnlyMatrix(2, 1, [1, 2]).is_anti_symmetric() is False
m = PropertiesOnlyMatrix(3, 3, [0, x**2 + 2*x + 1, y, -(x + 1)**2, 0, x*y, -y, -x*y, 0])
assert m.is_anti_symmetric() is True
assert m.is_anti_symmetric(simplify=False) is False
assert m.is_anti_symmetric(simplify=lambda x: x) is False
m = PropertiesOnlyMatrix(3, 3, [x.expand() for x in m])
assert m.is_anti_symmetric(simplify=False) is True
m = PropertiesOnlyMatrix(3, 3, [x.expand() for x in [S.One] + list(m)[1:]])
assert m.is_anti_symmetric() is False
def test_diagonal_symmetrical():
m = PropertiesOnlyMatrix(2, 2, [0, 1, 1, 0])
assert not m.is_diagonal()
assert m.is_symmetric()
assert m.is_symmetric(simplify=False)
m = PropertiesOnlyMatrix(2, 2, [1, 0, 0, 1])
assert m.is_diagonal()
m = PropertiesOnlyMatrix(3, 3, diag(1, 2, 3))
assert m.is_diagonal()
assert m.is_symmetric()
m = PropertiesOnlyMatrix(3, 3, [1, 0, 0, 0, 2, 0, 0, 0, 3])
assert m == diag(1, 2, 3)
m = PropertiesOnlyMatrix(2, 3, zeros(2, 3))
assert not m.is_symmetric()
assert m.is_diagonal()
m = PropertiesOnlyMatrix(((5, 0), (0, 6), (0, 0)))
assert m.is_diagonal()
m = PropertiesOnlyMatrix(((5, 0, 0), (0, 6, 0)))
assert m.is_diagonal()
m = Matrix(3, 3, [1, x**2 + 2*x + 1, y, (x + 1)**2, 2, 0, y, 0, 3])
assert m.is_symmetric()
assert not m.is_symmetric(simplify=False)
assert m.expand().is_symmetric(simplify=False)
def test_is_hermitian():
a = PropertiesOnlyMatrix([[1, I], [-I, 1]])
assert a.is_hermitian
a = PropertiesOnlyMatrix([[2*I, I], [-I, 1]])
assert a.is_hermitian is False
a = PropertiesOnlyMatrix([[x, I], [-I, 1]])
assert a.is_hermitian is None
a = PropertiesOnlyMatrix([[x, 1], [-I, 1]])
assert a.is_hermitian is False
def test_is_Identity():
assert eye_Properties(3).is_Identity
assert not PropertiesOnlyMatrix(zeros(3)).is_Identity
assert not PropertiesOnlyMatrix(ones(3)).is_Identity
# issue 6242
assert not PropertiesOnlyMatrix([[1, 0, 0]]).is_Identity
def test_is_symbolic():
a = PropertiesOnlyMatrix([[x, x], [x, x]])
assert a.is_symbolic() is True
a = PropertiesOnlyMatrix([[1, 2, 3, 4], [5, 6, 7, 8]])
assert a.is_symbolic() is False
a = PropertiesOnlyMatrix([[1, 2, 3, 4], [5, 6, x, 8]])
assert a.is_symbolic() is True
a = PropertiesOnlyMatrix([[1, x, 3]])
assert a.is_symbolic() is True
a = PropertiesOnlyMatrix([[1, 2, 3]])
assert a.is_symbolic() is False
a = PropertiesOnlyMatrix([[1], [x], [3]])
assert a.is_symbolic() is True
a = PropertiesOnlyMatrix([[1], [2], [3]])
assert a.is_symbolic() is False
def test_is_upper():
a = PropertiesOnlyMatrix([[1, 2, 3]])
assert a.is_upper is True
a = PropertiesOnlyMatrix([[1], [2], [3]])
assert a.is_upper is False
def test_is_lower():
a = PropertiesOnlyMatrix([[1, 2, 3]])
assert a.is_lower is False
a = PropertiesOnlyMatrix([[1], [2], [3]])
assert a.is_lower is True
def test_is_square():
m = PropertiesOnlyMatrix([[1], [1]])
m2 = PropertiesOnlyMatrix([[2, 2], [2, 2]])
assert not m.is_square
assert m2.is_square
def test_is_symmetric():
m = PropertiesOnlyMatrix(2, 2, [0, 1, 1, 0])
assert m.is_symmetric()
m = PropertiesOnlyMatrix(2, 2, [0, 1, 0, 1])
assert not m.is_symmetric()
def test_is_hessenberg():
A = PropertiesOnlyMatrix([[3, 4, 1], [2, 4, 5], [0, 1, 2]])
assert A.is_upper_hessenberg
A = PropertiesOnlyMatrix(3, 3, [3, 2, 0, 4, 4, 1, 1, 5, 2])
assert A.is_lower_hessenberg
A = PropertiesOnlyMatrix(3, 3, [3, 2, -1, 4, 4, 1, 1, 5, 2])
assert A.is_lower_hessenberg is False
assert A.is_upper_hessenberg is False
A = PropertiesOnlyMatrix([[3, 4, 1], [2, 4, 5], [3, 1, 2]])
assert not A.is_upper_hessenberg
def test_is_zero():
assert PropertiesOnlyMatrix(0, 0, []).is_zero_matrix
assert PropertiesOnlyMatrix([[0, 0], [0, 0]]).is_zero_matrix
assert PropertiesOnlyMatrix(zeros(3, 4)).is_zero_matrix
assert not PropertiesOnlyMatrix(eye(3)).is_zero_matrix
assert PropertiesOnlyMatrix([[x, 0], [0, 0]]).is_zero_matrix == None
assert PropertiesOnlyMatrix([[x, 1], [0, 0]]).is_zero_matrix == False
a = Symbol('a', nonzero=True)
assert PropertiesOnlyMatrix([[a, 0], [0, 0]]).is_zero_matrix == False
def test_values():
assert set(PropertiesOnlyMatrix(2, 2, [0, 1, 2, 3]
).values()) == {1, 2, 3}
x = Symbol('x', real=True)
assert set(PropertiesOnlyMatrix(2, 2, [x, 0, 0, 1]
).values()) == {x, 1}
# OperationsOnlyMatrix tests
def test_applyfunc():
m0 = OperationsOnlyMatrix(eye(3))
assert m0.applyfunc(lambda x: 2*x) == eye(3)*2
assert m0.applyfunc(lambda x: 0) == zeros(3)
assert m0.applyfunc(lambda x: 1) == ones(3)
def test_adjoint():
dat = [[0, I], [1, 0]]
ans = OperationsOnlyMatrix([[0, 1], [-I, 0]])
assert ans.adjoint() == Matrix(dat)
def test_as_real_imag():
m1 = OperationsOnlyMatrix(2, 2, [1, 2, 3, 4])
m3 = OperationsOnlyMatrix(2, 2,
[1 + S.ImaginaryUnit, 2 + 2*S.ImaginaryUnit,
3 + 3*S.ImaginaryUnit, 4 + 4*S.ImaginaryUnit])
a, b = m3.as_real_imag()
assert a == m1
assert b == m1
def test_conjugate():
M = OperationsOnlyMatrix([[0, I, 5],
[1, 2, 0]])
assert M.T == Matrix([[0, 1],
[I, 2],
[5, 0]])
assert M.C == Matrix([[0, -I, 5],
[1, 2, 0]])
assert M.C == M.conjugate()
assert M.H == M.T.C
assert M.H == Matrix([[ 0, 1],
[-I, 2],
[ 5, 0]])
def test_doit():
a = OperationsOnlyMatrix([[Add(x, x, evaluate=False)]])
assert a[0] != 2*x
assert a.doit() == Matrix([[2*x]])
def test_evalf():
a = OperationsOnlyMatrix(2, 1, [sqrt(5), 6])
assert all(a.evalf()[i] == a[i].evalf() for i in range(2))
assert all(a.evalf(2)[i] == a[i].evalf(2) for i in range(2))
assert all(a.n(2)[i] == a[i].n(2) for i in range(2))
def test_expand():
m0 = OperationsOnlyMatrix([[x*(x + y), 2], [((x + y)*y)*x, x*(y + x*(x + y))]])
# Test if expand() returns a matrix
m1 = m0.expand()
assert m1 == Matrix(
[[x*y + x**2, 2], [x*y**2 + y*x**2, x*y + y*x**2 + x**3]])
a = Symbol('a', real=True)
assert OperationsOnlyMatrix(1, 1, [exp(I*a)]).expand(complex=True) == \
Matrix([cos(a) + I*sin(a)])
def test_refine():
m0 = OperationsOnlyMatrix([[Abs(x)**2, sqrt(x**2)],
[sqrt(x**2)*Abs(y)**2, sqrt(y**2)*Abs(x)**2]])
m1 = m0.refine(Q.real(x) & Q.real(y))
assert m1 == Matrix([[x**2, Abs(x)], [y**2*Abs(x), x**2*Abs(y)]])
m1 = m0.refine(Q.positive(x) & Q.positive(y))
assert m1 == Matrix([[x**2, x], [x*y**2, x**2*y]])
m1 = m0.refine(Q.negative(x) & Q.negative(y))
assert m1 == Matrix([[x**2, -x], [-x*y**2, -x**2*y]])
def test_replace():
F, G = symbols('F, G', cls=Function)
K = OperationsOnlyMatrix(2, 2, lambda i, j: G(i+j))
M = OperationsOnlyMatrix(2, 2, lambda i, j: F(i+j))
N = M.replace(F, G)
assert N == K
def test_replace_map():
F, G = symbols('F, G', cls=Function)
K = OperationsOnlyMatrix(2, 2, [(G(0), {F(0): G(0)}), (G(1), {F(1): G(1)}), (G(1), {F(1) \
: G(1)}), (G(2), {F(2): G(2)})])
M = OperationsOnlyMatrix(2, 2, lambda i, j: F(i+j))
N = M.replace(F, G, True)
assert N == K
def test_rot90():
A = Matrix([[1, 2], [3, 4]])
assert A == A.rot90(0) == A.rot90(4)
assert A.rot90(2) == A.rot90(-2) == A.rot90(6) == Matrix(((4, 3), (2, 1)))
assert A.rot90(3) == A.rot90(-1) == A.rot90(7) == Matrix(((2, 4), (1, 3)))
assert A.rot90() == A.rot90(-7) == A.rot90(-3) == Matrix(((3, 1), (4, 2)))
def test_simplify():
n = Symbol('n')
f = Function('f')
M = OperationsOnlyMatrix([[ 1/x + 1/y, (x + x*y) / x ],
[ (f(x) + y*f(x))/f(x), 2 * (1/n - cos(n * pi)/n) / pi ]])
assert M.simplify() == Matrix([[ (x + y)/(x * y), 1 + y ],
[ 1 + y, 2*((1 - 1*cos(pi*n))/(pi*n)) ]])
eq = (1 + x)**2
M = OperationsOnlyMatrix([[eq]])
assert M.simplify() == Matrix([[eq]])
assert M.simplify(ratio=oo) == Matrix([[eq.simplify(ratio=oo)]])
def test_subs():
assert OperationsOnlyMatrix([[1, x], [x, 4]]).subs(x, 5) == Matrix([[1, 5], [5, 4]])
assert OperationsOnlyMatrix([[x, 2], [x + y, 4]]).subs([[x, -1], [y, -2]]) == \
Matrix([[-1, 2], [-3, 4]])
assert OperationsOnlyMatrix([[x, 2], [x + y, 4]]).subs([(x, -1), (y, -2)]) == \
Matrix([[-1, 2], [-3, 4]])
assert OperationsOnlyMatrix([[x, 2], [x + y, 4]]).subs({x: -1, y: -2}) == \
Matrix([[-1, 2], [-3, 4]])
assert OperationsOnlyMatrix([[x*y]]).subs({x: y - 1, y: x - 1}, simultaneous=True) == \
Matrix([[(x - 1)*(y - 1)]])
def test_trace():
M = OperationsOnlyMatrix([[1, 0, 0],
[0, 5, 0],
[0, 0, 8]])
assert M.trace() == 14
def test_xreplace():
assert OperationsOnlyMatrix([[1, x], [x, 4]]).xreplace({x: 5}) == \
Matrix([[1, 5], [5, 4]])
assert OperationsOnlyMatrix([[x, 2], [x + y, 4]]).xreplace({x: -1, y: -2}) == \
Matrix([[-1, 2], [-3, 4]])
def test_permute():
a = OperationsOnlyMatrix(3, 4, [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12])
raises(IndexError, lambda: a.permute([[0, 5]]))
raises(ValueError, lambda: a.permute(Symbol('x')))
b = a.permute_rows([[0, 2], [0, 1]])
assert a.permute([[0, 2], [0, 1]]) == b == Matrix([
[5, 6, 7, 8],
[9, 10, 11, 12],
[1, 2, 3, 4]])
b = a.permute_cols([[0, 2], [0, 1]])
assert a.permute([[0, 2], [0, 1]], orientation='cols') == b ==\
Matrix([
[ 2, 3, 1, 4],
[ 6, 7, 5, 8],
[10, 11, 9, 12]])
b = a.permute_cols([[0, 2], [0, 1]], direction='backward')
assert a.permute([[0, 2], [0, 1]], orientation='cols', direction='backward') == b ==\
Matrix([
[ 3, 1, 2, 4],
[ 7, 5, 6, 8],
[11, 9, 10, 12]])
assert a.permute([1, 2, 0, 3]) == Matrix([
[5, 6, 7, 8],
[9, 10, 11, 12],
[1, 2, 3, 4]])
from sympy.combinatorics import Permutation
assert a.permute(Permutation([1, 2, 0, 3])) == Matrix([
[5, 6, 7, 8],
[9, 10, 11, 12],
[1, 2, 3, 4]])
# ArithmeticOnlyMatrix tests
def test_abs():
m = ArithmeticOnlyMatrix([[1, -2], [x, y]])
assert abs(m) == ArithmeticOnlyMatrix([[1, 2], [Abs(x), Abs(y)]])
def test_add():
m = ArithmeticOnlyMatrix([[1, 2, 3], [x, y, x], [2*y, -50, z*x]])
assert m + m == ArithmeticOnlyMatrix([[2, 4, 6], [2*x, 2*y, 2*x], [4*y, -100, 2*z*x]])
n = ArithmeticOnlyMatrix(1, 2, [1, 2])
raises(ShapeError, lambda: m + n)
def test_multiplication():
a = ArithmeticOnlyMatrix((
(1, 2),
(3, 1),
(0, 6),
))
b = ArithmeticOnlyMatrix((
(1, 2),
(3, 0),
))
raises(ShapeError, lambda: b*a)
raises(TypeError, lambda: a*{})
c = a*b
assert c[0, 0] == 7
assert c[0, 1] == 2
assert c[1, 0] == 6
assert c[1, 1] == 6
assert c[2, 0] == 18
assert c[2, 1] == 0
try:
eval('c = a @ b')
except SyntaxError:
pass
else:
assert c[0, 0] == 7
assert c[0, 1] == 2
assert c[1, 0] == 6
assert c[1, 1] == 6
assert c[2, 0] == 18
assert c[2, 1] == 0
h = a.multiply_elementwise(c)
assert h == matrix_multiply_elementwise(a, c)
assert h[0, 0] == 7
assert h[0, 1] == 4
assert h[1, 0] == 18
assert h[1, 1] == 6
assert h[2, 0] == 0
assert h[2, 1] == 0
raises(ShapeError, lambda: a.multiply_elementwise(b))
c = b * Symbol("x")
assert isinstance(c, ArithmeticOnlyMatrix)
assert c[0, 0] == x
assert c[0, 1] == 2*x
assert c[1, 0] == 3*x
assert c[1, 1] == 0
c2 = x * b
assert c == c2
c = 5 * b
assert isinstance(c, ArithmeticOnlyMatrix)
assert c[0, 0] == 5
assert c[0, 1] == 2*5
assert c[1, 0] == 3*5
assert c[1, 1] == 0
try:
eval('c = 5 @ b')
except SyntaxError:
pass
else:
assert isinstance(c, ArithmeticOnlyMatrix)
assert c[0, 0] == 5
assert c[0, 1] == 2*5
assert c[1, 0] == 3*5
assert c[1, 1] == 0
def test_matmul():
a = Matrix([[1, 2], [3, 4]])
assert a.__matmul__(2) == NotImplemented
assert a.__rmatmul__(2) == NotImplemented
#This is done this way because @ is only supported in Python 3.5+
#To check 2@a case
try:
eval('2 @ a')
except SyntaxError:
pass
except TypeError: #TypeError is raised in case of NotImplemented is returned
pass
#Check a@2 case
try:
eval('a @ 2')
except SyntaxError:
pass
except TypeError: #TypeError is raised in case of NotImplemented is returned
pass
def test_power():
raises(NonSquareMatrixError, lambda: Matrix((1, 2))**2)
A = ArithmeticOnlyMatrix([[2, 3], [4, 5]])
assert (A**5)[:] == (6140, 8097, 10796, 14237)
A = ArithmeticOnlyMatrix([[2, 1, 3], [4, 2, 4], [6, 12, 1]])
assert (A**3)[:] == (290, 262, 251, 448, 440, 368, 702, 954, 433)
assert A**0 == eye(3)
assert A**1 == A
assert (ArithmeticOnlyMatrix([[2]]) ** 100)[0, 0] == 2**100
assert ArithmeticOnlyMatrix([[1, 2], [3, 4]])**Integer(2) == ArithmeticOnlyMatrix([[7, 10], [15, 22]])
A = Matrix([[1,2],[4,5]])
assert A.pow(20, method='cayley') == A.pow(20, method='multiply')
def test_neg():
n = ArithmeticOnlyMatrix(1, 2, [1, 2])
assert -n == ArithmeticOnlyMatrix(1, 2, [-1, -2])
def test_sub():
n = ArithmeticOnlyMatrix(1, 2, [1, 2])
assert n - n == ArithmeticOnlyMatrix(1, 2, [0, 0])
def test_div():
n = ArithmeticOnlyMatrix(1, 2, [1, 2])
assert n/2 == ArithmeticOnlyMatrix(1, 2, [S.Half, S(2)/2])
# SpecialOnlyMatrix tests
def test_eye():
assert list(SpecialOnlyMatrix.eye(2, 2)) == [1, 0, 0, 1]
assert list(SpecialOnlyMatrix.eye(2)) == [1, 0, 0, 1]
assert type(SpecialOnlyMatrix.eye(2)) == SpecialOnlyMatrix
assert type(SpecialOnlyMatrix.eye(2, cls=Matrix)) == Matrix
def test_ones():
assert list(SpecialOnlyMatrix.ones(2, 2)) == [1, 1, 1, 1]
assert list(SpecialOnlyMatrix.ones(2)) == [1, 1, 1, 1]
assert SpecialOnlyMatrix.ones(2, 3) == Matrix([[1, 1, 1], [1, 1, 1]])
assert type(SpecialOnlyMatrix.ones(2)) == SpecialOnlyMatrix
assert type(SpecialOnlyMatrix.ones(2, cls=Matrix)) == Matrix
def test_zeros():
assert list(SpecialOnlyMatrix.zeros(2, 2)) == [0, 0, 0, 0]
assert list(SpecialOnlyMatrix.zeros(2)) == [0, 0, 0, 0]
assert SpecialOnlyMatrix.zeros(2, 3) == Matrix([[0, 0, 0], [0, 0, 0]])
assert type(SpecialOnlyMatrix.zeros(2)) == SpecialOnlyMatrix
assert type(SpecialOnlyMatrix.zeros(2, cls=Matrix)) == Matrix
def test_diag_make():
diag = SpecialOnlyMatrix.diag
a = Matrix([[1, 2], [2, 3]])
b = Matrix([[3, x], [y, 3]])
c = Matrix([[3, x, 3], [y, 3, z], [x, y, z]])
assert diag(a, b, b) == Matrix([
[1, 2, 0, 0, 0, 0],
[2, 3, 0, 0, 0, 0],
[0, 0, 3, x, 0, 0],
[0, 0, y, 3, 0, 0],
[0, 0, 0, 0, 3, x],
[0, 0, 0, 0, y, 3],
])
assert diag(a, b, c) == Matrix([
[1, 2, 0, 0, 0, 0, 0],
[2, 3, 0, 0, 0, 0, 0],
[0, 0, 3, x, 0, 0, 0],
[0, 0, y, 3, 0, 0, 0],
[0, 0, 0, 0, 3, x, 3],
[0, 0, 0, 0, y, 3, z],
[0, 0, 0, 0, x, y, z],
])
assert diag(a, c, b) == Matrix([
[1, 2, 0, 0, 0, 0, 0],
[2, 3, 0, 0, 0, 0, 0],
[0, 0, 3, x, 3, 0, 0],
[0, 0, y, 3, z, 0, 0],
[0, 0, x, y, z, 0, 0],
[0, 0, 0, 0, 0, 3, x],
[0, 0, 0, 0, 0, y, 3],
])
a = Matrix([x, y, z])
b = Matrix([[1, 2], [3, 4]])
c = Matrix([[5, 6]])
# this "wandering diagonal" is what makes this
# a block diagonal where each block is independent
# of the others
assert diag(a, 7, b, c) == Matrix([
[x, 0, 0, 0, 0, 0],
[y, 0, 0, 0, 0, 0],
[z, 0, 0, 0, 0, 0],
[0, 7, 0, 0, 0, 0],
[0, 0, 1, 2, 0, 0],
[0, 0, 3, 4, 0, 0],
[0, 0, 0, 0, 5, 6]])
raises(ValueError, lambda: diag(a, 7, b, c, rows=5))
assert diag(1) == Matrix([[1]])
assert diag(1, rows=2) == Matrix([[1, 0], [0, 0]])
assert diag(1, cols=2) == Matrix([[1, 0], [0, 0]])
assert diag(1, rows=3, cols=2) == Matrix([[1, 0], [0, 0], [0, 0]])
assert diag(*[2, 3]) == Matrix([
[2, 0],
[0, 3]])
assert diag(Matrix([2, 3])) == Matrix([
[2],
[3]])
assert diag([1, [2, 3], 4], unpack=False) == \
diag([[1], [2, 3], [4]], unpack=False) == Matrix([
[1, 0],
[2, 3],
[4, 0]])
assert type(diag(1)) == SpecialOnlyMatrix
assert type(diag(1, cls=Matrix)) == Matrix
assert Matrix.diag([1, 2, 3]) == Matrix.diag(1, 2, 3)
assert Matrix.diag([1, 2, 3], unpack=False).shape == (3, 1)
assert Matrix.diag([[1, 2, 3]]).shape == (3, 1)
assert Matrix.diag([[1, 2, 3]], unpack=False).shape == (1, 3)
assert Matrix.diag([[[1, 2, 3]]]).shape == (1, 3)
# kerning can be used to move the starting point
assert Matrix.diag(ones(0, 2), 1, 2) == Matrix([
[0, 0, 1, 0],
[0, 0, 0, 2]])
assert Matrix.diag(ones(2, 0), 1, 2) == Matrix([
[0, 0],
[0, 0],
[1, 0],
[0, 2]])
def test_diagonal():
m = Matrix(3, 3, range(9))
d = m.diagonal()
assert d == m.diagonal(0)
assert tuple(d) == (0, 4, 8)
assert tuple(m.diagonal(1)) == (1, 5)
assert tuple(m.diagonal(-1)) == (3, 7)
assert tuple(m.diagonal(2)) == (2,)
assert type(m.diagonal()) == type(m)
s = SparseMatrix(3, 3, {(1, 1): 1})
assert type(s.diagonal()) == type(s)
assert type(m) != type(s)
raises(ValueError, lambda: m.diagonal(3))
raises(ValueError, lambda: m.diagonal(-3))
raises(ValueError, lambda: m.diagonal(pi))
M = ones(2, 3)
assert banded({i: list(M.diagonal(i))
for i in range(1-M.rows, M.cols)}) == M
def test_jordan_block():
assert SpecialOnlyMatrix.jordan_block(3, 2) == SpecialOnlyMatrix.jordan_block(3, eigenvalue=2) \
== SpecialOnlyMatrix.jordan_block(size=3, eigenvalue=2) \
== SpecialOnlyMatrix.jordan_block(3, 2, band='upper') \
== SpecialOnlyMatrix.jordan_block(
size=3, eigenval=2, eigenvalue=2) \
== Matrix([
[2, 1, 0],
[0, 2, 1],
[0, 0, 2]])
assert SpecialOnlyMatrix.jordan_block(3, 2, band='lower') == Matrix([
[2, 0, 0],
[1, 2, 0],
[0, 1, 2]])
# missing eigenvalue
raises(ValueError, lambda: SpecialOnlyMatrix.jordan_block(2))
# non-integral size
raises(ValueError, lambda: SpecialOnlyMatrix.jordan_block(3.5, 2))
# size not specified
raises(ValueError, lambda: SpecialOnlyMatrix.jordan_block(eigenvalue=2))
# inconsistent eigenvalue
raises(ValueError,
lambda: SpecialOnlyMatrix.jordan_block(
eigenvalue=2, eigenval=4))
# Deprecated feature
with warns_deprecated_sympy():
assert (SpecialOnlyMatrix.jordan_block(cols=3, eigenvalue=2) ==
SpecialOnlyMatrix(3, 3, (2, 1, 0, 0, 2, 1, 0, 0, 2)))
with warns_deprecated_sympy():
assert (SpecialOnlyMatrix.jordan_block(rows=3, eigenvalue=2) ==
SpecialOnlyMatrix(3, 3, (2, 1, 0, 0, 2, 1, 0, 0, 2)))
with warns_deprecated_sympy():
assert SpecialOnlyMatrix.jordan_block(3, 2) == \
SpecialOnlyMatrix.jordan_block(cols=3, eigenvalue=2) == \
SpecialOnlyMatrix.jordan_block(rows=3, eigenvalue=2)
with warns_deprecated_sympy():
assert SpecialOnlyMatrix.jordan_block(
rows=4, cols=3, eigenvalue=2) == \
Matrix([
[2, 1, 0],
[0, 2, 1],
[0, 0, 2],
[0, 0, 0]])
# Using alias keyword
assert SpecialOnlyMatrix.jordan_block(size=3, eigenvalue=2) == \
SpecialOnlyMatrix.jordan_block(size=3, eigenval=2)
def test_orthogonalize():
m = Matrix([[1, 2], [3, 4]])
assert m.orthogonalize(Matrix([[2], [1]])) == [Matrix([[2], [1]])]
assert m.orthogonalize(Matrix([[2], [1]]), normalize=True) == \
[Matrix([[2*sqrt(5)/5], [sqrt(5)/5]])]
assert m.orthogonalize(Matrix([[1], [2]]), Matrix([[-1], [4]])) == \
[Matrix([[1], [2]]), Matrix([[Rational(-12, 5)], [Rational(6, 5)]])]
assert m.orthogonalize(Matrix([[0], [0]]), Matrix([[-1], [4]])) == \
[Matrix([[-1], [4]])]
assert m.orthogonalize(Matrix([[0], [0]])) == []
n = Matrix([[9, 1, 9], [3, 6, 10], [8, 5, 2]])
vecs = [Matrix([[-5], [1]]), Matrix([[-5], [2]]), Matrix([[-5], [-2]])]
assert n.orthogonalize(*vecs) == \
[Matrix([[-5], [1]]), Matrix([[Rational(5, 26)], [Rational(25, 26)]])]
vecs = [Matrix([0, 0, 0]), Matrix([1, 2, 3]), Matrix([1, 4, 5])]
raises(ValueError, lambda: Matrix.orthogonalize(*vecs, rankcheck=True))
vecs = [Matrix([1, 2, 3]), Matrix([4, 5, 6]), Matrix([7, 8, 9])]
raises(ValueError, lambda: Matrix.orthogonalize(*vecs, rankcheck=True))
# CalculusOnlyMatrix tests
@XFAIL
def test_diff():
x, y = symbols('x y')
m = CalculusOnlyMatrix(2, 1, [x, y])
# TODO: currently not working as ``_MinimalMatrix`` cannot be sympified:
assert m.diff(x) == Matrix(2, 1, [1, 0])
def test_integrate():
x, y = symbols('x y')
m = CalculusOnlyMatrix(2, 1, [x, y])
assert m.integrate(x) == Matrix(2, 1, [x**2/2, y*x])
def test_jacobian2():
rho, phi = symbols("rho,phi")
X = CalculusOnlyMatrix(3, 1, [rho*cos(phi), rho*sin(phi), rho**2])
Y = CalculusOnlyMatrix(2, 1, [rho, phi])
J = Matrix([
[cos(phi), -rho*sin(phi)],
[sin(phi), rho*cos(phi)],
[ 2*rho, 0],
])
assert X.jacobian(Y) == J
m = CalculusOnlyMatrix(2, 2, [1, 2, 3, 4])
m2 = CalculusOnlyMatrix(4, 1, [1, 2, 3, 4])
raises(TypeError, lambda: m.jacobian(Matrix([1, 2])))
raises(TypeError, lambda: m2.jacobian(m))
def test_limit():
x, y = symbols('x y')
m = CalculusOnlyMatrix(2, 1, [1/x, y])
assert m.limit(x, 5) == Matrix(2, 1, [Rational(1, 5), y])
def test_issue_13774():
M = Matrix([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
v = [1, 1, 1]
raises(TypeError, lambda: M*v)
raises(TypeError, lambda: v*M)
def test_companion():
x = Symbol('x')
y = Symbol('y')
raises(ValueError, lambda: Matrix.companion(1))
raises(ValueError, lambda: Matrix.companion(Poly([1], x)))
raises(ValueError, lambda: Matrix.companion(Poly([2, 1], x)))
raises(ValueError, lambda: Matrix.companion(Poly(x*y, [x, y])))
c0, c1, c2 = symbols('c0:3')
assert Matrix.companion(Poly([1, c0], x)) == Matrix([-c0])
assert Matrix.companion(Poly([1, c1, c0], x)) == \
Matrix([[0, -c0], [1, -c1]])
assert Matrix.companion(Poly([1, c2, c1, c0], x)) == \
Matrix([[0, 0, -c0], [1, 0, -c1], [0, 1, -c2]])
|
5b83b07e663e9a48c3654844f6b220d9d4bf3179c9557b3bf0a3b9deddf51ca7 | from sympy.matrices.expressions import MatrixExpr
from sympy import MatrixBase, Dummy, Lambda, Function, FunctionClass
from sympy.core.sympify import sympify, _sympify
class ElementwiseApplyFunction(MatrixExpr):
r"""
Apply function to a matrix elementwise without evaluating.
Examples
========
It can be created by calling ``.applyfunc(<function>)`` on a matrix
expression:
>>> from sympy.matrices.expressions import MatrixSymbol
>>> from sympy.matrices.expressions.applyfunc import ElementwiseApplyFunction
>>> from sympy import exp
>>> X = MatrixSymbol("X", 3, 3)
>>> X.applyfunc(exp)
Lambda(_d, exp(_d)).(X)
Otherwise using the class constructor:
>>> from sympy import eye
>>> expr = ElementwiseApplyFunction(exp, eye(3))
>>> expr
Lambda(_d, exp(_d)).(Matrix([
[1, 0, 0],
[0, 1, 0],
[0, 0, 1]]))
>>> expr.doit()
Matrix([
[E, 1, 1],
[1, E, 1],
[1, 1, E]])
Notice the difference with the real mathematical functions:
>>> exp(eye(3))
Matrix([
[E, 0, 0],
[0, E, 0],
[0, 0, E]])
"""
def __new__(cls, function, expr):
expr = _sympify(expr)
if not expr.is_Matrix:
raise ValueError("{} must be a matrix instance.".format(expr))
if not isinstance(function, (FunctionClass, Lambda)):
d = Dummy('d')
function = Lambda(d, function(d))
function = sympify(function)
if not isinstance(function, (FunctionClass, Lambda)):
raise ValueError(
"{} should be compatible with SymPy function classes."
.format(function))
if 1 not in function.nargs:
raise ValueError(
'{} should be able to accept 1 arguments.'.format(function))
if not isinstance(function, Lambda):
d = Dummy('d')
function = Lambda(d, function(d))
obj = MatrixExpr.__new__(cls, function, expr)
return obj
@property
def function(self):
return self.args[0]
@property
def expr(self):
return self.args[1]
@property
def shape(self):
return self.expr.shape
def doit(self, **kwargs):
deep = kwargs.get("deep", True)
expr = self.expr
if deep:
expr = expr.doit(**kwargs)
function = self.function
if isinstance(function, Lambda) and function.is_identity:
# This is a Lambda containing the identity function.
return expr
if isinstance(expr, MatrixBase):
return expr.applyfunc(self.function)
elif isinstance(expr, ElementwiseApplyFunction):
return ElementwiseApplyFunction(
lambda x: self.function(expr.function(x)),
expr.expr
).doit()
else:
return self
def _entry(self, i, j, **kwargs):
return self.function(self.expr._entry(i, j, **kwargs))
def _get_function_fdiff(self):
d = Dummy("d")
function = self.function(d)
fdiff = function.diff(d)
if isinstance(fdiff, Function):
fdiff = type(fdiff)
else:
fdiff = Lambda(d, fdiff)
return fdiff
def _eval_derivative(self, x):
from sympy import hadamard_product
dexpr = self.expr.diff(x)
fdiff = self._get_function_fdiff()
return hadamard_product(
dexpr,
ElementwiseApplyFunction(fdiff, self.expr)
)
def _eval_derivative_matrix_lines(self, x):
from sympy import Identity
from sympy.codegen.array_utils import CodegenArrayContraction, CodegenArrayTensorProduct, CodegenArrayDiagonal
from sympy.core.expr import ExprBuilder
fdiff = self._get_function_fdiff()
lr = self.expr._eval_derivative_matrix_lines(x)
ewdiff = ElementwiseApplyFunction(fdiff, self.expr)
if 1 in x.shape:
# Vector:
iscolumn = self.shape[1] == 1
for i in lr:
if iscolumn:
ptr1 = i.first_pointer
ptr2 = Identity(self.shape[1])
else:
ptr1 = Identity(self.shape[0])
ptr2 = i.second_pointer
subexpr = ExprBuilder(
CodegenArrayDiagonal,
[
ExprBuilder(
CodegenArrayTensorProduct,
[
ewdiff,
ptr1,
ptr2,
]
),
(0, 2) if iscolumn else (1, 4)
],
validator=CodegenArrayDiagonal._validate
)
i._lines = [subexpr]
i._first_pointer_parent = subexpr.args[0].args
i._first_pointer_index = 1
i._second_pointer_parent = subexpr.args[0].args
i._second_pointer_index = 2
else:
# Matrix case:
for i in lr:
ptr1 = i.first_pointer
ptr2 = i.second_pointer
newptr1 = Identity(ptr1.shape[1])
newptr2 = Identity(ptr2.shape[1])
subexpr = ExprBuilder(
CodegenArrayContraction,
[
ExprBuilder(
CodegenArrayTensorProduct,
[ptr1, newptr1, ewdiff, ptr2, newptr2]
),
(1, 2, 4),
(5, 7, 8),
],
validator=CodegenArrayContraction._validate
)
i._first_pointer_parent = subexpr.args[0].args
i._first_pointer_index = 1
i._second_pointer_parent = subexpr.args[0].args
i._second_pointer_index = 4
i._lines = [subexpr]
return lr
|
48e364c529404756eb7bbfe227ca2c7efac25cbc2b3f9206584d810036c06caf | from sympy import Basic, Expr, S, sympify
from sympy.matrices.common import NonSquareMatrixError
class Determinant(Expr):
"""Matrix Determinant
Represents the determinant of a matrix expression.
Examples
========
>>> from sympy import MatrixSymbol, Determinant, eye
>>> A = MatrixSymbol('A', 3, 3)
>>> Determinant(A)
Determinant(A)
>>> Determinant(eye(3)).doit()
1
"""
is_commutative = True
def __new__(cls, mat):
mat = sympify(mat)
if not mat.is_Matrix:
raise TypeError("Input to Determinant, %s, not a matrix" % str(mat))
if not mat.is_square:
raise NonSquareMatrixError("Det of a non-square matrix")
return Basic.__new__(cls, mat)
@property
def arg(self):
return self.args[0]
def doit(self, expand=False):
try:
return self.arg._eval_determinant()
except (AttributeError, NotImplementedError):
return self
def det(matexpr):
""" Matrix Determinant
Examples
========
>>> from sympy import MatrixSymbol, det, eye
>>> A = MatrixSymbol('A', 3, 3)
>>> det(A)
Determinant(A)
>>> det(eye(3))
1
"""
return Determinant(matexpr).doit()
from sympy.assumptions.ask import ask, Q
from sympy.assumptions.refine import handlers_dict
def refine_Determinant(expr, assumptions):
"""
>>> from sympy import MatrixSymbol, Q, assuming, refine, det
>>> X = MatrixSymbol('X', 2, 2)
>>> det(X)
Determinant(X)
>>> with assuming(Q.orthogonal(X)):
... print(refine(det(X)))
1
"""
if ask(Q.orthogonal(expr.arg), assumptions):
return S.One
elif ask(Q.singular(expr.arg), assumptions):
return S.Zero
elif ask(Q.unit_triangular(expr.arg), assumptions):
return S.One
return expr
handlers_dict['Determinant'] = refine_Determinant
|
aa4f1b662f9f0861ab6ed938acf2374e5347188804efd5b2eec4655372dbe9b0 | """ A module which handles Matrix Expressions """
from .slice import MatrixSlice
from .blockmatrix import BlockMatrix, BlockDiagMatrix, block_collapse, blockcut
from .companion import CompanionMatrix
from .funcmatrix import FunctionMatrix
from .inverse import Inverse
from .matadd import MatAdd
from .matexpr import (Identity, MatrixExpr, MatrixSymbol, ZeroMatrix, OneMatrix,
matrix_symbols)
from .matmul import MatMul
from .matpow import MatPow
from .trace import Trace, trace
from .determinant import Determinant, det
from .transpose import Transpose
from .adjoint import Adjoint
from .hadamard import hadamard_product, HadamardProduct, hadamard_power, HadamardPower
from .diagonal import DiagonalMatrix, DiagonalOf, DiagMatrix, diagonalize_vector
from .dotproduct import DotProduct
from .kronecker import kronecker_product, KroneckerProduct, combine_kronecker
from .permutation import PermutationMatrix, MatrixPermute
__all__ = [
'MatrixSlice',
'BlockMatrix', 'BlockDiagMatrix', 'block_collapse', 'blockcut',
'FunctionMatrix',
'CompanionMatrix',
'Inverse',
'MatAdd',
'Identity', 'MatrixExpr', 'MatrixSymbol', 'ZeroMatrix', 'OneMatrix',
'matrix_symbols',
'MatMul',
'MatPow',
'Trace', 'trace',
'Determinant', 'det',
'Transpose',
'Adjoint',
'hadamard_product', 'HadamardProduct', 'hadamard_power', 'HadamardPower',
'DiagonalMatrix', 'DiagonalOf', 'DiagMatrix', 'diagonalize_vector',
'DotProduct',
'kronecker_product', 'KroneckerProduct', 'combine_kronecker',
'PermutationMatrix', 'MatrixPermute',
]
|
5e42bbc576b0428463ed7d6640074f77e11a31d40579c27675e93c748a79963d | from sympy import Number
from sympy.core import Mul, Basic, sympify, S
from sympy.functions import adjoint
from sympy.strategies import (rm_id, unpack, typed, flatten, exhaust,
do_one, new)
from sympy.matrices.common import ShapeError, NonInvertibleMatrixError
from sympy.matrices.matrices import MatrixBase
from .inverse import Inverse
from .matexpr import \
MatrixExpr, Identity, ZeroMatrix, OneMatrix, GenericIdentity
from .matpow import MatPow
from .transpose import transpose
from .permutation import PermutationMatrix
# XXX: MatMul should perhaps not subclass directly from Mul
class MatMul(MatrixExpr, Mul):
"""
A product of matrix expressions
Examples
========
>>> from sympy import MatMul, MatrixSymbol
>>> A = MatrixSymbol('A', 5, 4)
>>> B = MatrixSymbol('B', 4, 3)
>>> C = MatrixSymbol('C', 3, 6)
>>> MatMul(A, B, C)
A*B*C
"""
is_MatMul = True
identity = GenericIdentity()
def __new__(cls, *args, evaluate=False, **kwargs):
check = kwargs.get('check', True)
if not args:
return cls.identity
# This must be removed aggressively in the constructor to avoid
# TypeErrors from GenericIdentity().shape
args = filter(lambda i: cls.identity != i, args)
args = list(map(sympify, args))
obj = Basic.__new__(cls, *args)
factor, matrices = obj.as_coeff_matrices()
if check:
validate(*matrices)
if not matrices:
# Should it be
#
# return Basic.__neq__(cls, factor, GenericIdentity()) ?
return factor
if evaluate:
return canonicalize(obj)
return obj
@property
def shape(self):
matrices = [arg for arg in self.args if arg.is_Matrix]
return (matrices[0].rows, matrices[-1].cols)
def _entry(self, i, j, expand=True, **kwargs):
from sympy import Dummy, Sum, Mul, ImmutableMatrix, Integer
coeff, matrices = self.as_coeff_matrices()
if len(matrices) == 1: # situation like 2*X, matmul is just X
return coeff * matrices[0][i, j]
indices = [None]*(len(matrices) + 1)
ind_ranges = [None]*(len(matrices) - 1)
indices[0] = i
indices[-1] = j
def f():
counter = 1
while True:
yield Dummy("i_%i" % counter)
counter += 1
dummy_generator = kwargs.get("dummy_generator", f())
for i in range(1, len(matrices)):
indices[i] = next(dummy_generator)
for i, arg in enumerate(matrices[:-1]):
ind_ranges[i] = arg.shape[1] - 1
matrices = [arg._entry(indices[i], indices[i+1], dummy_generator=dummy_generator) for i, arg in enumerate(matrices)]
expr_in_sum = Mul.fromiter(matrices)
if any(v.has(ImmutableMatrix) for v in matrices):
expand = True
result = coeff*Sum(
expr_in_sum,
*zip(indices[1:-1], [0]*len(ind_ranges), ind_ranges)
)
# Don't waste time in result.doit() if the sum bounds are symbolic
if not any(isinstance(v, (Integer, int)) for v in ind_ranges):
expand = False
return result.doit() if expand else result
def as_coeff_matrices(self):
scalars = [x for x in self.args if not x.is_Matrix]
matrices = [x for x in self.args if x.is_Matrix]
coeff = Mul(*scalars)
if coeff.is_commutative is False:
raise NotImplementedError("noncommutative scalars in MatMul are not supported.")
return coeff, matrices
def as_coeff_mmul(self):
coeff, matrices = self.as_coeff_matrices()
return coeff, MatMul(*matrices)
def _eval_transpose(self):
"""Transposition of matrix multiplication.
Notes
=====
The following rules are applied.
Transposition for matrix multiplied with another matrix:
`\\left(A B\\right)^{T} = B^{T} A^{T}`
Transposition for matrix multiplied with scalar:
`\\left(c A\\right)^{T} = c A^{T}`
References
==========
.. [1] https://en.wikipedia.org/wiki/Transpose
"""
coeff, matrices = self.as_coeff_matrices()
return MatMul(
coeff, *[transpose(arg) for arg in matrices[::-1]]).doit()
def _eval_adjoint(self):
return MatMul(*[adjoint(arg) for arg in self.args[::-1]]).doit()
def _eval_trace(self):
factor, mmul = self.as_coeff_mmul()
if factor != 1:
from .trace import trace
return factor * trace(mmul.doit())
else:
raise NotImplementedError("Can't simplify any further")
def _eval_determinant(self):
from sympy.matrices.expressions.determinant import Determinant
factor, matrices = self.as_coeff_matrices()
square_matrices = only_squares(*matrices)
return factor**self.rows * Mul(*list(map(Determinant, square_matrices)))
def _eval_inverse(self):
try:
return MatMul(*[
arg.inverse() if isinstance(arg, MatrixExpr) else arg**-1
for arg in self.args[::-1]]).doit()
except ShapeError:
return Inverse(self)
def doit(self, **kwargs):
deep = kwargs.get('deep', True)
if deep:
args = [arg.doit(**kwargs) for arg in self.args]
else:
args = self.args
# treat scalar*MatrixSymbol or scalar*MatPow separately
expr = canonicalize(MatMul(*args))
return expr
# Needed for partial compatibility with Mul
def args_cnc(self, **kwargs):
coeff_c = [x for x in self.args if x.is_commutative]
coeff_nc = [x for x in self.args if not x.is_commutative]
return [coeff_c, coeff_nc]
def _eval_derivative_matrix_lines(self, x):
from .transpose import Transpose
with_x_ind = [i for i, arg in enumerate(self.args) if arg.has(x)]
lines = []
for ind in with_x_ind:
left_args = self.args[:ind]
right_args = self.args[ind+1:]
if right_args:
right_mat = MatMul.fromiter(right_args)
else:
right_mat = Identity(self.shape[1])
if left_args:
left_rev = MatMul.fromiter([Transpose(i).doit() if i.is_Matrix else i for i in reversed(left_args)])
else:
left_rev = Identity(self.shape[0])
d = self.args[ind]._eval_derivative_matrix_lines(x)
for i in d:
i.append_first(left_rev)
i.append_second(right_mat)
lines.append(i)
return lines
def validate(*matrices):
""" Checks for valid shapes for args of MatMul """
for i in range(len(matrices)-1):
A, B = matrices[i:i+2]
if A.cols != B.rows:
raise ShapeError("Matrices %s and %s are not aligned"%(A, B))
# Rules
def newmul(*args):
if args[0] == 1:
args = args[1:]
return new(MatMul, *args)
def any_zeros(mul):
if any([arg.is_zero or (arg.is_Matrix and arg.is_ZeroMatrix)
for arg in mul.args]):
matrices = [arg for arg in mul.args if arg.is_Matrix]
return ZeroMatrix(matrices[0].rows, matrices[-1].cols)
return mul
def merge_explicit(matmul):
""" Merge explicit MatrixBase arguments
>>> from sympy import MatrixSymbol, Matrix, MatMul, pprint
>>> from sympy.matrices.expressions.matmul import merge_explicit
>>> A = MatrixSymbol('A', 2, 2)
>>> B = Matrix([[1, 1], [1, 1]])
>>> C = Matrix([[1, 2], [3, 4]])
>>> X = MatMul(A, B, C)
>>> pprint(X)
[1 1] [1 2]
A*[ ]*[ ]
[1 1] [3 4]
>>> pprint(merge_explicit(X))
[4 6]
A*[ ]
[4 6]
>>> X = MatMul(B, A, C)
>>> pprint(X)
[1 1] [1 2]
[ ]*A*[ ]
[1 1] [3 4]
>>> pprint(merge_explicit(X))
[1 1] [1 2]
[ ]*A*[ ]
[1 1] [3 4]
"""
if not any(isinstance(arg, MatrixBase) for arg in matmul.args):
return matmul
newargs = []
last = matmul.args[0]
for arg in matmul.args[1:]:
if isinstance(arg, (MatrixBase, Number)) and isinstance(last, (MatrixBase, Number)):
last = last * arg
else:
newargs.append(last)
last = arg
newargs.append(last)
return MatMul(*newargs)
def remove_ids(mul):
""" Remove Identities from a MatMul
This is a modified version of sympy.strategies.rm_id.
This is necesssary because MatMul may contain both MatrixExprs and Exprs
as args.
See Also
========
sympy.strategies.rm_id
"""
# Separate Exprs from MatrixExprs in args
factor, mmul = mul.as_coeff_mmul()
# Apply standard rm_id for MatMuls
result = rm_id(lambda x: x.is_Identity is True)(mmul)
if result != mmul:
return newmul(factor, *result.args) # Recombine and return
else:
return mul
def factor_in_front(mul):
factor, matrices = mul.as_coeff_matrices()
if factor != 1:
return newmul(factor, *matrices)
return mul
def combine_powers(mul):
"""Combine consecutive powers with the same base into one
e.g. A*A**2 -> A**3
This also cancels out the possible matrix inverses using the
knowledgebase of ``Inverse``.
e.g. Y * X * X.I -> Y
"""
factor, args = mul.as_coeff_matrices()
new_args = [args[0]]
for B in args[1:]:
A = new_args[-1]
if A.is_square == False or B.is_square == False:
new_args.append(B)
continue
if isinstance(A, MatPow):
A_base, A_exp = A.args
else:
A_base, A_exp = A, S.One
if isinstance(B, MatPow):
B_base, B_exp = B.args
else:
B_base, B_exp = B, S.One
if A_base == B_base:
new_exp = A_exp + B_exp
new_args[-1] = MatPow(A_base, new_exp).doit(deep=False)
continue
elif not isinstance(B_base, MatrixBase):
try:
B_base_inv = B_base.inverse()
except NonInvertibleMatrixError:
B_base_inv = None
if B_base_inv is not None and A_base == B_base_inv:
new_exp = A_exp - B_exp
new_args[-1] = MatPow(A_base, new_exp).doit(deep=False)
continue
new_args.append(B)
return newmul(factor, *new_args)
def combine_permutations(mul):
"""Refine products of permutation matrices as the products of cycles.
"""
args = mul.args
l = len(args)
if l < 2:
return mul
result = [args[0]]
for i in range(1, l):
A = result[-1]
B = args[i]
if isinstance(A, PermutationMatrix) and \
isinstance(B, PermutationMatrix):
cycle_1 = A.args[0]
cycle_2 = B.args[0]
result[-1] = PermutationMatrix(cycle_1 * cycle_2)
else:
result.append(B)
return MatMul(*result)
def combine_one_matrices(mul):
"""
Combine products of OneMatrix
e.g. OneMatrix(2, 3) * OneMatrix(3, 4) -> 3 * OneMatrix(2, 4)
"""
factor, args = mul.as_coeff_matrices()
new_args = [args[0]]
for B in args[1:]:
A = new_args[-1]
if not isinstance(A, OneMatrix) or not isinstance(B, OneMatrix):
new_args.append(B)
continue
new_args.pop()
new_args.append(OneMatrix(A.shape[0], B.shape[1]))
factor *= A.shape[1]
return newmul(factor, *new_args)
rules = (
any_zeros, remove_ids, combine_one_matrices, combine_powers, unpack, rm_id(lambda x: x == 1),
merge_explicit, factor_in_front, flatten, combine_permutations)
canonicalize = exhaust(typed({MatMul: do_one(*rules)}))
def only_squares(*matrices):
"""factor matrices only if they are square"""
if matrices[0].rows != matrices[-1].cols:
raise RuntimeError("Invalid matrices being multiplied")
out = []
start = 0
for i, M in enumerate(matrices):
if M.cols == matrices[start].rows:
out.append(MatMul(*matrices[start:i+1]).doit())
start = i+1
return out
from sympy.assumptions.ask import ask, Q
from sympy.assumptions.refine import handlers_dict
def refine_MatMul(expr, assumptions):
"""
>>> from sympy import MatrixSymbol, Q, assuming, refine
>>> X = MatrixSymbol('X', 2, 2)
>>> expr = X * X.T
>>> print(expr)
X*X.T
>>> with assuming(Q.orthogonal(X)):
... print(refine(expr))
I
"""
newargs = []
exprargs = []
for args in expr.args:
if args.is_Matrix:
exprargs.append(args)
else:
newargs.append(args)
last = exprargs[0]
for arg in exprargs[1:]:
if arg == last.T and ask(Q.orthogonal(arg), assumptions):
last = Identity(arg.shape[0])
elif arg == last.conjugate() and ask(Q.unitary(arg), assumptions):
last = Identity(arg.shape[0])
else:
newargs.append(last)
last = arg
newargs.append(last)
return MatMul(*newargs)
handlers_dict['MatMul'] = refine_MatMul
|
a2829489a2ba93eec925e10cb09aef67c644c245316b4c8765470c6a9686812f | from typing import Any, Callable
from sympy.core.logic import FuzzyBool
from functools import wraps, reduce
import collections
from sympy.core import S, Symbol, Tuple, Integer, Basic, Expr, Eq, Mul, Add
from sympy.core.decorators import call_highest_priority
from sympy.core.compatibility import SYMPY_INTS, default_sort_key
from sympy.core.sympify import SympifyError, _sympify
from sympy.functions import conjugate, adjoint
from sympy.functions.special.tensor_functions import KroneckerDelta
from sympy.matrices.common import NonSquareMatrixError, NonInvertibleMatrixError
from sympy.simplify import simplify
from sympy.utilities.misc import filldedent
from sympy.assumptions.ask import ask, Q
def _sympifyit(arg, retval=None):
# This version of _sympifyit sympifies MutableMatrix objects
def deco(func):
@wraps(func)
def __sympifyit_wrapper(a, b):
try:
b = _sympify(b)
return func(a, b)
except SympifyError:
return retval
return __sympifyit_wrapper
return deco
class MatrixExpr(Expr):
"""Superclass for Matrix Expressions
MatrixExprs represent abstract matrices, linear transformations represented
within a particular basis.
Examples
========
>>> from sympy import MatrixSymbol
>>> A = MatrixSymbol('A', 3, 3)
>>> y = MatrixSymbol('y', 3, 1)
>>> x = (A.T*A).I * A * y
See Also
========
MatrixSymbol, MatAdd, MatMul, Transpose, Inverse
"""
# Should not be considered iterable by the
# sympy.core.compatibility.iterable function. Subclass that actually are
# iterable (i.e., explicit matrices) should set this to True.
_iterable = False
_op_priority = 11.0
is_Matrix = True # type: bool
is_MatrixExpr = True # type: bool
is_Identity = None # type: FuzzyBool
is_Inverse = False
is_Transpose = False
is_ZeroMatrix = False
is_MatAdd = False
is_MatMul = False
is_commutative = False
is_number = False
is_symbol = False
is_scalar = False
def __new__(cls, *args, **kwargs):
args = map(_sympify, args)
return Basic.__new__(cls, *args, **kwargs)
# The following is adapted from the core Expr object
def __neg__(self):
return MatMul(S.NegativeOne, self).doit()
def __abs__(self):
raise NotImplementedError
@_sympifyit('other', NotImplemented)
@call_highest_priority('__radd__')
def __add__(self, other):
return MatAdd(self, other, check=True).doit()
@_sympifyit('other', NotImplemented)
@call_highest_priority('__add__')
def __radd__(self, other):
return MatAdd(other, self, check=True).doit()
@_sympifyit('other', NotImplemented)
@call_highest_priority('__rsub__')
def __sub__(self, other):
return MatAdd(self, -other, check=True).doit()
@_sympifyit('other', NotImplemented)
@call_highest_priority('__sub__')
def __rsub__(self, other):
return MatAdd(other, -self, check=True).doit()
@_sympifyit('other', NotImplemented)
@call_highest_priority('__rmul__')
def __mul__(self, other):
return MatMul(self, other).doit()
@_sympifyit('other', NotImplemented)
@call_highest_priority('__rmul__')
def __matmul__(self, other):
return MatMul(self, other).doit()
@_sympifyit('other', NotImplemented)
@call_highest_priority('__mul__')
def __rmul__(self, other):
return MatMul(other, self).doit()
@_sympifyit('other', NotImplemented)
@call_highest_priority('__mul__')
def __rmatmul__(self, other):
return MatMul(other, self).doit()
@_sympifyit('other', NotImplemented)
@call_highest_priority('__rpow__')
def __pow__(self, other):
return MatPow(self, other).doit()
@_sympifyit('other', NotImplemented)
@call_highest_priority('__pow__')
def __rpow__(self, other):
raise NotImplementedError("Matrix Power not defined")
@_sympifyit('other', NotImplemented)
@call_highest_priority('__rdiv__')
def __div__(self, other):
return self * other**S.NegativeOne
@_sympifyit('other', NotImplemented)
@call_highest_priority('__div__')
def __rdiv__(self, other):
raise NotImplementedError()
#return MatMul(other, Pow(self, S.NegativeOne))
__truediv__ = __div__ # type: Callable[[MatrixExpr, Any], Any]
__rtruediv__ = __rdiv__ # type: Callable[[MatrixExpr, Any], Any]
@property
def rows(self):
return self.shape[0]
@property
def cols(self):
return self.shape[1]
@property
def is_square(self):
return self.rows == self.cols
def _eval_conjugate(self):
from sympy.matrices.expressions.adjoint import Adjoint
from sympy.matrices.expressions.transpose import Transpose
return Adjoint(Transpose(self))
def as_real_imag(self, deep=True, **hints):
from sympy import I
real = S.Half * (self + self._eval_conjugate())
im = (self - self._eval_conjugate())/(2*I)
return (real, im)
def _eval_inverse(self):
from sympy.matrices.expressions.inverse import Inverse
return Inverse(self)
def _eval_transpose(self):
return Transpose(self)
def _eval_power(self, exp):
"""
Override this in sub-classes to implement simplification of powers. The cases where the exponent
is -1, 0, 1 are already covered in MatPow.doit(), so implementations can exclude these cases.
"""
return MatPow(self, exp)
def _eval_simplify(self, **kwargs):
if self.is_Atom:
return self
else:
return self.func(*[simplify(x, **kwargs) for x in self.args])
def _eval_adjoint(self):
from sympy.matrices.expressions.adjoint import Adjoint
return Adjoint(self)
def _eval_derivative_array(self, x):
if isinstance(x, MatrixExpr):
return _matrix_derivative(self, x)
else:
return self._eval_derivative(x)
def _eval_derivative_n_times(self, x, n):
return Basic._eval_derivative_n_times(self, x, n)
def _visit_eval_derivative_scalar(self, x):
# `x` is a scalar:
if x.has(self):
return _matrix_derivative(x, self)
else:
return ZeroMatrix(*self.shape)
def _visit_eval_derivative_array(self, x):
if x.has(self):
return _matrix_derivative(x, self)
else:
from sympy import Derivative
return Derivative(x, self)
def _accept_eval_derivative(self, s):
from sympy import MatrixBase, NDimArray
if isinstance(s, (MatrixBase, NDimArray, MatrixExpr)):
return s._visit_eval_derivative_array(self)
else:
return s._visit_eval_derivative_scalar(self)
@classmethod
def _check_dim(cls, dim):
"""Helper function to check invalid matrix dimensions"""
from sympy.core.assumptions import check_assumptions
ok = check_assumptions(dim, integer=True, nonnegative=True)
if ok is False:
raise ValueError(
"The dimension specification {} should be "
"a nonnegative integer.".format(dim))
def _entry(self, i, j, **kwargs):
raise NotImplementedError(
"Indexing not implemented for %s" % self.__class__.__name__)
def adjoint(self):
return adjoint(self)
def as_coeff_Mul(self, rational=False):
"""Efficiently extract the coefficient of a product. """
return S.One, self
def conjugate(self):
return conjugate(self)
def transpose(self):
from sympy.matrices.expressions.transpose import transpose
return transpose(self)
@property
def T(self):
'''Matrix transposition'''
return self.transpose()
def inverse(self):
if not self.is_square:
raise NonSquareMatrixError('Inverse of non-square matrix')
return self._eval_inverse()
def inv(self):
return self.inverse()
@property
def I(self):
return self.inverse()
def valid_index(self, i, j):
def is_valid(idx):
return isinstance(idx, (int, Integer, Symbol, Expr))
return (is_valid(i) and is_valid(j) and
(self.rows is None or
(0 <= i) != False and (i < self.rows) != False) and
(0 <= j) != False and (j < self.cols) != False)
def __getitem__(self, key):
if not isinstance(key, tuple) and isinstance(key, slice):
from sympy.matrices.expressions.slice import MatrixSlice
return MatrixSlice(self, key, (0, None, 1))
if isinstance(key, tuple) and len(key) == 2:
i, j = key
if isinstance(i, slice) or isinstance(j, slice):
from sympy.matrices.expressions.slice import MatrixSlice
return MatrixSlice(self, i, j)
i, j = _sympify(i), _sympify(j)
if self.valid_index(i, j) != False:
return self._entry(i, j)
else:
raise IndexError("Invalid indices (%s, %s)" % (i, j))
elif isinstance(key, (SYMPY_INTS, Integer)):
# row-wise decomposition of matrix
rows, cols = self.shape
# allow single indexing if number of columns is known
if not isinstance(cols, Integer):
raise IndexError(filldedent('''
Single indexing is only supported when the number
of columns is known.'''))
key = _sympify(key)
i = key // cols
j = key % cols
if self.valid_index(i, j) != False:
return self._entry(i, j)
else:
raise IndexError("Invalid index %s" % key)
elif isinstance(key, (Symbol, Expr)):
raise IndexError(filldedent('''
Only integers may be used when addressing the matrix
with a single index.'''))
raise IndexError("Invalid index, wanted %s[i,j]" % self)
def as_explicit(self):
"""
Returns a dense Matrix with elements represented explicitly
Returns an object of type ImmutableDenseMatrix.
Examples
========
>>> from sympy import Identity
>>> I = Identity(3)
>>> I
I
>>> I.as_explicit()
Matrix([
[1, 0, 0],
[0, 1, 0],
[0, 0, 1]])
See Also
========
as_mutable: returns mutable Matrix type
"""
if (not isinstance(self.rows, (SYMPY_INTS, Integer))
or not isinstance(self.cols, (SYMPY_INTS, Integer))):
raise ValueError(
'Matrix with symbolic shape '
'cannot be represented explicitly.')
from sympy.matrices.immutable import ImmutableDenseMatrix
return ImmutableDenseMatrix([[self[i, j]
for j in range(self.cols)]
for i in range(self.rows)])
def as_mutable(self):
"""
Returns a dense, mutable matrix with elements represented explicitly
Examples
========
>>> from sympy import Identity
>>> I = Identity(3)
>>> I
I
>>> I.shape
(3, 3)
>>> I.as_mutable()
Matrix([
[1, 0, 0],
[0, 1, 0],
[0, 0, 1]])
See Also
========
as_explicit: returns ImmutableDenseMatrix
"""
return self.as_explicit().as_mutable()
def __array__(self):
from numpy import empty
a = empty(self.shape, dtype=object)
for i in range(self.rows):
for j in range(self.cols):
a[i, j] = self[i, j]
return a
def equals(self, other):
"""
Test elementwise equality between matrices, potentially of different
types
>>> from sympy import Identity, eye
>>> Identity(3).equals(eye(3))
True
"""
return self.as_explicit().equals(other)
def canonicalize(self):
return self
def as_coeff_mmul(self):
return 1, MatMul(self)
@staticmethod
def from_index_summation(expr, first_index=None, last_index=None, dimensions=None):
r"""
Parse expression of matrices with explicitly summed indices into a
matrix expression without indices, if possible.
This transformation expressed in mathematical notation:
`\sum_{j=0}^{N-1} A_{i,j} B_{j,k} \Longrightarrow \mathbf{A}\cdot \mathbf{B}`
Optional parameter ``first_index``: specify which free index to use as
the index starting the expression.
Examples
========
>>> from sympy import MatrixSymbol, MatrixExpr, Sum
>>> from sympy.abc import i, j, k, l, N
>>> A = MatrixSymbol("A", N, N)
>>> B = MatrixSymbol("B", N, N)
>>> expr = Sum(A[i, j]*B[j, k], (j, 0, N-1))
>>> MatrixExpr.from_index_summation(expr)
A*B
Transposition is detected:
>>> expr = Sum(A[j, i]*B[j, k], (j, 0, N-1))
>>> MatrixExpr.from_index_summation(expr)
A.T*B
Detect the trace:
>>> expr = Sum(A[i, i], (i, 0, N-1))
>>> MatrixExpr.from_index_summation(expr)
Trace(A)
More complicated expressions:
>>> expr = Sum(A[i, j]*B[k, j]*A[l, k], (j, 0, N-1), (k, 0, N-1))
>>> MatrixExpr.from_index_summation(expr)
A*B.T*A.T
"""
from sympy import Sum, Mul, Add, MatMul, transpose, trace
from sympy.strategies.traverse import bottom_up
def remove_matelement(expr, i1, i2):
def repl_match(pos):
def func(x):
if not isinstance(x, MatrixElement):
return False
if x.args[pos] != i1:
return False
if x.args[3-pos] == 0:
if x.args[0].shape[2-pos] == 1:
return True
else:
return False
return True
return func
expr = expr.replace(repl_match(1),
lambda x: x.args[0])
expr = expr.replace(repl_match(2),
lambda x: transpose(x.args[0]))
# Make sure that all Mul are transformed to MatMul and that they
# are flattened:
rule = bottom_up(lambda x: reduce(lambda a, b: a*b, x.args) if isinstance(x, (Mul, MatMul)) else x)
return rule(expr)
def recurse_expr(expr, index_ranges={}):
if expr.is_Mul:
nonmatargs = []
pos_arg = []
pos_ind = []
dlinks = {}
link_ind = []
counter = 0
args_ind = []
for arg in expr.args:
retvals = recurse_expr(arg, index_ranges)
assert isinstance(retvals, list)
if isinstance(retvals, list):
for i in retvals:
args_ind.append(i)
else:
args_ind.append(retvals)
for arg_symbol, arg_indices in args_ind:
if arg_indices is None:
nonmatargs.append(arg_symbol)
continue
if isinstance(arg_symbol, MatrixElement):
arg_symbol = arg_symbol.args[0]
pos_arg.append(arg_symbol)
pos_ind.append(arg_indices)
link_ind.append([None]*len(arg_indices))
for i, ind in enumerate(arg_indices):
if ind in dlinks:
other_i = dlinks[ind]
link_ind[counter][i] = other_i
link_ind[other_i[0]][other_i[1]] = (counter, i)
dlinks[ind] = (counter, i)
counter += 1
counter2 = 0
lines = {}
while counter2 < len(link_ind):
for i, e in enumerate(link_ind):
if None in e:
line_start_index = (i, e.index(None))
break
cur_ind_pos = line_start_index
cur_line = []
index1 = pos_ind[cur_ind_pos[0]][cur_ind_pos[1]]
while True:
d, r = cur_ind_pos
if pos_arg[d] != 1:
if r % 2 == 1:
cur_line.append(transpose(pos_arg[d]))
else:
cur_line.append(pos_arg[d])
next_ind_pos = link_ind[d][1-r]
counter2 += 1
# Mark as visited, there will be no `None` anymore:
link_ind[d] = (-1, -1)
if next_ind_pos is None:
index2 = pos_ind[d][1-r]
lines[(index1, index2)] = cur_line
break
cur_ind_pos = next_ind_pos
lines = {k: MatMul.fromiter(v) if len(v) != 1 else v[0] for k, v in lines.items()}
return [(Mul.fromiter(nonmatargs), None)] + [
(MatrixElement(a, i, j), (i, j)) for (i, j), a in lines.items()
]
elif expr.is_Add:
res = [recurse_expr(i) for i in expr.args]
d = collections.defaultdict(list)
for res_addend in res:
scalar = 1
for elem, indices in res_addend:
if indices is None:
scalar = elem
continue
indices = tuple(sorted(indices, key=default_sort_key))
d[indices].append(scalar*remove_matelement(elem, *indices))
scalar = 1
return [(MatrixElement(Add.fromiter(v), *k), k) for k, v in d.items()]
elif isinstance(expr, KroneckerDelta):
i1, i2 = expr.args
if dimensions is not None:
identity = Identity(dimensions[0])
else:
identity = S.One
return [(MatrixElement(identity, i1, i2), (i1, i2))]
elif isinstance(expr, MatrixElement):
matrix_symbol, i1, i2 = expr.args
if i1 in index_ranges:
r1, r2 = index_ranges[i1]
if r1 != 0 or matrix_symbol.shape[0] != r2+1:
raise ValueError("index range mismatch: {} vs. (0, {})".format(
(r1, r2), matrix_symbol.shape[0]))
if i2 in index_ranges:
r1, r2 = index_ranges[i2]
if r1 != 0 or matrix_symbol.shape[1] != r2+1:
raise ValueError("index range mismatch: {} vs. (0, {})".format(
(r1, r2), matrix_symbol.shape[1]))
if (i1 == i2) and (i1 in index_ranges):
return [(trace(matrix_symbol), None)]
return [(MatrixElement(matrix_symbol, i1, i2), (i1, i2))]
elif isinstance(expr, Sum):
return recurse_expr(
expr.args[0],
index_ranges={i[0]: i[1:] for i in expr.args[1:]}
)
else:
return [(expr, None)]
retvals = recurse_expr(expr)
factors, indices = zip(*retvals)
retexpr = Mul.fromiter(factors)
if len(indices) == 0 or list(set(indices)) == [None]:
return retexpr
if first_index is None:
for i in indices:
if i is not None:
ind0 = i
break
return remove_matelement(retexpr, *ind0)
else:
return remove_matelement(retexpr, first_index, last_index)
def applyfunc(self, func):
from .applyfunc import ElementwiseApplyFunction
return ElementwiseApplyFunction(func, self)
def _eval_Eq(self, other):
if not isinstance(other, MatrixExpr):
return False
if self.shape != other.shape:
return False
if (self - other).is_ZeroMatrix:
return True
return Eq(self, other, evaluate=False)
def get_postprocessor(cls):
def _postprocessor(expr):
# To avoid circular imports, we can't have MatMul/MatAdd on the top level
mat_class = {Mul: MatMul, Add: MatAdd}[cls]
nonmatrices = []
matrices = []
for term in expr.args:
if isinstance(term, MatrixExpr):
matrices.append(term)
else:
nonmatrices.append(term)
if not matrices:
return cls._from_args(nonmatrices)
if nonmatrices:
if cls == Mul:
for i in range(len(matrices)):
if not matrices[i].is_MatrixExpr:
# If one of the matrices explicit, absorb the scalar into it
# (doit will combine all explicit matrices into one, so it
# doesn't matter which)
matrices[i] = matrices[i].__mul__(cls._from_args(nonmatrices))
nonmatrices = []
break
else:
# Maintain the ability to create Add(scalar, matrix) without
# raising an exception. That way different algorithms can
# replace matrix expressions with non-commutative symbols to
# manipulate them like non-commutative scalars.
return cls._from_args(nonmatrices + [mat_class(*matrices).doit(deep=False)])
if mat_class == MatAdd:
return mat_class(*matrices).doit(deep=False)
return mat_class(cls._from_args(nonmatrices), *matrices).doit(deep=False)
return _postprocessor
Basic._constructor_postprocessor_mapping[MatrixExpr] = {
"Mul": [get_postprocessor(Mul)],
"Add": [get_postprocessor(Add)],
}
def _matrix_derivative(expr, x):
from sympy import Derivative
lines = expr._eval_derivative_matrix_lines(x)
parts = [i.build() for i in lines]
from sympy.codegen.array_utils import recognize_matrix_expression
parts = [[recognize_matrix_expression(j).doit() for j in i] for i in parts]
def _get_shape(elem):
if isinstance(elem, MatrixExpr):
return elem.shape
return 1, 1
def get_rank(parts):
return sum([j not in (1, None) for i in parts for j in _get_shape(i)])
ranks = [get_rank(i) for i in parts]
rank = ranks[0]
def contract_one_dims(parts):
if len(parts) == 1:
return parts[0]
else:
p1, p2 = parts[:2]
if p2.is_Matrix:
p2 = p2.T
if p1 == Identity(1):
pbase = p2
elif p2 == Identity(1):
pbase = p1
else:
pbase = p1*p2
if len(parts) == 2:
return pbase
else: # len(parts) > 2
if pbase.is_Matrix:
raise ValueError("")
return pbase*Mul.fromiter(parts[2:])
if rank <= 2:
return Add.fromiter([contract_one_dims(i) for i in parts])
return Derivative(expr, x)
class MatrixElement(Expr):
parent = property(lambda self: self.args[0])
i = property(lambda self: self.args[1])
j = property(lambda self: self.args[2])
_diff_wrt = True
is_symbol = True
is_commutative = True
def __new__(cls, name, n, m):
n, m = map(_sympify, (n, m))
from sympy import MatrixBase
if isinstance(name, (MatrixBase,)):
if n.is_Integer and m.is_Integer:
return name[n, m]
if isinstance(name, str):
name = Symbol(name)
name = _sympify(name)
obj = Expr.__new__(cls, name, n, m)
return obj
def doit(self, **kwargs):
deep = kwargs.get('deep', True)
if deep:
args = [arg.doit(**kwargs) for arg in self.args]
else:
args = self.args
return args[0][args[1], args[2]]
@property
def indices(self):
return self.args[1:]
def _eval_derivative(self, v):
from sympy import Sum, symbols, Dummy
if not isinstance(v, MatrixElement):
from sympy import MatrixBase
if isinstance(self.parent, MatrixBase):
return self.parent.diff(v)[self.i, self.j]
return S.Zero
M = self.args[0]
m, n = self.parent.shape
if M == v.args[0]:
return KroneckerDelta(self.args[1], v.args[1], (0, m-1)) * \
KroneckerDelta(self.args[2], v.args[2], (0, n-1))
if isinstance(M, Inverse):
i, j = self.args[1:]
i1, i2 = symbols("z1, z2", cls=Dummy)
Y = M.args[0]
r1, r2 = Y.shape
return -Sum(M[i, i1]*Y[i1, i2].diff(v)*M[i2, j], (i1, 0, r1-1), (i2, 0, r2-1))
if self.has(v.args[0]):
return None
return S.Zero
class MatrixSymbol(MatrixExpr):
"""Symbolic representation of a Matrix object
Creates a SymPy Symbol to represent a Matrix. This matrix has a shape and
can be included in Matrix Expressions
Examples
========
>>> from sympy import MatrixSymbol, Identity
>>> A = MatrixSymbol('A', 3, 4) # A 3 by 4 Matrix
>>> B = MatrixSymbol('B', 4, 3) # A 4 by 3 Matrix
>>> A.shape
(3, 4)
>>> 2*A*B + Identity(3)
I + 2*A*B
"""
is_commutative = False
is_symbol = True
_diff_wrt = True
def __new__(cls, name, n, m):
n, m = _sympify(n), _sympify(m)
cls._check_dim(m)
cls._check_dim(n)
if isinstance(name, str):
name = Symbol(name)
obj = Basic.__new__(cls, name, n, m)
return obj
def _hashable_content(self):
return (self.name, self.shape)
@property
def shape(self):
return self.args[1:3]
@property
def name(self):
return self.args[0].name
def _eval_subs(self, old, new):
# only do substitutions in shape
shape = Tuple(*self.shape)._subs(old, new)
return MatrixSymbol(self.args[0], *shape)
def __call__(self, *args):
raise TypeError("%s object is not callable" % self.__class__)
def _entry(self, i, j, **kwargs):
return MatrixElement(self, i, j)
@property
def free_symbols(self):
return {self}
def doit(self, **hints):
if hints.get('deep', True):
return type(self)(self.args[0], self.args[1].doit(**hints),
self.args[2].doit(**hints))
else:
return self
def _eval_simplify(self, **kwargs):
return self
def _eval_derivative(self, x):
# x is a scalar:
return ZeroMatrix(self.shape[0], self.shape[1])
def _eval_derivative_matrix_lines(self, x):
if self != x:
first = ZeroMatrix(x.shape[0], self.shape[0]) if self.shape[0] != 1 else S.Zero
second = ZeroMatrix(x.shape[1], self.shape[1]) if self.shape[1] != 1 else S.Zero
return [_LeftRightArgs(
[first, second],
)]
else:
first = Identity(self.shape[0]) if self.shape[0] != 1 else S.One
second = Identity(self.shape[1]) if self.shape[1] != 1 else S.One
return [_LeftRightArgs(
[first, second],
)]
class Identity(MatrixExpr):
"""The Matrix Identity I - multiplicative identity
Examples
========
>>> from sympy.matrices import Identity, MatrixSymbol
>>> A = MatrixSymbol('A', 3, 5)
>>> I = Identity(3)
>>> I*A
A
"""
is_Identity = True
def __new__(cls, n):
n = _sympify(n)
cls._check_dim(n)
return super().__new__(cls, n)
@property
def rows(self):
return self.args[0]
@property
def cols(self):
return self.args[0]
@property
def shape(self):
return (self.args[0], self.args[0])
@property
def is_square(self):
return True
def _eval_transpose(self):
return self
def _eval_trace(self):
return self.rows
def _eval_inverse(self):
return self
def conjugate(self):
return self
def _entry(self, i, j, **kwargs):
eq = Eq(i, j)
if eq is S.true:
return S.One
elif eq is S.false:
return S.Zero
return KroneckerDelta(i, j, (0, self.cols-1))
def _eval_determinant(self):
return S.One
def _eval_power(self, exp):
return self
class GenericIdentity(Identity):
"""
An identity matrix without a specified shape
This exists primarily so MatMul() with no arguments can return something
meaningful.
"""
def __new__(cls):
# super(Identity, cls) instead of super(GenericIdentity, cls) because
# Identity.__new__ doesn't have the same signature
return super(Identity, cls).__new__(cls)
@property
def rows(self):
raise TypeError("GenericIdentity does not have a specified shape")
@property
def cols(self):
raise TypeError("GenericIdentity does not have a specified shape")
@property
def shape(self):
raise TypeError("GenericIdentity does not have a specified shape")
# Avoid Matrix.__eq__ which might call .shape
def __eq__(self, other):
return isinstance(other, GenericIdentity)
def __ne__(self, other):
return not (self == other)
def __hash__(self):
return super().__hash__()
class ZeroMatrix(MatrixExpr):
"""The Matrix Zero 0 - additive identity
Examples
========
>>> from sympy import MatrixSymbol, ZeroMatrix
>>> A = MatrixSymbol('A', 3, 5)
>>> Z = ZeroMatrix(3, 5)
>>> A + Z
A
>>> Z*A.T
0
"""
is_ZeroMatrix = True
def __new__(cls, m, n):
m, n = _sympify(m), _sympify(n)
cls._check_dim(m)
cls._check_dim(n)
return super().__new__(cls, m, n)
@property
def shape(self):
return (self.args[0], self.args[1])
def _eval_power(self, exp):
# exp = -1, 0, 1 are already handled at this stage
if (exp < 0) == True:
raise NonInvertibleMatrixError("Matrix det == 0; not invertible")
return self
def _eval_transpose(self):
return ZeroMatrix(self.cols, self.rows)
def _eval_trace(self):
return S.Zero
def _eval_determinant(self):
return S.Zero
def _eval_inverse(self):
raise NonInvertibleMatrixError("Matrix det == 0; not invertible.")
def conjugate(self):
return self
def _entry(self, i, j, **kwargs):
return S.Zero
class GenericZeroMatrix(ZeroMatrix):
"""
A zero matrix without a specified shape
This exists primarily so MatAdd() with no arguments can return something
meaningful.
"""
def __new__(cls):
# super(ZeroMatrix, cls) instead of super(GenericZeroMatrix, cls)
# because ZeroMatrix.__new__ doesn't have the same signature
return super(ZeroMatrix, cls).__new__(cls)
@property
def rows(self):
raise TypeError("GenericZeroMatrix does not have a specified shape")
@property
def cols(self):
raise TypeError("GenericZeroMatrix does not have a specified shape")
@property
def shape(self):
raise TypeError("GenericZeroMatrix does not have a specified shape")
# Avoid Matrix.__eq__ which might call .shape
def __eq__(self, other):
return isinstance(other, GenericZeroMatrix)
def __ne__(self, other):
return not (self == other)
def __hash__(self):
return super().__hash__()
class OneMatrix(MatrixExpr):
"""
Matrix whose all entries are ones.
"""
def __new__(cls, m, n, evaluate=False):
m, n = _sympify(m), _sympify(n)
cls._check_dim(m)
cls._check_dim(n)
if evaluate:
condition = Eq(m, 1) & Eq(n, 1)
if condition == True:
return Identity(1)
obj = super().__new__(cls, m, n)
return obj
@property
def shape(self):
return self._args
@property
def is_Identity(self):
return self._is_1x1() == True
def as_explicit(self):
from sympy import ImmutableDenseMatrix
return ImmutableDenseMatrix.ones(*self.shape)
def doit(self, **hints):
args = self.args
if hints.get('deep', True):
args = [a.doit(**hints) for a in args]
return self.func(*args, evaluate=True)
def _eval_power(self, exp):
# exp = -1, 0, 1 are already handled at this stage
if self._is_1x1() == True:
return Identity(1)
if (exp < 0) == True:
raise NonInvertibleMatrixError("Matrix det == 0; not invertible")
if ask(Q.integer(exp)):
return self.shape[0] ** (exp - 1) * OneMatrix(*self.shape)
return super()._eval_power(exp)
def _eval_transpose(self):
return OneMatrix(self.cols, self.rows)
def _eval_trace(self):
return S.One*self.rows
def _is_1x1(self):
"""Returns true if the matrix is known to be 1x1"""
shape = self.shape
return Eq(shape[0], 1) & Eq(shape[1], 1)
def _eval_determinant(self):
condition = self._is_1x1()
if condition == True:
return S.One
elif condition == False:
return S.Zero
else:
from sympy import Determinant
return Determinant(self)
def _eval_inverse(self):
condition = self._is_1x1()
if condition == True:
return Identity(1)
elif condition == False:
raise NonInvertibleMatrixError("Matrix det == 0; not invertible.")
else:
return Inverse(self)
def conjugate(self):
return self
def _entry(self, i, j, **kwargs):
return S.One
def matrix_symbols(expr):
return [sym for sym in expr.free_symbols if sym.is_Matrix]
class _LeftRightArgs:
r"""
Helper class to compute matrix derivatives.
The logic: when an expression is derived by a matrix `X_{mn}`, two lines of
matrix multiplications are created: the one contracted to `m` (first line),
and the one contracted to `n` (second line).
Transposition flips the side by which new matrices are connected to the
lines.
The trace connects the end of the two lines.
"""
def __init__(self, lines, higher=S.One):
self._lines = [i for i in lines]
self._first_pointer_parent = self._lines
self._first_pointer_index = 0
self._first_line_index = 0
self._second_pointer_parent = self._lines
self._second_pointer_index = 1
self._second_line_index = 1
self.higher = higher
@property
def first_pointer(self):
return self._first_pointer_parent[self._first_pointer_index]
@first_pointer.setter
def first_pointer(self, value):
self._first_pointer_parent[self._first_pointer_index] = value
@property
def second_pointer(self):
return self._second_pointer_parent[self._second_pointer_index]
@second_pointer.setter
def second_pointer(self, value):
self._second_pointer_parent[self._second_pointer_index] = value
def __repr__(self):
built = [self._build(i) for i in self._lines]
return "_LeftRightArgs(lines=%s, higher=%s)" % (
built,
self.higher,
)
def transpose(self):
self._first_pointer_parent, self._second_pointer_parent = self._second_pointer_parent, self._first_pointer_parent
self._first_pointer_index, self._second_pointer_index = self._second_pointer_index, self._first_pointer_index
self._first_line_index, self._second_line_index = self._second_line_index, self._first_line_index
return self
@staticmethod
def _build(expr):
from sympy.core.expr import ExprBuilder
if isinstance(expr, ExprBuilder):
return expr.build()
if isinstance(expr, list):
if len(expr) == 1:
return expr[0]
else:
return expr[0](*[_LeftRightArgs._build(i) for i in expr[1]])
else:
return expr
def build(self):
data = [self._build(i) for i in self._lines]
if self.higher != 1:
data += [self._build(self.higher)]
data = [i.doit() for i in data]
return data
def matrix_form(self):
if self.first != 1 and self.higher != 1:
raise ValueError("higher dimensional array cannot be represented")
def _get_shape(elem):
if isinstance(elem, MatrixExpr):
return elem.shape
return (None, None)
if _get_shape(self.first)[1] != _get_shape(self.second)[1]:
# Remove one-dimensional identity matrices:
# (this is needed by `a.diff(a)` where `a` is a vector)
if _get_shape(self.second) == (1, 1):
return self.first*self.second[0, 0]
if _get_shape(self.first) == (1, 1):
return self.first[1, 1]*self.second.T
raise ValueError("incompatible shapes")
if self.first != 1:
return self.first*self.second.T
else:
return self.higher
def rank(self):
"""
Number of dimensions different from trivial (warning: not related to
matrix rank).
"""
rank = 0
if self.first != 1:
rank += sum([i != 1 for i in self.first.shape])
if self.second != 1:
rank += sum([i != 1 for i in self.second.shape])
if self.higher != 1:
rank += 2
return rank
def _multiply_pointer(self, pointer, other):
from sympy.core.expr import ExprBuilder
from sympy.codegen.array_utils import CodegenArrayContraction, CodegenArrayTensorProduct
subexpr = ExprBuilder(
CodegenArrayContraction,
[
ExprBuilder(
CodegenArrayTensorProduct,
[
pointer,
other
]
),
(1, 2)
],
validator=CodegenArrayContraction._validate
)
return subexpr
def append_first(self, other):
self.first_pointer *= other
def append_second(self, other):
self.second_pointer *= other
def __hash__(self):
return hash((self.first, self.second))
def __eq__(self, other):
if not isinstance(other, _LeftRightArgs):
return False
return (self.first == other.first) and (self.second == other.second)
def _make_matrix(x):
from sympy import ImmutableDenseMatrix
if isinstance(x, MatrixExpr):
return x
return ImmutableDenseMatrix([[x]])
from .matmul import MatMul
from .matadd import MatAdd
from .matpow import MatPow
from .transpose import Transpose
from .inverse import Inverse
|
95750945b8d047a7b599d1d0749c0d2095cc5bcd1798a6ce56a36f5178c1e7b5 | from sympy.core.singleton import S
from sympy.core.sympify import _sympify
from sympy.polys.polytools import Poly
from .matexpr import MatrixExpr
class CompanionMatrix(MatrixExpr):
"""A symbolic companion matrix of a polynomial.
Examples
========
>>> from sympy import Poly, Symbol, symbols
>>> from sympy.matrices.expressions import CompanionMatrix
>>> x = Symbol('x')
>>> c0, c1, c2, c3, c4 = symbols('c0:5')
>>> p = Poly(c0 + c1*x + c2*x**2 + c3*x**3 + c4*x**4 + x**5, x)
>>> CompanionMatrix(p)
CompanionMatrix(Poly(x**5 + c4*x**4 + c3*x**3 + c2*x**2 + c1*x + c0,
x, domain='ZZ[c0,c1,c2,c3,c4]'))
"""
def __new__(cls, poly):
poly = _sympify(poly)
if not isinstance(poly, Poly):
raise ValueError("{} must be a Poly instance.".format(poly))
if not poly.is_monic:
raise ValueError("{} must be a monic polynomial.".format(poly))
if not poly.is_univariate:
raise ValueError(
"{} must be a univariate polynomial.".format(poly))
if not poly.degree() >= 1:
raise ValueError(
"{} must have degree not less than 1.".format(poly))
return super().__new__(cls, poly)
@property
def shape(self):
poly = self.args[0]
size = poly.degree()
return size, size
def _entry(self, i, j):
if j == self.cols - 1:
return -self.args[0].all_coeffs()[-1 - i]
elif i == j + 1:
return S.One
return S.Zero
def as_explicit(self):
from sympy.matrices.immutable import ImmutableDenseMatrix
return ImmutableDenseMatrix.companion(self.args[0])
|
a61792b42176791487dbf5523479676faa6f92b64c6be72085dfa803c06f6776 | from sympy import Basic, Expr, sympify, S
from sympy.matrices.matrices import MatrixBase
from sympy.matrices.common import NonSquareMatrixError
class Trace(Expr):
"""Matrix Trace
Represents the trace of a matrix expression.
Examples
========
>>> from sympy import MatrixSymbol, Trace, eye
>>> A = MatrixSymbol('A', 3, 3)
>>> Trace(A)
Trace(A)
>>> Trace(eye(3))
Trace(Matrix([
[1, 0, 0],
[0, 1, 0],
[0, 0, 1]]))
>>> Trace(eye(3)).simplify()
3
"""
is_Trace = True
is_commutative = True
def __new__(cls, mat):
mat = sympify(mat)
if not mat.is_Matrix:
raise TypeError("input to Trace, %s, is not a matrix" % str(mat))
if not mat.is_square:
raise NonSquareMatrixError("Trace of a non-square matrix")
return Basic.__new__(cls, mat)
def _eval_transpose(self):
return self
def _eval_derivative(self, v):
from sympy import Sum
from .matexpr import MatrixElement
if isinstance(v, MatrixElement):
return self.rewrite(Sum).diff(v)
expr = self.doit()
if isinstance(expr, Trace):
# Avoid looping infinitely:
raise NotImplementedError
return expr._eval_derivative(v)
def _eval_derivative_matrix_lines(self, x):
from sympy.codegen.array_utils import CodegenArrayContraction, CodegenArrayTensorProduct
from sympy.core.expr import ExprBuilder
r = self.args[0]._eval_derivative_matrix_lines(x)
for lr in r:
if lr.higher == 1:
lr.higher = ExprBuilder(
CodegenArrayContraction,
[
ExprBuilder(
CodegenArrayTensorProduct,
[
lr._lines[0],
lr._lines[1],
]
),
(1, 3),
],
validator=CodegenArrayContraction._validate
)
else:
# This is not a matrix line:
lr.higher = ExprBuilder(
CodegenArrayContraction,
[
ExprBuilder(
CodegenArrayTensorProduct,
[
lr._lines[0],
lr._lines[1],
lr.higher,
]
),
(1, 3), (0, 2)
]
)
lr._lines = [S.One, S.One]
lr._first_pointer_parent = lr._lines
lr._second_pointer_parent = lr._lines
lr._first_pointer_index = 0
lr._second_pointer_index = 1
return r
@property
def arg(self):
return self.args[0]
def doit(self, **kwargs):
if kwargs.get('deep', True):
arg = self.arg.doit(**kwargs)
try:
return arg._eval_trace()
except (AttributeError, NotImplementedError):
return Trace(arg)
else:
# _eval_trace would go too deep here
if isinstance(self.arg, MatrixBase):
return trace(self.arg)
else:
return Trace(self.arg)
def _eval_rewrite_as_Sum(self, expr, **kwargs):
from sympy import Sum, Dummy
i = Dummy('i')
return Sum(self.arg[i, i], (i, 0, self.arg.rows-1)).doit()
def trace(expr):
"""Trace of a Matrix. Sum of the diagonal elements.
Examples
========
>>> from sympy import trace, Symbol, MatrixSymbol, eye
>>> n = Symbol('n')
>>> X = MatrixSymbol('X', n, n) # A square matrix
>>> trace(2*X)
2*Trace(X)
>>> trace(eye(3))
3
"""
return Trace(expr).doit()
|
f6af64692466a3f9a084c4f965592f8079a3c1b9db6c18d41d507c56ac4e43df | from sympy import ask, Q
from sympy.core import Basic, Add, Mul, S
from sympy.core.sympify import _sympify
from sympy.matrices.common import NonInvertibleMatrixError
from sympy.strategies import typed, exhaust, condition, do_one, unpack
from sympy.strategies.traverse import bottom_up
from sympy.utilities import sift
from sympy.utilities.misc import filldedent
from sympy.matrices.expressions.matexpr import MatrixExpr, ZeroMatrix, Identity, MatrixElement
from sympy.matrices.expressions.matmul import MatMul
from sympy.matrices.expressions.matadd import MatAdd
from sympy.matrices.expressions.matpow import MatPow
from sympy.matrices.expressions.transpose import Transpose, transpose
from sympy.matrices.expressions.trace import trace
from sympy.matrices.expressions.determinant import det, Determinant
from sympy.matrices.expressions.slice import MatrixSlice
from sympy.matrices.expressions.inverse import Inverse
from sympy.matrices import Matrix, ShapeError
from sympy.functions.elementary.complexes import re, im
class BlockMatrix(MatrixExpr):
"""A BlockMatrix is a Matrix comprised of other matrices.
The submatrices are stored in a SymPy Matrix object but accessed as part of
a Matrix Expression
>>> from sympy import (MatrixSymbol, BlockMatrix, symbols,
... Identity, ZeroMatrix, block_collapse)
>>> n,m,l = symbols('n m l')
>>> X = MatrixSymbol('X', n, n)
>>> Y = MatrixSymbol('Y', m ,m)
>>> Z = MatrixSymbol('Z', n, m)
>>> B = BlockMatrix([[X, Z], [ZeroMatrix(m,n), Y]])
>>> print(B)
Matrix([
[X, Z],
[0, Y]])
>>> C = BlockMatrix([[Identity(n), Z]])
>>> print(C)
Matrix([[I, Z]])
>>> print(block_collapse(C*B))
Matrix([[X, Z + Z*Y]])
Some matrices might be comprised of rows of blocks with
the matrices in each row having the same height and the
rows all having the same total number of columns but
not having the same number of columns for each matrix
in each row. In this case, the matrix is not a block
matrix and should be instantiated by Matrix.
>>> from sympy import ones, Matrix
>>> dat = [
... [ones(3,2), ones(3,3)*2],
... [ones(2,3)*3, ones(2,2)*4]]
...
>>> BlockMatrix(dat)
Traceback (most recent call last):
...
ValueError:
Although this matrix is comprised of blocks, the blocks do not fill
the matrix in a size-symmetric fashion. To create a full matrix from
these arguments, pass them directly to Matrix.
>>> Matrix(dat)
Matrix([
[1, 1, 2, 2, 2],
[1, 1, 2, 2, 2],
[1, 1, 2, 2, 2],
[3, 3, 3, 4, 4],
[3, 3, 3, 4, 4]])
See Also
========
sympy.matrices.matrices.MatrixBase.irregular
"""
def __new__(cls, *args, **kwargs):
from sympy.matrices.immutable import ImmutableDenseMatrix
from sympy.utilities.iterables import is_sequence
isMat = lambda i: getattr(i, 'is_Matrix', False)
if len(args) != 1 or \
not is_sequence(args[0]) or \
len({isMat(r) for r in args[0]}) != 1:
raise ValueError(filldedent('''
expecting a sequence of 1 or more rows
containing Matrices.'''))
rows = args[0] if args else []
if not isMat(rows):
if rows and isMat(rows[0]):
rows = [rows] # rows is not list of lists or []
# regularity check
# same number of matrices in each row
blocky = ok = len({len(r) for r in rows}) == 1
if ok:
# same number of rows for each matrix in a row
for r in rows:
ok = len({i.rows for i in r}) == 1
if not ok:
break
blocky = ok
if ok:
# same number of cols for each matrix in each col
for c in range(len(rows[0])):
ok = len({rows[i][c].cols
for i in range(len(rows))}) == 1
if not ok:
break
if not ok:
# same total cols in each row
ok = len({
sum([i.cols for i in r]) for r in rows}) == 1
if blocky and ok:
raise ValueError(filldedent('''
Although this matrix is comprised of blocks,
the blocks do not fill the matrix in a
size-symmetric fashion. To create a full matrix
from these arguments, pass them directly to
Matrix.'''))
raise ValueError(filldedent('''
When there are not the same number of rows in each
row's matrices or there are not the same number of
total columns in each row, the matrix is not a
block matrix. If this matrix is known to consist of
blocks fully filling a 2-D space then see
Matrix.irregular.'''))
mat = ImmutableDenseMatrix(rows, evaluate=False)
obj = Basic.__new__(cls, mat)
return obj
@property
def shape(self):
numrows = numcols = 0
M = self.blocks
for i in range(M.shape[0]):
numrows += M[i, 0].shape[0]
for i in range(M.shape[1]):
numcols += M[0, i].shape[1]
return (numrows, numcols)
@property
def blockshape(self):
return self.blocks.shape
@property
def blocks(self):
return self.args[0]
@property
def rowblocksizes(self):
return [self.blocks[i, 0].rows for i in range(self.blockshape[0])]
@property
def colblocksizes(self):
return [self.blocks[0, i].cols for i in range(self.blockshape[1])]
def structurally_equal(self, other):
return (isinstance(other, BlockMatrix)
and self.shape == other.shape
and self.blockshape == other.blockshape
and self.rowblocksizes == other.rowblocksizes
and self.colblocksizes == other.colblocksizes)
def _blockmul(self, other):
if (isinstance(other, BlockMatrix) and
self.colblocksizes == other.rowblocksizes):
return BlockMatrix(self.blocks*other.blocks)
return self * other
def _blockadd(self, other):
if (isinstance(other, BlockMatrix)
and self.structurally_equal(other)):
return BlockMatrix(self.blocks + other.blocks)
return self + other
def _eval_transpose(self):
# Flip all the individual matrices
matrices = [transpose(matrix) for matrix in self.blocks]
# Make a copy
M = Matrix(self.blockshape[0], self.blockshape[1], matrices)
# Transpose the block structure
M = M.transpose()
return BlockMatrix(M)
def _eval_trace(self):
if self.rowblocksizes == self.colblocksizes:
return Add(*[trace(self.blocks[i, i])
for i in range(self.blockshape[0])])
raise NotImplementedError(
"Can't perform trace of irregular blockshape")
def _eval_determinant(self):
if self.blockshape == (1, 1):
return det(self.blocks[0, 0])
if self.blockshape == (2, 2):
[[A, B],
[C, D]] = self.blocks.tolist()
if ask(Q.invertible(A)):
return det(A)*det(D - C*A.I*B)
elif ask(Q.invertible(D)):
return det(D)*det(A - B*D.I*C)
return Determinant(self)
def as_real_imag(self):
real_matrices = [re(matrix) for matrix in self.blocks]
real_matrices = Matrix(self.blockshape[0], self.blockshape[1], real_matrices)
im_matrices = [im(matrix) for matrix in self.blocks]
im_matrices = Matrix(self.blockshape[0], self.blockshape[1], im_matrices)
return (real_matrices, im_matrices)
def transpose(self):
"""Return transpose of matrix.
Examples
========
>>> from sympy import MatrixSymbol, BlockMatrix, ZeroMatrix
>>> from sympy.abc import m, n
>>> X = MatrixSymbol('X', n, n)
>>> Y = MatrixSymbol('Y', m ,m)
>>> Z = MatrixSymbol('Z', n, m)
>>> B = BlockMatrix([[X, Z], [ZeroMatrix(m,n), Y]])
>>> B.transpose()
Matrix([
[X.T, 0],
[Z.T, Y.T]])
>>> _.transpose()
Matrix([
[X, Z],
[0, Y]])
"""
return self._eval_transpose()
def _entry(self, i, j, **kwargs):
# Find row entry
orig_i, orig_j = i, j
for row_block, numrows in enumerate(self.rowblocksizes):
cmp = i < numrows
if cmp == True:
break
elif cmp == False:
i -= numrows
elif row_block < self.blockshape[0] - 1:
# Can't tell which block and it's not the last one, return unevaluated
return MatrixElement(self, orig_i, orig_j)
for col_block, numcols in enumerate(self.colblocksizes):
cmp = j < numcols
if cmp == True:
break
elif cmp == False:
j -= numcols
elif col_block < self.blockshape[1] - 1:
return MatrixElement(self, orig_i, orig_j)
return self.blocks[row_block, col_block][i, j]
@property
def is_Identity(self):
if self.blockshape[0] != self.blockshape[1]:
return False
for i in range(self.blockshape[0]):
for j in range(self.blockshape[1]):
if i==j and not self.blocks[i, j].is_Identity:
return False
if i!=j and not self.blocks[i, j].is_ZeroMatrix:
return False
return True
@property
def is_structurally_symmetric(self):
return self.rowblocksizes == self.colblocksizes
def equals(self, other):
if self == other:
return True
if (isinstance(other, BlockMatrix) and self.blocks == other.blocks):
return True
return super().equals(other)
class BlockDiagMatrix(BlockMatrix):
"""A sparse matrix with block matrices along its diagonals
Examples
========
>>> from sympy import MatrixSymbol, BlockDiagMatrix, symbols
>>> n, m, l = symbols('n m l')
>>> X = MatrixSymbol('X', n, n)
>>> Y = MatrixSymbol('Y', m ,m)
>>> BlockDiagMatrix(X, Y)
Matrix([
[X, 0],
[0, Y]])
Notes
=====
If you want to get the individual diagonal blocks, use
:meth:`get_diag_blocks`.
See Also
========
sympy.matrices.dense.diag
"""
def __new__(cls, *mats):
return Basic.__new__(BlockDiagMatrix, *[_sympify(m) for m in mats])
@property
def diag(self):
return self.args
@property
def blocks(self):
from sympy.matrices.immutable import ImmutableDenseMatrix
mats = self.args
data = [[mats[i] if i == j else ZeroMatrix(mats[i].rows, mats[j].cols)
for j in range(len(mats))]
for i in range(len(mats))]
return ImmutableDenseMatrix(data, evaluate=False)
@property
def shape(self):
return (sum(block.rows for block in self.args),
sum(block.cols for block in self.args))
@property
def blockshape(self):
n = len(self.args)
return (n, n)
@property
def rowblocksizes(self):
return [block.rows for block in self.args]
@property
def colblocksizes(self):
return [block.cols for block in self.args]
def _all_square_blocks(self):
"""Returns true if all blocks are square"""
return all(mat.is_square for mat in self.args)
def _eval_determinant(self):
if self._all_square_blocks():
return Mul(*[det(mat) for mat in self.args])
# At least one block is non-square. Since the entire matrix must be square we know there must
# be at least two blocks in this matrix, in which case the entire matrix is necessarily rank-deficient
return S.Zero
def _eval_inverse(self, expand='ignored'):
if self._all_square_blocks():
return BlockDiagMatrix(*[mat.inverse() for mat in self.args])
# See comment in _eval_determinant()
raise NonInvertibleMatrixError('Matrix det == 0; not invertible.')
def _eval_transpose(self):
return BlockDiagMatrix(*[mat.transpose() for mat in self.args])
def _blockmul(self, other):
if (isinstance(other, BlockDiagMatrix) and
self.colblocksizes == other.rowblocksizes):
return BlockDiagMatrix(*[a*b for a, b in zip(self.args, other.args)])
else:
return BlockMatrix._blockmul(self, other)
def _blockadd(self, other):
if (isinstance(other, BlockDiagMatrix) and
self.blockshape == other.blockshape and
self.rowblocksizes == other.rowblocksizes and
self.colblocksizes == other.colblocksizes):
return BlockDiagMatrix(*[a + b for a, b in zip(self.args, other.args)])
else:
return BlockMatrix._blockadd(self, other)
def get_diag_blocks(self):
"""Return the list of diagonal blocks of the matrix.
Examples
========
>>> from sympy.matrices import BlockDiagMatrix, Matrix
>>> A = Matrix([[1, 2], [3, 4]])
>>> B = Matrix([[5, 6], [7, 8]])
>>> M = BlockDiagMatrix(A, B)
How to get diagonal blocks from the block diagonal matrix:
>>> diag_blocks = M.get_diag_blocks()
>>> diag_blocks[0]
Matrix([
[1, 2],
[3, 4]])
>>> diag_blocks[1]
Matrix([
[5, 6],
[7, 8]])
"""
return self.args
def block_collapse(expr):
"""Evaluates a block matrix expression
>>> from sympy import MatrixSymbol, BlockMatrix, symbols, \
Identity, ZeroMatrix, block_collapse
>>> n,m,l = symbols('n m l')
>>> X = MatrixSymbol('X', n, n)
>>> Y = MatrixSymbol('Y', m ,m)
>>> Z = MatrixSymbol('Z', n, m)
>>> B = BlockMatrix([[X, Z], [ZeroMatrix(m, n), Y]])
>>> print(B)
Matrix([
[X, Z],
[0, Y]])
>>> C = BlockMatrix([[Identity(n), Z]])
>>> print(C)
Matrix([[I, Z]])
>>> print(block_collapse(C*B))
Matrix([[X, Z + Z*Y]])
"""
from sympy.strategies.util import expr_fns
hasbm = lambda expr: isinstance(expr, MatrixExpr) and expr.has(BlockMatrix)
conditioned_rl = condition(
hasbm,
typed(
{MatAdd: do_one(bc_matadd, bc_block_plus_ident),
MatMul: do_one(bc_matmul, bc_dist),
MatPow: bc_matmul,
Transpose: bc_transpose,
Inverse: bc_inverse,
BlockMatrix: do_one(bc_unpack, deblock)}
)
)
rule = exhaust(
bottom_up(
exhaust(conditioned_rl),
fns=expr_fns
)
)
result = rule(expr)
doit = getattr(result, 'doit', None)
if doit is not None:
return doit()
else:
return result
def bc_unpack(expr):
if expr.blockshape == (1, 1):
return expr.blocks[0, 0]
return expr
def bc_matadd(expr):
args = sift(expr.args, lambda M: isinstance(M, BlockMatrix))
blocks = args[True]
if not blocks:
return expr
nonblocks = args[False]
block = blocks[0]
for b in blocks[1:]:
block = block._blockadd(b)
if nonblocks:
return MatAdd(*nonblocks) + block
else:
return block
def bc_block_plus_ident(expr):
idents = [arg for arg in expr.args if arg.is_Identity]
if not idents:
return expr
blocks = [arg for arg in expr.args if isinstance(arg, BlockMatrix)]
if (blocks and all(b.structurally_equal(blocks[0]) for b in blocks)
and blocks[0].is_structurally_symmetric):
block_id = BlockDiagMatrix(*[Identity(k)
for k in blocks[0].rowblocksizes])
rest = [arg for arg in expr.args if not arg.is_Identity and not isinstance(arg, BlockMatrix)]
return MatAdd(block_id * len(idents), *blocks, *rest).doit()
return expr
def bc_dist(expr):
""" Turn a*[X, Y] into [a*X, a*Y] """
factor, mat = expr.as_coeff_mmul()
if factor == 1:
return expr
unpacked = unpack(mat)
if isinstance(unpacked, BlockDiagMatrix):
B = unpacked.diag
new_B = [factor * mat for mat in B]
return BlockDiagMatrix(*new_B)
elif isinstance(unpacked, BlockMatrix):
B = unpacked.blocks
new_B = [
[factor * B[i, j] for j in range(B.cols)] for i in range(B.rows)]
return BlockMatrix(new_B)
return unpacked
def bc_matmul(expr):
if isinstance(expr, MatPow):
if expr.args[1].is_Integer:
factor, matrices = (1, [expr.args[0]]*expr.args[1])
else:
return expr
else:
factor, matrices = expr.as_coeff_matrices()
i = 0
while (i+1 < len(matrices)):
A, B = matrices[i:i+2]
if isinstance(A, BlockMatrix) and isinstance(B, BlockMatrix):
matrices[i] = A._blockmul(B)
matrices.pop(i+1)
elif isinstance(A, BlockMatrix):
matrices[i] = A._blockmul(BlockMatrix([[B]]))
matrices.pop(i+1)
elif isinstance(B, BlockMatrix):
matrices[i] = BlockMatrix([[A]])._blockmul(B)
matrices.pop(i+1)
else:
i+=1
return MatMul(factor, *matrices).doit()
def bc_transpose(expr):
collapse = block_collapse(expr.arg)
return collapse._eval_transpose()
def bc_inverse(expr):
if isinstance(expr.arg, BlockDiagMatrix):
return expr.inverse()
expr2 = blockinverse_1x1(expr)
if expr != expr2:
return expr2
return blockinverse_2x2(Inverse(reblock_2x2(expr.arg)))
def blockinverse_1x1(expr):
if isinstance(expr.arg, BlockMatrix) and expr.arg.blockshape == (1, 1):
mat = Matrix([[expr.arg.blocks[0].inverse()]])
return BlockMatrix(mat)
return expr
def blockinverse_2x2(expr):
if isinstance(expr.arg, BlockMatrix) and expr.arg.blockshape == (2, 2):
# See: Inverses of 2x2 Block Matrices, Tzon-Tzer Lu and Sheng-Hua Shiou
[[A, B],
[C, D]] = expr.arg.blocks.tolist()
formula = _choose_2x2_inversion_formula(A, B, C, D)
if formula == 'A':
AI = A.I
MI = (D - C * AI * B).I
return BlockMatrix([[AI + AI * B * MI * C * AI, -AI * B * MI], [-MI * C * AI, MI]])
if formula == 'B':
BI = B.I
MI = (C - D * BI * A).I
return BlockMatrix([[-MI * D * BI, MI], [BI + BI * A * MI * D * BI, -BI * A * MI]])
if formula == 'C':
CI = C.I
MI = (B - A * CI * D).I
return BlockMatrix([[-CI * D * MI, CI + CI * D * MI * A * CI], [MI, -MI * A * CI]])
if formula == 'D':
DI = D.I
MI = (A - B * DI * C).I
return BlockMatrix([[MI, -MI * B * DI], [-DI * C * MI, DI + DI * C * MI * B * DI]])
return expr
def _choose_2x2_inversion_formula(A, B, C, D):
"""
Assuming [[A, B], [C, D]] would form a valid square block matrix, find
which of the classical 2x2 block matrix inversion formulas would be
best suited.
Returns 'A', 'B', 'C', 'D' to represent the algorithm involving inversion
of the given argument or None if the matrix cannot be inverted using
any of those formulas.
"""
# Try to find a known invertible matrix. Note that the Schur complement
# is currently not being considered for this
A_inv = ask(Q.invertible(A))
if A_inv == True:
return 'A'
B_inv = ask(Q.invertible(B))
if B_inv == True:
return 'B'
C_inv = ask(Q.invertible(C))
if C_inv == True:
return 'C'
D_inv = ask(Q.invertible(D))
if D_inv == True:
return 'D'
# Otherwise try to find a matrix that isn't known to be non-invertible
if A_inv != False:
return 'A'
if B_inv != False:
return 'B'
if C_inv != False:
return 'C'
if D_inv != False:
return 'D'
return None
def deblock(B):
""" Flatten a BlockMatrix of BlockMatrices """
if not isinstance(B, BlockMatrix) or not B.blocks.has(BlockMatrix):
return B
wrap = lambda x: x if isinstance(x, BlockMatrix) else BlockMatrix([[x]])
bb = B.blocks.applyfunc(wrap) # everything is a block
from sympy import Matrix
try:
MM = Matrix(0, sum(bb[0, i].blocks.shape[1] for i in range(bb.shape[1])), [])
for row in range(0, bb.shape[0]):
M = Matrix(bb[row, 0].blocks)
for col in range(1, bb.shape[1]):
M = M.row_join(bb[row, col].blocks)
MM = MM.col_join(M)
return BlockMatrix(MM)
except ShapeError:
return B
def reblock_2x2(expr):
"""
Reblock a BlockMatrix so that it has 2x2 blocks of block matrices. If
possible in such a way that the matrix continues to be invertible using the
classical 2x2 block inversion formulas.
"""
if not isinstance(expr, BlockMatrix) or not all(d > 2 for d in expr.blockshape):
return expr
BM = BlockMatrix # for brevity's sake
rowblocks, colblocks = expr.blockshape
blocks = expr.blocks
for i in range(1, rowblocks):
for j in range(1, colblocks):
# try to split rows at i and cols at j
A = bc_unpack(BM(blocks[:i, :j]))
B = bc_unpack(BM(blocks[:i, j:]))
C = bc_unpack(BM(blocks[i:, :j]))
D = bc_unpack(BM(blocks[i:, j:]))
formula = _choose_2x2_inversion_formula(A, B, C, D)
if formula is not None:
return BlockMatrix([[A, B], [C, D]])
# else: nothing worked, just split upper left corner
return BM([[blocks[0, 0], BM(blocks[0, 1:])],
[BM(blocks[1:, 0]), BM(blocks[1:, 1:])]])
def bounds(sizes):
""" Convert sequence of numbers into pairs of low-high pairs
>>> from sympy.matrices.expressions.blockmatrix import bounds
>>> bounds((1, 10, 50))
[(0, 1), (1, 11), (11, 61)]
"""
low = 0
rv = []
for size in sizes:
rv.append((low, low + size))
low += size
return rv
def blockcut(expr, rowsizes, colsizes):
""" Cut a matrix expression into Blocks
>>> from sympy import ImmutableMatrix, blockcut
>>> M = ImmutableMatrix(4, 4, range(16))
>>> B = blockcut(M, (1, 3), (1, 3))
>>> type(B).__name__
'BlockMatrix'
>>> ImmutableMatrix(B.blocks[0, 1])
Matrix([[1, 2, 3]])
"""
rowbounds = bounds(rowsizes)
colbounds = bounds(colsizes)
return BlockMatrix([[MatrixSlice(expr, rowbound, colbound)
for colbound in colbounds]
for rowbound in rowbounds])
|
aab179589ce6dc49f8ba3699e88caaf3a06a36f70a32987af61d5da130d60402 | from .matexpr import MatrixExpr
from sympy.core.function import FunctionClass, Lambda
from sympy.core.symbol import Dummy
from sympy.core.sympify import _sympify, sympify
from sympy.matrices import Matrix
from sympy.functions.elementary.complexes import re, im
class FunctionMatrix(MatrixExpr):
"""Represents a matrix using a function (``Lambda``) which gives
outputs according to the coordinates of each matrix entries.
Parameters
==========
rows : nonnegative integer. Can be symbolic.
cols : nonnegative integer. Can be symbolic.
lamda : Function, Lambda or str
If it is a SymPy ``Function`` or ``Lambda`` instance,
it should be able to accept two arguments which represents the
matrix coordinates.
If it is a pure string containing python ``lambda`` semantics,
it is interpreted by the SymPy parser and casted into a SymPy
``Lambda`` instance.
Examples
========
Creating a ``FunctionMatrix`` from ``Lambda``:
>>> from sympy import FunctionMatrix, symbols, Lambda, MatPow
>>> i, j, n, m = symbols('i,j,n,m')
>>> FunctionMatrix(n, m, Lambda((i, j), i + j))
FunctionMatrix(n, m, Lambda((i, j), i + j))
Creating a ``FunctionMatrix`` from a sympy function:
>>> from sympy.functions import KroneckerDelta
>>> X = FunctionMatrix(3, 3, KroneckerDelta)
>>> X.as_explicit()
Matrix([
[1, 0, 0],
[0, 1, 0],
[0, 0, 1]])
Creating a ``FunctionMatrix`` from a sympy undefined function:
>>> from sympy.core.function import Function
>>> f = Function('f')
>>> X = FunctionMatrix(3, 3, f)
>>> X.as_explicit()
Matrix([
[f(0, 0), f(0, 1), f(0, 2)],
[f(1, 0), f(1, 1), f(1, 2)],
[f(2, 0), f(2, 1), f(2, 2)]])
Creating a ``FunctionMatrix`` from python ``lambda``:
>>> FunctionMatrix(n, m, 'lambda i, j: i + j')
FunctionMatrix(n, m, Lambda((i, j), i + j))
Example of lazy evaluation of matrix product:
>>> Y = FunctionMatrix(1000, 1000, Lambda((i, j), i + j))
>>> isinstance(Y*Y, MatPow) # this is an expression object
True
>>> (Y**2)[10,10] # So this is evaluated lazily
342923500
Notes
=====
This class provides an alternative way to represent an extremely
dense matrix with entries in some form of a sequence, in a most
sparse way.
"""
def __new__(cls, rows, cols, lamda):
rows, cols = _sympify(rows), _sympify(cols)
cls._check_dim(rows)
cls._check_dim(cols)
lamda = sympify(lamda)
if not isinstance(lamda, (FunctionClass, Lambda)):
raise ValueError(
"{} should be compatible with SymPy function classes."
.format(lamda))
if 2 not in lamda.nargs:
raise ValueError(
'{} should be able to accept 2 arguments.'.format(lamda))
if not isinstance(lamda, Lambda):
i, j = Dummy('i'), Dummy('j')
lamda = Lambda((i, j), lamda(i, j))
return super().__new__(cls, rows, cols, lamda)
@property
def shape(self):
return self.args[0:2]
@property
def lamda(self):
return self.args[2]
def _entry(self, i, j, **kwargs):
return self.lamda(i, j)
def _eval_trace(self):
from sympy.matrices.expressions.trace import Trace
from sympy import Sum
return Trace(self).rewrite(Sum).doit()
def as_real_imag(self):
return (re(Matrix(self)), im(Matrix(self)))
|
39fd18a405968854e05720d4938f71d324872e824145f10094f237095d9ef92f | from sympy.core import symbols, Lambda
from sympy.functions import KroneckerDelta
from sympy.matrices import Matrix
from sympy.matrices.expressions import FunctionMatrix, MatrixExpr, Identity
from sympy.testing.pytest import raises, warns_deprecated_sympy
def test_funcmatrix_creation():
i, j, k = symbols('i j k')
assert FunctionMatrix(2, 2, Lambda((i, j), 0))
assert FunctionMatrix(0, 0, Lambda((i, j), 0))
raises(ValueError, lambda: FunctionMatrix(-1, 0, Lambda((i, j), 0)))
raises(ValueError, lambda: FunctionMatrix(2.0, 0, Lambda((i, j), 0)))
raises(ValueError, lambda: FunctionMatrix(2j, 0, Lambda((i, j), 0)))
raises(ValueError, lambda: FunctionMatrix(0, -1, Lambda((i, j), 0)))
raises(ValueError, lambda: FunctionMatrix(0, 2.0, Lambda((i, j), 0)))
raises(ValueError, lambda: FunctionMatrix(0, 2j, Lambda((i, j), 0)))
raises(ValueError, lambda: FunctionMatrix(2, 2, Lambda(i, 0)))
with warns_deprecated_sympy():
raises(ValueError, lambda: FunctionMatrix(2, 2, lambda i, j: 0))
raises(ValueError, lambda: FunctionMatrix(2, 2, Lambda((i,), 0)))
raises(ValueError, lambda: FunctionMatrix(2, 2, Lambda((i, j, k), 0)))
raises(ValueError, lambda: FunctionMatrix(2, 2, i+j))
assert FunctionMatrix(2, 2, "lambda i, j: 0") == \
FunctionMatrix(2, 2, Lambda((i, j), 0))
m = FunctionMatrix(2, 2, KroneckerDelta)
assert m.as_explicit() == Identity(2).as_explicit()
assert m.args[2].dummy_eq(Lambda((i, j), KroneckerDelta(i, j)))
n = symbols('n')
assert FunctionMatrix(n, n, Lambda((i, j), 0))
n = symbols('n', integer=False)
raises(ValueError, lambda: FunctionMatrix(n, n, Lambda((i, j), 0)))
n = symbols('n', negative=True)
raises(ValueError, lambda: FunctionMatrix(n, n, Lambda((i, j), 0)))
def test_funcmatrix():
i, j = symbols('i,j')
X = FunctionMatrix(3, 3, Lambda((i, j), i - j))
assert X[1, 1] == 0
assert X[1, 2] == -1
assert X.shape == (3, 3)
assert X.rows == X.cols == 3
assert Matrix(X) == Matrix(3, 3, lambda i, j: i - j)
assert isinstance(X*X + X, MatrixExpr)
def test_replace_issue():
X = FunctionMatrix(3, 3, KroneckerDelta)
assert X.replace(lambda x: True, lambda x: x) == X
|
86013c1099cd6e027776d975ca8962688bb46a8bdc019f813d6fd201935f667f | from sympy.core import S, symbols
from sympy.matrices import eye, Matrix, ShapeError
from sympy.matrices.expressions import (
Identity, MatrixExpr, MatrixSymbol, Determinant,
det, ZeroMatrix, Transpose
)
from sympy.matrices.expressions.matexpr import OneMatrix
from sympy.testing.pytest import raises
from sympy import refine, Q
n = symbols('n', integer=True)
A = MatrixSymbol('A', n, n)
B = MatrixSymbol('B', n, n)
C = MatrixSymbol('C', 3, 4)
def test_det():
assert isinstance(Determinant(A), Determinant)
assert not isinstance(Determinant(A), MatrixExpr)
raises(ShapeError, lambda: Determinant(C))
assert det(eye(3)) == 1
assert det(Matrix(3, 3, [1, 3, 2, 4, 1, 3, 2, 5, 2])) == 17
A / det(A) # Make sure this is possible
raises(TypeError, lambda: Determinant(S.One))
assert Determinant(A).arg is A
def test_eval_determinant():
assert det(Identity(n)) == 1
assert det(ZeroMatrix(n, n)) == 0
assert det(OneMatrix(n, n)) == Determinant(OneMatrix(n, n))
assert det(OneMatrix(1, 1)) == 1
assert det(OneMatrix(2, 2)) == 0
assert det(Transpose(A)) == det(A)
def test_refine():
assert refine(det(A), Q.orthogonal(A)) == 1
assert refine(det(A), Q.singular(A)) == 0
def test_commutative():
det_a = Determinant(A)
det_b = Determinant(B)
assert det_a.is_commutative
assert det_b.is_commutative
assert det_a * det_b == det_b * det_a
|
1732647d0a99bbcb0c946f3ce8da7606c499004032b6b2f72fc58560569ac697 | """
Some examples have been taken from:
http://www.math.uwaterloo.ca/~hwolkowi//matrixcookbook.pdf
"""
from sympy import (MatrixSymbol, Inverse, symbols, Determinant, Trace,
Derivative, sin, exp, cos, tan, log, S, sqrt,
hadamard_product, DiagMatrix, OneMatrix,
HadamardProduct, HadamardPower, KroneckerDelta, Sum,
Rational)
from sympy import MatAdd, Identity, MatMul, ZeroMatrix
from sympy.matrices.expressions import hadamard_power
k = symbols("k")
i, j = symbols("i j")
m, n = symbols("m n")
X = MatrixSymbol("X", k, k)
x = MatrixSymbol("x", k, 1)
y = MatrixSymbol("y", k, 1)
A = MatrixSymbol("A", k, k)
B = MatrixSymbol("B", k, k)
C = MatrixSymbol("C", k, k)
D = MatrixSymbol("D", k, k)
a = MatrixSymbol("a", k, 1)
b = MatrixSymbol("b", k, 1)
c = MatrixSymbol("c", k, 1)
d = MatrixSymbol("d", k, 1)
KDelta = lambda i, j: KroneckerDelta(i, j, (0, k-1))
def _check_derivative_with_explicit_matrix(expr, x, diffexpr, dim=2):
# TODO: this is commented because it slows down the tests.
return
expr = expr.xreplace({k: dim})
x = x.xreplace({k: dim})
diffexpr = diffexpr.xreplace({k: dim})
expr = expr.as_explicit()
x = x.as_explicit()
diffexpr = diffexpr.as_explicit()
assert expr.diff(x).reshape(*diffexpr.shape).tomatrix() == diffexpr
def test_matrix_derivative_by_scalar():
assert A.diff(i) == ZeroMatrix(k, k)
assert (A*(X + B)*c).diff(i) == ZeroMatrix(k, 1)
assert x.diff(i) == ZeroMatrix(k, 1)
assert (x.T*y).diff(i) == ZeroMatrix(1, 1)
assert (x*x.T).diff(i) == ZeroMatrix(k, k)
assert (x + y).diff(i) == ZeroMatrix(k, 1)
assert hadamard_power(x, 2).diff(i) == ZeroMatrix(k, 1)
assert hadamard_power(x, i).diff(i).dummy_eq(
HadamardProduct(x.applyfunc(log), HadamardPower(x, i)))
assert hadamard_product(x, y).diff(i) == ZeroMatrix(k, 1)
assert hadamard_product(i*OneMatrix(k, 1), x, y).diff(i) == hadamard_product(x, y)
assert (i*x).diff(i) == x
assert (sin(i)*A*B*x).diff(i) == cos(i)*A*B*x
assert x.applyfunc(sin).diff(i) == ZeroMatrix(k, 1)
assert Trace(i**2*X).diff(i) == 2*i*Trace(X)
mu = symbols("mu")
expr = (2*mu*x)
assert expr.diff(x) == 2*mu*Identity(k)
def test_matrix_derivative_non_matrix_result():
# This is a 4-dimensional array:
assert A.diff(A) == Derivative(A, A)
assert A.T.diff(A) == Derivative(A.T, A)
assert (2*A).diff(A) == Derivative(2*A, A)
assert MatAdd(A, A).diff(A) == Derivative(MatAdd(A, A), A)
assert (A + B).diff(A) == Derivative(A + B, A) # TODO: `B` can be removed.
def test_matrix_derivative_trivial_cases():
# Cookbook example 33:
# TODO: find a way to represent a four-dimensional zero-array:
assert X.diff(A) == Derivative(X, A)
def test_matrix_derivative_with_inverse():
# Cookbook example 61:
expr = a.T*Inverse(X)*b
assert expr.diff(X) == -Inverse(X).T*a*b.T*Inverse(X).T
# Cookbook example 62:
expr = Determinant(Inverse(X))
# Not implemented yet:
# assert expr.diff(X) == -Determinant(X.inv())*(X.inv()).T
# Cookbook example 63:
expr = Trace(A*Inverse(X)*B)
assert expr.diff(X) == -(X**(-1)*B*A*X**(-1)).T
# Cookbook example 64:
expr = Trace(Inverse(X + A))
assert expr.diff(X) == -(Inverse(X + A)).T**2
def test_matrix_derivative_vectors_and_scalars():
assert x.diff(x) == Identity(k)
assert x[i, 0].diff(x[m, 0]).doit() == KDelta(m, i)
assert x.T.diff(x) == Identity(k)
# Cookbook example 69:
expr = x.T*a
assert expr.diff(x) == a
assert expr[0, 0].diff(x[m, 0]).doit() == a[m, 0]
expr = a.T*x
assert expr.diff(x) == a
# Cookbook example 70:
expr = a.T*X*b
assert expr.diff(X) == a*b.T
# Cookbook example 71:
expr = a.T*X.T*b
assert expr.diff(X) == b*a.T
# Cookbook example 72:
expr = a.T*X*a
assert expr.diff(X) == a*a.T
expr = a.T*X.T*a
assert expr.diff(X) == a*a.T
# Cookbook example 77:
expr = b.T*X.T*X*c
assert expr.diff(X) == X*b*c.T + X*c*b.T
# Cookbook example 78:
expr = (B*x + b).T*C*(D*x + d)
assert expr.diff(x) == B.T*C*(D*x + d) + D.T*C.T*(B*x + b)
# Cookbook example 81:
expr = x.T*B*x
assert expr.diff(x) == B*x + B.T*x
# Cookbook example 82:
expr = b.T*X.T*D*X*c
assert expr.diff(X) == D.T*X*b*c.T + D*X*c*b.T
# Cookbook example 83:
expr = (X*b + c).T*D*(X*b + c)
assert expr.diff(X) == D*(X*b + c)*b.T + D.T*(X*b + c)*b.T
assert str(expr[0, 0].diff(X[m, n]).doit()) == \
'b[n, 0]*Sum((c[_i_1, 0] + Sum(X[_i_1, _i_3]*b[_i_3, 0], (_i_3, 0, k - 1)))*D[_i_1, m], (_i_1, 0, k - 1)) + Sum((c[_i_2, 0] + Sum(X[_i_2, _i_4]*b[_i_4, 0], (_i_4, 0, k - 1)))*D[m, _i_2]*b[n, 0], (_i_2, 0, k - 1))'
def test_matrix_derivatives_of_traces():
expr = Trace(A)*A
assert expr.diff(A) == Derivative(Trace(A)*A, A)
assert expr[i, j].diff(A[m, n]).doit() == (
KDelta(i, m)*KDelta(j, n)*Trace(A) +
KDelta(m, n)*A[i, j]
)
## First order:
# Cookbook example 99:
expr = Trace(X)
assert expr.diff(X) == Identity(k)
assert expr.rewrite(Sum).diff(X[m, n]).doit() == KDelta(m, n)
# Cookbook example 100:
expr = Trace(X*A)
assert expr.diff(X) == A.T
assert expr.rewrite(Sum).diff(X[m, n]).doit() == A[n, m]
# Cookbook example 101:
expr = Trace(A*X*B)
assert expr.diff(X) == A.T*B.T
assert expr.rewrite(Sum).diff(X[m, n]).doit().dummy_eq((A.T*B.T)[m, n])
# Cookbook example 102:
expr = Trace(A*X.T*B)
assert expr.diff(X) == B*A
# Cookbook example 103:
expr = Trace(X.T*A)
assert expr.diff(X) == A
# Cookbook example 104:
expr = Trace(A*X.T)
assert expr.diff(X) == A
# Cookbook example 105:
# TODO: TensorProduct is not supported
#expr = Trace(TensorProduct(A, X))
#assert expr.diff(X) == Trace(A)*Identity(k)
## Second order:
# Cookbook example 106:
expr = Trace(X**2)
assert expr.diff(X) == 2*X.T
# Cookbook example 107:
expr = Trace(X**2*B)
assert expr.diff(X) == (X*B + B*X).T
expr = Trace(MatMul(X, X, B))
assert expr.diff(X) == (X*B + B*X).T
# Cookbook example 108:
expr = Trace(X.T*B*X)
assert expr.diff(X) == B*X + B.T*X
# Cookbook example 109:
expr = Trace(B*X*X.T)
assert expr.diff(X) == B*X + B.T*X
# Cookbook example 110:
expr = Trace(X*X.T*B)
assert expr.diff(X) == B*X + B.T*X
# Cookbook example 111:
expr = Trace(X*B*X.T)
assert expr.diff(X) == X*B.T + X*B
# Cookbook example 112:
expr = Trace(B*X.T*X)
assert expr.diff(X) == X*B.T + X*B
# Cookbook example 113:
expr = Trace(X.T*X*B)
assert expr.diff(X) == X*B.T + X*B
# Cookbook example 114:
expr = Trace(A*X*B*X)
assert expr.diff(X) == A.T*X.T*B.T + B.T*X.T*A.T
# Cookbook example 115:
expr = Trace(X.T*X)
assert expr.diff(X) == 2*X
expr = Trace(X*X.T)
assert expr.diff(X) == 2*X
# Cookbook example 116:
expr = Trace(B.T*X.T*C*X*B)
assert expr.diff(X) == C.T*X*B*B.T + C*X*B*B.T
# Cookbook example 117:
expr = Trace(X.T*B*X*C)
assert expr.diff(X) == B*X*C + B.T*X*C.T
# Cookbook example 118:
expr = Trace(A*X*B*X.T*C)
assert expr.diff(X) == A.T*C.T*X*B.T + C*A*X*B
# Cookbook example 119:
expr = Trace((A*X*B + C)*(A*X*B + C).T)
assert expr.diff(X) == 2*A.T*(A*X*B + C)*B.T
# Cookbook example 120:
# TODO: no support for TensorProduct.
# expr = Trace(TensorProduct(X, X))
# expr = Trace(X)*Trace(X)
# expr.diff(X) == 2*Trace(X)*Identity(k)
# Higher Order
# Cookbook example 121:
expr = Trace(X**k)
#assert expr.diff(X) == k*(X**(k-1)).T
# Cookbook example 122:
expr = Trace(A*X**k)
#assert expr.diff(X) == # Needs indices
# Cookbook example 123:
expr = Trace(B.T*X.T*C*X*X.T*C*X*B)
assert expr.diff(X) == C*X*X.T*C*X*B*B.T + C.T*X*B*B.T*X.T*C.T*X + C*X*B*B.T*X.T*C*X + C.T*X*X.T*C.T*X*B*B.T
# Other
# Cookbook example 124:
expr = Trace(A*X**(-1)*B)
assert expr.diff(X) == -Inverse(X).T*A.T*B.T*Inverse(X).T
# Cookbook example 125:
expr = Trace(Inverse(X.T*C*X)*A)
# Warning: result in the cookbook is equivalent if B and C are symmetric:
assert expr.diff(X) == - X.inv().T*A.T*X.inv()*C.inv().T*X.inv().T - X.inv().T*A*X.inv()*C.inv()*X.inv().T
# Cookbook example 126:
expr = Trace((X.T*C*X).inv()*(X.T*B*X))
assert expr.diff(X) == -2*C*X*(X.T*C*X).inv()*X.T*B*X*(X.T*C*X).inv() + 2*B*X*(X.T*C*X).inv()
# Cookbook example 127:
expr = Trace((A + X.T*C*X).inv()*(X.T*B*X))
# Warning: result in the cookbook is equivalent if B and C are symmetric:
assert expr.diff(X) == B*X*Inverse(A + X.T*C*X) - C*X*Inverse(A + X.T*C*X)*X.T*B*X*Inverse(A + X.T*C*X) - C.T*X*Inverse(A.T + (C*X).T*X)*X.T*B.T*X*Inverse(A.T + (C*X).T*X) + B.T*X*Inverse(A.T + (C*X).T*X)
def test_derivatives_of_complicated_matrix_expr():
expr = a.T*(A*X*(X.T*B + X*A) + B.T*X.T*(a*b.T*(X*D*X.T + X*(X.T*B + A*X)*D*B - X.T*C.T*A)*B + B*(X*D.T + B*A*X*A.T - 3*X*D))*B + 42*X*B*X.T*A.T*(X + X.T))*b
result = (B*(B*A*X*A.T - 3*X*D + X*D.T) + a*b.T*(X*(A*X + X.T*B)*D*B + X*D*X.T - X.T*C.T*A)*B)*B*b*a.T*B.T + B**2*b*a.T*B.T*X.T*a*b.T*X*D + 42*A*X*B.T*X.T*a*b.T + B*D*B**3*b*a.T*B.T*X.T*a*b.T*X + B*b*a.T*A*X + 42*a*b.T*(X + X.T)*A*X*B.T + b*a.T*X*B*a*b.T*B.T**2*X*D.T + b*a.T*X*B*a*b.T*B.T**3*D.T*(B.T*X + X.T*A.T) + 42*b*a.T*X*B*X.T*A.T + 42*A.T*(X + X.T)*b*a.T*X*B + A.T*B.T**2*X*B*a*b.T*B.T*A + A.T*a*b.T*(A.T*X.T + B.T*X) + A.T*X.T*b*a.T*X*B*a*b.T*B.T**3*D.T + B.T*X*B*a*b.T*B.T*D - 3*B.T*X*B*a*b.T*B.T*D.T - C.T*A*B**2*b*a.T*B.T*X.T*a*b.T + X.T*A.T*a*b.T*A.T
assert expr.diff(X) == result
def test_mixed_deriv_mixed_expressions():
expr = 3*Trace(A)
assert expr.diff(A) == 3*Identity(k)
expr = k
deriv = expr.diff(A)
assert isinstance(deriv, ZeroMatrix)
assert deriv == ZeroMatrix(k, k)
expr = Trace(A)**2
assert expr.diff(A) == (2*Trace(A))*Identity(k)
expr = Trace(A)*A
# TODO: this is not yet supported:
assert expr.diff(A) == Derivative(expr, A)
expr = Trace(Trace(A)*A)
assert expr.diff(A) == (2*Trace(A))*Identity(k)
expr = Trace(Trace(Trace(A)*A)*A)
assert expr.diff(A) == (3*Trace(A)**2)*Identity(k)
def test_derivatives_matrix_norms():
expr = x.T*y
assert expr.diff(x) == y
assert expr[0, 0].diff(x[m, 0]).doit() == y[m, 0]
expr = (x.T*y)**S.Half
assert expr.diff(x) == y/(2*sqrt(x.T*y))
expr = (x.T*x)**S.Half
assert expr.diff(x) == x*(x.T*x)**Rational(-1, 2)
expr = (c.T*a*x.T*b)**S.Half
assert expr.diff(x) == b/(2*sqrt(c.T*a*x.T*b))*c.T*a
expr = (c.T*a*x.T*b)**Rational(1, 3)
assert expr.diff(x) == b*(c.T*a*x.T*b)**Rational(-2, 3)*c.T*a/3
expr = (a.T*X*b)**S.Half
assert expr.diff(X) == a/(2*sqrt(a.T*X*b))*b.T
expr = d.T*x*(a.T*X*b)**S.Half*y.T*c
assert expr.diff(X) == a*x.T*d/(2*sqrt(a.T*X*b))*y.T*c*b.T
def test_derivatives_elementwise_applyfunc():
from sympy.matrices.expressions.diagonal import DiagMatrix
expr = x.applyfunc(tan)
assert expr.diff(x).dummy_eq(
DiagMatrix(x.applyfunc(lambda x: tan(x)**2 + 1)))
assert expr[i, 0].diff(x[m, 0]).doit() == (tan(x[i, 0])**2 + 1)*KDelta(i, m)
_check_derivative_with_explicit_matrix(expr, x, expr.diff(x))
expr = (i**2*x).applyfunc(sin)
assert expr.diff(i).dummy_eq(
HadamardProduct((2*i)*x, (i**2*x).applyfunc(cos)))
assert expr[i, 0].diff(i).doit() == 2*i*x[i, 0]*cos(i**2*x[i, 0])
_check_derivative_with_explicit_matrix(expr, i, expr.diff(i))
expr = (log(i)*A*B).applyfunc(sin)
assert expr.diff(i).dummy_eq(
HadamardProduct(A*B/i, (log(i)*A*B).applyfunc(cos)))
_check_derivative_with_explicit_matrix(expr, i, expr.diff(i))
expr = A*x.applyfunc(exp)
assert expr.diff(x).dummy_eq(DiagMatrix(x.applyfunc(exp))*A.T)
_check_derivative_with_explicit_matrix(expr, x, expr.diff(x))
expr = x.T*A*x + k*y.applyfunc(sin).T*x
assert expr.diff(x).dummy_eq(A.T*x + A*x + k*y.applyfunc(sin))
_check_derivative_with_explicit_matrix(expr, x, expr.diff(x))
expr = x.applyfunc(sin).T*y
assert expr.diff(x).dummy_eq(DiagMatrix(x.applyfunc(cos))*y)
_check_derivative_with_explicit_matrix(expr, x, expr.diff(x))
expr = (a.T * X * b).applyfunc(sin)
assert expr.diff(X).dummy_eq(a*(a.T*X*b).applyfunc(cos)*b.T)
_check_derivative_with_explicit_matrix(expr, X, expr.diff(X))
expr = a.T * X.applyfunc(sin) * b
assert expr.diff(X).dummy_eq(
DiagMatrix(a)*X.applyfunc(cos)*DiagMatrix(b))
_check_derivative_with_explicit_matrix(expr, X, expr.diff(X))
expr = a.T * (A*X*B).applyfunc(sin) * b
assert expr.diff(X).dummy_eq(
A.T*DiagMatrix(a)*(A*X*B).applyfunc(cos)*DiagMatrix(b)*B.T)
_check_derivative_with_explicit_matrix(expr, X, expr.diff(X))
expr = a.T * (A*X*b).applyfunc(sin) * b.T
# TODO: not implemented
#assert expr.diff(X) == ...
#_check_derivative_with_explicit_matrix(expr, X, expr.diff(X))
expr = a.T*A*X.applyfunc(sin)*B*b
assert expr.diff(X).dummy_eq(
DiagMatrix(A.T*a)*X.applyfunc(cos)*DiagMatrix(B*b))
expr = a.T * (A*X.applyfunc(sin)*B).applyfunc(log) * b
# TODO: wrong
# assert expr.diff(X) == A.T*DiagMatrix(a)*(A*X.applyfunc(sin)*B).applyfunc(Lambda(k, 1/k))*DiagMatrix(b)*B.T
expr = a.T * (X.applyfunc(sin)).applyfunc(log) * b
# TODO: wrong
# assert expr.diff(X) == DiagMatrix(a)*X.applyfunc(sin).applyfunc(Lambda(k, 1/k))*DiagMatrix(b)
def test_derivatives_of_hadamard_expressions():
# Hadamard Product
expr = hadamard_product(a, x, b)
assert expr.diff(x) == DiagMatrix(hadamard_product(b, a))
expr = a.T*hadamard_product(A, X, B)*b
assert expr.diff(X) == DiagMatrix(a)*hadamard_product(B, A)*DiagMatrix(b)
# Hadamard Power
expr = hadamard_power(x, 2)
assert expr.diff(x).doit() == 2*DiagMatrix(x)
expr = hadamard_power(x.T, 2)
assert expr.diff(x).doit() == 2*DiagMatrix(x)
expr = hadamard_power(x, S.Half)
assert expr.diff(x) == S.Half*DiagMatrix(hadamard_power(x, Rational(-1, 2)))
expr = hadamard_power(a.T*X*b, 2)
assert expr.diff(X) == 2*a*a.T*X*b*b.T
expr = hadamard_power(a.T*X*b, S.Half)
assert expr.diff(X) == a/2*hadamard_power(a.T*X*b, Rational(-1, 2))*b.T
|
40f233c217e361a6243c298503728383e6c9ae2ce4763df59465ebee357a78fe | from sympy.core.symbol import symbols, Dummy
from sympy.matrices.expressions.applyfunc import ElementwiseApplyFunction
from sympy import Matrix, Lambda, MatrixSymbol, exp, MatMul, sin, simplify
from sympy.testing.pytest import raises
from sympy.matrices.common import ShapeError
X = MatrixSymbol("X", 3, 3)
Y = MatrixSymbol("Y", 3, 3)
k = symbols("k")
Xk = MatrixSymbol("X", k, k)
Xd = X.as_explicit()
x, y, z, t = symbols("x y z t")
def test_applyfunc_matrix():
x = Dummy('x')
double = Lambda(x, x**2)
expr = ElementwiseApplyFunction(double, Xd)
assert isinstance(expr, ElementwiseApplyFunction)
assert expr.doit() == Xd.applyfunc(lambda x: x**2)
assert expr.shape == (3, 3)
assert expr.func(*expr.args) == expr
assert simplify(expr) == expr
assert expr[0, 0] == double(Xd[0, 0])
expr = ElementwiseApplyFunction(double, X)
assert isinstance(expr, ElementwiseApplyFunction)
assert isinstance(expr.doit(), ElementwiseApplyFunction)
assert expr == X.applyfunc(double)
assert expr.func(*expr.args) == expr
expr = ElementwiseApplyFunction(exp, X*Y)
assert expr.expr == X*Y
assert expr.function.dummy_eq(Lambda(x, exp(x)))
assert expr.dummy_eq((X*Y).applyfunc(exp))
assert expr.func(*expr.args) == expr
assert isinstance(X*expr, MatMul)
assert (X*expr).shape == (3, 3)
Z = MatrixSymbol("Z", 2, 3)
assert (Z*expr).shape == (2, 3)
expr = ElementwiseApplyFunction(exp, Z.T)*ElementwiseApplyFunction(exp, Z)
assert expr.shape == (3, 3)
expr = ElementwiseApplyFunction(exp, Z)*ElementwiseApplyFunction(exp, Z.T)
assert expr.shape == (2, 2)
raises(ShapeError, lambda: ElementwiseApplyFunction(exp, Z)*ElementwiseApplyFunction(exp, Z))
M = Matrix([[x, y], [z, t]])
expr = ElementwiseApplyFunction(sin, M)
assert isinstance(expr, ElementwiseApplyFunction)
assert expr.function.dummy_eq(Lambda(x, sin(x)))
assert expr.expr == M
assert expr.doit() == M.applyfunc(sin)
assert expr.doit() == Matrix([[sin(x), sin(y)], [sin(z), sin(t)]])
assert expr.func(*expr.args) == expr
expr = ElementwiseApplyFunction(double, Xk)
assert expr.doit() == expr
assert expr.subs(k, 2).shape == (2, 2)
assert (expr*expr).shape == (k, k)
M = MatrixSymbol("M", k, t)
expr2 = M.T*expr*M
assert isinstance(expr2, MatMul)
assert expr2.args[1] == expr
assert expr2.shape == (t, t)
expr3 = expr*M
assert expr3.shape == (k, t)
raises(ShapeError, lambda: M*expr)
expr1 = ElementwiseApplyFunction(lambda x: x+1, Xk)
expr2 = ElementwiseApplyFunction(lambda x: x, Xk)
assert expr1 != expr2
def test_applyfunc_entry():
af = X.applyfunc(sin)
assert af[0, 0] == sin(X[0, 0])
af = Xd.applyfunc(sin)
assert af[0, 0] == sin(X[0, 0])
def test_applyfunc_as_explicit():
af = X.applyfunc(sin)
assert af.as_explicit() == Matrix([
[sin(X[0, 0]), sin(X[0, 1]), sin(X[0, 2])],
[sin(X[1, 0]), sin(X[1, 1]), sin(X[1, 2])],
[sin(X[2, 0]), sin(X[2, 1]), sin(X[2, 2])],
])
|
cd7199b383a8d729185ba27c9de2b8ed98d4afaa10f426b468b3d856750e2121 | from sympy.core.expr import unchanged
from sympy.core.symbol import Symbol, symbols
from sympy.matrices.immutable import ImmutableDenseMatrix
from sympy.matrices.expressions.companion import CompanionMatrix
from sympy.polys.polytools import Poly
from sympy.testing.pytest import raises
def test_creation():
x = Symbol('x')
y = Symbol('y')
raises(ValueError, lambda: CompanionMatrix(1))
raises(ValueError, lambda: CompanionMatrix(Poly([1], x)))
raises(ValueError, lambda: CompanionMatrix(Poly([2, 1], x)))
raises(ValueError, lambda: CompanionMatrix(Poly(x*y, [x, y])))
unchanged(CompanionMatrix, Poly([1, 2, 3], x))
def test_shape():
c0, c1, c2 = symbols('c0:3')
x = Symbol('x')
assert CompanionMatrix(Poly([1, c0], x)).shape == (1, 1)
assert CompanionMatrix(Poly([1, c1, c0], x)).shape == (2, 2)
assert CompanionMatrix(Poly([1, c2, c1, c0], x)).shape == (3, 3)
def test_entry():
c0, c1, c2 = symbols('c0:3')
x = Symbol('x')
A = CompanionMatrix(Poly([1, c2, c1, c0], x))
assert A[0, 0] == 0
assert A[1, 0] == 1
assert A[1, 1] == 0
assert A[2, 1] == 1
assert A[0, 2] == -c0
assert A[1, 2] == -c1
assert A[2, 2] == -c2
def test_as_explicit():
c0, c1, c2 = symbols('c0:3')
x = Symbol('x')
assert CompanionMatrix(Poly([1, c0], x)).as_explicit() == \
ImmutableDenseMatrix([-c0])
assert CompanionMatrix(Poly([1, c1, c0], x)).as_explicit() == \
ImmutableDenseMatrix([[0, -c0], [1, -c1]])
assert CompanionMatrix(Poly([1, c2, c1, c0], x)).as_explicit() == \
ImmutableDenseMatrix([[0, 0, -c0], [1, 0, -c1], [0, 1, -c2]])
|
4cadc590e530c762cfd7fc4da9022e694f7a6c431fc1c593328162d5ed20a8db | from sympy.sets.setexpr import SetExpr
from sympy.sets import Interval, FiniteSet, Intersection, ImageSet, Union
from sympy import (Expr, Set, exp, log, cos, Symbol, Min, Max, S, oo, I,
symbols, Lambda, Dummy, Rational)
a, x = symbols("a, x")
_d = Dummy("d")
def test_setexpr():
se = SetExpr(Interval(0, 1))
assert isinstance(se.set, Set)
assert isinstance(se, Expr)
def test_scalar_funcs():
assert SetExpr(Interval(0, 1)).set == Interval(0, 1)
a, b = Symbol('a', real=True), Symbol('b', real=True)
a, b = 1, 2
# TODO: add support for more functions in the future:
for f in [exp, log]:
input_se = f(SetExpr(Interval(a, b)))
output = input_se.set
expected = Interval(Min(f(a), f(b)), Max(f(a), f(b)))
assert output == expected
def test_Add_Mul():
assert (SetExpr(Interval(0, 1)) + 1).set == Interval(1, 2)
assert (SetExpr(Interval(0, 1))*2).set == Interval(0, 2)
def test_Pow():
assert (SetExpr(Interval(0, 2))**2).set == Interval(0, 4)
def test_compound():
assert (exp(SetExpr(Interval(0, 1))*2 + 1)).set == \
Interval(exp(1), exp(3))
def test_Interval_Interval():
assert (SetExpr(Interval(1, 2)) + SetExpr(Interval(10, 20))).set == \
Interval(11, 22)
assert (SetExpr(Interval(1, 2))*SetExpr(Interval(10, 20))).set == \
Interval(10, 40)
def test_FiniteSet_FiniteSet():
assert (SetExpr(FiniteSet(1, 2, 3)) + SetExpr(FiniteSet(1, 2))).set == \
FiniteSet(2, 3, 4, 5)
assert (SetExpr(FiniteSet(1, 2, 3))*SetExpr(FiniteSet(1, 2))).set == \
FiniteSet(1, 2, 3, 4, 6)
def test_Interval_FiniteSet():
assert (SetExpr(FiniteSet(1, 2)) + SetExpr(Interval(0, 10))).set == \
Interval(1, 12)
def test_Many_Sets():
assert (SetExpr(Interval(0, 1)) +
SetExpr(Interval(2, 3)) +
SetExpr(FiniteSet(10, 11, 12))).set == Interval(12, 16)
def test_same_setexprs_are_not_identical():
a = SetExpr(FiniteSet(0, 1))
b = SetExpr(FiniteSet(0, 1))
assert (a + b).set == FiniteSet(0, 1, 2)
# Cannont detect the set being the same:
# assert (a + a).set == FiniteSet(0, 2)
def test_Interval_arithmetic():
i12cc = SetExpr(Interval(1, 2))
i12lo = SetExpr(Interval.Lopen(1, 2))
i12ro = SetExpr(Interval.Ropen(1, 2))
i12o = SetExpr(Interval.open(1, 2))
n23cc = SetExpr(Interval(-2, 3))
n23lo = SetExpr(Interval.Lopen(-2, 3))
n23ro = SetExpr(Interval.Ropen(-2, 3))
n23o = SetExpr(Interval.open(-2, 3))
n3n2cc = SetExpr(Interval(-3, -2))
assert i12cc + i12cc == SetExpr(Interval(2, 4))
assert i12cc - i12cc == SetExpr(Interval(-1, 1))
assert i12cc*i12cc == SetExpr(Interval(1, 4))
assert i12cc/i12cc == SetExpr(Interval(S.Half, 2))
assert i12cc**2 == SetExpr(Interval(1, 4))
assert i12cc**3 == SetExpr(Interval(1, 8))
assert i12lo + i12ro == SetExpr(Interval.open(2, 4))
assert i12lo - i12ro == SetExpr(Interval.Lopen(-1, 1))
assert i12lo*i12ro == SetExpr(Interval.open(1, 4))
assert i12lo/i12ro == SetExpr(Interval.Lopen(S.Half, 2))
assert i12lo + i12lo == SetExpr(Interval.Lopen(2, 4))
assert i12lo - i12lo == SetExpr(Interval.open(-1, 1))
assert i12lo*i12lo == SetExpr(Interval.Lopen(1, 4))
assert i12lo/i12lo == SetExpr(Interval.open(S.Half, 2))
assert i12lo + i12cc == SetExpr(Interval.Lopen(2, 4))
assert i12lo - i12cc == SetExpr(Interval.Lopen(-1, 1))
assert i12lo*i12cc == SetExpr(Interval.Lopen(1, 4))
assert i12lo/i12cc == SetExpr(Interval.Lopen(S.Half, 2))
assert i12lo + i12o == SetExpr(Interval.open(2, 4))
assert i12lo - i12o == SetExpr(Interval.open(-1, 1))
assert i12lo*i12o == SetExpr(Interval.open(1, 4))
assert i12lo/i12o == SetExpr(Interval.open(S.Half, 2))
assert i12lo**2 == SetExpr(Interval.Lopen(1, 4))
assert i12lo**3 == SetExpr(Interval.Lopen(1, 8))
assert i12ro + i12ro == SetExpr(Interval.Ropen(2, 4))
assert i12ro - i12ro == SetExpr(Interval.open(-1, 1))
assert i12ro*i12ro == SetExpr(Interval.Ropen(1, 4))
assert i12ro/i12ro == SetExpr(Interval.open(S.Half, 2))
assert i12ro + i12cc == SetExpr(Interval.Ropen(2, 4))
assert i12ro - i12cc == SetExpr(Interval.Ropen(-1, 1))
assert i12ro*i12cc == SetExpr(Interval.Ropen(1, 4))
assert i12ro/i12cc == SetExpr(Interval.Ropen(S.Half, 2))
assert i12ro + i12o == SetExpr(Interval.open(2, 4))
assert i12ro - i12o == SetExpr(Interval.open(-1, 1))
assert i12ro*i12o == SetExpr(Interval.open(1, 4))
assert i12ro/i12o == SetExpr(Interval.open(S.Half, 2))
assert i12ro**2 == SetExpr(Interval.Ropen(1, 4))
assert i12ro**3 == SetExpr(Interval.Ropen(1, 8))
assert i12o + i12lo == SetExpr(Interval.open(2, 4))
assert i12o - i12lo == SetExpr(Interval.open(-1, 1))
assert i12o*i12lo == SetExpr(Interval.open(1, 4))
assert i12o/i12lo == SetExpr(Interval.open(S.Half, 2))
assert i12o + i12ro == SetExpr(Interval.open(2, 4))
assert i12o - i12ro == SetExpr(Interval.open(-1, 1))
assert i12o*i12ro == SetExpr(Interval.open(1, 4))
assert i12o/i12ro == SetExpr(Interval.open(S.Half, 2))
assert i12o + i12cc == SetExpr(Interval.open(2, 4))
assert i12o - i12cc == SetExpr(Interval.open(-1, 1))
assert i12o*i12cc == SetExpr(Interval.open(1, 4))
assert i12o/i12cc == SetExpr(Interval.open(S.Half, 2))
assert i12o**2 == SetExpr(Interval.open(1, 4))
assert i12o**3 == SetExpr(Interval.open(1, 8))
assert n23cc + n23cc == SetExpr(Interval(-4, 6))
assert n23cc - n23cc == SetExpr(Interval(-5, 5))
assert n23cc*n23cc == SetExpr(Interval(-6, 9))
assert n23cc/n23cc == SetExpr(Interval.open(-oo, oo))
assert n23cc + n23ro == SetExpr(Interval.Ropen(-4, 6))
assert n23cc - n23ro == SetExpr(Interval.Lopen(-5, 5))
assert n23cc*n23ro == SetExpr(Interval.Ropen(-6, 9))
assert n23cc/n23ro == SetExpr(Interval.Lopen(-oo, oo))
assert n23cc + n23lo == SetExpr(Interval.Lopen(-4, 6))
assert n23cc - n23lo == SetExpr(Interval.Ropen(-5, 5))
assert n23cc*n23lo == SetExpr(Interval(-6, 9))
assert n23cc/n23lo == SetExpr(Interval.open(-oo, oo))
assert n23cc + n23o == SetExpr(Interval.open(-4, 6))
assert n23cc - n23o == SetExpr(Interval.open(-5, 5))
assert n23cc*n23o == SetExpr(Interval.open(-6, 9))
assert n23cc/n23o == SetExpr(Interval.open(-oo, oo))
assert n23cc**2 == SetExpr(Interval(0, 9))
assert n23cc**3 == SetExpr(Interval(-8, 27))
n32cc = SetExpr(Interval(-3, 2))
n32lo = SetExpr(Interval.Lopen(-3, 2))
n32ro = SetExpr(Interval.Ropen(-3, 2))
assert n32cc*n32lo == SetExpr(Interval.Ropen(-6, 9))
assert n32cc*n32cc == SetExpr(Interval(-6, 9))
assert n32lo*n32cc == SetExpr(Interval.Ropen(-6, 9))
assert n32cc*n32ro == SetExpr(Interval(-6, 9))
assert n32lo*n32ro == SetExpr(Interval.Ropen(-6, 9))
assert n32cc/n32lo == SetExpr(Interval.Ropen(-oo, oo))
assert i12cc/n32lo == SetExpr(Interval.Ropen(-oo, oo))
assert n3n2cc**2 == SetExpr(Interval(4, 9))
assert n3n2cc**3 == SetExpr(Interval(-27, -8))
assert n23cc + i12cc == SetExpr(Interval(-1, 5))
assert n23cc - i12cc == SetExpr(Interval(-4, 2))
assert n23cc*i12cc == SetExpr(Interval(-4, 6))
assert n23cc/i12cc == SetExpr(Interval(-2, 3))
def test_SetExpr_Intersection():
x, y, z, w = symbols("x y z w")
set1 = Interval(x, y)
set2 = Interval(w, z)
inter = Intersection(set1, set2)
se = SetExpr(inter)
assert exp(se).set == Intersection(
ImageSet(Lambda(x, exp(x)), set1),
ImageSet(Lambda(x, exp(x)), set2))
assert cos(se).set == ImageSet(Lambda(x, cos(x)), inter)
def test_SetExpr_Interval_div():
# TODO: some expressions cannot be calculated due to bugs (currently
# commented):
assert SetExpr(Interval(-3, -2))/SetExpr(Interval(-2, 1)) == SetExpr(Interval(-oo, oo))
assert SetExpr(Interval(2, 3))/SetExpr(Interval(-2, 2)) == SetExpr(Interval(-oo, oo))
assert SetExpr(Interval(-3, -2))/SetExpr(Interval(0, 4)) == SetExpr(Interval(-oo, Rational(-1, 2)))
assert SetExpr(Interval(2, 4))/SetExpr(Interval(-3, 0)) == SetExpr(Interval(-oo, Rational(-2, 3)))
assert SetExpr(Interval(2, 4))/SetExpr(Interval(0, 3)) == SetExpr(Interval(Rational(2, 3), oo))
# assert SetExpr(Interval(0, 1))/SetExpr(Interval(0, 1)) == SetExpr(Interval(0, oo))
# assert SetExpr(Interval(-1, 0))/SetExpr(Interval(0, 1)) == SetExpr(Interval(-oo, 0))
assert SetExpr(Interval(-1, 2))/SetExpr(Interval(-2, 2)) == SetExpr(Interval(-oo, oo))
assert 1/SetExpr(Interval(-1, 2)) == SetExpr(Union(Interval(-oo, -1), Interval(S.Half, oo)))
assert 1/SetExpr(Interval(0, 2)) == SetExpr(Interval(S.Half, oo))
assert (-1)/SetExpr(Interval(0, 2)) == SetExpr(Interval(-oo, Rational(-1, 2)))
# assert 1/SetExpr(Interval(-oo, 0)) == SetExpr(Interval.open(-oo, 0))
assert 1/SetExpr(Interval(-1, 0)) == SetExpr(Interval(-oo, -1))
# assert (-2)/SetExpr(Interval(-oo, 0)) == SetExpr(Interval(0, oo))
# assert 1/SetExpr(Interval(-oo, -1)) == SetExpr(Interval(-1, 0))
# assert SetExpr(Interval(1, 2))/a == Mul(SetExpr(Interval(1, 2)), 1/a, evaluate=False)
# assert SetExpr(Interval(1, 2))/0 == SetExpr(Interval(1, 2))*zoo
# assert SetExpr(Interval(1, oo))/oo == SetExpr(Interval(0, oo))
# assert SetExpr(Interval(1, oo))/(-oo) == SetExpr(Interval(-oo, 0))
# assert SetExpr(Interval(-oo, -1))/oo == SetExpr(Interval(-oo, 0))
# assert SetExpr(Interval(-oo, -1))/(-oo) == SetExpr(Interval(0, oo))
# assert SetExpr(Interval(-oo, oo))/oo == SetExpr(Interval(-oo, oo))
# assert SetExpr(Interval(-oo, oo))/(-oo) == SetExpr(Interval(-oo, oo))
# assert SetExpr(Interval(-1, oo))/oo == SetExpr(Interval(0, oo))
# assert SetExpr(Interval(-1, oo))/(-oo) == SetExpr(Interval(-oo, 0))
# assert SetExpr(Interval(-oo, 1))/oo == SetExpr(Interval(-oo, 0))
# assert SetExpr(Interval(-oo, 1))/(-oo) == SetExpr(Interval(0, oo))
def test_SetExpr_Interval_pow():
assert SetExpr(Interval(0, 2))**2 == SetExpr(Interval(0, 4))
assert SetExpr(Interval(-1, 1))**2 == SetExpr(Interval(0, 1))
assert SetExpr(Interval(1, 2))**2 == SetExpr(Interval(1, 4))
assert SetExpr(Interval(-1, 2))**3 == SetExpr(Interval(-1, 8))
assert SetExpr(Interval(-1, 1))**0 == SetExpr(FiniteSet(1))
#assert SetExpr(Interval(1, 2))**Rational(5, 2) == SetExpr(Interval(1, 4*sqrt(2)))
#assert SetExpr(Interval(-1, 2))**Rational(1, 3) == SetExpr(Interval(-1, 2**Rational(1, 3)))
#assert SetExpr(Interval(0, 2))**S.Half == SetExpr(Interval(0, sqrt(2)))
#assert SetExpr(Interval(-4, 2))**Rational(2, 3) == SetExpr(Interval(0, 2*2**Rational(1, 3)))
#assert SetExpr(Interval(-1, 5))**S.Half == SetExpr(Interval(0, sqrt(5)))
#assert SetExpr(Interval(-oo, 2))**S.Half == SetExpr(Interval(0, sqrt(2)))
#assert SetExpr(Interval(-2, 3))**(Rational(-1, 4)) == SetExpr(Interval(0, oo))
assert SetExpr(Interval(1, 5))**(-2) == SetExpr(Interval(Rational(1, 25), 1))
assert SetExpr(Interval(-1, 3))**(-2) == SetExpr(Interval(0, oo))
assert SetExpr(Interval(0, 2))**(-2) == SetExpr(Interval(Rational(1, 4), oo))
assert SetExpr(Interval(-1, 2))**(-3) == SetExpr(Union(Interval(-oo, -1), Interval(Rational(1, 8), oo)))
assert SetExpr(Interval(-3, -2))**(-3) == SetExpr(Interval(Rational(-1, 8), Rational(-1, 27)))
assert SetExpr(Interval(-3, -2))**(-2) == SetExpr(Interval(Rational(1, 9), Rational(1, 4)))
#assert SetExpr(Interval(0, oo))**S.Half == SetExpr(Interval(0, oo))
#assert SetExpr(Interval(-oo, -1))**Rational(1, 3) == SetExpr(Interval(-oo, -1))
#assert SetExpr(Interval(-2, 3))**(Rational(-1, 3)) == SetExpr(Interval(-oo, oo))
assert SetExpr(Interval(-oo, 0))**(-2) == SetExpr(Interval.open(0, oo))
assert SetExpr(Interval(-2, 0))**(-2) == SetExpr(Interval(Rational(1, 4), oo))
assert SetExpr(Interval(Rational(1, 3), S.Half))**oo == SetExpr(FiniteSet(0))
assert SetExpr(Interval(0, S.Half))**oo == SetExpr(FiniteSet(0))
assert SetExpr(Interval(S.Half, 1))**oo == SetExpr(Interval(0, oo))
assert SetExpr(Interval(0, 1))**oo == SetExpr(Interval(0, oo))
assert SetExpr(Interval(2, 3))**oo == SetExpr(FiniteSet(oo))
assert SetExpr(Interval(1, 2))**oo == SetExpr(Interval(0, oo))
assert SetExpr(Interval(S.Half, 3))**oo == SetExpr(Interval(0, oo))
assert SetExpr(Interval(Rational(-1, 3), Rational(-1, 4)))**oo == SetExpr(FiniteSet(0))
assert SetExpr(Interval(-1, Rational(-1, 2)))**oo == SetExpr(Interval(-oo, oo))
assert SetExpr(Interval(-3, -2))**oo == SetExpr(FiniteSet(-oo, oo))
assert SetExpr(Interval(-2, -1))**oo == SetExpr(Interval(-oo, oo))
assert SetExpr(Interval(-2, Rational(-1, 2)))**oo == SetExpr(Interval(-oo, oo))
assert SetExpr(Interval(Rational(-1, 2), S.Half))**oo == SetExpr(FiniteSet(0))
assert SetExpr(Interval(Rational(-1, 2), 1))**oo == SetExpr(Interval(0, oo))
assert SetExpr(Interval(Rational(-2, 3), 2))**oo == SetExpr(Interval(0, oo))
assert SetExpr(Interval(-1, 1))**oo == SetExpr(Interval(-oo, oo))
assert SetExpr(Interval(-1, S.Half))**oo == SetExpr(Interval(-oo, oo))
assert SetExpr(Interval(-1, 2))**oo == SetExpr(Interval(-oo, oo))
assert SetExpr(Interval(-2, S.Half))**oo == SetExpr(Interval(-oo, oo))
assert (SetExpr(Interval(1, 2))**x).dummy_eq(SetExpr(ImageSet(Lambda(_d, _d**x), Interval(1, 2))))
assert SetExpr(Interval(2, 3))**(-oo) == SetExpr(FiniteSet(0))
assert SetExpr(Interval(0, 2))**(-oo) == SetExpr(Interval(0, oo))
assert (SetExpr(Interval(-1, 2))**(-oo)).dummy_eq(SetExpr(ImageSet(Lambda(_d, _d**(-oo)), Interval(-1, 2))))
def test_SetExpr_Integers():
assert SetExpr(S.Integers) + 1 == SetExpr(S.Integers)
assert (SetExpr(S.Integers) + I).dummy_eq(
SetExpr(ImageSet(Lambda(_d, _d + I), S.Integers)))
assert SetExpr(S.Integers)*(-1) == SetExpr(S.Integers)
assert (SetExpr(S.Integers)*2).dummy_eq(
SetExpr(ImageSet(Lambda(_d, 2*_d), S.Integers)))
assert (SetExpr(S.Integers)*I).dummy_eq(
SetExpr(ImageSet(Lambda(_d, I*_d), S.Integers)))
# issue #18050:
assert SetExpr(S.Integers)._eval_func(Lambda(x, I*x + 1)).dummy_eq(
SetExpr(ImageSet(Lambda(_d, I*_d + 1), S.Integers)))
# needs improvement:
assert (SetExpr(S.Integers)*I + 1).dummy_eq(
SetExpr(ImageSet(Lambda(x, x + 1),
ImageSet(Lambda(_d, _d*I), S.Integers))))
|
ecc397d453e8235abaf8581f8f13cc8d8b3938a3e77db66d8bc7a96ae9de20c5 | from sympy.core.expr import unchanged
from sympy.sets.fancysets import (ImageSet, Range, normalize_theta_set,
ComplexRegion)
from sympy.sets.sets import (FiniteSet, Interval, imageset, Union,
Intersection, ProductSet, Contains)
from sympy.simplify.simplify import simplify
from sympy import (S, Symbol, Lambda, symbols, cos, sin, pi, oo, Basic,
Rational, sqrt, tan, log, exp, Abs, I, Tuple, eye,
Dummy, floor, And, Eq)
from sympy.utilities.iterables import cartes
from sympy.testing.pytest import XFAIL, raises
from sympy.abc import x, y, t
import itertools
def test_naturals():
N = S.Naturals
assert 5 in N
assert -5 not in N
assert 5.5 not in N
ni = iter(N)
a, b, c, d = next(ni), next(ni), next(ni), next(ni)
assert (a, b, c, d) == (1, 2, 3, 4)
assert isinstance(a, Basic)
assert N.intersect(Interval(-5, 5)) == Range(1, 6)
assert N.intersect(Interval(-5, 5, True, True)) == Range(1, 5)
assert N.boundary == N
assert N.is_open == False
assert N.is_closed == True
assert N.inf == 1
assert N.sup is oo
assert not N.contains(oo)
for s in (S.Naturals0, S.Naturals):
assert s.intersection(S.Reals) is s
assert s.is_subset(S.Reals)
assert N.as_relational(x) == And(Eq(floor(x), x), x >= 1, x < oo)
def test_naturals0():
N = S.Naturals0
assert 0 in N
assert -1 not in N
assert next(iter(N)) == 0
assert not N.contains(oo)
assert N.contains(sin(x)) == Contains(sin(x), N)
def test_integers():
Z = S.Integers
assert 5 in Z
assert -5 in Z
assert 5.5 not in Z
assert not Z.contains(oo)
assert not Z.contains(-oo)
zi = iter(Z)
a, b, c, d = next(zi), next(zi), next(zi), next(zi)
assert (a, b, c, d) == (0, 1, -1, 2)
assert isinstance(a, Basic)
assert Z.intersect(Interval(-5, 5)) == Range(-5, 6)
assert Z.intersect(Interval(-5, 5, True, True)) == Range(-4, 5)
assert Z.intersect(Interval(5, S.Infinity)) == Range(5, S.Infinity)
assert Z.intersect(Interval.Lopen(5, S.Infinity)) == Range(6, S.Infinity)
assert Z.inf is -oo
assert Z.sup is oo
assert Z.boundary == Z
assert Z.is_open == False
assert Z.is_closed == True
assert Z.as_relational(x) == And(Eq(floor(x), x), -oo < x, x < oo)
def test_ImageSet():
raises(ValueError, lambda: ImageSet(x, S.Integers))
assert ImageSet(Lambda(x, 1), S.Integers) == FiniteSet(1)
assert ImageSet(Lambda(x, y), S.Integers) == {y}
assert ImageSet(Lambda(x, 1), S.EmptySet) == S.EmptySet
empty = Intersection(FiniteSet(log(2)/pi), S.Integers)
assert unchanged(ImageSet, Lambda(x, 1), empty) # issue #17471
squares = ImageSet(Lambda(x, x**2), S.Naturals)
assert 4 in squares
assert 5 not in squares
assert FiniteSet(*range(10)).intersect(squares) == FiniteSet(1, 4, 9)
assert 16 not in squares.intersect(Interval(0, 10))
si = iter(squares)
a, b, c, d = next(si), next(si), next(si), next(si)
assert (a, b, c, d) == (1, 4, 9, 16)
harmonics = ImageSet(Lambda(x, 1/x), S.Naturals)
assert Rational(1, 5) in harmonics
assert Rational(.25) in harmonics
assert 0.25 not in harmonics
assert Rational(.3) not in harmonics
assert (1, 2) not in harmonics
assert harmonics.is_iterable
assert imageset(x, -x, Interval(0, 1)) == Interval(-1, 0)
assert ImageSet(Lambda(x, x**2), Interval(0, 2)).doit() == Interval(0, 4)
assert ImageSet(Lambda((x, y), 2*x), {4}, {3}).doit() == FiniteSet(8)
assert (ImageSet(Lambda((x, y), x+y), {1, 2, 3}, {10, 20, 30}).doit() ==
FiniteSet(11, 12, 13, 21, 22, 23, 31, 32, 33))
c = Interval(1, 3) * Interval(1, 3)
assert Tuple(2, 6) in ImageSet(Lambda(((x, y),), (x, 2*y)), c)
assert Tuple(2, S.Half) in ImageSet(Lambda(((x, y),), (x, 1/y)), c)
assert Tuple(2, -2) not in ImageSet(Lambda(((x, y),), (x, y**2)), c)
assert Tuple(2, -2) in ImageSet(Lambda(((x, y),), (x, -2)), c)
c3 = ProductSet(Interval(3, 7), Interval(8, 11), Interval(5, 9))
assert Tuple(8, 3, 9) in ImageSet(Lambda(((t, y, x),), (y, t, x)), c3)
assert Tuple(Rational(1, 8), 3, 9) in ImageSet(Lambda(((t, y, x),), (1/y, t, x)), c3)
assert 2/pi not in ImageSet(Lambda(((x, y),), 2/x), c)
assert 2/S(100) not in ImageSet(Lambda(((x, y),), 2/x), c)
assert Rational(2, 3) in ImageSet(Lambda(((x, y),), 2/x), c)
S1 = imageset(lambda x, y: x + y, S.Integers, S.Naturals)
assert S1.base_pset == ProductSet(S.Integers, S.Naturals)
assert S1.base_sets == (S.Integers, S.Naturals)
# Passing a set instead of a FiniteSet shouldn't raise
assert unchanged(ImageSet, Lambda(x, x**2), {1, 2, 3})
S2 = ImageSet(Lambda(((x, y),), x+y), {(1, 2), (3, 4)})
assert 3 in S2.doit()
# FIXME: This doesn't yet work:
#assert 3 in S2
assert S2._contains(3) is None
raises(TypeError, lambda: ImageSet(Lambda(x, x**2), 1))
def test_image_is_ImageSet():
assert isinstance(imageset(x, sqrt(sin(x)), Range(5)), ImageSet)
def test_halfcircle():
r, th = symbols('r, theta', real=True)
L = Lambda(((r, th),), (r*cos(th), r*sin(th)))
halfcircle = ImageSet(L, Interval(0, 1)*Interval(0, pi))
assert (1, 0) in halfcircle
assert (0, -1) not in halfcircle
assert (0, 0) in halfcircle
assert halfcircle._contains((r, 0)) is None
# This one doesn't work:
#assert (r, 2*pi) not in halfcircle
assert not halfcircle.is_iterable
def test_ImageSet_iterator_not_injective():
L = Lambda(x, x - x % 2) # produces 0, 2, 2, 4, 4, 6, 6, ...
evens = ImageSet(L, S.Naturals)
i = iter(evens)
# No repeats here
assert (next(i), next(i), next(i), next(i)) == (0, 2, 4, 6)
def test_inf_Range_len():
raises(ValueError, lambda: len(Range(0, oo, 2)))
assert Range(0, oo, 2).size is S.Infinity
assert Range(0, -oo, -2).size is S.Infinity
assert Range(oo, 0, -2).size is S.Infinity
assert Range(-oo, 0, 2).size is S.Infinity
def test_Range_set():
empty = Range(0)
assert Range(5) == Range(0, 5) == Range(0, 5, 1)
r = Range(10, 20, 2)
assert 12 in r
assert 8 not in r
assert 11 not in r
assert 30 not in r
assert list(Range(0, 5)) == list(range(5))
assert list(Range(5, 0, -1)) == list(range(5, 0, -1))
assert Range(5, 15).sup == 14
assert Range(5, 15).inf == 5
assert Range(15, 5, -1).sup == 15
assert Range(15, 5, -1).inf == 6
assert Range(10, 67, 10).sup == 60
assert Range(60, 7, -10).inf == 10
assert len(Range(10, 38, 10)) == 3
assert Range(0, 0, 5) == empty
assert Range(oo, oo, 1) == empty
assert Range(oo, 1, 1) == empty
assert Range(-oo, 1, -1) == empty
assert Range(1, oo, -1) == empty
assert Range(1, -oo, 1) == empty
assert Range(1, -4, oo) == empty
assert Range(1, -4, -oo) == Range(1, 2)
assert Range(1, 4, oo) == Range(1, 2)
assert Range(-oo, oo).size == oo
assert Range(oo, -oo, -1).size == oo
raises(ValueError, lambda: Range(-oo, oo, 2))
raises(ValueError, lambda: Range(x, pi, y))
raises(ValueError, lambda: Range(x, y, 0))
assert 5 in Range(0, oo, 5)
assert -5 in Range(-oo, 0, 5)
assert oo not in Range(0, oo)
ni = symbols('ni', integer=False)
assert ni not in Range(oo)
u = symbols('u', integer=None)
assert Range(oo).contains(u) is not False
inf = symbols('inf', infinite=True)
assert inf not in Range(-oo, oo)
raises(ValueError, lambda: Range(0, oo, 2)[-1])
raises(ValueError, lambda: Range(0, -oo, -2)[-1])
assert Range(-oo, 1, 1)[-1] is S.Zero
assert Range(oo, 1, -1)[-1] == 2
assert inf not in Range(oo)
inf = symbols('inf', infinite=True)
assert inf not in Range(oo)
assert Range(-oo, 1, 1)[-1] is S.Zero
assert Range(oo, 1, -1)[-1] == 2
assert Range(1, 10, 1)[-1] == 9
assert all(i.is_Integer for i in Range(0, -1, 1))
it = iter(Range(-oo, 0, 2))
raises(TypeError, lambda: next(it))
assert empty.intersect(S.Integers) == empty
assert Range(-1, 10, 1).intersect(S.Integers) == Range(-1, 10, 1)
assert Range(-1, 10, 1).intersect(S.Naturals) == Range(1, 10, 1)
assert Range(-1, 10, 1).intersect(S.Naturals0) == Range(0, 10, 1)
# test slicing
assert Range(1, 10, 1)[5] == 6
assert Range(1, 12, 2)[5] == 11
assert Range(1, 10, 1)[-1] == 9
assert Range(1, 10, 3)[-1] == 7
raises(ValueError, lambda: Range(oo,0,-1)[1:3:0])
raises(ValueError, lambda: Range(oo,0,-1)[:1])
raises(ValueError, lambda: Range(1, oo)[-2])
raises(ValueError, lambda: Range(-oo, 1)[2])
raises(IndexError, lambda: Range(10)[-20])
raises(IndexError, lambda: Range(10)[20])
raises(ValueError, lambda: Range(2, -oo, -2)[2:2:0])
assert Range(2, -oo, -2)[2:2:2] == empty
assert Range(2, -oo, -2)[:2:2] == Range(2, -2, -4)
raises(ValueError, lambda: Range(-oo, 4, 2)[:2:2])
assert Range(-oo, 4, 2)[::-2] == Range(2, -oo, -4)
raises(ValueError, lambda: Range(-oo, 4, 2)[::2])
assert Range(oo, 2, -2)[::] == Range(oo, 2, -2)
assert Range(-oo, 4, 2)[:-2:-2] == Range(2, 0, -4)
assert Range(-oo, 4, 2)[:-2:2] == Range(-oo, 0, 4)
raises(ValueError, lambda: Range(-oo, 4, 2)[:0:-2])
raises(ValueError, lambda: Range(-oo, 4, 2)[:2:-2])
assert Range(-oo, 4, 2)[-2::-2] == Range(0, -oo, -4)
raises(ValueError, lambda: Range(-oo, 4, 2)[-2:0:-2])
raises(ValueError, lambda: Range(-oo, 4, 2)[0::2])
assert Range(oo, 2, -2)[0::] == Range(oo, 2, -2)
raises(ValueError, lambda: Range(-oo, 4, 2)[0:-2:2])
assert Range(oo, 2, -2)[0:-2:] == Range(oo, 6, -2)
raises(ValueError, lambda: Range(oo, 2, -2)[0:2:])
raises(ValueError, lambda: Range(-oo, 4, 2)[2::-1])
assert Range(-oo, 4, 2)[-2::2] == Range(0, 4, 4)
assert Range(oo, 0, -2)[-10:0:2] == empty
raises(ValueError, lambda: Range(oo, 0, -2)[-10:10:2])
raises(ValueError, lambda: Range(oo, 0, -2)[0::-2])
assert Range(oo, 0, -2)[0:-4:-2] == empty
assert Range(oo, 0, -2)[:0:2] == empty
raises(ValueError, lambda: Range(oo, 0, -2)[:1:-1])
# test empty Range
assert Range(x, x, y) == empty
assert empty.reversed == empty
assert 0 not in empty
assert list(empty) == []
assert len(empty) == 0
assert empty.size is S.Zero
assert empty.intersect(FiniteSet(0)) is S.EmptySet
assert bool(empty) is False
raises(IndexError, lambda: empty[0])
assert empty[:0] == empty
raises(NotImplementedError, lambda: empty.inf)
raises(NotImplementedError, lambda: empty.sup)
AB = [None] + list(range(12))
for R in [
Range(1, 10),
Range(1, 10, 2),
]:
r = list(R)
for a, b, c in cartes(AB, AB, [-3, -1, None, 1, 3]):
for reverse in range(2):
r = list(reversed(r))
R = R.reversed
result = list(R[a:b:c])
ans = r[a:b:c]
txt = ('\n%s[%s:%s:%s] = %s -> %s' % (
R, a, b, c, result, ans))
check = ans == result
assert check, txt
assert Range(1, 10, 1).boundary == Range(1, 10, 1)
for r in (Range(1, 10, 2), Range(1, oo, 2)):
rev = r.reversed
assert r.inf == rev.inf and r.sup == rev.sup
assert r.step == -rev.step
builtin_range = range
raises(TypeError, lambda: Range(builtin_range(1)))
assert S(builtin_range(10)) == Range(10)
assert S(builtin_range(1000000000000)) == Range(1000000000000)
# test Range.as_relational
assert Range(1, 4).as_relational(x) == (x >= 1) & (x <= 3) & Eq(x, floor(x))
assert Range(oo, 1, -2).as_relational(x) == (x >= 3) & (x < oo) & Eq(x, floor(x))
def test_Range_symbolic():
# symbolic Range
sr = Range(x, y, t)
i = Symbol('i', integer=True)
ip = Symbol('i', integer=True, positive=True)
ir = Range(i, i + 20, 2)
inf = symbols('inf', infinite=True)
# args
assert sr.args == (x, y, t)
assert ir.args == (i, i + 20, 2)
# reversed
raises(ValueError, lambda: sr.reversed)
assert ir.reversed == Range(i + 18, i - 2, -2)
# contains
assert inf not in sr
assert inf not in ir
assert .1 not in sr
assert .1 not in ir
assert i + 1 not in ir
assert i + 2 in ir
raises(TypeError, lambda: 1 in sr) # XXX is this what contains is supposed to do?
# iter
raises(ValueError, lambda: next(iter(sr)))
assert next(iter(ir)) == i
assert sr.intersect(S.Integers) == sr
assert sr.intersect(FiniteSet(x)) == Intersection({x}, sr)
raises(ValueError, lambda: sr[:2])
raises(ValueError, lambda: sr[0])
raises(ValueError, lambda: sr.as_relational(x))
# len
assert len(ir) == ir.size == 10
raises(ValueError, lambda: len(sr))
raises(ValueError, lambda: sr.size)
# bool
assert bool(ir) == bool(sr) == True
# getitem
raises(ValueError, lambda: sr[0])
raises(ValueError, lambda: sr[-1])
raises(ValueError, lambda: sr[:2])
assert ir[:2] == Range(i, i + 4, 2)
assert ir[0] == i
assert ir[-2] == i + 16
assert ir[-1] == i + 18
raises(ValueError, lambda: Range(i)[-1])
assert Range(ip)[-1] == ip - 1
assert ir.inf == i
assert ir.sup == i + 18
assert Range(ip).inf == 0
assert Range(ip).sup == ip - 1
raises(ValueError, lambda: Range(i).inf)
# as_relational
raises(ValueError, lambda: sr.as_relational(x))
assert ir.as_relational(x) == (
x >= i) & Eq(x, floor(x)) & (x <= i + 18)
assert Range(i, i + 1).as_relational(x) == Eq(x, i)
# contains() for symbolic values (issue #18146)
e = Symbol('e', integer=True, even=True)
o = Symbol('o', integer=True, odd=True)
assert Range(5).contains(i) == And(i >= 0, i <= 4)
assert Range(1).contains(i) == Eq(i, 0)
assert Range(-oo, 5, 1).contains(i) == (i <= 4)
assert Range(-oo, oo).contains(i) == True
assert Range(0, 8, 2).contains(i) == Contains(i, Range(0, 8, 2))
assert Range(0, 8, 2).contains(e) == And(e >= 0, e <= 6)
assert Range(0, 8, 2).contains(2*i) == And(2*i >= 0, 2*i <= 6)
assert Range(0, 8, 2).contains(o) == False
assert Range(1, 9, 2).contains(e) == False
assert Range(1, 9, 2).contains(o) == And(o >= 1, o <= 7)
assert Range(8, 0, -2).contains(o) == False
assert Range(9, 1, -2).contains(o) == And(o >= 3, o <= 9)
assert Range(-oo, 8, 2).contains(i) == Contains(i, Range(-oo, 8, 2))
def test_range_range_intersection():
for a, b, r in [
(Range(0), Range(1), S.EmptySet),
(Range(3), Range(4, oo), S.EmptySet),
(Range(3), Range(-3, -1), S.EmptySet),
(Range(1, 3), Range(0, 3), Range(1, 3)),
(Range(1, 3), Range(1, 4), Range(1, 3)),
(Range(1, oo, 2), Range(2, oo, 2), S.EmptySet),
(Range(0, oo, 2), Range(oo), Range(0, oo, 2)),
(Range(0, oo, 2), Range(100), Range(0, 100, 2)),
(Range(2, oo, 2), Range(oo), Range(2, oo, 2)),
(Range(0, oo, 2), Range(5, 6), S.EmptySet),
(Range(2, 80, 1), Range(55, 71, 4), Range(55, 71, 4)),
(Range(0, 6, 3), Range(-oo, 5, 3), S.EmptySet),
(Range(0, oo, 2), Range(5, oo, 3), Range(8, oo, 6)),
(Range(4, 6, 2), Range(2, 16, 7), S.EmptySet),]:
assert a.intersect(b) == r
assert a.intersect(b.reversed) == r
assert a.reversed.intersect(b) == r
assert a.reversed.intersect(b.reversed) == r
a, b = b, a
assert a.intersect(b) == r
assert a.intersect(b.reversed) == r
assert a.reversed.intersect(b) == r
assert a.reversed.intersect(b.reversed) == r
def test_range_interval_intersection():
p = symbols('p', positive=True)
assert isinstance(Range(3).intersect(Interval(p, p + 2)), Intersection)
assert Range(4).intersect(Interval(0, 3)) == Range(4)
assert Range(4).intersect(Interval(-oo, oo)) == Range(4)
assert Range(4).intersect(Interval(1, oo)) == Range(1, 4)
assert Range(4).intersect(Interval(1.1, oo)) == Range(2, 4)
assert Range(4).intersect(Interval(0.1, 3)) == Range(1, 4)
assert Range(4).intersect(Interval(0.1, 3.1)) == Range(1, 4)
assert Range(4).intersect(Interval.open(0, 3)) == Range(1, 3)
assert Range(4).intersect(Interval.open(0.1, 0.5)) is S.EmptySet
# Null Range intersections
assert Range(0).intersect(Interval(0.2, 0.8)) is S.EmptySet
assert Range(0).intersect(Interval(-oo, oo)) is S.EmptySet
def test_range_is_finite_set():
assert Range(-100, 100).is_finite_set is True
assert Range(2, oo).is_finite_set is False
assert Range(-oo, 50).is_finite_set is False
assert Range(-oo, oo).is_finite_set is False
assert Range(oo, -oo).is_finite_set is True
assert Range(0, 0).is_finite_set is True
assert Range(oo, oo).is_finite_set is True
assert Range(-oo, -oo).is_finite_set is True
n = Symbol('n', integer=True)
m = Symbol('m', integer=True)
assert Range(n, n + 49).is_finite_set is True
assert Range(n, 0).is_finite_set is True
assert Range(-3, n + 7).is_finite_set is True
assert Range(n, m).is_finite_set is True
assert Range(n + m, m - n).is_finite_set is True
assert Range(n, n + m + n).is_finite_set is True
assert Range(n, oo).is_finite_set is False
assert Range(-oo, n).is_finite_set is False
# assert Range(n, -oo).is_finite_set is True
# assert Range(oo, n).is_finite_set is True
# Above tests fail due to a (potential) bug in sympy.sets.fancysets.Range.size (See issue #18999)
def test_Integers_eval_imageset():
ans = ImageSet(Lambda(x, 2*x + Rational(3, 7)), S.Integers)
im = imageset(Lambda(x, -2*x + Rational(3, 7)), S.Integers)
assert im == ans
im = imageset(Lambda(x, -2*x - Rational(11, 7)), S.Integers)
assert im == ans
y = Symbol('y')
L = imageset(x, 2*x + y, S.Integers)
assert y + 4 in L
_x = symbols('x', negative=True)
eq = _x**2 - _x + 1
assert imageset(_x, eq, S.Integers).lamda.expr == _x**2 + _x + 1
eq = 3*_x - 1
assert imageset(_x, eq, S.Integers).lamda.expr == 3*_x + 2
assert imageset(x, (x, 1/x), S.Integers) == \
ImageSet(Lambda(x, (x, 1/x)), S.Integers)
def test_Range_eval_imageset():
a, b, c = symbols('a b c')
assert imageset(x, a*(x + b) + c, Range(3)) == \
imageset(x, a*x + a*b + c, Range(3))
eq = (x + 1)**2
assert imageset(x, eq, Range(3)).lamda.expr == eq
eq = a*(x + b) + c
r = Range(3, -3, -2)
imset = imageset(x, eq, r)
assert imset.lamda.expr != eq
assert list(imset) == [eq.subs(x, i).expand() for i in list(r)]
def test_fun():
assert (FiniteSet(*ImageSet(Lambda(x, sin(pi*x/4)),
Range(-10, 11))) == FiniteSet(-1, -sqrt(2)/2, 0, sqrt(2)/2, 1))
def test_Reals():
assert 5 in S.Reals
assert S.Pi in S.Reals
assert -sqrt(2) in S.Reals
assert (2, 5) not in S.Reals
assert sqrt(-1) not in S.Reals
assert S.Reals == Interval(-oo, oo)
assert S.Reals != Interval(0, oo)
assert S.Reals.is_subset(Interval(-oo, oo))
assert S.Reals.intersect(Range(-oo, oo)) == Range(-oo, oo)
def test_Complex():
assert 5 in S.Complexes
assert 5 + 4*I in S.Complexes
assert S.Pi in S.Complexes
assert -sqrt(2) in S.Complexes
assert -I in S.Complexes
assert sqrt(-1) in S.Complexes
assert S.Complexes.intersect(S.Reals) == S.Reals
assert S.Complexes.union(S.Reals) == S.Complexes
assert S.Complexes == ComplexRegion(S.Reals*S.Reals)
assert (S.Complexes == ComplexRegion(Interval(1, 2)*Interval(3, 4))) == False
assert str(S.Complexes) == "S.Complexes"
assert repr(S.Complexes) == "S.Complexes"
def take(n, iterable):
"Return first n items of the iterable as a list"
return list(itertools.islice(iterable, n))
def test_intersections():
assert S.Integers.intersect(S.Reals) == S.Integers
assert 5 in S.Integers.intersect(S.Reals)
assert 5 in S.Integers.intersect(S.Reals)
assert -5 not in S.Naturals.intersect(S.Reals)
assert 5.5 not in S.Integers.intersect(S.Reals)
assert 5 in S.Integers.intersect(Interval(3, oo))
assert -5 in S.Integers.intersect(Interval(-oo, 3))
assert all(x.is_Integer
for x in take(10, S.Integers.intersect(Interval(3, oo)) ))
def test_infinitely_indexed_set_1():
from sympy.abc import n, m, t
assert imageset(Lambda(n, n), S.Integers) == imageset(Lambda(m, m), S.Integers)
assert imageset(Lambda(n, 2*n), S.Integers).intersect(
imageset(Lambda(m, 2*m + 1), S.Integers)) is S.EmptySet
assert imageset(Lambda(n, 2*n), S.Integers).intersect(
imageset(Lambda(n, 2*n + 1), S.Integers)) is S.EmptySet
assert imageset(Lambda(m, 2*m), S.Integers).intersect(
imageset(Lambda(n, 3*n), S.Integers)).dummy_eq(
ImageSet(Lambda(t, 6*t), S.Integers))
assert imageset(x, x/2 + Rational(1, 3), S.Integers).intersect(S.Integers) is S.EmptySet
assert imageset(x, x/2 + S.Half, S.Integers).intersect(S.Integers) is S.Integers
# https://github.com/sympy/sympy/issues/17355
S53 = ImageSet(Lambda(n, 5*n + 3), S.Integers)
assert S53.intersect(S.Integers) == S53
def test_infinitely_indexed_set_2():
from sympy.abc import n
a = Symbol('a', integer=True)
assert imageset(Lambda(n, n), S.Integers) == \
imageset(Lambda(n, n + a), S.Integers)
assert imageset(Lambda(n, n + pi), S.Integers) == \
imageset(Lambda(n, n + a + pi), S.Integers)
assert imageset(Lambda(n, n), S.Integers) == \
imageset(Lambda(n, -n + a), S.Integers)
assert imageset(Lambda(n, -6*n), S.Integers) == \
ImageSet(Lambda(n, 6*n), S.Integers)
assert imageset(Lambda(n, 2*n + pi), S.Integers) == \
ImageSet(Lambda(n, 2*n + pi - 2), S.Integers)
def test_imageset_intersect_real():
from sympy import I
from sympy.abc import n
assert imageset(Lambda(n, n + (n - 1)*(n + 1)*I), S.Integers).intersect(S.Reals) == \
FiniteSet(-1, 1)
s = ImageSet(
Lambda(n, -I*(I*(2*pi*n - pi/4) + log(Abs(sqrt(-I))))),
S.Integers)
# s is unevaluated, but after intersection the result
# should be canonical
assert s.intersect(S.Reals) == imageset(
Lambda(n, 2*n*pi - pi/4), S.Integers) == ImageSet(
Lambda(n, 2*pi*n + pi*Rational(7, 4)), S.Integers)
def test_imageset_intersect_interval():
from sympy.abc import n
f1 = ImageSet(Lambda(n, n*pi), S.Integers)
f2 = ImageSet(Lambda(n, 2*n), Interval(0, pi))
f3 = ImageSet(Lambda(n, 2*n*pi + pi/2), S.Integers)
# complex expressions
f4 = ImageSet(Lambda(n, n*I*pi), S.Integers)
f5 = ImageSet(Lambda(n, 2*I*n*pi + pi/2), S.Integers)
# non-linear expressions
f6 = ImageSet(Lambda(n, log(n)), S.Integers)
f7 = ImageSet(Lambda(n, n**2), S.Integers)
f8 = ImageSet(Lambda(n, Abs(n)), S.Integers)
f9 = ImageSet(Lambda(n, exp(n)), S.Naturals0)
assert f1.intersect(Interval(-1, 1)) == FiniteSet(0)
assert f1.intersect(Interval(0, 2*pi, False, True)) == FiniteSet(0, pi)
assert f2.intersect(Interval(1, 2)) == Interval(1, 2)
assert f3.intersect(Interval(-1, 1)) == S.EmptySet
assert f3.intersect(Interval(-5, 5)) == FiniteSet(pi*Rational(-3, 2), pi/2)
assert f4.intersect(Interval(-1, 1)) == FiniteSet(0)
assert f4.intersect(Interval(1, 2)) == S.EmptySet
assert f5.intersect(Interval(0, 1)) == S.EmptySet
assert f6.intersect(Interval(0, 1)) == FiniteSet(S.Zero, log(2))
assert f7.intersect(Interval(0, 10)) == Intersection(f7, Interval(0, 10))
assert f8.intersect(Interval(0, 2)) == Intersection(f8, Interval(0, 2))
assert f9.intersect(Interval(1, 2)) == Intersection(f9, Interval(1, 2))
def test_imageset_intersect_diophantine():
from sympy.abc import m, n
# Check that same lambda variable for both ImageSets is handled correctly
img1 = ImageSet(Lambda(n, 2*n + 1), S.Integers)
img2 = ImageSet(Lambda(n, 4*n + 1), S.Integers)
assert img1.intersect(img2) == img2
# Empty solution set returned by diophantine:
assert ImageSet(Lambda(n, 2*n), S.Integers).intersect(
ImageSet(Lambda(n, 2*n + 1), S.Integers)) == S.EmptySet
# Check intersection with S.Integers:
assert ImageSet(Lambda(n, 9/n + 20*n/3), S.Integers).intersect(
S.Integers) == FiniteSet(-61, -23, 23, 61)
# Single solution (2, 3) for diophantine solution:
assert ImageSet(Lambda(n, (n - 2)**2), S.Integers).intersect(
ImageSet(Lambda(n, -(n - 3)**2), S.Integers)) == FiniteSet(0)
# Single parametric solution for diophantine solution:
assert ImageSet(Lambda(n, n**2 + 5), S.Integers).intersect(
ImageSet(Lambda(m, 2*m), S.Integers)).dummy_eq(ImageSet(
Lambda(n, 4*n**2 + 4*n + 6), S.Integers))
# 4 non-parametric solution couples for dioph. equation:
assert ImageSet(Lambda(n, n**2 - 9), S.Integers).intersect(
ImageSet(Lambda(m, -m**2), S.Integers)) == FiniteSet(-9, 0)
# Double parametric solution for diophantine solution:
assert ImageSet(Lambda(m, m**2 + 40), S.Integers).intersect(
ImageSet(Lambda(n, 41*n), S.Integers)).dummy_eq(Intersection(
ImageSet(Lambda(m, m**2 + 40), S.Integers),
ImageSet(Lambda(n, 41*n), S.Integers)))
# Check that diophantine returns *all* (8) solutions (permute=True)
assert ImageSet(Lambda(n, n**4 - 2**4), S.Integers).intersect(
ImageSet(Lambda(m, -m**4 + 3**4), S.Integers)) == FiniteSet(0, 65)
assert ImageSet(Lambda(n, pi/12 + n*5*pi/12), S.Integers).intersect(
ImageSet(Lambda(n, 7*pi/12 + n*11*pi/12), S.Integers)).dummy_eq(ImageSet(
Lambda(n, 55*pi*n/12 + 17*pi/4), S.Integers))
# TypeError raised by diophantine (#18081)
assert ImageSet(Lambda(n, n*log(2)), S.Integers).intersection(
S.Integers).dummy_eq(Intersection(ImageSet(
Lambda(n, n*log(2)), S.Integers), S.Integers))
# NotImplementedError raised by diophantine (no solver for cubic_thue)
assert ImageSet(Lambda(n, n**3 + 1), S.Integers).intersect(
ImageSet(Lambda(n, n**3), S.Integers)).dummy_eq(Intersection(
ImageSet(Lambda(n, n**3 + 1), S.Integers),
ImageSet(Lambda(n, n**3), S.Integers)))
def test_infinitely_indexed_set_3():
from sympy.abc import n, m, t
assert imageset(Lambda(m, 2*pi*m), S.Integers).intersect(
imageset(Lambda(n, 3*pi*n), S.Integers)).dummy_eq(
ImageSet(Lambda(t, 6*pi*t), S.Integers))
assert imageset(Lambda(n, 2*n + 1), S.Integers) == \
imageset(Lambda(n, 2*n - 1), S.Integers)
assert imageset(Lambda(n, 3*n + 2), S.Integers) == \
imageset(Lambda(n, 3*n - 1), S.Integers)
def test_ImageSet_simplification():
from sympy.abc import n, m
assert imageset(Lambda(n, n), S.Integers) == S.Integers
assert imageset(Lambda(n, sin(n)),
imageset(Lambda(m, tan(m)), S.Integers)) == \
imageset(Lambda(m, sin(tan(m))), S.Integers)
assert imageset(n, 1 + 2*n, S.Naturals) == Range(3, oo, 2)
assert imageset(n, 1 + 2*n, S.Naturals0) == Range(1, oo, 2)
assert imageset(n, 1 - 2*n, S.Naturals) == Range(-1, -oo, -2)
def test_ImageSet_contains():
from sympy.abc import x
assert (2, S.Half) in imageset(x, (x, 1/x), S.Integers)
assert imageset(x, x + I*3, S.Integers).intersection(S.Reals) is S.EmptySet
i = Dummy(integer=True)
q = imageset(x, x + I*y, S.Integers).intersection(S.Reals)
assert q.subs(y, I*i).intersection(S.Integers) is S.Integers
q = imageset(x, x + I*y/x, S.Integers).intersection(S.Reals)
assert q.subs(y, 0) is S.Integers
assert q.subs(y, I*i*x).intersection(S.Integers) is S.Integers
z = cos(1)**2 + sin(1)**2 - 1
q = imageset(x, x + I*z, S.Integers).intersection(S.Reals)
assert q is not S.EmptySet
def test_ComplexRegion_contains():
r = Symbol('r', real=True)
# contains in ComplexRegion
a = Interval(2, 3)
b = Interval(4, 6)
c = Interval(7, 9)
c1 = ComplexRegion(a*b)
c2 = ComplexRegion(Union(a*b, c*a))
assert 2.5 + 4.5*I in c1
assert 2 + 4*I in c1
assert 3 + 4*I in c1
assert 8 + 2.5*I in c2
assert 2.5 + 6.1*I not in c1
assert 4.5 + 3.2*I not in c1
assert c1.contains(x) == Contains(x, c1, evaluate=False)
assert c1.contains(r) == False
assert c2.contains(x) == Contains(x, c2, evaluate=False)
assert c2.contains(r) == False
r1 = Interval(0, 1)
theta1 = Interval(0, 2*S.Pi)
c3 = ComplexRegion(r1*theta1, polar=True)
assert (0.5 + I*Rational(6, 10)) in c3
assert (S.Half + I*Rational(6, 10)) in c3
assert (S.Half + .6*I) in c3
assert (0.5 + .6*I) in c3
assert I in c3
assert 1 in c3
assert 0 in c3
assert 1 + I not in c3
assert 1 - I not in c3
assert c3.contains(x) == Contains(x, c3, evaluate=False)
assert c3.contains(r + 2*I) == Contains(
r + 2*I, c3, evaluate=False) # is in fact False
assert c3.contains(1/(1 + r**2)) == Contains(
1/(1 + r**2), c3, evaluate=False) # is in fact True
r2 = Interval(0, 3)
theta2 = Interval(pi, 2*pi, left_open=True)
c4 = ComplexRegion(r2*theta2, polar=True)
assert c4.contains(0) == True
assert c4.contains(2 + I) == False
assert c4.contains(-2 + I) == False
assert c4.contains(-2 - I) == True
assert c4.contains(2 - I) == True
assert c4.contains(-2) == False
assert c4.contains(2) == True
assert c4.contains(x) == Contains(x, c4, evaluate=False)
assert c4.contains(3/(1 + r**2)) == Contains(
3/(1 + r**2), c4, evaluate=False) # is in fact True
raises(ValueError, lambda: ComplexRegion(r1*theta1, polar=2))
def test_ComplexRegion_intersect():
# Polar form
X_axis = ComplexRegion(Interval(0, oo)*FiniteSet(0, S.Pi), polar=True)
unit_disk = ComplexRegion(Interval(0, 1)*Interval(0, 2*S.Pi), polar=True)
upper_half_unit_disk = ComplexRegion(Interval(0, 1)*Interval(0, S.Pi), polar=True)
upper_half_disk = ComplexRegion(Interval(0, oo)*Interval(0, S.Pi), polar=True)
lower_half_disk = ComplexRegion(Interval(0, oo)*Interval(S.Pi, 2*S.Pi), polar=True)
right_half_disk = ComplexRegion(Interval(0, oo)*Interval(-S.Pi/2, S.Pi/2), polar=True)
first_quad_disk = ComplexRegion(Interval(0, oo)*Interval(0, S.Pi/2), polar=True)
assert upper_half_disk.intersect(unit_disk) == upper_half_unit_disk
assert right_half_disk.intersect(first_quad_disk) == first_quad_disk
assert upper_half_disk.intersect(right_half_disk) == first_quad_disk
assert upper_half_disk.intersect(lower_half_disk) == X_axis
c1 = ComplexRegion(Interval(0, 4)*Interval(0, 2*S.Pi), polar=True)
assert c1.intersect(Interval(1, 5)) == Interval(1, 4)
assert c1.intersect(Interval(4, 9)) == FiniteSet(4)
assert c1.intersect(Interval(5, 12)) is S.EmptySet
# Rectangular form
X_axis = ComplexRegion(Interval(-oo, oo)*FiniteSet(0))
unit_square = ComplexRegion(Interval(-1, 1)*Interval(-1, 1))
upper_half_unit_square = ComplexRegion(Interval(-1, 1)*Interval(0, 1))
upper_half_plane = ComplexRegion(Interval(-oo, oo)*Interval(0, oo))
lower_half_plane = ComplexRegion(Interval(-oo, oo)*Interval(-oo, 0))
right_half_plane = ComplexRegion(Interval(0, oo)*Interval(-oo, oo))
first_quad_plane = ComplexRegion(Interval(0, oo)*Interval(0, oo))
assert upper_half_plane.intersect(unit_square) == upper_half_unit_square
assert right_half_plane.intersect(first_quad_plane) == first_quad_plane
assert upper_half_plane.intersect(right_half_plane) == first_quad_plane
assert upper_half_plane.intersect(lower_half_plane) == X_axis
c1 = ComplexRegion(Interval(-5, 5)*Interval(-10, 10))
assert c1.intersect(Interval(2, 7)) == Interval(2, 5)
assert c1.intersect(Interval(5, 7)) == FiniteSet(5)
assert c1.intersect(Interval(6, 9)) is S.EmptySet
# unevaluated object
C1 = ComplexRegion(Interval(0, 1)*Interval(0, 2*S.Pi), polar=True)
C2 = ComplexRegion(Interval(-1, 1)*Interval(-1, 1))
assert C1.intersect(C2) == Intersection(C1, C2, evaluate=False)
def test_ComplexRegion_union():
# Polar form
c1 = ComplexRegion(Interval(0, 1)*Interval(0, 2*S.Pi), polar=True)
c2 = ComplexRegion(Interval(0, 1)*Interval(0, S.Pi), polar=True)
c3 = ComplexRegion(Interval(0, oo)*Interval(0, S.Pi), polar=True)
c4 = ComplexRegion(Interval(0, oo)*Interval(S.Pi, 2*S.Pi), polar=True)
p1 = Union(Interval(0, 1)*Interval(0, 2*S.Pi), Interval(0, 1)*Interval(0, S.Pi))
p2 = Union(Interval(0, oo)*Interval(0, S.Pi), Interval(0, oo)*Interval(S.Pi, 2*S.Pi))
assert c1.union(c2) == ComplexRegion(p1, polar=True)
assert c3.union(c4) == ComplexRegion(p2, polar=True)
# Rectangular form
c5 = ComplexRegion(Interval(2, 5)*Interval(6, 9))
c6 = ComplexRegion(Interval(4, 6)*Interval(10, 12))
c7 = ComplexRegion(Interval(0, 10)*Interval(-10, 0))
c8 = ComplexRegion(Interval(12, 16)*Interval(14, 20))
p3 = Union(Interval(2, 5)*Interval(6, 9), Interval(4, 6)*Interval(10, 12))
p4 = Union(Interval(0, 10)*Interval(-10, 0), Interval(12, 16)*Interval(14, 20))
assert c5.union(c6) == ComplexRegion(p3)
assert c7.union(c8) == ComplexRegion(p4)
assert c1.union(Interval(2, 4)) == Union(c1, Interval(2, 4), evaluate=False)
assert c5.union(Interval(2, 4)) == Union(c5, ComplexRegion.from_real(Interval(2, 4)))
def test_ComplexRegion_from_real():
c1 = ComplexRegion(Interval(0, 1) * Interval(0, 2 * S.Pi), polar=True)
raises(ValueError, lambda: c1.from_real(c1))
assert c1.from_real(Interval(-1, 1)) == ComplexRegion(Interval(-1, 1) * FiniteSet(0), False)
def test_ComplexRegion_measure():
a, b = Interval(2, 5), Interval(4, 8)
theta1, theta2 = Interval(0, 2*S.Pi), Interval(0, S.Pi)
c1 = ComplexRegion(a*b)
c2 = ComplexRegion(Union(a*theta1, b*theta2), polar=True)
assert c1.measure == 12
assert c2.measure == 9*pi
def test_normalize_theta_set():
# Interval
assert normalize_theta_set(Interval(pi, 2*pi)) == \
Union(FiniteSet(0), Interval.Ropen(pi, 2*pi))
assert normalize_theta_set(Interval(pi*Rational(9, 2), 5*pi)) == Interval(pi/2, pi)
assert normalize_theta_set(Interval(pi*Rational(-3, 2), pi/2)) == Interval.Ropen(0, 2*pi)
assert normalize_theta_set(Interval.open(pi*Rational(-3, 2), pi/2)) == \
Union(Interval.Ropen(0, pi/2), Interval.open(pi/2, 2*pi))
assert normalize_theta_set(Interval.open(pi*Rational(-7, 2), pi*Rational(-3, 2))) == \
Union(Interval.Ropen(0, pi/2), Interval.open(pi/2, 2*pi))
assert normalize_theta_set(Interval(-pi/2, pi/2)) == \
Union(Interval(0, pi/2), Interval.Ropen(pi*Rational(3, 2), 2*pi))
assert normalize_theta_set(Interval.open(-pi/2, pi/2)) == \
Union(Interval.Ropen(0, pi/2), Interval.open(pi*Rational(3, 2), 2*pi))
assert normalize_theta_set(Interval(-4*pi, 3*pi)) == Interval.Ropen(0, 2*pi)
assert normalize_theta_set(Interval(pi*Rational(-3, 2), -pi/2)) == Interval(pi/2, pi*Rational(3, 2))
assert normalize_theta_set(Interval.open(0, 2*pi)) == Interval.open(0, 2*pi)
assert normalize_theta_set(Interval.Ropen(-pi/2, pi/2)) == \
Union(Interval.Ropen(0, pi/2), Interval.Ropen(pi*Rational(3, 2), 2*pi))
assert normalize_theta_set(Interval.Lopen(-pi/2, pi/2)) == \
Union(Interval(0, pi/2), Interval.open(pi*Rational(3, 2), 2*pi))
assert normalize_theta_set(Interval(-pi/2, pi/2)) == \
Union(Interval(0, pi/2), Interval.Ropen(pi*Rational(3, 2), 2*pi))
assert normalize_theta_set(Interval.open(4*pi, pi*Rational(9, 2))) == Interval.open(0, pi/2)
assert normalize_theta_set(Interval.Lopen(4*pi, pi*Rational(9, 2))) == Interval.Lopen(0, pi/2)
assert normalize_theta_set(Interval.Ropen(4*pi, pi*Rational(9, 2))) == Interval.Ropen(0, pi/2)
assert normalize_theta_set(Interval.open(3*pi, 5*pi)) == \
Union(Interval.Ropen(0, pi), Interval.open(pi, 2*pi))
# FiniteSet
assert normalize_theta_set(FiniteSet(0, pi, 3*pi)) == FiniteSet(0, pi)
assert normalize_theta_set(FiniteSet(0, pi/2, pi, 2*pi)) == FiniteSet(0, pi/2, pi)
assert normalize_theta_set(FiniteSet(0, -pi/2, -pi, -2*pi)) == FiniteSet(0, pi, pi*Rational(3, 2))
assert normalize_theta_set(FiniteSet(pi*Rational(-3, 2), pi/2)) == \
FiniteSet(pi/2)
assert normalize_theta_set(FiniteSet(2*pi)) == FiniteSet(0)
# Unions
assert normalize_theta_set(Union(Interval(0, pi/3), Interval(pi/2, pi))) == \
Union(Interval(0, pi/3), Interval(pi/2, pi))
assert normalize_theta_set(Union(Interval(0, pi), Interval(2*pi, pi*Rational(7, 3)))) == \
Interval(0, pi)
# ValueError for non-real sets
raises(ValueError, lambda: normalize_theta_set(S.Complexes))
# NotImplementedError for subset of reals
raises(NotImplementedError, lambda: normalize_theta_set(Interval(0, 1)))
# NotImplementedError without pi as coefficient
raises(NotImplementedError, lambda: normalize_theta_set(Interval(1, 2*pi)))
raises(NotImplementedError, lambda: normalize_theta_set(Interval(2*pi, 10)))
raises(NotImplementedError, lambda: normalize_theta_set(FiniteSet(0, 3, 3*pi)))
def test_ComplexRegion_FiniteSet():
x, y, z, a, b, c = symbols('x y z a b c')
# Issue #9669
assert ComplexRegion(FiniteSet(a, b, c)*FiniteSet(x, y, z)) == \
FiniteSet(a + I*x, a + I*y, a + I*z, b + I*x, b + I*y,
b + I*z, c + I*x, c + I*y, c + I*z)
assert ComplexRegion(FiniteSet(2)*FiniteSet(3)) == FiniteSet(2 + 3*I)
def test_union_RealSubSet():
assert (S.Complexes).union(Interval(1, 2)) == S.Complexes
assert (S.Complexes).union(S.Integers) == S.Complexes
def test_issue_9980():
c1 = ComplexRegion(Interval(1, 2)*Interval(2, 3))
c2 = ComplexRegion(Interval(1, 5)*Interval(1, 3))
R = Union(c1, c2)
assert simplify(R) == ComplexRegion(Union(Interval(1, 2)*Interval(2, 3), \
Interval(1, 5)*Interval(1, 3)), False)
assert c1.func(*c1.args) == c1
assert R.func(*R.args) == R
def test_issue_11732():
interval12 = Interval(1, 2)
finiteset1234 = FiniteSet(1, 2, 3, 4)
pointComplex = Tuple(1, 5)
assert (interval12 in S.Naturals) == False
assert (interval12 in S.Naturals0) == False
assert (interval12 in S.Integers) == False
assert (interval12 in S.Complexes) == False
assert (finiteset1234 in S.Naturals) == False
assert (finiteset1234 in S.Naturals0) == False
assert (finiteset1234 in S.Integers) == False
assert (finiteset1234 in S.Complexes) == False
assert (pointComplex in S.Naturals) == False
assert (pointComplex in S.Naturals0) == False
assert (pointComplex in S.Integers) == False
assert (pointComplex in S.Complexes) == True
def test_issue_11730():
unit = Interval(0, 1)
square = ComplexRegion(unit ** 2)
assert Union(S.Complexes, FiniteSet(oo)) != S.Complexes
assert Union(S.Complexes, FiniteSet(eye(4))) != S.Complexes
assert Union(unit, square) == square
assert Intersection(S.Reals, square) == unit
def test_issue_11938():
unit = Interval(0, 1)
ival = Interval(1, 2)
cr1 = ComplexRegion(ival * unit)
assert Intersection(cr1, S.Reals) == ival
assert Intersection(cr1, unit) == FiniteSet(1)
arg1 = Interval(0, S.Pi)
arg2 = FiniteSet(S.Pi)
arg3 = Interval(S.Pi / 4, 3 * S.Pi / 4)
cp1 = ComplexRegion(unit * arg1, polar=True)
cp2 = ComplexRegion(unit * arg2, polar=True)
cp3 = ComplexRegion(unit * arg3, polar=True)
assert Intersection(cp1, S.Reals) == Interval(-1, 1)
assert Intersection(cp2, S.Reals) == Interval(-1, 0)
assert Intersection(cp3, S.Reals) == FiniteSet(0)
def test_issue_11914():
a, b = Interval(0, 1), Interval(0, pi)
c, d = Interval(2, 3), Interval(pi, 3 * pi / 2)
cp1 = ComplexRegion(a * b, polar=True)
cp2 = ComplexRegion(c * d, polar=True)
assert -3 in cp1.union(cp2)
assert -3 in cp2.union(cp1)
assert -5 not in cp1.union(cp2)
def test_issue_9543():
assert ImageSet(Lambda(x, x**2), S.Naturals).is_subset(S.Reals)
def test_issue_16871():
assert ImageSet(Lambda(x, x), FiniteSet(1)) == {1}
assert ImageSet(Lambda(x, x - 3), S.Integers
).intersection(S.Integers) is S.Integers
@XFAIL
def test_issue_16871b():
assert ImageSet(Lambda(x, x - 3), S.Integers).is_subset(S.Integers)
def test_issue_18050():
assert imageset(Lambda(x, I*x + 1), S.Integers
) == ImageSet(Lambda(x, I*x + 1), S.Integers)
assert imageset(Lambda(x, 3*I*x + 4 + 8*I), S.Integers
) == ImageSet(Lambda(x, 3*I*x + 4 + 2*I), S.Integers)
# no 'Mod' for next 2 tests:
assert imageset(Lambda(x, 2*x + 3*I), S.Integers
) == ImageSet(Lambda(x, 2*x + 3*I), S.Integers)
r = Symbol('r', positive=True)
assert imageset(Lambda(x, r*x + 10), S.Integers
) == ImageSet(Lambda(x, r*x + 10), S.Integers)
# reduce real part:
assert imageset(Lambda(x, 3*x + 8 + 5*I), S.Integers
) == ImageSet(Lambda(x, 3*x + 2 + 5*I), S.Integers)
def test_Rationals():
assert S.Integers.is_subset(S.Rationals)
assert S.Naturals.is_subset(S.Rationals)
assert S.Naturals0.is_subset(S.Rationals)
assert S.Rationals.is_subset(S.Reals)
assert S.Rationals.inf is -oo
assert S.Rationals.sup is oo
it = iter(S.Rationals)
assert [next(it) for i in range(12)] == [
0, 1, -1, S.Half, 2, Rational(-1, 2), -2,
Rational(1, 3), 3, Rational(-1, 3), -3, Rational(2, 3)]
assert Basic() not in S.Rationals
assert S.Half in S.Rationals
assert 1.0 not in S.Rationals
assert 2 in S.Rationals
r = symbols('r', rational=True)
assert r in S.Rationals
raises(TypeError, lambda: x in S.Rationals)
# issue #18134:
assert S.Rationals.boundary == S.Reals
assert S.Rationals.closure == S.Reals
assert S.Rationals.is_open == False
assert S.Rationals.is_closed == False
def test_NZQRC_unions():
# check that all trivial number set unions are simplified:
nbrsets = (S.Naturals, S.Naturals0, S.Integers, S.Rationals,
S.Reals, S.Complexes)
unions = (Union(a, b) for a in nbrsets for b in nbrsets)
assert all(u.is_Union is False for u in unions)
def test_imageset_intersection():
n = Dummy()
s = ImageSet(Lambda(n, -I*(I*(2*pi*n - pi/4) +
log(Abs(sqrt(-I))))), S.Integers)
assert s.intersect(S.Reals) == ImageSet(
Lambda(n, 2*pi*n + pi*Rational(7, 4)), S.Integers)
def test_issue_17858():
assert 1 in Range(-oo, oo)
assert 0 in Range(oo, -oo, -1)
assert oo not in Range(-oo, oo)
assert -oo not in Range(-oo, oo)
def test_issue_17859():
r = Range(-oo,oo)
raises(ValueError,lambda: r[::2])
raises(ValueError, lambda: r[::-2])
r = Range(oo,-oo,-1)
raises(ValueError,lambda: r[::2])
raises(ValueError, lambda: r[::-2])
|
391165c9a1333d2ce77fc97254780bc0395d2e2b697cb9f3c376c7039358f08f | from sympy import (Symbol, Set, Union, Interval, oo, S, sympify, nan,
Max, Min, Float, DisjointUnion,
FiniteSet, Intersection, imageset, I, true, false, ProductSet,
sqrt, Complement, EmptySet, sin, cos, Lambda, ImageSet, pi,
Pow, Contains, Sum, rootof, SymmetricDifference, Piecewise,
Matrix, Range, Add, symbols, zoo, Rational)
from mpmath import mpi
from sympy.core.expr import unchanged
from sympy.core.relational import Eq, Ne, Le, Lt, LessThan
from sympy.logic import And, Or, Xor
from sympy.testing.pytest import raises, XFAIL, warns_deprecated_sympy
from sympy.abc import x, y, z, m, n
def test_imageset():
ints = S.Integers
assert imageset(x, x - 1, S.Naturals) is S.Naturals0
assert imageset(x, x + 1, S.Naturals0) is S.Naturals
assert imageset(x, abs(x), S.Naturals0) is S.Naturals0
assert imageset(x, abs(x), S.Naturals) is S.Naturals
assert imageset(x, abs(x), S.Integers) is S.Naturals0
# issue 16878a
r = symbols('r', real=True)
assert imageset(x, (x, x), S.Reals)._contains((1, r)) == None
assert imageset(x, (x, x), S.Reals)._contains((1, 2)) == False
assert (r, r) in imageset(x, (x, x), S.Reals)
assert 1 + I in imageset(x, x + I, S.Reals)
assert {1} not in imageset(x, (x,), S.Reals)
assert (1, 1) not in imageset(x, (x,) , S.Reals)
raises(TypeError, lambda: imageset(x, ints))
raises(ValueError, lambda: imageset(x, y, z, ints))
raises(ValueError, lambda: imageset(Lambda(x, cos(x)), y))
assert (1, 2) in imageset(Lambda((x, y), (x, y)), ints, ints)
raises(ValueError, lambda: imageset(Lambda(x, x), ints, ints))
assert imageset(cos, ints) == ImageSet(Lambda(x, cos(x)), ints)
def f(x):
return cos(x)
assert imageset(f, ints) == imageset(x, cos(x), ints)
f = lambda x: cos(x)
assert imageset(f, ints) == ImageSet(Lambda(x, cos(x)), ints)
assert imageset(x, 1, ints) == FiniteSet(1)
assert imageset(x, y, ints) == {y}
assert imageset((x, y), (1, z), ints, S.Reals) == {(1, z)}
clash = Symbol('x', integer=true)
assert (str(imageset(lambda x: x + clash, Interval(-2, 1)).lamda.expr)
in ('x0 + x', 'x + x0'))
x1, x2 = symbols("x1, x2")
assert imageset(lambda x, y:
Add(x, y), Interval(1, 2), Interval(2, 3)).dummy_eq(
ImageSet(Lambda((x1, x2), x1 + x2),
Interval(1, 2), Interval(2, 3)))
def test_is_empty():
for s in [S.Naturals, S.Naturals0, S.Integers, S.Rationals, S.Reals,
S.UniversalSet]:
assert s.is_empty is False
assert S.EmptySet.is_empty is True
def test_is_finiteset():
for s in [S.Naturals, S.Naturals0, S.Integers, S.Rationals, S.Reals,
S.UniversalSet]:
assert s.is_finite_set is False
assert S.EmptySet.is_finite_set is True
assert FiniteSet(1, 2).is_finite_set is True
assert Interval(1, 2).is_finite_set is False
assert Interval(x, y).is_finite_set is None
assert ProductSet(FiniteSet(1), FiniteSet(2)).is_finite_set is True
assert ProductSet(FiniteSet(1), Interval(1, 2)).is_finite_set is False
assert ProductSet(FiniteSet(1), Interval(x, y)).is_finite_set is None
assert Union(Interval(0, 1), Interval(2, 3)).is_finite_set is False
assert Union(FiniteSet(1), Interval(2, 3)).is_finite_set is False
assert Union(FiniteSet(1), FiniteSet(2)).is_finite_set is True
assert Union(FiniteSet(1), Interval(x, y)).is_finite_set is None
assert Intersection(Interval(x, y), FiniteSet(1)).is_finite_set is True
assert Intersection(Interval(x, y), Interval(1, 2)).is_finite_set is None
assert Intersection(FiniteSet(x), FiniteSet(y)).is_finite_set is True
assert Complement(FiniteSet(1), Interval(x, y)).is_finite_set is True
assert Complement(Interval(x, y), FiniteSet(1)).is_finite_set is None
assert Complement(Interval(1, 2), FiniteSet(x)).is_finite_set is False
assert DisjointUnion(Interval(-5, 3), FiniteSet(x, y)).is_finite_set is False
assert DisjointUnion(S.EmptySet, FiniteSet(x, y), S.EmptySet).is_finite_set is True
def test_deprecated_is_EmptySet():
with warns_deprecated_sympy():
S.EmptySet.is_EmptySet
def test_interval_arguments():
assert Interval(0, oo) == Interval(0, oo, False, True)
assert Interval(0, oo).right_open is true
assert Interval(-oo, 0) == Interval(-oo, 0, True, False)
assert Interval(-oo, 0).left_open is true
assert Interval(oo, -oo) == S.EmptySet
assert Interval(oo, oo) == S.EmptySet
assert Interval(-oo, -oo) == S.EmptySet
assert Interval(oo, x) == S.EmptySet
assert Interval(oo, oo) == S.EmptySet
assert Interval(x, -oo) == S.EmptySet
assert Interval(x, x) == {x}
assert isinstance(Interval(1, 1), FiniteSet)
e = Sum(x, (x, 1, 3))
assert isinstance(Interval(e, e), FiniteSet)
assert Interval(1, 0) == S.EmptySet
assert Interval(1, 1).measure == 0
assert Interval(1, 1, False, True) == S.EmptySet
assert Interval(1, 1, True, False) == S.EmptySet
assert Interval(1, 1, True, True) == S.EmptySet
assert isinstance(Interval(0, Symbol('a')), Interval)
assert Interval(Symbol('a', real=True, positive=True), 0) == S.EmptySet
raises(ValueError, lambda: Interval(0, S.ImaginaryUnit))
raises(ValueError, lambda: Interval(0, Symbol('z', extended_real=False)))
raises(ValueError, lambda: Interval(x, x + S.ImaginaryUnit))
raises(NotImplementedError, lambda: Interval(0, 1, And(x, y)))
raises(NotImplementedError, lambda: Interval(0, 1, False, And(x, y)))
raises(NotImplementedError, lambda: Interval(0, 1, z, And(x, y)))
def test_interval_symbolic_end_points():
a = Symbol('a', real=True)
assert Union(Interval(0, a), Interval(0, 3)).sup == Max(a, 3)
assert Union(Interval(a, 0), Interval(-3, 0)).inf == Min(-3, a)
assert Interval(0, a).contains(1) == LessThan(1, a)
def test_interval_is_empty():
x, y = symbols('x, y')
r = Symbol('r', real=True)
p = Symbol('p', positive=True)
n = Symbol('n', negative=True)
nn = Symbol('nn', nonnegative=True)
assert Interval(1, 2).is_empty == False
assert Interval(3, 3).is_empty == False # FiniteSet
assert Interval(r, r).is_empty == False # FiniteSet
assert Interval(r, r + nn).is_empty == False
assert Interval(x, x).is_empty == False
assert Interval(1, oo).is_empty == False
assert Interval(-oo, oo).is_empty == False
assert Interval(-oo, 1).is_empty == False
assert Interval(x, y).is_empty == None
assert Interval(r, oo).is_empty == False # real implies finite
assert Interval(n, 0).is_empty == False
assert Interval(n, 0, left_open=True).is_empty == False
assert Interval(p, 0).is_empty == True # EmptySet
assert Interval(nn, 0).is_empty == None
assert Interval(n, p).is_empty == False
assert Interval(0, p, left_open=True).is_empty == False
assert Interval(0, p, right_open=True).is_empty == False
assert Interval(0, nn, left_open=True).is_empty == None
assert Interval(0, nn, right_open=True).is_empty == None
def test_union():
assert Union(Interval(1, 2), Interval(2, 3)) == Interval(1, 3)
assert Union(Interval(1, 2), Interval(2, 3, True)) == Interval(1, 3)
assert Union(Interval(1, 3), Interval(2, 4)) == Interval(1, 4)
assert Union(Interval(1, 2), Interval(1, 3)) == Interval(1, 3)
assert Union(Interval(1, 3), Interval(1, 2)) == Interval(1, 3)
assert Union(Interval(1, 3, False, True), Interval(1, 2)) == \
Interval(1, 3, False, True)
assert Union(Interval(1, 3), Interval(1, 2, False, True)) == Interval(1, 3)
assert Union(Interval(1, 2, True), Interval(1, 3)) == Interval(1, 3)
assert Union(Interval(1, 2, True), Interval(1, 3, True)) == \
Interval(1, 3, True)
assert Union(Interval(1, 2, True), Interval(1, 3, True, True)) == \
Interval(1, 3, True, True)
assert Union(Interval(1, 2, True, True), Interval(1, 3, True)) == \
Interval(1, 3, True)
assert Union(Interval(1, 3), Interval(2, 3)) == Interval(1, 3)
assert Union(Interval(1, 3, False, True), Interval(2, 3)) == \
Interval(1, 3)
assert Union(Interval(1, 2, False, True), Interval(2, 3, True)) != \
Interval(1, 3)
assert Union(Interval(1, 2), S.EmptySet) == Interval(1, 2)
assert Union(S.EmptySet) == S.EmptySet
assert Union(Interval(0, 1), *[FiniteSet(1.0/n) for n in range(1, 10)]) == \
Interval(0, 1)
# issue #18241:
x = Symbol('x')
assert Union(Interval(0, 1), FiniteSet(1, x)) == Union(
Interval(0, 1), FiniteSet(x))
assert unchanged(Union, Interval(0, 1), FiniteSet(2, x))
assert Interval(1, 2).union(Interval(2, 3)) == \
Interval(1, 2) + Interval(2, 3)
assert Interval(1, 2).union(Interval(2, 3)) == Interval(1, 3)
assert Union(Set()) == Set()
assert FiniteSet(1) + FiniteSet(2) + FiniteSet(3) == FiniteSet(1, 2, 3)
assert FiniteSet('ham') + FiniteSet('eggs') == FiniteSet('ham', 'eggs')
assert FiniteSet(1, 2, 3) + S.EmptySet == FiniteSet(1, 2, 3)
assert FiniteSet(1, 2, 3) & FiniteSet(2, 3, 4) == FiniteSet(2, 3)
assert FiniteSet(1, 2, 3) | FiniteSet(2, 3, 4) == FiniteSet(1, 2, 3, 4)
assert FiniteSet(1, 2, 3) & S.EmptySet == S.EmptySet
assert FiniteSet(1, 2, 3) | S.EmptySet == FiniteSet(1, 2, 3)
x = Symbol("x")
y = Symbol("y")
z = Symbol("z")
assert S.EmptySet | FiniteSet(x, FiniteSet(y, z)) == \
FiniteSet(x, FiniteSet(y, z))
# Test that Intervals and FiniteSets play nicely
assert Interval(1, 3) + FiniteSet(2) == Interval(1, 3)
assert Interval(1, 3, True, True) + FiniteSet(3) == \
Interval(1, 3, True, False)
X = Interval(1, 3) + FiniteSet(5)
Y = Interval(1, 2) + FiniteSet(3)
XandY = X.intersect(Y)
assert 2 in X and 3 in X and 3 in XandY
assert XandY.is_subset(X) and XandY.is_subset(Y)
raises(TypeError, lambda: Union(1, 2, 3))
assert X.is_iterable is False
# issue 7843
assert Union(S.EmptySet, FiniteSet(-sqrt(-I), sqrt(-I))) == \
FiniteSet(-sqrt(-I), sqrt(-I))
assert Union(S.Reals, S.Integers) == S.Reals
def test_union_iter():
# Use Range because it is ordered
u = Union(Range(3), Range(5), Range(4), evaluate=False)
# Round robin
assert list(u) == [0, 0, 0, 1, 1, 1, 2, 2, 2, 3, 3, 4]
def test_union_is_empty():
assert (Interval(x, y) + FiniteSet(1)).is_empty == False
assert (Interval(x, y) + Interval(-x, y)).is_empty == None
def test_difference():
assert Interval(1, 3) - Interval(1, 2) == Interval(2, 3, True)
assert Interval(1, 3) - Interval(2, 3) == Interval(1, 2, False, True)
assert Interval(1, 3, True) - Interval(2, 3) == Interval(1, 2, True, True)
assert Interval(1, 3, True) - Interval(2, 3, True) == \
Interval(1, 2, True, False)
assert Interval(0, 2) - FiniteSet(1) == \
Union(Interval(0, 1, False, True), Interval(1, 2, True, False))
# issue #18119
assert S.Reals - FiniteSet(I) == S.Reals
assert S.Reals - FiniteSet(-I, I) == S.Reals
assert Interval(0, 10) - FiniteSet(-I, I) == Interval(0, 10)
assert Interval(0, 10) - FiniteSet(1, I) == Union(
Interval.Ropen(0, 1), Interval.Lopen(1, 10))
assert S.Reals - FiniteSet(1, 2 + I, x, y**2) == Complement(
Union(Interval.open(-oo, 1), Interval.open(1, oo)), FiniteSet(x, y**2),
evaluate=False)
assert FiniteSet(1, 2, 3) - FiniteSet(2) == FiniteSet(1, 3)
assert FiniteSet('ham', 'eggs') - FiniteSet('eggs') == FiniteSet('ham')
assert FiniteSet(1, 2, 3, 4) - Interval(2, 10, True, False) == \
FiniteSet(1, 2)
assert FiniteSet(1, 2, 3, 4) - S.EmptySet == FiniteSet(1, 2, 3, 4)
assert Union(Interval(0, 2), FiniteSet(2, 3, 4)) - Interval(1, 3) == \
Union(Interval(0, 1, False, True), FiniteSet(4))
assert -1 in S.Reals - S.Naturals
def test_Complement():
A = FiniteSet(1, 3, 4)
B = FiniteSet(3, 4)
C = Interval(1, 3)
D = Interval(1, 2)
assert Complement(A, B, evaluate=False).is_iterable is True
assert Complement(A, C, evaluate=False).is_iterable is True
assert Complement(C, D, evaluate=False).is_iterable is None
assert FiniteSet(*Complement(A, B, evaluate=False)) == FiniteSet(1)
assert FiniteSet(*Complement(A, C, evaluate=False)) == FiniteSet(4)
raises(TypeError, lambda: FiniteSet(*Complement(C, A, evaluate=False)))
assert Complement(Interval(1, 3), Interval(1, 2)) == Interval(2, 3, True)
assert Complement(FiniteSet(1, 3, 4), FiniteSet(3, 4)) == FiniteSet(1)
assert Complement(Union(Interval(0, 2), FiniteSet(2, 3, 4)),
Interval(1, 3)) == \
Union(Interval(0, 1, False, True), FiniteSet(4))
assert not 3 in Complement(Interval(0, 5), Interval(1, 4), evaluate=False)
assert -1 in Complement(S.Reals, S.Naturals, evaluate=False)
assert not 1 in Complement(S.Reals, S.Naturals, evaluate=False)
assert Complement(S.Integers, S.UniversalSet) == EmptySet
assert S.UniversalSet.complement(S.Integers) == EmptySet
assert (not 0 in S.Reals.intersect(S.Integers - FiniteSet(0)))
assert S.EmptySet - S.Integers == S.EmptySet
assert (S.Integers - FiniteSet(0)) - FiniteSet(1) == S.Integers - FiniteSet(0, 1)
assert S.Reals - Union(S.Naturals, FiniteSet(pi)) == \
Intersection(S.Reals - S.Naturals, S.Reals - FiniteSet(pi))
# issue 12712
assert Complement(FiniteSet(x, y, 2), Interval(-10, 10)) == \
Complement(FiniteSet(x, y), Interval(-10, 10))
A = FiniteSet(*symbols('a:c'))
B = FiniteSet(*symbols('d:f'))
assert unchanged(Complement, ProductSet(A, A), B)
A2 = ProductSet(A, A)
B3 = ProductSet(B, B, B)
assert A2 - B3 == A2
assert B3 - A2 == B3
def test_set_operations_nonsets():
'''Tests that e.g. FiniteSet(1) * 2 raises TypeError'''
ops = [
lambda a, b: a + b,
lambda a, b: a - b,
lambda a, b: a * b,
lambda a, b: a / b,
lambda a, b: a // b,
lambda a, b: a | b,
lambda a, b: a & b,
lambda a, b: a ^ b,
# FiniteSet(1) ** 2 gives a ProductSet
#lambda a, b: a ** b,
]
Sx = FiniteSet(x)
Sy = FiniteSet(y)
sets = [
{1},
FiniteSet(1),
Interval(1, 2),
Union(Sx, Interval(1, 2)),
Intersection(Sx, Sy),
Complement(Sx, Sy),
ProductSet(Sx, Sy),
S.EmptySet,
]
nums = [0, 1, 2, S(0), S(1), S(2)]
for si in sets:
for ni in nums:
for op in ops:
raises(TypeError, lambda : op(si, ni))
raises(TypeError, lambda : op(ni, si))
raises(TypeError, lambda: si ** object())
raises(TypeError, lambda: si ** {1})
def test_complement():
assert Interval(0, 1).complement(S.Reals) == \
Union(Interval(-oo, 0, True, True), Interval(1, oo, True, True))
assert Interval(0, 1, True, False).complement(S.Reals) == \
Union(Interval(-oo, 0, True, False), Interval(1, oo, True, True))
assert Interval(0, 1, False, True).complement(S.Reals) == \
Union(Interval(-oo, 0, True, True), Interval(1, oo, False, True))
assert Interval(0, 1, True, True).complement(S.Reals) == \
Union(Interval(-oo, 0, True, False), Interval(1, oo, False, True))
assert S.UniversalSet.complement(S.EmptySet) == S.EmptySet
assert S.UniversalSet.complement(S.Reals) == S.EmptySet
assert S.UniversalSet.complement(S.UniversalSet) == S.EmptySet
assert S.EmptySet.complement(S.Reals) == S.Reals
assert Union(Interval(0, 1), Interval(2, 3)).complement(S.Reals) == \
Union(Interval(-oo, 0, True, True), Interval(1, 2, True, True),
Interval(3, oo, True, True))
assert FiniteSet(0).complement(S.Reals) == \
Union(Interval(-oo, 0, True, True), Interval(0, oo, True, True))
assert (FiniteSet(5) + Interval(S.NegativeInfinity,
0)).complement(S.Reals) == \
Interval(0, 5, True, True) + Interval(5, S.Infinity, True, True)
assert FiniteSet(1, 2, 3).complement(S.Reals) == \
Interval(S.NegativeInfinity, 1, True, True) + \
Interval(1, 2, True, True) + Interval(2, 3, True, True) +\
Interval(3, S.Infinity, True, True)
assert FiniteSet(x).complement(S.Reals) == Complement(S.Reals, FiniteSet(x))
assert FiniteSet(0, x).complement(S.Reals) == Complement(Interval(-oo, 0, True, True) +
Interval(0, oo, True, True)
, FiniteSet(x), evaluate=False)
square = Interval(0, 1) * Interval(0, 1)
notsquare = square.complement(S.Reals*S.Reals)
assert all(pt in square for pt in [(0, 0), (.5, .5), (1, 0), (1, 1)])
assert not any(
pt in notsquare for pt in [(0, 0), (.5, .5), (1, 0), (1, 1)])
assert not any(pt in square for pt in [(-1, 0), (1.5, .5), (10, 10)])
assert all(pt in notsquare for pt in [(-1, 0), (1.5, .5), (10, 10)])
def test_intersect1():
assert all(S.Integers.intersection(i) is i for i in
(S.Naturals, S.Naturals0))
assert all(i.intersection(S.Integers) is i for i in
(S.Naturals, S.Naturals0))
s = S.Naturals0
assert S.Naturals.intersection(s) is S.Naturals
assert s.intersection(S.Naturals) is S.Naturals
x = Symbol('x')
assert Interval(0, 2).intersect(Interval(1, 2)) == Interval(1, 2)
assert Interval(0, 2).intersect(Interval(1, 2, True)) == \
Interval(1, 2, True)
assert Interval(0, 2, True).intersect(Interval(1, 2)) == \
Interval(1, 2, False, False)
assert Interval(0, 2, True, True).intersect(Interval(1, 2)) == \
Interval(1, 2, False, True)
assert Interval(0, 2).intersect(Union(Interval(0, 1), Interval(2, 3))) == \
Union(Interval(0, 1), Interval(2, 2))
assert FiniteSet(1, 2).intersect(FiniteSet(1, 2, 3)) == FiniteSet(1, 2)
assert FiniteSet(1, 2, x).intersect(FiniteSet(x)) == FiniteSet(x)
assert FiniteSet('ham', 'eggs').intersect(FiniteSet('ham')) == \
FiniteSet('ham')
assert FiniteSet(1, 2, 3, 4, 5).intersect(S.EmptySet) == S.EmptySet
assert Interval(0, 5).intersect(FiniteSet(1, 3)) == FiniteSet(1, 3)
assert Interval(0, 1, True, True).intersect(FiniteSet(1)) == S.EmptySet
assert Union(Interval(0, 1), Interval(2, 3)).intersect(Interval(1, 2)) == \
Union(Interval(1, 1), Interval(2, 2))
assert Union(Interval(0, 1), Interval(2, 3)).intersect(Interval(0, 2)) == \
Union(Interval(0, 1), Interval(2, 2))
assert Union(Interval(0, 1), Interval(2, 3)).intersect(Interval(1, 2, True, True)) == \
S.EmptySet
assert Union(Interval(0, 1), Interval(2, 3)).intersect(S.EmptySet) == \
S.EmptySet
assert Union(Interval(0, 5), FiniteSet('ham')).intersect(FiniteSet(2, 3, 4, 5, 6)) == \
Intersection(FiniteSet(2, 3, 4, 5, 6), Union(FiniteSet('ham'), Interval(0, 5)))
assert Intersection(FiniteSet(1, 2, 3), Interval(2, x), Interval(3, y)) == \
Intersection(FiniteSet(3), Interval(2, x), Interval(3, y), evaluate=False)
assert Intersection(FiniteSet(1, 2), Interval(0, 3), Interval(x, y)) == \
Intersection({1, 2}, Interval(x, y), evaluate=False)
assert Intersection(FiniteSet(1, 2, 4), Interval(0, 3), Interval(x, y)) == \
Intersection({1, 2}, Interval(x, y), evaluate=False)
# XXX: Is the real=True necessary here?
# https://github.com/sympy/sympy/issues/17532
m, n = symbols('m, n', real=True)
assert Intersection(FiniteSet(m), FiniteSet(m, n), Interval(m, m+1)) == \
FiniteSet(m)
# issue 8217
assert Intersection(FiniteSet(x), FiniteSet(y)) == \
Intersection(FiniteSet(x), FiniteSet(y), evaluate=False)
assert FiniteSet(x).intersect(S.Reals) == \
Intersection(S.Reals, FiniteSet(x), evaluate=False)
# tests for the intersection alias
assert Interval(0, 5).intersection(FiniteSet(1, 3)) == FiniteSet(1, 3)
assert Interval(0, 1, True, True).intersection(FiniteSet(1)) == S.EmptySet
assert Union(Interval(0, 1), Interval(2, 3)).intersection(Interval(1, 2)) == \
Union(Interval(1, 1), Interval(2, 2))
def test_intersection():
# iterable
i = Intersection(FiniteSet(1, 2, 3), Interval(2, 5), evaluate=False)
assert i.is_iterable
assert set(i) == {S(2), S(3)}
# challenging intervals
x = Symbol('x', real=True)
i = Intersection(Interval(0, 3), Interval(x, 6))
assert (5 in i) is False
raises(TypeError, lambda: 2 in i)
# Singleton special cases
assert Intersection(Interval(0, 1), S.EmptySet) == S.EmptySet
assert Intersection(Interval(-oo, oo), Interval(-oo, x)) == Interval(-oo, x)
# Products
line = Interval(0, 5)
i = Intersection(line**2, line**3, evaluate=False)
assert (2, 2) not in i
assert (2, 2, 2) not in i
raises(TypeError, lambda: list(i))
a = Intersection(Intersection(S.Integers, S.Naturals, evaluate=False), S.Reals, evaluate=False)
assert a._argset == frozenset([Intersection(S.Naturals, S.Integers, evaluate=False), S.Reals])
assert Intersection(S.Complexes, FiniteSet(S.ComplexInfinity)) == S.EmptySet
# issue 12178
assert Intersection() == S.UniversalSet
# issue 16987
assert Intersection({1}, {1}, {x}) == Intersection({1}, {x})
def test_issue_9623():
n = Symbol('n')
a = S.Reals
b = Interval(0, oo)
c = FiniteSet(n)
assert Intersection(a, b, c) == Intersection(b, c)
assert Intersection(Interval(1, 2), Interval(3, 4), FiniteSet(n)) == EmptySet
def test_is_disjoint():
assert Interval(0, 2).is_disjoint(Interval(1, 2)) == False
assert Interval(0, 2).is_disjoint(Interval(3, 4)) == True
def test_ProductSet__len__():
A = FiniteSet(1, 2)
B = FiniteSet(1, 2, 3)
assert ProductSet(A).__len__() == 2
assert ProductSet(A).__len__() is not S(2)
assert ProductSet(A, B).__len__() == 6
assert ProductSet(A, B).__len__() is not S(6)
def test_ProductSet():
# ProductSet is always a set of Tuples
assert ProductSet(S.Reals) == S.Reals ** 1
assert ProductSet(S.Reals, S.Reals) == S.Reals ** 2
assert ProductSet(S.Reals, S.Reals, S.Reals) == S.Reals ** 3
assert ProductSet(S.Reals) != S.Reals
assert ProductSet(S.Reals, S.Reals) == S.Reals * S.Reals
assert ProductSet(S.Reals, S.Reals, S.Reals) != S.Reals * S.Reals * S.Reals
assert ProductSet(S.Reals, S.Reals, S.Reals) == (S.Reals * S.Reals * S.Reals).flatten()
assert 1 not in ProductSet(S.Reals)
assert (1,) in ProductSet(S.Reals)
assert 1 not in ProductSet(S.Reals, S.Reals)
assert (1, 2) in ProductSet(S.Reals, S.Reals)
assert (1, I) not in ProductSet(S.Reals, S.Reals)
assert (1, 2, 3) in ProductSet(S.Reals, S.Reals, S.Reals)
assert (1, 2, 3) in S.Reals ** 3
assert (1, 2, 3) not in S.Reals * S.Reals * S.Reals
assert ((1, 2), 3) in S.Reals * S.Reals * S.Reals
assert (1, (2, 3)) not in S.Reals * S.Reals * S.Reals
assert (1, (2, 3)) in S.Reals * (S.Reals * S.Reals)
assert ProductSet() == FiniteSet(())
assert ProductSet(S.Reals, S.EmptySet) == S.EmptySet
# See GH-17458
for ni in range(5):
Rn = ProductSet(*(S.Reals,) * ni)
assert (1,) * ni in Rn
assert 1 not in Rn
assert (S.Reals * S.Reals) * S.Reals != S.Reals * (S.Reals * S.Reals)
S1 = S.Reals
S2 = S.Integers
x1 = pi
x2 = 3
assert x1 in S1
assert x2 in S2
assert (x1, x2) in S1 * S2
S3 = S1 * S2
x3 = (x1, x2)
assert x3 in S3
assert (x3, x3) in S3 * S3
assert x3 + x3 not in S3 * S3
raises(ValueError, lambda: S.Reals**-1)
with warns_deprecated_sympy():
ProductSet(FiniteSet(s) for s in range(2))
raises(TypeError, lambda: ProductSet(None))
S1 = FiniteSet(1, 2)
S2 = FiniteSet(3, 4)
S3 = ProductSet(S1, S2)
assert (S3.as_relational(x, y)
== And(S1.as_relational(x), S2.as_relational(y))
== And(Or(Eq(x, 1), Eq(x, 2)), Or(Eq(y, 3), Eq(y, 4))))
raises(ValueError, lambda: S3.as_relational(x))
raises(ValueError, lambda: S3.as_relational(x, 1))
raises(ValueError, lambda: ProductSet(Interval(0, 1)).as_relational(x, y))
Z2 = ProductSet(S.Integers, S.Integers)
assert Z2.contains((1, 2)) is S.true
assert Z2.contains((1,)) is S.false
assert Z2.contains(x) == Contains(x, Z2, evaluate=False)
assert Z2.contains(x).subs(x, 1) is S.false
assert Z2.contains((x, 1)).subs(x, 2) is S.true
assert Z2.contains((x, y)) == Contains((x, y), Z2, evaluate=False)
assert unchanged(Contains, (x, y), Z2)
assert Contains((1, 2), Z2) is S.true
def test_ProductSet_of_single_arg_is_not_arg():
assert unchanged(ProductSet, Interval(0, 1))
assert ProductSet(Interval(0, 1)) != Interval(0, 1)
def test_ProductSet_is_empty():
assert ProductSet(S.Integers, S.Reals).is_empty == False
assert ProductSet(Interval(x, 1), S.Reals).is_empty == None
def test_interval_subs():
a = Symbol('a', real=True)
assert Interval(0, a).subs(a, 2) == Interval(0, 2)
assert Interval(a, 0).subs(a, 2) == S.EmptySet
def test_interval_to_mpi():
assert Interval(0, 1).to_mpi() == mpi(0, 1)
assert Interval(0, 1, True, False).to_mpi() == mpi(0, 1)
assert type(Interval(0, 1).to_mpi()) == type(mpi(0, 1))
def test_set_evalf():
assert Interval(S(11)/64, S.Half).evalf() == Interval(
Float('0.171875'), Float('0.5'))
assert Interval(x, S.Half, right_open=True).evalf() == Interval(
x, Float('0.5'), right_open=True)
assert Interval(-oo, S.Half).evalf() == Interval(-oo, Float('0.5'))
assert FiniteSet(2, x).evalf() == FiniteSet(Float('2.0'), x)
def test_measure():
a = Symbol('a', real=True)
assert Interval(1, 3).measure == 2
assert Interval(0, a).measure == a
assert Interval(1, a).measure == a - 1
assert Union(Interval(1, 2), Interval(3, 4)).measure == 2
assert Union(Interval(1, 2), Interval(3, 4), FiniteSet(5, 6, 7)).measure \
== 2
assert FiniteSet(1, 2, oo, a, -oo, -5).measure == 0
assert S.EmptySet.measure == 0
square = Interval(0, 10) * Interval(0, 10)
offsetsquare = Interval(5, 15) * Interval(5, 15)
band = Interval(-oo, oo) * Interval(2, 4)
assert square.measure == offsetsquare.measure == 100
assert (square + offsetsquare).measure == 175 # there is some overlap
assert (square - offsetsquare).measure == 75
assert (square * FiniteSet(1, 2, 3)).measure == 0
assert (square.intersect(band)).measure == 20
assert (square + band).measure is oo
assert (band * FiniteSet(1, 2, 3)).measure is nan
def test_is_subset():
assert Interval(0, 1).is_subset(Interval(0, 2)) is True
assert Interval(0, 3).is_subset(Interval(0, 2)) is False
assert Interval(0, 1).is_subset(FiniteSet(0, 1)) is False
assert FiniteSet(1, 2).is_subset(FiniteSet(1, 2, 3, 4))
assert FiniteSet(4, 5).is_subset(FiniteSet(1, 2, 3, 4)) is False
assert FiniteSet(1).is_subset(Interval(0, 2))
assert FiniteSet(1, 2).is_subset(Interval(0, 2, True, True)) is False
assert (Interval(1, 2) + FiniteSet(3)).is_subset(
(Interval(0, 2, False, True) + FiniteSet(2, 3)))
assert Interval(3, 4).is_subset(Union(Interval(0, 1), Interval(2, 5))) is True
assert Interval(3, 6).is_subset(Union(Interval(0, 1), Interval(2, 5))) is False
assert FiniteSet(1, 2, 3, 4).is_subset(Interval(0, 5)) is True
assert S.EmptySet.is_subset(FiniteSet(1, 2, 3)) is True
assert Interval(0, 1).is_subset(S.EmptySet) is False
assert S.EmptySet.is_subset(S.EmptySet) is True
raises(ValueError, lambda: S.EmptySet.is_subset(1))
# tests for the issubset alias
assert FiniteSet(1, 2, 3, 4).issubset(Interval(0, 5)) is True
assert S.EmptySet.issubset(FiniteSet(1, 2, 3)) is True
assert S.Naturals.is_subset(S.Integers)
assert S.Naturals0.is_subset(S.Integers)
assert FiniteSet(x).is_subset(FiniteSet(y)) is None
assert FiniteSet(x).is_subset(FiniteSet(y).subs(y, x)) is True
assert FiniteSet(x).is_subset(FiniteSet(y).subs(y, x+1)) is False
assert Interval(0, 1).is_subset(Interval(0, 1, left_open=True)) is False
assert Interval(-2, 3).is_subset(Union(Interval(-oo, -2), Interval(3, oo))) is False
n = Symbol('n', integer=True)
assert Range(-3, 4, 1).is_subset(FiniteSet(-10, 10)) is False
assert Range(S(10)**100).is_subset(FiniteSet(0, 1, 2)) is False
assert Range(6, 0, -2).is_subset(FiniteSet(2, 4, 6)) is True
assert Range(1, oo).is_subset(FiniteSet(1, 2)) is False
assert Range(-oo, 1).is_subset(FiniteSet(1)) is False
assert Range(3).is_subset(FiniteSet(0, 1, n)) is None
assert Range(n, n + 2).is_subset(FiniteSet(n, n + 1)) is True
assert Range(5).is_subset(Interval(0, 4, right_open=True)) is False
def test_is_proper_subset():
assert Interval(0, 1).is_proper_subset(Interval(0, 2)) is True
assert Interval(0, 3).is_proper_subset(Interval(0, 2)) is False
assert S.EmptySet.is_proper_subset(FiniteSet(1, 2, 3)) is True
raises(ValueError, lambda: Interval(0, 1).is_proper_subset(0))
def test_is_superset():
assert Interval(0, 1).is_superset(Interval(0, 2)) == False
assert Interval(0, 3).is_superset(Interval(0, 2))
assert FiniteSet(1, 2).is_superset(FiniteSet(1, 2, 3, 4)) == False
assert FiniteSet(4, 5).is_superset(FiniteSet(1, 2, 3, 4)) == False
assert FiniteSet(1).is_superset(Interval(0, 2)) == False
assert FiniteSet(1, 2).is_superset(Interval(0, 2, True, True)) == False
assert (Interval(1, 2) + FiniteSet(3)).is_superset(
(Interval(0, 2, False, True) + FiniteSet(2, 3))) == False
assert Interval(3, 4).is_superset(Union(Interval(0, 1), Interval(2, 5))) == False
assert FiniteSet(1, 2, 3, 4).is_superset(Interval(0, 5)) == False
assert S.EmptySet.is_superset(FiniteSet(1, 2, 3)) == False
assert Interval(0, 1).is_superset(S.EmptySet) == True
assert S.EmptySet.is_superset(S.EmptySet) == True
raises(ValueError, lambda: S.EmptySet.is_superset(1))
# tests for the issuperset alias
assert Interval(0, 1).issuperset(S.EmptySet) == True
assert S.EmptySet.issuperset(S.EmptySet) == True
def test_is_proper_superset():
assert Interval(0, 1).is_proper_superset(Interval(0, 2)) is False
assert Interval(0, 3).is_proper_superset(Interval(0, 2)) is True
assert FiniteSet(1, 2, 3).is_proper_superset(S.EmptySet) is True
raises(ValueError, lambda: Interval(0, 1).is_proper_superset(0))
def test_contains():
assert Interval(0, 2).contains(1) is S.true
assert Interval(0, 2).contains(3) is S.false
assert Interval(0, 2, True, False).contains(0) is S.false
assert Interval(0, 2, True, False).contains(2) is S.true
assert Interval(0, 2, False, True).contains(0) is S.true
assert Interval(0, 2, False, True).contains(2) is S.false
assert Interval(0, 2, True, True).contains(0) is S.false
assert Interval(0, 2, True, True).contains(2) is S.false
assert (Interval(0, 2) in Interval(0, 2)) is False
assert FiniteSet(1, 2, 3).contains(2) is S.true
assert FiniteSet(1, 2, Symbol('x')).contains(Symbol('x')) is S.true
assert FiniteSet(y)._contains(x) is None
raises(TypeError, lambda: x in FiniteSet(y))
assert FiniteSet({x, y})._contains({x}) is None
assert FiniteSet({x, y}).subs(y, x)._contains({x}) is True
assert FiniteSet({x, y}).subs(y, x+1)._contains({x}) is False
# issue 8197
from sympy.abc import a, b
assert isinstance(FiniteSet(b).contains(-a), Contains)
assert isinstance(FiniteSet(b).contains(a), Contains)
assert isinstance(FiniteSet(a).contains(1), Contains)
raises(TypeError, lambda: 1 in FiniteSet(a))
# issue 8209
rad1 = Pow(Pow(2, Rational(1, 3)) - 1, Rational(1, 3))
rad2 = Pow(Rational(1, 9), Rational(1, 3)) - Pow(Rational(2, 9), Rational(1, 3)) + Pow(Rational(4, 9), Rational(1, 3))
s1 = FiniteSet(rad1)
s2 = FiniteSet(rad2)
assert s1 - s2 == S.EmptySet
items = [1, 2, S.Infinity, S('ham'), -1.1]
fset = FiniteSet(*items)
assert all(item in fset for item in items)
assert all(fset.contains(item) is S.true for item in items)
assert Union(Interval(0, 1), Interval(2, 5)).contains(3) is S.true
assert Union(Interval(0, 1), Interval(2, 5)).contains(6) is S.false
assert Union(Interval(0, 1), FiniteSet(2, 5)).contains(3) is S.false
assert S.EmptySet.contains(1) is S.false
assert FiniteSet(rootof(x**3 + x - 1, 0)).contains(S.Infinity) is S.false
assert rootof(x**5 + x**3 + 1, 0) in S.Reals
assert not rootof(x**5 + x**3 + 1, 1) in S.Reals
# non-bool results
assert Union(Interval(1, 2), Interval(3, 4)).contains(x) == \
Or(And(S.One <= x, x <= 2), And(S(3) <= x, x <= 4))
assert Intersection(Interval(1, x), Interval(2, 3)).contains(y) == \
And(y <= 3, y <= x, S.One <= y, S(2) <= y)
assert (S.Complexes).contains(S.ComplexInfinity) == S.false
def test_interval_symbolic():
x = Symbol('x')
e = Interval(0, 1)
assert e.contains(x) == And(S.Zero <= x, x <= 1)
raises(TypeError, lambda: x in e)
e = Interval(0, 1, True, True)
assert e.contains(x) == And(S.Zero < x, x < 1)
c = Symbol('c', real=False)
assert Interval(x, x + 1).contains(c) == False
e = Symbol('e', extended_real=True)
assert Interval(-oo, oo).contains(e) == And(
S.NegativeInfinity < e, e < S.Infinity)
def test_union_contains():
x = Symbol('x')
i1 = Interval(0, 1)
i2 = Interval(2, 3)
i3 = Union(i1, i2)
assert i3.as_relational(x) == Or(And(S.Zero <= x, x <= 1), And(S(2) <= x, x <= 3))
raises(TypeError, lambda: x in i3)
e = i3.contains(x)
assert e == i3.as_relational(x)
assert e.subs(x, -0.5) is false
assert e.subs(x, 0.5) is true
assert e.subs(x, 1.5) is false
assert e.subs(x, 2.5) is true
assert e.subs(x, 3.5) is false
U = Interval(0, 2, True, True) + Interval(10, oo) + FiniteSet(-1, 2, 5, 6)
assert all(el not in U for el in [0, 4, -oo])
assert all(el in U for el in [2, 5, 10])
def test_is_number():
assert Interval(0, 1).is_number is False
assert Set().is_number is False
def test_Interval_is_left_unbounded():
assert Interval(3, 4).is_left_unbounded is False
assert Interval(-oo, 3).is_left_unbounded is True
assert Interval(Float("-inf"), 3).is_left_unbounded is True
def test_Interval_is_right_unbounded():
assert Interval(3, 4).is_right_unbounded is False
assert Interval(3, oo).is_right_unbounded is True
assert Interval(3, Float("+inf")).is_right_unbounded is True
def test_Interval_as_relational():
x = Symbol('x')
assert Interval(-1, 2, False, False).as_relational(x) == \
And(Le(-1, x), Le(x, 2))
assert Interval(-1, 2, True, False).as_relational(x) == \
And(Lt(-1, x), Le(x, 2))
assert Interval(-1, 2, False, True).as_relational(x) == \
And(Le(-1, x), Lt(x, 2))
assert Interval(-1, 2, True, True).as_relational(x) == \
And(Lt(-1, x), Lt(x, 2))
assert Interval(-oo, 2, right_open=False).as_relational(x) == And(Lt(-oo, x), Le(x, 2))
assert Interval(-oo, 2, right_open=True).as_relational(x) == And(Lt(-oo, x), Lt(x, 2))
assert Interval(-2, oo, left_open=False).as_relational(x) == And(Le(-2, x), Lt(x, oo))
assert Interval(-2, oo, left_open=True).as_relational(x) == And(Lt(-2, x), Lt(x, oo))
assert Interval(-oo, oo).as_relational(x) == And(Lt(-oo, x), Lt(x, oo))
x = Symbol('x', real=True)
y = Symbol('y', real=True)
assert Interval(x, y).as_relational(x) == (x <= y)
assert Interval(y, x).as_relational(x) == (y <= x)
def test_Finite_as_relational():
x = Symbol('x')
y = Symbol('y')
assert FiniteSet(1, 2).as_relational(x) == Or(Eq(x, 1), Eq(x, 2))
assert FiniteSet(y, -5).as_relational(x) == Or(Eq(x, y), Eq(x, -5))
def test_Union_as_relational():
x = Symbol('x')
assert (Interval(0, 1) + FiniteSet(2)).as_relational(x) == \
Or(And(Le(0, x), Le(x, 1)), Eq(x, 2))
assert (Interval(0, 1, True, True) + FiniteSet(1)).as_relational(x) == \
And(Lt(0, x), Le(x, 1))
def test_Intersection_as_relational():
x = Symbol('x')
assert (Intersection(Interval(0, 1), FiniteSet(2),
evaluate=False).as_relational(x)
== And(And(Le(0, x), Le(x, 1)), Eq(x, 2)))
def test_Complement_as_relational():
x = Symbol('x')
expr = Complement(Interval(0, 1), FiniteSet(2), evaluate=False)
assert expr.as_relational(x) == \
And(Le(0, x), Le(x, 1), Ne(x, 2))
@XFAIL
def test_Complement_as_relational_fail():
x = Symbol('x')
expr = Complement(Interval(0, 1), FiniteSet(2), evaluate=False)
# XXX This example fails because 0 <= x changes to x >= 0
# during the evaluation.
assert expr.as_relational(x) == \
(0 <= x) & (x <= 1) & Ne(x, 2)
def test_SymmetricDifference_as_relational():
x = Symbol('x')
expr = SymmetricDifference(Interval(0, 1), FiniteSet(2), evaluate=False)
assert expr.as_relational(x) == Xor(Eq(x, 2), Le(0, x) & Le(x, 1))
def test_EmptySet():
assert S.EmptySet.as_relational(Symbol('x')) is S.false
assert S.EmptySet.intersect(S.UniversalSet) == S.EmptySet
assert S.EmptySet.boundary == S.EmptySet
def test_finite_basic():
x = Symbol('x')
A = FiniteSet(1, 2, 3)
B = FiniteSet(3, 4, 5)
AorB = Union(A, B)
AandB = A.intersect(B)
assert A.is_subset(AorB) and B.is_subset(AorB)
assert AandB.is_subset(A)
assert AandB == FiniteSet(3)
assert A.inf == 1 and A.sup == 3
assert AorB.inf == 1 and AorB.sup == 5
assert FiniteSet(x, 1, 5).sup == Max(x, 5)
assert FiniteSet(x, 1, 5).inf == Min(x, 1)
# issue 7335
assert FiniteSet(S.EmptySet) != S.EmptySet
assert FiniteSet(FiniteSet(1, 2, 3)) != FiniteSet(1, 2, 3)
assert FiniteSet((1, 2, 3)) != FiniteSet(1, 2, 3)
# Ensure a variety of types can exist in a FiniteSet
assert FiniteSet((1, 2), Float, A, -5, x, 'eggs', x**2, Interval)
assert (A > B) is False
assert (A >= B) is False
assert (A < B) is False
assert (A <= B) is False
assert AorB > A and AorB > B
assert AorB >= A and AorB >= B
assert A >= A and A <= A
assert A >= AandB and B >= AandB
assert A > AandB and B > AandB
assert FiniteSet(1.0) == FiniteSet(1)
def test_product_basic():
H, T = 'H', 'T'
unit_line = Interval(0, 1)
d6 = FiniteSet(1, 2, 3, 4, 5, 6)
d4 = FiniteSet(1, 2, 3, 4)
coin = FiniteSet(H, T)
square = unit_line * unit_line
assert (0, 0) in square
assert 0 not in square
assert (H, T) in coin ** 2
assert (.5, .5, .5) in (square * unit_line).flatten()
assert ((.5, .5), .5) in square * unit_line
assert (H, 3, 3) in (coin * d6 * d6).flatten()
assert ((H, 3), 3) in coin * d6 * d6
HH, TT = sympify(H), sympify(T)
assert set(coin**2) == set(((HH, HH), (HH, TT), (TT, HH), (TT, TT)))
assert (d4*d4).is_subset(d6*d6)
assert square.complement(Interval(-oo, oo)*Interval(-oo, oo)) == Union(
(Interval(-oo, 0, True, True) +
Interval(1, oo, True, True))*Interval(-oo, oo),
Interval(-oo, oo)*(Interval(-oo, 0, True, True) +
Interval(1, oo, True, True)))
assert (Interval(-5, 5)**3).is_subset(Interval(-10, 10)**3)
assert not (Interval(-10, 10)**3).is_subset(Interval(-5, 5)**3)
assert not (Interval(-5, 5)**2).is_subset(Interval(-10, 10)**3)
assert (Interval(.2, .5)*FiniteSet(.5)).is_subset(square) # segment in square
assert len(coin*coin*coin) == 8
assert len(S.EmptySet*S.EmptySet) == 0
assert len(S.EmptySet*coin) == 0
raises(TypeError, lambda: len(coin*Interval(0, 2)))
def test_real():
x = Symbol('x', real=True, finite=True)
I = Interval(0, 5)
J = Interval(10, 20)
A = FiniteSet(1, 2, 30, x, S.Pi)
B = FiniteSet(-4, 0)
C = FiniteSet(100)
D = FiniteSet('Ham', 'Eggs')
assert all(s.is_subset(S.Reals) for s in [I, J, A, B, C])
assert not D.is_subset(S.Reals)
assert all((a + b).is_subset(S.Reals) for a in [I, J, A, B, C] for b in [I, J, A, B, C])
assert not any((a + D).is_subset(S.Reals) for a in [I, J, A, B, C, D])
assert not (I + A + D).is_subset(S.Reals)
def test_supinf():
x = Symbol('x', real=True)
y = Symbol('y', real=True)
assert (Interval(0, 1) + FiniteSet(2)).sup == 2
assert (Interval(0, 1) + FiniteSet(2)).inf == 0
assert (Interval(0, 1) + FiniteSet(x)).sup == Max(1, x)
assert (Interval(0, 1) + FiniteSet(x)).inf == Min(0, x)
assert FiniteSet(5, 1, x).sup == Max(5, x)
assert FiniteSet(5, 1, x).inf == Min(1, x)
assert FiniteSet(5, 1, x, y).sup == Max(5, x, y)
assert FiniteSet(5, 1, x, y).inf == Min(1, x, y)
assert FiniteSet(5, 1, x, y, S.Infinity, S.NegativeInfinity).sup == \
S.Infinity
assert FiniteSet(5, 1, x, y, S.Infinity, S.NegativeInfinity).inf == \
S.NegativeInfinity
assert FiniteSet('Ham', 'Eggs').sup == Max('Ham', 'Eggs')
def test_universalset():
U = S.UniversalSet
x = Symbol('x')
assert U.as_relational(x) is S.true
assert U.union(Interval(2, 4)) == U
assert U.intersect(Interval(2, 4)) == Interval(2, 4)
assert U.measure is S.Infinity
assert U.boundary == S.EmptySet
assert U.contains(0) is S.true
def test_Union_of_ProductSets_shares():
line = Interval(0, 2)
points = FiniteSet(0, 1, 2)
assert Union(line * line, line * points) == line * line
def test_Interval_free_symbols():
# issue 6211
assert Interval(0, 1).free_symbols == set()
x = Symbol('x', real=True)
assert Interval(0, x).free_symbols == {x}
def test_image_interval():
from sympy.core.numbers import Rational
x = Symbol('x', real=True)
a = Symbol('a', real=True)
assert imageset(x, 2*x, Interval(-2, 1)) == Interval(-4, 2)
assert imageset(x, 2*x, Interval(-2, 1, True, False)) == \
Interval(-4, 2, True, False)
assert imageset(x, x**2, Interval(-2, 1, True, False)) == \
Interval(0, 4, False, True)
assert imageset(x, x**2, Interval(-2, 1)) == Interval(0, 4)
assert imageset(x, x**2, Interval(-2, 1, True, False)) == \
Interval(0, 4, False, True)
assert imageset(x, x**2, Interval(-2, 1, True, True)) == \
Interval(0, 4, False, True)
assert imageset(x, (x - 2)**2, Interval(1, 3)) == Interval(0, 1)
assert imageset(x, 3*x**4 - 26*x**3 + 78*x**2 - 90*x, Interval(0, 4)) == \
Interval(-35, 0) # Multiple Maxima
assert imageset(x, x + 1/x, Interval(-oo, oo)) == Interval(-oo, -2) \
+ Interval(2, oo) # Single Infinite discontinuity
assert imageset(x, 1/x + 1/(x-1)**2, Interval(0, 2, True, False)) == \
Interval(Rational(3, 2), oo, False) # Multiple Infinite discontinuities
# Test for Python lambda
assert imageset(lambda x: 2*x, Interval(-2, 1)) == Interval(-4, 2)
assert imageset(Lambda(x, a*x), Interval(0, 1)) == \
ImageSet(Lambda(x, a*x), Interval(0, 1))
assert imageset(Lambda(x, sin(cos(x))), Interval(0, 1)) == \
ImageSet(Lambda(x, sin(cos(x))), Interval(0, 1))
def test_image_piecewise():
f = Piecewise((x, x <= -1), (1/x**2, x <= 5), (x**3, True))
f1 = Piecewise((0, x <= 1), (1, x <= 2), (2, True))
assert imageset(x, f, Interval(-5, 5)) == Union(Interval(-5, -1), Interval(Rational(1, 25), oo))
assert imageset(x, f1, Interval(1, 2)) == FiniteSet(0, 1)
@XFAIL # See: https://github.com/sympy/sympy/pull/2723#discussion_r8659826
def test_image_Intersection():
x = Symbol('x', real=True)
y = Symbol('y', real=True)
assert imageset(x, x**2, Interval(-2, 0).intersect(Interval(x, y))) == \
Interval(0, 4).intersect(Interval(Min(x**2, y**2), Max(x**2, y**2)))
def test_image_FiniteSet():
x = Symbol('x', real=True)
assert imageset(x, 2*x, FiniteSet(1, 2, 3)) == FiniteSet(2, 4, 6)
def test_image_Union():
x = Symbol('x', real=True)
assert imageset(x, x**2, Interval(-2, 0) + FiniteSet(1, 2, 3)) == \
(Interval(0, 4) + FiniteSet(9))
def test_image_EmptySet():
x = Symbol('x', real=True)
assert imageset(x, 2*x, S.EmptySet) == S.EmptySet
def test_issue_5724_7680():
assert I not in S.Reals # issue 7680
assert Interval(-oo, oo).contains(I) is S.false
def test_boundary():
assert FiniteSet(1).boundary == FiniteSet(1)
assert all(Interval(0, 1, left_open, right_open).boundary == FiniteSet(0, 1)
for left_open in (true, false) for right_open in (true, false))
def test_boundary_Union():
assert (Interval(0, 1) + Interval(2, 3)).boundary == FiniteSet(0, 1, 2, 3)
assert ((Interval(0, 1, False, True)
+ Interval(1, 2, True, False)).boundary == FiniteSet(0, 1, 2))
assert (Interval(0, 1) + FiniteSet(2)).boundary == FiniteSet(0, 1, 2)
assert Union(Interval(0, 10), Interval(5, 15), evaluate=False).boundary \
== FiniteSet(0, 15)
assert Union(Interval(0, 10), Interval(0, 1), evaluate=False).boundary \
== FiniteSet(0, 10)
assert Union(Interval(0, 10, True, True),
Interval(10, 15, True, True), evaluate=False).boundary \
== FiniteSet(0, 10, 15)
@XFAIL
def test_union_boundary_of_joining_sets():
""" Testing the boundary of unions is a hard problem """
assert Union(Interval(0, 10), Interval(10, 15), evaluate=False).boundary \
== FiniteSet(0, 15)
def test_boundary_ProductSet():
open_square = Interval(0, 1, True, True) ** 2
assert open_square.boundary == (FiniteSet(0, 1) * Interval(0, 1)
+ Interval(0, 1) * FiniteSet(0, 1))
second_square = Interval(1, 2, True, True) * Interval(0, 1, True, True)
assert (open_square + second_square).boundary == (
FiniteSet(0, 1) * Interval(0, 1)
+ FiniteSet(1, 2) * Interval(0, 1)
+ Interval(0, 1) * FiniteSet(0, 1)
+ Interval(1, 2) * FiniteSet(0, 1))
def test_boundary_ProductSet_line():
line_in_r2 = Interval(0, 1) * FiniteSet(0)
assert line_in_r2.boundary == line_in_r2
def test_is_open():
assert Interval(0, 1, False, False).is_open is False
assert Interval(0, 1, True, False).is_open is False
assert Interval(0, 1, True, True).is_open is True
assert FiniteSet(1, 2, 3).is_open is False
def test_is_closed():
assert Interval(0, 1, False, False).is_closed is True
assert Interval(0, 1, True, False).is_closed is False
assert FiniteSet(1, 2, 3).is_closed is True
def test_closure():
assert Interval(0, 1, False, True).closure == Interval(0, 1, False, False)
def test_interior():
assert Interval(0, 1, False, True).interior == Interval(0, 1, True, True)
def test_issue_7841():
raises(TypeError, lambda: x in S.Reals)
def test_Eq():
assert Eq(Interval(0, 1), Interval(0, 1))
assert Eq(Interval(0, 1), Interval(0, 2)) == False
s1 = FiniteSet(0, 1)
s2 = FiniteSet(1, 2)
assert Eq(s1, s1)
assert Eq(s1, s2) == False
assert Eq(s1*s2, s1*s2)
assert Eq(s1*s2, s2*s1) == False
assert unchanged(Eq, FiniteSet({x, y}), FiniteSet({x}))
assert Eq(FiniteSet({x, y}).subs(y, x), FiniteSet({x})) is S.true
assert Eq(FiniteSet({x, y}), FiniteSet({x})).subs(y, x) is S.true
assert Eq(FiniteSet({x, y}).subs(y, x+1), FiniteSet({x})) is S.false
assert Eq(FiniteSet({x, y}), FiniteSet({x})).subs(y, x+1) is S.false
assert Eq(ProductSet({1}, {2}), Interval(1, 2)) not in (S.true, S.false)
assert Eq(ProductSet({1}), ProductSet({1}, {2})) is S.false
assert Eq(FiniteSet(()), FiniteSet(1)) is S.false
assert Eq(ProductSet(), FiniteSet(1)) is S.false
i1 = Interval(0, 1)
i2 = Interval(x, y)
assert unchanged(Eq, ProductSet(i1, i1), ProductSet(i2, i2))
def test_SymmetricDifference():
A = FiniteSet(0, 1, 2, 3, 4, 5)
B = FiniteSet(2, 4, 6, 8, 10)
C = Interval(8, 10)
assert SymmetricDifference(A, B, evaluate=False).is_iterable is True
assert SymmetricDifference(A, C, evaluate=False).is_iterable is None
assert FiniteSet(*SymmetricDifference(A, B, evaluate=False)) == \
FiniteSet(0, 1, 3, 5, 6, 8, 10)
raises(TypeError,
lambda: FiniteSet(*SymmetricDifference(A, C, evaluate=False)))
assert SymmetricDifference(FiniteSet(0, 1, 2, 3, 4, 5), \
FiniteSet(2, 4, 6, 8, 10)) == FiniteSet(0, 1, 3, 5, 6, 8, 10)
assert SymmetricDifference(FiniteSet(2, 3, 4), FiniteSet(2, 3 , 4 , 5)) \
== FiniteSet(5)
assert FiniteSet(1, 2, 3, 4, 5) ^ FiniteSet(1, 2, 5, 6) == \
FiniteSet(3, 4, 6)
assert Set(1, 2 , 3) ^ Set(2, 3, 4) == Union(Set(1, 2, 3) - Set(2, 3, 4), \
Set(2, 3, 4) - Set(1, 2, 3))
assert Interval(0, 4) ^ Interval(2, 5) == Union(Interval(0, 4) - \
Interval(2, 5), Interval(2, 5) - Interval(0, 4))
def test_issue_9536():
from sympy.functions.elementary.exponential import log
a = Symbol('a', real=True)
assert FiniteSet(log(a)).intersect(S.Reals) == Intersection(S.Reals, FiniteSet(log(a)))
def test_issue_9637():
n = Symbol('n')
a = FiniteSet(n)
b = FiniteSet(2, n)
assert Complement(S.Reals, a) == Complement(S.Reals, a, evaluate=False)
assert Complement(Interval(1, 3), a) == Complement(Interval(1, 3), a, evaluate=False)
assert Complement(Interval(1, 3), b) == \
Complement(Union(Interval(1, 2, False, True), Interval(2, 3, True, False)), a)
assert Complement(a, S.Reals) == Complement(a, S.Reals, evaluate=False)
assert Complement(a, Interval(1, 3)) == Complement(a, Interval(1, 3), evaluate=False)
def test_issue_9808():
# See https://github.com/sympy/sympy/issues/16342
assert Complement(FiniteSet(y), FiniteSet(1)) == Complement(FiniteSet(y), FiniteSet(1), evaluate=False)
assert Complement(FiniteSet(1, 2, x), FiniteSet(x, y, 2, 3)) == \
Complement(FiniteSet(1), FiniteSet(y), evaluate=False)
def test_issue_9956():
assert Union(Interval(-oo, oo), FiniteSet(1)) == Interval(-oo, oo)
assert Interval(-oo, oo).contains(1) is S.true
def test_issue_Symbol_inter():
i = Interval(0, oo)
r = S.Reals
mat = Matrix([0, 0, 0])
assert Intersection(r, i, FiniteSet(m), FiniteSet(m, n)) == \
Intersection(i, FiniteSet(m))
assert Intersection(FiniteSet(1, m, n), FiniteSet(m, n, 2), i) == \
Intersection(i, FiniteSet(m, n))
assert Intersection(FiniteSet(m, n, x), FiniteSet(m, z), r) == \
Intersection(Intersection({m, z}, {m, n, x}), r)
assert Intersection(FiniteSet(m, n, 3), FiniteSet(m, n, x), r) == \
Intersection(FiniteSet(3, m, n), FiniteSet(m, n, x), r, evaluate=False)
assert Intersection(FiniteSet(m, n, 3), FiniteSet(m, n, 2, 3), r) == \
Intersection(FiniteSet(3, m, n), r)
assert Intersection(r, FiniteSet(mat, 2, n), FiniteSet(0, mat, n)) == \
Intersection(r, FiniteSet(n))
assert Intersection(FiniteSet(sin(x), cos(x)), FiniteSet(sin(x), cos(x), 1), r) == \
Intersection(r, FiniteSet(sin(x), cos(x)))
assert Intersection(FiniteSet(x**2, 1, sin(x)), FiniteSet(x**2, 2, sin(x)), r) == \
Intersection(r, FiniteSet(x**2, sin(x)))
def test_issue_11827():
assert S.Naturals0**4
def test_issue_10113():
f = x**2/(x**2 - 4)
assert imageset(x, f, S.Reals) == Union(Interval(-oo, 0), Interval(1, oo, True, True))
assert imageset(x, f, Interval(-2, 2)) == Interval(-oo, 0)
assert imageset(x, f, Interval(-2, 3)) == Union(Interval(-oo, 0), Interval(Rational(9, 5), oo))
def test_issue_10248():
raises(
TypeError, lambda: list(Intersection(S.Reals, FiniteSet(x)))
)
A = Symbol('A', real=True)
assert list(Intersection(S.Reals, FiniteSet(A))) == [A]
def test_issue_9447():
a = Interval(0, 1) + Interval(2, 3)
assert Complement(S.UniversalSet, a) == Complement(
S.UniversalSet, Union(Interval(0, 1), Interval(2, 3)), evaluate=False)
assert Complement(S.Naturals, a) == Complement(
S.Naturals, Union(Interval(0, 1), Interval(2, 3)), evaluate=False)
def test_issue_10337():
assert (FiniteSet(2) == 3) is False
assert (FiniteSet(2) != 3) is True
raises(TypeError, lambda: FiniteSet(2) < 3)
raises(TypeError, lambda: FiniteSet(2) <= 3)
raises(TypeError, lambda: FiniteSet(2) > 3)
raises(TypeError, lambda: FiniteSet(2) >= 3)
def test_issue_10326():
bad = [
EmptySet,
FiniteSet(1),
Interval(1, 2),
S.ComplexInfinity,
S.ImaginaryUnit,
S.Infinity,
S.NaN,
S.NegativeInfinity,
]
interval = Interval(0, 5)
for i in bad:
assert i not in interval
x = Symbol('x', real=True)
nr = Symbol('nr', extended_real=False)
assert x + 1 in Interval(x, x + 4)
assert nr not in Interval(x, x + 4)
assert Interval(1, 2) in FiniteSet(Interval(0, 5), Interval(1, 2))
assert Interval(-oo, oo).contains(oo) is S.false
assert Interval(-oo, oo).contains(-oo) is S.false
def test_issue_2799():
U = S.UniversalSet
a = Symbol('a', real=True)
inf_interval = Interval(a, oo)
R = S.Reals
assert U + inf_interval == inf_interval + U
assert U + R == R + U
assert R + inf_interval == inf_interval + R
def test_issue_9706():
assert Interval(-oo, 0).closure == Interval(-oo, 0, True, False)
assert Interval(0, oo).closure == Interval(0, oo, False, True)
assert Interval(-oo, oo).closure == Interval(-oo, oo)
def test_issue_8257():
reals_plus_infinity = Union(Interval(-oo, oo), FiniteSet(oo))
reals_plus_negativeinfinity = Union(Interval(-oo, oo), FiniteSet(-oo))
assert Interval(-oo, oo) + FiniteSet(oo) == reals_plus_infinity
assert FiniteSet(oo) + Interval(-oo, oo) == reals_plus_infinity
assert Interval(-oo, oo) + FiniteSet(-oo) == reals_plus_negativeinfinity
assert FiniteSet(-oo) + Interval(-oo, oo) == reals_plus_negativeinfinity
def test_issue_10931():
assert S.Integers - S.Integers == EmptySet
assert S.Integers - S.Reals == EmptySet
def test_issue_11174():
soln = Intersection(Interval(-oo, oo), FiniteSet(-x), evaluate=False)
assert Intersection(FiniteSet(-x), S.Reals) == soln
soln = Intersection(S.Reals, FiniteSet(x), evaluate=False)
assert Intersection(FiniteSet(x), S.Reals) == soln
def test_issue_18505():
assert ImageSet(Lambda(n, sqrt(pi*n/2 - 1 + pi/2)), S.Integers).contains(0) == \
Contains(0, ImageSet(Lambda(n, sqrt(pi*n/2 - 1 + pi/2)), S.Integers))
def test_finite_set_intersection():
# The following should not produce recursion errors
# Note: some of these are not completely correct. See
# https://github.com/sympy/sympy/issues/16342.
assert Intersection(FiniteSet(-oo, x), FiniteSet(x)) == FiniteSet(x)
assert Intersection._handle_finite_sets([FiniteSet(-oo, x), FiniteSet(0, x)]) == FiniteSet(x)
assert Intersection._handle_finite_sets([FiniteSet(-oo, x), FiniteSet(x)]) == FiniteSet(x)
assert Intersection._handle_finite_sets([FiniteSet(2, 3, x, y), FiniteSet(1, 2, x)]) == \
Intersection._handle_finite_sets([FiniteSet(1, 2, x), FiniteSet(2, 3, x, y)]) == \
Intersection(FiniteSet(1, 2, x), FiniteSet(2, 3, x, y)) == \
Intersection(FiniteSet(1, 2, x), FiniteSet(2, x, y))
assert FiniteSet(1+x-y) & FiniteSet(1) == \
FiniteSet(1) & FiniteSet(1+x-y) == \
Intersection(FiniteSet(1+x-y), FiniteSet(1), evaluate=False)
assert FiniteSet(1) & FiniteSet(x) == FiniteSet(x) & FiniteSet(1) == \
Intersection(FiniteSet(1), FiniteSet(x), evaluate=False)
assert FiniteSet({x}) & FiniteSet({x, y}) == \
Intersection(FiniteSet({x}), FiniteSet({x, y}), evaluate=False)
def test_union_intersection_constructor():
# The actual exception does not matter here, so long as these fail
sets = [FiniteSet(1), FiniteSet(2)]
raises(Exception, lambda: Union(sets))
raises(Exception, lambda: Intersection(sets))
raises(Exception, lambda: Union(tuple(sets)))
raises(Exception, lambda: Intersection(tuple(sets)))
raises(Exception, lambda: Union(i for i in sets))
raises(Exception, lambda: Intersection(i for i in sets))
# Python sets are treated the same as FiniteSet
# The union of a single set (of sets) is the set (of sets) itself
assert Union(set(sets)) == FiniteSet(*sets)
assert Intersection(set(sets)) == FiniteSet(*sets)
assert Union({1}, {2}) == FiniteSet(1, 2)
assert Intersection({1, 2}, {2, 3}) == FiniteSet(2)
def test_Union_contains():
assert zoo not in Union(
Interval.open(-oo, 0), Interval.open(0, oo))
@XFAIL
def test_issue_16878b():
# in intersection_sets for (ImageSet, Set) there is no code
# that handles the base_set of S.Reals like there is
# for Integers
assert imageset(x, (x, x), S.Reals).is_subset(S.Reals**2) is True
def test_DisjointUnion():
assert DisjointUnion(FiniteSet(1, 2, 3), FiniteSet(1, 2, 3), FiniteSet(1, 2, 3)).rewrite(Union) == (FiniteSet(1, 2, 3) * FiniteSet(0, 1, 2))
assert DisjointUnion(Interval(1, 3), Interval(2, 4)).rewrite(Union) == Union(Interval(1, 3) * FiniteSet(0), Interval(2, 4) * FiniteSet(1))
assert DisjointUnion(Interval(0, 5), Interval(0, 5)).rewrite(Union) == Union(Interval(0, 5) * FiniteSet(0), Interval(0, 5) * FiniteSet(1))
assert DisjointUnion(Interval(-1, 2), S.EmptySet, S.EmptySet).rewrite(Union) == Interval(-1, 2) * FiniteSet(0)
assert DisjointUnion(Interval(-1, 2)).rewrite(Union) == Interval(-1, 2) * FiniteSet(0)
assert DisjointUnion(S.EmptySet, Interval(-1, 2), S.EmptySet).rewrite(Union) == Interval(-1, 2) * FiniteSet(1)
assert DisjointUnion(Interval(-oo, oo)).rewrite(Union) == Interval(-oo, oo) * FiniteSet(0)
assert DisjointUnion(S.EmptySet).rewrite(Union) == S.EmptySet
assert DisjointUnion().rewrite(Union) == S.EmptySet
raises(TypeError, lambda: DisjointUnion(Symbol('n')))
x = Symbol("x")
y = Symbol("y")
z = Symbol("z")
assert DisjointUnion(FiniteSet(x), FiniteSet(y, z)).rewrite(Union) == (FiniteSet(x) * FiniteSet(0)) + (FiniteSet(y, z) * FiniteSet(1))
def test_DisjointUnion_is_empty():
assert DisjointUnion(S.EmptySet).is_empty is True
assert DisjointUnion(S.EmptySet, S.EmptySet).is_empty is True
assert DisjointUnion(S.EmptySet, FiniteSet(1, 2, 3)).is_empty is False
def test_DisjointUnion_is_iterable():
assert DisjointUnion(S.Integers, S.Naturals, S.Rationals).is_iterable is True
assert DisjointUnion(S.EmptySet, S.Reals).is_iterable is False
assert DisjointUnion(FiniteSet(1, 2, 3), S.EmptySet, FiniteSet(x, y)).is_iterable is True
assert DisjointUnion(S.EmptySet, S.EmptySet).is_iterable is False
def test_DisjointUnion_contains():
assert (0, 0) in DisjointUnion(FiniteSet(0, 1, 2), FiniteSet(0, 1, 2), FiniteSet(0, 1, 2))
assert (0, 1) in DisjointUnion(FiniteSet(0, 1, 2), FiniteSet(0, 1, 2), FiniteSet(0, 1, 2))
assert (0, 2) in DisjointUnion(FiniteSet(0, 1, 2), FiniteSet(0, 1, 2), FiniteSet(0, 1, 2))
assert (1, 0) in DisjointUnion(FiniteSet(0, 1, 2), FiniteSet(0, 1, 2), FiniteSet(0, 1, 2))
assert (1, 1) in DisjointUnion(FiniteSet(0, 1, 2), FiniteSet(0, 1, 2), FiniteSet(0, 1, 2))
assert (1, 2) in DisjointUnion(FiniteSet(0, 1, 2), FiniteSet(0, 1, 2), FiniteSet(0, 1, 2))
assert (2, 0) in DisjointUnion(FiniteSet(0, 1, 2), FiniteSet(0, 1, 2), FiniteSet(0, 1, 2))
assert (2, 1) in DisjointUnion(FiniteSet(0, 1, 2), FiniteSet(0, 1, 2), FiniteSet(0, 1, 2))
assert (2, 2) in DisjointUnion(FiniteSet(0, 1, 2), FiniteSet(0, 1, 2), FiniteSet(0, 1, 2))
assert (0, 1, 2) not in DisjointUnion(FiniteSet(0, 1, 2), FiniteSet(0, 1, 2), FiniteSet(0, 1, 2))
assert (0, 0.5) not in DisjointUnion(FiniteSet(0.5))
assert (0, 5) not in DisjointUnion(FiniteSet(0, 1, 2), FiniteSet(0, 1, 2), FiniteSet(0, 1, 2))
assert (x, 0) in DisjointUnion(FiniteSet(x, y, z), S.EmptySet, FiniteSet(y))
assert (y, 0) in DisjointUnion(FiniteSet(x, y, z), S.EmptySet, FiniteSet(y))
assert (z, 0) in DisjointUnion(FiniteSet(x, y, z), S.EmptySet, FiniteSet(y))
assert (y, 2) in DisjointUnion(FiniteSet(x, y, z), S.EmptySet, FiniteSet(y))
assert (0.5, 0) in DisjointUnion(Interval(0, 1), Interval(0, 2))
assert (0.5, 1) in DisjointUnion(Interval(0, 1), Interval(0, 2))
assert (1.5, 0) not in DisjointUnion(Interval(0, 1), Interval(0, 2))
assert (1.5, 1) in DisjointUnion(Interval(0, 1), Interval(0, 2))
def test_DisjointUnion_iter():
D = DisjointUnion(FiniteSet(3, 5, 7, 9), FiniteSet(x, y, z))
it = iter(D)
L1 = [(x, 1), (y, 1), (z, 1)]
L2 = [(3, 0), (5, 0), (7, 0), (9, 0)]
nxt = next(it)
assert nxt in L2
L2.remove(nxt)
nxt = next(it)
assert nxt in L1
L1.remove(nxt)
nxt = next(it)
assert nxt in L2
L2.remove(nxt)
nxt = next(it)
assert nxt in L1
L1.remove(nxt)
nxt = next(it)
assert nxt in L2
L2.remove(nxt)
nxt = next(it)
assert nxt in L1
L1.remove(nxt)
nxt = next(it)
assert nxt in L2
L2.remove(nxt)
raises(StopIteration, lambda: next(it))
raises(ValueError, lambda: iter(DisjointUnion(Interval(0, 1), S.EmptySet)))
def test_DisjointUnion_len():
assert len(DisjointUnion(FiniteSet(3, 5, 7, 9), FiniteSet(x, y, z))) == 7
assert len(DisjointUnion(S.EmptySet, S.EmptySet, FiniteSet(x, y, z), S.EmptySet)) == 3
raises(ValueError, lambda: len(DisjointUnion(Interval(0, 1), S.EmptySet)))
|
0ccec651141b7527dc75d1f3db733139285ab4115cdbab4ffec7771bca460901 | """Plotting module that can plot 2D and 3D functions
"""
from sympy.utilities.decorator import doctest_depends_on
@doctest_depends_on(modules=('pyglet',))
def PygletPlot(*args, **kwargs):
"""
Plot Examples
=============
See examples/advanced/pyglet_plotting.py for many more examples.
>>> from sympy.plotting.pygletplot import PygletPlot as Plot
>>> from sympy.abc import x, y, z
>>> Plot(x*y**3-y*x**3)
[0]: -x**3*y + x*y**3, 'mode=cartesian'
>>> p = Plot()
>>> p[1] = x*y
>>> p[1].color = z, (0.4,0.4,0.9), (0.9,0.4,0.4)
>>> p = Plot()
>>> p[1] = x**2+y**2
>>> p[2] = -x**2-y**2
Variable Intervals
==================
The basic format is [var, min, max, steps], but the
syntax is flexible and arguments left out are taken
from the defaults for the current coordinate mode:
>>> Plot(x**2) # implies [x,-5,5,100]
[0]: x**2, 'mode=cartesian'
>>> Plot(x**2, [], []) # [x,-1,1,40], [y,-1,1,40]
[0]: x**2, 'mode=cartesian'
>>> Plot(x**2-y**2, [100], [100]) # [x,-1,1,100], [y,-1,1,100]
[0]: x**2 - y**2, 'mode=cartesian'
>>> Plot(x**2, [x,-13,13,100])
[0]: x**2, 'mode=cartesian'
>>> Plot(x**2, [-13,13]) # [x,-13,13,100]
[0]: x**2, 'mode=cartesian'
>>> Plot(x**2, [x,-13,13]) # [x,-13,13,100]
[0]: x**2, 'mode=cartesian'
>>> Plot(1*x, [], [x], mode='cylindrical')
... # [unbound_theta,0,2*Pi,40], [x,-1,1,20]
[0]: x, 'mode=cartesian'
Coordinate Modes
================
Plot supports several curvilinear coordinate modes, and
they independent for each plotted function. You can specify
a coordinate mode explicitly with the 'mode' named argument,
but it can be automatically determined for Cartesian or
parametric plots, and therefore must only be specified for
polar, cylindrical, and spherical modes.
Specifically, Plot(function arguments) and Plot[n] =
(function arguments) will interpret your arguments as a
Cartesian plot if you provide one function and a parametric
plot if you provide two or three functions. Similarly, the
arguments will be interpreted as a curve if one variable is
used, and a surface if two are used.
Supported mode names by number of variables:
1: parametric, cartesian, polar
2: parametric, cartesian, cylindrical = polar, spherical
>>> Plot(1, mode='spherical')
Calculator-like Interface
=========================
>>> p = Plot(visible=False)
>>> f = x**2
>>> p[1] = f
>>> p[2] = f.diff(x)
>>> p[3] = f.diff(x).diff(x)
>>> p
[1]: x**2, 'mode=cartesian'
[2]: 2*x, 'mode=cartesian'
[3]: 2, 'mode=cartesian'
>>> p.show()
>>> p.clear()
>>> p
<blank plot>
>>> p[1] = x**2+y**2
>>> p[1].style = 'solid'
>>> p[2] = -x**2-y**2
>>> p[2].style = 'wireframe'
>>> p[1].color = z, (0.4,0.4,0.9), (0.9,0.4,0.4)
>>> p[1].style = 'both'
>>> p[2].style = 'both'
>>> p.close()
Plot Window Keyboard Controls
=============================
Screen Rotation:
X,Y axis Arrow Keys, A,S,D,W, Numpad 4,6,8,2
Z axis Q,E, Numpad 7,9
Model Rotation:
Z axis Z,C, Numpad 1,3
Zoom: R,F, PgUp,PgDn, Numpad +,-
Reset Camera: X, Numpad 5
Camera Presets:
XY F1
XZ F2
YZ F3
Perspective F4
Sensitivity Modifier: SHIFT
Axes Toggle:
Visible F5
Colors F6
Close Window: ESCAPE
=============================
"""
from sympy.plotting.pygletplot.plot import PygletPlot
return PygletPlot(*args, **kwargs)
|
3d73f226e23a171de81304f2153d18a557b42be0c5925398b677afd76286fc08 | from __future__ import print_function, division
from threading import RLock
# it is sufficient to import "pyglet" here once
try:
import pyglet.gl as pgl
except ImportError:
raise ImportError("pyglet is required for plotting.\n "
"visit http://www.pyglet.org/")
from sympy.core.compatibility import is_sequence, SYMPY_INTS
from sympy.core.numbers import Integer
from sympy.geometry.entity import GeometryEntity
from sympy.plotting.pygletplot.plot_axes import PlotAxes
from sympy.plotting.pygletplot.plot_mode import PlotMode
from sympy.plotting.pygletplot.plot_object import PlotObject
from sympy.plotting.pygletplot.plot_window import PlotWindow
from sympy.plotting.pygletplot.util import parse_option_string
from sympy.utilities.decorator import doctest_depends_on
from time import sleep
from os import getcwd, listdir
import ctypes
@doctest_depends_on(modules=('pyglet',))
class PygletPlot(object):
"""
Plot Examples
=============
See examples/advanced/pyglet_plotting.py for many more examples.
>>> from sympy.plotting.pygletplot import PygletPlot as Plot
>>> from sympy.abc import x, y, z
>>> Plot(x*y**3-y*x**3)
[0]: -x**3*y + x*y**3, 'mode=cartesian'
>>> p = Plot()
>>> p[1] = x*y
>>> p[1].color = z, (0.4,0.4,0.9), (0.9,0.4,0.4)
>>> p = Plot()
>>> p[1] = x**2+y**2
>>> p[2] = -x**2-y**2
Variable Intervals
==================
The basic format is [var, min, max, steps], but the
syntax is flexible and arguments left out are taken
from the defaults for the current coordinate mode:
>>> Plot(x**2) # implies [x,-5,5,100]
[0]: x**2, 'mode=cartesian'
>>> Plot(x**2, [], []) # [x,-1,1,40], [y,-1,1,40]
[0]: x**2, 'mode=cartesian'
>>> Plot(x**2-y**2, [100], [100]) # [x,-1,1,100], [y,-1,1,100]
[0]: x**2 - y**2, 'mode=cartesian'
>>> Plot(x**2, [x,-13,13,100])
[0]: x**2, 'mode=cartesian'
>>> Plot(x**2, [-13,13]) # [x,-13,13,100]
[0]: x**2, 'mode=cartesian'
>>> Plot(x**2, [x,-13,13]) # [x,-13,13,10]
[0]: x**2, 'mode=cartesian'
>>> Plot(1*x, [], [x], mode='cylindrical')
... # [unbound_theta,0,2*Pi,40], [x,-1,1,20]
[0]: x, 'mode=cartesian'
Coordinate Modes
================
Plot supports several curvilinear coordinate modes, and
they independent for each plotted function. You can specify
a coordinate mode explicitly with the 'mode' named argument,
but it can be automatically determined for Cartesian or
parametric plots, and therefore must only be specified for
polar, cylindrical, and spherical modes.
Specifically, Plot(function arguments) and Plot[n] =
(function arguments) will interpret your arguments as a
Cartesian plot if you provide one function and a parametric
plot if you provide two or three functions. Similarly, the
arguments will be interpreted as a curve if one variable is
used, and a surface if two are used.
Supported mode names by number of variables:
1: parametric, cartesian, polar
2: parametric, cartesian, cylindrical = polar, spherical
>>> Plot(1, mode='spherical')
Calculator-like Interface
=========================
>>> p = Plot(visible=False)
>>> f = x**2
>>> p[1] = f
>>> p[2] = f.diff(x)
>>> p[3] = f.diff(x).diff(x)
>>> p
[1]: x**2, 'mode=cartesian'
[2]: 2*x, 'mode=cartesian'
[3]: 2, 'mode=cartesian'
>>> p.show()
>>> p.clear()
>>> p
<blank plot>
>>> p[1] = x**2+y**2
>>> p[1].style = 'solid'
>>> p[2] = -x**2-y**2
>>> p[2].style = 'wireframe'
>>> p[1].color = z, (0.4,0.4,0.9), (0.9,0.4,0.4)
>>> p[1].style = 'both'
>>> p[2].style = 'both'
>>> p.close()
Plot Window Keyboard Controls
=============================
Screen Rotation:
X,Y axis Arrow Keys, A,S,D,W, Numpad 4,6,8,2
Z axis Q,E, Numpad 7,9
Model Rotation:
Z axis Z,C, Numpad 1,3
Zoom: R,F, PgUp,PgDn, Numpad +,-
Reset Camera: X, Numpad 5
Camera Presets:
XY F1
XZ F2
YZ F3
Perspective F4
Sensitivity Modifier: SHIFT
Axes Toggle:
Visible F5
Colors F6
Close Window: ESCAPE
=============================
"""
@doctest_depends_on(modules=('pyglet',))
def __init__(self, *fargs, **win_args):
"""
Positional Arguments
====================
Any given positional arguments are used to
initialize a plot function at index 1. In
other words...
>>> from sympy.plotting.pygletplot import PygletPlot as Plot
>>> from sympy.abc import x
>>> p = Plot(x**2, visible=False)
...is equivalent to...
>>> p = Plot(visible=False)
>>> p[1] = x**2
Note that in earlier versions of the plotting
module, you were able to specify multiple
functions in the initializer. This functionality
has been dropped in favor of better automatic
plot plot_mode detection.
Named Arguments
===============
axes
An option string of the form
"key1=value1; key2 = value2" which
can use the following options:
style = ordinate
none OR frame OR box OR ordinate
stride = 0.25
val OR (val_x, val_y, val_z)
overlay = True (draw on top of plot)
True OR False
colored = False (False uses Black,
True uses colors
R,G,B = X,Y,Z)
True OR False
label_axes = False (display axis names
at endpoints)
True OR False
visible = True (show immediately
True OR False
The following named arguments are passed as
arguments to window initialization:
antialiasing = True
True OR False
ortho = False
True OR False
invert_mouse_zoom = False
True OR False
"""
# Register the plot modes
from . import plot_modes # noqa
self._win_args = win_args
self._window = None
self._render_lock = RLock()
self._functions = {}
self._pobjects = []
self._screenshot = ScreenShot(self)
axe_options = parse_option_string(win_args.pop('axes', ''))
self.axes = PlotAxes(**axe_options)
self._pobjects.append(self.axes)
self[0] = fargs
if win_args.get('visible', True):
self.show()
## Window Interfaces
def show(self):
"""
Creates and displays a plot window, or activates it
(gives it focus) if it has already been created.
"""
if self._window and not self._window.has_exit:
self._window.activate()
else:
self._win_args['visible'] = True
self.axes.reset_resources()
#if hasattr(self, '_doctest_depends_on'):
# self._win_args['runfromdoctester'] = True
self._window = PlotWindow(self, **self._win_args)
def close(self):
"""
Closes the plot window.
"""
if self._window:
self._window.close()
def saveimage(self, outfile=None, format='', size=(600, 500)):
"""
Saves a screen capture of the plot window to an
image file.
If outfile is given, it can either be a path
or a file object. Otherwise a png image will
be saved to the current working directory.
If the format is omitted, it is determined from
the filename extension.
"""
self._screenshot.save(outfile, format, size)
## Function List Interfaces
def clear(self):
"""
Clears the function list of this plot.
"""
self._render_lock.acquire()
self._functions = {}
self.adjust_all_bounds()
self._render_lock.release()
def __getitem__(self, i):
"""
Returns the function at position i in the
function list.
"""
return self._functions[i]
def __setitem__(self, i, args):
"""
Parses and adds a PlotMode to the function
list.
"""
if not (isinstance(i, (SYMPY_INTS, Integer)) and i >= 0):
raise ValueError("Function index must "
"be an integer >= 0.")
if isinstance(args, PlotObject):
f = args
else:
if (not is_sequence(args)) or isinstance(args, GeometryEntity):
args = [args]
if len(args) == 0:
return # no arguments given
kwargs = dict(bounds_callback=self.adjust_all_bounds)
f = PlotMode(*args, **kwargs)
if f:
self._render_lock.acquire()
self._functions[i] = f
self._render_lock.release()
else:
raise ValueError("Failed to parse '%s'."
% ', '.join(str(a) for a in args))
def __delitem__(self, i):
"""
Removes the function in the function list at
position i.
"""
self._render_lock.acquire()
del self._functions[i]
self.adjust_all_bounds()
self._render_lock.release()
def firstavailableindex(self):
"""
Returns the first unused index in the function list.
"""
i = 0
self._render_lock.acquire()
while i in self._functions:
i += 1
self._render_lock.release()
return i
def append(self, *args):
"""
Parses and adds a PlotMode to the function
list at the first available index.
"""
self.__setitem__(self.firstavailableindex(), args)
def __len__(self):
"""
Returns the number of functions in the function list.
"""
return len(self._functions)
def __iter__(self):
"""
Allows iteration of the function list.
"""
return self._functions.itervalues()
def __repr__(self):
return str(self)
def __str__(self):
"""
Returns a string containing a new-line separated
list of the functions in the function list.
"""
s = ""
if len(self._functions) == 0:
s += "<blank plot>"
else:
self._render_lock.acquire()
s += "\n".join(["%s[%i]: %s" % ("", i, str(self._functions[i]))
for i in self._functions])
self._render_lock.release()
return s
def adjust_all_bounds(self):
self._render_lock.acquire()
self.axes.reset_bounding_box()
for f in self._functions:
self.axes.adjust_bounds(self._functions[f].bounds)
self._render_lock.release()
def wait_for_calculations(self):
sleep(0)
self._render_lock.acquire()
for f in self._functions:
a = self._functions[f]._get_calculating_verts
b = self._functions[f]._get_calculating_cverts
while a() or b():
sleep(0)
self._render_lock.release()
class ScreenShot:
def __init__(self, plot):
self._plot = plot
self.screenshot_requested = False
self.outfile = None
self.format = ''
self.invisibleMode = False
self.flag = 0
def __nonzero__(self):
return self.screenshot_requested
__bool__ = __nonzero__
def _execute_saving(self):
if self.flag < 3:
self.flag += 1
return
size_x, size_y = self._plot._window.get_size()
size = size_x*size_y*4*ctypes.sizeof(ctypes.c_ubyte)
image = ctypes.create_string_buffer(size)
pgl.glReadPixels(0, 0, size_x, size_y, pgl.GL_RGBA, pgl.GL_UNSIGNED_BYTE, image)
from PIL import Image
im = Image.frombuffer('RGBA', (size_x, size_y),
image.raw, 'raw', 'RGBA', 0, 1)
im.transpose(Image.FLIP_TOP_BOTTOM).save(self.outfile, self.format)
self.flag = 0
self.screenshot_requested = False
if self.invisibleMode:
self._plot._window.close()
def save(self, outfile=None, format='', size=(600, 500)):
self.outfile = outfile
self.format = format
self.size = size
self.screenshot_requested = True
if not self._plot._window or self._plot._window.has_exit:
self._plot._win_args['visible'] = False
self._plot._win_args['width'] = size[0]
self._plot._win_args['height'] = size[1]
self._plot.axes.reset_resources()
self._plot._window = PlotWindow(self._plot, **self._plot._win_args)
self.invisibleMode = True
if self.outfile is None:
self.outfile = self._create_unique_path()
print(self.outfile)
def _create_unique_path(self):
cwd = getcwd()
l = listdir(cwd)
path = ''
i = 0
while True:
if not 'plot_%s.png' % i in l:
path = cwd + '/plot_%s.png' % i
break
i += 1
return path
|
5701fe0649f24384cbfc683c1ac82c1265012bf38c64c9b759037b753726591d | #!/usr/bin/env python
"""
Test that
from sympy import *
Doesn't import anything other than SymPy, it's hard dependencies (mpmath), and
hard optional dependencies (gmpy2). Importing unnecessary libraries
can accidentally add hard dependencies to SymPy in the worst case, or at best
slow down the SymPy import time when they are installed.
Note, for this test to be effective, every external library that could
potentially be imported by SymPy must be installed.
TODO: Monkeypatch the importer to detect non-standard library imports even
when they aren't installed.
Based on code from
https://stackoverflow.com/questions/22195382/how-to-check-if-a-module-library-package-is-part-of-the-python-standard-library.
"""
# These libraries will always be imported with SymPy
hard_dependencies = ['mpmath']
# These libraries are optional, but are always imported at SymPy import time
# when they are installed. External libraries should only be added to this
# list if they are required for core SymPy functionality.
hard_optional_dependencies = ['gmpy', 'gmpy2', 'pycosat']
import sys
import os
stdlib = {p for p in sys.path if p.startswith(sys.prefix) and
'site-packages' not in p}
existing_modules = list(sys.modules.keys())
# hook in-tree SymPy into Python path, if possible
this_path = os.path.abspath(__file__)
this_dir = os.path.dirname(this_path)
sympy_top = os.path.split(this_dir)[0]
sympy_dir = os.path.join(sympy_top, 'sympy')
if os.path.isdir(sympy_dir):
sys.path.insert(0, sympy_top)
def test_external_imports():
exec("from sympy import *", {})
bad = []
for mod in sys.modules:
if '.' in mod and mod.split('.')[0] in sys.modules:
# Only worry about the top-level modules
continue
if mod in existing_modules:
continue
if any(mod == i or mod.startswith(i + '.') for i in ['sympy'] +
hard_dependencies + hard_optional_dependencies):
continue
if mod in sys.builtin_module_names:
continue
fname = getattr(sys.modules[mod], "__file__", None)
if fname is None:
bad.append(mod)
continue
if fname.endswith(('__init__.py', '__init__.pyc', '__init__.pyo')):
fname = os.path.dirname(fname)
if os.path.dirname(fname) in stdlib:
continue
bad.append(mod)
if bad:
raise RuntimeError("""Unexpected external modules found when running 'from sympy import *':
""" + '\n '.join(bad))
print("No unexpected external modules were imported with 'from sympy import *'!")
if __name__ == '__main__':
test_external_imports()
|
79b927ca209d2b3bd9b4f3ac6dc7c6dc76139963a5563bba46d827ff73bb4992 | """
SymPy is a Python library for symbolic mathematics. It aims to become a
full-featured computer algebra system (CAS) while keeping the code as simple
as possible in order to be comprehensible and easily extensible. SymPy is
written entirely in Python. It depends on mpmath, and other external libraries
may be optionally for things like plotting support.
See the webpage for more information and documentation:
https://sympy.org
"""
import sys
if sys.version_info < (3, 5):
raise ImportError("Python version 3.5 or above is required for SymPy.")
del sys
try:
import mpmath
except ImportError:
raise ImportError("SymPy now depends on mpmath as an external library. "
"See https://docs.sympy.org/latest/install.html#mpmath for more information.")
del mpmath
from sympy.release import __version__
if 'dev' in __version__:
def enable_warnings():
import warnings
warnings.filterwarnings('default', '.*', DeprecationWarning, module='sympy.*')
del warnings
enable_warnings()
del enable_warnings
def __sympy_debug():
# helper function so we don't import os globally
import os
debug_str = os.getenv('SYMPY_DEBUG', 'False')
if debug_str in ('True', 'False'):
return eval(debug_str)
else:
raise RuntimeError("unrecognized value for SYMPY_DEBUG: %s" %
debug_str)
SYMPY_DEBUG = __sympy_debug() # type: bool
from .core import (sympify, SympifyError, cacheit, Basic, Atom,
preorder_traversal, S, Expr, AtomicExpr, UnevaluatedExpr, Symbol,
Wild, Dummy, symbols, var, Number, Float, Rational, Integer,
NumberSymbol, RealNumber, igcd, ilcm, seterr, E, I, nan, oo, pi, zoo,
AlgebraicNumber, comp, mod_inverse, Pow, integer_nthroot, integer_log,
Mul, prod, Add, Mod, Rel, Eq, Ne, Lt, Le, Gt, Ge, Equality,
GreaterThan, LessThan, Unequality, StrictGreaterThan, StrictLessThan,
vectorize, Lambda, WildFunction, Derivative, diff, FunctionClass,
Function, Subs, expand, PoleError, count_ops, expand_mul, expand_log,
expand_func, expand_trig, expand_complex, expand_multinomial, nfloat,
expand_power_base, expand_power_exp, arity, PrecisionExhausted, N,
evalf, Tuple, Dict, gcd_terms, factor_terms, factor_nc, evaluate,
Catalan, EulerGamma, GoldenRatio, TribonacciConstant)
from .logic import (to_cnf, to_dnf, to_nnf, And, Or, Not, Xor, Nand, Nor,
Implies, Equivalent, ITE, POSform, SOPform, simplify_logic, bool_map,
true, false, satisfiable)
from .assumptions import (AppliedPredicate, Predicate, AssumptionsContext,
assuming, Q, ask, register_handler, remove_handler, refine)
from .polys import (Poly, PurePoly, poly_from_expr, parallel_poly_from_expr,
degree, total_degree, degree_list, LC, LM, LT, pdiv, prem, pquo,
pexquo, div, rem, quo, exquo, half_gcdex, gcdex, invert,
subresultants, resultant, discriminant, cofactors, gcd_list, gcd,
lcm_list, lcm, terms_gcd, trunc, monic, content, primitive, compose,
decompose, sturm, gff_list, gff, sqf_norm, sqf_part, sqf_list, sqf,
factor_list, factor, intervals, refine_root, count_roots, real_roots,
nroots, ground_roots, nth_power_roots_poly, cancel, reduced, groebner,
is_zero_dimensional, GroebnerBasis, poly, symmetrize, horner,
interpolate, rational_interpolate, viete, together,
BasePolynomialError, ExactQuotientFailed, PolynomialDivisionFailed,
OperationNotSupported, HeuristicGCDFailed, HomomorphismFailed,
IsomorphismFailed, ExtraneousFactors, EvaluationFailed,
RefinementFailed, CoercionFailed, NotInvertible, NotReversible,
NotAlgebraic, DomainError, PolynomialError, UnificationFailed,
GeneratorsError, GeneratorsNeeded, ComputationFailed,
UnivariatePolynomialError, MultivariatePolynomialError,
PolificationFailed, OptionError, FlagError, minpoly,
minimal_polynomial, primitive_element, field_isomorphism,
to_number_field, isolate, itermonomials, Monomial, lex, grlex,
grevlex, ilex, igrlex, igrevlex, CRootOf, rootof, RootOf,
ComplexRootOf, RootSum, roots, Domain, FiniteField, IntegerRing,
RationalField, RealField, ComplexField, PythonFiniteField,
GMPYFiniteField, PythonIntegerRing, GMPYIntegerRing, PythonRational,
GMPYRationalField, AlgebraicField, PolynomialRing, FractionField,
ExpressionDomain, FF_python, FF_gmpy, ZZ_python, ZZ_gmpy, QQ_python,
QQ_gmpy, GF, FF, ZZ, QQ, ZZ_I, QQ_I, RR, CC, EX, construct_domain,
swinnerton_dyer_poly, cyclotomic_poly, symmetric_poly, random_poly,
interpolating_poly, jacobi_poly, chebyshevt_poly, chebyshevu_poly,
hermite_poly, legendre_poly, laguerre_poly, apart, apart_list,
assemble_partfrac_list, Options, ring, xring, vring, sring, field,
xfield, vfield, sfield)
from .series import (Order, O, limit, Limit, gruntz, series, approximants,
residue, EmptySequence, SeqPer, SeqFormula, sequence, SeqAdd, SeqMul,
fourier_series, fps, difference_delta, limit_seq)
from .functions import (factorial, factorial2, rf, ff, binomial,
RisingFactorial, FallingFactorial, subfactorial, carmichael,
fibonacci, lucas, tribonacci, harmonic, bernoulli, bell, euler,
catalan, genocchi, partition, sqrt, root, Min, Max, Id, real_root,
cbrt, re, im, sign, Abs, conjugate, arg, polar_lift,
periodic_argument, unbranched_argument, principal_branch, transpose,
adjoint, polarify, unpolarify, sin, cos, tan, sec, csc, cot, sinc,
asin, acos, atan, asec, acsc, acot, atan2, exp_polar, exp, ln, log,
LambertW, sinh, cosh, tanh, coth, sech, csch, asinh, acosh, atanh,
acoth, asech, acsch, floor, ceiling, frac, Piecewise, piecewise_fold,
erf, erfc, erfi, erf2, erfinv, erfcinv, erf2inv, Ei, expint, E1, li,
Li, Si, Ci, Shi, Chi, fresnels, fresnelc, gamma, lowergamma,
uppergamma, polygamma, loggamma, digamma, trigamma, multigamma,
dirichlet_eta, zeta, lerchphi, polylog, stieltjes, Eijk, LeviCivita,
KroneckerDelta, SingularityFunction, DiracDelta, Heaviside,
bspline_basis, bspline_basis_set, interpolating_spline, besselj,
bessely, besseli, besselk, hankel1, hankel2, jn, yn, jn_zeros, hn1,
hn2, airyai, airybi, airyaiprime, airybiprime, marcumq, hyper,
meijerg, appellf1, legendre, assoc_legendre, hermite, chebyshevt,
chebyshevu, chebyshevu_root, chebyshevt_root, laguerre,
assoc_laguerre, gegenbauer, jacobi, jacobi_normalized, Ynm, Ynm_c,
Znm, elliptic_k, elliptic_f, elliptic_e, elliptic_pi, beta, mathieus,
mathieuc, mathieusprime, mathieucprime)
from .ntheory import (nextprime, prevprime, prime, primepi, primerange,
randprime, Sieve, sieve, primorial, cycle_length, composite,
compositepi, isprime, divisors, proper_divisors, factorint,
multiplicity, perfect_power, pollard_pm1, pollard_rho, primefactors,
totient, trailing, divisor_count, proper_divisor_count, divisor_sigma,
factorrat, reduced_totient, primenu, primeomega,
mersenne_prime_exponent, is_perfect, is_mersenne_prime, is_abundant,
is_deficient, is_amicable, abundance, npartitions, is_primitive_root,
is_quad_residue, legendre_symbol, jacobi_symbol, n_order, sqrt_mod,
quadratic_residues, primitive_root, nthroot_mod, is_nthpow_residue,
sqrt_mod_iter, mobius, discrete_log, quadratic_congruence,
binomial_coefficients, binomial_coefficients_list,
multinomial_coefficients, continued_fraction_periodic,
continued_fraction_iterator, continued_fraction_reduce,
continued_fraction_convergents, continued_fraction, egyptian_fraction)
from .concrete import product, Product, summation, Sum
from .discrete import (fft, ifft, ntt, intt, fwht, ifwht, mobius_transform,
inverse_mobius_transform, convolution, covering_product,
intersecting_product)
from .simplify import (simplify, hypersimp, hypersimilar, logcombine,
separatevars, posify, besselsimp, kroneckersimp, signsimp, bottom_up,
nsimplify, FU, fu, sqrtdenest, cse, use, epath, EPath, hyperexpand,
collect, rcollect, radsimp, collect_const, fraction, numer, denom,
trigsimp, exptrigsimp, powsimp, powdenest, combsimp, gammasimp,
ratsimp, ratsimpmodprime)
from .sets import (Set, Interval, Union, EmptySet, FiniteSet, ProductSet,
Intersection, DisjointUnion, imageset, Complement, SymmetricDifference, ImageSet,
Range, ComplexRegion, Reals, Contains, ConditionSet, Ordinal,
OmegaPower, ord0, PowerSet, Naturals, Naturals0, UniversalSet,
Integers, Rationals)
from .solvers import (solve, solve_linear_system, solve_linear_system_LU,
solve_undetermined_coeffs, nsolve, solve_linear, checksol, det_quick,
inv_quick, check_assumptions, failing_assumptions, diophantine,
rsolve, rsolve_poly, rsolve_ratio, rsolve_hyper, checkodesol,
classify_ode, dsolve, homogeneous_order, solve_poly_system,
solve_triangulated, pde_separate, pde_separate_add, pde_separate_mul,
pdsolve, classify_pde, checkpdesol, ode_order, reduce_inequalities,
reduce_abs_inequality, reduce_abs_inequalities, solve_poly_inequality,
solve_rational_inequalities, solve_univariate_inequality, decompogen,
solveset, linsolve, linear_eq_to_matrix, nonlinsolve, substitution,
Complexes)
from .matrices import (ShapeError, NonSquareMatrixError, GramSchmidt,
casoratian, diag, eye, hessian, jordan_cell, list2numpy, matrix2numpy,
matrix_multiply_elementwise, ones, randMatrix, rot_axis1, rot_axis2,
rot_axis3, symarray, wronskian, zeros, MutableDenseMatrix,
DeferredVector, MatrixBase, Matrix, MutableMatrix,
MutableSparseMatrix, banded, ImmutableDenseMatrix,
ImmutableSparseMatrix, ImmutableMatrix, SparseMatrix, MatrixSlice,
BlockDiagMatrix, BlockMatrix, FunctionMatrix, Identity, Inverse,
MatAdd, MatMul, MatPow, MatrixExpr, MatrixSymbol, Trace, Transpose,
ZeroMatrix, OneMatrix, blockcut, block_collapse, matrix_symbols,
Adjoint, hadamard_product, HadamardProduct, HadamardPower,
Determinant, det, diagonalize_vector, DiagMatrix, DiagonalMatrix,
DiagonalOf, trace, DotProduct, kronecker_product, KroneckerProduct,
PermutationMatrix, MatrixPermute)
from .geometry import (Point, Point2D, Point3D, Line, Ray, Segment, Line2D,
Segment2D, Ray2D, Line3D, Segment3D, Ray3D, Plane, Ellipse, Circle,
Polygon, RegularPolygon, Triangle, rad, deg, are_similar, centroid,
convex_hull, idiff, intersection, closest_points, farthest_points,
GeometryError, Curve, Parabola)
from .utilities import (flatten, group, take, subsets, variations,
numbered_symbols, cartes, capture, dict_merge, postorder_traversal,
interactive_traversal, prefixes, postfixes, sift, topological_sort,
unflatten, has_dups, has_variety, reshape, default_sort_key, ordered,
rotations, filldedent, lambdify, source, threaded, xthreaded, public,
memoize_property, timed)
from .integrals import (integrate, Integral, line_integrate, mellin_transform,
inverse_mellin_transform, MellinTransform, InverseMellinTransform,
laplace_transform, inverse_laplace_transform, LaplaceTransform,
InverseLaplaceTransform, fourier_transform, inverse_fourier_transform,
FourierTransform, InverseFourierTransform, sine_transform,
inverse_sine_transform, SineTransform, InverseSineTransform,
cosine_transform, inverse_cosine_transform, CosineTransform,
InverseCosineTransform, hankel_transform, inverse_hankel_transform,
HankelTransform, InverseHankelTransform, singularityintegrate)
from .tensor import (IndexedBase, Idx, Indexed, get_contraction_structure,
get_indices, MutableDenseNDimArray, ImmutableDenseNDimArray,
MutableSparseNDimArray, ImmutableSparseNDimArray, NDimArray,
tensorproduct, tensorcontraction, derive_by_array, permutedims, Array,
DenseNDimArray, SparseNDimArray)
from .parsing import parse_expr
from .calculus import (euler_equations, singularities, is_increasing,
is_strictly_increasing, is_decreasing, is_strictly_decreasing,
is_monotonic, finite_diff_weights, apply_finite_diff, as_finite_diff,
differentiate_finite, periodicity, not_empty_in, AccumBounds,
is_convex, stationary_points, minimum, maximum)
from .algebras import Quaternion
from .printing import (pager_print, pretty, pretty_print, pprint,
pprint_use_unicode, pprint_try_use_unicode, latex, print_latex,
multiline_latex, mathml, print_mathml, python, print_python, pycode,
ccode, print_ccode, glsl_code, print_glsl, cxxcode, fcode,
print_fcode, rcode, print_rcode, jscode, print_jscode, julia_code,
mathematica_code, octave_code, rust_code, print_gtk, preview, srepr,
print_tree, StrPrinter, sstr, sstrrepr, TableForm, dotprint,
maple_code, print_maple_code)
from .testing import test, doctest
# This module causes conflicts with other modules:
# from .stats import *
# Adds about .04-.05 seconds of import time
# from combinatorics import *
# This module is slow to import:
#from physics import units
from .plotting import plot, textplot, plot_backends, plot_implicit, plot_parametric
from .interactive import init_session, init_printing
evalf._create_evalf_table()
# This is slow to import:
#import abc
from .deprecated import C, ClassRegistry, class_registry
__all__ = [
# sympy.core
'sympify', 'SympifyError', 'cacheit', 'Basic', 'Atom',
'preorder_traversal', 'S', 'Expr', 'AtomicExpr', 'UnevaluatedExpr',
'Symbol', 'Wild', 'Dummy', 'symbols', 'var', 'Number', 'Float',
'Rational', 'Integer', 'NumberSymbol', 'RealNumber', 'igcd', 'ilcm',
'seterr', 'E', 'I', 'nan', 'oo', 'pi', 'zoo', 'AlgebraicNumber', 'comp',
'mod_inverse', 'Pow', 'integer_nthroot', 'integer_log', 'Mul', 'prod',
'Add', 'Mod', 'Rel', 'Eq', 'Ne', 'Lt', 'Le', 'Gt', 'Ge', 'Equality',
'GreaterThan', 'LessThan', 'Unequality', 'StrictGreaterThan',
'StrictLessThan', 'vectorize', 'Lambda', 'WildFunction', 'Derivative',
'diff', 'FunctionClass', 'Function', 'Subs', 'expand', 'PoleError',
'count_ops', 'expand_mul', 'expand_log', 'expand_func', 'expand_trig',
'expand_complex', 'expand_multinomial', 'nfloat', 'expand_power_base',
'expand_power_exp', 'arity', 'PrecisionExhausted', 'N', 'evalf', 'Tuple',
'Dict', 'gcd_terms', 'factor_terms', 'factor_nc', 'evaluate', 'Catalan',
'EulerGamma', 'GoldenRatio', 'TribonacciConstant',
# sympy.logic
'to_cnf', 'to_dnf', 'to_nnf', 'And', 'Or', 'Not', 'Xor', 'Nand', 'Nor',
'Implies', 'Equivalent', 'ITE', 'POSform', 'SOPform', 'simplify_logic',
'bool_map', 'true', 'false', 'satisfiable',
# sympy.assumptions
'AppliedPredicate', 'Predicate', 'AssumptionsContext', 'assuming', 'Q',
'ask', 'register_handler', 'remove_handler', 'refine',
# sympy.polys
'Poly', 'PurePoly', 'poly_from_expr', 'parallel_poly_from_expr', 'degree',
'total_degree', 'degree_list', 'LC', 'LM', 'LT', 'pdiv', 'prem', 'pquo',
'pexquo', 'div', 'rem', 'quo', 'exquo', 'half_gcdex', 'gcdex', 'invert',
'subresultants', 'resultant', 'discriminant', 'cofactors', 'gcd_list',
'gcd', 'lcm_list', 'lcm', 'terms_gcd', 'trunc', 'monic', 'content',
'primitive', 'compose', 'decompose', 'sturm', 'gff_list', 'gff',
'sqf_norm', 'sqf_part', 'sqf_list', 'sqf', 'factor_list', 'factor',
'intervals', 'refine_root', 'count_roots', 'real_roots', 'nroots',
'ground_roots', 'nth_power_roots_poly', 'cancel', 'reduced', 'groebner',
'is_zero_dimensional', 'GroebnerBasis', 'poly', 'symmetrize', 'horner',
'interpolate', 'rational_interpolate', 'viete', 'together',
'BasePolynomialError', 'ExactQuotientFailed', 'PolynomialDivisionFailed',
'OperationNotSupported', 'HeuristicGCDFailed', 'HomomorphismFailed',
'IsomorphismFailed', 'ExtraneousFactors', 'EvaluationFailed',
'RefinementFailed', 'CoercionFailed', 'NotInvertible', 'NotReversible',
'NotAlgebraic', 'DomainError', 'PolynomialError', 'UnificationFailed',
'GeneratorsError', 'GeneratorsNeeded', 'ComputationFailed',
'UnivariatePolynomialError', 'MultivariatePolynomialError',
'PolificationFailed', 'OptionError', 'FlagError', 'minpoly',
'minimal_polynomial', 'primitive_element', 'field_isomorphism',
'to_number_field', 'isolate', 'itermonomials', 'Monomial', 'lex', 'grlex',
'grevlex', 'ilex', 'igrlex', 'igrevlex', 'CRootOf', 'rootof', 'RootOf',
'ComplexRootOf', 'RootSum', 'roots', 'Domain', 'FiniteField',
'IntegerRing', 'RationalField', 'RealField', 'ComplexField',
'PythonFiniteField', 'GMPYFiniteField', 'PythonIntegerRing',
'GMPYIntegerRing', 'PythonRational', 'GMPYRationalField',
'AlgebraicField', 'PolynomialRing', 'FractionField', 'ExpressionDomain',
'FF_python', 'FF_gmpy', 'ZZ_python', 'ZZ_gmpy', 'QQ_python', 'QQ_gmpy',
'GF', 'FF', 'ZZ', 'QQ', 'ZZ_I', 'QQ_I', 'RR', 'CC', 'EX', 'construct_domain',
'swinnerton_dyer_poly', 'cyclotomic_poly', 'symmetric_poly',
'random_poly', 'interpolating_poly', 'jacobi_poly', 'chebyshevt_poly',
'chebyshevu_poly', 'hermite_poly', 'legendre_poly', 'laguerre_poly',
'apart', 'apart_list', 'assemble_partfrac_list', 'Options', 'ring',
'xring', 'vring', 'sring', 'field', 'xfield', 'vfield', 'sfield',
# sympy.series
'Order', 'O', 'limit', 'Limit', 'gruntz', 'series', 'approximants',
'residue', 'EmptySequence', 'SeqPer', 'SeqFormula', 'sequence', 'SeqAdd',
'SeqMul', 'fourier_series', 'fps', 'difference_delta', 'limit_seq',
# sympy.functions
'factorial', 'factorial2', 'rf', 'ff', 'binomial', 'RisingFactorial',
'FallingFactorial', 'subfactorial', 'carmichael', 'fibonacci', 'lucas',
'tribonacci', 'harmonic', 'bernoulli', 'bell', 'euler', 'catalan',
'genocchi', 'partition', 'sqrt', 'root', 'Min', 'Max', 'Id', 'real_root',
'cbrt', 're', 'im', 'sign', 'Abs', 'conjugate', 'arg', 'polar_lift',
'periodic_argument', 'unbranched_argument', 'principal_branch',
'transpose', 'adjoint', 'polarify', 'unpolarify', 'sin', 'cos', 'tan',
'sec', 'csc', 'cot', 'sinc', 'asin', 'acos', 'atan', 'asec', 'acsc',
'acot', 'atan2', 'exp_polar', 'exp', 'ln', 'log', 'LambertW', 'sinh',
'cosh', 'tanh', 'coth', 'sech', 'csch', 'asinh', 'acosh', 'atanh',
'acoth', 'asech', 'acsch', 'floor', 'ceiling', 'frac', 'Piecewise',
'piecewise_fold', 'erf', 'erfc', 'erfi', 'erf2', 'erfinv', 'erfcinv',
'erf2inv', 'Ei', 'expint', 'E1', 'li', 'Li', 'Si', 'Ci', 'Shi', 'Chi',
'fresnels', 'fresnelc', 'gamma', 'lowergamma', 'uppergamma', 'polygamma',
'loggamma', 'digamma', 'trigamma', 'multigamma', 'dirichlet_eta', 'zeta',
'lerchphi', 'polylog', 'stieltjes', 'Eijk', 'LeviCivita',
'KroneckerDelta', 'SingularityFunction', 'DiracDelta', 'Heaviside',
'bspline_basis', 'bspline_basis_set', 'interpolating_spline', 'besselj',
'bessely', 'besseli', 'besselk', 'hankel1', 'hankel2', 'jn', 'yn',
'jn_zeros', 'hn1', 'hn2', 'airyai', 'airybi', 'airyaiprime',
'airybiprime', 'marcumq', 'hyper', 'meijerg', 'appellf1', 'legendre',
'assoc_legendre', 'hermite', 'chebyshevt', 'chebyshevu',
'chebyshevu_root', 'chebyshevt_root', 'laguerre', 'assoc_laguerre',
'gegenbauer', 'jacobi', 'jacobi_normalized', 'Ynm', 'Ynm_c', 'Znm',
'elliptic_k', 'elliptic_f', 'elliptic_e', 'elliptic_pi', 'beta',
'mathieus', 'mathieuc', 'mathieusprime', 'mathieucprime',
# sympy.ntheory
'nextprime', 'prevprime', 'prime', 'primepi', 'primerange', 'randprime',
'Sieve', 'sieve', 'primorial', 'cycle_length', 'composite', 'compositepi',
'isprime', 'divisors', 'proper_divisors', 'factorint', 'multiplicity',
'perfect_power', 'pollard_pm1', 'pollard_rho', 'primefactors', 'totient',
'trailing', 'divisor_count', 'proper_divisor_count', 'divisor_sigma',
'factorrat', 'reduced_totient', 'primenu', 'primeomega',
'mersenne_prime_exponent', 'is_perfect', 'is_mersenne_prime',
'is_abundant', 'is_deficient', 'is_amicable', 'abundance', 'npartitions',
'is_primitive_root', 'is_quad_residue', 'legendre_symbol',
'jacobi_symbol', 'n_order', 'sqrt_mod', 'quadratic_residues',
'primitive_root', 'nthroot_mod', 'is_nthpow_residue', 'sqrt_mod_iter',
'mobius', 'discrete_log', 'quadratic_congruence', 'binomial_coefficients',
'binomial_coefficients_list', 'multinomial_coefficients',
'continued_fraction_periodic', 'continued_fraction_iterator',
'continued_fraction_reduce', 'continued_fraction_convergents',
'continued_fraction', 'egyptian_fraction',
# sympy.concrete
'product', 'Product', 'summation', 'Sum',
# sympy.discrete
'fft', 'ifft', 'ntt', 'intt', 'fwht', 'ifwht', 'mobius_transform',
'inverse_mobius_transform', 'convolution', 'covering_product',
'intersecting_product',
# sympy.simplify
'simplify', 'hypersimp', 'hypersimilar', 'logcombine', 'separatevars',
'posify', 'besselsimp', 'kroneckersimp', 'signsimp', 'bottom_up',
'nsimplify', 'FU', 'fu', 'sqrtdenest', 'cse', 'use', 'epath', 'EPath',
'hyperexpand', 'collect', 'rcollect', 'radsimp', 'collect_const',
'fraction', 'numer', 'denom', 'trigsimp', 'exptrigsimp', 'powsimp',
'powdenest', 'combsimp', 'gammasimp', 'ratsimp', 'ratsimpmodprime',
# sympy.sets
'Set', 'Interval', 'Union', 'EmptySet', 'FiniteSet', 'ProductSet',
'Intersection', 'imageset', 'DisjointUnion', 'Complement', 'SymmetricDifference',
'ImageSet', 'Range', 'ComplexRegion', 'Reals', 'Contains', 'ConditionSet',
'Ordinal', 'OmegaPower', 'ord0', 'PowerSet', 'Reals', 'Naturals',
'Naturals0', 'UniversalSet', 'Integers', 'Rationals',
# sympy.solvers
'solve', 'solve_linear_system', 'solve_linear_system_LU',
'solve_undetermined_coeffs', 'nsolve', 'solve_linear', 'checksol',
'det_quick', 'inv_quick', 'check_assumptions', 'failing_assumptions',
'diophantine', 'rsolve', 'rsolve_poly', 'rsolve_ratio', 'rsolve_hyper',
'checkodesol', 'classify_ode', 'dsolve', 'homogeneous_order',
'solve_poly_system', 'solve_triangulated', 'pde_separate',
'pde_separate_add', 'pde_separate_mul', 'pdsolve', 'classify_pde',
'checkpdesol', 'ode_order', 'reduce_inequalities',
'reduce_abs_inequality', 'reduce_abs_inequalities',
'solve_poly_inequality', 'solve_rational_inequalities',
'solve_univariate_inequality', 'decompogen', 'solveset', 'linsolve',
'linear_eq_to_matrix', 'nonlinsolve', 'substitution', 'Complexes',
# sympy.matrices
'ShapeError', 'NonSquareMatrixError', 'GramSchmidt', 'casoratian', 'diag',
'eye', 'hessian', 'jordan_cell', 'list2numpy', 'matrix2numpy',
'matrix_multiply_elementwise', 'ones', 'randMatrix', 'rot_axis1',
'rot_axis2', 'rot_axis3', 'symarray', 'wronskian', 'zeros',
'MutableDenseMatrix', 'DeferredVector', 'MatrixBase', 'Matrix',
'MutableMatrix', 'MutableSparseMatrix', 'banded', 'ImmutableDenseMatrix',
'ImmutableSparseMatrix', 'ImmutableMatrix', 'SparseMatrix', 'MatrixSlice',
'BlockDiagMatrix', 'BlockMatrix', 'FunctionMatrix', 'Identity', 'Inverse',
'MatAdd', 'MatMul', 'MatPow', 'MatrixExpr', 'MatrixSymbol', 'Trace',
'Transpose', 'ZeroMatrix', 'OneMatrix', 'blockcut', 'block_collapse',
'matrix_symbols', 'Adjoint', 'hadamard_product', 'HadamardProduct',
'HadamardPower', 'Determinant', 'det', 'diagonalize_vector', 'DiagMatrix',
'DiagonalMatrix', 'DiagonalOf', 'trace', 'DotProduct',
'kronecker_product', 'KroneckerProduct', 'PermutationMatrix',
'MatrixPermute',
# sympy.geometry
'Point', 'Point2D', 'Point3D', 'Line', 'Ray', 'Segment', 'Line2D',
'Segment2D', 'Ray2D', 'Line3D', 'Segment3D', 'Ray3D', 'Plane', 'Ellipse',
'Circle', 'Polygon', 'RegularPolygon', 'Triangle', 'rad', 'deg',
'are_similar', 'centroid', 'convex_hull', 'idiff', 'intersection',
'closest_points', 'farthest_points', 'GeometryError', 'Curve', 'Parabola',
# sympy.utilities
'flatten', 'group', 'take', 'subsets', 'variations', 'numbered_symbols',
'cartes', 'capture', 'dict_merge', 'postorder_traversal',
'interactive_traversal', 'prefixes', 'postfixes', 'sift',
'topological_sort', 'unflatten', 'has_dups', 'has_variety', 'reshape',
'default_sort_key', 'ordered', 'rotations', 'filldedent', 'lambdify',
'source', 'threaded', 'xthreaded', 'public', 'memoize_property', 'test',
'doctest', 'timed',
# sympy.integrals
'integrate', 'Integral', 'line_integrate', 'mellin_transform',
'inverse_mellin_transform', 'MellinTransform', 'InverseMellinTransform',
'laplace_transform', 'inverse_laplace_transform', 'LaplaceTransform',
'InverseLaplaceTransform', 'fourier_transform',
'inverse_fourier_transform', 'FourierTransform',
'InverseFourierTransform', 'sine_transform', 'inverse_sine_transform',
'SineTransform', 'InverseSineTransform', 'cosine_transform',
'inverse_cosine_transform', 'CosineTransform', 'InverseCosineTransform',
'hankel_transform', 'inverse_hankel_transform', 'HankelTransform',
'InverseHankelTransform', 'singularityintegrate',
# sympy.tensor
'IndexedBase', 'Idx', 'Indexed', 'get_contraction_structure',
'get_indices', 'MutableDenseNDimArray', 'ImmutableDenseNDimArray',
'MutableSparseNDimArray', 'ImmutableSparseNDimArray', 'NDimArray',
'tensorproduct', 'tensorcontraction', 'derive_by_array', 'permutedims',
'Array', 'DenseNDimArray', 'SparseNDimArray',
# sympy.parsing
'parse_expr',
# sympy.calculus
'euler_equations', 'singularities', 'is_increasing',
'is_strictly_increasing', 'is_decreasing', 'is_strictly_decreasing',
'is_monotonic', 'finite_diff_weights', 'apply_finite_diff',
'as_finite_diff', 'differentiate_finite', 'periodicity', 'not_empty_in',
'AccumBounds', 'is_convex', 'stationary_points', 'minimum', 'maximum',
# sympy.algebras
'Quaternion',
# sympy.printing
'pager_print', 'pretty', 'pretty_print', 'pprint', 'pprint_use_unicode',
'pprint_try_use_unicode', 'latex', 'print_latex', 'multiline_latex',
'mathml', 'print_mathml', 'python', 'print_python', 'pycode', 'ccode',
'print_ccode', 'glsl_code', 'print_glsl', 'cxxcode', 'fcode',
'print_fcode', 'rcode', 'print_rcode', 'jscode', 'print_jscode',
'julia_code', 'mathematica_code', 'octave_code', 'rust_code', 'print_gtk',
'preview', 'srepr', 'print_tree', 'StrPrinter', 'sstr', 'sstrrepr',
'TableForm', 'dotprint', 'maple_code', 'print_maple_code',
# sympy.plotting
'plot', 'textplot', 'plot_backends', 'plot_implicit', 'plot_parametric',
# sympy.interactive
'init_session', 'init_printing',
# sympy.testing
'test', 'doctest',
# sympy.deprecated:
'C', 'ClassRegistry', 'class_registry',
]
#===========================================================================#
# #
# XXX: The names below were importable before sympy 1.6 using #
# #
# from sympy import * #
# #
# This happened implicitly because there was no __all__ defined in this #
# __init__.py file. Not every package is imported. The list matches what #
# would have been imported before. It is possible that these packages will #
# not be imported by a star-import from sympy in future. #
# #
#===========================================================================#
__all__.extend([
'algebras',
'assumptions',
'calculus',
'codegen',
'combinatorics',
'concrete',
'deprecated',
'discrete',
'external',
'functions',
'geometry',
'interactive',
'multipledispatch',
'ntheory',
'parsing',
'plotting',
'polys',
'printing',
'release',
'strategies',
'tensor',
'utilities',
])
#===========================================================================#
# #
# XXX: The names listed in _DEPRECATED_IMPORTS below were importable before #
# sympy 1.6 using #
# #
# from sympy import * #
# #
# This happened implicitly because there was no __all__ defined in this #
# __init__.py file. The plan is to remove them but for now they remain #
# importable but will give a deprecation warning when used. In future these #
# names will be removed and will not be importable from here. #
# #
#===========================================================================#
class DeprecatedImportModule:
# Add a docstring that someone can see if calling help on these objects
"""Deprecated imported module object.
See https://github.com/sympy/sympy/pull/19316
This is a wrapper around a module that has been imported incorrectly.
Previously this module was importable using
from sympy import *
or (for example)
from sympy import add
However it was unintentional that this module would be imported in that
way and it will be removed in a future sympy version. If you do need to
use this module then the correct way to import it is to give its full
module path e.g.
import sympy.core.add as add
"""
from sympy.utilities.exceptions import SymPyDeprecationWarning as Warn
import sys
sympy = sys.modules[__name__]
_DEPRECATED_IMPORTS = [
'sympy.concrete.expr_with_intlimits',
'sympy.concrete.expr_with_limits',
'sympy.concrete.gosper',
'sympy.concrete.products',
'sympy.concrete.summations',
'sympy.core.add',
'sympy.core.basic',
'sympy.core.cache',
'sympy.core.compatibility',
'sympy.core.containers',
'sympy.core.coreerrors',
'sympy.core.decorators',
'sympy.core.expr',
'sympy.core.exprtools',
'sympy.core.facts',
'sympy.core.function',
'sympy.core.logic',
'sympy.core.mod',
'sympy.core.mul',
'sympy.core.multidimensional',
'sympy.core.numbers',
'sympy.core.operations',
'sympy.core.power',
'sympy.core.relational',
'sympy.core.rules',
'sympy.core.singleton',
'sympy.core.symbol',
'sympy.discrete.convolutions',
'sympy.geometry.curve',
'sympy.geometry.ellipse',
'sympy.geometry.entity',
'sympy.geometry.exceptions',
'sympy.geometry.line',
'sympy.geometry.parabola',
'sympy.geometry.plane',
'sympy.geometry.point',
'sympy.geometry.polygon',
'sympy.geometry.util',
'sympy.integrals.integrals',
'sympy.integrals.manualintegrate',
'sympy.integrals.meijerint',
'sympy.integrals.singularityfunctions',
'sympy.integrals.transforms',
'sympy.integrals.trigonometry',
'sympy.logic.boolalg',
'sympy.logic.inference',
'sympy.matrices.common',
'sympy.matrices.dense',
'sympy.matrices.expressions',
'sympy.matrices.immutable',
'sympy.matrices.matrices',
'sympy.matrices.sparse',
'sympy.matrices.sparsetools',
'sympy.ntheory.factor_',
'sympy.ntheory.generate',
'sympy.ntheory.multinomial',
'sympy.ntheory.partitions_',
'sympy.ntheory.primetest',
'sympy.ntheory.residue_ntheory',
'sympy.sets.conditionset',
'sympy.sets.contains',
'sympy.sets.fancysets',
'sympy.sets.ordinals',
'sympy.sets.powerset',
'sympy.sets.sets',
'sympy.simplify.cse_main',
'sympy.simplify.cse_opts',
'sympy.simplify.epathtools',
'sympy.simplify.traversaltools',
'sympy.solvers.bivariate',
'sympy.solvers.deutils',
'sympy.solvers.inequalities',
'sympy.solvers.ode',
'sympy.solvers.pde',
'sympy.solvers.polysys',
'sympy.solvers.recurr',
'sympy.solvers.solvers',
'sympy.tensor.array',
'sympy.tensor.index_methods',
'sympy.tensor.indexed'
]
def __init__(self, modname):
from importlib import import_module
self.modname = modname
self.mod = import_module(modname)
def __getattr__(self, name):
self.Warn(
feature="importing %s with 'from sympy import *'" % self.modname,
useinstead="import %s" % self.modname,
issue=18245,
deprecated_since_version="1.6").warn()
return getattr(self.mod, name)
@classmethod
def inject_imports(cls):
for modname in cls._DEPRECATED_IMPORTS:
name = modname.split('.')[-1]
deprecated_mod = cls(modname)
setattr(cls.sympy, name, deprecated_mod)
__all__.append(name)
DeprecatedImportModule.inject_imports()
del DeprecatedImportModule
|
11ea83913d7b97476fadf1cedc82fd0936e9173c82b11ed51863c082b4c8e97e | #!/usr/bin/env python
"""FEM library
Demonstrates some simple finite element definitions, and computes a mass
matrix
$ python fem.py
[ 1/60, 0, -1/360, 0, -1/90, -1/360]
[ 0, 4/45, 0, 2/45, 2/45, -1/90]
[-1/360, 0, 1/60, -1/90, 0, -1/360]
[ 0, 2/45, -1/90, 4/45, 2/45, 0]
[ -1/90, 2/45, 0, 2/45, 4/45, 0]
[-1/360, -1/90, -1/360, 0, 0, 1/60]
"""
from sympy import symbols, Symbol, factorial, Rational, zeros, eye, \
integrate, diff, pprint, reduced
x, y, z = symbols('x,y,z')
class ReferenceSimplex:
def __init__(self, nsd):
self.nsd = nsd
if nsd <= 3:
coords = symbols('x,y,z')[:nsd]
else:
coords = [Symbol("x_%d" % d) for d in range(nsd)]
self.coords = coords
def integrate(self, f):
coords = self.coords
nsd = self.nsd
limit = 1
for p in coords:
limit -= p
intf = f
for d in range(0, nsd):
p = coords[d]
limit += p
intf = integrate(intf, (p, 0, limit))
return intf
def bernstein_space(order, nsd):
if nsd > 3:
raise RuntimeError("Bernstein only implemented in 1D, 2D, and 3D")
sum = 0
basis = []
coeff = []
if nsd == 1:
b1, b2 = x, 1 - x
for o1 in range(0, order + 1):
for o2 in range(0, order + 1):
if o1 + o2 == order:
aij = Symbol("a_%d_%d" % (o1, o2))
sum += aij*binomial(order, o1)*pow(b1, o1)*pow(b2, o2)
basis.append(binomial(order, o1)*pow(b1, o1)*pow(b2, o2))
coeff.append(aij)
if nsd == 2:
b1, b2, b3 = x, y, 1 - x - y
for o1 in range(0, order + 1):
for o2 in range(0, order + 1):
for o3 in range(0, order + 1):
if o1 + o2 + o3 == order:
aij = Symbol("a_%d_%d_%d" % (o1, o2, o3))
fac = factorial(order) / (factorial(o1)*factorial(o2)*factorial(o3))
sum += aij*fac*pow(b1, o1)*pow(b2, o2)*pow(b3, o3)
basis.append(fac*pow(b1, o1)*pow(b2, o2)*pow(b3, o3))
coeff.append(aij)
if nsd == 3:
b1, b2, b3, b4 = x, y, z, 1 - x - y - z
for o1 in range(0, order + 1):
for o2 in range(0, order + 1):
for o3 in range(0, order + 1):
for o4 in range(0, order + 1):
if o1 + o2 + o3 + o4 == order:
aij = Symbol("a_%d_%d_%d_%d" % (o1, o2, o3, o4))
fac = factorial(order)/(factorial(o1)*factorial(o2)*factorial(o3)*factorial(o4))
sum += aij*fac*pow(b1, o1)*pow(b2, o2)*pow(b3, o3)*pow(b4, o4)
basis.append(fac*pow(b1, o1)*pow(b2, o2)*pow(b3, o3)*pow(b4, o4))
coeff.append(aij)
return sum, coeff, basis
def create_point_set(order, nsd):
h = Rational(1, order)
set = []
if nsd == 1:
for i in range(0, order + 1):
x = i*h
if x <= 1:
set.append((x, y))
if nsd == 2:
for i in range(0, order + 1):
x = i*h
for j in range(0, order + 1):
y = j*h
if x + y <= 1:
set.append((x, y))
if nsd == 3:
for i in range(0, order + 1):
x = i*h
for j in range(0, order + 1):
y = j*h
for k in range(0, order + 1):
z = k*h
if x + y + z <= 1:
set.append((x, y, z))
return set
def create_matrix(equations, coeffs):
A = zeros(len(equations))
i = 0
j = 0
for j in range(0, len(coeffs)):
c = coeffs[j]
for i in range(0, len(equations)):
e = equations[i]
d, _ = reduced(e, [c])
A[i, j] = d[0]
return A
class Lagrange:
def __init__(self, nsd, order):
self.nsd = nsd
self.order = order
self.compute_basis()
def nbf(self):
return len(self.N)
def compute_basis(self):
order = self.order
nsd = self.nsd
N = []
pol, coeffs, basis = bernstein_space(order, nsd)
points = create_point_set(order, nsd)
equations = []
for p in points:
ex = pol.subs(x, p[0])
if nsd > 1:
ex = ex.subs(y, p[1])
if nsd > 2:
ex = ex.subs(z, p[2])
equations.append(ex)
A = create_matrix(equations, coeffs)
Ainv = A.inv()
b = eye(len(equations))
xx = Ainv*b
for i in range(0, len(equations)):
Ni = pol
for j in range(0, len(coeffs)):
Ni = Ni.subs(coeffs[j], xx[j, i])
N.append(Ni)
self.N = N
def main():
t = ReferenceSimplex(2)
fe = Lagrange(2, 2)
u = 0
# compute u = sum_i u_i N_i
us = []
for i in range(0, fe.nbf()):
ui = Symbol("u_%d" % i)
us.append(ui)
u += ui*fe.N[i]
J = zeros(fe.nbf())
for i in range(0, fe.nbf()):
Fi = u*fe.N[i]
print(Fi)
for j in range(0, fe.nbf()):
uj = us[j]
integrands = diff(Fi, uj)
print(integrands)
J[j, i] = t.integrate(integrands)
pprint(J)
if __name__ == "__main__":
main()
|
a87475b69f38fe22975a421497fc4a6e33e68cd37dd18511397184747dbea475 | from __future__ import print_function, division
import random
from sympy import (Matrix, MatrixSymbol, S, Indexed, Basic,
Set, And, Eq, FiniteSet, ImmutableMatrix,
Lambda, Mul, Dummy, IndexedBase, Add,
linsolve, eye, Or, Not, Intersection,
Union, Expr, Function, exp, cacheit,
Ge, Piecewise, Symbol, NonSquareMatrixError)
from sympy.core.relational import Relational
from sympy.logic.boolalg import Boolean
from sympy.stats.joint_rv import JointDistributionHandmade, JointDistribution
from sympy.stats.rv import (RandomIndexedSymbol, random_symbols, RandomSymbol,
_symbol_converter, _value_check, pspace, given,
dependent, is_random, sample_iter)
from sympy.stats.stochastic_process import StochasticPSpace
from sympy.stats.symbolic_probability import Probability, Expectation
from sympy.stats.frv_types import Bernoulli, BernoulliDistribution, FiniteRV
from sympy.core.sympify import _sympify
__all__ = [
'StochasticProcess',
'DiscreteTimeStochasticProcess',
'DiscreteMarkovChain',
'TransitionMatrixOf',
'StochasticStateSpaceOf',
'GeneratorMatrixOf',
'ContinuousMarkovChain',
'BernoulliProcess'
]
@is_random.register(Indexed)
def _(x):
return is_random(x.base)
@is_random.register(RandomIndexedSymbol)
def _(x):
return True
def _set_converter(itr):
"""
Helper function for converting list/tuple/set to Set.
If parameter is not an instance of list/tuple/set then
no operation is performed.
Returns
=======
Set
The argument converted to Set.
Raises
======
TypeError
If the argument is not an instance of list/tuple/set.
"""
if isinstance(itr, (list, tuple, set)):
itr = FiniteSet(*itr)
if not isinstance(itr, Set):
raise TypeError("%s is not an instance of list/tuple/set."%(itr))
return itr
def _sym_sympify(arg):
"""
Converts an arbitrary expression to a type that can be used inside SymPy.
As generally strings are unwise to use in the expressions,
it returns the Symbol of argument if the string type argument is passed.
Parameters
=========
arg: The parameter to be converted to be used in Sympy.
Returns
=======
The converted parameter.
"""
if isinstance(arg, str):
return Symbol(arg)
else:
return _sympify(arg)
def _matrix_checks(matrix):
if not isinstance(matrix, (Matrix, MatrixSymbol, ImmutableMatrix)):
raise TypeError("Transition probabilities either should "
"be a Matrix or a MatrixSymbol.")
if matrix.shape[0] != matrix.shape[1]:
raise NonSquareMatrixError("%s is not a square matrix"%(matrix))
if isinstance(matrix, Matrix):
matrix = ImmutableMatrix(matrix.tolist())
return matrix
class StochasticProcess(Basic):
"""
Base class for all the stochastic processes whether
discrete or continuous.
Parameters
==========
sym: Symbol or str
state_space: Set
The state space of the stochastic process, by default S.Reals.
For discrete sets it is zero indexed.
See Also
========
DiscreteTimeStochasticProcess
"""
index_set = S.Reals
def __new__(cls, sym, state_space=S.Reals, **kwargs):
sym = _symbol_converter(sym)
state_space = _set_converter(state_space)
return Basic.__new__(cls, sym, state_space)
@property
def symbol(self):
return self.args[0]
@property
def state_space(self):
return self.args[1]
def __call__(self, time):
"""
Overridden in ContinuousTimeStochasticProcess.
"""
raise NotImplementedError("Use [] for indexing discrete time stochastic process.")
def __getitem__(self, time):
"""
Overridden in DiscreteTimeStochasticProcess.
"""
raise NotImplementedError("Use () for indexing continuous time stochastic process.")
def probability(self, condition):
raise NotImplementedError()
def joint_distribution(self, *args):
"""
Computes the joint distribution of the random indexed variables.
Parameters
==========
args: iterable
The finite list of random indexed variables/the key of a stochastic
process whose joint distribution has to be computed.
Returns
=======
JointDistribution
The joint distribution of the list of random indexed variables.
An unevaluated object is returned if it is not possible to
compute the joint distribution.
Raises
======
ValueError: When the arguments passed are not of type RandomIndexSymbol
or Number.
"""
args = list(args)
for i, arg in enumerate(args):
if S(arg).is_Number:
if self.index_set.is_subset(S.Integers):
args[i] = self.__getitem__(arg)
else:
args[i] = self.__call__(arg)
elif not isinstance(arg, RandomIndexedSymbol):
raise ValueError("Expected a RandomIndexedSymbol or "
"key not %s"%(type(arg)))
if args[0].pspace.distribution == None: # checks if there is any distribution available
return JointDistribution(*args)
pdf = Lambda(tuple(args),
expr=Mul.fromiter(arg.pspace.process.density(arg) for arg in args))
return JointDistributionHandmade(pdf)
def expectation(self, condition, given_condition):
raise NotImplementedError("Abstract method for expectation queries.")
def sample(self):
raise NotImplementedError("Abstract method for sampling queries.")
class DiscreteTimeStochasticProcess(StochasticProcess):
"""
Base class for all discrete stochastic processes.
"""
def __getitem__(self, time):
"""
For indexing discrete time stochastic processes.
Returns
=======
RandomIndexedSymbol
"""
if time not in self.index_set:
raise IndexError("%s is not in the index set of %s"%(time, self.symbol))
idx_obj = Indexed(self.symbol, time)
distribution = getattr(self, 'distribution', None)
pspace_obj = StochasticPSpace(self.symbol, self, distribution)
return RandomIndexedSymbol(idx_obj, pspace_obj)
class ContinuousTimeStochasticProcess(StochasticProcess):
"""
Base class for all continuous time stochastic process.
"""
def __call__(self, time):
"""
For indexing continuous time stochastic processes.
Returns
=======
RandomIndexedSymbol
"""
if time not in self.index_set:
raise IndexError("%s is not in the index set of %s"%(time, self.symbol))
func_obj = Function(self.symbol)(time)
pspace_obj = StochasticPSpace(self.symbol, self)
return RandomIndexedSymbol(func_obj, pspace_obj)
class TransitionMatrixOf(Boolean):
"""
Assumes that the matrix is the transition matrix
of the process.
"""
def __new__(cls, process, matrix):
if not isinstance(process, DiscreteMarkovChain):
raise ValueError("Currently only DiscreteMarkovChain "
"support TransitionMatrixOf.")
matrix = _matrix_checks(matrix)
return Basic.__new__(cls, process, matrix)
process = property(lambda self: self.args[0])
matrix = property(lambda self: self.args[1])
class GeneratorMatrixOf(TransitionMatrixOf):
"""
Assumes that the matrix is the generator matrix
of the process.
"""
def __new__(cls, process, matrix):
if not isinstance(process, ContinuousMarkovChain):
raise ValueError("Currently only ContinuousMarkovChain "
"support GeneratorMatrixOf.")
matrix = _matrix_checks(matrix)
return Basic.__new__(cls, process, matrix)
class StochasticStateSpaceOf(Boolean):
def __new__(cls, process, state_space):
if not isinstance(process, (DiscreteMarkovChain, ContinuousMarkovChain)):
raise ValueError("Currently only DiscreteMarkovChain and ContinuousMarkovChain "
"support StochasticStateSpaceOf.")
state_space = _set_converter(state_space)
return Basic.__new__(cls, process, state_space)
process = property(lambda self: self.args[0])
state_space = property(lambda self: self.args[1])
class MarkovProcess(StochasticProcess):
"""
Contains methods that handle queries
common to Markov processes.
"""
def _extract_information(self, given_condition):
"""
Helper function to extract information, like,
transition matrix/generator matrix, state space, etc.
"""
if isinstance(self, DiscreteMarkovChain):
trans_probs = self.transition_probabilities
elif isinstance(self, ContinuousMarkovChain):
trans_probs = self.generator_matrix
state_space = self.state_space
if isinstance(given_condition, And):
gcs = given_condition.args
given_condition = S.true
for gc in gcs:
if isinstance(gc, TransitionMatrixOf):
trans_probs = gc.matrix
if isinstance(gc, StochasticStateSpaceOf):
state_space = gc.state_space
if isinstance(gc, Relational):
given_condition = given_condition & gc
if isinstance(given_condition, TransitionMatrixOf):
trans_probs = given_condition.matrix
given_condition = S.true
if isinstance(given_condition, StochasticStateSpaceOf):
state_space = given_condition.state_space
given_condition = S.true
return trans_probs, state_space, given_condition
def _check_trans_probs(self, trans_probs, row_sum=1):
"""
Helper function for checking the validity of transition
probabilities.
"""
if not isinstance(trans_probs, MatrixSymbol):
rows = trans_probs.tolist()
for row in rows:
if (sum(row) - row_sum) != 0:
raise ValueError("Values in a row must sum to %s. "
"If you are using Float or floats then please use Rational."%(row_sum))
def _work_out_state_space(self, state_space, given_condition, trans_probs):
"""
Helper function to extract state space if there
is a random symbol in the given condition.
"""
# if given condition is None, then there is no need to work out
# state_space from random variables
if given_condition != None:
rand_var = list(given_condition.atoms(RandomSymbol) -
given_condition.atoms(RandomIndexedSymbol))
if len(rand_var) == 1:
state_space = rand_var[0].pspace.set
if not FiniteSet(*[i for i in range(trans_probs.shape[0])]).is_subset(state_space):
raise ValueError("state space is not compatible with the transition probabilites.")
state_space = FiniteSet(*[i for i in range(trans_probs.shape[0])])
return state_space
@cacheit
def _preprocess(self, given_condition, evaluate):
"""
Helper function for pre-processing the information.
"""
is_insufficient = False
if not evaluate: # avoid pre-processing if the result is not to be evaluated
return (True, None, None, None)
# extracting transition matrix and state space
trans_probs, state_space, given_condition = self._extract_information(given_condition)
# given_condition does not have sufficient information
# for computations
if trans_probs == None or \
given_condition == None:
is_insufficient = True
else:
# checking transition probabilities
if isinstance(self, DiscreteMarkovChain):
self._check_trans_probs(trans_probs, row_sum=1)
elif isinstance(self, ContinuousMarkovChain):
self._check_trans_probs(trans_probs, row_sum=0)
# working out state space
state_space = self._work_out_state_space(state_space, given_condition, trans_probs)
return is_insufficient, trans_probs, state_space, given_condition
def probability(self, condition, given_condition=None, evaluate=True, **kwargs):
"""
Handles probability queries for Markov process.
Parameters
==========
condition: Relational
given_condition: Relational/And
Returns
=======
Probability
If the information is not sufficient.
Expr
In all other cases.
Note
====
Any information passed at the time of query overrides
any information passed at the time of object creation like
transition probabilities, state space.
Pass the transition matrix using TransitionMatrixOf,
generator matrix using GeneratorMatrixOf and state space
using StochasticStateSpaceOf in given_condition using & or And.
"""
check, mat, state_space, new_given_condition = \
self._preprocess(given_condition, evaluate)
if check:
return Probability(condition, new_given_condition)
if isinstance(self, ContinuousMarkovChain):
trans_probs = self.transition_probabilities(mat)
elif isinstance(self, DiscreteMarkovChain):
trans_probs = mat
if isinstance(condition, Relational):
rv, states = (list(condition.atoms(RandomIndexedSymbol))[0], condition.as_set())
if isinstance(new_given_condition, And):
gcs = new_given_condition.args
else:
gcs = (new_given_condition, )
grvs = new_given_condition.atoms(RandomIndexedSymbol)
min_key_rv = None
for grv in grvs:
if grv.key <= rv.key:
min_key_rv = grv
if min_key_rv == None:
return Probability(condition)
prob, gstate = dict(), None
for gc in gcs:
if gc.has(min_key_rv):
if gc.has(Probability):
p, gp = (gc.rhs, gc.lhs) if isinstance(gc.lhs, Probability) \
else (gc.lhs, gc.rhs)
gr = gp.args[0]
gset = Intersection(gr.as_set(), state_space)
gstate = list(gset)[0]
prob[gset] = p
else:
_, gstate = (gc.lhs.key, gc.rhs) if isinstance(gc.lhs, RandomIndexedSymbol) \
else (gc.rhs.key, gc.lhs)
if any((k not in self.index_set) for k in (rv.key, min_key_rv.key)):
raise IndexError("The timestamps of the process are not in it's index set.")
states = Intersection(states, state_space)
for state in Union(states, FiniteSet(gstate)):
if Ge(state, mat.shape[0]) == True:
raise IndexError("No information is available for (%s, %s) in "
"transition probabilities of shape, (%s, %s). "
"State space is zero indexed."
%(gstate, state, mat.shape[0], mat.shape[1]))
if prob:
gstates = Union(*prob.keys())
if len(gstates) == 1:
gstate = list(gstates)[0]
gprob = list(prob.values())[0]
prob[gstates] = gprob
elif len(gstates) == len(state_space) - 1:
gstate = list(state_space - gstates)[0]
gprob = S.One - sum(prob.values())
prob[state_space - gstates] = gprob
else:
raise ValueError("Conflicting information.")
else:
gprob = S.One
if min_key_rv == rv:
return sum([prob[FiniteSet(state)] for state in states])
if isinstance(self, ContinuousMarkovChain):
return gprob * sum([trans_probs(rv.key - min_key_rv.key).__getitem__((gstate, state))
for state in states])
if isinstance(self, DiscreteMarkovChain):
return gprob * sum([(trans_probs**(rv.key - min_key_rv.key)).__getitem__((gstate, state))
for state in states])
if isinstance(condition, Not):
expr = condition.args[0]
return S.One - self.probability(expr, given_condition, evaluate, **kwargs)
if isinstance(condition, And):
compute_later, state2cond, conds = [], dict(), condition.args
for expr in conds:
if isinstance(expr, Relational):
ris = list(expr.atoms(RandomIndexedSymbol))[0]
if state2cond.get(ris, None) is None:
state2cond[ris] = S.true
state2cond[ris] &= expr
else:
compute_later.append(expr)
ris = []
for ri in state2cond:
ris.append(ri)
cset = Intersection(state2cond[ri].as_set(), state_space)
if len(cset) == 0:
return S.Zero
state2cond[ri] = cset.as_relational(ri)
sorted_ris = sorted(ris, key=lambda ri: ri.key)
prod = self.probability(state2cond[sorted_ris[0]], given_condition, evaluate, **kwargs)
for i in range(1, len(sorted_ris)):
ri, prev_ri = sorted_ris[i], sorted_ris[i-1]
if not isinstance(state2cond[ri], Eq):
raise ValueError("The process is in multiple states at %s, unable to determine the probability."%(ri))
mat_of = TransitionMatrixOf(self, mat) if isinstance(self, DiscreteMarkovChain) else GeneratorMatrixOf(self, mat)
prod *= self.probability(state2cond[ri], state2cond[prev_ri]
& mat_of
& StochasticStateSpaceOf(self, state_space),
evaluate, **kwargs)
for expr in compute_later:
prod *= self.probability(expr, given_condition, evaluate, **kwargs)
return prod
if isinstance(condition, Or):
return sum([self.probability(expr, given_condition, evaluate, **kwargs)
for expr in condition.args])
raise NotImplementedError("Mechanism for handling (%s, %s) queries hasn't been "
"implemented yet."%(expr, condition))
def expectation(self, expr, condition=None, evaluate=True, **kwargs):
"""
Handles expectation queries for markov process.
Parameters
==========
expr: RandomIndexedSymbol, Relational, Logic
Condition for which expectation has to be computed. Must
contain a RandomIndexedSymbol of the process.
condition: Relational, Logic
The given conditions under which computations should be done.
Returns
=======
Expectation
Unevaluated object if computations cannot be done due to
insufficient information.
Expr
In all other cases when the computations are successful.
Note
====
Any information passed at the time of query overrides
any information passed at the time of object creation like
transition probabilities, state space.
Pass the transition matrix using TransitionMatrixOf,
generator matrix using GeneratorMatrixOf and state space
using StochasticStateSpaceOf in given_condition using & or And.
"""
check, mat, state_space, condition = \
self._preprocess(condition, evaluate)
if check:
return Expectation(expr, condition)
rvs = random_symbols(expr)
if isinstance(expr, Expr) and isinstance(condition, Eq) \
and len(rvs) == 1:
# handle queries similar to E(f(X[i]), Eq(X[i-m], <some-state>))
rv = list(rvs)[0]
lhsg, rhsg = condition.lhs, condition.rhs
if not isinstance(lhsg, RandomIndexedSymbol):
lhsg, rhsg = (rhsg, lhsg)
if rhsg not in self.state_space:
raise ValueError("%s state is not in the state space."%(rhsg))
if rv.key < lhsg.key:
raise ValueError("Incorrect given condition is given, expectation "
"time %s < time %s"%(rv.key, rv.key))
mat_of = TransitionMatrixOf(self, mat) if isinstance(self, DiscreteMarkovChain) else GeneratorMatrixOf(self, mat)
cond = condition & mat_of & \
StochasticStateSpaceOf(self, state_space)
func = lambda s: self.probability(Eq(rv, s), cond)*expr.subs(rv, s)
return sum([func(s) for s in state_space])
raise NotImplementedError("Mechanism for handling (%s, %s) queries hasn't been "
"implemented yet."%(expr, condition))
class DiscreteMarkovChain(DiscreteTimeStochasticProcess, MarkovProcess):
"""
Represents discrete time Markov chain.
Parameters
==========
sym: Symbol/str
state_space: Set
Optional, by default, S.Reals
trans_probs: Matrix/ImmutableMatrix/MatrixSymbol
Optional, by default, None
Examples
========
>>> from sympy.stats import DiscreteMarkovChain, TransitionMatrixOf
>>> from sympy import Matrix, MatrixSymbol, Eq
>>> from sympy.stats import P
>>> T = Matrix([[0.5, 0.2, 0.3],[0.2, 0.5, 0.3],[0.2, 0.3, 0.5]])
>>> Y = DiscreteMarkovChain("Y", [0, 1, 2], T)
>>> YS = DiscreteMarkovChain("Y")
>>> Y.state_space
FiniteSet(0, 1, 2)
>>> Y.transition_probabilities
Matrix([
[0.5, 0.2, 0.3],
[0.2, 0.5, 0.3],
[0.2, 0.3, 0.5]])
>>> TS = MatrixSymbol('T', 3, 3)
>>> P(Eq(YS[3], 2), Eq(YS[1], 1) & TransitionMatrixOf(YS, TS))
T[0, 2]*T[1, 0] + T[1, 1]*T[1, 2] + T[1, 2]*T[2, 2]
>>> P(Eq(Y[3], 2), Eq(Y[1], 1)).round(2)
0.36
References
==========
.. [1] https://en.wikipedia.org/wiki/Markov_chain#Discrete-time_Markov_chain
.. [2] https://www.dartmouth.edu/~chance/teaching_aids/books_articles/probability_book/Chapter11.pdf
"""
index_set = S.Naturals0
def __new__(cls, sym, state_space=S.Reals, trans_probs=None):
sym = _symbol_converter(sym)
state_space = _set_converter(state_space)
if trans_probs != None:
trans_probs = _matrix_checks(trans_probs)
return Basic.__new__(cls, sym, state_space, trans_probs)
@property
def transition_probabilities(self):
"""
Transition probabilities of discrete Markov chain,
either an instance of Matrix or MatrixSymbol.
"""
return self.args[2]
def _transient2transient(self):
"""
Computes the one step probabilities of transient
states to transient states. Used in finding
fundamental matrix, absorbing probabilties.
"""
trans_probs = self.transition_probabilities
if not isinstance(trans_probs, ImmutableMatrix):
return None
m = trans_probs.shape[0]
trans_states = [i for i in range(m) if trans_probs[i, i] != 1]
t2t = [[trans_probs[si, sj] for sj in trans_states] for si in trans_states]
return ImmutableMatrix(t2t)
def _transient2absorbing(self):
"""
Computes the one step probabilities of transient
states to absorbing states. Used in finding
fundamental matrix, absorbing probabilties.
"""
trans_probs = self.transition_probabilities
if not isinstance(trans_probs, ImmutableMatrix):
return None
m, trans_states, absorb_states = \
trans_probs.shape[0], [], []
for i in range(m):
if trans_probs[i, i] == 1:
absorb_states.append(i)
else:
trans_states.append(i)
if not absorb_states or not trans_states:
return None
t2a = [[trans_probs[si, sj] for sj in absorb_states]
for si in trans_states]
return ImmutableMatrix(t2a)
def fundamental_matrix(self):
Q = self._transient2transient()
if Q == None:
return None
I = eye(Q.shape[0])
if (I - Q).det() == 0:
raise ValueError("Fundamental matrix doesn't exists.")
return ImmutableMatrix((I - Q).inv().tolist())
def absorbing_probabilites(self):
"""
Computes the absorbing probabilities, i.e.,
the ij-th entry of the matrix denotes the
probability of Markov chain being absorbed
in state j starting from state i.
"""
R = self._transient2absorbing()
N = self.fundamental_matrix()
if R == None or N == None:
return None
return N*R
def is_regular(self):
w = self.fixed_row_vector()
if w is None or isinstance(w, (Lambda)):
return None
return all((wi > 0) == True for wi in w.row(0))
def is_absorbing_state(self, state):
trans_probs = self.transition_probabilities
if isinstance(trans_probs, ImmutableMatrix) and \
state < trans_probs.shape[0]:
return S(trans_probs[state, state]) is S.One
def is_absorbing_chain(self):
trans_probs = self.transition_probabilities
return any(self.is_absorbing_state(state) == True
for state in range(trans_probs.shape[0]))
def fixed_row_vector(self):
trans_probs = self.transition_probabilities
if trans_probs == None:
return None
if isinstance(trans_probs, MatrixSymbol):
wm = MatrixSymbol('wm', 1, trans_probs.shape[0])
return Lambda((wm, trans_probs), Eq(wm*trans_probs, wm))
w = IndexedBase('w')
wi = [w[i] for i in range(trans_probs.shape[0])]
wm = Matrix([wi])
eqs = (wm*trans_probs - wm).tolist()[0]
eqs.append(sum(wi) - 1)
soln = list(linsolve(eqs, wi))[0]
return ImmutableMatrix([[sol for sol in soln]])
@property
def limiting_distribution(self):
"""
The fixed row vector is the limiting
distribution of a discrete Markov chain.
"""
return self.fixed_row_vector()
def sample(self):
"""
Returns
=======
sample: iterator object
iterator object containing the sample
"""
if not isinstance(self.transition_probabilities, (Matrix, ImmutableMatrix)):
raise ValueError("Transition Matrix must be provided for sampling")
Tlist = self.transition_probabilities.tolist()
samps = [random.choice(list(self.state_space))]
yield samps[0]
time = 1
densities = {}
for state in self.state_space:
states = list(self.state_space)
densities[state] = {states[i]: Tlist[state][i]
for i in range(len(states))}
while time < S.Infinity:
samps.append((next(sample_iter(FiniteRV("_", densities[samps[time - 1]])))))
yield samps[time]
time += 1
class ContinuousMarkovChain(ContinuousTimeStochasticProcess, MarkovProcess):
"""
Represents continuous time Markov chain.
Parameters
==========
sym: Symbol/str
state_space: Set
Optional, by default, S.Reals
gen_mat: Matrix/ImmutableMatrix/MatrixSymbol
Optional, by default, None
Examples
========
>>> from sympy.stats import ContinuousMarkovChain
>>> from sympy import Matrix, S
>>> G = Matrix([[-S(1), S(1)], [S(1), -S(1)]])
>>> C = ContinuousMarkovChain('C', state_space=[0, 1], gen_mat=G)
>>> C.limiting_distribution()
Matrix([[1/2, 1/2]])
References
==========
.. [1] https://en.wikipedia.org/wiki/Markov_chain#Continuous-time_Markov_chain
.. [2] http://u.math.biu.ac.il/~amirgi/CTMCnotes.pdf
"""
index_set = S.Reals
def __new__(cls, sym, state_space=S.Reals, gen_mat=None):
sym = _symbol_converter(sym)
state_space = _set_converter(state_space)
if gen_mat != None:
gen_mat = _matrix_checks(gen_mat)
return Basic.__new__(cls, sym, state_space, gen_mat)
@property
def generator_matrix(self):
return self.args[2]
@cacheit
def transition_probabilities(self, gen_mat=None):
t = Dummy('t')
if isinstance(gen_mat, (Matrix, ImmutableMatrix)) and \
gen_mat.is_diagonalizable():
# for faster computation use diagonalized generator matrix
Q, D = gen_mat.diagonalize()
return Lambda(t, Q*exp(t*D)*Q.inv())
if gen_mat != None:
return Lambda(t, exp(t*gen_mat))
def limiting_distribution(self):
gen_mat = self.generator_matrix
if gen_mat == None:
return None
if isinstance(gen_mat, MatrixSymbol):
wm = MatrixSymbol('wm', 1, gen_mat.shape[0])
return Lambda((wm, gen_mat), Eq(wm*gen_mat, wm))
w = IndexedBase('w')
wi = [w[i] for i in range(gen_mat.shape[0])]
wm = Matrix([wi])
eqs = (wm*gen_mat).tolist()[0]
eqs.append(sum(wi) - 1)
soln = list(linsolve(eqs, wi))[0]
return ImmutableMatrix([[sol for sol in soln]])
class BernoulliProcess(DiscreteTimeStochasticProcess):
"""
The Bernoulli process consists of repeated
independent Bernoulli process trials with the same parameter `p`.
It's assumed that the probability `p` applies to every
trial and that the outcomes of each trial
are independent of all the rest. Therefore Bernoulli Processs
is Discrete State and Discrete Time Stochastic Process.
Parameters
==========
sym: Symbol/str
success: Integer/str
The event which is considered to be success, by default is 1.
failure: Integer/str
The event which is considered to be failure, by default is 0.
p: Real Number between 0 and 1
Represents the probability of getting success.
Examples
========
>>> from sympy.stats import BernoulliProcess, P, E
>>> from sympy import Eq, Gt
>>> B = BernoulliProcess("B", p=0.7, success=1, failure=0)
>>> B.state_space
FiniteSet(0, 1)
>>> (B.p).round(2)
0.70
>>> B.success
1
>>> B.failure
0
>>> X = B[1] + B[2] + B[3]
>>> P(Eq(X, 0)).round(2)
0.03
>>> P(Eq(X, 2)).round(2)
0.44
>>> P(Eq(X, 4)).round(2)
0
>>> P(Gt(X, 1)).round(2)
0.78
>>> P(Eq(B[1], 0) & Eq(B[2], 1) & Eq(B[3], 0) & Eq(B[4], 1)).round(2)
0.04
>>> B.joint_distribution(B[1], B[2])
JointDistributionHandmade(Lambda((B[1], B[2]), Piecewise((0.7, Eq(B[1], 1)),
(0.3, Eq(B[1], 0)), (0, True))*Piecewise((0.7, Eq(B[2], 1)), (0.3, Eq(B[2], 0)),
(0, True))))
>>> E(2*B[1] + B[2]).round(2)
2.10
>>> P(B[1] < 1).round(2)
0.30
References
==========
.. [1] https://en.wikipedia.org/wiki/Bernoulli_process
.. [2] https://mathcs.clarku.edu/~djoyce/ma217/bernoulli.pdf
"""
index_set = S.Naturals0
def __new__(cls, sym, p, success=1, failure=0):
_value_check(p >= 0 and p <= 1, 'Value of p must be between 0 and 1.')
sym = _symbol_converter(sym)
p = _sympify(p)
success = _sym_sympify(success)
failure = _sym_sympify(failure)
return Basic.__new__(cls, sym, p, success, failure)
@property
def symbol(self):
return self.args[0]
@property
def p(self):
return self.args[1]
@property
def success(self):
return self.args[2]
@property
def failure(self):
return self.args[3]
@property
def state_space(self):
return _set_converter([self.success, self.failure])
@property
def distribution(self):
return BernoulliDistribution(self.p)
def _rvindexed_subs(self, expr, condition=None):
"""
Substitutes the RandomIndexedSymbol with the RandomSymbol with
same name, distribution and probability as RandomIndexedSymbol.
"""
rvs_expr = random_symbols(expr)
if len(rvs_expr) != 0:
swapdict_expr = {}
for rv in rvs_expr:
if isinstance(rv, RandomIndexedSymbol):
newrv = Bernoulli(rv.name, p=rv.pspace.process.p,
succ=self.success, fail=self.failure)
swapdict_expr[rv] = newrv
expr = expr.subs(swapdict_expr)
rvs_cond = random_symbols(condition)
if len(rvs_cond)!=0:
swapdict_cond = {}
if condition is not None:
for rv in rvs_cond:
if isinstance(rv, RandomIndexedSymbol):
newrv = Bernoulli(rv.name, p=rv.pspace.process.p,
succ=self.success, fail=self.failure)
swapdict_cond[rv] = newrv
condition = condition.subs(swapdict_cond)
return expr, condition
def expectation(self, expr, condition=None, evaluate=True, **kwargs):
"""
Computes expectation.
Parameters
==========
expr: RandomIndexedSymbol, Relational, Logic
Condition for which expectation has to be computed. Must
contain a RandomIndexedSymbol of the process.
condition: Relational, Logic
The given conditions under which computations should be done.
Returns
=======
Expectation of the RandomIndexedSymbol.
"""
new_expr, new_condition = self._rvindexed_subs(expr, condition)
new_pspace = pspace(new_expr)
if new_condition is not None:
new_expr = given(new_expr, new_condition)
if new_expr.is_Add: # As E is Linear
return Add(*[new_pspace.compute_expectation(
expr=arg, evaluate=evaluate, **kwargs)
for arg in new_expr.args])
return new_pspace.compute_expectation(
new_expr, evaluate=evaluate, **kwargs)
def probability(self, condition, given_condition=None, evaluate=True, **kwargs):
"""
Computes probability.
Parameters
==========
condition: Relational
Condition for which probability has to be computed. Must
contain a RandomIndexedSymbol of the process.
given_condition: Relational/And
The given conditions under which computations should be done.
Returns
=======
Probability of the condition.
"""
new_condition, new_givencondition = self._rvindexed_subs(condition, given_condition)
if isinstance(new_givencondition, RandomSymbol):
condrv = random_symbols(new_condition)
if len(condrv) == 1 and condrv[0] == new_givencondition:
return BernoulliDistribution(self.probability(new_condition), 0, 1)
if any([dependent(rv, new_givencondition) for rv in condrv]):
return Probability(new_condition, new_givencondition)
else:
return self.probability(new_condition)
if new_givencondition is not None and \
not isinstance(new_givencondition, (Relational, Boolean)):
raise ValueError("%s is not a relational or combination of relationals"
% (new_givencondition))
if new_givencondition == False:
return S.Zero
if new_condition == True:
return S.One
if new_condition == False:
return S.Zero
if not isinstance(new_condition, (Relational, Boolean)):
raise ValueError("%s is not a relational or combination of relationals"
% (new_condition))
if new_givencondition is not None: # If there is a condition
# Recompute on new conditional expr
return self.probability(given(new_condition, new_givencondition, **kwargs), **kwargs)
return pspace(new_condition).probability(new_condition, **kwargs)
def density(self, x):
return Piecewise((self.p, Eq(x, self.success)),
(1 - self.p, Eq(x, self.failure)),
(S.Zero, True))
|
5e910313b236dfecef1cae89b2508df345df8ac39ec993daacbe53be807fb667 | """
SymPy statistics module
Introduces a random variable type into the SymPy language.
Random variables may be declared using prebuilt functions such as
Normal, Exponential, Coin, Die, etc... or built with functions like FiniteRV.
Queries on random expressions can be made using the functions
========================= =============================
Expression Meaning
------------------------- -----------------------------
``P(condition)`` Probability
``E(expression)`` Expected value
``H(expression)`` Entropy
``variance(expression)`` Variance
``density(expression)`` Probability Density Function
``sample(expression)`` Produce a realization
``where(condition)`` Where the condition is true
========================= =============================
Examples
========
>>> from sympy.stats import P, E, variance, Die, Normal
>>> from sympy import Eq, simplify
>>> X, Y = Die('X', 6), Die('Y', 6) # Define two six sided dice
>>> Z = Normal('Z', 0, 1) # Declare a Normal random variable with mean 0, std 1
>>> P(X>3) # Probability X is greater than 3
1/2
>>> E(X+Y) # Expectation of the sum of two dice
7
>>> variance(X+Y) # Variance of the sum of two dice
35/6
>>> simplify(P(Z>1)) # Probability of Z being greater than 1
1/2 - erf(sqrt(2)/2)/2
One could also create custom distribution and define custom random variables
as follows:
1. If the you want to create a Continuous Random Variable:
>>> from sympy.stats import ContinuousRV, P, E
>>> from sympy import exp, Symbol, Interval, oo
>>> x = Symbol('x')
>>> pdf = exp(-x) # pdf of the Continuous Distribution
>>> Z = ContinuousRV(x, pdf, set=Interval(0, oo))
>>> E(Z)
1
>>> P(Z > 5)
exp(-5)
1.1 To create an instance of Continuous Distribution:
>>> from sympy.stats import ContinuousDistributionHandmade
>>> from sympy import Lambda
>>> dist = ContinuousDistributionHandmade(Lambda(x, pdf), set=Interval(0, oo))
>>> dist.pdf(x)
exp(-x)
2. If you want to create a Discrete Random Variable:
>>> from sympy.stats import DiscreteRV, P, E
>>> from sympy import Symbol, S
>>> p = S(1)/2
>>> x = Symbol('x', integer=True, positive=True)
>>> pdf = p*(1 - p)**(x - 1)
>>> D = DiscreteRV(x, pdf, set=S.Naturals)
>>> E(D)
2
>>> P(D > 3)
1/8
2.1 To create an instance of Discrete Distribution:
>>> from sympy.stats import DiscreteDistributionHandmade
>>> from sympy import Lambda
>>> dist = DiscreteDistributionHandmade(Lambda(x, pdf), set=S.Naturals)
>>> dist.pdf(x)
2**(1 - x)/2
3. If the you want to create a Finite Random Variable:
>>> from sympy.stats import FiniteRV, P, E
>>> from sympy import Rational
>>> pmf = {1: Rational(1, 3), 2: Rational(1, 6), 3: Rational(1, 4), 4: Rational(1, 4)}
>>> X = FiniteRV('X', pmf)
>>> E(X)
29/12
>>> P(X > 3)
1/4
3.1 To create an instance of Finite Distribution:
>>> from sympy.stats import FiniteDistributionHandmade
>>> dist = FiniteDistributionHandmade(pmf)
>>> dist.pmf(x)
Lambda(x, Piecewise((1/3, Eq(x, 1)), (1/6, Eq(x, 2)), (1/4, Eq(x, 3) | Eq(x, 4)), (0, True)))
"""
__all__ = [
'P', 'E', 'H', 'density', 'where', 'given', 'sample', 'cdf','median',
'characteristic_function', 'pspace', 'sample_iter', 'variance', 'std',
'skewness', 'kurtosis', 'covariance', 'dependent', 'entropy', 'independent',
'random_symbols', 'correlation', 'factorial_moment', 'moment', 'cmoment',
'sampling_density', 'moment_generating_function', 'smoment', 'quantile',
'coskewness', 'sample_stochastic_process',
'FiniteRV', 'DiscreteUniform', 'Die', 'Bernoulli', 'Coin', 'Binomial',
'BetaBinomial', 'Hypergeometric', 'Rademacher',
'FiniteDistributionHandmade',
'ContinuousRV', 'Arcsin', 'Benini', 'Beta', 'BetaNoncentral', 'BetaPrime',
'BoundedPareto', 'Cauchy', 'Chi', 'ChiNoncentral', 'ChiSquared', 'Dagum', 'Erlang',
'ExGaussian', 'Exponential', 'ExponentialPower', 'FDistribution',
'FisherZ', 'Frechet', 'Gamma', 'GammaInverse', 'Gompertz', 'Gumbel',
'Kumaraswamy', 'Laplace', 'Levy', 'Logistic', 'LogLogistic', 'LogNormal', 'Lomax',
'Moyal', 'Maxwell', 'Nakagami', 'Normal', 'GaussianInverse', 'Pareto', 'PowerFunction',
'QuadraticU', 'RaisedCosine', 'Rayleigh','Reciprocal', 'StudentT', 'ShiftedGompertz',
'Trapezoidal', 'Triangular', 'Uniform', 'UniformSum', 'VonMises', 'Wald',
'Weibull', 'WignerSemicircle', 'ContinuousDistributionHandmade',
'Geometric','Hermite', 'Logarithmic', 'NegativeBinomial', 'Poisson', 'Skellam',
'YuleSimon', 'Zeta', 'DiscreteRV', 'DiscreteDistributionHandmade',
'JointRV', 'Dirichlet', 'GeneralizedMultivariateLogGamma',
'GeneralizedMultivariateLogGammaOmega', 'Multinomial', 'MultivariateBeta',
'MultivariateEwens', 'MultivariateT', 'NegativeMultinomial',
'NormalGamma',
'StochasticProcess', 'DiscreteTimeStochasticProcess',
'DiscreteMarkovChain', 'TransitionMatrixOf', 'StochasticStateSpaceOf',
'GeneratorMatrixOf', 'ContinuousMarkovChain', 'BernoulliProcess',
'CircularEnsemble', 'CircularUnitaryEnsemble',
'CircularOrthogonalEnsemble', 'CircularSymplecticEnsemble',
'GaussianEnsemble', 'GaussianUnitaryEnsemble',
'GaussianOrthogonalEnsemble', 'GaussianSymplecticEnsemble',
'joint_eigen_distribution', 'JointEigenDistribution',
'level_spacing_distribution',
'Probability', 'Expectation', 'Variance', 'Covariance',
'ExpectationMatrix', 'VarianceMatrix', 'CrossCovarianceMatrix'
]
from .rv_interface import (P, E, H, density, where, given, sample, cdf, median,
characteristic_function, pspace, sample_iter, variance, std, skewness,
kurtosis, covariance, dependent, entropy, independent, random_symbols,
correlation, factorial_moment, moment, cmoment, sampling_density,
moment_generating_function, smoment, quantile, coskewness,
sample_stochastic_process)
from .frv_types import (FiniteRV, DiscreteUniform, Die, Bernoulli, Coin,
Binomial, BetaBinomial, Hypergeometric, Rademacher,
FiniteDistributionHandmade)
from .crv_types import (ContinuousRV, Arcsin, Benini, Beta, BetaNoncentral,
BetaPrime, BoundedPareto, Cauchy, Chi, ChiNoncentral, ChiSquared, Dagum, Erlang,
ExGaussian, Exponential, ExponentialPower, FDistribution, FisherZ,
Frechet, Gamma, GammaInverse, Gompertz, Gumbel, Kumaraswamy, Laplace,
Levy, Logistic, LogLogistic, LogNormal, Lomax, Maxwell, Moyal, Nakagami, Normal,
GaussianInverse, Pareto, QuadraticU, RaisedCosine, Rayleigh, Reciprocal, StudentT,
PowerFunction, ShiftedGompertz, Trapezoidal, Triangular, Uniform, UniformSum,
VonMises, Wald, Weibull, WignerSemicircle, ContinuousDistributionHandmade)
from .drv_types import (Geometric, Hermite, Logarithmic, NegativeBinomial, Poisson,
Skellam, YuleSimon, Zeta, DiscreteRV, DiscreteDistributionHandmade)
from .joint_rv_types import (JointRV, Dirichlet,
GeneralizedMultivariateLogGamma, GeneralizedMultivariateLogGammaOmega,
Multinomial, MultivariateBeta, MultivariateEwens, MultivariateT,
NegativeMultinomial, NormalGamma)
from .stochastic_process_types import (StochasticProcess,
DiscreteTimeStochasticProcess, DiscreteMarkovChain,
TransitionMatrixOf, StochasticStateSpaceOf, GeneratorMatrixOf,
ContinuousMarkovChain, BernoulliProcess)
from .random_matrix_models import (CircularEnsemble, CircularUnitaryEnsemble,
CircularOrthogonalEnsemble, CircularSymplecticEnsemble,
GaussianEnsemble, GaussianUnitaryEnsemble, GaussianOrthogonalEnsemble,
GaussianSymplecticEnsemble, joint_eigen_distribution,
JointEigenDistribution, level_spacing_distribution)
from .symbolic_probability import (Probability, Expectation, Variance,
Covariance)
from .symbolic_multivariate_probability import (ExpectationMatrix, VarianceMatrix,
CrossCovarianceMatrix)
|
3501c3ba719d8487c862bd6a46aa0ba893ea5ffea380896c8e74d26d69e2f87c | import itertools
from sympy import (MatrixExpr, Expr, ShapeError, ZeroMatrix,
Add, Mul, MatMul, S, expand as _expand)
from sympy.stats.rv import RandomSymbol, is_random
from sympy.core.sympify import _sympify
from sympy.stats.symbolic_probability import Variance, Covariance, Expectation
class ExpectationMatrix(Expectation, MatrixExpr):
"""
Expectation of a random matrix expression.
Examples
========
>>> from sympy.stats import ExpectationMatrix, Normal
>>> from sympy.stats.rv import RandomMatrixSymbol
>>> from sympy import symbols, MatrixSymbol, Matrix
>>> k = symbols("k")
>>> A, B = MatrixSymbol("A", k, k), MatrixSymbol("B", k, k)
>>> X, Y = RandomMatrixSymbol("X", k, 1), RandomMatrixSymbol("Y", k, 1)
>>> ExpectationMatrix(X)
ExpectationMatrix(X)
>>> ExpectationMatrix(A*X).shape
(k, 1)
To expand the expectation in its expression, use ``expand()``:
>>> ExpectationMatrix(A*X + B*Y).expand()
A*ExpectationMatrix(X) + B*ExpectationMatrix(Y)
>>> ExpectationMatrix((X + Y)*(X - Y).T).expand()
ExpectationMatrix(X*X.T) - ExpectationMatrix(X*Y.T) + ExpectationMatrix(Y*X.T) - ExpectationMatrix(Y*Y.T)
To evaluate the ``ExpectationMatrix``, use ``doit()``:
>>> N11, N12 = Normal('N11', 11, 1), Normal('N12', 12, 1)
>>> N21, N22 = Normal('N21', 21, 1), Normal('N22', 22, 1)
>>> M11, M12 = Normal('M11', 1, 1), Normal('M12', 2, 1)
>>> M21, M22 = Normal('M21', 3, 1), Normal('M22', 4, 1)
>>> x1 = Matrix([[N11, N12], [N21, N22]])
>>> x2 = Matrix([[M11, M12], [M21, M22]])
>>> ExpectationMatrix(x1 + x2).doit()
Matrix([
[12, 14],
[24, 26]])
"""
def __new__(cls, expr, condition=None):
expr = _sympify(expr)
if condition is None:
if not is_random(expr):
return expr
obj = Expr.__new__(cls, expr)
else:
condition = _sympify(condition)
obj = Expr.__new__(cls, expr, condition)
obj._shape = expr.shape
obj._condition = condition
return obj
@property
def shape(self):
return self._shape
def expand(self, **hints):
expr = self.args[0]
condition = self._condition
if not is_random(expr):
return expr
if isinstance(expr, Add):
return Add(*[Expectation(a, condition=condition).expand() for a in expr.args])
elif isinstance(expr, (Mul, MatMul)):
if isinstance(_expand(expr), Add):
return Expectation(_expand(expr)).expand()
rv = []
nonrv = []
postnon = []
for a in expr.args:
if is_random(a):
if rv:
rv.extend(postnon)
else:
nonrv.extend(postnon)
postnon = []
rv.append(a)
elif a.is_Matrix:
postnon.append(a)
else:
nonrv.append(a)
# In order to avoid infinite-looping (MatMul may call .doit() again),
# do not rebuild
if len(nonrv) == 0:
return self
return Mul.fromiter(nonrv)*Expectation(Mul.fromiter(rv),
condition=condition)*Mul.fromiter(postnon)
return self
class VarianceMatrix(Variance, MatrixExpr):
"""
Variance of a random matrix probability expression. Also known as
Covariance matrix, auto-covariance matrix, dispersion matrix,
or variance-covariance matrix
Examples
========
>>> from sympy.stats import VarianceMatrix
>>> from sympy.stats.rv import RandomMatrixSymbol
>>> from sympy import symbols, MatrixSymbol
>>> k = symbols("k")
>>> A, B = MatrixSymbol("A", k, k), MatrixSymbol("B", k, k)
>>> X, Y = RandomMatrixSymbol("X", k, 1), RandomMatrixSymbol("Y", k, 1)
>>> VarianceMatrix(X)
VarianceMatrix(X)
>>> VarianceMatrix(X).shape
(k, k)
To expand the variance in its expression, use ``expand()``:
>>> VarianceMatrix(A*X).expand()
A*VarianceMatrix(X)*A.T
>>> VarianceMatrix(A*X + B*Y).expand()
2*A*CrossCovarianceMatrix(X, Y)*B.T + A*VarianceMatrix(X)*A.T + B*VarianceMatrix(Y)*B.T
"""
def __new__(cls, arg, condition=None):
arg = _sympify(arg)
if 1 not in arg.shape:
raise ShapeError("Expression is not a vector")
shape = (arg.shape[0], arg.shape[0]) if arg.shape[1] == 1 else (arg.shape[1], arg.shape[1])
if condition:
obj = Expr.__new__(cls, arg, condition)
else:
obj = Expr.__new__(cls, arg)
obj._shape = shape
obj._condition = condition
return obj
@property
def shape(self):
return self._shape
def expand(self, **hints):
arg = self.args[0]
condition = self._condition
if not is_random(arg):
return ZeroMatrix(*self.shape)
if isinstance(arg, RandomSymbol):
return self
elif isinstance(arg, Add):
rv = []
for a in arg.args:
if is_random(a):
rv.append(a)
variances = Add(*map(lambda xv: Variance(xv, condition).expand(), rv))
map_to_covar = lambda x: 2*Covariance(*x, condition=condition).expand()
covariances = Add(*map(map_to_covar, itertools.combinations(rv, 2)))
return variances + covariances
elif isinstance(arg, (Mul, MatMul)):
nonrv = []
rv = []
for a in arg.args:
if is_random(a):
rv.append(a)
else:
nonrv.append(a)
if len(rv) == 0:
return ZeroMatrix(*self.shape)
# Avoid possible infinite loops with MatMul:
if len(nonrv) == 0:
return self
# Variance of many multiple matrix products is not implemented:
if len(rv) > 1:
return self
return Mul.fromiter(nonrv)*Variance(Mul.fromiter(rv),
condition)*(Mul.fromiter(nonrv)).transpose()
# this expression contains a RandomSymbol somehow:
return self
class CrossCovarianceMatrix(Covariance, MatrixExpr):
"""
Covariance of a random matrix probability expression.
Examples
========
>>> from sympy.stats import CrossCovarianceMatrix
>>> from sympy.stats.rv import RandomMatrixSymbol
>>> from sympy import symbols, MatrixSymbol
>>> k = symbols("k")
>>> A, B = MatrixSymbol("A", k, k), MatrixSymbol("B", k, k)
>>> C, D = MatrixSymbol("C", k, k), MatrixSymbol("D", k, k)
>>> X, Y = RandomMatrixSymbol("X", k, 1), RandomMatrixSymbol("Y", k, 1)
>>> Z, W = RandomMatrixSymbol("Z", k, 1), RandomMatrixSymbol("W", k, 1)
>>> CrossCovarianceMatrix(X, Y)
CrossCovarianceMatrix(X, Y)
>>> CrossCovarianceMatrix(X, Y).shape
(k, k)
To expand the covariance in its expression, use ``expand()``:
>>> CrossCovarianceMatrix(X + Y, Z).expand()
CrossCovarianceMatrix(X, Z) + CrossCovarianceMatrix(Y, Z)
>>> CrossCovarianceMatrix(A*X , Y).expand()
A*CrossCovarianceMatrix(X, Y)
>>> CrossCovarianceMatrix(A*X, B.T*Y).expand()
A*CrossCovarianceMatrix(X, Y)*B
>>> CrossCovarianceMatrix(A*X + B*Y, C.T*Z + D.T*W).expand()
A*CrossCovarianceMatrix(X, W)*D + A*CrossCovarianceMatrix(X, Z)*C + B*CrossCovarianceMatrix(Y, W)*D + B*CrossCovarianceMatrix(Y, Z)*C
"""
def __new__(cls, arg1, arg2, condition=None):
arg1 = _sympify(arg1)
arg2 = _sympify(arg2)
if (1 not in arg1.shape) or (1 not in arg2.shape) or (arg1.shape[1] != arg2.shape[1]):
raise ShapeError("Expression is not a vector")
shape = (arg1.shape[0], arg2.shape[0]) if arg1.shape[1] == 1 and arg2.shape[1] == 1 \
else (1, 1)
if condition:
obj = Expr.__new__(cls, arg1, arg2, condition)
else:
obj = Expr.__new__(cls, arg1, arg2)
obj._shape = shape
obj._condition = condition
return obj
@property
def shape(self):
return self._shape
def expand(self, **hints):
arg1 = self.args[0]
arg2 = self.args[1]
condition = self._condition
if arg1 == arg2:
return VarianceMatrix(arg1, condition).expand()
if not is_random(arg1) or not is_random(arg2):
return ZeroMatrix(*self.shape)
if isinstance(arg1, RandomSymbol) and isinstance(arg2, RandomSymbol):
return CrossCovarianceMatrix(arg1, arg2, condition)
coeff_rv_list1 = self._expand_single_argument(arg1.expand())
coeff_rv_list2 = self._expand_single_argument(arg2.expand())
addends = [a*CrossCovarianceMatrix(r1, r2, condition=condition)*b.transpose()
for (a, r1) in coeff_rv_list1 for (b, r2) in coeff_rv_list2]
return Add(*addends)
@classmethod
def _expand_single_argument(cls, expr):
# return (coefficient, random_symbol) pairs:
if isinstance(expr, RandomSymbol):
return [(S.One, expr)]
elif isinstance(expr, Add):
outval = []
for a in expr.args:
if isinstance(a, (Mul, MatMul)):
outval.append(cls._get_mul_nonrv_rv_tuple(a))
elif is_random(a):
outval.append((S.One, a))
return outval
elif isinstance(expr, (Mul, MatMul)):
return [cls._get_mul_nonrv_rv_tuple(expr)]
elif is_random(expr):
return [(S.One, expr)]
@classmethod
def _get_mul_nonrv_rv_tuple(cls, m):
rv = []
nonrv = []
for a in m.args:
if is_random(a):
rv.append(a)
else:
nonrv.append(a)
return (Mul(*nonrv), Mul(*rv))
|
96b0e6ffc2bd85abdaff2f186c6b5860ddf40d1333c039f61c194d61635a68a6 | from __future__ import print_function, division
from sympy.sets import FiniteSet
from sympy import sqrt, log, exp, FallingFactorial, Rational, Eq, Dummy, piecewise_fold, solveset
from .rv import (probability, expectation, density, where, given, pspace, cdf, PSpace,
characteristic_function, sample, sample_iter, random_symbols, independent, dependent,
sampling_density, moment_generating_function, quantile, is_random,
sample_stochastic_process)
__all__ = ['P', 'E', 'H', 'density', 'where', 'given', 'sample', 'cdf',
'characteristic_function', 'pspace', 'sample_iter', 'variance', 'std',
'skewness', 'kurtosis', 'covariance', 'dependent', 'entropy', 'median',
'independent', 'random_symbols', 'correlation', 'factorial_moment',
'moment', 'cmoment', 'sampling_density', 'moment_generating_function',
'smoment', 'quantile', 'sample_stochastic_process']
def moment(X, n, c=0, condition=None, **kwargs):
"""
Return the nth moment of a random expression about c i.e. E((X-c)**n)
Default value of c is 0.
Examples
========
>>> from sympy.stats import Die, moment, E
>>> X = Die('X', 6)
>>> moment(X, 1, 6)
-5/2
>>> moment(X, 2)
91/6
>>> moment(X, 1) == E(X)
True
"""
return expectation((X - c)**n, condition, **kwargs)
def variance(X, condition=None, **kwargs):
"""
Variance of a random expression
Expectation of (X-E(X))**2
Examples
========
>>> from sympy.stats import Die, Bernoulli, variance
>>> from sympy import simplify, Symbol
>>> X = Die('X', 6)
>>> p = Symbol('p')
>>> B = Bernoulli('B', p, 1, 0)
>>> variance(2*X)
35/3
>>> simplify(variance(B))
p*(1 - p)
"""
if is_random(X) and pspace(X) == PSpace():
from sympy.stats.symbolic_probability import Variance
return Variance(X, condition)
return cmoment(X, 2, condition, **kwargs)
def standard_deviation(X, condition=None, **kwargs):
"""
Standard Deviation of a random expression
Square root of the Expectation of (X-E(X))**2
Examples
========
>>> from sympy.stats import Bernoulli, std
>>> from sympy import Symbol, simplify
>>> p = Symbol('p')
>>> B = Bernoulli('B', p, 1, 0)
>>> simplify(std(B))
sqrt(p*(1 - p))
"""
return sqrt(variance(X, condition, **kwargs))
std = standard_deviation
def entropy(expr, condition=None, **kwargs):
"""
Calculuates entropy of a probability distribution
Parameters
==========
expression : the random expression whose entropy is to be calculated
condition : optional, to specify conditions on random expression
b: base of the logarithm, optional
By default, it is taken as Euler's number
Returns
=======
result : Entropy of the expression, a constant
Examples
========
>>> from sympy.stats import Normal, Die, entropy
>>> X = Normal('X', 0, 1)
>>> entropy(X)
log(2)/2 + 1/2 + log(pi)/2
>>> D = Die('D', 4)
>>> entropy(D)
log(4)
References
==========
.. [1] https://en.wikipedia.org/wiki/Entropy_(information_theory)
.. [2] https://www.crmarsh.com/static/pdf/Charles_Marsh_Continuous_Entropy.pdf
.. [3] http://www.math.uconn.edu/~kconrad/blurbs/analysis/entropypost.pdf
"""
pdf = density(expr, condition, **kwargs)
base = kwargs.get('b', exp(1))
if hasattr(pdf, 'dict'):
return sum([-prob*log(prob, base) for prob in pdf.dict.values()])
return expectation(-log(pdf(expr), base))
def covariance(X, Y, condition=None, **kwargs):
"""
Covariance of two random expressions
The expectation that the two variables will rise and fall together
Covariance(X,Y) = E( (X-E(X)) * (Y-E(Y)) )
Examples
========
>>> from sympy.stats import Exponential, covariance
>>> from sympy import Symbol
>>> rate = Symbol('lambda', positive=True, real=True, finite=True)
>>> X = Exponential('X', rate)
>>> Y = Exponential('Y', rate)
>>> covariance(X, X)
lambda**(-2)
>>> covariance(X, Y)
0
>>> covariance(X, Y + rate*X)
1/lambda
"""
if (is_random(X) and pspace(X) == PSpace()) or (is_random(Y) and pspace(Y) == PSpace()):
from sympy.stats.symbolic_probability import Covariance
return Covariance(X, Y, condition)
return expectation(
(X - expectation(X, condition, **kwargs)) *
(Y - expectation(Y, condition, **kwargs)),
condition, **kwargs)
def correlation(X, Y, condition=None, **kwargs):
"""
Correlation of two random expressions, also known as correlation
coefficient or Pearson's correlation
The normalized expectation that the two variables will rise
and fall together
Correlation(X,Y) = E( (X-E(X)) * (Y-E(Y)) / (sigma(X) * sigma(Y)) )
Examples
========
>>> from sympy.stats import Exponential, correlation
>>> from sympy import Symbol
>>> rate = Symbol('lambda', positive=True, real=True, finite=True)
>>> X = Exponential('X', rate)
>>> Y = Exponential('Y', rate)
>>> correlation(X, X)
1
>>> correlation(X, Y)
0
>>> correlation(X, Y + rate*X)
1/sqrt(1 + lambda**(-2))
"""
return covariance(X, Y, condition, **kwargs)/(std(X, condition, **kwargs)
* std(Y, condition, **kwargs))
def cmoment(X, n, condition=None, **kwargs):
"""
Return the nth central moment of a random expression about its mean
i.e. E((X - E(X))**n)
Examples
========
>>> from sympy.stats import Die, cmoment, variance
>>> X = Die('X', 6)
>>> cmoment(X, 3)
0
>>> cmoment(X, 2)
35/12
>>> cmoment(X, 2) == variance(X)
True
"""
mu = expectation(X, condition, **kwargs)
return moment(X, n, mu, condition, **kwargs)
def smoment(X, n, condition=None, **kwargs):
"""
Return the nth Standardized moment of a random expression i.e.
E(((X - mu)/sigma(X))**n)
Examples
========
>>> from sympy.stats import skewness, Exponential, smoment
>>> from sympy import Symbol
>>> rate = Symbol('lambda', positive=True, real=True, finite=True)
>>> Y = Exponential('Y', rate)
>>> smoment(Y, 4)
9
>>> smoment(Y, 4) == smoment(3*Y, 4)
True
>>> smoment(Y, 3) == skewness(Y)
True
"""
sigma = std(X, condition, **kwargs)
return (1/sigma)**n*cmoment(X, n, condition, **kwargs)
def skewness(X, condition=None, **kwargs):
"""
Measure of the asymmetry of the probability distribution.
Positive skew indicates that most of the values lie to the right of
the mean.
skewness(X) = E(((X - E(X))/sigma)**3)
Parameters
==========
condition : Expr containing RandomSymbols
A conditional expression. skewness(X, X>0) is skewness of X given X > 0
Examples
========
>>> from sympy.stats import skewness, Exponential, Normal
>>> from sympy import Symbol
>>> X = Normal('X', 0, 1)
>>> skewness(X)
0
>>> skewness(X, X > 0) # find skewness given X > 0
(-sqrt(2)/sqrt(pi) + 4*sqrt(2)/pi**(3/2))/(1 - 2/pi)**(3/2)
>>> rate = Symbol('lambda', positive=True, real=True, finite=True)
>>> Y = Exponential('Y', rate)
>>> skewness(Y)
2
"""
return smoment(X, 3, condition=condition, **kwargs)
def kurtosis(X, condition=None, **kwargs):
"""
Characterizes the tails/outliers of a probability distribution.
Kurtosis of any univariate normal distribution is 3. Kurtosis less than
3 means that the distribution produces fewer and less extreme outliers
than the normal distribution.
kurtosis(X) = E(((X - E(X))/sigma)**4)
Parameters
==========
condition : Expr containing RandomSymbols
A conditional expression. kurtosis(X, X>0) is kurtosis of X given X > 0
Examples
========
>>> from sympy.stats import kurtosis, Exponential, Normal
>>> from sympy import Symbol
>>> X = Normal('X', 0, 1)
>>> kurtosis(X)
3
>>> kurtosis(X, X > 0) # find kurtosis given X > 0
(-4/pi - 12/pi**2 + 3)/(1 - 2/pi)**2
>>> rate = Symbol('lamda', positive=True, real=True, finite=True)
>>> Y = Exponential('Y', rate)
>>> kurtosis(Y)
9
References
==========
.. [1] https://en.wikipedia.org/wiki/Kurtosis
.. [2] http://mathworld.wolfram.com/Kurtosis.html
"""
return smoment(X, 4, condition=condition, **kwargs)
def factorial_moment(X, n, condition=None, **kwargs):
"""
The factorial moment is a mathematical quantity defined as the expectation
or average of the falling factorial of a random variable.
factorial_moment(X, n) = E(X*(X - 1)*(X - 2)*...*(X - n + 1))
Parameters
==========
n: A natural number, n-th factorial moment.
condition : Expr containing RandomSymbols
A conditional expression.
Examples
========
>>> from sympy.stats import factorial_moment, Poisson, Binomial
>>> from sympy import Symbol, S
>>> lamda = Symbol('lamda')
>>> X = Poisson('X', lamda)
>>> factorial_moment(X, 2)
lamda**2
>>> Y = Binomial('Y', 2, S.Half)
>>> factorial_moment(Y, 2)
1/2
>>> factorial_moment(Y, 2, Y > 1) # find factorial moment for Y > 1
2
References
==========
.. [1] https://en.wikipedia.org/wiki/Factorial_moment
.. [2] http://mathworld.wolfram.com/FactorialMoment.html
"""
return expectation(FallingFactorial(X, n), condition=condition, **kwargs)
def median(X, evaluate=True, **kwargs):
r"""
Calculuates the median of the probability distribution.
Mathematically, median of Probability distribution is defined as all those
values of `m` for which the following condition is satisfied
.. math::
P(X\geq m)\geq 1/2 \hspace{5} \text{and} \hspace{5} P(X\leq m)\geq 1/2
Parameters
==========
X: The random expression whose median is to be calculated.
Returns
=======
The FiniteSet or an Interval which contains the median of the
random expression.
Examples
========
>>> from sympy.stats import Normal, Die, median
>>> N = Normal('N', 3, 1)
>>> median(N)
FiniteSet(3)
>>> D = Die('D')
>>> median(D)
FiniteSet(3, 4)
References
==========
.. [1] https://en.wikipedia.org/wiki/Median#Probability_distributions
"""
from sympy.stats.crv import ContinuousPSpace
from sympy.stats.drv import DiscretePSpace
from sympy.stats.frv import FinitePSpace
if isinstance(pspace(X), FinitePSpace):
cdf = pspace(X).compute_cdf(X)
result = []
for key, value in cdf.items():
if value>= Rational(1, 2) and (1 - value) + \
pspace(X).probability(Eq(X, key)) >= Rational(1, 2):
result.append(key)
return FiniteSet(*result)
if isinstance(pspace(X), ContinuousPSpace) or isinstance(pspace(X), DiscretePSpace):
cdf = pspace(X).compute_cdf(X)
x = Dummy('x')
result = solveset(piecewise_fold(cdf(x) - Rational(1, 2)), x, pspace(X).set)
return result
raise NotImplementedError("The median of %s is not implemeted."%str(pspace(X)))
def coskewness(X, Y, Z, condition=None, **kwargs):
r"""
Calculates the co-skewness of three random variables.
Mathematically Coskewness is defined as
.. math::
coskewness(X,Y,Z)=\frac{E[(X-E[X]) * (Y-E[Y]) * (Z-E[Z])]} {\sigma_{X}\sigma_{Y}\sigma_{Z}}
Parameters
==========
X : RandomSymbol
Random Variable used to calculate coskewness
Y : RandomSymbol
Random Variable used to calculate coskewness
Z : RandomSymbol
Random Variable used to calculate coskewness
condition : Expr containing RandomSymbols
A conditional expression
Examples
========
>>> from sympy.stats import coskewness, Exponential, skewness
>>> from sympy import symbols
>>> p = symbols('p', positive=True)
>>> X = Exponential('X', p)
>>> Y = Exponential('Y', 2*p)
>>> coskewness(X, Y, Y)
0
>>> coskewness(X, Y + X, Y + 2*X)
16*sqrt(85)/85
>>> coskewness(X + 2*Y, Y + X, Y + 2*X, X > 3)
9*sqrt(170)/85
>>> coskewness(Y, Y, Y) == skewness(Y)
True
>>> coskewness(X, Y + p*X, Y + 2*p*X)
4/(sqrt(1 + 1/(4*p**2))*sqrt(4 + 1/(4*p**2)))
Returns
=======
coskewness : The coskewness of the three random variables
References
==========
.. [1] https://en.wikipedia.org/wiki/Coskewness
"""
num = expectation((X - expectation(X, condition, **kwargs)) \
* (Y - expectation(Y, condition, **kwargs)) \
* (Z - expectation(Z, condition, **kwargs)), condition, **kwargs)
den = std(X, condition, **kwargs) * std(Y, condition, **kwargs) \
* std(Z, condition, **kwargs)
return num/den
P = probability
E = expectation
H = entropy
|
43e60d480dadf261d460b504487d47c01a82433a8dba7acbafaa927a46277852 | """
Main Random Variables Module
Defines abstract random variable type.
Contains interfaces for probability space object (PSpace) as well as standard
operators, P, E, sample, density, where, quantile
See Also
========
sympy.stats.crv
sympy.stats.frv
sympy.stats.rv_interface
"""
from __future__ import print_function, division
from functools import singledispatch
from typing import Tuple as tTuple
from sympy import (Basic, S, Expr, Symbol, Tuple, And, Add, Eq, lambdify,
Equality, Lambda, sympify, Dummy, Ne, KroneckerDelta,
DiracDelta, Mul, Indexed, MatrixSymbol, Function, Integral)
from sympy.core.relational import Relational
from sympy.core.sympify import _sympify
from sympy.logic.boolalg import Boolean
from sympy.sets.sets import FiniteSet, ProductSet, Intersection
from sympy.solvers.solveset import solveset
from sympy.external import import_module
from sympy.utilities.misc import filldedent
import warnings
x = Symbol('x')
@singledispatch
def is_random(x):
return False
@is_random.register(Basic)
def _(x):
atoms = x.free_symbols
return any([is_random(i) for i in atoms])
class RandomDomain(Basic):
"""
Represents a set of variables and the values which they can take
See Also
========
sympy.stats.crv.ContinuousDomain
sympy.stats.frv.FiniteDomain
"""
is_ProductDomain = False
is_Finite = False
is_Continuous = False
is_Discrete = False
def __new__(cls, symbols, *args):
symbols = FiniteSet(*symbols)
return Basic.__new__(cls, symbols, *args)
@property
def symbols(self):
return self.args[0]
@property
def set(self):
return self.args[1]
def __contains__(self, other):
raise NotImplementedError()
def compute_expectation(self, expr):
raise NotImplementedError()
class SingleDomain(RandomDomain):
"""
A single variable and its domain
See Also
========
sympy.stats.crv.SingleContinuousDomain
sympy.stats.frv.SingleFiniteDomain
"""
def __new__(cls, symbol, set):
assert symbol.is_Symbol
return Basic.__new__(cls, symbol, set)
@property
def symbol(self):
return self.args[0]
@property
def symbols(self):
return FiniteSet(self.symbol)
def __contains__(self, other):
if len(other) != 1:
return False
sym, val = tuple(other)[0]
return self.symbol == sym and val in self.set
class ConditionalDomain(RandomDomain):
"""
A RandomDomain with an attached condition
See Also
========
sympy.stats.crv.ConditionalContinuousDomain
sympy.stats.frv.ConditionalFiniteDomain
"""
def __new__(cls, fulldomain, condition):
condition = condition.xreplace(dict((rs, rs.symbol)
for rs in random_symbols(condition)))
return Basic.__new__(cls, fulldomain, condition)
@property
def symbols(self):
return self.fulldomain.symbols
@property
def fulldomain(self):
return self.args[0]
@property
def condition(self):
return self.args[1]
@property
def set(self):
raise NotImplementedError("Set of Conditional Domain not Implemented")
def as_boolean(self):
return And(self.fulldomain.as_boolean(), self.condition)
class PSpace(Basic):
"""
A Probability Space
Probability Spaces encode processes that equal different values
probabilistically. These underly Random Symbols which occur in SymPy
expressions and contain the mechanics to evaluate statistical statements.
See Also
========
sympy.stats.crv.ContinuousPSpace
sympy.stats.frv.FinitePSpace
"""
is_Finite = None # type: bool
is_Continuous = None # type: bool
is_Discrete = None # type: bool
is_real = None # type: bool
@property
def domain(self):
return self.args[0]
@property
def density(self):
return self.args[1]
@property
def values(self):
return frozenset(RandomSymbol(sym, self) for sym in self.symbols)
@property
def symbols(self):
return self.domain.symbols
def where(self, condition):
raise NotImplementedError()
def compute_density(self, expr):
raise NotImplementedError()
def sample(self):
raise NotImplementedError()
def probability(self, condition):
raise NotImplementedError()
def compute_expectation(self, expr):
raise NotImplementedError()
class SinglePSpace(PSpace):
"""
Represents the probabilities of a set of random events that can be
attributed to a single variable/symbol.
"""
def __new__(cls, s, distribution):
if isinstance(s, str):
s = Symbol(s)
if not isinstance(s, Symbol):
raise TypeError("s should have been string or Symbol")
return Basic.__new__(cls, s, distribution)
@property
def value(self):
return RandomSymbol(self.symbol, self)
@property
def symbol(self):
return self.args[0]
@property
def distribution(self):
return self.args[1]
@property
def pdf(self):
return self.distribution.pdf(self.symbol)
class RandomSymbol(Expr):
"""
Random Symbols represent ProbabilitySpaces in SymPy Expressions
In principle they can take on any value that their symbol can take on
within the associated PSpace with probability determined by the PSpace
Density.
Random Symbols contain pspace and symbol properties.
The pspace property points to the represented Probability Space
The symbol is a standard SymPy Symbol that is used in that probability space
for example in defining a density.
You can form normal SymPy expressions using RandomSymbols and operate on
those expressions with the Functions
E - Expectation of a random expression
P - Probability of a condition
density - Probability Density of an expression
given - A new random expression (with new random symbols) given a condition
An object of the RandomSymbol type should almost never be created by the
user. They tend to be created instead by the PSpace class's value method.
Traditionally a user doesn't even do this but instead calls one of the
convenience functions Normal, Exponential, Coin, Die, FiniteRV, etc....
"""
def __new__(cls, symbol, pspace=None):
from sympy.stats.joint_rv import JointRandomSymbol
if pspace is None:
# Allow single arg, representing pspace == PSpace()
pspace = PSpace()
if isinstance(symbol, str):
symbol = Symbol(symbol)
if not isinstance(symbol, Symbol):
raise TypeError("symbol should be of type Symbol or string")
if not isinstance(pspace, PSpace):
raise TypeError("pspace variable should be of type PSpace")
if cls == JointRandomSymbol and isinstance(pspace, SinglePSpace):
cls = RandomSymbol
return Basic.__new__(cls, symbol, pspace)
is_finite = True
is_symbol = True
is_Atom = True
_diff_wrt = True
pspace = property(lambda self: self.args[1])
symbol = property(lambda self: self.args[0])
name = property(lambda self: self.symbol.name)
def _eval_is_positive(self):
return self.symbol.is_positive
def _eval_is_integer(self):
return self.symbol.is_integer
def _eval_is_real(self):
return self.symbol.is_real or self.pspace.is_real
@property
def is_commutative(self):
return self.symbol.is_commutative
@property
def free_symbols(self):
return {self}
class RandomIndexedSymbol(RandomSymbol):
def __new__(cls, idx_obj, pspace=None):
if pspace is None:
# Allow single arg, representing pspace == PSpace()
pspace = PSpace()
if not isinstance(idx_obj, (Indexed, Function)):
raise TypeError("An Function or Indexed object is expected not %s"%(idx_obj))
return Basic.__new__(cls, idx_obj, pspace)
symbol = property(lambda self: self.args[0])
name = property(lambda self: str(self.args[0]))
@property
def key(self):
if isinstance(self.symbol, Indexed):
return self.symbol.args[1]
elif isinstance(self.symbol, Function):
return self.symbol.args[0]
@property
def free_symbols(self):
if self.key.free_symbols:
free_syms = self.key.free_symbols
free_syms.add(self)
return free_syms
return {self}
class RandomMatrixSymbol(RandomSymbol, MatrixSymbol):
def __new__(cls, symbol, n, m, pspace=None):
n, m = _sympify(n), _sympify(m)
symbol = _symbol_converter(symbol)
if pspace is None:
# Allow single arg, representing pspace == PSpace()
pspace = PSpace()
return Basic.__new__(cls, symbol, n, m, pspace)
symbol = property(lambda self: self.args[0])
pspace = property(lambda self: self.args[3])
class ProductPSpace(PSpace):
"""
Abstract class for representing probability spaces with multiple random
variables.
See Also
========
sympy.stats.rv.IndependentProductPSpace
sympy.stats.joint_rv.JointPSpace
"""
pass
class IndependentProductPSpace(ProductPSpace):
"""
A probability space resulting from the merger of two independent probability
spaces.
Often created using the function, pspace
"""
def __new__(cls, *spaces):
rs_space_dict = {}
for space in spaces:
for value in space.values:
rs_space_dict[value] = space
symbols = FiniteSet(*[val.symbol for val in rs_space_dict.keys()])
# Overlapping symbols
from sympy.stats.joint_rv import MarginalDistribution, CompoundDistribution
if len(symbols) < sum(len(space.symbols) for space in spaces if not
isinstance(space.distribution, (
CompoundDistribution, MarginalDistribution))):
raise ValueError("Overlapping Random Variables")
if all(space.is_Finite for space in spaces):
from sympy.stats.frv import ProductFinitePSpace
cls = ProductFinitePSpace
obj = Basic.__new__(cls, *FiniteSet(*spaces))
return obj
@property
def pdf(self):
p = Mul(*[space.pdf for space in self.spaces])
return p.subs(dict((rv, rv.symbol) for rv in self.values))
@property
def rs_space_dict(self):
d = {}
for space in self.spaces:
for value in space.values:
d[value] = space
return d
@property
def symbols(self):
return FiniteSet(*[val.symbol for val in self.rs_space_dict.keys()])
@property
def spaces(self):
return FiniteSet(*self.args)
@property
def values(self):
return sumsets(space.values for space in self.spaces)
def compute_expectation(self, expr, rvs=None, evaluate=False, **kwargs):
rvs = rvs or self.values
rvs = frozenset(rvs)
for space in self.spaces:
expr = space.compute_expectation(expr, rvs & space.values, evaluate=False, **kwargs)
if evaluate and hasattr(expr, 'doit'):
return expr.doit(**kwargs)
return expr
@property
def domain(self):
return ProductDomain(*[space.domain for space in self.spaces])
@property
def density(self):
raise NotImplementedError("Density not available for ProductSpaces")
def sample(self, size=(), library='scipy'):
return {k: v for space in self.spaces
for k, v in space.sample(size=size, library=library).items()}
def probability(self, condition, **kwargs):
cond_inv = False
if isinstance(condition, Ne):
condition = Eq(condition.args[0], condition.args[1])
cond_inv = True
expr = condition.lhs - condition.rhs
rvs = random_symbols(expr)
dens = self.compute_density(expr)
if any([pspace(rv).is_Continuous for rv in rvs]):
from sympy.stats.crv import SingleContinuousPSpace
from sympy.stats.crv_types import ContinuousDistributionHandmade
if expr in self.values:
# Marginalize all other random symbols out of the density
randomsymbols = tuple(set(self.values) - frozenset([expr]))
symbols = tuple(rs.symbol for rs in randomsymbols)
pdf = self.domain.integrate(self.pdf, symbols, **kwargs)
return Lambda(expr.symbol, pdf)
dens = ContinuousDistributionHandmade(dens)
z = Dummy('z', real=True)
space = SingleContinuousPSpace(z, dens)
result = space.probability(condition.__class__(space.value, 0))
else:
from sympy.stats.drv import SingleDiscretePSpace
from sympy.stats.drv_types import DiscreteDistributionHandmade
dens = DiscreteDistributionHandmade(dens)
z = Dummy('z', integer=True)
space = SingleDiscretePSpace(z, dens)
result = space.probability(condition.__class__(space.value, 0))
return result if not cond_inv else S.One - result
def compute_density(self, expr, **kwargs):
rvs = random_symbols(expr)
if any(pspace(rv).is_Continuous for rv in rvs):
z = Dummy('z', real=True)
expr = self.compute_expectation(DiracDelta(expr - z),
**kwargs)
else:
z = Dummy('z', integer=True)
expr = self.compute_expectation(KroneckerDelta(expr, z),
**kwargs)
return Lambda(z, expr)
def compute_cdf(self, expr, **kwargs):
raise ValueError("CDF not well defined on multivariate expressions")
def conditional_space(self, condition, normalize=True, **kwargs):
rvs = random_symbols(condition)
condition = condition.xreplace(dict((rv, rv.symbol) for rv in self.values))
if any([pspace(rv).is_Continuous for rv in rvs]):
from sympy.stats.crv import (ConditionalContinuousDomain,
ContinuousPSpace)
space = ContinuousPSpace
domain = ConditionalContinuousDomain(self.domain, condition)
elif any([pspace(rv).is_Discrete for rv in rvs]):
from sympy.stats.drv import (ConditionalDiscreteDomain,
DiscretePSpace)
space = DiscretePSpace
domain = ConditionalDiscreteDomain(self.domain, condition)
elif all([pspace(rv).is_Finite for rv in rvs]):
from sympy.stats.frv import FinitePSpace
return FinitePSpace.conditional_space(self, condition)
if normalize:
replacement = {rv: Dummy(str(rv)) for rv in self.symbols}
norm = domain.compute_expectation(self.pdf, **kwargs)
pdf = self.pdf / norm.xreplace(replacement)
# XXX: Converting symbols from set to tuple. The order matters to
# Lambda though so we shouldn't be starting with a set here...
density = Lambda(tuple(domain.symbols), pdf)
return space(domain, density)
class ProductDomain(RandomDomain):
"""
A domain resulting from the merger of two independent domains
See Also
========
sympy.stats.crv.ProductContinuousDomain
sympy.stats.frv.ProductFiniteDomain
"""
is_ProductDomain = True
def __new__(cls, *domains):
# Flatten any product of products
domains2 = []
for domain in domains:
if not domain.is_ProductDomain:
domains2.append(domain)
else:
domains2.extend(domain.domains)
domains2 = FiniteSet(*domains2)
if all(domain.is_Finite for domain in domains2):
from sympy.stats.frv import ProductFiniteDomain
cls = ProductFiniteDomain
if all(domain.is_Continuous for domain in domains2):
from sympy.stats.crv import ProductContinuousDomain
cls = ProductContinuousDomain
if all(domain.is_Discrete for domain in domains2):
from sympy.stats.drv import ProductDiscreteDomain
cls = ProductDiscreteDomain
return Basic.__new__(cls, *domains2)
@property
def sym_domain_dict(self):
return dict((symbol, domain) for domain in self.domains
for symbol in domain.symbols)
@property
def symbols(self):
return FiniteSet(*[sym for domain in self.domains
for sym in domain.symbols])
@property
def domains(self):
return self.args
@property
def set(self):
return ProductSet(*(domain.set for domain in self.domains))
def __contains__(self, other):
# Split event into each subdomain
for domain in self.domains:
# Collect the parts of this event which associate to this domain
elem = frozenset([item for item in other
if sympify(domain.symbols.contains(item[0]))
is S.true])
# Test this sub-event
if elem not in domain:
return False
# All subevents passed
return True
def as_boolean(self):
return And(*[domain.as_boolean() for domain in self.domains])
def random_symbols(expr):
"""
Returns all RandomSymbols within a SymPy Expression.
"""
atoms = getattr(expr, 'atoms', None)
if atoms is not None:
comp = lambda rv: rv.symbol.name
l = list(atoms(RandomSymbol))
return sorted(l, key=comp)
else:
return []
def pspace(expr):
"""
Returns the underlying Probability Space of a random expression.
For internal use.
Examples
========
>>> from sympy.stats import pspace, Normal
>>> X = Normal('X', 0, 1)
>>> pspace(2*X + 1) == X.pspace
True
"""
expr = sympify(expr)
if isinstance(expr, RandomSymbol) and expr.pspace is not None:
return expr.pspace
if expr.has(RandomMatrixSymbol):
rm = list(expr.atoms(RandomMatrixSymbol))[0]
return rm.pspace
rvs = random_symbols(expr)
if not rvs:
raise ValueError("Expression containing Random Variable expected, not %s" % (expr))
# If only one space present
if all(rv.pspace == rvs[0].pspace for rv in rvs):
return rvs[0].pspace
# Otherwise make a product space
return IndependentProductPSpace(*[rv.pspace for rv in rvs])
def sumsets(sets):
"""
Union of sets
"""
return frozenset().union(*sets)
def rs_swap(a, b):
"""
Build a dictionary to swap RandomSymbols based on their underlying symbol.
i.e.
if ``X = ('x', pspace1)``
and ``Y = ('x', pspace2)``
then ``X`` and ``Y`` match and the key, value pair
``{X:Y}`` will appear in the result
Inputs: collections a and b of random variables which share common symbols
Output: dict mapping RVs in a to RVs in b
"""
d = {}
for rsa in a:
d[rsa] = [rsb for rsb in b if rsa.symbol == rsb.symbol][0]
return d
def given(expr, condition=None, **kwargs):
r""" Conditional Random Expression
From a random expression and a condition on that expression creates a new
probability space from the condition and returns the same expression on that
conditional probability space.
Examples
========
>>> from sympy.stats import given, density, Die
>>> X = Die('X', 6)
>>> Y = given(X, X > 3)
>>> density(Y).dict
{4: 1/3, 5: 1/3, 6: 1/3}
Following convention, if the condition is a random symbol then that symbol
is considered fixed.
>>> from sympy.stats import Normal
>>> from sympy import pprint
>>> from sympy.abc import z
>>> X = Normal('X', 0, 1)
>>> Y = Normal('Y', 0, 1)
>>> pprint(density(X + Y, Y)(z), use_unicode=False)
2
-(-Y + z)
-----------
___ 2
\/ 2 *e
------------------
____
2*\/ pi
"""
if not is_random(condition) or pspace_independent(expr, condition):
return expr
if isinstance(condition, RandomSymbol):
condition = Eq(condition, condition.symbol)
condsymbols = random_symbols(condition)
if (isinstance(condition, Equality) and len(condsymbols) == 1 and
not isinstance(pspace(expr).domain, ConditionalDomain)):
rv = tuple(condsymbols)[0]
results = solveset(condition, rv)
if isinstance(results, Intersection) and S.Reals in results.args:
results = list(results.args[1])
sums = 0
for res in results:
temp = expr.subs(rv, res)
if temp == True:
return True
if temp != False:
# XXX: This seems nonsensical but preserves existing behaviour
# after the change that Relational is no longer a subclass of
# Expr. Here expr is sometimes Relational and sometimes Expr
# but we are trying to add them with +=. This needs to be
# fixed somehow.
if sums == 0 and isinstance(expr, Relational):
sums = expr.subs(rv, res)
else:
sums += expr.subs(rv, res)
if sums == 0:
return False
return sums
# Get full probability space of both the expression and the condition
fullspace = pspace(Tuple(expr, condition))
# Build new space given the condition
space = fullspace.conditional_space(condition, **kwargs)
# Dictionary to swap out RandomSymbols in expr with new RandomSymbols
# That point to the new conditional space
swapdict = rs_swap(fullspace.values, space.values)
# Swap random variables in the expression
expr = expr.xreplace(swapdict)
return expr
def expectation(expr, condition=None, numsamples=None, evaluate=True, **kwargs):
"""
Returns the expected value of a random expression
Parameters
==========
expr : Expr containing RandomSymbols
The expression of which you want to compute the expectation value
given : Expr containing RandomSymbols
A conditional expression. E(X, X>0) is expectation of X given X > 0
numsamples : int
Enables sampling and approximates the expectation with this many samples
evalf : Bool (defaults to True)
If sampling return a number rather than a complex expression
evaluate : Bool (defaults to True)
In case of continuous systems return unevaluated integral
Examples
========
>>> from sympy.stats import E, Die
>>> X = Die('X', 6)
>>> E(X)
7/2
>>> E(2*X + 1)
8
>>> E(X, X > 3) # Expectation of X given that it is above 3
5
"""
if not is_random(expr): # expr isn't random?
return expr
kwargs['numsamples'] = numsamples
from sympy.stats.symbolic_probability import Expectation
if evaluate:
return Expectation(expr, condition).doit(**kwargs)
return Expectation(expr, condition).rewrite(Integral) # will return Sum in case of discrete RV
def probability(condition, given_condition=None, numsamples=None,
evaluate=True, **kwargs):
"""
Probability that a condition is true, optionally given a second condition
Parameters
==========
condition : Combination of Relationals containing RandomSymbols
The condition of which you want to compute the probability
given_condition : Combination of Relationals containing RandomSymbols
A conditional expression. P(X > 1, X > 0) is expectation of X > 1
given X > 0
numsamples : int
Enables sampling and approximates the probability with this many samples
evaluate : Bool (defaults to True)
In case of continuous systems return unevaluated integral
Examples
========
>>> from sympy.stats import P, Die
>>> from sympy import Eq
>>> X, Y = Die('X', 6), Die('Y', 6)
>>> P(X > 3)
1/2
>>> P(Eq(X, 5), X > 2) # Probability that X == 5 given that X > 2
1/4
>>> P(X > Y)
5/12
"""
condition = sympify(condition)
given_condition = sympify(given_condition)
if condition.has(RandomIndexedSymbol):
return pspace(condition).probability(condition, given_condition, evaluate, **kwargs)
if isinstance(given_condition, RandomSymbol):
condrv = random_symbols(condition)
if len(condrv) == 1 and condrv[0] == given_condition:
from sympy.stats.frv_types import BernoulliDistribution
return BernoulliDistribution(probability(condition), 0, 1)
if any([dependent(rv, given_condition) for rv in condrv]):
from sympy.stats.symbolic_probability import Probability
return Probability(condition, given_condition)
else:
return probability(condition)
if given_condition is not None and \
not isinstance(given_condition, (Relational, Boolean)):
raise ValueError("%s is not a relational or combination of relationals"
% (given_condition))
if given_condition == False:
return S.Zero
if not isinstance(condition, (Relational, Boolean)):
raise ValueError("%s is not a relational or combination of relationals"
% (condition))
if condition is S.true:
return S.One
if condition is S.false:
return S.Zero
if numsamples:
return sampling_P(condition, given_condition, numsamples=numsamples,
**kwargs)
if given_condition is not None: # If there is a condition
# Recompute on new conditional expr
return probability(given(condition, given_condition, **kwargs), **kwargs)
# Otherwise pass work off to the ProbabilitySpace
if pspace(condition) == PSpace():
from sympy.stats.symbolic_probability import Probability
return Probability(condition, given_condition)
result = pspace(condition).probability(condition, **kwargs)
if evaluate and hasattr(result, 'doit'):
return result.doit()
else:
return result
class Density(Basic):
expr = property(lambda self: self.args[0])
@property
def condition(self):
if len(self.args) > 1:
return self.args[1]
else:
return None
def doit(self, evaluate=True, **kwargs):
from sympy.stats.joint_rv import JointPSpace
from sympy.stats.frv import SingleFiniteDistribution
expr, condition = self.expr, self.condition
if _sympify(expr).has(RandomMatrixSymbol):
return pspace(expr).compute_density(expr)
if isinstance(expr, SingleFiniteDistribution):
return expr.dict
if condition is not None:
# Recompute on new conditional expr
expr = given(expr, condition, **kwargs)
if isinstance(expr, RandomSymbol) and \
isinstance(expr.pspace, JointPSpace):
return expr.pspace.distribution
if not random_symbols(expr):
return Lambda(x, DiracDelta(x - expr))
if (isinstance(expr, RandomSymbol) and
hasattr(expr.pspace, 'distribution') and
isinstance(pspace(expr), (SinglePSpace))):
return expr.pspace.distribution
result = pspace(expr).compute_density(expr, **kwargs)
if evaluate and hasattr(result, 'doit'):
return result.doit()
else:
return result
def density(expr, condition=None, evaluate=True, numsamples=None, **kwargs):
"""
Probability density of a random expression, optionally given a second
condition.
This density will take on different forms for different types of
probability spaces. Discrete variables produce Dicts. Continuous
variables produce Lambdas.
Parameters
==========
expr : Expr containing RandomSymbols
The expression of which you want to compute the density value
condition : Relational containing RandomSymbols
A conditional expression. density(X > 1, X > 0) is density of X > 1
given X > 0
numsamples : int
Enables sampling and approximates the density with this many samples
Examples
========
>>> from sympy.stats import density, Die, Normal
>>> from sympy import Symbol
>>> x = Symbol('x')
>>> D = Die('D', 6)
>>> X = Normal(x, 0, 1)
>>> density(D).dict
{1: 1/6, 2: 1/6, 3: 1/6, 4: 1/6, 5: 1/6, 6: 1/6}
>>> density(2*D).dict
{2: 1/6, 4: 1/6, 6: 1/6, 8: 1/6, 10: 1/6, 12: 1/6}
>>> density(X)(x)
sqrt(2)*exp(-x**2/2)/(2*sqrt(pi))
"""
if numsamples:
return sampling_density(expr, condition, numsamples=numsamples,
**kwargs)
return Density(expr, condition).doit(evaluate=evaluate, **kwargs)
def cdf(expr, condition=None, evaluate=True, **kwargs):
"""
Cumulative Distribution Function of a random expression.
optionally given a second condition
This density will take on different forms for different types of
probability spaces.
Discrete variables produce Dicts.
Continuous variables produce Lambdas.
Examples
========
>>> from sympy.stats import density, Die, Normal, cdf
>>> D = Die('D', 6)
>>> X = Normal('X', 0, 1)
>>> density(D).dict
{1: 1/6, 2: 1/6, 3: 1/6, 4: 1/6, 5: 1/6, 6: 1/6}
>>> cdf(D)
{1: 1/6, 2: 1/3, 3: 1/2, 4: 2/3, 5: 5/6, 6: 1}
>>> cdf(3*D, D > 2)
{9: 1/4, 12: 1/2, 15: 3/4, 18: 1}
>>> cdf(X)
Lambda(_z, erf(sqrt(2)*_z/2)/2 + 1/2)
"""
if condition is not None: # If there is a condition
# Recompute on new conditional expr
return cdf(given(expr, condition, **kwargs), **kwargs)
# Otherwise pass work off to the ProbabilitySpace
result = pspace(expr).compute_cdf(expr, **kwargs)
if evaluate and hasattr(result, 'doit'):
return result.doit()
else:
return result
def characteristic_function(expr, condition=None, evaluate=True, **kwargs):
"""
Characteristic function of a random expression, optionally given a second condition
Returns a Lambda
Examples
========
>>> from sympy.stats import Normal, DiscreteUniform, Poisson, characteristic_function
>>> X = Normal('X', 0, 1)
>>> characteristic_function(X)
Lambda(_t, exp(-_t**2/2))
>>> Y = DiscreteUniform('Y', [1, 2, 7])
>>> characteristic_function(Y)
Lambda(_t, exp(7*_t*I)/3 + exp(2*_t*I)/3 + exp(_t*I)/3)
>>> Z = Poisson('Z', 2)
>>> characteristic_function(Z)
Lambda(_t, exp(2*exp(_t*I) - 2))
"""
if condition is not None:
return characteristic_function(given(expr, condition, **kwargs), **kwargs)
result = pspace(expr).compute_characteristic_function(expr, **kwargs)
if evaluate and hasattr(result, 'doit'):
return result.doit()
else:
return result
def moment_generating_function(expr, condition=None, evaluate=True, **kwargs):
if condition is not None:
return moment_generating_function(given(expr, condition, **kwargs), **kwargs)
result = pspace(expr).compute_moment_generating_function(expr, **kwargs)
if evaluate and hasattr(result, 'doit'):
return result.doit()
else:
return result
def where(condition, given_condition=None, **kwargs):
"""
Returns the domain where a condition is True.
Examples
========
>>> from sympy.stats import where, Die, Normal
>>> from sympy import And
>>> D1, D2 = Die('a', 6), Die('b', 6)
>>> a, b = D1.symbol, D2.symbol
>>> X = Normal('x', 0, 1)
>>> where(X**2<1)
Domain: (-1 < x) & (x < 1)
>>> where(X**2<1).set
Interval.open(-1, 1)
>>> where(And(D1<=D2 , D2<3))
Domain: (Eq(a, 1) & Eq(b, 1)) | (Eq(a, 1) & Eq(b, 2)) | (Eq(a, 2) & Eq(b, 2))
"""
if given_condition is not None: # If there is a condition
# Recompute on new conditional expr
return where(given(condition, given_condition, **kwargs), **kwargs)
# Otherwise pass work off to the ProbabilitySpace
return pspace(condition).where(condition, **kwargs)
def sample(expr, condition=None, size=(), library='scipy', numsamples=1,
**kwargs):
"""
A realization of the random expression
Parameters
==========
expr : Expression of random variables
Expression from which sample is extracted
condition : Expr containing RandomSymbols
A conditional expression
size : int, tuple
Represents size of each sample in numsamples
library : str
- 'scipy' : Sample using scipy
- 'numpy' : Sample using numpy
- 'pymc3' : Sample using PyMC3
Choose any of the available options to sample from as string,
by default is 'scipy'
numsamples : int
Number of samples, each with size as ``size``
Examples
========
>>> from sympy.stats import Die, sample, Normal
>>> X, Y, Z = Die('X', 6), Die('Y', 6), Die('Z', 6)
>>> die_roll = sample(X + Y + Z) # doctest: +SKIP
>>> N = Normal('N', 3, 4)
>>> samp = next(sample(N)) # doctest: +SKIP
>>> samp in N.pspace.domain.set # doctest: +SKIP
True
>>> samp = next(sample(N, N>0)) # doctest: +SKIP
>>> samp > 0 # doctest: +SKIP
True
>>> samp_list = next(sample(N, size=4)) # doctest: +SKIP
>>> [sam in N.pspace.domain.set for sam in samp_list] # doctest: +SKIP
[True, True, True, True]
Returns
=======
sample: iterator object
iterator object containing the sample/samples of given expr
"""
message = ("The return type of sample has been changed to return an "
"iterator object since version 1.7. For more information see "
"https://github.com/sympy/sympy/issues/19061")
warnings.warn(filldedent(message))
return sample_iter(expr, condition, size=size, library=library,
numsamples=numsamples)
def quantile(expr, evaluate=True, **kwargs):
r"""
Return the :math:`p^{th}` order quantile of a probability distribution.
Quantile is defined as the value at which the probability of the random
variable is less than or equal to the given probability.
..math::
Q(p) = inf{x \in (-\infty, \infty) such that p <= F(x)}
Examples
========
>>> from sympy.stats import quantile, Die, Exponential
>>> from sympy import Symbol, pprint
>>> p = Symbol("p")
>>> l = Symbol("lambda", positive=True)
>>> X = Exponential("x", l)
>>> quantile(X)(p)
-log(1 - p)/lambda
>>> D = Die("d", 6)
>>> pprint(quantile(D)(p), use_unicode=False)
/nan for Or(p > 1, p < 0)
|
| 1 for p <= 1/6
|
| 2 for p <= 1/3
|
< 3 for p <= 1/2
|
| 4 for p <= 2/3
|
| 5 for p <= 5/6
|
\ 6 for p <= 1
"""
result = pspace(expr).compute_quantile(expr, **kwargs)
if evaluate and hasattr(result, 'doit'):
return result.doit()
else:
return result
def sample_iter(expr, condition=None, size=(), library='scipy',
numsamples=S.Infinity, **kwargs):
"""
Returns an iterator of realizations from the expression given a condition
Parameters
==========
expr: Expr
Random expression to be realized
condition: Expr, optional
A conditional expression
size : int, tuple
Represents size of each sample in numsamples
numsamples: integer, optional
Length of the iterator (defaults to infinity)
Examples
========
>>> from sympy.stats import Normal, sample_iter
>>> X = Normal('X', 0, 1)
>>> expr = X*X + 3
>>> iterator = sample_iter(expr, numsamples=3) # doctest: +SKIP
>>> list(iterator) # doctest: +SKIP
[12, 4, 7]
Returns
=======
sample_iter: iterator object
iterator object containing the sample/samples of given expr
See Also
========
sample
sampling_P
sampling_E
"""
if not import_module(library):
raise ValueError("Failed to import %s" % library)
if condition is not None:
ps = pspace(Tuple(expr, condition))
else:
ps = pspace(expr)
rvs = list(ps.values)
if library == 'pymc3':
# Currently unable to lambdify in pymc3
# TODO : Remove 'pymc3' when lambdify accepts 'pymc3' as module
fn = lambdify(rvs, expr, **kwargs)
else:
fn = lambdify(rvs, expr, modules=library, **kwargs)
if condition is not None:
given_fn = lambdify(rvs, condition, **kwargs)
def return_generator():
count = 0
while count < numsamples:
d = ps.sample(size=size, library=library) # a dictionary that maps RVs to values
args = [d[rv] for rv in rvs]
if condition is not None: # Check that these values satisfy the condition
gd = given_fn(*args)
if gd != True and gd != False:
raise ValueError(
"Conditions must not contain free symbols")
if not gd: # If the values don't satisfy then try again
continue
yield fn(*args)
count += 1
return return_generator()
def sample_iter_lambdify(expr, condition=None, size=(), numsamples=S.Infinity,
**kwargs):
return sample_iter(expr, condition=condition, size=size, numsamples=numsamples,
**kwargs)
def sample_iter_subs(expr, condition=None, size=(), numsamples=S.Infinity,
**kwargs):
return sample_iter(expr, condition=condition, size=size, numsamples=numsamples,
**kwargs)
def sampling_P(condition, given_condition=None, library='scipy', numsamples=1,
evalf=True, **kwargs):
"""
Sampling version of P
See Also
========
P
sampling_E
sampling_density
"""
count_true = 0
count_false = 0
samples = sample_iter(condition, given_condition, library=library,
numsamples=numsamples, **kwargs)
for sample in samples:
if sample:
count_true += 1
else:
count_false += 1
result = S(count_true) / numsamples
if evalf:
return result.evalf()
else:
return result
def sampling_E(expr, given_condition=None, library='scipy', numsamples=1,
evalf=True, **kwargs):
"""
Sampling version of E
See Also
========
P
sampling_P
sampling_density
"""
samples = list(sample_iter(expr, given_condition, library=library,
numsamples=numsamples, **kwargs))
result = Add(*[samp for samp in samples]) / numsamples
if evalf:
return result.evalf()
else:
return result
def sampling_density(expr, given_condition=None, library='scipy',
numsamples=1, **kwargs):
"""
Sampling version of density
See Also
========
density
sampling_P
sampling_E
"""
results = {}
for result in sample_iter(expr, given_condition, library=library,
numsamples=numsamples, **kwargs):
results[result] = results.get(result, 0) + 1
return results
def dependent(a, b):
"""
Dependence of two random expressions
Two expressions are independent if knowledge of one does not change
computations on the other.
Examples
========
>>> from sympy.stats import Normal, dependent, given
>>> from sympy import Tuple, Eq
>>> X, Y = Normal('X', 0, 1), Normal('Y', 0, 1)
>>> dependent(X, Y)
False
>>> dependent(2*X + Y, -Y)
True
>>> X, Y = given(Tuple(X, Y), Eq(X + Y, 3))
>>> dependent(X, Y)
True
See Also
========
independent
"""
if pspace_independent(a, b):
return False
z = Symbol('z', real=True)
# Dependent if density is unchanged when one is given information about
# the other
return (density(a, Eq(b, z)) != density(a) or
density(b, Eq(a, z)) != density(b))
def independent(a, b):
"""
Independence of two random expressions
Two expressions are independent if knowledge of one does not change
computations on the other.
Examples
========
>>> from sympy.stats import Normal, independent, given
>>> from sympy import Tuple, Eq
>>> X, Y = Normal('X', 0, 1), Normal('Y', 0, 1)
>>> independent(X, Y)
True
>>> independent(2*X + Y, -Y)
False
>>> X, Y = given(Tuple(X, Y), Eq(X + Y, 3))
>>> independent(X, Y)
False
See Also
========
dependent
"""
return not dependent(a, b)
def pspace_independent(a, b):
"""
Tests for independence between a and b by checking if their PSpaces have
overlapping symbols. This is a sufficient but not necessary condition for
independence and is intended to be used internally.
Notes
=====
pspace_independent(a, b) implies independent(a, b)
independent(a, b) does not imply pspace_independent(a, b)
"""
a_symbols = set(pspace(b).symbols)
b_symbols = set(pspace(a).symbols)
if len(set(random_symbols(a)).intersection(random_symbols(b))) != 0:
return False
if len(a_symbols.intersection(b_symbols)) == 0:
return True
return None
def rv_subs(expr, symbols=None):
"""
Given a random expression replace all random variables with their symbols.
If symbols keyword is given restrict the swap to only the symbols listed.
"""
if symbols is None:
symbols = random_symbols(expr)
if not symbols:
return expr
swapdict = {rv: rv.symbol for rv in symbols}
return expr.subs(swapdict)
class NamedArgsMixin(object):
_argnames = () # type: tTuple[str, ...]
def __getattr__(self, attr):
try:
return self.args[self._argnames.index(attr)]
except ValueError:
raise AttributeError("'%s' object has no attribute '%s'" % (
type(self).__name__, attr))
def _value_check(condition, message):
"""
Raise a ValueError with message if condition is False, else
return True if all conditions were True, else False.
Examples
========
>>> from sympy.stats.rv import _value_check
>>> from sympy.abc import a, b, c
>>> from sympy import And, Dummy
>>> _value_check(2 < 3, '')
True
Here, the condition is not False, but it doesn't evaluate to True
so False is returned (but no error is raised). So checking if the
return value is True or False will tell you if all conditions were
evaluated.
>>> _value_check(a < b, '')
False
In this case the condition is False so an error is raised:
>>> r = Dummy(real=True)
>>> _value_check(r < r - 1, 'condition is not true')
Traceback (most recent call last):
...
ValueError: condition is not true
If no condition of many conditions must be False, they can be
checked by passing them as an iterable:
>>> _value_check((a < 0, b < 0, c < 0), '')
False
The iterable can be a generator, too:
>>> _value_check((i < 0 for i in (a, b, c)), '')
False
The following are equivalent to the above but do not pass
an iterable:
>>> all(_value_check(i < 0, '') for i in (a, b, c))
False
>>> _value_check(And(a < 0, b < 0, c < 0), '')
False
"""
from sympy.core.compatibility import iterable
from sympy.core.logic import fuzzy_and
if not iterable(condition):
condition = [condition]
truth = fuzzy_and(condition)
if truth == False:
raise ValueError(message)
return truth == True
def _symbol_converter(sym):
"""
Casts the parameter to Symbol if it is 'str'
otherwise no operation is performed on it.
Parameters
==========
sym
The parameter to be converted.
Returns
=======
Symbol
the parameter converted to Symbol.
Raises
======
TypeError
If the parameter is not an instance of both str and
Symbol.
Examples
========
>>> from sympy import Symbol
>>> from sympy.stats.rv import _symbol_converter
>>> s = _symbol_converter('s')
>>> isinstance(s, Symbol)
True
>>> _symbol_converter(1)
Traceback (most recent call last):
...
TypeError: 1 is neither a Symbol nor a string
>>> r = Symbol('r')
>>> isinstance(r, Symbol)
True
"""
if isinstance(sym, str):
sym = Symbol(sym)
if not isinstance(sym, Symbol):
raise TypeError("%s is neither a Symbol nor a string"%(sym))
return sym
def sample_stochastic_process(process):
"""
This function is used to sample from stochastic process.
Parameters
==========
process: StochasticProcess
Process used to extract the samples. It must be an instance of
StochasticProcess
Examples
========
>>> from sympy.stats import sample_stochastic_process, DiscreteMarkovChain
>>> from sympy import Matrix
>>> T = Matrix([[0.5, 0.2, 0.3],[0.2, 0.5, 0.3],[0.2, 0.3, 0.5]])
>>> Y = DiscreteMarkovChain("Y", [0, 1, 2], T)
>>> next(sample_stochastic_process(Y)) in Y.state_space # doctest: +SKIP
True
>>> next(sample_stochastic_process(Y)) # doctest: +SKIP
0
>>> next(sample_stochastic_process(Y)) # doctest: +SKIP
2
Returns
=======
sample: iterator object
iterator object containing the sample of given process
"""
from sympy.stats.stochastic_process_types import StochasticProcess
if not isinstance(process, StochasticProcess):
raise ValueError("Process must be an instance of Stochastic Process")
return process.sample()
|
6c50fdefce25b0dbc76ef770bf87e75355cda96c9a544baa099a467b5fd69d09 | import itertools
from sympy import Expr, Add, Mul, S, Integral, Eq, Sum, Symbol, expand as _expand
from sympy.core.compatibility import default_sort_key
from sympy.core.parameters import global_parameters
from sympy.core.sympify import _sympify
from sympy.stats import variance, covariance
from sympy.stats.rv import (RandomSymbol, probability, pspace,
given, sampling_E, RandomIndexedSymbol, is_random,
PSpace)
__all__ = ['Probability', 'Expectation', 'Variance', 'Covariance']
@is_random.register(Expr)
def _(x):
atoms = x.free_symbols
if len(atoms) == 1 and next(iter(atoms)) == x:
return False
return any([is_random(i) for i in atoms])
@is_random.register(RandomSymbol)
def _(x):
return True
class Probability(Expr):
"""
Symbolic expression for the probability.
Examples
========
>>> from sympy.stats import Probability, Normal
>>> from sympy import Integral
>>> X = Normal("X", 0, 1)
>>> prob = Probability(X > 1)
>>> prob
Probability(X > 1)
Integral representation:
>>> prob.rewrite(Integral)
Integral(sqrt(2)*exp(-_z**2/2)/(2*sqrt(pi)), (_z, 1, oo))
Evaluation of the integral:
>>> prob.evaluate_integral()
sqrt(2)*(-sqrt(2)*sqrt(pi)*erf(sqrt(2)/2) + sqrt(2)*sqrt(pi))/(4*sqrt(pi))
"""
def __new__(cls, prob, condition=None, **kwargs):
prob = _sympify(prob)
if condition is None:
obj = Expr.__new__(cls, prob)
else:
condition = _sympify(condition)
obj = Expr.__new__(cls, prob, condition)
obj._condition = condition
return obj
def _eval_rewrite_as_Integral(self, arg, condition=None, **kwargs):
return probability(arg, condition, evaluate=False)
_eval_rewrite_as_Sum = _eval_rewrite_as_Integral
def evaluate_integral(self):
return self.rewrite(Integral).doit()
class Expectation(Expr):
"""
Symbolic expression for the expectation.
Examples
========
>>> from sympy.stats import Expectation, Normal, Probability, Poisson
>>> from sympy import symbols, Integral, Sum
>>> mu = symbols("mu")
>>> sigma = symbols("sigma", positive=True)
>>> X = Normal("X", mu, sigma)
>>> Expectation(X)
Expectation(X)
>>> Expectation(X).evaluate_integral().simplify()
mu
To get the integral expression of the expectation:
>>> Expectation(X).rewrite(Integral)
Integral(sqrt(2)*X*exp(-(X - mu)**2/(2*sigma**2))/(2*sqrt(pi)*sigma), (X, -oo, oo))
The same integral expression, in more abstract terms:
>>> Expectation(X).rewrite(Probability)
Integral(x*Probability(Eq(X, x)), (x, -oo, oo))
To get the Summation expression of the expectation for discrete random variables:
>>> lamda = symbols('lamda', positive=True)
>>> Z = Poisson('Z', lamda)
>>> Expectation(Z).rewrite(Sum)
Sum(Z*lamda**Z*exp(-lamda)/factorial(Z), (Z, 0, oo))
This class is aware of some properties of the expectation:
>>> from sympy.abc import a
>>> Expectation(a*X)
Expectation(a*X)
>>> Y = Normal("Y", 1, 2)
>>> Expectation(X + Y)
Expectation(X + Y)
To expand the ``Expectation`` into its expression, use ``expand()``:
>>> Expectation(X + Y).expand()
Expectation(X) + Expectation(Y)
>>> Expectation(a*X + Y).expand()
a*Expectation(X) + Expectation(Y)
>>> Expectation(a*X + Y)
Expectation(a*X + Y)
>>> Expectation((X + Y)*(X - Y)).expand()
Expectation(X**2) - Expectation(Y**2)
To evaluate the ``Expectation``, use ``doit()``:
>>> Expectation(X + Y).doit()
mu + 1
>>> Expectation(X + Expectation(Y + Expectation(2*X))).doit()
3*mu + 1
To prevent evaluating nested ``Expectation``, use ``doit(deep=False)``
>>> Expectation(X + Expectation(Y)).doit(deep=False)
mu + Expectation(Expectation(Y))
>>> Expectation(X + Expectation(Y + Expectation(2*X))).doit(deep=False)
mu + Expectation(Expectation(Y + Expectation(2*X)))
"""
def __new__(cls, expr, condition=None, **kwargs):
expr = _sympify(expr)
if expr.is_Matrix:
from sympy.stats.symbolic_multivariate_probability import ExpectationMatrix
return ExpectationMatrix(expr, condition)
if condition is None:
if not is_random(expr):
return expr
obj = Expr.__new__(cls, expr)
else:
condition = _sympify(condition)
obj = Expr.__new__(cls, expr, condition)
obj._condition = condition
return obj
def expand(self, **hints):
expr = self.args[0]
condition = self._condition
if not is_random(expr):
return expr
if isinstance(expr, Add):
return Add(*[Expectation(a, condition=condition).expand()
for a in expr.args])
elif isinstance(expr, Mul):
if isinstance(_expand(expr), Add):
return Expectation(_expand(expr)).expand()
rv = []
nonrv = []
for a in expr.args:
if is_random(a):
rv.append(a)
else:
nonrv.append(a)
return Mul(*nonrv)*Expectation(Mul(*rv), condition=condition)
return self
def doit(self, **hints):
deep = hints.get('deep', True)
condition = self._condition
expr = self.args[0]
numsamples = hints.get('numsamples', False)
for_rewrite = not hints.get('for_rewrite', False)
if deep:
expr = expr.doit(**hints)
if not is_random(expr) or isinstance(expr, Expectation): # expr isn't random?
return expr
if numsamples: # Computing by monte carlo sampling?
evalf = hints.get('evalf', True)
return sampling_E(expr, condition, numsamples=numsamples, evalf=evalf)
if expr.has(RandomIndexedSymbol):
return pspace(expr).compute_expectation(expr, condition)
# Create new expr and recompute E
if condition is not None: # If there is a condition
return self.func(given(expr, condition)).doit(**hints)
# A few known statements for efficiency
if expr.is_Add: # We know that E is Linear
return Add(*[self.func(arg, condition).doit(**hints)
if not isinstance(arg, Expectation) else self.func(arg, condition)
for arg in expr.args])
if expr.is_Mul:
if expr.atoms(Expectation):
return expr
if pspace(expr) == PSpace():
return self.func(expr)
# Otherwise case is simple, pass work off to the ProbabilitySpace
result = pspace(expr).compute_expectation(expr, evaluate=for_rewrite)
if hasattr(result, 'doit') and for_rewrite:
return result.doit(**hints)
else:
return result
def _eval_rewrite_as_Probability(self, arg, condition=None, **kwargs):
rvs = arg.atoms(RandomSymbol)
if len(rvs) > 1:
raise NotImplementedError()
if len(rvs) == 0:
return arg
rv = rvs.pop()
if rv.pspace is None:
raise ValueError("Probability space not known")
symbol = rv.symbol
if symbol.name[0].isupper():
symbol = Symbol(symbol.name.lower())
else :
symbol = Symbol(symbol.name + "_1")
if rv.pspace.is_Continuous:
return Integral(arg.replace(rv, symbol)*Probability(Eq(rv, symbol), condition), (symbol, rv.pspace.domain.set.inf, rv.pspace.domain.set.sup))
else:
if rv.pspace.is_Finite:
raise NotImplementedError
else:
return Sum(arg.replace(rv, symbol)*Probability(Eq(rv, symbol), condition), (symbol, rv.pspace.domain.set.inf, rv.pspace.set.sup))
def _eval_rewrite_as_Integral(self, arg, condition=None, **kwargs):
return self.func(arg, condition=condition).doit(deep=False, for_rewrite=True)
_eval_rewrite_as_Sum = _eval_rewrite_as_Integral # For discrete this will be Sum
def evaluate_integral(self):
return self.rewrite(Integral).doit()
evaluate_sum = evaluate_integral
class Variance(Expr):
"""
Symbolic expression for the variance.
Examples
========
>>> from sympy import symbols, Integral
>>> from sympy.stats import Normal, Expectation, Variance, Probability
>>> mu = symbols("mu", positive=True)
>>> sigma = symbols("sigma", positive=True)
>>> X = Normal("X", mu, sigma)
>>> Variance(X)
Variance(X)
>>> Variance(X).evaluate_integral()
sigma**2
Integral representation of the underlying calculations:
>>> Variance(X).rewrite(Integral)
Integral(sqrt(2)*(X - Integral(sqrt(2)*X*exp(-(X - mu)**2/(2*sigma**2))/(2*sqrt(pi)*sigma), (X, -oo, oo)))**2*exp(-(X - mu)**2/(2*sigma**2))/(2*sqrt(pi)*sigma), (X, -oo, oo))
Integral representation, without expanding the PDF:
>>> Variance(X).rewrite(Probability)
-Integral(x*Probability(Eq(X, x)), (x, -oo, oo))**2 + Integral(x**2*Probability(Eq(X, x)), (x, -oo, oo))
Rewrite the variance in terms of the expectation
>>> Variance(X).rewrite(Expectation)
-Expectation(X)**2 + Expectation(X**2)
Some transformations based on the properties of the variance may happen:
>>> from sympy.abc import a
>>> Y = Normal("Y", 0, 1)
>>> Variance(a*X)
Variance(a*X)
To expand the variance in its expression, use ``expand()``:
>>> Variance(a*X).expand()
a**2*Variance(X)
>>> Variance(X + Y)
Variance(X + Y)
>>> Variance(X + Y).expand()
2*Covariance(X, Y) + Variance(X) + Variance(Y)
"""
def __new__(cls, arg, condition=None, **kwargs):
arg = _sympify(arg)
if arg.is_Matrix:
from sympy.stats.symbolic_multivariate_probability import VarianceMatrix
return VarianceMatrix(arg, condition)
if condition is None:
obj = Expr.__new__(cls, arg)
else:
condition = _sympify(condition)
obj = Expr.__new__(cls, arg, condition)
obj._condition = condition
return obj
def expand(self, **hints):
arg = self.args[0]
condition = self._condition
if not is_random(arg):
return S.Zero
if isinstance(arg, RandomSymbol):
return self
elif isinstance(arg, Add):
rv = []
for a in arg.args:
if is_random(a):
rv.append(a)
variances = Add(*map(lambda xv: Variance(xv, condition).expand(), rv))
map_to_covar = lambda x: 2*Covariance(*x, condition=condition).expand()
covariances = Add(*map(map_to_covar, itertools.combinations(rv, 2)))
return variances + covariances
elif isinstance(arg, Mul):
nonrv = []
rv = []
for a in arg.args:
if is_random(a):
rv.append(a)
else:
nonrv.append(a**2)
if len(rv) == 0:
return S.Zero
return Mul(*nonrv)*Variance(Mul(*rv), condition)
# this expression contains a RandomSymbol somehow:
return self
def _eval_rewrite_as_Expectation(self, arg, condition=None, **kwargs):
e1 = Expectation(arg**2, condition)
e2 = Expectation(arg, condition)**2
return e1 - e2
def _eval_rewrite_as_Probability(self, arg, condition=None, **kwargs):
return self.rewrite(Expectation).rewrite(Probability)
def _eval_rewrite_as_Integral(self, arg, condition=None, **kwargs):
return variance(self.args[0], self._condition, evaluate=False)
_eval_rewrite_as_Sum = _eval_rewrite_as_Integral
def evaluate_integral(self):
return self.rewrite(Integral).doit()
class Covariance(Expr):
"""
Symbolic expression for the covariance.
Examples
========
>>> from sympy.stats import Covariance
>>> from sympy.stats import Normal
>>> X = Normal("X", 3, 2)
>>> Y = Normal("Y", 0, 1)
>>> Z = Normal("Z", 0, 1)
>>> W = Normal("W", 0, 1)
>>> cexpr = Covariance(X, Y)
>>> cexpr
Covariance(X, Y)
Evaluate the covariance, `X` and `Y` are independent,
therefore zero is the result:
>>> cexpr.evaluate_integral()
0
Rewrite the covariance expression in terms of expectations:
>>> from sympy.stats import Expectation
>>> cexpr.rewrite(Expectation)
Expectation(X*Y) - Expectation(X)*Expectation(Y)
In order to expand the argument, use ``expand()``:
>>> from sympy.abc import a, b, c, d
>>> Covariance(a*X + b*Y, c*Z + d*W)
Covariance(a*X + b*Y, c*Z + d*W)
>>> Covariance(a*X + b*Y, c*Z + d*W).expand()
a*c*Covariance(X, Z) + a*d*Covariance(W, X) + b*c*Covariance(Y, Z) + b*d*Covariance(W, Y)
This class is aware of some properties of the covariance:
>>> Covariance(X, X).expand()
Variance(X)
>>> Covariance(a*X, b*Y).expand()
a*b*Covariance(X, Y)
"""
def __new__(cls, arg1, arg2, condition=None, **kwargs):
arg1 = _sympify(arg1)
arg2 = _sympify(arg2)
if arg1.is_Matrix or arg2.is_Matrix:
from sympy.stats.symbolic_multivariate_probability import CrossCovarianceMatrix
return CrossCovarianceMatrix(arg1, arg2, condition)
if kwargs.pop('evaluate', global_parameters.evaluate):
arg1, arg2 = sorted([arg1, arg2], key=default_sort_key)
if condition is None:
obj = Expr.__new__(cls, arg1, arg2)
else:
condition = _sympify(condition)
obj = Expr.__new__(cls, arg1, arg2, condition)
obj._condition = condition
return obj
def expand(self, **hints):
arg1 = self.args[0]
arg2 = self.args[1]
condition = self._condition
if arg1 == arg2:
return Variance(arg1, condition).expand()
if not is_random(arg1):
return S.Zero
if not is_random(arg2):
return S.Zero
arg1, arg2 = sorted([arg1, arg2], key=default_sort_key)
if isinstance(arg1, RandomSymbol) and isinstance(arg2, RandomSymbol):
return Covariance(arg1, arg2, condition)
coeff_rv_list1 = self._expand_single_argument(arg1.expand())
coeff_rv_list2 = self._expand_single_argument(arg2.expand())
addends = [a*b*Covariance(*sorted([r1, r2], key=default_sort_key), condition=condition)
for (a, r1) in coeff_rv_list1 for (b, r2) in coeff_rv_list2]
return Add(*addends)
@classmethod
def _expand_single_argument(cls, expr):
# return (coefficient, random_symbol) pairs:
if isinstance(expr, RandomSymbol):
return [(S.One, expr)]
elif isinstance(expr, Add):
outval = []
for a in expr.args:
if isinstance(a, Mul):
outval.append(cls._get_mul_nonrv_rv_tuple(a))
elif is_random(a):
outval.append((S.One, a))
return outval
elif isinstance(expr, Mul):
return [cls._get_mul_nonrv_rv_tuple(expr)]
elif is_random(expr):
return [(S.One, expr)]
@classmethod
def _get_mul_nonrv_rv_tuple(cls, m):
rv = []
nonrv = []
for a in m.args:
if is_random(a):
rv.append(a)
else:
nonrv.append(a)
return (Mul(*nonrv), Mul(*rv))
def _eval_rewrite_as_Expectation(self, arg1, arg2, condition=None, **kwargs):
e1 = Expectation(arg1*arg2, condition)
e2 = Expectation(arg1, condition)*Expectation(arg2, condition)
return e1 - e2
def _eval_rewrite_as_Probability(self, arg1, arg2, condition=None, **kwargs):
return self.rewrite(Expectation).rewrite(Probability)
def _eval_rewrite_as_Integral(self, arg1, arg2, condition=None, **kwargs):
return covariance(self.args[0], self.args[1], self._condition, evaluate=False)
_eval_rewrite_as_Sum = _eval_rewrite_as_Integral
def evaluate_integral(self):
return self.rewrite(Integral).doit()
|
e90782ee4a9a94415734eff208cc5a8558a05779921972edaaa9026348190e7b | """
Number theory module (primes, etc)
"""
from .generate import nextprime, prevprime, prime, primepi, primerange, \
randprime, Sieve, sieve, primorial, cycle_length, composite, compositepi
from .primetest import isprime, is_gaussian_prime
from .factor_ import divisors, proper_divisors, factorint, multiplicity, \
multiplicity_in_factorial, perfect_power, pollard_pm1, pollard_rho, \
primefactors, totient, trailing, \
divisor_count, proper_divisor_count, divisor_sigma, factorrat, \
reduced_totient, primenu, primeomega, mersenne_prime_exponent, \
is_perfect, is_mersenne_prime, is_abundant, is_deficient, is_amicable, \
abundance, dra, drm
from .partitions_ import npartitions
from .residue_ntheory import is_primitive_root, is_quad_residue, \
legendre_symbol, jacobi_symbol, n_order, sqrt_mod, quadratic_residues, \
primitive_root, nthroot_mod, is_nthpow_residue, sqrt_mod_iter, mobius, \
discrete_log, quadratic_congruence, polynomial_congruence
from .multinomial import binomial_coefficients, binomial_coefficients_list, \
multinomial_coefficients
from .continued_fraction import continued_fraction_periodic, \
continued_fraction_iterator, continued_fraction_reduce, \
continued_fraction_convergents, continued_fraction
from .digits import count_digits, digits, is_palindromic
from .egyptian_fraction import egyptian_fraction
from .ecm import ecm
__all__ = [
'nextprime', 'prevprime', 'prime', 'primepi', 'primerange', 'randprime',
'Sieve', 'sieve', 'primorial', 'cycle_length', 'composite', 'compositepi',
'isprime', 'is_gaussian_prime',
'divisors', 'proper_divisors', 'factorint', 'multiplicity', 'perfect_power',
'pollard_pm1', 'pollard_rho', 'primefactors', 'totient', 'trailing',
'divisor_count', 'proper_divisor_count', 'divisor_sigma', 'factorrat',
'reduced_totient', 'primenu', 'primeomega', 'mersenne_prime_exponent',
'is_perfect', 'is_mersenne_prime', 'is_abundant', 'is_deficient', 'is_amicable',
'abundance', 'dra', 'drm', 'multiplicity_in_factorial',
'npartitions',
'is_primitive_root', 'is_quad_residue', 'legendre_symbol',
'jacobi_symbol', 'n_order', 'sqrt_mod', 'quadratic_residues',
'primitive_root', 'nthroot_mod', 'is_nthpow_residue', 'sqrt_mod_iter',
'mobius', 'discrete_log', 'quadratic_congruence', 'polynomial_congruence',
'binomial_coefficients', 'binomial_coefficients_list',
'multinomial_coefficients',
'continued_fraction_periodic', 'continued_fraction_iterator',
'continued_fraction_reduce', 'continued_fraction_convergents',
'continued_fraction',
'digits',
'count_digits',
'is_palindromic',
'egyptian_fraction',
'ecm',
]
|
967a926dfa45e6515ed2702020bf31c2c9ee0e058cbd70ea7d95889e6492ef4e | from sympy.ntheory import sieve, isprime
from sympy.core.power import integer_log
from sympy.core.compatibility import as_int
import random
#----------------------------------------------------------------------------#
# #
# Lenstra's Elliptic Curve Factorization #
# #
#----------------------------------------------------------------------------#
class Point:
"""Montgomery form of Points in an elliptic curve.
In this form, the addition and doubling of points
does not need any y-coordinate information thus
decreasing the number of operations.
Using Montgomery form we try to perform point addition
and doubling in least amount of multiplications.
The elliptic curve used here is of the form
(E : b*y**2*z = x**3 + a*x**2*z + x*z**2).
The a_24 parameter is equal to (a + 2)/4.
References
==========
.. [1] http://www.hyperelliptic.org/tanja/SHARCS/talks06/Gaj.pdf
"""
def __init__(self, x_cord, z_cord, a_24, mod):
"""
Initial parameters for the Point class.
Parameters:
===========
x_cord : X coordinate of the Point
z_cord : Z coordinate of the Point
a_24 : Parameter of the elliptic curve in Montgomery form
mod : modulus
"""
self.x_cord = x_cord
self.z_cord = z_cord
self.a_24 = a_24
self.mod = mod
def __eq__(self, other):
"""Two points are equal if X/Z of both points are equal
"""
from sympy import mod_inverse
if self.a_24 != other.a_24 or self.mod != other.mod:
return False
return self.x_cord * mod_inverse(self.z_cord, self.mod) % self.mod ==\
other.x_cord * mod_inverse(other.z_cord, self.mod) % self.mod
def add(self, Q, diff):
"""
Add two points self and Q where diff = self - Q. Moreover the assumption
is self.x_cord*Q.x_cord*(self.x_cord - Q.x_cord) != 0. This algorithm
requires 6 multiplications. Here the difference between the points
is already known and using this algorihtm speeds up the addition
by reducing the number of multiplication required. Also in the
mont_ladder algorithm is constructed in a way so that the difference
between intermediate points is always equal to the initial point.
So, we always know what the difference between the point is.
Parameters:
===========
Q : point on the curve in Montgomery form
diff : self - Q
Examples
========
>>> from sympy.ntheory.ecm import Point
>>> p1 = Point(11, 16, 7, 29)
>>> p2 = Point(13, 10, 7, 29)
>>> p3 = p2.add(p1, p1)
>>> p3.x_cord
23
>>> p3.z_cord
17
"""
u = (self.x_cord - self.z_cord)*(Q.x_cord + Q.z_cord)
v = (self.x_cord + self.z_cord)*(Q.x_cord - Q.z_cord)
add, subt = u + v, u - v
x_cord = diff.z_cord * add * add % self.mod
z_cord = diff.x_cord * subt * subt % self.mod
return Point(x_cord, z_cord, self.a_24, self.mod)
def double(self):
"""
Doubles a point in an elliptic curve in Montgomery form.
This algorithm requires 5 multiplications.
Examples
========
>>> from sympy.ntheory.ecm import Point
>>> p1 = Point(11, 16, 7, 29)
>>> p2 = p1.double()
>>> p2.x_cord
13
>>> p2.z_cord
10
"""
u, v = self.x_cord + self.z_cord, self.x_cord - self.z_cord
u, v = u*u, v*v
diff = u - v
x_cord = u*v % self.mod
z_cord = diff*(v + self.a_24*diff) % self.mod
return Point(x_cord, z_cord, self.a_24, self.mod)
def mont_ladder(self, k):
"""
Scalar multiplication of a point in Montgomery form
using Montgomery Ladder Algorithm.
A total of 11 multiplications are required in each step of this
algorithm.
Parameters:
==========
k : The positive integer multiplier
Examples
========
>>> from sympy.ntheory.ecm import Point
>>> p1 = Point(11, 16, 7, 29)
>>> p3 = p1.mont_ladder(3)
>>> p3.x_cord
23
>>> p3.z_cord
17
"""
Q = self
R = self.double()
for i in bin(k)[3:]:
if i == '1':
Q = R.add(Q, self)
R = R.double()
else:
R = Q.add(R, self)
Q = Q.double()
return Q
def ecm_one_factor(n, B1=10000, B2=100000, max_curve=200):
"""Returns one factor of n using
Lenstra's 2 Stage Elliptic curve Factorization
with Suyama's Parameterization. Here Montgomery
arithmetic is used for fast computation of addition
and doubling of points in elliptic curve.
This ECM method considers elliptic curves in Montgomery
form (E : b*y**2*z = x**3 + a*x**2*z + x*z**2) and involves
elliptic curve operations (mod N), where the elements in
Z are reduced (mod N). Since N is not a prime, E over FF(N)
is not really an elliptic curve but we can still do point additions
and doubling as if FF(N) was a field.
Stage 1 : The basic algorithm involves taking a random point (P) on an
elliptic curve in FF(N). The compute k*P using Montgomery ladder algorithm.
Let q be an unknown factor of N. Then the order of the curve E, |E(FF(q))|,
might be a smooth number that divides k. Then we have k = l * |E(FF(q))|
for some l. For any point belonging to the curve E, |E(FF(q))|*P = O,
hence k*P = l*|E(FF(q))|*P. Thus kP.z_cord = 0 (mod q), and the unknownn
factor of N (q) can be recovered by taking gcd(kP.z_cord, N).
Stage 2 : This is a continuation of Stage 1 if k*P != O. The idea utilize
the fact that even if kP != 0, the value of k might miss just one large
prime divisor of |E(FF(q))|. In this case we only need to compute the
scalar multiplication by p to get p*k*P = O. Here a second bound B2
restrict the size of possible values of p.
Parameters:
===========
n : Number to be Factored
B1 : Stage 1 Bound
B2 : Stage 2 Bound
max_curve : Maximum number of curves generated
References
==========
.. [1] Carl Pomerance and Richard Crandall "Prime Numbers:
A Computational Perspective" (2nd Ed.), page 344
"""
from sympy import gcd, mod_inverse, sqrt
n = as_int(n)
if B1 % 2 != 0 or B2 % 2 != 0:
raise ValueError("The Bounds should be an even integer")
sieve.extend(B2)
if isprime(n):
return n
curve = 0
D = int(sqrt(B2))
beta = [0]*(D + 1)
S = [0]*(D + 1)
k = 1
for p in sieve.primerange(1, B1 + 1):
k *= pow(p, integer_log(B1, p)[0])
g = 1
while(curve <= max_curve):
curve += 1
#Suyama's Paramatrization
sigma = random.randint(6, n - 1)
u = (sigma*sigma - 5) % n
v = (4*sigma) % n
diff = v - u
u_3 = pow(u, 3, n)
try:
C = (pow(diff, 3, n)*(3*u + v)*mod_inverse(4*u_3*v, n) - 2) % n
except ValueError:
#If the mod_inverse(4*u_3*v, n) doesn't exist
return gcd(4*u_3*v, n)
a24 = (C + 2)*mod_inverse(4, n) % n
Q = Point(u_3 , pow(v, 3, n), a24, n)
Q = Q.mont_ladder(k)
g = gcd(Q.z_cord, n)
#Stage 1 factor
if g != 1 and g != n:
return g
#Stage 1 failure. Q.z = 0, Try another curve
elif g == n:
continue
#Stage 2 - Improved Standard Continuation
S[1] = Q.double()
S[2] = S[1].double()
beta[1] = (S[1].x_cord*S[1].z_cord) % n
beta[2] = (S[2].x_cord*S[2].z_cord) % n
for d in range(3, D + 1):
S[d] = S[d - 1].add(S[1], S[d - 2])
beta[d] = (S[d].x_cord*S[d].z_cord) % n
g = 1
B = B1 - 1
T = Q.mont_ladder(B - 2*D)
R = Q.mont_ladder(B)
for r in range(B, B2, 2*D):
alpha = (R.x_cord*R.z_cord) % n
for q in sieve.primerange(r + 2, r + 2*D + 1):
delta = (q - r) // 2
f = (R.x_cord - S[d].x_cord)*(R.z_cord + S[d].z_cord) -\
alpha + beta[delta]
g = (g*f) % n
#Swap
T, R = R, R.add(S[D], T)
g = gcd(n, g)
#Stage 2 Factor found
if g != 1 and g != n:
return g
#ECM failed, Increase the bounds
raise ValueError("Increase the bounds")
def ecm(n, B1=10000, B2=100000, max_curve=200, seed=1234):
"""Performs factorization using Lenstra's Elliptic curve method.
This function repeatedly calls `ecm_one_factor` to compute the factors
of n. First all the small factors are taken out using trial division.
Then `ecm_one_factor` is used to compute one factor at a time.
Parameters:
===========
n : Number to be Factored
B1 : Stage 1 Bound
B2 : Stage 2 Bound
max_curve : Maximum number of curves generated
seed : Initialize pseudorandom generator
Examples
========
>>> from sympy.ntheory import ecm
>>> ecm(25645121643901801)
{5394769, 4753701529}
>>> ecm(9804659461513846513)
{4641991, 2112166839943}
"""
_factors = set()
for prime in sieve.primerange(1, 100000):
if n % prime == 0:
_factors.add(prime)
while(n % prime == 0):
n //= prime
random.seed(seed)
while(n > 1):
try:
factor = ecm_one_factor(n, B1, B2, max_curve)
except ValueError:
raise ValueError("Increase the bounds")
_factors.add(factor)
n //= factor
factors = set()
for factor in _factors:
if isprime(factor):
factors.add(factor)
continue
factors |= ecm(factor)
return factors
|
05a38b5f9acfec05975d1323bd376c81d292a94368d9966d681d4bf05ecd0924 | from __future__ import print_function, division
from sympy.core import S, Symbol, Add, sympify, Expr, PoleError, Mul
from sympy.core.exprtools import factor_terms
from sympy.core.symbol import Dummy
from sympy.functions.combinatorial.factorials import factorial
from sympy.functions.special.gamma_functions import gamma
from sympy.polys import PolynomialError, factor
from sympy.series.order import Order
from sympy.simplify.ratsimp import ratsimp
from sympy.simplify.simplify import together
from .gruntz import gruntz
def limit(e, z, z0, dir="+"):
"""Computes the limit of ``e(z)`` at the point ``z0``.
Parameters
==========
e : expression, the limit of which is to be taken
z : symbol representing the variable in the limit.
Other symbols are treated as constants. Multivariate limits
are not supported.
z0 : the value toward which ``z`` tends. Can be any expression,
including ``oo`` and ``-oo``.
dir : string, optional (default: "+")
The limit is bi-directional if ``dir="+-"``, from the right
(z->z0+) if ``dir="+"``, and from the left (z->z0-) if
``dir="-"``. For infinite ``z0`` (``oo`` or ``-oo``), the ``dir``
argument is determined from the direction of the infinity
(i.e., ``dir="-"`` for ``oo``).
Examples
========
>>> from sympy import limit, sin, oo
>>> from sympy.abc import x
>>> limit(sin(x)/x, x, 0)
1
>>> limit(1/x, x, 0) # default dir='+'
oo
>>> limit(1/x, x, 0, dir="-")
-oo
>>> limit(1/x, x, 0, dir='+-')
zoo
>>> limit(1/x, x, oo)
0
Notes
=====
First we try some heuristics for easy and frequent cases like "x", "1/x",
"x**2" and similar, so that it's fast. For all other cases, we use the
Gruntz algorithm (see the gruntz() function).
See Also
========
limit_seq : returns the limit of a sequence.
"""
return Limit(e, z, z0, dir).doit(deep=False)
def heuristics(e, z, z0, dir):
"""Computes the limit of an expression term-wise.
Parameters are the same as for the ``limit`` function.
Works with the arguments of expression ``e`` one by one, computing
the limit of each and then combining the results. This approach
works only for simple limits, but it is fast.
"""
from sympy.calculus.util import AccumBounds
rv = None
if abs(z0) is S.Infinity:
rv = limit(e.subs(z, 1/z), z, S.Zero, "+" if z0 is S.Infinity else "-")
if isinstance(rv, Limit):
return
elif e.is_Mul or e.is_Add or e.is_Pow or e.is_Function:
r = []
for a in e.args:
l = limit(a, z, z0, dir)
if l.has(S.Infinity) and l.is_finite is None:
if isinstance(e, Add):
m = factor_terms(e)
if not isinstance(m, Mul): # try together
m = together(m)
if not isinstance(m, Mul): # try factor if the previous methods failed
m = factor(e)
if isinstance(m, Mul):
return heuristics(m, z, z0, dir)
return
return
elif isinstance(l, Limit):
return
elif l is S.NaN:
return
else:
r.append(l)
if r:
rv = e.func(*r)
if rv is S.NaN and e.is_Mul and any(isinstance(rr, AccumBounds) for rr in r):
r2 = []
e2 = []
for ii in range(len(r)):
if isinstance(r[ii], AccumBounds):
r2.append(r[ii])
else:
e2.append(e.args[ii])
if len(e2) > 0:
e3 = Mul(*e2).simplify()
l = limit(e3, z, z0, dir)
rv = l * Mul(*r2)
if rv is S.NaN:
try:
rat_e = ratsimp(e)
except PolynomialError:
return
if rat_e is S.NaN or rat_e == e:
return
return limit(rat_e, z, z0, dir)
return rv
class Limit(Expr):
"""Represents an unevaluated limit.
Examples
========
>>> from sympy import Limit, sin
>>> from sympy.abc import x
>>> Limit(sin(x)/x, x, 0)
Limit(sin(x)/x, x, 0)
>>> Limit(1/x, x, 0, dir="-")
Limit(1/x, x, 0, dir='-')
"""
def __new__(cls, e, z, z0, dir="+"):
e = sympify(e)
z = sympify(z)
z0 = sympify(z0)
if z0 is S.Infinity:
dir = "-"
elif z0 is S.NegativeInfinity:
dir = "+"
if isinstance(dir, str):
dir = Symbol(dir)
elif not isinstance(dir, Symbol):
raise TypeError("direction must be of type basestring or "
"Symbol, not %s" % type(dir))
if str(dir) not in ('+', '-', '+-'):
raise ValueError("direction must be one of '+', '-' "
"or '+-', not %s" % dir)
obj = Expr.__new__(cls)
obj._args = (e, z, z0, dir)
return obj
@property
def free_symbols(self):
e = self.args[0]
isyms = e.free_symbols
isyms.difference_update(self.args[1].free_symbols)
isyms.update(self.args[2].free_symbols)
return isyms
def doit(self, **hints):
"""Evaluates the limit.
Parameters
==========
deep : bool, optional (default: True)
Invoke the ``doit`` method of the expressions involved before
taking the limit.
hints : optional keyword arguments
To be passed to ``doit`` methods; only used if deep is True.
"""
from sympy import sign
from sympy.functions import RisingFactorial
e, z, z0, dir = self.args
if z0 is S.ComplexInfinity:
raise NotImplementedError("Limits at complex "
"infinity are not implemented")
if hints.get('deep', True):
e = e.doit(**hints)
z = z.doit(**hints)
z0 = z0.doit(**hints)
if e == z:
return z0
if not e.has(z):
return e
cdir = 0
if str(dir) == "+":
cdir = 1
elif str(dir) == "-":
cdir = -1
if e.is_meromorphic(z, z0):
if abs(z0) is S.Infinity:
newe = e.subs(z, -1/z)
else:
newe = e.subs(z, z + z0)
try:
coeff, exp = newe.leadterm(z, cdir)
except ValueError:
pass
else:
if exp > 0:
return S.Zero
elif exp == 0:
return coeff
if str(dir) == "+" or not(int(exp) & 1):
return S.Infinity*sign(coeff)
elif str(dir) == "-":
return S.NegativeInfinity*sign(coeff)
else:
return S.ComplexInfinity
# gruntz fails on factorials but works with the gamma function
# If no factorial term is present, e should remain unchanged.
# factorial is defined to be zero for negative inputs (which
# differs from gamma) so only rewrite for positive z0.
if z0.is_extended_positive:
e = e.rewrite([factorial, RisingFactorial], gamma)
if e.is_Mul and abs(z0) is S.Infinity:
e = factor_terms(e)
u = Dummy('u', positive=True)
if z0 is S.NegativeInfinity:
inve = e.subs(z, -1/u)
else:
inve = e.subs(z, 1/u)
try:
f = inve.as_leading_term(u)
if f.is_meromorphic(u, S.Zero):
r = limit(f, u, S.Zero, "+")
if isinstance(r, Limit):
return self
else:
return r
except (ValueError, NotImplementedError, PoleError):
pass
if e.is_Order:
return Order(limit(e.expr, z, z0), *e.args[1:])
l = None
try:
if str(dir) == '+-':
r = gruntz(e, z, z0, '+')
l = gruntz(e, z, z0, '-')
if l != r:
raise ValueError("The limit does not exist since "
"left hand limit = %s and right hand limit = %s"
% (l, r))
else:
r = gruntz(e, z, z0, dir)
if r is S.NaN or l is S.NaN:
raise PoleError()
except (PoleError, ValueError):
if l is not None:
raise
r = heuristics(e, z, z0, dir)
if r is None:
return self
return r
|
2a14d9034e56b4eb09d3564fbfff42d45c5abc3666637dc78136d023e84da165 | """Limits of sequences"""
from __future__ import print_function, division
from sympy.core.add import Add
from sympy.core.function import PoleError
from sympy.core.power import Pow
from sympy.core.singleton import S
from sympy.core.symbol import Dummy
from sympy.core.sympify import sympify
from sympy.functions.combinatorial.numbers import fibonacci
from sympy.functions.combinatorial.factorials import factorial, subfactorial
from sympy.functions.special.gamma_functions import gamma
from sympy.functions.elementary.complexes import Abs
from sympy.functions.elementary.miscellaneous import Max, Min
from sympy.functions.elementary.trigonometric import cos, sin
from sympy.series.limits import Limit
def difference_delta(expr, n=None, step=1):
"""Difference Operator.
Discrete analog of differential operator. Given a sequence x[n],
returns the sequence x[n + step] - x[n].
Examples
========
>>> from sympy import difference_delta as dd
>>> from sympy.abc import n
>>> dd(n*(n + 1), n)
2*n + 2
>>> dd(n*(n + 1), n, 2)
4*n + 6
References
==========
.. [1] https://reference.wolfram.com/language/ref/DifferenceDelta.html
"""
expr = sympify(expr)
if n is None:
f = expr.free_symbols
if len(f) == 1:
n = f.pop()
elif len(f) == 0:
return S.Zero
else:
raise ValueError("Since there is more than one variable in the"
" expression, a variable must be supplied to"
" take the difference of %s" % expr)
step = sympify(step)
if step.is_number is False or step.is_finite is False:
raise ValueError("Step should be a finite number.")
if hasattr(expr, '_eval_difference_delta'):
result = expr._eval_difference_delta(n, step)
if result:
return result
return expr.subs(n, n + step) - expr
def dominant(expr, n):
"""Finds the dominant term in a sum, that is a term that dominates
every other term.
If limit(a/b, n, oo) is oo then a dominates b.
If limit(a/b, n, oo) is 0 then b dominates a.
Otherwise, a and b are comparable.
If there is no unique dominant term, then returns ``None``.
Examples
========
>>> from sympy import Sum
>>> from sympy.series.limitseq import dominant
>>> from sympy.abc import n, k
>>> dominant(5*n**3 + 4*n**2 + n + 1, n)
5*n**3
>>> dominant(2**n + Sum(k, (k, 0, n)), n)
2**n
See Also
========
sympy.series.limitseq.dominant
"""
terms = Add.make_args(expr.expand(func=True))
term0 = terms[-1]
comp = [term0] # comparable terms
for t in terms[:-1]:
e = (term0 / t).gammasimp()
l = limit_seq(e, n)
if l is None:
return None
elif l.is_zero:
term0 = t
comp = [term0]
elif l not in [S.Infinity, S.NegativeInfinity]:
comp.append(t)
if len(comp) > 1:
return None
return term0
def _limit_inf(expr, n):
try:
return Limit(expr, n, S.Infinity).doit(deep=False)
except (NotImplementedError, PoleError):
return None
def _limit_seq(expr, n, trials):
from sympy.concrete.summations import Sum
for i in range(trials):
if not expr.has(Sum):
result = _limit_inf(expr, n)
if result is not None:
return result
num, den = expr.as_numer_denom()
if not den.has(n) or not num.has(n):
result = _limit_inf(expr.doit(), n)
if result is not None:
return result
return None
num, den = (difference_delta(t.expand(), n) for t in [num, den])
expr = (num / den).gammasimp()
if not expr.has(Sum):
result = _limit_inf(expr, n)
if result is not None:
return result
num, den = expr.as_numer_denom()
num = dominant(num, n)
if num is None:
return None
den = dominant(den, n)
if den is None:
return None
expr = (num / den).gammasimp()
def limit_seq(expr, n=None, trials=5):
"""Finds the limit of a sequence as index n tends to infinity.
Parameters
==========
expr : Expr
SymPy expression for the n-th term of the sequence
n : Symbol, optional
The index of the sequence, an integer that tends to positive
infinity. If None, inferred from the expression unless it has
multiple symbols.
trials: int, optional
The algorithm is highly recursive. ``trials`` is a safeguard from
infinite recursion in case the limit is not easily computed by the
algorithm. Try increasing ``trials`` if the algorithm returns ``None``.
Admissible Terms
================
The algorithm is designed for sequences built from rational functions,
indefinite sums, and indefinite products over an indeterminate n. Terms of
alternating sign are also allowed, but more complex oscillatory behavior is
not supported.
Examples
========
>>> from sympy import limit_seq, Sum, binomial
>>> from sympy.abc import n, k, m
>>> limit_seq((5*n**3 + 3*n**2 + 4) / (3*n**3 + 4*n - 5), n)
5/3
>>> limit_seq(binomial(2*n, n) / Sum(binomial(2*k, k), (k, 1, n)), n)
3/4
>>> limit_seq(Sum(k**2 * Sum(2**m/m, (m, 1, k)), (k, 1, n)) / (2**n*n), n)
4
See Also
========
sympy.series.limitseq.dominant
References
==========
.. [1] Computing Limits of Sequences - Manuel Kauers
"""
from sympy.concrete.summations import Sum
from sympy.calculus.util import AccumulationBounds
if n is None:
free = expr.free_symbols
if len(free) == 1:
n = free.pop()
elif not free:
return expr
else:
raise ValueError("Expression has more than one variable. "
"Please specify a variable.")
elif n not in expr.free_symbols:
return expr
expr = expr.rewrite(fibonacci, S.GoldenRatio)
expr = expr.rewrite(factorial, subfactorial, gamma)
n_ = Dummy("n", integer=True, positive=True)
n1 = Dummy("n", odd=True, positive=True)
n2 = Dummy("n", even=True, positive=True)
# If there is a negative term raised to a power involving n, or a
# trigonometric function, then consider even and odd n separately.
powers = (p.as_base_exp() for p in expr.atoms(Pow))
if (any(b.is_negative and e.has(n) for b, e in powers) or
expr.has(cos, sin)):
L1 = _limit_seq(expr.xreplace({n: n1}), n1, trials)
if L1 is not None:
L2 = _limit_seq(expr.xreplace({n: n2}), n2, trials)
if L1 != L2:
if L1.is_comparable and L2.is_comparable:
return AccumulationBounds(Min(L1, L2), Max(L1, L2))
else:
return None
else:
L1 = _limit_seq(expr.xreplace({n: n_}), n_, trials)
if L1 is not None:
return L1
else:
if expr.is_Add:
limits = [limit_seq(term, n, trials) for term in expr.args]
if any(result is None for result in limits):
return None
else:
return Add(*limits)
# Maybe the absolute value is easier to deal with (though not if
# it has a Sum). If it tends to 0, the limit is 0.
elif not expr.has(Sum):
lim = _limit_seq(Abs(expr.xreplace({n: n_})), n_, trials)
if lim is not None and lim.is_zero:
return S.Zero
|
ad24839459dae19841c5c586eb057e2da93f3e52c7d97b2e55891e0da745596f | """Fourier Series"""
from __future__ import print_function, division
from sympy import pi, oo, Wild
from sympy.core.expr import Expr
from sympy.core.add import Add
from sympy.core.compatibility import is_sequence
from sympy.core.containers import Tuple
from sympy.core.singleton import S
from sympy.core.symbol import Dummy, Symbol
from sympy.core.sympify import sympify
from sympy.functions.elementary.trigonometric import sin, cos, sinc
from sympy.series.series_class import SeriesBase
from sympy.series.sequences import SeqFormula
from sympy.sets.sets import Interval
from sympy.simplify.fu import TR2, TR1, TR10, sincos_to_sum
def fourier_cos_seq(func, limits, n):
"""Returns the cos sequence in a Fourier series"""
from sympy.integrals import integrate
x, L = limits[0], limits[2] - limits[1]
cos_term = cos(2*n*pi*x / L)
formula = 2 * cos_term * integrate(func * cos_term, limits) / L
a0 = formula.subs(n, S.Zero) / 2
return a0, SeqFormula(2 * cos_term * integrate(func * cos_term, limits)
/ L, (n, 1, oo))
def fourier_sin_seq(func, limits, n):
"""Returns the sin sequence in a Fourier series"""
from sympy.integrals import integrate
x, L = limits[0], limits[2] - limits[1]
sin_term = sin(2*n*pi*x / L)
return SeqFormula(2 * sin_term * integrate(func * sin_term, limits)
/ L, (n, 1, oo))
def _process_limits(func, limits):
"""
Limits should be of the form (x, start, stop).
x should be a symbol. Both start and stop should be bounded.
* If x is not given, x is determined from func.
* If limits is None. Limit of the form (x, -pi, pi) is returned.
Examples
========
>>> from sympy.series.fourier import _process_limits as pari
>>> from sympy.abc import x
>>> pari(x**2, (x, -2, 2))
(x, -2, 2)
>>> pari(x**2, (-2, 2))
(x, -2, 2)
>>> pari(x**2, None)
(x, -pi, pi)
"""
def _find_x(func):
free = func.free_symbols
if len(free) == 1:
return free.pop()
elif not free:
return Dummy('k')
else:
raise ValueError(
" specify dummy variables for %s. If the function contains"
" more than one free symbol, a dummy variable should be"
" supplied explicitly e.g. FourierSeries(m*n**2, (n, -pi, pi))"
% func)
x, start, stop = None, None, None
if limits is None:
x, start, stop = _find_x(func), -pi, pi
if is_sequence(limits, Tuple):
if len(limits) == 3:
x, start, stop = limits
elif len(limits) == 2:
x = _find_x(func)
start, stop = limits
if not isinstance(x, Symbol) or start is None or stop is None:
raise ValueError('Invalid limits given: %s' % str(limits))
unbounded = [S.NegativeInfinity, S.Infinity]
if start in unbounded or stop in unbounded:
raise ValueError("Both the start and end value should be bounded")
return sympify((x, start, stop))
def finite_check(f, x, L):
def check_fx(exprs, x):
return x not in exprs.free_symbols
def check_sincos(_expr, x, L):
if isinstance(_expr, (sin, cos)):
sincos_args = _expr.args[0]
if sincos_args.match(a*(pi/L)*x + b) is not None:
return True
else:
return False
_expr = sincos_to_sum(TR2(TR1(f)))
add_coeff = _expr.as_coeff_add()
a = Wild('a', properties=[lambda k: k.is_Integer, lambda k: k != S.Zero, ])
b = Wild('b', properties=[lambda k: x not in k.free_symbols, ])
for s in add_coeff[1]:
mul_coeffs = s.as_coeff_mul()[1]
for t in mul_coeffs:
if not (check_fx(t, x) or check_sincos(t, x, L)):
return False, f
return True, _expr
class FourierSeries(SeriesBase):
r"""Represents Fourier sine/cosine series.
This class only represents a fourier series.
No computation is performed.
For how to compute Fourier series, see the :func:`fourier_series`
docstring.
See Also
========
sympy.series.fourier.fourier_series
"""
def __new__(cls, *args):
args = map(sympify, args)
return Expr.__new__(cls, *args)
@property
def function(self):
return self.args[0]
@property
def x(self):
return self.args[1][0]
@property
def period(self):
return (self.args[1][1], self.args[1][2])
@property
def a0(self):
return self.args[2][0]
@property
def an(self):
return self.args[2][1]
@property
def bn(self):
return self.args[2][2]
@property
def interval(self):
return Interval(0, oo)
@property
def start(self):
return self.interval.inf
@property
def stop(self):
return self.interval.sup
@property
def length(self):
return oo
@property
def L(self):
return abs(self.period[1] - self.period[0]) / 2
def _eval_subs(self, old, new):
x = self.x
if old.has(x):
return self
def truncate(self, n=3):
"""
Return the first n nonzero terms of the series.
If n is None return an iterator.
Parameters
==========
n : int or None
Amount of non-zero terms in approximation or None.
Returns
=======
Expr or iterator
Approximation of function expanded into Fourier series.
Examples
========
>>> from sympy import fourier_series, pi
>>> from sympy.abc import x
>>> s = fourier_series(x, (x, -pi, pi))
>>> s.truncate(4)
2*sin(x) - sin(2*x) + 2*sin(3*x)/3 - sin(4*x)/2
See Also
========
sympy.series.fourier.FourierSeries.sigma_approximation
"""
if n is None:
return iter(self)
terms = []
for t in self:
if len(terms) == n:
break
if t is not S.Zero:
terms.append(t)
return Add(*terms)
def sigma_approximation(self, n=3):
r"""
Return :math:`\sigma`-approximation of Fourier series with respect
to order n.
Sigma approximation adjusts a Fourier summation to eliminate the Gibbs
phenomenon which would otherwise occur at discontinuities.
A sigma-approximated summation for a Fourier series of a T-periodical
function can be written as
.. math::
s(\theta) = \frac{1}{2} a_0 + \sum _{k=1}^{m-1}
\operatorname{sinc} \Bigl( \frac{k}{m} \Bigr) \cdot
\left[ a_k \cos \Bigl( \frac{2\pi k}{T} \theta \Bigr)
+ b_k \sin \Bigl( \frac{2\pi k}{T} \theta \Bigr) \right],
where :math:`a_0, a_k, b_k, k=1,\ldots,{m-1}` are standard Fourier
series coefficients and
:math:`\operatorname{sinc} \Bigl( \frac{k}{m} \Bigr)` is a Lanczos
:math:`\sigma` factor (expressed in terms of normalized
:math:`\operatorname{sinc}` function).
Parameters
==========
n : int
Highest order of the terms taken into account in approximation.
Returns
=======
Expr
Sigma approximation of function expanded into Fourier series.
Examples
========
>>> from sympy import fourier_series, pi
>>> from sympy.abc import x
>>> s = fourier_series(x, (x, -pi, pi))
>>> s.sigma_approximation(4)
2*sin(x)*sinc(pi/4) - 2*sin(2*x)/pi + 2*sin(3*x)*sinc(3*pi/4)/3
See Also
========
sympy.series.fourier.FourierSeries.truncate
Notes
=====
The behaviour of
:meth:`~sympy.series.fourier.FourierSeries.sigma_approximation`
is different from :meth:`~sympy.series.fourier.FourierSeries.truncate`
- it takes all nonzero terms of degree smaller than n, rather than
first n nonzero ones.
References
==========
.. [1] https://en.wikipedia.org/wiki/Gibbs_phenomenon
.. [2] https://en.wikipedia.org/wiki/Sigma_approximation
"""
terms = [sinc(pi * i / n) * t for i, t in enumerate(self[:n])
if t is not S.Zero]
return Add(*terms)
def shift(self, s):
"""Shift the function by a term independent of x.
f(x) -> f(x) + s
This is fast, if Fourier series of f(x) is already
computed.
Examples
========
>>> from sympy import fourier_series, pi
>>> from sympy.abc import x
>>> s = fourier_series(x**2, (x, -pi, pi))
>>> s.shift(1).truncate()
-4*cos(x) + cos(2*x) + 1 + pi**2/3
"""
s, x = sympify(s), self.x
if x in s.free_symbols:
raise ValueError("'%s' should be independent of %s" % (s, x))
a0 = self.a0 + s
sfunc = self.function + s
return self.func(sfunc, self.args[1], (a0, self.an, self.bn))
def shiftx(self, s):
"""Shift x by a term independent of x.
f(x) -> f(x + s)
This is fast, if Fourier series of f(x) is already
computed.
Examples
========
>>> from sympy import fourier_series, pi
>>> from sympy.abc import x
>>> s = fourier_series(x**2, (x, -pi, pi))
>>> s.shiftx(1).truncate()
-4*cos(x + 1) + cos(2*x + 2) + pi**2/3
"""
s, x = sympify(s), self.x
if x in s.free_symbols:
raise ValueError("'%s' should be independent of %s" % (s, x))
an = self.an.subs(x, x + s)
bn = self.bn.subs(x, x + s)
sfunc = self.function.subs(x, x + s)
return self.func(sfunc, self.args[1], (self.a0, an, bn))
def scale(self, s):
"""Scale the function by a term independent of x.
f(x) -> s * f(x)
This is fast, if Fourier series of f(x) is already
computed.
Examples
========
>>> from sympy import fourier_series, pi
>>> from sympy.abc import x
>>> s = fourier_series(x**2, (x, -pi, pi))
>>> s.scale(2).truncate()
-8*cos(x) + 2*cos(2*x) + 2*pi**2/3
"""
s, x = sympify(s), self.x
if x in s.free_symbols:
raise ValueError("'%s' should be independent of %s" % (s, x))
an = self.an.coeff_mul(s)
bn = self.bn.coeff_mul(s)
a0 = self.a0 * s
sfunc = self.args[0] * s
return self.func(sfunc, self.args[1], (a0, an, bn))
def scalex(self, s):
"""Scale x by a term independent of x.
f(x) -> f(s*x)
This is fast, if Fourier series of f(x) is already
computed.
Examples
========
>>> from sympy import fourier_series, pi
>>> from sympy.abc import x
>>> s = fourier_series(x**2, (x, -pi, pi))
>>> s.scalex(2).truncate()
-4*cos(2*x) + cos(4*x) + pi**2/3
"""
s, x = sympify(s), self.x
if x in s.free_symbols:
raise ValueError("'%s' should be independent of %s" % (s, x))
an = self.an.subs(x, x * s)
bn = self.bn.subs(x, x * s)
sfunc = self.function.subs(x, x * s)
return self.func(sfunc, self.args[1], (self.a0, an, bn))
def _eval_as_leading_term(self, x, cdir=0):
for t in self:
if t is not S.Zero:
return t
def _eval_term(self, pt):
if pt == 0:
return self.a0
return self.an.coeff(pt) + self.bn.coeff(pt)
def __neg__(self):
return self.scale(-1)
def __add__(self, other):
if isinstance(other, FourierSeries):
if self.period != other.period:
raise ValueError("Both the series should have same periods")
x, y = self.x, other.x
function = self.function + other.function.subs(y, x)
if self.x not in function.free_symbols:
return function
an = self.an + other.an
bn = self.bn + other.bn
a0 = self.a0 + other.a0
return self.func(function, self.args[1], (a0, an, bn))
return Add(self, other)
def __sub__(self, other):
return self.__add__(-other)
class FiniteFourierSeries(FourierSeries):
r"""Represents Finite Fourier sine/cosine series.
For how to compute Fourier series, see the :func:`fourier_series`
docstring.
Parameters
==========
f : Expr
Expression for finding fourier_series
limits : ( x, start, stop)
x is the independent variable for the expression f
(start, stop) is the period of the fourier series
exprs: (a0, an, bn) or Expr
a0 is the constant term a0 of the fourier series
an is a dictionary of coefficients of cos terms
an[k] = coefficient of cos(pi*(k/L)*x)
bn is a dictionary of coefficients of sin terms
bn[k] = coefficient of sin(pi*(k/L)*x)
or exprs can be an expression to be converted to fourier form
Methods
=======
This class is an extension of FourierSeries class.
Please refer to sympy.series.fourier.FourierSeries for
further information.
See Also
========
sympy.series.fourier.FourierSeries
sympy.series.fourier.fourier_series
"""
def __new__(cls, f, limits, exprs):
f = sympify(f)
limits = sympify(limits)
exprs = sympify(exprs)
if not (type(exprs) == Tuple and len(exprs) == 3): # exprs is not of form (a0, an, bn)
# Converts the expression to fourier form
c, e = exprs.as_coeff_add()
rexpr = c + Add(*[TR10(i) for i in e])
a0, exp_ls = rexpr.expand(trig=False, power_base=False, power_exp=False, log=False).as_coeff_add()
x = limits[0]
L = abs(limits[2] - limits[1]) / 2
a = Wild('a', properties=[lambda k: k.is_Integer, lambda k: k is not S.Zero, ])
b = Wild('b', properties=[lambda k: x not in k.free_symbols, ])
an = dict()
bn = dict()
# separates the coefficients of sin and cos terms in dictionaries an, and bn
for p in exp_ls:
t = p.match(b * cos(a * (pi / L) * x))
q = p.match(b * sin(a * (pi / L) * x))
if t:
an[t[a]] = t[b] + an.get(t[a], S.Zero)
elif q:
bn[q[a]] = q[b] + bn.get(q[a], S.Zero)
else:
a0 += p
exprs = Tuple(a0, an, bn)
return Expr.__new__(cls, f, limits, exprs)
@property
def interval(self):
_length = 1 if self.a0 else 0
_length += max(set(self.an.keys()).union(set(self.bn.keys()))) + 1
return Interval(0, _length)
@property
def length(self):
return self.stop - self.start
def shiftx(self, s):
s, x = sympify(s), self.x
if x in s.free_symbols:
raise ValueError("'%s' should be independent of %s" % (s, x))
_expr = self.truncate().subs(x, x + s)
sfunc = self.function.subs(x, x + s)
return self.func(sfunc, self.args[1], _expr)
def scale(self, s):
s, x = sympify(s), self.x
if x in s.free_symbols:
raise ValueError("'%s' should be independent of %s" % (s, x))
_expr = self.truncate() * s
sfunc = self.function * s
return self.func(sfunc, self.args[1], _expr)
def scalex(self, s):
s, x = sympify(s), self.x
if x in s.free_symbols:
raise ValueError("'%s' should be independent of %s" % (s, x))
_expr = self.truncate().subs(x, x * s)
sfunc = self.function.subs(x, x * s)
return self.func(sfunc, self.args[1], _expr)
def _eval_term(self, pt):
if pt == 0:
return self.a0
_term = self.an.get(pt, S.Zero) * cos(pt * (pi / self.L) * self.x) \
+ self.bn.get(pt, S.Zero) * sin(pt * (pi / self.L) * self.x)
return _term
def __add__(self, other):
if isinstance(other, FourierSeries):
return other.__add__(fourier_series(self.function, self.args[1],\
finite=False))
elif isinstance(other, FiniteFourierSeries):
if self.period != other.period:
raise ValueError("Both the series should have same periods")
x, y = self.x, other.x
function = self.function + other.function.subs(y, x)
if self.x not in function.free_symbols:
return function
return fourier_series(function, limits=self.args[1])
def fourier_series(f, limits=None, finite=True):
r"""Computes the Fourier trigonometric series expansion.
Explanation
===========
Fourier trigonometric series of $f(x)$ over the interval $(a, b)$
is defined as:
.. math::
\frac{a_0}{2} + \sum_{n=1}^{\infty}
(a_n \cos(\frac{2n \pi x}{L}) + b_n \sin(\frac{2n \pi x}{L}))
where the coefficients are:
.. math::
L = b - a
.. math::
a_0 = \frac{2}{L} \int_{a}^{b}{f(x) dx}
.. math::
a_n = \frac{2}{L} \int_{a}^{b}{f(x) \cos(\frac{2n \pi x}{L}) dx}
.. math::
b_n = \frac{2}{L} \int_{a}^{b}{f(x) \sin(\frac{2n \pi x}{L}) dx}
The condition whether the function $f(x)$ given should be periodic
or not is more than necessary, because it is sufficient to consider
the series to be converging to $f(x)$ only in the given interval,
not throughout the whole real line.
This also brings a lot of ease for the computation because
you don't have to make $f(x)$ artificially periodic by
wrapping it with piecewise, modulo operations,
but you can shape the function to look like the desired periodic
function only in the interval $(a, b)$, and the computed series will
automatically become the series of the periodic version of $f(x)$.
This property is illustrated in the examples section below.
Parameters
==========
limits : (sym, start, end), optional
*sym* denotes the symbol the series is computed with respect to.
*start* and *end* denotes the start and the end of the interval
where the fourier series converges to the given function.
Default range is specified as $-\pi$ and $\pi$.
Returns
=======
FourierSeries
A symbolic object representing the Fourier trigonometric series.
Examples
========
Computing the Fourier series of $f(x) = x^2$:
>>> from sympy import fourier_series, pi
>>> from sympy.abc import x
>>> f = x**2
>>> s = fourier_series(f, (x, -pi, pi))
>>> s1 = s.truncate(n=3)
>>> s1
-4*cos(x) + cos(2*x) + pi**2/3
Shifting of the Fourier series:
>>> s.shift(1).truncate()
-4*cos(x) + cos(2*x) + 1 + pi**2/3
>>> s.shiftx(1).truncate()
-4*cos(x + 1) + cos(2*x + 2) + pi**2/3
Scaling of the Fourier series:
>>> s.scale(2).truncate()
-8*cos(x) + 2*cos(2*x) + 2*pi**2/3
>>> s.scalex(2).truncate()
-4*cos(2*x) + cos(4*x) + pi**2/3
Computing the Fourier series of $f(x) = x$:
This illustrates how truncating to the higher order gives better
convergence.
.. plot::
:context: reset
:format: doctest
:include-source: True
>>> from sympy import fourier_series, pi, plot
>>> from sympy.abc import x
>>> f = x
>>> s = fourier_series(f, (x, -pi, pi))
>>> s1 = s.truncate(n = 3)
>>> s2 = s.truncate(n = 5)
>>> s3 = s.truncate(n = 7)
>>> p = plot(f, s1, s2, s3, (x, -pi, pi), show=False, legend=True)
>>> p[0].line_color = (0, 0, 0)
>>> p[0].label = 'x'
>>> p[1].line_color = (0.7, 0.7, 0.7)
>>> p[1].label = 'n=3'
>>> p[2].line_color = (0.5, 0.5, 0.5)
>>> p[2].label = 'n=5'
>>> p[3].line_color = (0.3, 0.3, 0.3)
>>> p[3].label = 'n=7'
>>> p.show()
This illustrates how the series converges to different sawtooth
waves if the different ranges are specified.
.. plot::
:context: close-figs
:format: doctest
:include-source: True
>>> s1 = fourier_series(x, (x, -1, 1)).truncate(10)
>>> s2 = fourier_series(x, (x, -pi, pi)).truncate(10)
>>> s3 = fourier_series(x, (x, 0, 1)).truncate(10)
>>> p = plot(x, s1, s2, s3, (x, -5, 5), show=False, legend=True)
>>> p[0].line_color = (0, 0, 0)
>>> p[0].label = 'x'
>>> p[1].line_color = (0.7, 0.7, 0.7)
>>> p[1].label = '[-1, 1]'
>>> p[2].line_color = (0.5, 0.5, 0.5)
>>> p[2].label = '[-pi, pi]'
>>> p[3].line_color = (0.3, 0.3, 0.3)
>>> p[3].label = '[0, 1]'
>>> p.show()
Notes
=====
Computing Fourier series can be slow
due to the integration required in computing
an, bn.
It is faster to compute Fourier series of a function
by using shifting and scaling on an already
computed Fourier series rather than computing
again.
e.g. If the Fourier series of ``x**2`` is known
the Fourier series of ``x**2 - 1`` can be found by shifting by ``-1``.
See Also
========
sympy.series.fourier.FourierSeries
References
==========
.. [1] https://mathworld.wolfram.com/FourierSeries.html
"""
f = sympify(f)
limits = _process_limits(f, limits)
x = limits[0]
if x not in f.free_symbols:
return f
if finite:
L = abs(limits[2] - limits[1]) / 2
is_finite, res_f = finite_check(f, x, L)
if is_finite:
return FiniteFourierSeries(f, limits, res_f)
n = Dummy('n')
center = (limits[1] + limits[2]) / 2
if center.is_zero:
neg_f = f.subs(x, -x)
if f == neg_f:
a0, an = fourier_cos_seq(f, limits, n)
bn = SeqFormula(0, (1, oo))
return FourierSeries(f, limits, (a0, an, bn))
elif f == -neg_f:
a0 = S.Zero
an = SeqFormula(0, (1, oo))
bn = fourier_sin_seq(f, limits, n)
return FourierSeries(f, limits, (a0, an, bn))
a0, an = fourier_cos_seq(f, limits, n)
bn = fourier_sin_seq(f, limits, n)
return FourierSeries(f, limits, (a0, an, bn))
|
de685393562a26e102a3699003dd9130aebaf5b9426bd2413f213bbe2cad8f3e | """Formal Power Series"""
from __future__ import print_function, division
from collections import defaultdict
from sympy import oo, zoo, nan
from sympy.core.add import Add
from sympy.core.compatibility import iterable
from sympy.core.expr import Expr
from sympy.core.function import Derivative, Function, expand
from sympy.core.mul import Mul
from sympy.core.numbers import Rational
from sympy.core.relational import Eq
from sympy.sets.sets import Interval
from sympy.core.singleton import S
from sympy.core.symbol import Wild, Dummy, symbols, Symbol
from sympy.core.sympify import sympify
from sympy.discrete.convolutions import convolution
from sympy.functions.combinatorial.factorials import binomial, factorial, rf
from sympy.functions.combinatorial.numbers import bell
from sympy.functions.elementary.integers import floor, frac, ceiling
from sympy.functions.elementary.miscellaneous import Min, Max
from sympy.functions.elementary.piecewise import Piecewise
from sympy.series.limits import Limit
from sympy.series.order import Order
from sympy.simplify.powsimp import powsimp
from sympy.series.sequences import sequence
from sympy.series.series_class import SeriesBase
def rational_algorithm(f, x, k, order=4, full=False):
"""
Rational algorithm for computing
formula of coefficients of Formal Power Series
of a function.
Applicable when f(x) or some derivative of f(x)
is a rational function in x.
:func:`rational_algorithm` uses :func:`~.apart` function for partial fraction
decomposition. :func:`~.apart` by default uses 'undetermined coefficients
method'. By setting ``full=True``, 'Bronstein's algorithm' can be used
instead.
Looks for derivative of a function up to 4'th order (by default).
This can be overridden using order option.
Returns
=======
formula : Expr
ind : Expr
Independent terms.
order : int
Examples
========
>>> from sympy import log, atan
>>> from sympy.series.formal import rational_algorithm as ra
>>> from sympy.abc import x, k
>>> ra(1 / (1 - x), x, k)
(1, 0, 0)
>>> ra(log(1 + x), x, k)
(-(-1)**(-k)/k, 0, 1)
>>> ra(atan(x), x, k, full=True)
((-I*(-I)**(-k)/2 + I*I**(-k)/2)/k, 0, 1)
Notes
=====
By setting ``full=True``, range of admissible functions to be solved using
``rational_algorithm`` can be increased. This option should be used
carefully as it can significantly slow down the computation as ``doit`` is
performed on the :class:`~.RootSum` object returned by the :func:`~.apart`
function. Use ``full=False`` whenever possible.
See Also
========
sympy.polys.partfrac.apart
References
==========
.. [1] Formal Power Series - Dominik Gruntz, Wolfram Koepf
.. [2] Power Series in Computer Algebra - Wolfram Koepf
"""
from sympy.polys import RootSum, apart
from sympy.integrals import integrate
diff = f
ds = [] # list of diff
for i in range(order + 1):
if i:
diff = diff.diff(x)
if diff.is_rational_function(x):
coeff, sep = S.Zero, S.Zero
terms = apart(diff, x, full=full)
if terms.has(RootSum):
terms = terms.doit()
for t in Add.make_args(terms):
num, den = t.as_numer_denom()
if not den.has(x):
sep += t
else:
if isinstance(den, Mul):
# m*(n*x - a)**j -> (n*x - a)**j
ind = den.as_independent(x)
den = ind[1]
num /= ind[0]
# (n*x - a)**j -> (x - b)
den, j = den.as_base_exp()
a, xterm = den.as_coeff_add(x)
# term -> m/x**n
if not a:
sep += t
continue
xc = xterm[0].coeff(x)
a /= -xc
num /= xc**j
ak = ((-1)**j * num *
binomial(j + k - 1, k).rewrite(factorial) /
a**(j + k))
coeff += ak
# Hacky, better way?
if coeff.is_zero:
return None
if (coeff.has(x) or coeff.has(zoo) or coeff.has(oo) or
coeff.has(nan)):
return None
for j in range(i):
coeff = (coeff / (k + j + 1))
sep = integrate(sep, x)
sep += (ds.pop() - sep).limit(x, 0) # constant of integration
return (coeff.subs(k, k - i), sep, i)
else:
ds.append(diff)
return None
def rational_independent(terms, x):
"""Returns a list of all the rationally independent terms.
Examples
========
>>> from sympy import sin, cos
>>> from sympy.series.formal import rational_independent
>>> from sympy.abc import x
>>> rational_independent([cos(x), sin(x)], x)
[cos(x), sin(x)]
>>> rational_independent([x**2, sin(x), x*sin(x), x**3], x)
[x**3 + x**2, x*sin(x) + sin(x)]
"""
if not terms:
return []
ind = terms[0:1]
for t in terms[1:]:
n = t.as_independent(x)[1]
for i, term in enumerate(ind):
d = term.as_independent(x)[1]
q = (n / d).cancel()
if q.is_rational_function(x):
ind[i] += t
break
else:
ind.append(t)
return ind
def simpleDE(f, x, g, order=4):
r"""Generates simple DE.
DE is of the form
.. math::
f^k(x) + \sum\limits_{j=0}^{k-1} A_j f^j(x) = 0
where :math:`A_j` should be rational function in x.
Generates DE's upto order 4 (default). DE's can also have free parameters.
By increasing order, higher order DE's can be found.
Yields a tuple of (DE, order).
"""
from sympy.solvers.solveset import linsolve
a = symbols('a:%d' % (order))
def _makeDE(k):
eq = f.diff(x, k) + Add(*[a[i]*f.diff(x, i) for i in range(0, k)])
DE = g(x).diff(x, k) + Add(*[a[i]*g(x).diff(x, i) for i in range(0, k)])
return eq, DE
found = False
for k in range(1, order + 1):
eq, DE = _makeDE(k)
eq = eq.expand()
terms = eq.as_ordered_terms()
ind = rational_independent(terms, x)
if found or len(ind) == k:
sol = dict(zip(a, (i for s in linsolve(ind, a[:k]) for i in s)))
if sol:
found = True
DE = DE.subs(sol)
DE = DE.as_numer_denom()[0]
DE = DE.factor().as_coeff_mul(Derivative)[1][0]
yield DE.collect(Derivative(g(x))), k
def exp_re(DE, r, k):
"""Converts a DE with constant coefficients (explike) into a RE.
Performs the substitution:
.. math::
f^j(x) \\to r(k + j)
Normalises the terms so that lowest order of a term is always r(k).
Examples
========
>>> from sympy import Function, Derivative
>>> from sympy.series.formal import exp_re
>>> from sympy.abc import x, k
>>> f, r = Function('f'), Function('r')
>>> exp_re(-f(x) + Derivative(f(x)), r, k)
-r(k) + r(k + 1)
>>> exp_re(Derivative(f(x), x) + Derivative(f(x), (x, 2)), r, k)
r(k) + r(k + 1)
See Also
========
sympy.series.formal.hyper_re
"""
RE = S.Zero
g = DE.atoms(Function).pop()
mini = None
for t in Add.make_args(DE):
coeff, d = t.as_independent(g)
if isinstance(d, Derivative):
j = d.derivative_count
else:
j = 0
if mini is None or j < mini:
mini = j
RE += coeff * r(k + j)
if mini:
RE = RE.subs(k, k - mini)
return RE
def hyper_re(DE, r, k):
"""Converts a DE into a RE.
Performs the substitution:
.. math::
x^l f^j(x) \\to (k + 1 - l)_j . a_{k + j - l}
Normalises the terms so that lowest order of a term is always r(k).
Examples
========
>>> from sympy import Function, Derivative
>>> from sympy.series.formal import hyper_re
>>> from sympy.abc import x, k
>>> f, r = Function('f'), Function('r')
>>> hyper_re(-f(x) + Derivative(f(x)), r, k)
(k + 1)*r(k + 1) - r(k)
>>> hyper_re(-x*f(x) + Derivative(f(x), (x, 2)), r, k)
(k + 2)*(k + 3)*r(k + 3) - r(k)
See Also
========
sympy.series.formal.exp_re
"""
RE = S.Zero
g = DE.atoms(Function).pop()
x = g.atoms(Symbol).pop()
mini = None
for t in Add.make_args(DE.expand()):
coeff, d = t.as_independent(g)
c, v = coeff.as_independent(x)
l = v.as_coeff_exponent(x)[1]
if isinstance(d, Derivative):
j = d.derivative_count
else:
j = 0
RE += c * rf(k + 1 - l, j) * r(k + j - l)
if mini is None or j - l < mini:
mini = j - l
RE = RE.subs(k, k - mini)
m = Wild('m')
return RE.collect(r(k + m))
def _transformation_a(f, x, P, Q, k, m, shift):
f *= x**(-shift)
P = P.subs(k, k + shift)
Q = Q.subs(k, k + shift)
return f, P, Q, m
def _transformation_c(f, x, P, Q, k, m, scale):
f = f.subs(x, x**scale)
P = P.subs(k, k / scale)
Q = Q.subs(k, k / scale)
m *= scale
return f, P, Q, m
def _transformation_e(f, x, P, Q, k, m):
f = f.diff(x)
P = P.subs(k, k + 1) * (k + m + 1)
Q = Q.subs(k, k + 1) * (k + 1)
return f, P, Q, m
def _apply_shift(sol, shift):
return [(res, cond + shift) for res, cond in sol]
def _apply_scale(sol, scale):
return [(res, cond / scale) for res, cond in sol]
def _apply_integrate(sol, x, k):
return [(res / ((cond + 1)*(cond.as_coeff_Add()[1].coeff(k))), cond + 1)
for res, cond in sol]
def _compute_formula(f, x, P, Q, k, m, k_max):
"""Computes the formula for f."""
from sympy.polys import roots
sol = []
for i in range(k_max + 1, k_max + m + 1):
if (i < 0) == True:
continue
r = f.diff(x, i).limit(x, 0) / factorial(i)
if r.is_zero:
continue
kterm = m*k + i
res = r
p = P.subs(k, kterm)
q = Q.subs(k, kterm)
c1 = p.subs(k, 1/k).leadterm(k)[0]
c2 = q.subs(k, 1/k).leadterm(k)[0]
res *= (-c1 / c2)**k
for r, mul in roots(p, k).items():
res *= rf(-r, k)**mul
for r, mul in roots(q, k).items():
res /= rf(-r, k)**mul
sol.append((res, kterm))
return sol
def _rsolve_hypergeometric(f, x, P, Q, k, m):
"""Recursive wrapper to rsolve_hypergeometric.
Returns a Tuple of (formula, series independent terms,
maximum power of x in independent terms) if successful
otherwise ``None``.
See :func:`rsolve_hypergeometric` for details.
"""
from sympy.polys import lcm, roots
from sympy.integrals import integrate
# transformation - c
proots, qroots = roots(P, k), roots(Q, k)
all_roots = dict(proots)
all_roots.update(qroots)
scale = lcm([r.as_numer_denom()[1] for r, t in all_roots.items()
if r.is_rational])
f, P, Q, m = _transformation_c(f, x, P, Q, k, m, scale)
# transformation - a
qroots = roots(Q, k)
if qroots:
k_min = Min(*qroots.keys())
else:
k_min = S.Zero
shift = k_min + m
f, P, Q, m = _transformation_a(f, x, P, Q, k, m, shift)
l = (x*f).limit(x, 0)
if not isinstance(l, Limit) and l != 0: # Ideally should only be l != 0
return None
qroots = roots(Q, k)
if qroots:
k_max = Max(*qroots.keys())
else:
k_max = S.Zero
ind, mp = S.Zero, -oo
for i in range(k_max + m + 1):
r = f.diff(x, i).limit(x, 0) / factorial(i)
if r.is_finite is False:
old_f = f
f, P, Q, m = _transformation_a(f, x, P, Q, k, m, i)
f, P, Q, m = _transformation_e(f, x, P, Q, k, m)
sol, ind, mp = _rsolve_hypergeometric(f, x, P, Q, k, m)
sol = _apply_integrate(sol, x, k)
sol = _apply_shift(sol, i)
ind = integrate(ind, x)
ind += (old_f - ind).limit(x, 0) # constant of integration
mp += 1
return sol, ind, mp
elif r:
ind += r*x**(i + shift)
pow_x = Rational((i + shift), scale)
if pow_x > mp:
mp = pow_x # maximum power of x
ind = ind.subs(x, x**(1/scale))
sol = _compute_formula(f, x, P, Q, k, m, k_max)
sol = _apply_shift(sol, shift)
sol = _apply_scale(sol, scale)
return sol, ind, mp
def rsolve_hypergeometric(f, x, P, Q, k, m):
"""Solves RE of hypergeometric type.
Attempts to solve RE of the form
Q(k)*a(k + m) - P(k)*a(k)
Transformations that preserve Hypergeometric type:
a. x**n*f(x): b(k + m) = R(k - n)*b(k)
b. f(A*x): b(k + m) = A**m*R(k)*b(k)
c. f(x**n): b(k + n*m) = R(k/n)*b(k)
d. f(x**(1/m)): b(k + 1) = R(k*m)*b(k)
e. f'(x): b(k + m) = ((k + m + 1)/(k + 1))*R(k + 1)*b(k)
Some of these transformations have been used to solve the RE.
Returns
=======
formula : Expr
ind : Expr
Independent terms.
order : int
Examples
========
>>> from sympy import exp, ln, S
>>> from sympy.series.formal import rsolve_hypergeometric as rh
>>> from sympy.abc import x, k
>>> rh(exp(x), x, -S.One, (k + 1), k, 1)
(Piecewise((1/factorial(k), Eq(Mod(k, 1), 0)), (0, True)), 1, 1)
>>> rh(ln(1 + x), x, k**2, k*(k + 1), k, 1)
(Piecewise(((-1)**(k - 1)*factorial(k - 1)/RisingFactorial(2, k - 1),
Eq(Mod(k, 1), 0)), (0, True)), x, 2)
References
==========
.. [1] Formal Power Series - Dominik Gruntz, Wolfram Koepf
.. [2] Power Series in Computer Algebra - Wolfram Koepf
"""
result = _rsolve_hypergeometric(f, x, P, Q, k, m)
if result is None:
return None
sol_list, ind, mp = result
sol_dict = defaultdict(lambda: S.Zero)
for res, cond in sol_list:
j, mk = cond.as_coeff_Add()
c = mk.coeff(k)
if j.is_integer is False:
res *= x**frac(j)
j = floor(j)
res = res.subs(k, (k - j) / c)
cond = Eq(k % c, j % c)
sol_dict[cond] += res # Group together formula for same conditions
sol = []
for cond, res in sol_dict.items():
sol.append((res, cond))
sol.append((S.Zero, True))
sol = Piecewise(*sol)
if mp is -oo:
s = S.Zero
elif mp.is_integer is False:
s = ceiling(mp)
else:
s = mp + 1
# save all the terms of
# form 1/x**k in ind
if s < 0:
ind += sum(sequence(sol * x**k, (k, s, -1)))
s = S.Zero
return (sol, ind, s)
def _solve_hyper_RE(f, x, RE, g, k):
"""See docstring of :func:`rsolve_hypergeometric` for details."""
terms = Add.make_args(RE)
if len(terms) == 2:
gs = list(RE.atoms(Function))
P, Q = map(RE.coeff, gs)
m = gs[1].args[0] - gs[0].args[0]
if m < 0:
P, Q = Q, P
m = abs(m)
return rsolve_hypergeometric(f, x, P, Q, k, m)
def _solve_explike_DE(f, x, DE, g, k):
"""Solves DE with constant coefficients."""
from sympy.solvers import rsolve
for t in Add.make_args(DE):
coeff, d = t.as_independent(g)
if coeff.free_symbols:
return
RE = exp_re(DE, g, k)
init = {}
for i in range(len(Add.make_args(RE))):
if i:
f = f.diff(x)
init[g(k).subs(k, i)] = f.limit(x, 0)
sol = rsolve(RE, g(k), init)
if sol:
return (sol / factorial(k), S.Zero, S.Zero)
def _solve_simple(f, x, DE, g, k):
"""Converts DE into RE and solves using :func:`rsolve`."""
from sympy.solvers import rsolve
RE = hyper_re(DE, g, k)
init = {}
for i in range(len(Add.make_args(RE))):
if i:
f = f.diff(x)
init[g(k).subs(k, i)] = f.limit(x, 0) / factorial(i)
sol = rsolve(RE, g(k), init)
if sol:
return (sol, S.Zero, S.Zero)
def _transform_explike_DE(DE, g, x, order, syms):
"""Converts DE with free parameters into DE with constant coefficients."""
from sympy.solvers.solveset import linsolve
eq = []
highest_coeff = DE.coeff(Derivative(g(x), x, order))
for i in range(order):
coeff = DE.coeff(Derivative(g(x), x, i))
coeff = (coeff / highest_coeff).expand().collect(x)
for t in Add.make_args(coeff):
eq.append(t)
temp = []
for e in eq:
if e.has(x):
break
elif e.has(Symbol):
temp.append(e)
else:
eq = temp
if eq:
sol = dict(zip(syms, (i for s in linsolve(eq, list(syms)) for i in s)))
if sol:
DE = DE.subs(sol)
DE = DE.factor().as_coeff_mul(Derivative)[1][0]
DE = DE.collect(Derivative(g(x)))
return DE
def _transform_DE_RE(DE, g, k, order, syms):
"""Converts DE with free parameters into RE of hypergeometric type."""
from sympy.solvers.solveset import linsolve
RE = hyper_re(DE, g, k)
eq = []
for i in range(1, order):
coeff = RE.coeff(g(k + i))
eq.append(coeff)
sol = dict(zip(syms, (i for s in linsolve(eq, list(syms)) for i in s)))
if sol:
m = Wild('m')
RE = RE.subs(sol)
RE = RE.factor().as_numer_denom()[0].collect(g(k + m))
RE = RE.as_coeff_mul(g)[1][0]
for i in range(order): # smallest order should be g(k)
if RE.coeff(g(k + i)) and i:
RE = RE.subs(k, k - i)
break
return RE
def solve_de(f, x, DE, order, g, k):
"""Solves the DE.
Tries to solve DE by either converting into a RE containing two terms or
converting into a DE having constant coefficients.
Returns
=======
formula : Expr
ind : Expr
Independent terms.
order : int
Examples
========
>>> from sympy import Derivative as D, Function
>>> from sympy import exp, ln
>>> from sympy.series.formal import solve_de
>>> from sympy.abc import x, k
>>> f = Function('f')
>>> solve_de(exp(x), x, D(f(x), x) - f(x), 1, f, k)
(Piecewise((1/factorial(k), Eq(Mod(k, 1), 0)), (0, True)), 1, 1)
>>> solve_de(ln(1 + x), x, (x + 1)*D(f(x), x, 2) + D(f(x)), 2, f, k)
(Piecewise(((-1)**(k - 1)*factorial(k - 1)/RisingFactorial(2, k - 1),
Eq(Mod(k, 1), 0)), (0, True)), x, 2)
"""
sol = None
syms = DE.free_symbols.difference({g, x})
if syms:
RE = _transform_DE_RE(DE, g, k, order, syms)
else:
RE = hyper_re(DE, g, k)
if not RE.free_symbols.difference({k}):
sol = _solve_hyper_RE(f, x, RE, g, k)
if sol:
return sol
if syms:
DE = _transform_explike_DE(DE, g, x, order, syms)
if not DE.free_symbols.difference({x}):
sol = _solve_explike_DE(f, x, DE, g, k)
if sol:
return sol
def hyper_algorithm(f, x, k, order=4):
"""Hypergeometric algorithm for computing Formal Power Series.
Steps:
* Generates DE
* Convert the DE into RE
* Solves the RE
Examples
========
>>> from sympy import exp, ln
>>> from sympy.series.formal import hyper_algorithm
>>> from sympy.abc import x, k
>>> hyper_algorithm(exp(x), x, k)
(Piecewise((1/factorial(k), Eq(Mod(k, 1), 0)), (0, True)), 1, 1)
>>> hyper_algorithm(ln(1 + x), x, k)
(Piecewise(((-1)**(k - 1)*factorial(k - 1)/RisingFactorial(2, k - 1),
Eq(Mod(k, 1), 0)), (0, True)), x, 2)
See Also
========
sympy.series.formal.simpleDE
sympy.series.formal.solve_de
"""
g = Function('g')
des = [] # list of DE's
sol = None
for DE, i in simpleDE(f, x, g, order):
if DE is not None:
sol = solve_de(f, x, DE, i, g, k)
if sol:
return sol
if not DE.free_symbols.difference({x}):
des.append(DE)
# If nothing works
# Try plain rsolve
for DE in des:
sol = _solve_simple(f, x, DE, g, k)
if sol:
return sol
def _compute_fps(f, x, x0, dir, hyper, order, rational, full):
"""Recursive wrapper to compute fps.
See :func:`compute_fps` for details.
"""
if x0 in [S.Infinity, S.NegativeInfinity]:
dir = S.One if x0 is S.Infinity else -S.One
temp = f.subs(x, 1/x)
result = _compute_fps(temp, x, 0, dir, hyper, order, rational, full)
if result is None:
return None
return (result[0], result[1].subs(x, 1/x), result[2].subs(x, 1/x))
elif x0 or dir == -S.One:
if dir == -S.One:
rep = -x + x0
rep2 = -x
rep2b = x0
else:
rep = x + x0
rep2 = x
rep2b = -x0
temp = f.subs(x, rep)
result = _compute_fps(temp, x, 0, S.One, hyper, order, rational, full)
if result is None:
return None
return (result[0], result[1].subs(x, rep2 + rep2b),
result[2].subs(x, rep2 + rep2b))
if f.is_polynomial(x):
k = Dummy('k')
ak = sequence(Coeff(f, x, k), (k, 1, oo))
xk = sequence(x**k, (k, 0, oo))
ind = f.coeff(x, 0)
return ak, xk, ind
# Break instances of Add
# this allows application of different
# algorithms on different terms increasing the
# range of admissible functions.
if isinstance(f, Add):
result = False
ak = sequence(S.Zero, (0, oo))
ind, xk = S.Zero, None
for t in Add.make_args(f):
res = _compute_fps(t, x, 0, S.One, hyper, order, rational, full)
if res:
if not result:
result = True
xk = res[1]
if res[0].start > ak.start:
seq = ak
s, f = ak.start, res[0].start
else:
seq = res[0]
s, f = res[0].start, ak.start
save = Add(*[z[0]*z[1] for z in zip(seq[0:(f - s)], xk[s:f])])
ak += res[0]
ind += res[2] + save
else:
ind += t
if result:
return ak, xk, ind
return None
# The symbolic term - symb, if present, is being separated from the function
# Otherwise symb is being set to S.One
syms = f.free_symbols.difference({x})
(f, symb) = expand(f).as_independent(*syms)
if symb.is_zero:
symb = S.One
symb = powsimp(symb)
result = None
# from here on it's x0=0 and dir=1 handling
k = Dummy('k')
if rational:
result = rational_algorithm(f, x, k, order, full)
if result is None and hyper:
result = hyper_algorithm(f, x, k, order)
if result is None:
return None
ak = sequence(result[0], (k, result[2], oo))
xk_formula = powsimp(x**k * symb)
xk = sequence(xk_formula, (k, 0, oo))
ind = powsimp(result[1] * symb)
return ak, xk, ind
def compute_fps(f, x, x0=0, dir=1, hyper=True, order=4, rational=True,
full=False):
"""Computes the formula for Formal Power Series of a function.
Tries to compute the formula by applying the following techniques
(in order):
* rational_algorithm
* Hypergeometric algorithm
Parameters
==========
x : Symbol
x0 : number, optional
Point to perform series expansion about. Default is 0.
dir : {1, -1, '+', '-'}, optional
If dir is 1 or '+' the series is calculated from the right and
for -1 or '-' the series is calculated from the left. For smooth
functions this flag will not alter the results. Default is 1.
hyper : {True, False}, optional
Set hyper to False to skip the hypergeometric algorithm.
By default it is set to False.
order : int, optional
Order of the derivative of ``f``, Default is 4.
rational : {True, False}, optional
Set rational to False to skip rational algorithm. By default it is set
to True.
full : {True, False}, optional
Set full to True to increase the range of rational algorithm.
See :func:`rational_algorithm` for details. By default it is set to
False.
Returns
=======
ak : sequence
Sequence of coefficients.
xk : sequence
Sequence of powers of x.
ind : Expr
Independent terms.
mul : Pow
Common terms.
See Also
========
sympy.series.formal.rational_algorithm
sympy.series.formal.hyper_algorithm
"""
f = sympify(f)
x = sympify(x)
if not f.has(x):
return None
x0 = sympify(x0)
if dir == '+':
dir = S.One
elif dir == '-':
dir = -S.One
elif dir not in [S.One, -S.One]:
raise ValueError("Dir must be '+' or '-'")
else:
dir = sympify(dir)
return _compute_fps(f, x, x0, dir, hyper, order, rational, full)
class Coeff(Function):
"""
Coeff(p, x, n) represents the nth coefficient of the polynomial p in x
"""
@classmethod
def eval(cls, p, x, n):
if p.is_polynomial(x) and n.is_integer:
return p.coeff(x, n)
class FormalPowerSeries(SeriesBase):
"""Represents Formal Power Series of a function.
No computation is performed. This class should only to be used to represent
a series. No checks are performed.
For computing a series use :func:`fps`.
See Also
========
sympy.series.formal.fps
"""
def __new__(cls, *args):
args = map(sympify, args)
return Expr.__new__(cls, *args)
def __init__(self, *args):
ak = args[4][0]
k = ak.variables[0]
self.ak_seq = sequence(ak.formula, (k, 1, oo))
self.fact_seq = sequence(factorial(k), (k, 1, oo))
self.bell_coeff_seq = self.ak_seq * self.fact_seq
self.sign_seq = sequence((-1, 1), (k, 1, oo))
@property
def function(self):
return self.args[0]
@property
def x(self):
return self.args[1]
@property
def x0(self):
return self.args[2]
@property
def dir(self):
return self.args[3]
@property
def ak(self):
return self.args[4][0]
@property
def xk(self):
return self.args[4][1]
@property
def ind(self):
return self.args[4][2]
@property
def interval(self):
return Interval(0, oo)
@property
def start(self):
return self.interval.inf
@property
def stop(self):
return self.interval.sup
@property
def length(self):
return oo
@property
def infinite(self):
"""Returns an infinite representation of the series"""
from sympy.concrete import Sum
ak, xk = self.ak, self.xk
k = ak.variables[0]
inf_sum = Sum(ak.formula * xk.formula, (k, ak.start, ak.stop))
return self.ind + inf_sum
def _get_pow_x(self, term):
"""Returns the power of x in a term."""
xterm, pow_x = term.as_independent(self.x)[1].as_base_exp()
if not xterm.has(self.x):
return S.Zero
return pow_x
def polynomial(self, n=6):
"""Truncated series as polynomial.
Returns series expansion of ``f`` upto order ``O(x**n)``
as a polynomial(without ``O`` term).
"""
terms = []
sym = self.free_symbols
for i, t in enumerate(self):
xp = self._get_pow_x(t)
if xp.has(*sym):
xp = xp.as_coeff_add(*sym)[0]
if xp >= n:
break
elif xp.is_integer is True and i == n + 1:
break
elif t is not S.Zero:
terms.append(t)
return Add(*terms)
def truncate(self, n=6):
"""Truncated series.
Returns truncated series expansion of f upto
order ``O(x**n)``.
If n is ``None``, returns an infinite iterator.
"""
if n is None:
return iter(self)
x, x0 = self.x, self.x0
pt_xk = self.xk.coeff(n)
if x0 is S.NegativeInfinity:
x0 = S.Infinity
return self.polynomial(n) + Order(pt_xk, (x, x0))
def zero_coeff(self):
return self._eval_term(0)
def _eval_term(self, pt):
try:
pt_xk = self.xk.coeff(pt)
pt_ak = self.ak.coeff(pt).simplify() # Simplify the coefficients
except IndexError:
term = S.Zero
else:
term = (pt_ak * pt_xk)
if self.ind:
ind = S.Zero
sym = self.free_symbols
for t in Add.make_args(self.ind):
pow_x = self._get_pow_x(t)
if pow_x.has(*sym):
pow_x = pow_x.as_coeff_add(*sym)[0]
if pt == 0 and pow_x < 1:
ind += t
elif pow_x >= pt and pow_x < pt + 1:
ind += t
term += ind
return term.collect(self.x)
def _eval_subs(self, old, new):
x = self.x
if old.has(x):
return self
def _eval_as_leading_term(self, x, cdir=0):
for t in self:
if t is not S.Zero:
return t
def _eval_derivative(self, x):
f = self.function.diff(x)
ind = self.ind.diff(x)
pow_xk = self._get_pow_x(self.xk.formula)
ak = self.ak
k = ak.variables[0]
if ak.formula.has(x):
form = []
for e, c in ak.formula.args:
temp = S.Zero
for t in Add.make_args(e):
pow_x = self._get_pow_x(t)
temp += t * (pow_xk + pow_x)
form.append((temp, c))
form = Piecewise(*form)
ak = sequence(form.subs(k, k + 1), (k, ak.start - 1, ak.stop))
else:
ak = sequence((ak.formula * pow_xk).subs(k, k + 1),
(k, ak.start - 1, ak.stop))
return self.func(f, self.x, self.x0, self.dir, (ak, self.xk, ind))
def integrate(self, x=None, **kwargs):
"""Integrate Formal Power Series.
Examples
========
>>> from sympy import fps, sin, integrate
>>> from sympy.abc import x
>>> f = fps(sin(x))
>>> f.integrate(x).truncate()
-1 + x**2/2 - x**4/24 + O(x**6)
>>> integrate(f, (x, 0, 1))
1 - cos(1)
"""
from sympy.integrals import integrate
if x is None:
x = self.x
elif iterable(x):
return integrate(self.function, x)
f = integrate(self.function, x)
ind = integrate(self.ind, x)
ind += (f - ind).limit(x, 0) # constant of integration
pow_xk = self._get_pow_x(self.xk.formula)
ak = self.ak
k = ak.variables[0]
if ak.formula.has(x):
form = []
for e, c in ak.formula.args:
temp = S.Zero
for t in Add.make_args(e):
pow_x = self._get_pow_x(t)
temp += t / (pow_xk + pow_x + 1)
form.append((temp, c))
form = Piecewise(*form)
ak = sequence(form.subs(k, k - 1), (k, ak.start + 1, ak.stop))
else:
ak = sequence((ak.formula / (pow_xk + 1)).subs(k, k - 1),
(k, ak.start + 1, ak.stop))
return self.func(f, self.x, self.x0, self.dir, (ak, self.xk, ind))
def product(self, other, x=None, n=6):
"""Multiplies two Formal Power Series, using discrete convolution and
return the truncated terms upto specified order.
Parameters
==========
n : Number, optional
Specifies the order of the term up to which the polynomial should
be truncated.
Examples
========
>>> from sympy import fps, sin, exp
>>> from sympy.abc import x
>>> f1 = fps(sin(x))
>>> f2 = fps(exp(x))
>>> f1.product(f2, x).truncate(4)
x + x**2 + x**3/3 + O(x**4)
See Also
========
sympy.discrete.convolutions
sympy.series.formal.FormalPowerSeriesProduct
"""
if x is None:
x = self.x
if n is None:
return iter(self)
other = sympify(other)
if not isinstance(other, FormalPowerSeries):
raise ValueError("Both series should be an instance of FormalPowerSeries"
" class.")
if self.dir != other.dir:
raise ValueError("Both series should be calculated from the"
" same direction.")
elif self.x0 != other.x0:
raise ValueError("Both series should be calculated about the"
" same point.")
elif self.x != other.x:
raise ValueError("Both series should have the same symbol.")
return FormalPowerSeriesProduct(self, other)
def coeff_bell(self, n):
r"""
self.coeff_bell(n) returns a sequence of Bell polynomials of the second kind.
Note that ``n`` should be a integer.
The second kind of Bell polynomials (are sometimes called "partial" Bell
polynomials or incomplete Bell polynomials) are defined as
.. math::
B_{n,k}(x_1, x_2,\dotsc x_{n-k+1}) =
\sum_{j_1+j_2+j_2+\dotsb=k \atop j_1+2j_2+3j_2+\dotsb=n}
\frac{n!}{j_1!j_2!\dotsb j_{n-k+1}!}
\left(\frac{x_1}{1!} \right)^{j_1}
\left(\frac{x_2}{2!} \right)^{j_2} \dotsb
\left(\frac{x_{n-k+1}}{(n-k+1)!} \right) ^{j_{n-k+1}}.
* ``bell(n, k, (x1, x2, ...))`` gives Bell polynomials of the second kind,
`B_{n,k}(x_1, x_2, \dotsc, x_{n-k+1})`.
See Also
========
sympy.functions.combinatorial.numbers.bell
"""
inner_coeffs = [bell(n, j, tuple(self.bell_coeff_seq[:n-j+1])) for j in range(1, n+1)]
k = Dummy('k')
return sequence(tuple(inner_coeffs), (k, 1, oo))
def compose(self, other, x=None, n=6):
r"""
Returns the truncated terms of the formal power series of the composed function,
up to specified `n`.
If `f` and `g` are two formal power series of two different functions,
then the coefficient sequence ``ak`` of the composed formal power series `fp`
will be as follows.
.. math::
\sum\limits_{k=0}^{n} b_k B_{n,k}(x_1, x_2, \dotsc, x_{n-k+1})
Parameters
==========
n : Number, optional
Specifies the order of the term up to which the polynomial should
be truncated.
Examples
========
>>> from sympy import fps, sin, exp
>>> from sympy.abc import x
>>> f1 = fps(exp(x))
>>> f2 = fps(sin(x))
>>> f1.compose(f2, x).truncate()
1 + x + x**2/2 - x**4/8 - x**5/15 + O(x**6)
>>> f1.compose(f2, x).truncate(8)
1 + x + x**2/2 - x**4/8 - x**5/15 - x**6/240 + x**7/90 + O(x**8)
See Also
========
sympy.functions.combinatorial.numbers.bell
sympy.series.formal.FormalPowerSeriesCompose
References
==========
.. [1] Comtet, Louis: Advanced combinatorics; the art of finite and infinite expansions. Reidel, 1974.
"""
if x is None:
x = self.x
if n is None:
return iter(self)
other = sympify(other)
if not isinstance(other, FormalPowerSeries):
raise ValueError("Both series should be an instance of FormalPowerSeries"
" class.")
if self.dir != other.dir:
raise ValueError("Both series should be calculated from the"
" same direction.")
elif self.x0 != other.x0:
raise ValueError("Both series should be calculated about the"
" same point.")
elif self.x != other.x:
raise ValueError("Both series should have the same symbol.")
if other._eval_term(0).as_coeff_mul(other.x)[0] is not S.Zero:
raise ValueError("The formal power series of the inner function should not have any "
"constant coefficient term.")
return FormalPowerSeriesCompose(self, other)
def inverse(self, x=None, n=6):
r"""
Returns the truncated terms of the inverse of the formal power series,
up to specified `n`.
If `f` and `g` are two formal power series of two different functions,
then the coefficient sequence ``ak`` of the composed formal power series `fp`
will be as follows.
.. math::
\sum\limits_{k=0}^{n} (-1)^{k} x_0^{-k-1} B_{n,k}(x_1, x_2, \dotsc, x_{n-k+1})
Parameters
==========
n : Number, optional
Specifies the order of the term up to which the polynomial should
be truncated.
Examples
========
>>> from sympy import fps, exp, cos
>>> from sympy.abc import x
>>> f1 = fps(exp(x))
>>> f2 = fps(cos(x))
>>> f1.inverse(x).truncate()
1 - x + x**2/2 - x**3/6 + x**4/24 - x**5/120 + O(x**6)
>>> f2.inverse(x).truncate(8)
1 + x**2/2 + 5*x**4/24 + 61*x**6/720 + O(x**8)
See Also
========
sympy.functions.combinatorial.numbers.bell
sympy.series.formal.FormalPowerSeriesInverse
References
==========
.. [1] Comtet, Louis: Advanced combinatorics; the art of finite and infinite expansions. Reidel, 1974.
"""
if x is None:
x = self.x
if n is None:
return iter(self)
if self._eval_term(0).is_zero:
raise ValueError("Constant coefficient should exist for an inverse of a formal"
" power series to exist.")
return FormalPowerSeriesInverse(self)
def __add__(self, other):
other = sympify(other)
if isinstance(other, FormalPowerSeries):
if self.dir != other.dir:
raise ValueError("Both series should be calculated from the"
" same direction.")
elif self.x0 != other.x0:
raise ValueError("Both series should be calculated about the"
" same point.")
x, y = self.x, other.x
f = self.function + other.function.subs(y, x)
if self.x not in f.free_symbols:
return f
ak = self.ak + other.ak
if self.ak.start > other.ak.start:
seq = other.ak
s, e = other.ak.start, self.ak.start
else:
seq = self.ak
s, e = self.ak.start, other.ak.start
save = Add(*[z[0]*z[1] for z in zip(seq[0:(e - s)], self.xk[s:e])])
ind = self.ind + other.ind + save
return self.func(f, x, self.x0, self.dir, (ak, self.xk, ind))
elif not other.has(self.x):
f = self.function + other
ind = self.ind + other
return self.func(f, self.x, self.x0, self.dir,
(self.ak, self.xk, ind))
return Add(self, other)
def __radd__(self, other):
return self.__add__(other)
def __neg__(self):
return self.func(-self.function, self.x, self.x0, self.dir,
(-self.ak, self.xk, -self.ind))
def __sub__(self, other):
return self.__add__(-other)
def __rsub__(self, other):
return (-self).__add__(other)
def __mul__(self, other):
other = sympify(other)
if other.has(self.x):
return Mul(self, other)
f = self.function * other
ak = self.ak.coeff_mul(other)
ind = self.ind * other
return self.func(f, self.x, self.x0, self.dir, (ak, self.xk, ind))
def __rmul__(self, other):
return self.__mul__(other)
class FiniteFormalPowerSeries(FormalPowerSeries):
"""Base Class for Product, Compose and Inverse classes"""
def __init__(self, *args):
pass
@property
def ffps(self):
return self.args[0]
@property
def gfps(self):
return self.args[1]
@property
def f(self):
return self.ffps.function
@property
def g(self):
return self.gfps.function
@property
def infinite(self):
raise NotImplementedError("No infinite version for an object of"
" FiniteFormalPowerSeries class.")
def _eval_terms(self, n):
raise NotImplementedError("(%s)._eval_terms()" % self)
def _eval_term(self, pt):
raise NotImplementedError("By the current logic, one can get terms"
"upto a certain order, instead of getting term by term.")
def polynomial(self, n):
return self._eval_terms(n)
def truncate(self, n=6):
ffps = self.ffps
pt_xk = ffps.xk.coeff(n)
x, x0 = ffps.x, ffps.x0
return self.polynomial(n) + Order(pt_xk, (x, x0))
def _eval_derivative(self, x):
raise NotImplementedError
def integrate(self, x):
raise NotImplementedError
class FormalPowerSeriesProduct(FiniteFormalPowerSeries):
"""Represents the product of two formal power series of two functions.
No computation is performed. Terms are calculated using a term by term logic,
instead of a point by point logic.
There are two differences between a :obj:`FormalPowerSeries` object and a
:obj:`FormalPowerSeriesProduct` object. The first argument contains the two
functions involved in the product. Also, the coefficient sequence contains
both the coefficient sequence of the formal power series of the involved functions.
See Also
========
sympy.series.formal.FormalPowerSeries
sympy.series.formal.FiniteFormalPowerSeries
"""
def __init__(self, *args):
ffps, gfps = self.ffps, self.gfps
k = ffps.ak.variables[0]
self.coeff1 = sequence(ffps.ak.formula, (k, 0, oo))
k = gfps.ak.variables[0]
self.coeff2 = sequence(gfps.ak.formula, (k, 0, oo))
@property
def function(self):
"""Function of the product of two formal power series."""
return self.f * self.g
def _eval_terms(self, n):
"""
Returns the first `n` terms of the product formal power series.
Term by term logic is implemented here.
Examples
========
>>> from sympy import fps, sin, exp
>>> from sympy.abc import x
>>> f1 = fps(sin(x))
>>> f2 = fps(exp(x))
>>> fprod = f1.product(f2, x)
>>> fprod._eval_terms(4)
x**3/3 + x**2 + x
See Also
========
sympy.series.formal.FormalPowerSeries.product
"""
coeff1, coeff2 = self.coeff1, self.coeff2
aks = convolution(coeff1[:n], coeff2[:n])
terms = []
for i in range(0, n):
terms.append(aks[i] * self.ffps.xk.coeff(i))
return Add(*terms)
class FormalPowerSeriesCompose(FiniteFormalPowerSeries):
"""Represents the composed formal power series of two functions.
No computation is performed. Terms are calculated using a term by term logic,
instead of a point by point logic.
There are two differences between a :obj:`FormalPowerSeries` object and a
:obj:`FormalPowerSeriesCompose` object. The first argument contains the outer
function and the inner function involved in the omposition. Also, the
coefficient sequence contains the generic sequence which is to be multiplied
by a custom ``bell_seq`` finite sequence. The finite terms will then be added up to
get the final terms.
See Also
========
sympy.series.formal.FormalPowerSeries
sympy.series.formal.FiniteFormalPowerSeries
"""
@property
def function(self):
"""Function for the composed formal power series."""
f, g, x = self.f, self.g, self.ffps.x
return f.subs(x, g)
def _eval_terms(self, n):
"""
Returns the first `n` terms of the composed formal power series.
Term by term logic is implemented here.
The coefficient sequence of the :obj:`FormalPowerSeriesCompose` object is the generic sequence.
It is multiplied by ``bell_seq`` to get a sequence, whose terms are added up to get
the final terms for the polynomial.
Examples
========
>>> from sympy import fps, sin, exp
>>> from sympy.abc import x
>>> f1 = fps(exp(x))
>>> f2 = fps(sin(x))
>>> fcomp = f1.compose(f2, x)
>>> fcomp._eval_terms(6)
-x**5/15 - x**4/8 + x**2/2 + x + 1
>>> fcomp._eval_terms(8)
x**7/90 - x**6/240 - x**5/15 - x**4/8 + x**2/2 + x + 1
See Also
========
sympy.series.formal.FormalPowerSeries.compose
sympy.series.formal.FormalPowerSeries.coeff_bell
"""
ffps, gfps = self.ffps, self.gfps
terms = [ffps.zero_coeff()]
for i in range(1, n):
bell_seq = gfps.coeff_bell(i)
seq = (ffps.bell_coeff_seq * bell_seq)
terms.append(Add(*(seq[:i])) / ffps.fact_seq[i-1] * ffps.xk.coeff(i))
return Add(*terms)
class FormalPowerSeriesInverse(FiniteFormalPowerSeries):
"""Represents the Inverse of a formal power series.
No computation is performed. Terms are calculated using a term by term logic,
instead of a point by point logic.
There is a single difference between a :obj:`FormalPowerSeries` object and a
:obj:`FormalPowerSeriesInverse` object. The coefficient sequence contains the
generic sequence which is to be multiplied by a custom ``bell_seq`` finite sequence.
The finite terms will then be added up to get the final terms.
See Also
========
sympy.series.formal.FormalPowerSeries
sympy.series.formal.FiniteFormalPowerSeries
"""
def __init__(self, *args):
ffps = self.ffps
k = ffps.xk.variables[0]
inv = ffps.zero_coeff()
inv_seq = sequence(inv ** (-(k + 1)), (k, 1, oo))
self.aux_seq = ffps.sign_seq * ffps.fact_seq * inv_seq
@property
def function(self):
"""Function for the inverse of a formal power series."""
f = self.f
return 1 / f
@property
def g(self):
raise ValueError("Only one function is considered while performing"
"inverse of a formal power series.")
@property
def gfps(self):
raise ValueError("Only one function is considered while performing"
"inverse of a formal power series.")
def _eval_terms(self, n):
"""
Returns the first `n` terms of the composed formal power series.
Term by term logic is implemented here.
The coefficient sequence of the `FormalPowerSeriesInverse` object is the generic sequence.
It is multiplied by ``bell_seq`` to get a sequence, whose terms are added up to get
the final terms for the polynomial.
Examples
========
>>> from sympy import fps, exp, cos
>>> from sympy.abc import x
>>> f1 = fps(exp(x))
>>> f2 = fps(cos(x))
>>> finv1, finv2 = f1.inverse(), f2.inverse()
>>> finv1._eval_terms(6)
-x**5/120 + x**4/24 - x**3/6 + x**2/2 - x + 1
>>> finv2._eval_terms(8)
61*x**6/720 + 5*x**4/24 + x**2/2 + 1
See Also
========
sympy.series.formal.FormalPowerSeries.inverse
sympy.series.formal.FormalPowerSeries.coeff_bell
"""
ffps = self.ffps
terms = [ffps.zero_coeff()]
for i in range(1, n):
bell_seq = ffps.coeff_bell(i)
seq = (self.aux_seq * bell_seq)
terms.append(Add(*(seq[:i])) / ffps.fact_seq[i-1] * ffps.xk.coeff(i))
return Add(*terms)
def fps(f, x=None, x0=0, dir=1, hyper=True, order=4, rational=True, full=False):
"""Generates Formal Power Series of f.
Returns the formal series expansion of ``f`` around ``x = x0``
with respect to ``x`` in the form of a ``FormalPowerSeries`` object.
Formal Power Series is represented using an explicit formula
computed using different algorithms.
See :func:`compute_fps` for the more details regarding the computation
of formula.
Parameters
==========
x : Symbol, optional
If x is None and ``f`` is univariate, the univariate symbols will be
supplied, otherwise an error will be raised.
x0 : number, optional
Point to perform series expansion about. Default is 0.
dir : {1, -1, '+', '-'}, optional
If dir is 1 or '+' the series is calculated from the right and
for -1 or '-' the series is calculated from the left. For smooth
functions this flag will not alter the results. Default is 1.
hyper : {True, False}, optional
Set hyper to False to skip the hypergeometric algorithm.
By default it is set to False.
order : int, optional
Order of the derivative of ``f``, Default is 4.
rational : {True, False}, optional
Set rational to False to skip rational algorithm. By default it is set
to True.
full : {True, False}, optional
Set full to True to increase the range of rational algorithm.
See :func:`rational_algorithm` for details. By default it is set to
False.
Examples
========
>>> from sympy import fps, ln, atan, sin
>>> from sympy.abc import x, n
Rational Functions
>>> fps(ln(1 + x)).truncate()
x - x**2/2 + x**3/3 - x**4/4 + x**5/5 + O(x**6)
>>> fps(atan(x), full=True).truncate()
x - x**3/3 + x**5/5 + O(x**6)
Symbolic Functions
>>> fps(x**n*sin(x**2), x).truncate(8)
-x**(n + 6)/6 + x**(n + 2) + O(x**(n + 8))
See Also
========
sympy.series.formal.FormalPowerSeries
sympy.series.formal.compute_fps
"""
f = sympify(f)
if x is None:
free = f.free_symbols
if len(free) == 1:
x = free.pop()
elif not free:
return f
else:
raise NotImplementedError("multivariate formal power series")
result = compute_fps(f, x, x0, dir, hyper, order, rational, full)
if result is None:
return f
return FormalPowerSeries(f, x, x0, dir, result)
|
999d2b7a6a24442b46293484706fd50547d305374a65c20e9c176075db79b95d | from __future__ import print_function, division
from sympy.core import S, sympify, Expr, Rational, Dummy
from sympy.core import Add, Mul, expand_power_base, expand_log
from sympy.core.cache import cacheit
from sympy.core.compatibility import default_sort_key, is_sequence
from sympy.core.containers import Tuple
from sympy.sets.sets import Complement
from sympy.utilities.iterables import uniq
class Order(Expr):
r""" Represents the limiting behavior of some function
The order of a function characterizes the function based on the limiting
behavior of the function as it goes to some limit. Only taking the limit
point to be a number is currently supported. This is expressed in
big O notation [1]_.
The formal definition for the order of a function `g(x)` about a point `a`
is such that `g(x) = O(f(x))` as `x \rightarrow a` if and only if for any
`\delta > 0` there exists a `M > 0` such that `|g(x)| \leq M|f(x)|` for
`|x-a| < \delta`. This is equivalent to `\lim_{x \rightarrow a}
\sup |g(x)/f(x)| < \infty`.
Let's illustrate it on the following example by taking the expansion of
`\sin(x)` about 0:
.. math ::
\sin(x) = x - x^3/3! + O(x^5)
where in this case `O(x^5) = x^5/5! - x^7/7! + \cdots`. By the definition
of `O`, for any `\delta > 0` there is an `M` such that:
.. math ::
|x^5/5! - x^7/7! + ....| <= M|x^5| \text{ for } |x| < \delta
or by the alternate definition:
.. math ::
\lim_{x \rightarrow 0} | (x^5/5! - x^7/7! + ....) / x^5| < \infty
which surely is true, because
.. math ::
\lim_{x \rightarrow 0} | (x^5/5! - x^7/7! + ....) / x^5| = 1/5!
As it is usually used, the order of a function can be intuitively thought
of representing all terms of powers greater than the one specified. For
example, `O(x^3)` corresponds to any terms proportional to `x^3,
x^4,\ldots` and any higher power. For a polynomial, this leaves terms
proportional to `x^2`, `x` and constants.
Examples
========
>>> from sympy import O, oo, cos, pi
>>> from sympy.abc import x, y
>>> O(x + x**2)
O(x)
>>> O(x + x**2, (x, 0))
O(x)
>>> O(x + x**2, (x, oo))
O(x**2, (x, oo))
>>> O(1 + x*y)
O(1, x, y)
>>> O(1 + x*y, (x, 0), (y, 0))
O(1, x, y)
>>> O(1 + x*y, (x, oo), (y, oo))
O(x*y, (x, oo), (y, oo))
>>> O(1) in O(1, x)
True
>>> O(1, x) in O(1)
False
>>> O(x) in O(1, x)
True
>>> O(x**2) in O(x)
True
>>> O(x)*x
O(x**2)
>>> O(x) - O(x)
O(x)
>>> O(cos(x))
O(1)
>>> O(cos(x), (x, pi/2))
O(x - pi/2, (x, pi/2))
References
==========
.. [1] `Big O notation <https://en.wikipedia.org/wiki/Big_O_notation>`_
Notes
=====
In ``O(f(x), x)`` the expression ``f(x)`` is assumed to have a leading
term. ``O(f(x), x)`` is automatically transformed to
``O(f(x).as_leading_term(x),x)``.
``O(expr*f(x), x)`` is ``O(f(x), x)``
``O(expr, x)`` is ``O(1)``
``O(0, x)`` is 0.
Multivariate O is also supported:
``O(f(x, y), x, y)`` is transformed to
``O(f(x, y).as_leading_term(x,y).as_leading_term(y), x, y)``
In the multivariate case, it is assumed the limits w.r.t. the various
symbols commute.
If no symbols are passed then all symbols in the expression are used
and the limit point is assumed to be zero.
"""
is_Order = True
__slots__ = ()
@cacheit
def __new__(cls, expr, *args, **kwargs):
expr = sympify(expr)
if not args:
if expr.is_Order:
variables = expr.variables
point = expr.point
else:
variables = list(expr.free_symbols)
point = [S.Zero]*len(variables)
else:
args = list(args if is_sequence(args) else [args])
variables, point = [], []
if is_sequence(args[0]):
for a in args:
v, p = list(map(sympify, a))
variables.append(v)
point.append(p)
else:
variables = list(map(sympify, args))
point = [S.Zero]*len(variables)
if not all(v.is_symbol for v in variables):
raise TypeError('Variables are not symbols, got %s' % variables)
if len(list(uniq(variables))) != len(variables):
raise ValueError('Variables are supposed to be unique symbols, got %s' % variables)
if expr.is_Order:
expr_vp = dict(expr.args[1:])
new_vp = dict(expr_vp)
vp = dict(zip(variables, point))
for v, p in vp.items():
if v in new_vp.keys():
if p != new_vp[v]:
raise NotImplementedError(
"Mixing Order at different points is not supported.")
else:
new_vp[v] = p
if set(expr_vp.keys()) == set(new_vp.keys()):
return expr
else:
variables = list(new_vp.keys())
point = [new_vp[v] for v in variables]
if expr is S.NaN:
return S.NaN
if any(x in p.free_symbols for x in variables for p in point):
raise ValueError('Got %s as a point.' % point)
if variables:
if any(p != point[0] for p in point):
raise NotImplementedError(
"Multivariable orders at different points are not supported.")
if point[0] is S.Infinity:
s = {k: 1/Dummy() for k in variables}
rs = {1/v: 1/k for k, v in s.items()}
elif point[0] is S.NegativeInfinity:
s = {k: -1/Dummy() for k in variables}
rs = {-1/v: -1/k for k, v in s.items()}
elif point[0] is not S.Zero:
s = dict((k, Dummy() + point[0]) for k in variables)
rs = dict((v - point[0], k - point[0]) for k, v in s.items())
else:
s = ()
rs = ()
expr = expr.subs(s)
if expr.is_Add:
expr = expr.factor()
if s:
args = tuple([r[0] for r in rs.items()])
else:
args = tuple(variables)
if len(variables) > 1:
# XXX: better way? We need this expand() to
# workaround e.g: expr = x*(x + y).
# (x*(x + y)).as_leading_term(x, y) currently returns
# x*y (wrong order term!). That's why we want to deal with
# expand()'ed expr (handled in "if expr.is_Add" branch below).
expr = expr.expand()
old_expr = None
while old_expr != expr:
old_expr = expr
if expr.is_Add:
lst = expr.extract_leading_order(args)
expr = Add(*[f.expr for (e, f) in lst])
elif expr:
expr = expr.as_leading_term(*args)
expr = expr.as_independent(*args, as_Add=False)[1]
expr = expand_power_base(expr)
expr = expand_log(expr)
if len(args) == 1:
# The definition of O(f(x)) symbol explicitly stated that
# the argument of f(x) is irrelevant. That's why we can
# combine some power exponents (only "on top" of the
# expression tree for f(x)), e.g.:
# x**p * (-x)**q -> x**(p+q) for real p, q.
x = args[0]
margs = list(Mul.make_args(
expr.as_independent(x, as_Add=False)[1]))
for i, t in enumerate(margs):
if t.is_Pow:
b, q = t.args
if b in (x, -x) and q.is_real and not q.has(x):
margs[i] = x**q
elif b.is_Pow and not b.exp.has(x):
b, r = b.args
if b in (x, -x) and r.is_real:
margs[i] = x**(r*q)
elif b.is_Mul and b.args[0] is S.NegativeOne:
b = -b
if b.is_Pow and not b.exp.has(x):
b, r = b.args
if b in (x, -x) and r.is_real:
margs[i] = x**(r*q)
expr = Mul(*margs)
expr = expr.subs(rs)
if expr.is_Order:
expr = expr.expr
if not expr.has(*variables) and not expr.is_zero:
expr = S.One
# create Order instance:
vp = dict(zip(variables, point))
variables.sort(key=default_sort_key)
point = [vp[v] for v in variables]
args = (expr,) + Tuple(*zip(variables, point))
obj = Expr.__new__(cls, *args)
return obj
def _eval_nseries(self, x, n, logx, cdir=0):
return self
@property
def expr(self):
return self.args[0]
@property
def variables(self):
if self.args[1:]:
return tuple(x[0] for x in self.args[1:])
else:
return ()
@property
def point(self):
if self.args[1:]:
return tuple(x[1] for x in self.args[1:])
else:
return ()
@property
def free_symbols(self):
return self.expr.free_symbols | set(self.variables)
def _eval_power(b, e):
if e.is_Number and e.is_nonnegative:
return b.func(b.expr ** e, *b.args[1:])
if e == O(1):
return b
return
def as_expr_variables(self, order_symbols):
if order_symbols is None:
order_symbols = self.args[1:]
else:
if (not all(o[1] == order_symbols[0][1] for o in order_symbols) and
not all(p == self.point[0] for p in self.point)): # pragma: no cover
raise NotImplementedError('Order at points other than 0 '
'or oo not supported, got %s as a point.' % self.point)
if order_symbols and order_symbols[0][1] != self.point[0]:
raise NotImplementedError(
"Multiplying Order at different points is not supported.")
order_symbols = dict(order_symbols)
for s, p in dict(self.args[1:]).items():
if s not in order_symbols.keys():
order_symbols[s] = p
order_symbols = sorted(order_symbols.items(), key=lambda x: default_sort_key(x[0]))
return self.expr, tuple(order_symbols)
def removeO(self):
return S.Zero
def getO(self):
return self
@cacheit
def contains(self, expr):
r"""
Return True if expr belongs to Order(self.expr, \*self.variables).
Return False if self belongs to expr.
Return None if the inclusion relation cannot be determined
(e.g. when self and expr have different symbols).
"""
from sympy import powsimp
if expr.is_zero:
return True
if expr is S.NaN:
return False
point = self.point[0] if self.point else S.Zero
if expr.is_Order:
if (any(p != point for p in expr.point) or
any(p != point for p in self.point)):
return None
if expr.expr == self.expr:
# O(1) + O(1), O(1) + O(1, x), etc.
return all([x in self.args[1:] for x in expr.args[1:]])
if expr.expr.is_Add:
return all([self.contains(x) for x in expr.expr.args])
if self.expr.is_Add and point.is_zero:
return any([self.func(x, *self.args[1:]).contains(expr)
for x in self.expr.args])
if self.variables and expr.variables:
common_symbols = tuple(
[s for s in self.variables if s in expr.variables])
elif self.variables:
common_symbols = self.variables
else:
common_symbols = expr.variables
if not common_symbols:
return None
if (self.expr.is_Pow and len(self.variables) == 1
and self.variables == expr.variables):
symbol = self.variables[0]
other = expr.expr.as_independent(symbol, as_Add=False)[1]
if (other.is_Pow and other.base == symbol and
self.expr.base == symbol):
if point.is_zero:
rv = (self.expr.exp - other.exp).is_nonpositive
if point.is_infinite:
rv = (self.expr.exp - other.exp).is_nonnegative
if rv is not None:
return rv
r = None
ratio = self.expr/expr.expr
ratio = powsimp(ratio, deep=True, combine='exp')
for s in common_symbols:
from sympy.series.limits import Limit
l = Limit(ratio, s, point).doit(heuristics=False)
if not isinstance(l, Limit):
l = l != 0
else:
l = None
if r is None:
r = l
else:
if r != l:
return
return r
if self.expr.is_Pow and len(self.variables) == 1:
symbol = self.variables[0]
other = expr.as_independent(symbol, as_Add=False)[1]
if (other.is_Pow and other.base == symbol and
self.expr.base == symbol):
if point.is_zero:
rv = (self.expr.exp - other.exp).is_nonpositive
if point.is_infinite:
rv = (self.expr.exp - other.exp).is_nonnegative
if rv is not None:
return rv
obj = self.func(expr, *self.args[1:])
return self.contains(obj)
def __contains__(self, other):
result = self.contains(other)
if result is None:
raise TypeError('contains did not evaluate to a bool')
return result
def _eval_subs(self, old, new):
if old in self.variables:
newexpr = self.expr.subs(old, new)
i = self.variables.index(old)
newvars = list(self.variables)
newpt = list(self.point)
if new.is_symbol:
newvars[i] = new
else:
syms = new.free_symbols
if len(syms) == 1 or old in syms:
if old in syms:
var = self.variables[i]
else:
var = syms.pop()
# First, try to substitute self.point in the "new"
# expr to see if this is a fixed point.
# E.g. O(y).subs(y, sin(x))
point = new.subs(var, self.point[i])
if point != self.point[i]:
from sympy.solvers.solveset import solveset
d = Dummy()
sol = solveset(old - new.subs(var, d), d)
if isinstance(sol, Complement):
e1 = sol.args[0]
e2 = sol.args[1]
sol = set(e1) - set(e2)
res = [dict(zip((d, ), sol))]
point = d.subs(res[0]).limit(old, self.point[i])
newvars[i] = var
newpt[i] = point
elif old not in syms:
del newvars[i], newpt[i]
if not syms and new == self.point[i]:
newvars.extend(syms)
newpt.extend([S.Zero]*len(syms))
else:
return
return Order(newexpr, *zip(newvars, newpt))
def _eval_conjugate(self):
expr = self.expr._eval_conjugate()
if expr is not None:
return self.func(expr, *self.args[1:])
def _eval_derivative(self, x):
return self.func(self.expr.diff(x), *self.args[1:]) or self
def _eval_transpose(self):
expr = self.expr._eval_transpose()
if expr is not None:
return self.func(expr, *self.args[1:])
def _sage_(self):
#XXX: SAGE doesn't have Order yet. Let's return 0 instead.
return Rational(0)._sage_()
def __neg__(self):
return self
O = Order
|
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