hash
stringlengths 64
64
| content
stringlengths 0
1.51M
|
---|---|
218a9b36e89d6f7aa788e7a6eb3bb43fe5a74ace1a22e66ef12f36ae9002cf33 | """Implementation of :class:`QuotientRing` class."""
from sympy.polys.agca.modules import FreeModuleQuotientRing
from sympy.polys.domains.ring import Ring
from sympy.polys.polyerrors import NotReversible, CoercionFailed
from sympy.utilities import public
# TODO
# - successive quotients (when quotient ideals are implemented)
# - poly rings over quotients?
# - division by non-units in integral domains?
@public
class QuotientRingElement:
"""
Class representing elements of (commutative) quotient rings.
Attributes:
- ring - containing ring
- data - element of ring.ring (i.e. base ring) representing self
"""
def __init__(self, ring, data):
self.ring = ring
self.data = data
def __str__(self):
from sympy import sstr
return sstr(self.data) + " + " + str(self.ring.base_ideal)
__repr__ = __str__
def __bool__(self):
return not self.ring.is_zero(self)
def __add__(self, om):
if not isinstance(om, self.__class__) or om.ring != self.ring:
try:
om = self.ring.convert(om)
except (NotImplementedError, CoercionFailed):
return NotImplemented
return self.ring(self.data + om.data)
__radd__ = __add__
def __neg__(self):
return self.ring(self.data*self.ring.ring.convert(-1))
def __sub__(self, om):
return self.__add__(-om)
def __rsub__(self, om):
return (-self).__add__(om)
def __mul__(self, o):
if not isinstance(o, self.__class__):
try:
o = self.ring.convert(o)
except (NotImplementedError, CoercionFailed):
return NotImplemented
return self.ring(self.data*o.data)
__rmul__ = __mul__
def __rtruediv__(self, o):
return self.ring.revert(self)*o
def __truediv__(self, o):
if not isinstance(o, self.__class__):
try:
o = self.ring.convert(o)
except (NotImplementedError, CoercionFailed):
return NotImplemented
return self.ring.revert(o)*self
def __pow__(self, oth):
if oth < 0:
return self.ring.revert(self) ** -oth
return self.ring(self.data ** oth)
def __eq__(self, om):
if not isinstance(om, self.__class__) or om.ring != self.ring:
return False
return self.ring.is_zero(self - om)
def __ne__(self, om):
return not self == om
class QuotientRing(Ring):
"""
Class representing (commutative) quotient rings.
You should not usually instantiate this by hand, instead use the constructor
from the base ring in the construction.
>>> from sympy.abc import x
>>> from sympy import QQ
>>> I = QQ.old_poly_ring(x).ideal(x**3 + 1)
>>> QQ.old_poly_ring(x).quotient_ring(I)
QQ[x]/<x**3 + 1>
Shorter versions are possible:
>>> QQ.old_poly_ring(x)/I
QQ[x]/<x**3 + 1>
>>> QQ.old_poly_ring(x)/[x**3 + 1]
QQ[x]/<x**3 + 1>
Attributes:
- ring - the base ring
- base_ideal - the ideal used to form the quotient
"""
has_assoc_Ring = True
has_assoc_Field = False
dtype = QuotientRingElement
def __init__(self, ring, ideal):
if not ideal.ring == ring:
raise ValueError('Ideal must belong to %s, got %s' % (ring, ideal))
self.ring = ring
self.base_ideal = ideal
self.zero = self(self.ring.zero)
self.one = self(self.ring.one)
def __str__(self):
return str(self.ring) + "/" + str(self.base_ideal)
def __hash__(self):
return hash((self.__class__.__name__, self.dtype, self.ring, self.base_ideal))
def new(self, a):
"""Construct an element of ``self`` domain from ``a``. """
if not isinstance(a, self.ring.dtype):
a = self.ring(a)
# TODO optionally disable reduction?
return self.dtype(self, self.base_ideal.reduce_element(a))
def __eq__(self, other):
"""Returns ``True`` if two domains are equivalent. """
return isinstance(other, QuotientRing) and \
self.ring == other.ring and self.base_ideal == other.base_ideal
def from_ZZ(K1, a, K0):
"""Convert a Python ``int`` object to ``dtype``. """
return K1(K1.ring.convert(a, K0))
from_ZZ_python = from_ZZ
from_QQ_python = from_ZZ_python
from_ZZ_gmpy = from_ZZ_python
from_QQ_gmpy = from_ZZ_python
from_RealField = from_ZZ_python
from_GlobalPolynomialRing = from_ZZ_python
from_FractionField = from_ZZ_python
def from_sympy(self, a):
return self(self.ring.from_sympy(a))
def to_sympy(self, a):
return self.ring.to_sympy(a.data)
def from_QuotientRing(self, a, K0):
if K0 == self:
return a
def poly_ring(self, *gens):
"""Returns a polynomial ring, i.e. ``K[X]``. """
raise NotImplementedError('nested domains not allowed')
def frac_field(self, *gens):
"""Returns a fraction field, i.e. ``K(X)``. """
raise NotImplementedError('nested domains not allowed')
def revert(self, a):
"""
Compute a**(-1), if possible.
"""
I = self.ring.ideal(a.data) + self.base_ideal
try:
return self(I.in_terms_of_generators(1)[0])
except ValueError: # 1 not in I
raise NotReversible('%s not a unit in %r' % (a, self))
def is_zero(self, a):
return self.base_ideal.contains(a.data)
def free_module(self, rank):
"""
Generate a free module of rank ``rank`` over ``self``.
>>> from sympy.abc import x
>>> from sympy import QQ
>>> (QQ.old_poly_ring(x)/[x**2 + 1]).free_module(2)
(QQ[x]/<x**2 + 1>)**2
"""
return FreeModuleQuotientRing(self, rank)
|
5ade7fcbb1d30def2ed6a19a9613614d3ed1c3709089592681db90ee37290fd7 | """Implementation of :class:`RealField` class. """
from sympy.core.numbers import Float
from sympy.polys.domains.field import Field
from sympy.polys.domains.simpledomain import SimpleDomain
from sympy.polys.domains.characteristiczero import CharacteristicZero
from sympy.polys.domains.mpelements import MPContext
from sympy.polys.polyerrors import CoercionFailed
from sympy.utilities import public
@public
class RealField(Field, CharacteristicZero, SimpleDomain):
"""Real numbers up to the given precision. """
rep = 'RR'
is_RealField = is_RR = True
is_Exact = False
is_Numerical = True
is_PID = False
has_assoc_Ring = False
has_assoc_Field = True
_default_precision = 53
@property
def has_default_precision(self):
return self.precision == self._default_precision
@property
def precision(self):
return self._context.prec
@property
def dps(self):
return self._context.dps
@property
def tolerance(self):
return self._context.tolerance
def __init__(self, prec=_default_precision, dps=None, tol=None):
context = MPContext(prec, dps, tol, True)
context._parent = self
self._context = context
self.dtype = context.mpf
self.zero = self.dtype(0)
self.one = self.dtype(1)
def __eq__(self, other):
return (isinstance(other, RealField)
and self.precision == other.precision
and self.tolerance == other.tolerance)
def __hash__(self):
return hash((self.__class__.__name__, self.dtype, self.precision, self.tolerance))
def to_sympy(self, element):
"""Convert ``element`` to SymPy number. """
return Float(element, self.dps)
def from_sympy(self, expr):
"""Convert SymPy's number to ``dtype``. """
number = expr.evalf(n=self.dps)
if number.is_Number:
return self.dtype(number)
else:
raise CoercionFailed("expected real number, got %s" % expr)
def from_ZZ(self, element, base):
return self.dtype(element)
def from_ZZ_python(self, element, base):
return self.dtype(element)
def from_QQ(self, element, base):
return self.dtype(element.numerator) / element.denominator
def from_QQ_python(self, element, base):
return self.dtype(element.numerator) / element.denominator
def from_ZZ_gmpy(self, element, base):
return self.dtype(int(element))
def from_QQ_gmpy(self, element, base):
return self.dtype(int(element.numerator)) / int(element.denominator)
def from_AlgebraicField(self, element, base):
return self.from_sympy(base.to_sympy(element).evalf(self.dps))
def from_RealField(self, element, base):
if self == base:
return element
else:
return self.dtype(element)
def from_ComplexField(self, element, base):
if not element.imag:
return self.dtype(element.real)
def to_rational(self, element, limit=True):
"""Convert a real number to rational number. """
return self._context.to_rational(element, limit)
def get_ring(self):
"""Returns a ring associated with ``self``. """
return self
def get_exact(self):
"""Returns an exact domain associated with ``self``. """
from sympy.polys.domains import QQ
return QQ
def gcd(self, a, b):
"""Returns GCD of ``a`` and ``b``. """
return self.one
def lcm(self, a, b):
"""Returns LCM of ``a`` and ``b``. """
return a*b
def almosteq(self, a, b, tolerance=None):
"""Check if ``a`` and ``b`` are almost equal. """
return self._context.almosteq(a, b, tolerance)
RR = RealField()
|
c7c04f9062a3d163eec09ccdda9cb997d6808c7698e1d8efeb9738b73513af76 | """Implementation of :class:`PythonRationalField` class. """
from sympy.polys.domains.groundtypes import PythonInteger, PythonRational, SymPyRational
from sympy.polys.domains.rationalfield import RationalField
from sympy.polys.polyerrors import CoercionFailed
from sympy.utilities import public
@public
class PythonRationalField(RationalField):
"""Rational field based on :ref:`MPQ`.
This will be used as :ref:`QQ` if ``gmpy`` and ``gmpy2`` are not
installed. Elements are instances of :ref:`MPQ`.
"""
dtype = PythonRational
zero = dtype(0)
one = dtype(1)
alias = 'QQ_python'
def __init__(self):
pass
def get_ring(self):
"""Returns ring associated with ``self``. """
from sympy.polys.domains import PythonIntegerRing
return PythonIntegerRing()
def to_sympy(self, a):
"""Convert `a` to a SymPy object. """
return SymPyRational(a.numerator, a.denominator)
def from_sympy(self, a):
"""Convert SymPy's Rational to `dtype`. """
if a.is_Rational:
return PythonRational(a.p, a.q)
elif a.is_Float:
from sympy.polys.domains import RR
p, q = RR.to_rational(a)
return PythonRational(int(p), int(q))
else:
raise CoercionFailed("expected `Rational` object, got %s" % a)
def from_ZZ_python(K1, a, K0):
"""Convert a Python `int` object to `dtype`. """
return PythonRational(a)
def from_QQ_python(K1, a, K0):
"""Convert a Python `Fraction` object to `dtype`. """
return a
def from_ZZ_gmpy(K1, a, K0):
"""Convert a GMPY `mpz` object to `dtype`. """
return PythonRational(PythonInteger(a))
def from_QQ_gmpy(K1, a, K0):
"""Convert a GMPY `mpq` object to `dtype`. """
return PythonRational(PythonInteger(a.numer()),
PythonInteger(a.denom()))
def from_RealField(K1, a, K0):
"""Convert a mpmath `mpf` object to `dtype`. """
p, q = K0.to_rational(a)
return PythonRational(int(p), int(q))
def numer(self, a):
"""Returns numerator of `a`. """
return a.numerator
def denom(self, a):
"""Returns denominator of `a`. """
return a.denominator
|
e477003608edb5c0661c1686353ebb153c0429ad516d7ae54d7f76d307e27c05 | """Implementation of :class:`PolynomialRing` class. """
from sympy.core.compatibility import iterable
from sympy.polys.agca.modules import FreeModulePolyRing
from sympy.polys.domains.characteristiczero import CharacteristicZero
from sympy.polys.domains.compositedomain import CompositeDomain
from sympy.polys.domains.old_fractionfield import FractionField
from sympy.polys.domains.ring import Ring
from sympy.polys.orderings import monomial_key, build_product_order
from sympy.polys.polyclasses import DMP, DMF
from sympy.polys.polyerrors import (GeneratorsNeeded, PolynomialError,
CoercionFailed, ExactQuotientFailed, NotReversible)
from sympy.polys.polyutils import dict_from_basic, basic_from_dict, _dict_reorder
from sympy.utilities import public
# XXX why does this derive from CharacteristicZero???
@public
class PolynomialRingBase(Ring, CharacteristicZero, CompositeDomain):
"""
Base class for generalized polynomial rings.
This base class should be used for uniform access to generalized polynomial
rings. Subclasses only supply information about the element storage etc.
Do not instantiate.
"""
has_assoc_Ring = True
has_assoc_Field = True
default_order = "grevlex"
def __init__(self, dom, *gens, **opts):
if not gens:
raise GeneratorsNeeded("generators not specified")
lev = len(gens) - 1
self.ngens = len(gens)
self.zero = self.dtype.zero(lev, dom, ring=self)
self.one = self.dtype.one(lev, dom, ring=self)
self.domain = self.dom = dom
self.symbols = self.gens = gens
# NOTE 'order' may not be set if inject was called through CompositeDomain
self.order = opts.get('order', monomial_key(self.default_order))
def new(self, element):
return self.dtype(element, self.dom, len(self.gens) - 1, ring=self)
def __str__(self):
s_order = str(self.order)
orderstr = (
" order=" + s_order) if s_order != self.default_order else ""
return str(self.dom) + '[' + ','.join(map(str, self.gens)) + orderstr + ']'
def __hash__(self):
return hash((self.__class__.__name__, self.dtype, self.dom,
self.gens, self.order))
def __eq__(self, other):
"""Returns ``True`` if two domains are equivalent. """
return isinstance(other, PolynomialRingBase) and \
self.dtype == other.dtype and self.dom == other.dom and \
self.gens == other.gens and self.order == other.order
def from_ZZ(K1, a, K0):
"""Convert a Python ``int`` object to ``dtype``. """
return K1(K1.dom.convert(a, K0))
def from_ZZ_python(K1, a, K0):
"""Convert a Python ``int`` object to ``dtype``. """
return K1(K1.dom.convert(a, K0))
def from_QQ(K1, a, K0):
"""Convert a Python ``Fraction`` object to ``dtype``. """
return K1(K1.dom.convert(a, K0))
def from_QQ_python(K1, a, K0):
"""Convert a Python ``Fraction`` object to ``dtype``. """
return K1(K1.dom.convert(a, K0))
def from_ZZ_gmpy(K1, a, K0):
"""Convert a GMPY ``mpz`` object to ``dtype``. """
return K1(K1.dom.convert(a, K0))
def from_QQ_gmpy(K1, a, K0):
"""Convert a GMPY ``mpq`` object to ``dtype``. """
return K1(K1.dom.convert(a, K0))
def from_RealField(K1, a, K0):
"""Convert a mpmath ``mpf`` object to ``dtype``. """
return K1(K1.dom.convert(a, K0))
def from_AlgebraicField(K1, a, K0):
"""Convert a ``ANP`` object to ``dtype``. """
if K1.dom == K0:
return K1(a)
def from_PolynomialRing(K1, a, K0):
"""Convert a ``PolyElement`` object to ``dtype``. """
if K1.gens == K0.symbols:
if K1.dom == K0.dom:
return K1(dict(a)) # set the correct ring
else:
convert_dom = lambda c: K1.dom.convert_from(c, K0.dom)
return K1({m: convert_dom(c) for m, c in a.items()})
else:
monoms, coeffs = _dict_reorder(a.to_dict(), K0.symbols, K1.gens)
if K1.dom != K0.dom:
coeffs = [ K1.dom.convert(c, K0.dom) for c in coeffs ]
return K1(dict(zip(monoms, coeffs)))
def from_GlobalPolynomialRing(K1, a, K0):
"""Convert a ``DMP`` object to ``dtype``. """
if K1.gens == K0.gens:
if K1.dom == K0.dom:
return K1(a.rep) # set the correct ring
else:
return K1(a.convert(K1.dom).rep)
else:
monoms, coeffs = _dict_reorder(a.to_dict(), K0.gens, K1.gens)
if K1.dom != K0.dom:
coeffs = [ K1.dom.convert(c, K0.dom) for c in coeffs ]
return K1(dict(zip(monoms, coeffs)))
def get_field(self):
"""Returns a field associated with ``self``. """
return FractionField(self.dom, *self.gens)
def poly_ring(self, *gens):
"""Returns a polynomial ring, i.e. ``K[X]``. """
raise NotImplementedError('nested domains not allowed')
def frac_field(self, *gens):
"""Returns a fraction field, i.e. ``K(X)``. """
raise NotImplementedError('nested domains not allowed')
def revert(self, a):
try:
return 1/a
except (ExactQuotientFailed, ZeroDivisionError):
raise NotReversible('%s is not a unit' % a)
def gcdex(self, a, b):
"""Extended GCD of ``a`` and ``b``. """
return a.gcdex(b)
def gcd(self, a, b):
"""Returns GCD of ``a`` and ``b``. """
return a.gcd(b)
def lcm(self, a, b):
"""Returns LCM of ``a`` and ``b``. """
return a.lcm(b)
def factorial(self, a):
"""Returns factorial of ``a``. """
return self.dtype(self.dom.factorial(a))
def _vector_to_sdm(self, v, order):
"""
For internal use by the modules class.
Convert an iterable of elements of this ring into a sparse distributed
module element.
"""
raise NotImplementedError
def _sdm_to_dics(self, s, n):
"""Helper for _sdm_to_vector."""
from sympy.polys.distributedmodules import sdm_to_dict
dic = sdm_to_dict(s)
res = [{} for _ in range(n)]
for k, v in dic.items():
res[k[0]][k[1:]] = v
return res
def _sdm_to_vector(self, s, n):
"""
For internal use by the modules class.
Convert a sparse distributed module into a list of length ``n``.
Examples
========
>>> from sympy import QQ, ilex
>>> from sympy.abc import x, y
>>> R = QQ.old_poly_ring(x, y, order=ilex)
>>> L = [((1, 1, 1), QQ(1)), ((0, 1, 0), QQ(1)), ((0, 0, 1), QQ(2))]
>>> R._sdm_to_vector(L, 2)
[x + 2*y, x*y]
"""
dics = self._sdm_to_dics(s, n)
# NOTE this works for global and local rings!
return [self(x) for x in dics]
def free_module(self, rank):
"""
Generate a free module of rank ``rank`` over ``self``.
Examples
========
>>> from sympy.abc import x
>>> from sympy import QQ
>>> QQ.old_poly_ring(x).free_module(2)
QQ[x]**2
"""
return FreeModulePolyRing(self, rank)
def _vector_to_sdm_helper(v, order):
"""Helper method for common code in Global and Local poly rings."""
from sympy.polys.distributedmodules import sdm_from_dict
d = {}
for i, e in enumerate(v):
for key, value in e.to_dict().items():
d[(i,) + key] = value
return sdm_from_dict(d, order)
@public
class GlobalPolynomialRing(PolynomialRingBase):
"""A true polynomial ring, with objects DMP. """
is_PolynomialRing = is_Poly = True
dtype = DMP
def from_FractionField(K1, a, K0):
"""
Convert a ``DMF`` object to ``DMP``.
Examples
========
>>> from sympy.polys.polyclasses import DMP, DMF
>>> from sympy.polys.domains import ZZ
>>> from sympy.abc import x
>>> f = DMF(([ZZ(1), ZZ(1)], [ZZ(1)]), ZZ)
>>> K = ZZ.old_frac_field(x)
>>> F = ZZ.old_poly_ring(x).from_FractionField(f, K)
>>> F == DMP([ZZ(1), ZZ(1)], ZZ)
True
>>> type(F)
<class 'sympy.polys.polyclasses.DMP'>
"""
if a.denom().is_one:
return K1.from_GlobalPolynomialRing(a.numer(), K0)
def to_sympy(self, a):
"""Convert ``a`` to a SymPy object. """
return basic_from_dict(a.to_sympy_dict(), *self.gens)
def from_sympy(self, a):
"""Convert SymPy's expression to ``dtype``. """
try:
rep, _ = dict_from_basic(a, gens=self.gens)
except PolynomialError:
raise CoercionFailed("can't convert %s to type %s" % (a, self))
for k, v in rep.items():
rep[k] = self.dom.from_sympy(v)
return self(rep)
def is_positive(self, a):
"""Returns True if ``LC(a)`` is positive. """
return self.dom.is_positive(a.LC())
def is_negative(self, a):
"""Returns True if ``LC(a)`` is negative. """
return self.dom.is_negative(a.LC())
def is_nonpositive(self, a):
"""Returns True if ``LC(a)`` is non-positive. """
return self.dom.is_nonpositive(a.LC())
def is_nonnegative(self, a):
"""Returns True if ``LC(a)`` is non-negative. """
return self.dom.is_nonnegative(a.LC())
def _vector_to_sdm(self, v, order):
"""
Examples
========
>>> from sympy import lex, QQ
>>> from sympy.abc import x, y
>>> R = QQ.old_poly_ring(x, y)
>>> f = R.convert(x + 2*y)
>>> g = R.convert(x * y)
>>> R._vector_to_sdm([f, g], lex)
[((1, 1, 1), 1), ((0, 1, 0), 1), ((0, 0, 1), 2)]
"""
return _vector_to_sdm_helper(v, order)
class GeneralizedPolynomialRing(PolynomialRingBase):
"""A generalized polynomial ring, with objects DMF. """
dtype = DMF
def new(self, a):
"""Construct an element of ``self`` domain from ``a``. """
res = self.dtype(a, self.dom, len(self.gens) - 1, ring=self)
# make sure res is actually in our ring
if res.denom().terms(order=self.order)[0][0] != (0,)*len(self.gens):
from sympy.printing.str import sstr
raise CoercionFailed("denominator %s not allowed in %s"
% (sstr(res), self))
return res
def __contains__(self, a):
try:
a = self.convert(a)
except CoercionFailed:
return False
return a.denom().terms(order=self.order)[0][0] == (0,)*len(self.gens)
def from_FractionField(K1, a, K0):
dmf = K1.get_field().from_FractionField(a, K0)
return K1((dmf.num, dmf.den))
def to_sympy(self, a):
"""Convert ``a`` to a SymPy object. """
return (basic_from_dict(a.numer().to_sympy_dict(), *self.gens) /
basic_from_dict(a.denom().to_sympy_dict(), *self.gens))
def from_sympy(self, a):
"""Convert SymPy's expression to ``dtype``. """
p, q = a.as_numer_denom()
num, _ = dict_from_basic(p, gens=self.gens)
den, _ = dict_from_basic(q, gens=self.gens)
for k, v in num.items():
num[k] = self.dom.from_sympy(v)
for k, v in den.items():
den[k] = self.dom.from_sympy(v)
return self((num, den)).cancel()
def _vector_to_sdm(self, v, order):
"""
Turn an iterable into a sparse distributed module.
Note that the vector is multiplied by a unit first to make all entries
polynomials.
Examples
========
>>> from sympy import ilex, QQ
>>> from sympy.abc import x, y
>>> R = QQ.old_poly_ring(x, y, order=ilex)
>>> f = R.convert((x + 2*y) / (1 + x))
>>> g = R.convert(x * y)
>>> R._vector_to_sdm([f, g], ilex)
[((0, 0, 1), 2), ((0, 1, 0), 1), ((1, 1, 1), 1), ((1,
2, 1), 1)]
"""
# NOTE this is quite inefficient...
u = self.one.numer()
for x in v:
u *= x.denom()
return _vector_to_sdm_helper([x.numer()*u/x.denom() for x in v], order)
@public
def PolynomialRing(dom, *gens, **opts):
r"""
Create a generalized multivariate polynomial ring.
A generalized polynomial ring is defined by a ground field `K`, a set
of generators (typically `x_1, \ldots, x_n`) and a monomial order `<`.
The monomial order can be global, local or mixed. In any case it induces
a total ordering on the monomials, and there exists for every (non-zero)
polynomial `f \in K[x_1, \ldots, x_n]` a well-defined "leading monomial"
`LM(f) = LM(f, >)`. One can then define a multiplicative subset
`S = S_> = \{f \in K[x_1, \ldots, x_n] | LM(f) = 1\}`. The generalized
polynomial ring corresponding to the monomial order is
`R = S^{-1}K[x_1, \ldots, x_n]`.
If `>` is a so-called global order, that is `1` is the smallest monomial,
then we just have `S = K` and `R = K[x_1, \ldots, x_n]`.
Examples
========
A few examples may make this clearer.
>>> from sympy.abc import x, y
>>> from sympy import QQ
Our first ring uses global lexicographic order.
>>> R1 = QQ.old_poly_ring(x, y, order=(("lex", x, y),))
The second ring uses local lexicographic order. Note that when using a
single (non-product) order, you can just specify the name and omit the
variables:
>>> R2 = QQ.old_poly_ring(x, y, order="ilex")
The third and fourth rings use a mixed orders:
>>> o1 = (("ilex", x), ("lex", y))
>>> o2 = (("lex", x), ("ilex", y))
>>> R3 = QQ.old_poly_ring(x, y, order=o1)
>>> R4 = QQ.old_poly_ring(x, y, order=o2)
We will investigate what elements of `K(x, y)` are contained in the various
rings.
>>> L = [x, 1/x, y/(1 + x), 1/(1 + y), 1/(1 + x*y)]
>>> test = lambda R: [f in R for f in L]
The first ring is just `K[x, y]`:
>>> test(R1)
[True, False, False, False, False]
The second ring is R1 localised at the maximal ideal (x, y):
>>> test(R2)
[True, False, True, True, True]
The third ring is R1 localised at the prime ideal (x):
>>> test(R3)
[True, False, True, False, True]
Finally the fourth ring is R1 localised at `S = K[x, y] \setminus yK[y]`:
>>> test(R4)
[True, False, False, True, False]
"""
order = opts.get("order", GeneralizedPolynomialRing.default_order)
if iterable(order):
order = build_product_order(order, gens)
order = monomial_key(order)
opts['order'] = order
if order.is_global:
return GlobalPolynomialRing(dom, *gens, **opts)
else:
return GeneralizedPolynomialRing(dom, *gens, **opts)
|
ea509c88ec0c6008e55048a6cef9f8b928d2153c9aa732a1b0572b65987eaf41 | """Implementation of :class:`PolynomialRing` class. """
from sympy.polys.domains.ring import Ring
from sympy.polys.domains.compositedomain import CompositeDomain
from sympy.polys.polyerrors import CoercionFailed, GeneratorsError
from sympy.utilities import public
@public
class PolynomialRing(Ring, CompositeDomain):
"""A class for representing multivariate polynomial rings. """
is_PolynomialRing = is_Poly = True
has_assoc_Ring = True
has_assoc_Field = True
def __init__(self, domain_or_ring, symbols=None, order=None):
from sympy.polys.rings import PolyRing
if isinstance(domain_or_ring, PolyRing) and symbols is None and order is None:
ring = domain_or_ring
else:
ring = PolyRing(symbols, domain_or_ring, order)
self.ring = ring
self.dtype = ring.dtype
self.gens = ring.gens
self.ngens = ring.ngens
self.symbols = ring.symbols
self.domain = ring.domain
if symbols:
if ring.domain.is_Field and ring.domain.is_Exact and len(symbols)==1:
self.is_PID = True
# TODO: remove this
self.dom = self.domain
def new(self, element):
return self.ring.ring_new(element)
@property
def zero(self):
return self.ring.zero
@property
def one(self):
return self.ring.one
@property
def order(self):
return self.ring.order
def __str__(self):
return str(self.domain) + '[' + ','.join(map(str, self.symbols)) + ']'
def __hash__(self):
return hash((self.__class__.__name__, self.dtype.ring, self.domain, self.symbols))
def __eq__(self, other):
"""Returns `True` if two domains are equivalent. """
return isinstance(other, PolynomialRing) and \
(self.dtype.ring, self.domain, self.symbols) == \
(other.dtype.ring, other.domain, other.symbols)
def is_unit(self, a):
"""Returns ``True`` if ``a`` is a unit of ``self``"""
if not a.is_ground:
return False
K = self.domain
return K.is_unit(K.convert_from(a, self))
def canonical_unit(self, a):
u = self.domain.canonical_unit(a.LC)
return self.ring.ground_new(u)
def to_sympy(self, a):
"""Convert `a` to a SymPy object. """
return a.as_expr()
def from_sympy(self, a):
"""Convert SymPy's expression to `dtype`. """
return self.ring.from_expr(a)
def from_ZZ(K1, a, K0):
"""Convert a Python `int` object to `dtype`. """
return K1(K1.domain.convert(a, K0))
def from_ZZ_python(K1, a, K0):
"""Convert a Python `int` object to `dtype`. """
return K1(K1.domain.convert(a, K0))
def from_QQ(K1, a, K0):
"""Convert a Python `Fraction` object to `dtype`. """
return K1(K1.domain.convert(a, K0))
def from_QQ_python(K1, a, K0):
"""Convert a Python `Fraction` object to `dtype`. """
return K1(K1.domain.convert(a, K0))
def from_ZZ_gmpy(K1, a, K0):
"""Convert a GMPY `mpz` object to `dtype`. """
return K1(K1.domain.convert(a, K0))
def from_QQ_gmpy(K1, a, K0):
"""Convert a GMPY `mpq` object to `dtype`. """
return K1(K1.domain.convert(a, K0))
def from_GaussianIntegerRing(K1, a, K0):
"""Convert a `GaussianInteger` object to `dtype`. """
return K1(K1.domain.convert(a, K0))
def from_GaussianRationalField(K1, a, K0):
"""Convert a `GaussianRational` object to `dtype`. """
return K1(K1.domain.convert(a, K0))
def from_RealField(K1, a, K0):
"""Convert a mpmath `mpf` object to `dtype`. """
return K1(K1.domain.convert(a, K0))
def from_ComplexField(K1, a, K0):
"""Convert a mpmath `mpf` object to `dtype`. """
return K1(K1.domain.convert(a, K0))
def from_AlgebraicField(K1, a, K0):
"""Convert an algebraic number to ``dtype``. """
if K1.domain != K0:
a = K1.domain.convert_from(a, K0)
if a is not None:
return K1.new(a)
def from_PolynomialRing(K1, a, K0):
"""Convert a polynomial to ``dtype``. """
try:
return a.set_ring(K1.ring)
except (CoercionFailed, GeneratorsError):
return None
def from_FractionField(K1, a, K0):
"""Convert a rational function to ``dtype``. """
q, r = K0.numer(a).div(K0.denom(a))
if r.is_zero:
return K1.from_PolynomialRing(q, K0.field.ring.to_domain())
else:
return None
def from_GlobalPolynomialRing(K1, a, K0):
"""Convert from old poly ring to ``dtype``. """
if K1.symbols == K0.gens:
ad = a.to_dict()
if K1.domain != K0.domain:
ad = {m: K1.domain.convert(c) for m, c in ad.items()}
return K1(ad)
elif a.is_ground and K0.domain == K1:
return K1.convert_from(a.to_list()[0], K0.domain)
def get_field(self):
"""Returns a field associated with `self`. """
return self.ring.to_field().to_domain()
def is_positive(self, a):
"""Returns True if `LC(a)` is positive. """
return self.domain.is_positive(a.LC)
def is_negative(self, a):
"""Returns True if `LC(a)` is negative. """
return self.domain.is_negative(a.LC)
def is_nonpositive(self, a):
"""Returns True if `LC(a)` is non-positive. """
return self.domain.is_nonpositive(a.LC)
def is_nonnegative(self, a):
"""Returns True if `LC(a)` is non-negative. """
return self.domain.is_nonnegative(a.LC)
def gcdex(self, a, b):
"""Extended GCD of `a` and `b`. """
return a.gcdex(b)
def gcd(self, a, b):
"""Returns GCD of `a` and `b`. """
return a.gcd(b)
def lcm(self, a, b):
"""Returns LCM of `a` and `b`. """
return a.lcm(b)
def factorial(self, a):
"""Returns factorial of `a`. """
return self.dtype(self.domain.factorial(a))
|
5034f1e28b00604dc2b2977140d02204d37feef1a79a75ff33aa4752de9a6e17 | """Implementation of :class:`IntegerRing` class. """
from sympy.external.gmpy import MPZ, HAS_GMPY
from sympy.polys.domains.groundtypes import (
SymPyInteger,
factorial,
gcdex, gcd, lcm, sqrt,
)
from sympy.polys.domains.characteristiczero import CharacteristicZero
from sympy.polys.domains.ring import Ring
from sympy.polys.domains.simpledomain import SimpleDomain
from sympy.polys.polyerrors import CoercionFailed
from sympy.utilities import public
import math
@public
class IntegerRing(Ring, CharacteristicZero, SimpleDomain):
r"""The domain ``ZZ`` representing the integers `\mathbb{Z}`.
The :py:class:`IntegerRing` class represents the ring of integers as a
:py:class:`~.Domain` in the domain system. :py:class:`IntegerRing` is a
super class of :py:class:`PythonIntegerRing` and
:py:class:`GMPYIntegerRing` one of which will be the implementation for
:ref:`ZZ` depending on whether or not ``gmpy`` or ``gmpy2`` is installed.
See also
========
Domain
"""
rep = 'ZZ'
alias = 'ZZ'
dtype = MPZ
zero = dtype(0)
one = dtype(1)
tp = type(one)
is_IntegerRing = is_ZZ = True
is_Numerical = True
is_PID = True
has_assoc_Ring = True
has_assoc_Field = True
def __init__(self):
"""Allow instantiation of this domain. """
def to_sympy(self, a):
"""Convert ``a`` to a SymPy object. """
return SymPyInteger(int(a))
def from_sympy(self, a):
"""Convert SymPy's Integer to ``dtype``. """
if a.is_Integer:
return MPZ(a.p)
elif a.is_Float and int(a) == a:
return MPZ(int(a))
else:
raise CoercionFailed("expected an integer, got %s" % a)
def get_field(self):
r"""Return the associated field of fractions :ref:`QQ`
Returns
=======
:ref:`QQ`:
The associated field of fractions :ref:`QQ`, a
:py:class:`~.Domain` representing the rational numbers
`\mathbb{Q}`.
Examples
========
>>> from sympy import ZZ
>>> ZZ.get_field()
QQ
"""
from sympy.polys.domains import QQ
return QQ
def algebraic_field(self, *extension):
r"""Returns an algebraic field, i.e. `\mathbb{Q}(\alpha, \ldots)`.
Parameters
==========
*extension: One or more Expr.
Generators of the extension. These should be expressions that are
algebraic over `\mathbb{Q}`.
Returns
=======
:py:class:`~.AlgebraicField`
A :py:class:`~.Domain` representing the algebraic field extension.
Examples
========
>>> from sympy import ZZ, sqrt
>>> ZZ.algebraic_field(sqrt(2))
QQ<sqrt(2)>
"""
return self.get_field().algebraic_field(*extension)
def from_AlgebraicField(K1, a, K0):
"""Convert a :py:class:`~.ANP` object to :ref:`ZZ`.
See :py:meth:`~.Domain.convert`.
"""
if a.is_ground:
return K1.convert(a.LC(), K0.dom)
def log(self, a, b):
r"""logarithm of *a* to the base *b*
Parameters
==========
a: number
b: number
Returns
=======
$\\lfloor\log(a, b)\\rfloor$:
Floor of the logarithm of *a* to the base *b*
Examples
========
>>> from sympy import ZZ
>>> ZZ.log(ZZ(8), ZZ(2))
3
>>> ZZ.log(ZZ(9), ZZ(2))
3
Notes
=====
This function uses ``math.log`` which is based on ``float`` so it will
fail for large integer arguments.
"""
return self.dtype(math.log(int(a), b))
def from_FF(K1, a, K0):
"""Convert ``ModularInteger(int)`` to GMPY's ``mpz``. """
return MPZ(a.to_int())
def from_FF_python(K1, a, K0):
"""Convert ``ModularInteger(int)`` to GMPY's ``mpz``. """
return MPZ(a.to_int())
def from_ZZ(K1, a, K0):
"""Convert Python's ``int`` to GMPY's ``mpz``. """
return MPZ(a)
def from_ZZ_python(K1, a, K0):
"""Convert Python's ``int`` to GMPY's ``mpz``. """
return MPZ(a)
def from_QQ(K1, a, K0):
"""Convert Python's ``Fraction`` to GMPY's ``mpz``. """
if a.denominator == 1:
return MPZ(a.numerator)
def from_QQ_python(K1, a, K0):
"""Convert Python's ``Fraction`` to GMPY's ``mpz``. """
if a.denominator == 1:
return MPZ(a.numerator)
def from_FF_gmpy(K1, a, K0):
"""Convert ``ModularInteger(mpz)`` to GMPY's ``mpz``. """
return a.to_int()
def from_ZZ_gmpy(K1, a, K0):
"""Convert GMPY's ``mpz`` to GMPY's ``mpz``. """
return a
def from_QQ_gmpy(K1, a, K0):
"""Convert GMPY ``mpq`` to GMPY's ``mpz``. """
if a.denominator == 1:
return a.numerator
def from_RealField(K1, a, K0):
"""Convert mpmath's ``mpf`` to GMPY's ``mpz``. """
p, q = K0.to_rational(a)
if q == 1:
return MPZ(p)
def from_GaussianIntegerRing(K1, a, K0):
if a.y == 0:
return a.x
def gcdex(self, a, b):
"""Compute extended GCD of ``a`` and ``b``. """
h, s, t = gcdex(a, b)
if HAS_GMPY:
return s, t, h
else:
return h, s, t
def gcd(self, a, b):
"""Compute GCD of ``a`` and ``b``. """
return gcd(a, b)
def lcm(self, a, b):
"""Compute LCM of ``a`` and ``b``. """
return lcm(a, b)
def sqrt(self, a):
"""Compute square root of ``a``. """
return sqrt(a)
def factorial(self, a):
"""Compute factorial of ``a``. """
return factorial(a)
ZZ = IntegerRing()
|
d868de34c79b4d7c4b1c390a62a6eb3fe859a815c7a1cfd4bfb924ef12323dcc | """Implementation of :class:`ComplexField` class. """
from sympy.core.numbers import Float, I
from sympy.polys.domains.characteristiczero import CharacteristicZero
from sympy.polys.domains.field import Field
from sympy.polys.domains.mpelements import MPContext
from sympy.polys.domains.simpledomain import SimpleDomain
from sympy.polys.polyerrors import DomainError, CoercionFailed
from sympy.utilities import public
@public
class ComplexField(Field, CharacteristicZero, SimpleDomain):
"""Complex numbers up to the given precision. """
rep = 'CC'
is_ComplexField = is_CC = True
is_Exact = False
is_Numerical = True
has_assoc_Ring = False
has_assoc_Field = True
_default_precision = 53
@property
def has_default_precision(self):
return self.precision == self._default_precision
@property
def precision(self):
return self._context.prec
@property
def dps(self):
return self._context.dps
@property
def tolerance(self):
return self._context.tolerance
def __init__(self, prec=_default_precision, dps=None, tol=None):
context = MPContext(prec, dps, tol, False)
context._parent = self
self._context = context
self.dtype = context.mpc
self.zero = self.dtype(0)
self.one = self.dtype(1)
def __eq__(self, other):
return (isinstance(other, ComplexField)
and self.precision == other.precision
and self.tolerance == other.tolerance)
def __hash__(self):
return hash((self.__class__.__name__, self.dtype, self.precision, self.tolerance))
def to_sympy(self, element):
"""Convert ``element`` to SymPy number. """
return Float(element.real, self.dps) + I*Float(element.imag, self.dps)
def from_sympy(self, expr):
"""Convert SymPy's number to ``dtype``. """
number = expr.evalf(n=self.dps)
real, imag = number.as_real_imag()
if real.is_Number and imag.is_Number:
return self.dtype(real, imag)
else:
raise CoercionFailed("expected complex number, got %s" % expr)
def from_ZZ(self, element, base):
return self.dtype(element)
def from_QQ(self, element, base):
return self.dtype(int(element.numerator)) / int(element.denominator)
def from_ZZ_python(self, element, base):
return self.dtype(element)
def from_QQ_python(self, element, base):
return self.dtype(element.numerator) / element.denominator
def from_ZZ_gmpy(self, element, base):
return self.dtype(int(element))
def from_QQ_gmpy(self, element, base):
return self.dtype(int(element.numerator)) / int(element.denominator)
def from_GaussianIntegerRing(self, element, base):
return self.dtype(int(element.x), int(element.y))
def from_GaussianRationalField(self, element, base):
x = element.x
y = element.y
return (self.dtype(int(x.numerator)) / int(x.denominator) +
self.dtype(0, int(y.numerator)) / int(y.denominator))
def from_AlgebraicField(self, element, base):
return self.from_sympy(base.to_sympy(element).evalf(self.dps))
def from_RealField(self, element, base):
return self.dtype(element)
def from_ComplexField(self, element, base):
if self == base:
return element
else:
return self.dtype(element)
def get_ring(self):
"""Returns a ring associated with ``self``. """
raise DomainError("there is no ring associated with %s" % self)
def get_exact(self):
"""Returns an exact domain associated with ``self``. """
raise DomainError("there is no exact domain associated with %s" % self)
def is_negative(self, element):
"""Returns ``False`` for any ``ComplexElement``. """
return False
def is_positive(self, element):
"""Returns ``False`` for any ``ComplexElement``. """
return False
def is_nonnegative(self, element):
"""Returns ``False`` for any ``ComplexElement``. """
return False
def is_nonpositive(self, element):
"""Returns ``False`` for any ``ComplexElement``. """
return False
def gcd(self, a, b):
"""Returns GCD of ``a`` and ``b``. """
return self.one
def lcm(self, a, b):
"""Returns LCM of ``a`` and ``b``. """
return a*b
def almosteq(self, a, b, tolerance=None):
"""Check if ``a`` and ``b`` are almost equal. """
return self._context.almosteq(a, b, tolerance)
CC = ComplexField()
|
c37709713edaa8b2c79f77e01710b9fe51b0b96b0214978b7e8abefc43c3d8f6 | """Implementation of :class:`FractionField` class. """
from sympy.polys.domains.compositedomain import CompositeDomain
from sympy.polys.domains.field import Field
from sympy.polys.polyerrors import CoercionFailed, GeneratorsError
from sympy.utilities import public
@public
class FractionField(Field, CompositeDomain):
"""A class for representing multivariate rational function fields. """
is_FractionField = is_Frac = True
has_assoc_Ring = True
has_assoc_Field = True
def __init__(self, domain_or_field, symbols=None, order=None):
from sympy.polys.fields import FracField
if isinstance(domain_or_field, FracField) and symbols is None and order is None:
field = domain_or_field
else:
field = FracField(symbols, domain_or_field, order)
self.field = field
self.dtype = field.dtype
self.gens = field.gens
self.ngens = field.ngens
self.symbols = field.symbols
self.domain = field.domain
# TODO: remove this
self.dom = self.domain
def new(self, element):
return self.field.field_new(element)
@property
def zero(self):
return self.field.zero
@property
def one(self):
return self.field.one
@property
def order(self):
return self.field.order
@property
def is_Exact(self):
return self.domain.is_Exact
def get_exact(self):
return FractionField(self.domain.get_exact(), self.symbols)
def __str__(self):
return str(self.domain) + '(' + ','.join(map(str, self.symbols)) + ')'
def __hash__(self):
return hash((self.__class__.__name__, self.dtype.field, self.domain, self.symbols))
def __eq__(self, other):
"""Returns ``True`` if two domains are equivalent. """
return isinstance(other, FractionField) and \
(self.dtype.field, self.domain, self.symbols) ==\
(other.dtype.field, other.domain, other.symbols)
def to_sympy(self, a):
"""Convert ``a`` to a SymPy object. """
return a.as_expr()
def from_sympy(self, a):
"""Convert SymPy's expression to ``dtype``. """
return self.field.from_expr(a)
def from_ZZ(K1, a, K0):
"""Convert a Python ``int`` object to ``dtype``. """
return K1(K1.domain.convert(a, K0))
def from_ZZ_python(K1, a, K0):
"""Convert a Python ``int`` object to ``dtype``. """
return K1(K1.domain.convert(a, K0))
def from_QQ(K1, a, K0):
"""Convert a Python ``Fraction`` object to ``dtype``. """
return K1(K1.domain.convert(a, K0))
def from_QQ_python(K1, a, K0):
"""Convert a Python ``Fraction`` object to ``dtype``. """
return K1(K1.domain.convert(a, K0))
def from_ZZ_gmpy(K1, a, K0):
"""Convert a GMPY ``mpz`` object to ``dtype``. """
return K1(K1.domain.convert(a, K0))
def from_QQ_gmpy(K1, a, K0):
"""Convert a GMPY ``mpq`` object to ``dtype``. """
return K1(K1.domain.convert(a, K0))
def from_GaussianRationalField(K1, a, K0):
"""Convert a ``GaussianRational`` object to ``dtype``. """
return K1(K1.domain.convert(a, K0))
def from_GaussianIntegerRing(K1, a, K0):
"""Convert a ``GaussianInteger`` object to ``dtype``. """
return K1(K1.domain.convert(a, K0))
def from_RealField(K1, a, K0):
"""Convert a mpmath ``mpf`` object to ``dtype``. """
return K1(K1.domain.convert(a, K0))
def from_ComplexField(K1, a, K0):
"""Convert a mpmath ``mpf`` object to ``dtype``. """
return K1(K1.domain.convert(a, K0))
def from_AlgebraicField(K1, a, K0):
"""Convert an algebraic number to ``dtype``. """
if K1.domain != K0:
a = K1.domain.convert_from(a, K0)
if a is not None:
return K1.new(a)
def from_PolynomialRing(K1, a, K0):
"""Convert a polynomial to ``dtype``. """
try:
return K1.new(a.set_ring(K1.field.ring))
except (CoercionFailed, GeneratorsError):
# XXX: We get here if K1=ZZ(x,y) and K0=QQ[x,y]
# and the poly a in K0 has non-integer coefficients.
# It seems that K1.new can handle this but K1.new doesn't work
# when K0.domain is an algebraic field...
try:
return K1.new(a)
except (CoercionFailed, GeneratorsError):
return None
def from_FractionField(K1, a, K0):
"""Convert a rational function to ``dtype``. """
try:
return a.set_field(K1.field)
except (CoercionFailed, GeneratorsError):
return None
def get_ring(self):
"""Returns a field associated with ``self``. """
return self.field.to_ring().to_domain()
def is_positive(self, a):
"""Returns True if ``LC(a)`` is positive. """
return self.domain.is_positive(a.numer.LC)
def is_negative(self, a):
"""Returns True if ``LC(a)`` is negative. """
return self.domain.is_negative(a.numer.LC)
def is_nonpositive(self, a):
"""Returns True if ``LC(a)`` is non-positive. """
return self.domain.is_nonpositive(a.numer.LC)
def is_nonnegative(self, a):
"""Returns True if ``LC(a)`` is non-negative. """
return self.domain.is_nonnegative(a.numer.LC)
def numer(self, a):
"""Returns numerator of ``a``. """
return a.numer
def denom(self, a):
"""Returns denominator of ``a``. """
return a.denom
def factorial(self, a):
"""Returns factorial of ``a``. """
return self.dtype(self.domain.factorial(a))
|
81e47de0c7a489d9bbdb6578ed1a0a303af23505bc0cbffca5129a35f85f2742 | """Implementation of :class:`ExpressionDomain` class. """
from sympy.core import sympify, SympifyError
from sympy.polys.domains.characteristiczero import CharacteristicZero
from sympy.polys.domains.field import Field
from sympy.polys.domains.simpledomain import SimpleDomain
from sympy.polys.polyutils import PicklableWithSlots
from sympy.utilities import public
eflags = dict(deep=False, mul=True, power_exp=False, power_base=False,
basic=False, multinomial=False, log=False)
@public
class ExpressionDomain(Field, CharacteristicZero, SimpleDomain):
"""A class for arbitrary expressions. """
is_SymbolicDomain = is_EX = True
class Expression(PicklableWithSlots):
"""An arbitrary expression. """
__slots__ = ('ex',)
def __init__(self, ex):
if not isinstance(ex, self.__class__):
self.ex = sympify(ex)
else:
self.ex = ex.ex
def __repr__(f):
return 'EX(%s)' % repr(f.ex)
def __str__(f):
return 'EX(%s)' % str(f.ex)
def __hash__(self):
return hash((self.__class__.__name__, self.ex))
def as_expr(f):
return f.ex
def numer(f):
return f.__class__(f.ex.as_numer_denom()[0])
def denom(f):
return f.__class__(f.ex.as_numer_denom()[1])
def simplify(f, ex):
return f.__class__(ex.cancel().expand(**eflags))
def __abs__(f):
return f.__class__(abs(f.ex))
def __neg__(f):
return f.__class__(-f.ex)
def _to_ex(f, g):
try:
return f.__class__(g)
except SympifyError:
return None
def __add__(f, g):
g = f._to_ex(g)
if g is not None:
return f.simplify(f.ex + g.ex)
else:
return NotImplemented
def __radd__(f, g):
return f.simplify(f.__class__(g).ex + f.ex)
def __sub__(f, g):
g = f._to_ex(g)
if g is not None:
return f.simplify(f.ex - g.ex)
else:
return NotImplemented
def __rsub__(f, g):
return f.simplify(f.__class__(g).ex - f.ex)
def __mul__(f, g):
g = f._to_ex(g)
if g is not None:
return f.simplify(f.ex*g.ex)
else:
return NotImplemented
def __rmul__(f, g):
return f.simplify(f.__class__(g).ex*f.ex)
def __pow__(f, n):
n = f._to_ex(n)
if n is not None:
return f.simplify(f.ex**n.ex)
else:
return NotImplemented
def __truediv__(f, g):
g = f._to_ex(g)
if g is not None:
return f.simplify(f.ex/g.ex)
else:
return NotImplemented
def __rtruediv__(f, g):
return f.simplify(f.__class__(g).ex/f.ex)
def __eq__(f, g):
return f.ex == f.__class__(g).ex
def __ne__(f, g):
return not f == g
def __bool__(f):
return not f.ex.is_zero
def gcd(f, g):
from sympy.polys import gcd
return f.__class__(gcd(f.ex, f.__class__(g).ex))
def lcm(f, g):
from sympy.polys import lcm
return f.__class__(lcm(f.ex, f.__class__(g).ex))
dtype = Expression
zero = Expression(0)
one = Expression(1)
rep = 'EX'
has_assoc_Ring = False
has_assoc_Field = True
def __init__(self):
pass
def to_sympy(self, a):
"""Convert ``a`` to a SymPy object. """
return a.as_expr()
def from_sympy(self, a):
"""Convert SymPy's expression to ``dtype``. """
return self.dtype(a)
def from_ZZ(K1, a, K0):
"""Convert a Python ``int`` object to ``dtype``. """
return K1(K0.to_sympy(a))
def from_ZZ_python(K1, a, K0):
"""Convert a Python ``int`` object to ``dtype``. """
return K1(K0.to_sympy(a))
def from_QQ(K1, a, K0):
"""Convert a Python ``Fraction`` object to ``dtype``. """
return K1(K0.to_sympy(a))
def from_QQ_python(K1, a, K0):
"""Convert a Python ``Fraction`` object to ``dtype``. """
return K1(K0.to_sympy(a))
def from_ZZ_gmpy(K1, a, K0):
"""Convert a GMPY ``mpz`` object to ``dtype``. """
return K1(K0.to_sympy(a))
def from_QQ_gmpy(K1, a, K0):
"""Convert a GMPY ``mpq`` object to ``dtype``. """
return K1(K0.to_sympy(a))
def from_GaussianIntegerRing(K1, a, K0):
"""Convert a ``GaussianRational`` object to ``dtype``. """
return K1(K0.to_sympy(a))
def from_GaussianRationalField(K1, a, K0):
"""Convert a ``GaussianRational`` object to ``dtype``. """
return K1(K0.to_sympy(a))
def from_RealField(K1, a, K0):
"""Convert a mpmath ``mpf`` object to ``dtype``. """
return K1(K0.to_sympy(a))
def from_PolynomialRing(K1, a, K0):
"""Convert a ``DMP`` object to ``dtype``. """
return K1(K0.to_sympy(a))
def from_FractionField(K1, a, K0):
"""Convert a ``DMF`` object to ``dtype``. """
return K1(K0.to_sympy(a))
def from_ExpressionDomain(K1, a, K0):
"""Convert a ``EX`` object to ``dtype``. """
return a
def get_ring(self):
"""Returns a ring associated with ``self``. """
return self # XXX: EX is not a ring but we don't have much choice here.
def get_field(self):
"""Returns a field associated with ``self``. """
return self
def is_positive(self, a):
"""Returns True if ``a`` is positive. """
return a.ex.as_coeff_mul()[0].is_positive
def is_negative(self, a):
"""Returns True if ``a`` is negative. """
return a.ex.could_extract_minus_sign()
def is_nonpositive(self, a):
"""Returns True if ``a`` is non-positive. """
return a.ex.as_coeff_mul()[0].is_nonpositive
def is_nonnegative(self, a):
"""Returns True if ``a`` is non-negative. """
return a.ex.as_coeff_mul()[0].is_nonnegative
def numer(self, a):
"""Returns numerator of ``a``. """
return a.numer()
def denom(self, a):
"""Returns denominator of ``a``. """
return a.denom()
def gcd(self, a, b):
return self(1)
def lcm(self, a, b):
return a.lcm(b)
EX = ExpressionDomain()
|
d2f7764242e2daae27a62b38c11d703057c7185197abb2c97be90951df0e61cd | """Domains of Gaussian type."""
from sympy.core.numbers import I
from sympy.polys.polyerrors import CoercionFailed
from sympy.polys.domains.integerring import ZZ
from sympy.polys.domains.rationalfield import QQ
from sympy.polys.domains.algebraicfield import AlgebraicField
from sympy.polys.domains.domain import Domain
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 # type: Domain
_parent = None # type: Domain
__slots__ = ('x', 'y')
def __init__(self, x, y=0):
conv = self.base.convert
self.x = conv(x)
self.y = conv(y)
@classmethod
def new(cls, x, y):
"""Create a new GaussianElement of the same domain."""
return cls(x, y)
def parent(self):
"""The domain that this is an element of (ZZ_I or QQ_I)"""
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 __lt__(self, other):
if not isinstance(other, GaussianElement):
return NotImplemented
return [self.y, self.x] < [other.y, other.x]
def __pos__(self):
return self
def __neg__(self):
return self.new(-self.x, -self.y)
def __repr__(self):
return "%s(%s, %s)" % (self._parent.rep, 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.new(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.new(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.new(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.new(self.x*x - self.y*y, self.x*y + self.y*x)
else:
return NotImplemented
__rmul__ = __mul__
def __pow__(self, exp):
if exp == 0:
return self.new(1, 0)
if exp < 0:
self, exp = 1/self, -exp
if exp == 1:
return self
pow2 = self
prod = self if exp % 2 else self._parent.one
exp //= 2
while exp:
pow2 *= pow2
if exp % 2:
prod *= pow2
exp //= 2
return prod
def __bool__(self):
return bool(self.x) or bool(self.y)
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
def __rdivmod__(self, other):
try:
other = self._parent.convert(other)
except CoercionFailed:
return NotImplemented
else:
return other.__divmod__(self)
def __rtruediv__(self, other):
try:
other = QQ_I.convert(other)
except CoercionFailed:
return NotImplemented
else:
return other.__truediv__(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 GaussianInteger(GaussianElement):
"""Gaussian integer: domain element for :ref:`ZZ_I`
>>> from sympy import ZZ_I
>>> z = ZZ_I(2, 3)
>>> z
(2 + 3*I)
>>> type(z)
<class 'sympy.polys.domains.gaussiandomains.GaussianInteger'>
"""
base = ZZ
def __truediv__(self, other):
"""Return a Gaussian rational."""
return QQ_I.convert(self)/other
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|
class GaussianRational(GaussianElement):
"""Gaussian rational: domain element for :ref:`QQ_I`
>>> from sympy import QQ_I, QQ
>>> z = QQ_I(QQ(2, 3), QQ(4, 5))
>>> z
(2/3 + 4/5*I)
>>> type(z)
<class 'sympy.polys.domains.gaussiandomains.GaussianRational'>
"""
base = QQ
def __truediv__(self, other):
"""Return a Gaussian rational."""
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)
def __divmod__(self, other):
try:
other = self._parent.convert(other)
except CoercionFailed:
return NotImplemented
if not other:
raise ZeroDivisionError('{} % 0'.format(self))
else:
return self/other, QQ_I.zero
class GaussianDomain():
"""Base class for Gaussian domains."""
dom = None # type: Domain
is_Numerical = True
is_Exact = True
has_assoc_Ring = True
has_assoc_Field = True
def to_sympy(self, a):
"""Convert ``a`` to a SymPy object. """
conv = self.dom.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.dom.from_sympy(r) # may raise CoercionFailed
if not b:
return self.new(x, 0)
r, b = b.as_coeff_Mul()
y = self.dom.from_sympy(r)
if b is I:
return self.new(x, y)
else:
raise CoercionFailed("{} is not Gaussian".format(a))
def inject(self, *gens):
"""Inject generators into this domain. """
return self.poly_ring(*gens)
def canonical_unit(self, d):
unit = self.units[-d.quadrant()] # - for inverse power
return unit
def is_negative(self, element):
"""Returns ``False`` for any ``GaussianElement``. """
return False
def is_positive(self, element):
"""Returns ``False`` for any ``GaussianElement``. """
return False
def is_nonnegative(self, element):
"""Returns ``False`` for any ``GaussianElement``. """
return False
def is_nonpositive(self, element):
"""Returns ``False`` for any ``GaussianElement``. """
return False
def from_ZZ_gmpy(K1, a, K0):
"""Convert a GMPY mpz to ``self.dtype``."""
return K1(a)
def from_ZZ(K1, a, K0):
"""Convert a ZZ_python element to ``self.dtype``."""
return K1(a)
def from_ZZ_python(K1, a, K0):
"""Convert a ZZ_python element to ``self.dtype``."""
return K1(a)
def from_QQ(K1, a, K0):
"""Convert a GMPY mpq to ``self.dtype``."""
return K1(a)
def from_QQ_gmpy(K1, a, K0):
"""Convert a GMPY mpq to ``self.dtype``."""
return K1(a)
def from_QQ_python(K1, a, K0):
"""Convert a QQ_python element to ``self.dtype``."""
return K1(a)
def from_AlgebraicField(K1, a, K0):
"""Convert an element from ZZ<I> or QQ<I> to ``self.dtype``."""
if K0.ext.args[0] == I:
return K1.from_sympy(K0.to_sympy(a))
class GaussianIntegerRing(GaussianDomain, Ring):
r"""Ring of Gaussian integers ``ZZ_I``
The :ref:`ZZ_I` domain represents the `Gaussian integers`_ `\mathbb{Z}[i]`
as a :py:class:`~.Domain` in the domain system (see
:ref:`polys-domainsintro`).
By default a :py:class:`~.Poly` created from an expression with
coefficients that are combinations of integers and ``I`` (`\sqrt{-1}`)
will have the domain :ref:`ZZ_I`.
>>> from sympy import Poly, Symbol, I
>>> x = Symbol('x')
>>> p = Poly(x**2 + I)
>>> p
Poly(x**2 + I, x, domain='ZZ_I')
>>> p.domain
ZZ_I
The :ref:`ZZ_I` domain can be used to factorise polynomials that are
reducible over the Gaussian integers.
>>> from sympy import factor
>>> factor(x**2 + 1)
x**2 + 1
>>> factor(x**2 + 1, domain='ZZ_I')
(x - I)*(x + I)
The corresponding `field of fractions`_ is the domain of the Gaussian
rationals :ref:`QQ_I`. Conversely :ref:`ZZ_I` is the `ring of integers`_
of :ref:`QQ_I`.
>>> from sympy import ZZ_I, QQ_I
>>> ZZ_I.get_field()
QQ_I
>>> QQ_I.get_ring()
ZZ_I
When using the domain directly :ref:`ZZ_I` can be used as a constructor.
>>> ZZ_I(3, 4)
(3 + 4*I)
>>> ZZ_I(5)
(5 + 0*I)
The domain elements of :ref:`ZZ_I` are instances of
:py:class:`~.GaussianInteger` which support the rings operations
``+,-,*,**``.
>>> z1 = ZZ_I(5, 1)
>>> z2 = ZZ_I(2, 3)
>>> z1
(5 + 1*I)
>>> z2
(2 + 3*I)
>>> z1 + z2
(7 + 4*I)
>>> z1 * z2
(7 + 17*I)
>>> z1 ** 2
(24 + 10*I)
Both floor (``//``) and modulo (``%``) division work with
:py:class:`~.GaussianInteger` (see the :py:meth:`~.Domain.div` method).
>>> z3, z4 = ZZ_I(5), ZZ_I(1, 3)
>>> z3 // z4 # floor division
(1 + -1*I)
>>> z3 % z4 # modulo division (remainder)
(1 + -2*I)
>>> (z3//z4)*z4 + z3%z4 == z3
True
True division (``/``) in :ref:`ZZ_I` gives an element of :ref:`QQ_I`. The
:py:meth:`~.Domain.exquo` method can be used to divide in :ref:`ZZ_I` when
exact division is possible.
>>> z1 / z2
(1 + -1*I)
>>> ZZ_I.exquo(z1, z2)
(1 + -1*I)
>>> z3 / z4
(1/2 + -3/2*I)
>>> ZZ_I.exquo(z3, z4)
Traceback (most recent call last):
...
ExactQuotientFailed: (1 + 3*I) does not divide (5 + 0*I) in ZZ_I
The :py:meth:`~.Domain.gcd` method can be used to compute the `gcd`_ of any
two elements.
>>> ZZ_I.gcd(ZZ_I(10), ZZ_I(2))
(2 + 0*I)
>>> ZZ_I.gcd(ZZ_I(5), ZZ_I(2, 1))
(2 + 1*I)
.. _Gaussian integers: https://en.wikipedia.org/wiki/Gaussian_integer
.. _gcd: https://en.wikipedia.org/wiki/Greatest_common_divisor
"""
dom = ZZ
dtype = GaussianInteger
zero = dtype(ZZ(0), ZZ(0))
one = dtype(ZZ(1), ZZ(0))
imag_unit = dtype(ZZ(0), ZZ(1))
units = (one, imag_unit, -one, -imag_unit) # powers of i
rep = 'ZZ_I'
is_GaussianRing = True
is_ZZ_I = True
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.canonical_unit(d)
d *= unit
args = tuple(a*unit for a in args)
return (d,) + args if args else d
def gcd(self, a, b):
"""Greatest common divisor of a and b over ZZ_I."""
while b:
a, b = b, a % b
return self.normalize(a)
def lcm(self, a, b):
"""Least common multiple of a and b over ZZ_I."""
return (a * b) // self.gcd(a, b)
def from_GaussianIntegerRing(K1, a, K0):
"""Convert a ZZ_I element to ZZ_I."""
return a
def from_GaussianRationalField(K1, a, K0):
"""Convert a QQ_I element to ZZ_I."""
return K1.new(ZZ.convert(a.x), ZZ.convert(a.y))
ZZ_I = GaussianInteger._parent = GaussianIntegerRing()
class GaussianRationalField(GaussianDomain, Field):
r"""Field of Gaussian rationals ``QQ_I``
The :ref:`QQ_I` domain represents the `Gaussian rationals`_ `\mathbb{Q}(i)`
as a :py:class:`~.Domain` in the domain system (see
:ref:`polys-domainsintro`).
By default a :py:class:`~.Poly` created from an expression with
coefficients that are combinations of rationals and ``I`` (`\sqrt{-1}`)
will have the domain :ref:`QQ_I`.
>>> from sympy import Poly, Symbol, I
>>> x = Symbol('x')
>>> p = Poly(x**2 + I/2)
>>> p
Poly(x**2 + I/2, x, domain='QQ_I')
>>> p.domain
QQ_I
The polys option ``gaussian=True`` can be used to specify that the domain
should be :ref:`QQ_I` even if the coefficients do not contain ``I`` or are
all integers.
>>> Poly(x**2)
Poly(x**2, x, domain='ZZ')
>>> Poly(x**2 + I)
Poly(x**2 + I, x, domain='ZZ_I')
>>> Poly(x**2/2)
Poly(1/2*x**2, x, domain='QQ')
>>> Poly(x**2, gaussian=True)
Poly(x**2, x, domain='QQ_I')
>>> Poly(x**2 + I, gaussian=True)
Poly(x**2 + I, x, domain='QQ_I')
>>> Poly(x**2/2, gaussian=True)
Poly(1/2*x**2, x, domain='QQ_I')
The :ref:`QQ_I` domain can be used to factorise polynomials that are
reducible over the Gaussian rationals.
>>> from sympy import factor, QQ_I
>>> factor(x**2/4 + 1)
(x**2 + 4)/4
>>> factor(x**2/4 + 1, domain='QQ_I')
(x - 2*I)*(x + 2*I)/4
>>> factor(x**2/4 + 1, domain=QQ_I)
(x - 2*I)*(x + 2*I)/4
It is also possible to specify the :ref:`QQ_I` domain explicitly with
polys functions like :py:func:`~.apart`.
>>> from sympy import apart
>>> apart(1/(1 + x**2))
1/(x**2 + 1)
>>> apart(1/(1 + x**2), domain=QQ_I)
I/(2*(x + I)) - I/(2*(x - I))
The corresponding `ring of integers`_ is the domain of the Gaussian
integers :ref:`ZZ_I`. Conversely :ref:`QQ_I` is the `field of fractions`_
of :ref:`ZZ_I`.
>>> from sympy import ZZ_I, QQ_I, QQ
>>> ZZ_I.get_field()
QQ_I
>>> QQ_I.get_ring()
ZZ_I
When using the domain directly :ref:`QQ_I` can be used as a constructor.
>>> QQ_I(3, 4)
(3 + 4*I)
>>> QQ_I(5)
(5 + 0*I)
>>> QQ_I(QQ(2, 3), QQ(4, 5))
(2/3 + 4/5*I)
The domain elements of :ref:`QQ_I` are instances of
:py:class:`~.GaussianRational` which support the field operations
``+,-,*,**,/``.
>>> z1 = QQ_I(5, 1)
>>> z2 = QQ_I(2, QQ(1, 2))
>>> z1
(5 + 1*I)
>>> z2
(2 + 1/2*I)
>>> z1 + z2
(7 + 3/2*I)
>>> z1 * z2
(19/2 + 9/2*I)
>>> z2 ** 2
(15/4 + 2*I)
True division (``/``) in :ref:`QQ_I` gives an element of :ref:`QQ_I` and
is always exact.
>>> z1 / z2
(42/17 + -2/17*I)
>>> QQ_I.exquo(z1, z2)
(42/17 + -2/17*I)
>>> z1 == (z1/z2)*z2
True
Both floor (``//``) and modulo (``%``) division can be used with
:py:class:`~.GaussianRational` (see :py:meth:`~.Domain.div`)
but division is always exact so there is no remainder.
>>> z1 // z2
(42/17 + -2/17*I)
>>> z1 % z2
(0 + 0*I)
>>> QQ_I.div(z1, z2)
((42/17 + -2/17*I), (0 + 0*I))
>>> (z1//z2)*z2 + z1%z2 == z1
True
.. _Gaussian rationals: https://en.wikipedia.org/wiki/Gaussian_rational
"""
dom = QQ
dtype = GaussianRational
zero = dtype(QQ(0), QQ(0))
one = dtype(QQ(1), QQ(0))
imag_unit = dtype(QQ(0), QQ(1))
units = (one, imag_unit, -one, -imag_unit) # powers of i
rep = 'QQ_I'
is_GaussianField = True
is_QQ_I = True
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
def as_AlgebraicField(self):
"""Get equivalent domain as an ``AlgebraicField``. """
return AlgebraicField(self.dom, I)
def numer(self, a):
"""Get the numerator of ``a``."""
ZZ_I = self.get_ring()
return ZZ_I.convert(a * self.denom(a))
def denom(self, a):
"""Get the denominator of ``a``."""
ZZ = self.dom.get_ring()
QQ = self.dom
ZZ_I = self.get_ring()
denom_ZZ = ZZ.lcm(QQ.denom(a.x), QQ.denom(a.y))
return ZZ_I(denom_ZZ, ZZ.zero)
def from_GaussianIntegerRing(K1, a, K0):
"""Convert a ZZ_I element to QQ_I."""
return K1.new(a.x, a.y)
def from_GaussianRationalField(K1, a, K0):
"""Convert a QQ_I element to QQ_I."""
return a
QQ_I = GaussianRational._parent = GaussianRationalField()
|
7580b76333fa009aa668ca30d2adbdc5b847dd78e8b8a1639bb222f47d5a10d7 | """Ground types for various mathematical domains in SymPy. """
import builtins
from sympy.external.gmpy import HAS_GMPY, factorial, sqrt
PythonInteger = builtins.int
PythonReal = builtins.float
PythonComplex = builtins.complex
from .pythonrational import PythonRational
from sympy.core.numbers import (
igcdex as python_gcdex,
igcd2 as python_gcd,
ilcm as python_lcm,
)
from sympy import (
Float as SymPyReal,
Integer as SymPyInteger,
Rational as SymPyRational,
)
if HAS_GMPY == 2:
from gmpy2 import (
mpz as GMPYInteger,
mpq as GMPYRational,
numer as gmpy_numer,
denom as gmpy_denom,
gcdext as gmpy_gcdex,
gcd as gmpy_gcd,
lcm as gmpy_lcm,
qdiv as gmpy_qdiv,
)
gcdex = gmpy_gcdex
gcd = gmpy_gcd
lcm = gmpy_lcm
else:
class _GMPYInteger:
def __init__(self, obj):
pass
class _GMPYRational:
def __init__(self, obj):
pass
GMPYInteger = _GMPYInteger
GMPYRational = _GMPYRational
gmpy_numer = None
gmpy_denom = None
gmpy_gcdex = None
gmpy_gcd = None
gmpy_lcm = None
gmpy_qdiv = None
gcdex = python_gcdex
gcd = python_gcd
lcm = python_lcm
__all__ = [
'PythonInteger', 'PythonReal', 'PythonComplex',
'PythonRational',
'python_gcdex', 'python_gcd', 'python_lcm',
'SymPyReal', 'SymPyInteger', 'SymPyRational',
'GMPYInteger', 'GMPYRational', 'gmpy_numer',
'gmpy_denom', 'gmpy_gcdex', 'gmpy_gcd', 'gmpy_lcm',
'gmpy_qdiv',
'factorial', 'sqrt',
'GMPYInteger', 'GMPYRational',
]
|
53a3e1d7fe44cf96d737825c1764ecab51540c4c18b06c749d44fe8190ef3b95 | """Implementation of :class:`GMPYIntegerRing` class. """
from sympy.polys.domains.groundtypes import (
GMPYInteger, SymPyInteger,
factorial as gmpy_factorial,
gmpy_gcdex, gmpy_gcd, gmpy_lcm, sqrt as gmpy_sqrt,
)
from sympy.polys.domains.integerring import IntegerRing
from sympy.polys.polyerrors import CoercionFailed
from sympy.utilities import public
@public
class GMPYIntegerRing(IntegerRing):
"""Integer ring based on GMPY's ``mpz`` type.
This will be the implementation of :ref:`ZZ` if ``gmpy`` or ``gmpy2`` is
installed. Elements will be of type ``gmpy.mpz``.
"""
dtype = GMPYInteger
zero = dtype(0)
one = dtype(1)
tp = type(one)
alias = 'ZZ_gmpy'
def __init__(self):
"""Allow instantiation of this domain. """
def to_sympy(self, a):
"""Convert ``a`` to a SymPy object. """
return SymPyInteger(int(a))
def from_sympy(self, a):
"""Convert SymPy's Integer to ``dtype``. """
if a.is_Integer:
return GMPYInteger(a.p)
elif a.is_Float and int(a) == a:
return GMPYInteger(int(a))
else:
raise CoercionFailed("expected an integer, got %s" % a)
def from_FF_python(K1, a, K0):
"""Convert ``ModularInteger(int)`` to GMPY's ``mpz``. """
return GMPYInteger(a.to_int())
def from_ZZ_python(K1, a, K0):
"""Convert Python's ``int`` to GMPY's ``mpz``. """
return GMPYInteger(a)
def from_QQ(K1, a, K0):
"""Convert Python's ``Fraction`` to GMPY's ``mpz``. """
if a.denominator == 1:
return GMPYInteger(a.numerator)
def from_QQ_python(K1, a, K0):
"""Convert Python's ``Fraction`` to GMPY's ``mpz``. """
if a.denominator == 1:
return GMPYInteger(a.numerator)
def from_FF_gmpy(K1, a, K0):
"""Convert ``ModularInteger(mpz)`` to GMPY's ``mpz``. """
return a.to_int()
def from_ZZ_gmpy(K1, a, K0):
"""Convert GMPY's ``mpz`` to GMPY's ``mpz``. """
return a
def from_QQ_gmpy(K1, a, K0):
"""Convert GMPY ``mpq`` to GMPY's ``mpz``. """
if a.denominator == 1:
return a.numerator
def from_RealField(K1, a, K0):
"""Convert mpmath's ``mpf`` to GMPY's ``mpz``. """
p, q = K0.to_rational(a)
if q == 1:
return GMPYInteger(p)
def from_GaussianIntegerRing(K1, a, K0):
if a.y == 0:
return a.x
def gcdex(self, a, b):
"""Compute extended GCD of ``a`` and ``b``. """
h, s, t = gmpy_gcdex(a, b)
return s, t, h
def gcd(self, a, b):
"""Compute GCD of ``a`` and ``b``. """
return gmpy_gcd(a, b)
def lcm(self, a, b):
"""Compute LCM of ``a`` and ``b``. """
return gmpy_lcm(a, b)
def sqrt(self, a):
"""Compute square root of ``a``. """
return gmpy_sqrt(a)
def factorial(self, a):
"""Compute factorial of ``a``. """
return gmpy_factorial(a)
|
aa9b463dddb40f744cecae51473098df801cf7a775bd4a54b691f7ae6692efcf | """Implementation of :class:`Domain` class. """
from typing import Any, Optional, Type
from sympy.core import Basic, sympify
from sympy.core.compatibility import HAS_GMPY, is_sequence, ordered
from sympy.core.decorators import deprecated
from sympy.polys.domains.domainelement import DomainElement
from sympy.polys.orderings import lex
from sympy.polys.polyerrors import UnificationFailed, CoercionFailed, DomainError
from sympy.polys.polyutils import _unify_gens, _not_a_coeff
from sympy.utilities import default_sort_key, public
@public
class Domain:
"""Superclass for all domains in the polys domains system.
See :ref:`polys-domainsintro` for an introductory explanation of the
domains system.
The :py:class:`~.Domain` class is an abstract base class for all of the
concrete domain types. There are many different :py:class:`~.Domain`
subclasses each of which has an associated ``dtype`` which is a class
representing the elements of the domain. The coefficients of a
:py:class:`~.Poly` are elements of a domain which must be a subclass of
:py:class:`~.Domain`.
Examples
========
The most common example domains are the integers :ref:`ZZ` and the
rationals :ref:`QQ`.
>>> from sympy import Poly, symbols, Domain
>>> x, y = symbols('x, y')
>>> p = Poly(x**2 + y)
>>> p
Poly(x**2 + y, x, y, domain='ZZ')
>>> p.domain
ZZ
>>> isinstance(p.domain, Domain)
True
>>> Poly(x**2 + y/2)
Poly(x**2 + 1/2*y, x, y, domain='QQ')
The domains can be used directly in which case the domain object e.g.
(:ref:`ZZ` or :ref:`QQ`) can be used as a constructor for elements of
``dtype``.
>>> from sympy import ZZ, QQ
>>> ZZ(2)
2
>>> ZZ.dtype # doctest: +SKIP
<class 'int'>
>>> type(ZZ(2)) # doctest: +SKIP
<class 'int'>
>>> QQ(1, 2)
1/2
>>> type(QQ(1, 2)) # doctest: +SKIP
<class 'sympy.polys.domains.pythonrational.PythonRational'>
The corresponding domain elements can be used with the arithmetic
operations ``+,-,*,**`` and depending on the domain some combination of
``/,//,%`` might be usable. For example in :ref:`ZZ` both ``//`` (floor
division) and ``%`` (modulo division) can be used but ``/`` (true
division) can not. Since :ref:`QQ` is a :py:class:`~.Field` its elements
can be used with ``/`` but ``//`` and ``%`` should not be used. Some
domains have a :py:meth:`~.Domain.gcd` method.
>>> ZZ(2) + ZZ(3)
5
>>> ZZ(5) // ZZ(2)
2
>>> ZZ(5) % ZZ(2)
1
>>> QQ(1, 2) / QQ(2, 3)
3/4
>>> ZZ.gcd(ZZ(4), ZZ(2))
2
>>> QQ.gcd(QQ(2,7), QQ(5,3))
1/21
>>> ZZ.is_Field
False
>>> QQ.is_Field
True
There are also many other domains including:
1. :ref:`GF(p)` for finite fields of prime order.
2. :ref:`RR` for real (floating point) numbers.
3. :ref:`CC` for complex (floating point) numbers.
4. :ref:`QQ(a)` for algebraic number fields.
5. :ref:`K[x]` for polynomial rings.
6. :ref:`K(x)` for rational function fields.
7. :ref:`EX` for arbitrary expressions.
Each domain is represented by a domain object and also an implementation
class (``dtype``) for the elements of the domain. For example the
:ref:`K[x]` domains are represented by a domain object which is an
instance of :py:class:`~.PolynomialRing` and the elements are always
instances of :py:class:`~.PolyElement`. The implementation class
represents particular types of mathematical expressions in a way that is
more efficient than a normal SymPy expression which is of type
:py:class:`~.Expr`. The domain methods :py:meth:`~.Domain.from_sympy` and
:py:meth:`~.Domain.to_sympy` are used to convert from :py:class:`~.Expr`
to a domain element and vice versa.
>>> from sympy import Symbol, ZZ, Expr
>>> x = Symbol('x')
>>> K = ZZ[x] # polynomial ring domain
>>> K
ZZ[x]
>>> type(K) # class of the domain
<class 'sympy.polys.domains.polynomialring.PolynomialRing'>
>>> K.dtype # class of the elements
<class 'sympy.polys.rings.PolyElement'>
>>> p_expr = x**2 + 1 # Expr
>>> p_expr
x**2 + 1
>>> type(p_expr)
<class 'sympy.core.add.Add'>
>>> isinstance(p_expr, Expr)
True
>>> p_domain = K.from_sympy(p_expr)
>>> p_domain # domain element
x**2 + 1
>>> type(p_domain)
<class 'sympy.polys.rings.PolyElement'>
>>> K.to_sympy(p_domain) == p_expr
True
The :py:meth:`~.Domain.convert_from` method is used to convert domain
elements from one domain to another.
>>> from sympy import ZZ, QQ
>>> ez = ZZ(2)
>>> eq = QQ.convert_from(ez, ZZ)
>>> type(ez) # doctest: +SKIP
<class 'int'>
>>> type(eq) # doctest: +SKIP
<class 'sympy.polys.domains.pythonrational.PythonRational'>
Elements from different domains should not be mixed in arithmetic or other
operations: they should be converted to a common domain first. The domain
method :py:meth:`~.Domain.unify` is used to find a domain that can
represent all the elements of two given domains.
>>> from sympy import ZZ, QQ, symbols
>>> x, y = symbols('x, y')
>>> ZZ.unify(QQ)
QQ
>>> ZZ[x].unify(QQ)
QQ[x]
>>> ZZ[x].unify(QQ[y])
QQ[x,y]
If a domain is a :py:class:`~.Ring` then is might have an associated
:py:class:`~.Field` and vice versa. The :py:meth:`~.Domain.get_field` and
:py:meth:`~.Domain.get_ring` methods will find or create the associated
domain.
>>> from sympy import ZZ, QQ, Symbol
>>> x = Symbol('x')
>>> ZZ.has_assoc_Field
True
>>> ZZ.get_field()
QQ
>>> QQ.has_assoc_Ring
True
>>> QQ.get_ring()
ZZ
>>> K = QQ[x]
>>> K
QQ[x]
>>> K.get_field()
QQ(x)
See also
========
DomainElement: abstract base class for domain elements
construct_domain: construct a minimal domain for some expressions
"""
dtype = None # type: Optional[Type]
"""The type (class) of the elements of this :py:class:`~.Domain`:
>>> from sympy import ZZ, QQ, Symbol
>>> ZZ.dtype
<class 'int'>
>>> z = ZZ(2)
>>> z
2
>>> type(z)
<class 'int'>
>>> type(z) == ZZ.dtype
True
Every domain has an associated **dtype** ("datatype") which is the
class of the associated domain elements.
See also
========
of_type
"""
zero = None # type: Optional[Any]
"""The zero element of the :py:class:`~.Domain`:
>>> from sympy import QQ
>>> QQ.zero
0
>>> QQ.of_type(QQ.zero)
True
See also
========
of_type
one
"""
one = None # type: Optional[Any]
"""The one element of the :py:class:`~.Domain`:
>>> from sympy import QQ
>>> QQ.one
1
>>> QQ.of_type(QQ.one)
True
See also
========
of_type
zero
"""
is_Ring = False
"""Boolean flag indicating if the domain is a :py:class:`~.Ring`.
>>> from sympy import ZZ
>>> ZZ.is_Ring
True
Basically every :py:class:`~.Domain` represents a ring so this flag is
not that useful.
See also
========
is_PID
is_Field
get_ring
has_assoc_Ring
"""
is_Field = False
"""Boolean flag indicating if the domain is a :py:class:`~.Field`.
>>> from sympy import ZZ, QQ
>>> ZZ.is_Field
False
>>> QQ.is_Field
True
See also
========
is_PID
is_Ring
get_field
has_assoc_Field
"""
has_assoc_Ring = False
"""Boolean flag indicating if the domain has an associated
:py:class:`~.Ring`.
>>> from sympy import QQ
>>> QQ.has_assoc_Ring
True
>>> QQ.get_ring()
ZZ
See also
========
is_Field
get_ring
"""
has_assoc_Field = False
"""Boolean flag indicating if the domain has an associated
:py:class:`~.Field`.
>>> from sympy import ZZ
>>> ZZ.has_assoc_Field
True
>>> ZZ.get_field()
QQ
See also
========
is_Field
get_field
"""
is_FiniteField = is_FF = False
is_IntegerRing = is_ZZ = False
is_RationalField = is_QQ = False
is_GaussianRing = is_ZZ_I = False
is_GaussianField = is_QQ_I = False
is_RealField = is_RR = False
is_ComplexField = is_CC = False
is_AlgebraicField = is_Algebraic = False
is_PolynomialRing = is_Poly = False
is_FractionField = is_Frac = False
is_SymbolicDomain = is_EX = False
is_FiniteExtension = False
is_Exact = True
is_Numerical = False
is_Simple = False
is_Composite = False
is_PID = False
"""Boolean flag indicating if the domain is a `principal ideal domain`_.
>>> from sympy import ZZ
>>> ZZ.has_assoc_Field
True
>>> ZZ.get_field()
QQ
.. _principal ideal domain: https://en.wikipedia.org/wiki/Principal_ideal_domain
See also
========
is_Field
get_field
"""
has_CharacteristicZero = False
rep = None # type: Optional[str]
alias = None # type: Optional[str]
@property # type: ignore
@deprecated(useinstead="is_Field", issue=12723, deprecated_since_version="1.1")
def has_Field(self):
return self.is_Field
@property # type: ignore
@deprecated(useinstead="is_Ring", issue=12723, deprecated_since_version="1.1")
def has_Ring(self):
return self.is_Ring
def __init__(self):
raise NotImplementedError
def __str__(self):
return self.rep
def __repr__(self):
return str(self)
def __hash__(self):
return hash((self.__class__.__name__, self.dtype))
def new(self, *args):
return self.dtype(*args)
@property
def tp(self):
"""Alias for :py:attr:`~.Domain.dtype`"""
return self.dtype
def __call__(self, *args):
"""Construct an element of ``self`` domain from ``args``. """
return self.new(*args)
def normal(self, *args):
return self.dtype(*args)
def convert_from(self, element, base):
"""Convert ``element`` to ``self.dtype`` given the base domain. """
if base.alias is not None:
method = "from_" + base.alias
else:
method = "from_" + base.__class__.__name__
_convert = getattr(self, method)
if _convert is not None:
result = _convert(element, base)
if result is not None:
return result
raise CoercionFailed("can't convert %s of type %s from %s to %s" % (element, type(element), base, self))
def convert(self, element, base=None):
"""Convert ``element`` to ``self.dtype``. """
if _not_a_coeff(element):
raise CoercionFailed('%s is not in any domain' % element)
if base is not None:
return self.convert_from(element, base)
if self.of_type(element):
return element
from sympy.polys.domains import ZZ, QQ, RealField, ComplexField
if ZZ.of_type(element):
return self.convert_from(element, ZZ)
if isinstance(element, int):
return self.convert_from(ZZ(element), ZZ)
if HAS_GMPY:
integers = ZZ
if isinstance(element, integers.tp):
return self.convert_from(element, integers)
rationals = QQ
if isinstance(element, rationals.tp):
return self.convert_from(element, rationals)
if isinstance(element, float):
parent = RealField(tol=False)
return self.convert_from(parent(element), parent)
if isinstance(element, complex):
parent = ComplexField(tol=False)
return self.convert_from(parent(element), parent)
if isinstance(element, DomainElement):
return self.convert_from(element, element.parent())
# TODO: implement this in from_ methods
if self.is_Numerical and getattr(element, 'is_ground', False):
return self.convert(element.LC())
if isinstance(element, Basic):
try:
return self.from_sympy(element)
except (TypeError, ValueError):
pass
else: # TODO: remove this branch
if not is_sequence(element):
try:
element = sympify(element, strict=True)
if isinstance(element, Basic):
return self.from_sympy(element)
except (TypeError, ValueError):
pass
raise CoercionFailed("can't convert %s of type %s to %s" % (element, type(element), self))
def of_type(self, element):
"""Check if ``a`` is of type ``dtype``. """
return isinstance(element, self.tp) # XXX: this isn't correct, e.g. PolyElement
def __contains__(self, a):
"""Check if ``a`` belongs to this domain. """
try:
if _not_a_coeff(a):
raise CoercionFailed
self.convert(a) # this might raise, too
except CoercionFailed:
return False
return True
def to_sympy(self, a):
"""Convert domain element *a* to a SymPy expression (Expr).
Explanation
===========
Convert a :py:class:`~.Domain` element *a* to :py:class:`~.Expr`. Most
public SymPy functions work with objects of type :py:class:`~.Expr`.
The elements of a :py:class:`~.Domain` have a different internal
representation. It is not possible to mix domain elements with
:py:class:`~.Expr` so each domain has :py:meth:`~.Domain.to_sympy` and
:py:meth:`~.Domain.from_sympy` methods to convert its domain elements
to and from :py:class:`~.Expr`.
Parameters
==========
a: domain element
An element of this :py:class:`~.Domain`.
Returns
=======
expr: Expr
A normal sympy expression of type :py:class:`~.Expr`.
Examples
========
Construct an element of the :ref:`QQ` domain and then convert it to
:py:class:`~.Expr`.
>>> from sympy import QQ, Expr
>>> q_domain = QQ(2)
>>> q_domain
2
>>> q_expr = QQ.to_sympy(q_domain)
>>> q_expr
2
Although the printed forms look similar these objects are not of the
same type.
>>> isinstance(q_domain, Expr)
False
>>> isinstance(q_expr, Expr)
True
Construct an element of :ref:`K[x]` and convert to
:py:class:`~.Expr`.
>>> from sympy import Symbol
>>> x = Symbol('x')
>>> K = QQ[x]
>>> x_domain = K.gens[0] # generator x as a domain element
>>> p_domain = x_domain**2/3 + 1
>>> p_domain
1/3*x**2 + 1
>>> p_expr = K.to_sympy(p_domain)
>>> p_expr
x**2/3 + 1
The :py:meth:`~.Domain.from_sympy` method is used for the opposite
conversion from a normal SymPy expression to a domain element.
>>> p_domain == p_expr
False
>>> K.from_sympy(p_expr) == p_domain
True
>>> K.to_sympy(p_domain) == p_expr
True
>>> K.from_sympy(K.to_sympy(p_domain)) == p_domain
True
>>> K.to_sympy(K.from_sympy(p_expr)) == p_expr
True
The :py:meth:`~.Domain.from_sympy` method makes it easier to construct
domain elements interactively.
>>> from sympy import Symbol
>>> x = Symbol('x')
>>> K = QQ[x]
>>> K.from_sympy(x**2/3 + 1)
1/3*x**2 + 1
See also
========
from_sympy
convert_from
"""
raise NotImplementedError
def from_sympy(self, a):
"""Convert a SymPy expression to an element of this domain.
Explanation
===========
See :py:meth:`~.Domain.to_sympy` for explanation and examples.
Parameters
==========
expr: Expr
A normal sympy expression of type :py:class:`~.Expr`.
Returns
=======
a: domain element
An element of this :py:class:`~.Domain`.
See also
========
to_sympy
convert_from
"""
raise NotImplementedError
def from_FF(K1, a, K0):
"""Convert ``ModularInteger(int)`` to ``dtype``. """
return None
def from_FF_python(K1, a, K0):
"""Convert ``ModularInteger(int)`` to ``dtype``. """
return None
def from_ZZ_python(K1, a, K0):
"""Convert a Python ``int`` object to ``dtype``. """
return None
def from_QQ_python(K1, a, K0):
"""Convert a Python ``Fraction`` object to ``dtype``. """
return None
def from_FF_gmpy(K1, a, K0):
"""Convert ``ModularInteger(mpz)`` to ``dtype``. """
return None
def from_ZZ_gmpy(K1, a, K0):
"""Convert a GMPY ``mpz`` object to ``dtype``. """
return None
def from_QQ_gmpy(K1, a, K0):
"""Convert a GMPY ``mpq`` object to ``dtype``. """
return None
def from_RealField(K1, a, K0):
"""Convert a real element object to ``dtype``. """
return None
def from_ComplexField(K1, a, K0):
"""Convert a complex element to ``dtype``. """
return None
def from_AlgebraicField(K1, a, K0):
"""Convert an algebraic number to ``dtype``. """
return None
def from_PolynomialRing(K1, a, K0):
"""Convert a polynomial to ``dtype``. """
if a.is_ground:
return K1.convert(a.LC, K0.dom)
def from_FractionField(K1, a, K0):
"""Convert a rational function to ``dtype``. """
return None
def from_MonogenicFiniteExtension(K1, a, K0):
"""Convert an ``ExtensionElement`` to ``dtype``. """
return K1.convert_from(a.rep, K0.ring)
def from_ExpressionDomain(K1, a, K0):
"""Convert a ``EX`` object to ``dtype``. """
return K1.from_sympy(a.ex)
def from_GlobalPolynomialRing(K1, a, K0):
"""Convert a polynomial to ``dtype``. """
if a.degree() <= 0:
return K1.convert(a.LC(), K0.dom)
def from_GeneralizedPolynomialRing(K1, a, K0):
return K1.from_FractionField(a, K0)
def unify_with_symbols(K0, K1, symbols):
if (K0.is_Composite and (set(K0.symbols) & set(symbols))) or (K1.is_Composite and (set(K1.symbols) & set(symbols))):
raise UnificationFailed("can't unify %s with %s, given %s generators" % (K0, K1, tuple(symbols)))
return K0.unify(K1)
def unify(K0, K1, symbols=None):
"""
Construct a minimal domain that contains elements of ``K0`` and ``K1``.
Known domains (from smallest to largest):
- ``GF(p)``
- ``ZZ``
- ``QQ``
- ``RR(prec, tol)``
- ``CC(prec, tol)``
- ``ALG(a, b, c)``
- ``K[x, y, z]``
- ``K(x, y, z)``
- ``EX``
"""
if symbols is not None:
return K0.unify_with_symbols(K1, symbols)
if K0 == K1:
return K0
if K0.is_EX:
return K0
if K1.is_EX:
return K1
if K0.is_FiniteExtension or K1.is_FiniteExtension:
if K1.is_FiniteExtension:
K0, K1 = K1, K0
if K1.is_FiniteExtension:
# Unifying two extensions.
# Try to ensure that K0.unify(K1) == K1.unify(K0)
if list(ordered([K0.modulus, K1.modulus]))[1] == K0.modulus:
K0, K1 = K1, K0
return K1.set_domain(K0)
else:
# Drop the generator from other and unify with the base domain
K1 = K1.drop(K0.symbol)
K1 = K0.domain.unify(K1)
return K0.set_domain(K1)
if K0.is_Composite or K1.is_Composite:
K0_ground = K0.dom if K0.is_Composite else K0
K1_ground = K1.dom if K1.is_Composite else K1
K0_symbols = K0.symbols if K0.is_Composite else ()
K1_symbols = K1.symbols if K1.is_Composite else ()
domain = K0_ground.unify(K1_ground)
symbols = _unify_gens(K0_symbols, K1_symbols)
order = K0.order if K0.is_Composite else K1.order
if ((K0.is_FractionField and K1.is_PolynomialRing or
K1.is_FractionField and K0.is_PolynomialRing) and
(not K0_ground.is_Field or not K1_ground.is_Field) and domain.is_Field
and domain.has_assoc_Ring):
domain = domain.get_ring()
if K0.is_Composite and (not K1.is_Composite or K0.is_FractionField or K1.is_PolynomialRing):
cls = K0.__class__
else:
cls = K1.__class__
from sympy.polys.domains.old_polynomialring import GlobalPolynomialRing
if cls == GlobalPolynomialRing:
return cls(domain, symbols)
return cls(domain, symbols, order)
def mkinexact(cls, K0, K1):
prec = max(K0.precision, K1.precision)
tol = max(K0.tolerance, K1.tolerance)
return cls(prec=prec, tol=tol)
if K1.is_ComplexField:
K0, K1 = K1, K0
if K0.is_ComplexField:
if K1.is_ComplexField or K1.is_RealField:
return mkinexact(K0.__class__, K0, K1)
else:
return K0
if K1.is_RealField:
K0, K1 = K1, K0
if K0.is_RealField:
if K1.is_RealField:
return mkinexact(K0.__class__, K0, K1)
elif K1.is_GaussianRing or K1.is_GaussianField:
from sympy.polys.domains.complexfield import ComplexField
return ComplexField(prec=K0.precision, tol=K0.tolerance)
else:
return K0
if K1.is_AlgebraicField:
K0, K1 = K1, K0
if K0.is_AlgebraicField:
if K1.is_GaussianRing:
K1 = K1.get_field()
if K1.is_GaussianField:
K1 = K1.as_AlgebraicField()
if K1.is_AlgebraicField:
return K0.__class__(K0.dom.unify(K1.dom), *_unify_gens(K0.orig_ext, K1.orig_ext))
else:
return K0
if K0.is_GaussianField:
return K0
if K1.is_GaussianField:
return K1
if K0.is_GaussianRing:
if K1.is_RationalField:
K0 = K0.get_field()
return K0
if K1.is_GaussianRing:
if K0.is_RationalField:
K1 = K1.get_field()
return K1
if K0.is_RationalField:
return K0
if K1.is_RationalField:
return K1
if K0.is_IntegerRing:
return K0
if K1.is_IntegerRing:
return K1
if K0.is_FiniteField and K1.is_FiniteField:
return K0.__class__(max(K0.mod, K1.mod, key=default_sort_key))
from sympy.polys.domains import EX
return EX
def __eq__(self, other):
"""Returns ``True`` if two domains are equivalent. """
return isinstance(other, Domain) and self.dtype == other.dtype
def __ne__(self, other):
"""Returns ``False`` if two domains are equivalent. """
return not self == other
def map(self, seq):
"""Rersively apply ``self`` to all elements of ``seq``. """
result = []
for elt in seq:
if isinstance(elt, list):
result.append(self.map(elt))
else:
result.append(self(elt))
return result
def get_ring(self):
"""Returns a ring associated with ``self``. """
raise DomainError('there is no ring associated with %s' % self)
def get_field(self):
"""Returns a field associated with ``self``. """
raise DomainError('there is no field associated with %s' % self)
def get_exact(self):
"""Returns an exact domain associated with ``self``. """
return self
def __getitem__(self, symbols):
"""The mathematical way to make a polynomial ring. """
if hasattr(symbols, '__iter__'):
return self.poly_ring(*symbols)
else:
return self.poly_ring(symbols)
def poly_ring(self, *symbols, order=lex):
"""Returns a polynomial ring, i.e. `K[X]`. """
from sympy.polys.domains.polynomialring import PolynomialRing
return PolynomialRing(self, symbols, order)
def frac_field(self, *symbols, order=lex):
"""Returns a fraction field, i.e. `K(X)`. """
from sympy.polys.domains.fractionfield import FractionField
return FractionField(self, symbols, order)
def old_poly_ring(self, *symbols, **kwargs):
"""Returns a polynomial ring, i.e. `K[X]`. """
from sympy.polys.domains.old_polynomialring import PolynomialRing
return PolynomialRing(self, *symbols, **kwargs)
def old_frac_field(self, *symbols, **kwargs):
"""Returns a fraction field, i.e. `K(X)`. """
from sympy.polys.domains.old_fractionfield import FractionField
return FractionField(self, *symbols, **kwargs)
def algebraic_field(self, *extension):
r"""Returns an algebraic field, i.e. `K(\alpha, \ldots)`. """
raise DomainError("can't create algebraic field over %s" % self)
def inject(self, *symbols):
"""Inject generators into this domain. """
raise NotImplementedError
def drop(self, *symbols):
"""Drop generators from this domain. """
if self.is_Simple:
return self
raise NotImplementedError # pragma: no cover
def is_zero(self, a):
"""Returns True if ``a`` is zero. """
return not a
def is_one(self, a):
"""Returns True if ``a`` is one. """
return a == self.one
def is_positive(self, a):
"""Returns True if ``a`` is positive. """
return a > 0
def is_negative(self, a):
"""Returns True if ``a`` is negative. """
return a < 0
def is_nonpositive(self, a):
"""Returns True if ``a`` is non-positive. """
return a <= 0
def is_nonnegative(self, a):
"""Returns True if ``a`` is non-negative. """
return a >= 0
def canonical_unit(self, a):
if self.is_negative(a):
return -self.one
else:
return self.one
def abs(self, a):
"""Absolute value of ``a``, implies ``__abs__``. """
return abs(a)
def neg(self, a):
"""Returns ``a`` negated, implies ``__neg__``. """
return -a
def pos(self, a):
"""Returns ``a`` positive, implies ``__pos__``. """
return +a
def add(self, a, b):
"""Sum of ``a`` and ``b``, implies ``__add__``. """
return a + b
def sub(self, a, b):
"""Difference of ``a`` and ``b``, implies ``__sub__``. """
return a - b
def mul(self, a, b):
"""Product of ``a`` and ``b``, implies ``__mul__``. """
return a * b
def pow(self, a, b):
"""Raise ``a`` to power ``b``, implies ``__pow__``. """
return a ** b
def exquo(self, a, b):
"""Exact quotient of *a* and *b*. Analogue of ``a / b``.
Explanation
===========
This is essentially the same as ``a / b`` except that an error will be
raised if the division is inexact (if there is any remainder) and the
result will always be a domain element. When working in a
:py:class:`~.Domain` that is not a :py:class:`~.Field` (e.g. :ref:`ZZ`
or :ref:`K[x]`) ``exquo`` should be used instead of ``/``.
The key invariant is that if ``q = K.exquo(a, b)`` (and ``exquo`` does
not raise an exception) then ``a == b*q``.
Examples
========
We can use ``K.exquo`` instead of ``/`` for exact division.
>>> from sympy import ZZ
>>> ZZ.exquo(ZZ(4), ZZ(2))
2
>>> ZZ.exquo(ZZ(5), ZZ(2))
Traceback (most recent call last):
...
ExactQuotientFailed: 2 does not divide 5 in ZZ
Over a :py:class:`~.Field` such as :ref:`QQ`, division (with nonzero
divisor) is always exact so in that case ``/`` can be used instead of
:py:meth:`~.Domain.exquo`.
>>> from sympy import QQ
>>> QQ.exquo(QQ(5), QQ(2))
5/2
>>> QQ(5) / QQ(2)
5/2
Parameters
==========
a: domain element
The dividend
b: domain element
The divisor
Returns
=======
q: domain element
The exact quotient
Raises
======
ExactQuotientFailed: if exact division is not possible.
ZeroDivisionError: when the divisor is zero.
See also
========
quo: Analogue of ``a // b``
rem: Analogue of ``a % b``
div: Analogue of ``divmod(a, b)``
Notes
=====
Since the default :py:attr:`~.Domain.dtype` for :ref:`ZZ` is ``int``
(or ``mpz``) division as ``a / b`` should not be used as it would give
a ``float``.
>>> ZZ(4) / ZZ(2)
2.0
>>> ZZ(5) / ZZ(2)
2.5
Using ``/`` with :ref:`ZZ` will lead to incorrect results so
:py:meth:`~.Domain.exquo` should be used instead.
"""
raise NotImplementedError
def quo(self, a, b):
"""Quotient of *a* and *b*. Analogue of ``a // b``.
``K.quo(a, b)`` is equivalent to ``K.div(a, b)[0]``. See
:py:meth:`~.Domain.div` for more explanation.
See also
========
rem: Analogue of ``a % b``
div: Analogue of ``divmod(a, b)``
exquo: Analogue of ``a / b``
"""
raise NotImplementedError
def rem(self, a, b):
"""Modulo division of *a* and *b*. Analogue of ``a % b``.
``K.rem(a, b)`` is equivalent to ``K.div(a, b)[1]``. See
:py:meth:`~.Domain.div` for more explanation.
See also
========
quo: Analogue of ``a // b``
div: Analogue of ``divmod(a, b)``
exquo: Analogue of ``a / b``
"""
raise NotImplementedError
def div(self, a, b):
"""Quotient and remainder for *a* and *b*. Analogue of ``divmod(a, b)``
Explanation
===========
This is essentially the same as ``divmod(a, b)`` except that is more
consistent when working over some :py:class:`~.Field` domains such as
:ref:`QQ`. When working over an arbitrary :py:class:`~.Domain` the
:py:meth:`~.Domain.div` method should be used instead of ``divmod``.
The key invariant is that if ``q, r = K.div(a, b)`` then
``a == b*q + r``.
The result of ``K.div(a, b)`` is the same as the tuple
``(K.quo(a, b), K.rem(a, b))`` except that if both quotient and
remainder are needed then it is more efficient to use
:py:meth:`~.Domain.div`.
Examples
========
We can use ``K.div`` instead of ``divmod`` for floor division and
remainder.
>>> from sympy import ZZ, QQ
>>> ZZ.div(ZZ(5), ZZ(2))
(2, 1)
If ``K`` is a :py:class:`~.Field` then the division is always exact
with a remainder of :py:attr:`~.Domain.zero`.
>>> QQ.div(QQ(5), QQ(2))
(5/2, 0)
Parameters
==========
a: domain element
The dividend
b: domain element
The divisor
Returns
=======
(q, r): tuple of domain elements
The quotient and remainder
Raises
======
ZeroDivisionError: when the divisor is zero.
See also
========
quo: Analogue of ``a // b``
rem: Analogue of ``a % b``
exquo: Analogue of ``a / b``
Notes
=====
If ``gmpy`` is installed then the ``gmpy.mpq`` type will be used as
the :py:attr:`~.Domain.dtype` for :ref:`QQ`. The ``gmpy.mpq`` type
defines ``divmod`` in a way that is undesirable so
:py:meth:`~.Domain.div` should be used instead of ``divmod``.
>>> a = QQ(1)
>>> b = QQ(3, 2)
>>> a # doctest: +SKIP
mpq(1,1)
>>> b # doctest: +SKIP
mpq(3,2)
>>> divmod(a, b) # doctest: +SKIP
(mpz(0), mpq(1,1))
>>> QQ.div(a, b) # doctest: +SKIP
(mpq(2,3), mpq(0,1))
Using ``//`` or ``%`` with :ref:`QQ` will lead to incorrect results so
:py:meth:`~.Domain.div` should be used instead.
"""
raise NotImplementedError
def invert(self, a, b):
"""Returns inversion of ``a mod b``, implies something. """
raise NotImplementedError
def revert(self, a):
"""Returns ``a**(-1)`` if possible. """
raise NotImplementedError
def numer(self, a):
"""Returns numerator of ``a``. """
raise NotImplementedError
def denom(self, a):
"""Returns denominator of ``a``. """
raise NotImplementedError
def half_gcdex(self, a, b):
"""Half extended GCD of ``a`` and ``b``. """
s, t, h = self.gcdex(a, b)
return s, h
def gcdex(self, a, b):
"""Extended GCD of ``a`` and ``b``. """
raise NotImplementedError
def cofactors(self, a, b):
"""Returns GCD and cofactors of ``a`` and ``b``. """
gcd = self.gcd(a, b)
cfa = self.quo(a, gcd)
cfb = self.quo(b, gcd)
return gcd, cfa, cfb
def gcd(self, a, b):
"""Returns GCD of ``a`` and ``b``. """
raise NotImplementedError
def lcm(self, a, b):
"""Returns LCM of ``a`` and ``b``. """
raise NotImplementedError
def log(self, a, b):
"""Returns b-base logarithm of ``a``. """
raise NotImplementedError
def sqrt(self, a):
"""Returns square root of ``a``. """
raise NotImplementedError
def evalf(self, a, prec=None, **options):
"""Returns numerical approximation of ``a``. """
return self.to_sympy(a).evalf(prec, **options)
n = evalf
def real(self, a):
return a
def imag(self, a):
return self.zero
def almosteq(self, a, b, tolerance=None):
"""Check if ``a`` and ``b`` are almost equal. """
return a == b
def characteristic(self):
"""Return the characteristic of this domain. """
raise NotImplementedError('characteristic()')
__all__ = ['Domain']
|
362c97c49bcc9fc873a0d0f509fea6bc747651ce7fe5222909797d8351392e35 | """Tests for the implementation of RootOf class and related tools. """
from sympy.polys.polytools import Poly
from sympy.polys.rootoftools import (rootof, RootOf, CRootOf, RootSum,
_pure_key_dict as D)
from sympy.polys.polyerrors import (
MultivariatePolynomialError,
GeneratorsNeeded,
PolynomialError,
)
from sympy import (
S, sqrt, I, Rational, Float, Lambda, log, exp, tan, Function, Eq,
solve, legendre_poly, Integral
)
from sympy.testing.pytest import raises, slow
from sympy.core.expr import unchanged
from sympy.abc import a, b, x, y, z, r
def test_CRootOf___new__():
assert rootof(x, 0) == 0
assert rootof(x, -1) == 0
assert rootof(x, S.Zero) == 0
assert rootof(x - 1, 0) == 1
assert rootof(x - 1, -1) == 1
assert rootof(x + 1, 0) == -1
assert rootof(x + 1, -1) == -1
assert rootof(x**2 + 2*x + 3, 0) == -1 - I*sqrt(2)
assert rootof(x**2 + 2*x + 3, 1) == -1 + I*sqrt(2)
assert rootof(x**2 + 2*x + 3, -1) == -1 + I*sqrt(2)
assert rootof(x**2 + 2*x + 3, -2) == -1 - I*sqrt(2)
r = rootof(x**2 + 2*x + 3, 0, radicals=False)
assert isinstance(r, RootOf) is True
r = rootof(x**2 + 2*x + 3, 1, radicals=False)
assert isinstance(r, RootOf) is True
r = rootof(x**2 + 2*x + 3, -1, radicals=False)
assert isinstance(r, RootOf) is True
r = rootof(x**2 + 2*x + 3, -2, radicals=False)
assert isinstance(r, RootOf) is True
assert rootof((x - 1)*(x + 1), 0, radicals=False) == -1
assert rootof((x - 1)*(x + 1), 1, radicals=False) == 1
assert rootof((x - 1)*(x + 1), -1, radicals=False) == 1
assert rootof((x - 1)*(x + 1), -2, radicals=False) == -1
assert rootof((x - 1)*(x + 1), 0, radicals=True) == -1
assert rootof((x - 1)*(x + 1), 1, radicals=True) == 1
assert rootof((x - 1)*(x + 1), -1, radicals=True) == 1
assert rootof((x - 1)*(x + 1), -2, radicals=True) == -1
assert rootof((x - 1)*(x**3 + x + 3), 0) == rootof(x**3 + x + 3, 0)
assert rootof((x - 1)*(x**3 + x + 3), 1) == 1
assert rootof((x - 1)*(x**3 + x + 3), 2) == rootof(x**3 + x + 3, 1)
assert rootof((x - 1)*(x**3 + x + 3), 3) == rootof(x**3 + x + 3, 2)
assert rootof((x - 1)*(x**3 + x + 3), -1) == rootof(x**3 + x + 3, 2)
assert rootof((x - 1)*(x**3 + x + 3), -2) == rootof(x**3 + x + 3, 1)
assert rootof((x - 1)*(x**3 + x + 3), -3) == 1
assert rootof((x - 1)*(x**3 + x + 3), -4) == rootof(x**3 + x + 3, 0)
assert rootof(x**4 + 3*x**3, 0) == -3
assert rootof(x**4 + 3*x**3, 1) == 0
assert rootof(x**4 + 3*x**3, 2) == 0
assert rootof(x**4 + 3*x**3, 3) == 0
raises(GeneratorsNeeded, lambda: rootof(0, 0))
raises(GeneratorsNeeded, lambda: rootof(1, 0))
raises(PolynomialError, lambda: rootof(Poly(0, x), 0))
raises(PolynomialError, lambda: rootof(Poly(1, x), 0))
raises(PolynomialError, lambda: rootof(x - y, 0))
# issue 8617
raises(PolynomialError, lambda: rootof(exp(x), 0))
raises(NotImplementedError, lambda: rootof(x**3 - x + sqrt(2), 0))
raises(NotImplementedError, lambda: rootof(x**3 - x + I, 0))
raises(IndexError, lambda: rootof(x**2 - 1, -4))
raises(IndexError, lambda: rootof(x**2 - 1, -3))
raises(IndexError, lambda: rootof(x**2 - 1, 2))
raises(IndexError, lambda: rootof(x**2 - 1, 3))
raises(ValueError, lambda: rootof(x**2 - 1, x))
assert rootof(Poly(x - y, x), 0) == y
assert rootof(Poly(x**2 - y, x), 0) == -sqrt(y)
assert rootof(Poly(x**2 - y, x), 1) == sqrt(y)
assert rootof(Poly(x**3 - y, x), 0) == y**Rational(1, 3)
assert rootof(y*x**3 + y*x + 2*y, x, 0) == -1
raises(NotImplementedError, lambda: rootof(x**3 + x + 2*y, x, 0))
assert rootof(x**3 + x + 1, 0).is_commutative is True
def test_CRootOf_attributes():
r = rootof(x**3 + x + 3, 0)
assert r.is_number
assert r.free_symbols == set()
# if the following assertion fails then multivariate polynomials
# are apparently supported and the RootOf.free_symbols routine
# should be changed to return whatever symbols would not be
# the PurePoly dummy symbol
raises(NotImplementedError, lambda: rootof(Poly(x**3 + y*x + 1, x), 0))
def test_CRootOf___eq__():
assert (rootof(x**3 + x + 3, 0) == rootof(x**3 + x + 3, 0)) is True
assert (rootof(x**3 + x + 3, 0) == rootof(x**3 + x + 3, 1)) is False
assert (rootof(x**3 + x + 3, 1) == rootof(x**3 + x + 3, 1)) is True
assert (rootof(x**3 + x + 3, 1) == rootof(x**3 + x + 3, 2)) is False
assert (rootof(x**3 + x + 3, 2) == rootof(x**3 + x + 3, 2)) is True
assert (rootof(x**3 + x + 3, 0) == rootof(y**3 + y + 3, 0)) is True
assert (rootof(x**3 + x + 3, 0) == rootof(y**3 + y + 3, 1)) is False
assert (rootof(x**3 + x + 3, 1) == rootof(y**3 + y + 3, 1)) is True
assert (rootof(x**3 + x + 3, 1) == rootof(y**3 + y + 3, 2)) is False
assert (rootof(x**3 + x + 3, 2) == rootof(y**3 + y + 3, 2)) is True
def test_CRootOf___eval_Eq__():
f = Function('f')
eq = x**3 + x + 3
r = rootof(eq, 2)
r1 = rootof(eq, 1)
assert Eq(r, r1) is S.false
assert Eq(r, r) is S.true
assert unchanged(Eq, r, x)
assert Eq(r, 0) is S.false
assert Eq(r, S.Infinity) is S.false
assert Eq(r, I) is S.false
assert unchanged(Eq, r, f(0))
sol = solve(eq)
for s in sol:
if s.is_real:
assert Eq(r, s) is S.false
r = rootof(eq, 0)
for s in sol:
if s.is_real:
assert Eq(r, s) is S.true
eq = x**3 + x + 1
sol = solve(eq)
assert [Eq(rootof(eq, i), j) for i in range(3) for j in sol] == [
False, False, True, False, True, False, True, False, False]
assert Eq(rootof(eq, 0), 1 + S.ImaginaryUnit) == False
def test_CRootOf_is_real():
assert rootof(x**3 + x + 3, 0).is_real is True
assert rootof(x**3 + x + 3, 1).is_real is False
assert rootof(x**3 + x + 3, 2).is_real is False
def test_CRootOf_is_complex():
assert rootof(x**3 + x + 3, 0).is_complex is True
def test_CRootOf_subs():
assert rootof(x**3 + x + 1, 0).subs(x, y) == rootof(y**3 + y + 1, 0)
def test_CRootOf_diff():
assert rootof(x**3 + x + 1, 0).diff(x) == 0
assert rootof(x**3 + x + 1, 0).diff(y) == 0
@slow
def test_CRootOf_evalf():
real = rootof(x**3 + x + 3, 0).evalf(n=20)
assert real.epsilon_eq(Float("-1.2134116627622296341"))
re, im = rootof(x**3 + x + 3, 1).evalf(n=20).as_real_imag()
assert re.epsilon_eq( Float("0.60670583138111481707"))
assert im.epsilon_eq(-Float("1.45061224918844152650"))
re, im = rootof(x**3 + x + 3, 2).evalf(n=20).as_real_imag()
assert re.epsilon_eq(Float("0.60670583138111481707"))
assert im.epsilon_eq(Float("1.45061224918844152650"))
p = legendre_poly(4, x, polys=True)
roots = [str(r.n(17)) for r in p.real_roots()]
# magnitudes are given by
# sqrt(3/S(7) - 2*sqrt(6/S(5))/7)
# and
# sqrt(3/S(7) + 2*sqrt(6/S(5))/7)
assert roots == [
"-0.86113631159405258",
"-0.33998104358485626",
"0.33998104358485626",
"0.86113631159405258",
]
re = rootof(x**5 - 5*x + 12, 0).evalf(n=20)
assert re.epsilon_eq(Float("-1.84208596619025438271"))
re, im = rootof(x**5 - 5*x + 12, 1).evalf(n=20).as_real_imag()
assert re.epsilon_eq(Float("-0.351854240827371999559"))
assert im.epsilon_eq(Float("-1.709561043370328882010"))
re, im = rootof(x**5 - 5*x + 12, 2).evalf(n=20).as_real_imag()
assert re.epsilon_eq(Float("-0.351854240827371999559"))
assert im.epsilon_eq(Float("+1.709561043370328882010"))
re, im = rootof(x**5 - 5*x + 12, 3).evalf(n=20).as_real_imag()
assert re.epsilon_eq(Float("+1.272897223922499190910"))
assert im.epsilon_eq(Float("-0.719798681483861386681"))
re, im = rootof(x**5 - 5*x + 12, 4).evalf(n=20).as_real_imag()
assert re.epsilon_eq(Float("+1.272897223922499190910"))
assert im.epsilon_eq(Float("+0.719798681483861386681"))
# issue 6393
assert str(rootof(x**5 + 2*x**4 + x**3 - 68719476736, 0).n(3)) == '147.'
eq = (531441*x**11 + 3857868*x**10 + 13730229*x**9 + 32597882*x**8 +
55077472*x**7 + 60452000*x**6 + 32172064*x**5 - 4383808*x**4 -
11942912*x**3 - 1506304*x**2 + 1453312*x + 512)
a, b = rootof(eq, 1).n(2).as_real_imag()
c, d = rootof(eq, 2).n(2).as_real_imag()
assert a == c
assert b < d
assert b == -d
# issue 6451
r = rootof(legendre_poly(64, x), 7)
assert r.n(2) == r.n(100).n(2)
# issue 9019
r0 = rootof(x**2 + 1, 0, radicals=False)
r1 = rootof(x**2 + 1, 1, radicals=False)
assert r0.n(4) == -1.0*I
assert r1.n(4) == 1.0*I
# make sure verification is used in case a max/min traps the "root"
assert str(rootof(4*x**5 + 16*x**3 + 12*x**2 + 7, 0).n(3)) == '-0.976'
# watch out for UnboundLocalError
c = CRootOf(90720*x**6 - 4032*x**4 + 84*x**2 - 1, 0)
assert c._eval_evalf(2) # doesn't fail
# watch out for imaginary parts that don't want to evaluate
assert str(RootOf(x**16 + 32*x**14 + 508*x**12 + 5440*x**10 +
39510*x**8 + 204320*x**6 + 755548*x**4 + 1434496*x**2 +
877969, 10).n(2)) == '-3.4*I'
assert abs(RootOf(x**4 + 10*x**2 + 1, 0).n(2)) < 0.4
# check reset and args
r = [RootOf(x**3 + x + 3, i) for i in range(3)]
r[0]._reset()
for ri in r:
i = ri._get_interval()
ri.n(2)
assert i != ri._get_interval()
ri._reset()
assert i == ri._get_interval()
assert i == i.func(*i.args)
def test_CRootOf_evalf_caching_bug():
r = rootof(x**5 - 5*x + 12, 1)
r.n()
a = r._get_interval()
r = rootof(x**5 - 5*x + 12, 1)
r.n()
b = r._get_interval()
assert a == b
def test_CRootOf_real_roots():
assert Poly(x**5 + x + 1).real_roots() == [rootof(x**3 - x**2 + 1, 0)]
assert Poly(x**5 + x + 1).real_roots(radicals=False) == [rootof(
x**3 - x**2 + 1, 0)]
# https://github.com/sympy/sympy/issues/20902
p = Poly(-3*x**4 - 10*x**3 - 12*x**2 - 6*x - 1, x, domain='ZZ')
assert CRootOf.real_roots(p) == [S(-1), S(-1), S(-1), S(-1)/3]
def test_CRootOf_all_roots():
assert Poly(x**5 + x + 1).all_roots() == [
rootof(x**3 - x**2 + 1, 0),
Rational(-1, 2) - sqrt(3)*I/2,
Rational(-1, 2) + sqrt(3)*I/2,
rootof(x**3 - x**2 + 1, 1),
rootof(x**3 - x**2 + 1, 2),
]
assert Poly(x**5 + x + 1).all_roots(radicals=False) == [
rootof(x**3 - x**2 + 1, 0),
rootof(x**2 + x + 1, 0, radicals=False),
rootof(x**2 + x + 1, 1, radicals=False),
rootof(x**3 - x**2 + 1, 1),
rootof(x**3 - x**2 + 1, 2),
]
def test_CRootOf_eval_rational():
p = legendre_poly(4, x, polys=True)
roots = [r.eval_rational(n=18) for r in p.real_roots()]
for root in roots:
assert isinstance(root, Rational)
roots = [str(root.n(17)) for root in roots]
assert roots == [
"-0.86113631159405258",
"-0.33998104358485626",
"0.33998104358485626",
"0.86113631159405258",
]
def test_RootSum___new__():
f = x**3 + x + 3
g = Lambda(r, log(r*x))
s = RootSum(f, g)
assert isinstance(s, RootSum) is True
assert RootSum(f**2, g) == 2*RootSum(f, g)
assert RootSum((x - 7)*f**3, g) == log(7*x) + 3*RootSum(f, g)
# issue 5571
assert hash(RootSum((x - 7)*f**3, g)) == hash(log(7*x) + 3*RootSum(f, g))
raises(MultivariatePolynomialError, lambda: RootSum(x**3 + x + y))
raises(ValueError, lambda: RootSum(x**2 + 3, lambda x: x))
assert RootSum(f, exp) == RootSum(f, Lambda(x, exp(x)))
assert RootSum(f, log) == RootSum(f, Lambda(x, log(x)))
assert isinstance(RootSum(f, auto=False), RootSum) is True
assert RootSum(f) == 0
assert RootSum(f, Lambda(x, x)) == 0
assert RootSum(f, Lambda(x, x**2)) == -2
assert RootSum(f, Lambda(x, 1)) == 3
assert RootSum(f, Lambda(x, 2)) == 6
assert RootSum(f, auto=False).is_commutative is True
assert RootSum(f, Lambda(x, 1/(x + x**2))) == Rational(11, 3)
assert RootSum(f, Lambda(x, y/(x + x**2))) == Rational(11, 3)*y
assert RootSum(x**2 - 1, Lambda(x, 3*x**2), x) == 6
assert RootSum(x**2 - y, Lambda(x, 3*x**2), x) == 6*y
assert RootSum(x**2 - 1, Lambda(x, z*x**2), x) == 2*z
assert RootSum(x**2 - y, Lambda(x, z*x**2), x) == 2*z*y
assert RootSum(
x**2 - 1, Lambda(x, exp(x)), quadratic=True) == exp(-1) + exp(1)
assert RootSum(x**3 + a*x + a**3, tan, x) == \
RootSum(x**3 + x + 1, Lambda(x, tan(a*x)))
assert RootSum(a**3*x**3 + a*x + 1, tan, x) == \
RootSum(x**3 + x + 1, Lambda(x, tan(x/a)))
def test_RootSum_free_symbols():
assert RootSum(x**3 + x + 3, Lambda(r, exp(r))).free_symbols == set()
assert RootSum(x**3 + x + 3, Lambda(r, exp(a*r))).free_symbols == {a}
assert RootSum(
x**3 + x + y, Lambda(r, exp(a*r)), x).free_symbols == {a, y}
def test_RootSum___eq__():
f = Lambda(x, exp(x))
assert (RootSum(x**3 + x + 1, f) == RootSum(x**3 + x + 1, f)) is True
assert (RootSum(x**3 + x + 1, f) == RootSum(y**3 + y + 1, f)) is True
assert (RootSum(x**3 + x + 1, f) == RootSum(x**3 + x + 2, f)) is False
assert (RootSum(x**3 + x + 1, f) == RootSum(y**3 + y + 2, f)) is False
def test_RootSum_doit():
rs = RootSum(x**2 + 1, exp)
assert isinstance(rs, RootSum) is True
assert rs.doit() == exp(-I) + exp(I)
rs = RootSum(x**2 + a, exp, x)
assert isinstance(rs, RootSum) is True
assert rs.doit() == exp(-sqrt(-a)) + exp(sqrt(-a))
def test_RootSum_evalf():
rs = RootSum(x**2 + 1, exp)
assert rs.evalf(n=20, chop=True).epsilon_eq(Float("1.0806046117362794348"))
assert rs.evalf(n=15, chop=True).epsilon_eq(Float("1.08060461173628"))
rs = RootSum(x**2 + a, exp, x)
assert rs.evalf() == rs
def test_RootSum_diff():
f = x**3 + x + 3
g = Lambda(r, exp(r*x))
h = Lambda(r, r*exp(r*x))
assert RootSum(f, g).diff(x) == RootSum(f, h)
def test_RootSum_subs():
f = x**3 + x + 3
g = Lambda(r, exp(r*x))
F = y**3 + y + 3
G = Lambda(r, exp(r*y))
assert RootSum(f, g).subs(y, 1) == RootSum(f, g)
assert RootSum(f, g).subs(x, y) == RootSum(F, G)
def test_RootSum_rational():
assert RootSum(
z**5 - z + 1, Lambda(z, z/(x - z))) == (4*x - 5)/(x**5 - x + 1)
f = 161*z**3 + 115*z**2 + 19*z + 1
g = Lambda(z, z*log(
-3381*z**4/4 - 3381*z**3/4 - 625*z**2/2 - z*Rational(125, 2) - 5 + exp(x)))
assert RootSum(f, g).diff(x) == -(
(5*exp(2*x) - 6*exp(x) + 4)*exp(x)/(exp(3*x) - exp(2*x) + 1))/7
def test_RootSum_independent():
f = (x**3 - a)**2*(x**4 - b)**3
g = Lambda(x, 5*tan(x) + 7)
h = Lambda(x, tan(x))
r0 = RootSum(x**3 - a, h, x)
r1 = RootSum(x**4 - b, h, x)
assert RootSum(f, g, x).as_ordered_terms() == [10*r0, 15*r1, 126]
def test_issue_7876():
l1 = Poly(x**6 - x + 1, x).all_roots()
l2 = [rootof(x**6 - x + 1, i) for i in range(6)]
assert frozenset(l1) == frozenset(l2)
def test_issue_8316():
f = Poly(7*x**8 - 9)
assert len(f.all_roots()) == 8
f = Poly(7*x**8 - 10)
assert len(f.all_roots()) == 8
def test__imag_count():
from sympy.polys.rootoftools import _imag_count_of_factor
def imag_count(p):
return sum([_imag_count_of_factor(f)*m for f, m in
p.factor_list()[1]])
assert imag_count(Poly(x**6 + 10*x**2 + 1)) == 2
assert imag_count(Poly(x**2)) == 0
assert imag_count(Poly([1]*3 + [-1], x)) == 0
assert imag_count(Poly(x**3 + 1)) == 0
assert imag_count(Poly(x**2 + 1)) == 2
assert imag_count(Poly(x**2 - 1)) == 0
assert imag_count(Poly(x**4 - 1)) == 2
assert imag_count(Poly(x**4 + 1)) == 0
assert imag_count(Poly([1, 2, 3], x)) == 0
assert imag_count(Poly(x**3 + x + 1)) == 0
assert imag_count(Poly(x**4 + x + 1)) == 0
def q(r1, r2, p):
return Poly(((x - r1)*(x - r2)).subs(x, x**p), x)
assert imag_count(q(-1, -2, 2)) == 4
assert imag_count(q(-1, 2, 2)) == 2
assert imag_count(q(1, 2, 2)) == 0
assert imag_count(q(1, 2, 4)) == 4
assert imag_count(q(-1, 2, 4)) == 2
assert imag_count(q(-1, -2, 4)) == 0
def test_RootOf_is_imaginary():
r = RootOf(x**4 + 4*x**2 + 1, 1)
i = r._get_interval()
assert r.is_imaginary and i.ax*i.bx <= 0
def test_is_disjoint():
eq = x**3 + 5*x + 1
ir = rootof(eq, 0)._get_interval()
ii = rootof(eq, 1)._get_interval()
assert ir.is_disjoint(ii)
assert ii.is_disjoint(ir)
def test_pure_key_dict():
p = D()
assert (x in p) is False
assert (1 in p) is False
p[x] = 1
assert x in p
assert y in p
assert p[y] == 1
raises(KeyError, lambda: p[1])
def dont(k):
p[k] = 2
raises(ValueError, lambda: dont(1))
@slow
def test_eval_approx_relative():
CRootOf.clear_cache()
t = [CRootOf(x**3 + 10*x + 1, i) for i in range(3)]
assert [i.eval_rational(1e-1) for i in t] == [
Rational(-21, 220), Rational(15, 256) - I*Rational(805, 256),
Rational(15, 256) + I*Rational(805, 256)]
t[0]._reset()
assert [i.eval_rational(1e-1, 1e-4) for i in t] == [
Rational(-21, 220), Rational(3275, 65536) - I*Rational(414645, 131072),
Rational(3275, 65536) + I*Rational(414645, 131072)]
assert S(t[0]._get_interval().dx) < 1e-1
assert S(t[1]._get_interval().dx) < 1e-1
assert S(t[1]._get_interval().dy) < 1e-4
assert S(t[2]._get_interval().dx) < 1e-1
assert S(t[2]._get_interval().dy) < 1e-4
t[0]._reset()
assert [i.eval_rational(1e-4, 1e-4) for i in t] == [
Rational(-2001, 20020), Rational(6545, 131072) - I*Rational(414645, 131072),
Rational(6545, 131072) + I*Rational(414645, 131072)]
assert S(t[0]._get_interval().dx) < 1e-4
assert S(t[1]._get_interval().dx) < 1e-4
assert S(t[1]._get_interval().dy) < 1e-4
assert S(t[2]._get_interval().dx) < 1e-4
assert S(t[2]._get_interval().dy) < 1e-4
# in the following, the actual relative precision is
# less than tested, but it should never be greater
t[0]._reset()
assert [i.eval_rational(n=2) for i in t] == [
Rational(-202201, 2024022), Rational(104755, 2097152) - I*Rational(6634255, 2097152),
Rational(104755, 2097152) + I*Rational(6634255, 2097152)]
assert abs(S(t[0]._get_interval().dx)/t[0]) < 1e-2
assert abs(S(t[1]._get_interval().dx)/t[1]).n() < 1e-2
assert abs(S(t[1]._get_interval().dy)/t[1]).n() < 1e-2
assert abs(S(t[2]._get_interval().dx)/t[2]).n() < 1e-2
assert abs(S(t[2]._get_interval().dy)/t[2]).n() < 1e-2
t[0]._reset()
assert [i.eval_rational(n=3) for i in t] == [
Rational(-202201, 2024022), Rational(1676045, 33554432) - I*Rational(106148135, 33554432),
Rational(1676045, 33554432) + I*Rational(106148135, 33554432)]
assert abs(S(t[0]._get_interval().dx)/t[0]) < 1e-3
assert abs(S(t[1]._get_interval().dx)/t[1]).n() < 1e-3
assert abs(S(t[1]._get_interval().dy)/t[1]).n() < 1e-3
assert abs(S(t[2]._get_interval().dx)/t[2]).n() < 1e-3
assert abs(S(t[2]._get_interval().dy)/t[2]).n() < 1e-3
t[0]._reset()
a = [i.eval_approx(2) for i in t]
assert [str(i) for i in a] == [
'-0.10', '0.05 - 3.2*I', '0.05 + 3.2*I']
assert all(abs(((a[i] - t[i])/t[i]).n()) < 1e-2 for i in range(len(a)))
def test_issue_15920():
r = rootof(x**5 - x + 1, 0)
p = Integral(x, (x, 1, y))
assert unchanged(Eq, r, p)
def test_issue_19113():
eq = y**3 - y + 1
# generator is a canonical x in RootOf
assert str(Poly(eq).real_roots()) == '[CRootOf(x**3 - x + 1, 0)]'
assert str(Poly(eq.subs(y, tan(y))).real_roots()
) == '[CRootOf(x**3 - x + 1, 0)]'
assert str(Poly(eq.subs(y, tan(x))).real_roots()
) == '[CRootOf(x**3 - x + 1, 0)]'
|
c0d182603659e0bea6d4748f157f72a71503e754796730345e39ea25db4ee171 | """Tests for user-friendly public interface to polynomial functions. """
import pickle
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, ZZ_I, QQ_I, 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_from_domain_element():
dom = ZZ[x]
assert Poly(dom(x+1), y, domain=dom).rep == DMP([dom(x+1)], dom)
dom = dom.get_field()
assert Poly(dom(x+1), y, domain=dom).rep == DMP([dom(x+1)], dom)
dom = QQ[x]
assert Poly(dom(x+1), y, domain=dom).rep == DMP([dom(x+1)], dom)
dom = dom.get_field()
assert Poly(dom(x+1), y, domain=dom).rep == DMP([dom(x+1)], dom)
dom = ZZ.old_poly_ring(x)
assert Poly(dom([1, 1]), y, domain=dom).rep == DMP([dom([1, 1])], dom)
dom = dom.get_field()
assert Poly(dom([1, 1]), y, domain=dom).rep == DMP([dom([1, 1])], dom)
dom = QQ.old_poly_ring(x)
assert Poly(dom([1, 1]), y, domain=dom).rep == DMP([dom([1, 1])], dom)
dom = dom.get_field()
assert Poly(dom([1, 1]), y, domain=dom).rep == DMP([dom([1, 1])], dom)
dom = QQ.algebraic_field(I)
assert Poly(dom([1, 1]), x, domain=dom).rep == DMP([dom([1, 1])], dom)
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))
assert Poly(x**2 + I, x, domain=ZZ_I).unify(Poly(x**2 + sqrt(2), x, extension=True)) == \
(Poly(x**2 + I, x, domain='QQ<sqrt(2) + I>'), Poly(x**2 + sqrt(2), x, domain='QQ<sqrt(2) + I>'))
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_issue_20427():
f = Poly(-117968192370600*18**(S(1)/3)/(217603955769048*(24201 +
253*sqrt(9165))**(S(1)/3) + 2273005839412*sqrt(9165)*(24201 +
253*sqrt(9165))**(S(1)/3)) - 15720318185*2**(S(2)/3)*3**(S(1)/3)*(24201
+ 253*sqrt(9165))**(S(2)/3)/(217603955769048*(24201 + 253*sqrt(9165))**
(S(1)/3) + 2273005839412*sqrt(9165)*(24201 + 253*sqrt(9165))**(S(1)/3))
+ 15720318185*12**(S(1)/3)*(24201 + 253*sqrt(9165))**(S(2)/3)/(
217603955769048*(24201 + 253*sqrt(9165))**(S(1)/3) + 2273005839412*
sqrt(9165)*(24201 + 253*sqrt(9165))**(S(1)/3)) + 117968192370600*2**(
S(1)/3)*3**(S(2)/3)/(217603955769048*(24201 + 253*sqrt(9165))**(S(1)/3)
+ 2273005839412*sqrt(9165)*(24201 + 253*sqrt(9165))**(S(1)/3)), x)
assert f == Poly(0, x, domain='EX')
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
# https://github.com/sympy/sympy/issues/19579
p = Poly(2+3*I, x, domain=ZZ_I)
q = Poly(1-I, x, domain=ZZ_I)
assert p.div(q, auto=False) == \
(Poly(0, x, domain='ZZ_I'), Poly(2 + 3*I, x, domain='ZZ_I'))
assert p.div(q, auto=True) == \
(Poly(-S(1)/2 + 5*I/2, x, domain='QQ_I'), Poly(0, x, domain='QQ_I'))
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
a = sqrt(2)
assert gcd_list([a, -a]) == gcd_list([-a, a]) == a
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
a, b = sqrt(2), -sqrt(2)
assert gcd(a, b) == gcd(b, a) == sqrt(2)
a, b = sqrt(-2), -sqrt(-2)
assert gcd(a, b) == gcd(b, a) == sqrt(2)
assert gcd(Poly(x - 2, x), Poly(I*x, x)) == Poly(1, x, domain=ZZ_I)
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
# partial fix of 20597
assert gcd(Mul(2, 3, evaluate=False), 2) == 2
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)
assert factor(x**2 + 4*I*x - 4) == (x + 2*I)**2
f = x**2 + 2*I*x - 4
assert factor(f) == f
f = 8192*x**2 + x*(22656 + 175232*I) - 921416 + 242313*I
f_zzi = I*(x*(64 - 64*I) + 773 + 596*I)**2
f_qqi = 8192*(x + S(177)/128 + 1369*I/128)**2
assert factor(f) == f_zzi
assert factor(f, domain=ZZ_I) == f_zzi
assert factor(f, domain=QQ_I) == f_qqi
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)) == (1, x + 2, 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
assert cancel((x**2 + 1)/(x - I)) == x + I
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
# issue 21268
assert to_rational_coeffs(
Poly(y**3 + sqrt(2)*y**2*sin(x) + 1, y)) is None
assert to_rational_coeffs(Poly(x, y)) is None
assert to_rational_coeffs(Poly(sqrt(2)*y)) 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)
def test_poly_copy_equals_original():
poly = Poly(x + y, x, y, z)
copy = poly.copy()
assert poly == copy, (
"Copied polynomial not equal to original.")
assert poly.gens == copy.gens, (
"Copied polynomial has different generators than original.")
def test_deserialized_poly_equals_original():
poly = Poly(x + y, x, y, z)
deserialized = pickle.loads(pickle.dumps(poly))
assert poly == deserialized, (
"Deserialized polynomial not equal to original.")
assert poly.gens == deserialized.gens, (
"Deserialized polynomial has different generators than original.")
def test_issue_20389():
result = degree(x * (x + 1) - x ** 2 - x, x)
assert result == -oo
def test_issue_20985():
from sympy import symbols
w, R = symbols('w R')
poly = Poly(1.0 + I*w/R, w, 1/R)
assert poly.degree() == S(1)
|
bd8325393b98bdffb5e4fd27293166fae46b85473e9c214492a60500f2f1834f | """Tests for PythonRational type. """
from sympy.polys.domains import PythonRational as QQ
from sympy.testing.pytest import raises
def test_PythonRational__init__():
assert QQ(0).numerator == 0
assert QQ(0).denominator == 1
assert QQ(0, 1).numerator == 0
assert QQ(0, 1).denominator == 1
assert QQ(0, -1).numerator == 0
assert QQ(0, -1).denominator == 1
assert QQ(1).numerator == 1
assert QQ(1).denominator == 1
assert QQ(1, 1).numerator == 1
assert QQ(1, 1).denominator == 1
assert QQ(-1, -1).numerator == 1
assert QQ(-1, -1).denominator == 1
assert QQ(-1).numerator == -1
assert QQ(-1).denominator == 1
assert QQ(-1, 1).numerator == -1
assert QQ(-1, 1).denominator == 1
assert QQ( 1, -1).numerator == -1
assert QQ( 1, -1).denominator == 1
assert QQ(1, 2).numerator == 1
assert QQ(1, 2).denominator == 2
assert QQ(3, 4).numerator == 3
assert QQ(3, 4).denominator == 4
assert QQ(2, 2).numerator == 1
assert QQ(2, 2).denominator == 1
assert QQ(2, 4).numerator == 1
assert QQ(2, 4).denominator == 2
def test_PythonRational__hash__():
assert hash(QQ(0)) == hash(0)
assert hash(QQ(1)) == hash(1)
assert hash(QQ(117)) == hash(117)
def test_PythonRational__int__():
assert int(QQ(-1, 4)) == 0
assert int(QQ( 1, 4)) == 0
assert int(QQ(-5, 4)) == -1
assert int(QQ( 5, 4)) == 1
def test_PythonRational__float__():
assert float(QQ(-1, 2)) == -0.5
assert float(QQ( 1, 2)) == 0.5
def test_PythonRational__abs__():
assert abs(QQ(-1, 2)) == QQ(1, 2)
assert abs(QQ( 1, 2)) == QQ(1, 2)
def test_PythonRational__pos__():
assert +QQ(-1, 2) == QQ(-1, 2)
assert +QQ( 1, 2) == QQ( 1, 2)
def test_PythonRational__neg__():
assert -QQ(-1, 2) == QQ( 1, 2)
assert -QQ( 1, 2) == QQ(-1, 2)
def test_PythonRational__add__():
assert QQ(-1, 2) + QQ( 1, 2) == QQ(0)
assert QQ( 1, 2) + QQ(-1, 2) == QQ(0)
assert QQ(1, 2) + QQ(1, 2) == QQ(1)
assert QQ(1, 2) + QQ(3, 2) == QQ(2)
assert QQ(3, 2) + QQ(1, 2) == QQ(2)
assert QQ(3, 2) + QQ(3, 2) == QQ(3)
assert 1 + QQ(1, 2) == QQ(3, 2)
assert QQ(1, 2) + 1 == QQ(3, 2)
def test_PythonRational__sub__():
assert QQ(-1, 2) - QQ( 1, 2) == QQ(-1)
assert QQ( 1, 2) - QQ(-1, 2) == QQ( 1)
assert QQ(1, 2) - QQ(1, 2) == QQ( 0)
assert QQ(1, 2) - QQ(3, 2) == QQ(-1)
assert QQ(3, 2) - QQ(1, 2) == QQ( 1)
assert QQ(3, 2) - QQ(3, 2) == QQ( 0)
assert 1 - QQ(1, 2) == QQ( 1, 2)
assert QQ(1, 2) - 1 == QQ(-1, 2)
def test_PythonRational__mul__():
assert QQ(-1, 2) * QQ( 1, 2) == QQ(-1, 4)
assert QQ( 1, 2) * QQ(-1, 2) == QQ(-1, 4)
assert QQ(1, 2) * QQ(1, 2) == QQ(1, 4)
assert QQ(1, 2) * QQ(3, 2) == QQ(3, 4)
assert QQ(3, 2) * QQ(1, 2) == QQ(3, 4)
assert QQ(3, 2) * QQ(3, 2) == QQ(9, 4)
assert 2 * QQ(1, 2) == QQ(1)
assert QQ(1, 2) * 2 == QQ(1)
def test_PythonRational__truediv__():
assert QQ(-1, 2) / QQ( 1, 2) == QQ(-1)
assert QQ( 1, 2) / QQ(-1, 2) == QQ(-1)
assert QQ(1, 2) / QQ(1, 2) == QQ(1)
assert QQ(1, 2) / QQ(3, 2) == QQ(1, 3)
assert QQ(3, 2) / QQ(1, 2) == QQ(3)
assert QQ(3, 2) / QQ(3, 2) == QQ(1)
assert 2 / QQ(1, 2) == QQ(4)
assert QQ(1, 2) / 2 == QQ(1, 4)
raises(ZeroDivisionError, lambda: QQ(1, 2) / QQ(0))
raises(ZeroDivisionError, lambda: QQ(1, 2) / 0)
def test_PythonRational__pow__():
assert QQ(1)**10 == QQ(1)
assert QQ(2)**10 == QQ(1024)
assert QQ(1)**(-10) == QQ(1)
assert QQ(2)**(-10) == QQ(1, 1024)
def test_PythonRational__eq__():
assert (QQ(1, 2) == QQ(1, 2)) is True
assert (QQ(1, 2) != QQ(1, 2)) is False
assert (QQ(1, 2) == QQ(1, 3)) is False
assert (QQ(1, 2) != QQ(1, 3)) is True
def test_PythonRational__lt_le_gt_ge__():
assert (QQ(1, 2) < QQ(1, 4)) is False
assert (QQ(1, 2) <= QQ(1, 4)) is False
assert (QQ(1, 2) > QQ(1, 4)) is True
assert (QQ(1, 2) >= QQ(1, 4)) is True
assert (QQ(1, 4) < QQ(1, 2)) is True
assert (QQ(1, 4) <= QQ(1, 2)) is True
assert (QQ(1, 4) > QQ(1, 2)) is False
assert (QQ(1, 4) >= QQ(1, 2)) is False
|
aa19912a11188068ebd96a71c4a018295d7c036e08a7850c1ba4772d27ee5322 | """Tests for algorithms for computing symbolic roots of polynomials. """
from sympy import (S, symbols, Symbol, Wild, Rational, sqrt,
powsimp, sin, cos, pi, I, Interval, re, im, exp, ZZ, Piecewise,
acos, root, conjugate)
from sympy.polys import Poly, cyclotomic_poly, intervals, nroots, rootof
from sympy.polys.polyroots import (root_factors, roots_linear,
roots_quadratic, roots_cubic, roots_quartic, roots_cyclotomic,
roots_binomial, preprocess_roots, roots)
from sympy.polys.orthopolys import legendre_poly
from sympy.polys.polyerrors import PolynomialError
from sympy.polys.polyutils import _nsort
from sympy.utilities.iterables import cartes
from sympy.testing.pytest import raises, slow
from sympy.testing.randtest import verify_numerically
import mpmath
a, b, c, d, e, q, t, x, y, z = symbols('a,b,c,d,e,q,t,x,y,z')
def _check(roots):
# this is the desired invariant for roots returned
# by all_roots. It is trivially true for linear
# polynomials.
nreal = sum([1 if i.is_real else 0 for i in roots])
assert list(sorted(roots[:nreal])) == list(roots[:nreal])
for ix in range(nreal, len(roots), 2):
if not (
roots[ix + 1] == roots[ix] or
roots[ix + 1] == conjugate(roots[ix])):
return False
return True
def test_roots_linear():
assert roots_linear(Poly(2*x + 1, x)) == [Rational(-1, 2)]
def test_roots_quadratic():
assert roots_quadratic(Poly(2*x**2, x)) == [0, 0]
assert roots_quadratic(Poly(2*x**2 + 3*x, x)) == [Rational(-3, 2), 0]
assert roots_quadratic(Poly(2*x**2 + 3, x)) == [-I*sqrt(6)/2, I*sqrt(6)/2]
assert roots_quadratic(Poly(2*x**2 + 4*x + 3, x)) == [-1 - I*sqrt(2)/2, -1 + I*sqrt(2)/2]
_check(Poly(2*x**2 + 4*x + 3, x).all_roots())
f = x**2 + (2*a*e + 2*c*e)/(a - c)*x + (d - b + a*e**2 - c*e**2)/(a - c)
assert roots_quadratic(Poly(f, x)) == \
[-e*(a + c)/(a - c) - sqrt(a*b + c*d - a*d - b*c + 4*a*c*e**2)/(a - c),
-e*(a + c)/(a - c) + sqrt(a*b + c*d - a*d - b*c + 4*a*c*e**2)/(a - c)]
# check for simplification
f = Poly(y*x**2 - 2*x - 2*y, x)
assert roots_quadratic(f) == \
[-sqrt(2*y**2 + 1)/y + 1/y, sqrt(2*y**2 + 1)/y + 1/y]
f = Poly(x**2 + (-y**2 - 2)*x + y**2 + 1, x)
assert roots_quadratic(f) == \
[1,y**2 + 1]
f = Poly(sqrt(2)*x**2 - 1, x)
r = roots_quadratic(f)
assert r == _nsort(r)
# issue 8255
f = Poly(-24*x**2 - 180*x + 264)
assert [w.n(2) for w in f.all_roots(radicals=True)] == \
[w.n(2) for w in f.all_roots(radicals=False)]
for _a, _b, _c in cartes((-2, 2), (-2, 2), (0, -1)):
f = Poly(_a*x**2 + _b*x + _c)
roots = roots_quadratic(f)
assert roots == _nsort(roots)
def test_issue_7724():
eq = Poly(x**4*I + x**2 + I, x)
assert roots(eq) == {
sqrt(I/2 + sqrt(5)*I/2): 1,
sqrt(-sqrt(5)*I/2 + I/2): 1,
-sqrt(I/2 + sqrt(5)*I/2): 1,
-sqrt(-sqrt(5)*I/2 + I/2): 1}
def test_issue_8438():
p = Poly([1, y, -2, -3], x).as_expr()
roots = roots_cubic(Poly(p, x), x)
z = Rational(-3, 2) - I*Rational(7, 2) # this will fail in code given in commit msg
post = [r.subs(y, z) for r in roots]
assert set(post) == \
set(roots_cubic(Poly(p.subs(y, z), x)))
# /!\ if p is not made an expression, this is *very* slow
assert all(p.subs({y: z, x: i}).n(2, chop=True) == 0 for i in post)
def test_issue_8285():
roots = (Poly(4*x**8 - 1, x)*Poly(x**2 + 1)).all_roots()
assert _check(roots)
f = Poly(x**4 + 5*x**2 + 6, x)
ro = [rootof(f, i) for i in range(4)]
roots = Poly(x**4 + 5*x**2 + 6, x).all_roots()
assert roots == ro
assert _check(roots)
# more than 2 complex roots from which to identify the
# imaginary ones
roots = Poly(2*x**8 - 1).all_roots()
assert _check(roots)
assert len(Poly(2*x**10 - 1).all_roots()) == 10 # doesn't fail
def test_issue_8289():
roots = (Poly(x**2 + 2)*Poly(x**4 + 2)).all_roots()
assert _check(roots)
roots = Poly(x**6 + 3*x**3 + 2, x).all_roots()
assert _check(roots)
roots = Poly(x**6 - x + 1).all_roots()
assert _check(roots)
# all imaginary roots with multiplicity of 2
roots = Poly(x**4 + 4*x**2 + 4, x).all_roots()
assert _check(roots)
def test_issue_14291():
assert Poly(((x - 1)**2 + 1)*((x - 1)**2 + 2)*(x - 1)
).all_roots() == [1, 1 - I, 1 + I, 1 - sqrt(2)*I, 1 + sqrt(2)*I]
p = x**4 + 10*x**2 + 1
ans = [rootof(p, i) for i in range(4)]
assert Poly(p).all_roots() == ans
_check(ans)
def test_issue_13340():
eq = Poly(y**3 + exp(x)*y + x, y, domain='EX')
roots_d = roots(eq)
assert len(roots_d) == 3
def test_issue_14522():
eq = Poly(x**4 + x**3*(16 + 32*I) + x**2*(-285 + 386*I) + x*(-2824 - 448*I) - 2058 - 6053*I, x)
roots_eq = roots(eq)
assert all(eq(r) == 0 for r in roots_eq)
def test_issue_15076():
sol = roots_quartic(Poly(t**4 - 6*t**2 + t/x - 3, t))
assert sol[0].has(x)
def test_issue_16589():
eq = Poly(x**4 - 8*sqrt(2)*x**3 + 4*x**3 - 64*sqrt(2)*x**2 + 1024*x, x)
roots_eq = roots(eq)
assert 0 in roots_eq
def test_roots_cubic():
assert roots_cubic(Poly(2*x**3, x)) == [0, 0, 0]
assert roots_cubic(Poly(x**3 - 3*x**2 + 3*x - 1, x)) == [1, 1, 1]
# valid for arbitrary y (issue 21263)
r = root(y, 3)
assert roots_cubic(Poly(x**3 - y, x)) == [r,
r*(-S.Half + sqrt(3)*I/2),
r*(-S.Half - sqrt(3)*I/2)]
# simpler form when y is negative
assert roots_cubic(Poly(x**3 - -1, x)) == \
[-1, S.Half - I*sqrt(3)/2, S.Half + I*sqrt(3)/2]
assert roots_cubic(Poly(2*x**3 - 3*x**2 - 3*x - 1, x))[0] == \
S.Half + 3**Rational(1, 3)/2 + 3**Rational(2, 3)/2
eq = -x**3 + 2*x**2 + 3*x - 2
assert roots(eq, trig=True, multiple=True) == \
roots_cubic(Poly(eq, x), trig=True) == [
Rational(2, 3) + 2*sqrt(13)*cos(acos(8*sqrt(13)/169)/3)/3,
-2*sqrt(13)*sin(-acos(8*sqrt(13)/169)/3 + pi/6)/3 + Rational(2, 3),
-2*sqrt(13)*cos(-acos(8*sqrt(13)/169)/3 + pi/3)/3 + Rational(2, 3),
]
def test_roots_quartic():
assert roots_quartic(Poly(x**4, x)) == [0, 0, 0, 0]
assert roots_quartic(Poly(x**4 + x**3, x)) in [
[-1, 0, 0, 0],
[0, -1, 0, 0],
[0, 0, -1, 0],
[0, 0, 0, -1]
]
assert roots_quartic(Poly(x**4 - x**3, x)) in [
[1, 0, 0, 0],
[0, 1, 0, 0],
[0, 0, 1, 0],
[0, 0, 0, 1]
]
lhs = roots_quartic(Poly(x**4 + x, x))
rhs = [S.Half + I*sqrt(3)/2, S.Half - I*sqrt(3)/2, S.Zero, -S.One]
assert sorted(lhs, key=hash) == sorted(rhs, key=hash)
# test of all branches of roots quartic
for i, (a, b, c, d) in enumerate([(1, 2, 3, 0),
(3, -7, -9, 9),
(1, 2, 3, 4),
(1, 2, 3, 4),
(-7, -3, 3, -6),
(-3, 5, -6, -4),
(6, -5, -10, -3)]):
if i == 2:
c = -a*(a**2/S(8) - b/S(2))
elif i == 3:
d = a*(a*(a**2*Rational(3, 256) - b/S(16)) + c/S(4))
eq = x**4 + a*x**3 + b*x**2 + c*x + d
ans = roots_quartic(Poly(eq, x))
assert all(eq.subs(x, ai).n(chop=True) == 0 for ai in ans)
# not all symbolic quartics are unresolvable
eq = Poly(q*x + q/4 + x**4 + x**3 + 2*x**2 - Rational(1, 3), x)
sol = roots_quartic(eq)
assert all(verify_numerically(eq.subs(x, i), 0) for i in sol)
z = symbols('z', negative=True)
eq = x**4 + 2*x**3 + 3*x**2 + x*(z + 11) + 5
zans = roots_quartic(Poly(eq, x))
assert all([verify_numerically(eq.subs(((x, i), (z, -1))), 0) for i in zans])
# but some are (see also issue 4989)
# it's ok if the solution is not Piecewise, but the tests below should pass
eq = Poly(y*x**4 + x**3 - x + z, x)
ans = roots_quartic(eq)
assert all(type(i) == Piecewise for i in ans)
reps = (
dict(y=Rational(-1, 3), z=Rational(-1, 4)), # 4 real
dict(y=Rational(-1, 3), z=Rational(-1, 2)), # 2 real
dict(y=Rational(-1, 3), z=-2)) # 0 real
for rep in reps:
sol = roots_quartic(Poly(eq.subs(rep), x))
assert all([verify_numerically(w.subs(rep) - s, 0) for w, s in zip(ans, sol)])
def test_issue_21287():
assert not any(isinstance(i, Piecewise) for i in roots_quartic(
Poly(x**4 - x**2*(3 + 5*I) + 2*x*(-1 + I) - 1 + 3*I, x)))
def test_roots_cyclotomic():
assert roots_cyclotomic(cyclotomic_poly(1, x, polys=True)) == [1]
assert roots_cyclotomic(cyclotomic_poly(2, x, polys=True)) == [-1]
assert roots_cyclotomic(cyclotomic_poly(
3, x, polys=True)) == [Rational(-1, 2) - I*sqrt(3)/2, Rational(-1, 2) + I*sqrt(3)/2]
assert roots_cyclotomic(cyclotomic_poly(4, x, polys=True)) == [-I, I]
assert roots_cyclotomic(cyclotomic_poly(
6, x, polys=True)) == [S.Half - I*sqrt(3)/2, S.Half + I*sqrt(3)/2]
assert roots_cyclotomic(cyclotomic_poly(7, x, polys=True)) == [
-cos(pi/7) - I*sin(pi/7),
-cos(pi/7) + I*sin(pi/7),
-cos(pi*Rational(3, 7)) - I*sin(pi*Rational(3, 7)),
-cos(pi*Rational(3, 7)) + I*sin(pi*Rational(3, 7)),
cos(pi*Rational(2, 7)) - I*sin(pi*Rational(2, 7)),
cos(pi*Rational(2, 7)) + I*sin(pi*Rational(2, 7)),
]
assert roots_cyclotomic(cyclotomic_poly(8, x, polys=True)) == [
-sqrt(2)/2 - I*sqrt(2)/2,
-sqrt(2)/2 + I*sqrt(2)/2,
sqrt(2)/2 - I*sqrt(2)/2,
sqrt(2)/2 + I*sqrt(2)/2,
]
assert roots_cyclotomic(cyclotomic_poly(12, x, polys=True)) == [
-sqrt(3)/2 - I/2,
-sqrt(3)/2 + I/2,
sqrt(3)/2 - I/2,
sqrt(3)/2 + I/2,
]
assert roots_cyclotomic(
cyclotomic_poly(1, x, polys=True), factor=True) == [1]
assert roots_cyclotomic(
cyclotomic_poly(2, x, polys=True), factor=True) == [-1]
assert roots_cyclotomic(cyclotomic_poly(3, x, polys=True), factor=True) == \
[-root(-1, 3), -1 + root(-1, 3)]
assert roots_cyclotomic(cyclotomic_poly(4, x, polys=True), factor=True) == \
[-I, I]
assert roots_cyclotomic(cyclotomic_poly(5, x, polys=True), factor=True) == \
[-root(-1, 5), -root(-1, 5)**3, root(-1, 5)**2, -1 - root(-1, 5)**2 + root(-1, 5) + root(-1, 5)**3]
assert roots_cyclotomic(cyclotomic_poly(6, x, polys=True), factor=True) == \
[1 - root(-1, 3), root(-1, 3)]
def test_roots_binomial():
assert roots_binomial(Poly(5*x, x)) == [0]
assert roots_binomial(Poly(5*x**4, x)) == [0, 0, 0, 0]
assert roots_binomial(Poly(5*x + 2, x)) == [Rational(-2, 5)]
A = 10**Rational(3, 4)/10
assert roots_binomial(Poly(5*x**4 + 2, x)) == \
[-A - A*I, -A + A*I, A - A*I, A + A*I]
_check(roots_binomial(Poly(x**8 - 2)))
a1 = Symbol('a1', nonnegative=True)
b1 = Symbol('b1', nonnegative=True)
r0 = roots_quadratic(Poly(a1*x**2 + b1, x))
r1 = roots_binomial(Poly(a1*x**2 + b1, x))
assert powsimp(r0[0]) == powsimp(r1[0])
assert powsimp(r0[1]) == powsimp(r1[1])
for a, b, s, n in cartes((1, 2), (1, 2), (-1, 1), (2, 3, 4, 5)):
if a == b and a != 1: # a == b == 1 is sufficient
continue
p = Poly(a*x**n + s*b)
ans = roots_binomial(p)
assert ans == _nsort(ans)
# issue 8813
assert roots(Poly(2*x**3 - 16*y**3, x)) == {
2*y*(Rational(-1, 2) - sqrt(3)*I/2): 1,
2*y: 1,
2*y*(Rational(-1, 2) + sqrt(3)*I/2): 1}
def test_roots_preprocessing():
f = a*y*x**2 + y - b
coeff, poly = preprocess_roots(Poly(f, x))
assert coeff == 1
assert poly == Poly(a*y*x**2 + y - b, x)
f = c**3*x**3 + c**2*x**2 + c*x + a
coeff, poly = preprocess_roots(Poly(f, x))
assert coeff == 1/c
assert poly == Poly(x**3 + x**2 + x + a, x)
f = c**3*x**3 + c**2*x**2 + a
coeff, poly = preprocess_roots(Poly(f, x))
assert coeff == 1/c
assert poly == Poly(x**3 + x**2 + a, x)
f = c**3*x**3 + c*x + a
coeff, poly = preprocess_roots(Poly(f, x))
assert coeff == 1/c
assert poly == Poly(x**3 + x + a, x)
f = c**3*x**3 + a
coeff, poly = preprocess_roots(Poly(f, x))
assert coeff == 1/c
assert poly == Poly(x**3 + a, x)
E, F, J, L = symbols("E,F,J,L")
f = -21601054687500000000*E**8*J**8/L**16 + \
508232812500000000*F*x*E**7*J**7/L**14 - \
4269543750000000*E**6*F**2*J**6*x**2/L**12 + \
16194716250000*E**5*F**3*J**5*x**3/L**10 - \
27633173750*E**4*F**4*J**4*x**4/L**8 + \
14840215*E**3*F**5*J**3*x**5/L**6 + \
54794*E**2*F**6*J**2*x**6/(5*L**4) - \
1153*E*J*F**7*x**7/(80*L**2) + \
633*F**8*x**8/160000
coeff, poly = preprocess_roots(Poly(f, x))
assert coeff == 20*E*J/(F*L**2)
assert poly == 633*x**8 - 115300*x**7 + 4383520*x**6 + 296804300*x**5 - 27633173750*x**4 + \
809735812500*x**3 - 10673859375000*x**2 + 63529101562500*x - 135006591796875
f = Poly(-y**2 + x**2*exp(x), y, domain=ZZ[x, exp(x)])
g = Poly(-y**2 + exp(x), y, domain=ZZ[exp(x)])
assert preprocess_roots(f) == (x, g)
def test_roots0():
assert roots(1, x) == {}
assert roots(x, x) == {S.Zero: 1}
assert roots(x**9, x) == {S.Zero: 9}
assert roots(((x - 2)*(x + 3)*(x - 4)).expand(), x) == {-S(3): 1, S(2): 1, S(4): 1}
assert roots(2*x + 1, x) == {Rational(-1, 2): 1}
assert roots((2*x + 1)**2, x) == {Rational(-1, 2): 2}
assert roots((2*x + 1)**5, x) == {Rational(-1, 2): 5}
assert roots((2*x + 1)**10, x) == {Rational(-1, 2): 10}
assert roots(x**4 - 1, x) == {I: 1, S.One: 1, -S.One: 1, -I: 1}
assert roots((x**4 - 1)**2, x) == {I: 2, S.One: 2, -S.One: 2, -I: 2}
assert roots(((2*x - 3)**2).expand(), x) == {Rational( 3, 2): 2}
assert roots(((2*x + 3)**2).expand(), x) == {Rational(-3, 2): 2}
assert roots(((2*x - 3)**3).expand(), x) == {Rational( 3, 2): 3}
assert roots(((2*x + 3)**3).expand(), x) == {Rational(-3, 2): 3}
assert roots(((2*x - 3)**5).expand(), x) == {Rational( 3, 2): 5}
assert roots(((2*x + 3)**5).expand(), x) == {Rational(-3, 2): 5}
assert roots(((a*x - b)**5).expand(), x) == { b/a: 5}
assert roots(((a*x + b)**5).expand(), x) == {-b/a: 5}
assert roots(x**2 + (-a - 1)*x + a, x) == {a: 1, S.One: 1}
assert roots(x**4 - 2*x**2 + 1, x) == {S.One: 2, S.NegativeOne: 2}
assert roots(x**6 - 4*x**4 + 4*x**3 - x**2, x) == \
{S.One: 2, -1 - sqrt(2): 1, S.Zero: 2, -1 + sqrt(2): 1}
assert roots(x**8 - 1, x) == {
sqrt(2)/2 + I*sqrt(2)/2: 1,
sqrt(2)/2 - I*sqrt(2)/2: 1,
-sqrt(2)/2 + I*sqrt(2)/2: 1,
-sqrt(2)/2 - I*sqrt(2)/2: 1,
S.One: 1, -S.One: 1, I: 1, -I: 1
}
f = -2016*x**2 - 5616*x**3 - 2056*x**4 + 3324*x**5 + 2176*x**6 - \
224*x**7 - 384*x**8 - 64*x**9
assert roots(f) == {S.Zero: 2, -S(2): 2, S(2): 1, Rational(-7, 2): 1,
Rational(-3, 2): 1, Rational(-1, 2): 1, Rational(3, 2): 1}
assert roots((a + b + c)*x - (a + b + c + d), x) == {(a + b + c + d)/(a + b + c): 1}
assert roots(x**3 + x**2 - x + 1, x, cubics=False) == {}
assert roots(((x - 2)*(
x + 3)*(x - 4)).expand(), x, cubics=False) == {-S(3): 1, S(2): 1, S(4): 1}
assert roots(((x - 2)*(x + 3)*(x - 4)*(x - 5)).expand(), x, cubics=False) == \
{-S(3): 1, S(2): 1, S(4): 1, S(5): 1}
assert roots(x**3 + 2*x**2 + 4*x + 8, x) == {-S(2): 1, -2*I: 1, 2*I: 1}
assert roots(x**3 + 2*x**2 + 4*x + 8, x, cubics=True) == \
{-2*I: 1, 2*I: 1, -S(2): 1}
assert roots((x**2 - x)*(x**3 + 2*x**2 + 4*x + 8), x ) == \
{S.One: 1, S.Zero: 1, -S(2): 1, -2*I: 1, 2*I: 1}
r1_2, r1_3 = S.Half, Rational(1, 3)
x0 = (3*sqrt(33) + 19)**r1_3
x1 = 4/x0/3
x2 = x0/3
x3 = sqrt(3)*I/2
x4 = x3 - r1_2
x5 = -x3 - r1_2
assert roots(x**3 + x**2 - x + 1, x, cubics=True) == {
-x1 - x2 - r1_3: 1,
-x1/x4 - x2*x4 - r1_3: 1,
-x1/x5 - x2*x5 - r1_3: 1,
}
f = (x**2 + 2*x + 3).subs(x, 2*x**2 + 3*x).subs(x, 5*x - 4)
r13_20, r1_20 = [ Rational(*r)
for r in ((13, 20), (1, 20)) ]
s2 = sqrt(2)
assert roots(f, x) == {
r13_20 + r1_20*sqrt(1 - 8*I*s2): 1,
r13_20 - r1_20*sqrt(1 - 8*I*s2): 1,
r13_20 + r1_20*sqrt(1 + 8*I*s2): 1,
r13_20 - r1_20*sqrt(1 + 8*I*s2): 1,
}
f = x**4 + x**3 + x**2 + x + 1
r1_4, r1_8, r5_8 = [ Rational(*r) for r in ((1, 4), (1, 8), (5, 8)) ]
assert roots(f, x) == {
-r1_4 + r1_4*5**r1_2 + I*(r5_8 + r1_8*5**r1_2)**r1_2: 1,
-r1_4 + r1_4*5**r1_2 - I*(r5_8 + r1_8*5**r1_2)**r1_2: 1,
-r1_4 - r1_4*5**r1_2 + I*(r5_8 - r1_8*5**r1_2)**r1_2: 1,
-r1_4 - r1_4*5**r1_2 - I*(r5_8 - r1_8*5**r1_2)**r1_2: 1,
}
f = z**3 + (-2 - y)*z**2 + (1 + 2*y - 2*x**2)*z - y + 2*x**2
assert roots(f, z) == {
S.One: 1,
S.Half + S.Half*y + S.Half*sqrt(1 - 2*y + y**2 + 8*x**2): 1,
S.Half + S.Half*y - S.Half*sqrt(1 - 2*y + y**2 + 8*x**2): 1,
}
assert roots(a*b*c*x**3 + 2*x**2 + 4*x + 8, x, cubics=False) == {}
assert roots(a*b*c*x**3 + 2*x**2 + 4*x + 8, x, cubics=True) != {}
assert roots(x**4 - 1, x, filter='Z') == {S.One: 1, -S.One: 1}
assert roots(x**4 - 1, x, filter='I') == {I: 1, -I: 1}
assert roots((x - 1)*(x + 1), x) == {S.One: 1, -S.One: 1}
assert roots(
(x - 1)*(x + 1), x, predicate=lambda r: r.is_positive) == {S.One: 1}
assert roots(x**4 - 1, x, filter='Z', multiple=True) == [-S.One, S.One]
assert roots(x**4 - 1, x, filter='I', multiple=True) == [I, -I]
ar, br = symbols('a, b', real=True)
p = x**2*(ar-br)**2 + 2*x*(br-ar) + 1
assert roots(p, x, filter='R') == {1/(ar - br): 2}
assert roots(x**3, x, multiple=True) == [S.Zero, S.Zero, S.Zero]
assert roots(1234, x, multiple=True) == []
f = x**6 - x**5 + x**4 - x**3 + x**2 - x + 1
assert roots(f) == {
-I*sin(pi/7) + cos(pi/7): 1,
-I*sin(pi*Rational(2, 7)) - cos(pi*Rational(2, 7)): 1,
-I*sin(pi*Rational(3, 7)) + cos(pi*Rational(3, 7)): 1,
I*sin(pi/7) + cos(pi/7): 1,
I*sin(pi*Rational(2, 7)) - cos(pi*Rational(2, 7)): 1,
I*sin(pi*Rational(3, 7)) + cos(pi*Rational(3, 7)): 1,
}
g = ((x**2 + 1)*f**2).expand()
assert roots(g) == {
-I*sin(pi/7) + cos(pi/7): 2,
-I*sin(pi*Rational(2, 7)) - cos(pi*Rational(2, 7)): 2,
-I*sin(pi*Rational(3, 7)) + cos(pi*Rational(3, 7)): 2,
I*sin(pi/7) + cos(pi/7): 2,
I*sin(pi*Rational(2, 7)) - cos(pi*Rational(2, 7)): 2,
I*sin(pi*Rational(3, 7)) + cos(pi*Rational(3, 7)): 2,
-I: 1, I: 1,
}
r = roots(x**3 + 40*x + 64)
real_root = [rx for rx in r if rx.is_real][0]
cr = 108 + 6*sqrt(1074)
assert real_root == -2*root(cr, 3)/3 + 20/root(cr, 3)
eq = Poly((7 + 5*sqrt(2))*x**3 + (-6 - 4*sqrt(2))*x**2 + (-sqrt(2) - 1)*x + 2, x, domain='EX')
assert roots(eq) == {-1 + sqrt(2): 1, -2 + 2*sqrt(2): 1, -sqrt(2) + 1: 1}
eq = Poly(41*x**5 + 29*sqrt(2)*x**5 - 153*x**4 - 108*sqrt(2)*x**4 +
175*x**3 + 125*sqrt(2)*x**3 - 45*x**2 - 30*sqrt(2)*x**2 - 26*sqrt(2)*x -
26*x + 24, x, domain='EX')
assert roots(eq) == {-sqrt(2) + 1: 1, -2 + 2*sqrt(2): 1, -1 + sqrt(2): 1,
-4 + 4*sqrt(2): 1, -3 + 3*sqrt(2): 1}
eq = Poly(x**3 - 2*x**2 + 6*sqrt(2)*x**2 - 8*sqrt(2)*x + 23*x - 14 +
14*sqrt(2), x, domain='EX')
assert roots(eq) == {-2*sqrt(2) + 2: 1, -2*sqrt(2) + 1: 1, -2*sqrt(2) - 1: 1}
assert roots(Poly((x + sqrt(2))**3 - 7, x, domain='EX')) == \
{-sqrt(2) + root(7, 3)*(-S.Half - sqrt(3)*I/2): 1,
-sqrt(2) + root(7, 3)*(-S.Half + sqrt(3)*I/2): 1,
-sqrt(2) + root(7, 3): 1}
def test_roots_slow():
"""Just test that calculating these roots does not hang. """
a, b, c, d, x = symbols("a,b,c,d,x")
f1 = x**2*c + (a/b) + x*c*d - a
f2 = x**2*(a + b*(c - d)*a) + x*a*b*c/(b*d - d) + (a*d - c/d)
assert list(roots(f1, x).values()) == [1, 1]
assert list(roots(f2, x).values()) == [1, 1]
(zz, yy, xx, zy, zx, yx, k) = symbols("zz,yy,xx,zy,zx,yx,k")
e1 = (zz - k)*(yy - k)*(xx - k) + zy*yx*zx + zx - zy - yx
e2 = (zz - k)*yx*yx + zx*(yy - k)*zx + zy*zy*(xx - k)
assert list(roots(e1 - e2, k).values()) == [1, 1, 1]
f = x**3 + 2*x**2 + 8
R = list(roots(f).keys())
assert not any(i for i in [f.subs(x, ri).n(chop=True) for ri in R])
def test_roots_inexact():
R1 = roots(x**2 + x + 1, x, multiple=True)
R2 = roots(x**2 + x + 1.0, x, multiple=True)
for r1, r2 in zip(R1, R2):
assert abs(r1 - r2) < 1e-12
f = x**4 + 3.0*sqrt(2.0)*x**3 - (78.0 + 24.0*sqrt(3.0))*x**2 \
+ 144.0*(2*sqrt(3.0) + 9.0)
R1 = roots(f, multiple=True)
R2 = (-12.7530479110482, -3.85012393732929,
4.89897948556636, 7.46155167569183)
for r1, r2 in zip(R1, R2):
assert abs(r1 - r2) < 1e-10
def test_roots_preprocessed():
E, F, J, L = symbols("E,F,J,L")
f = -21601054687500000000*E**8*J**8/L**16 + \
508232812500000000*F*x*E**7*J**7/L**14 - \
4269543750000000*E**6*F**2*J**6*x**2/L**12 + \
16194716250000*E**5*F**3*J**5*x**3/L**10 - \
27633173750*E**4*F**4*J**4*x**4/L**8 + \
14840215*E**3*F**5*J**3*x**5/L**6 + \
54794*E**2*F**6*J**2*x**6/(5*L**4) - \
1153*E*J*F**7*x**7/(80*L**2) + \
633*F**8*x**8/160000
assert roots(f, x) == {}
R1 = roots(f.evalf(), x, multiple=True)
R2 = [-1304.88375606366, 97.1168816800648, 186.946430171876, 245.526792947065,
503.441004174773, 791.549343830097, 1273.16678129348, 1850.10650616851]
w = Wild('w')
p = w*E*J/(F*L**2)
assert len(R1) == len(R2)
for r1, r2 in zip(R1, R2):
match = r1.match(p)
assert match is not None and abs(match[w] - r2) < 1e-10
def test_roots_mixed():
f = -1936 - 5056*x - 7592*x**2 + 2704*x**3 - 49*x**4
_re, _im = intervals(f, all=True)
_nroots = nroots(f)
_sroots = roots(f, multiple=True)
_re = [ Interval(a, b) for (a, b), _ in _re ]
_im = [ Interval(re(a), re(b))*Interval(im(a), im(b)) for (a, b),
_ in _im ]
_intervals = _re + _im
_sroots = [ r.evalf() for r in _sroots ]
_nroots = sorted(_nroots, key=lambda x: x.sort_key())
_sroots = sorted(_sroots, key=lambda x: x.sort_key())
for _roots in (_nroots, _sroots):
for i, r in zip(_intervals, _roots):
if r.is_real:
assert r in i
else:
assert (re(r), im(r)) in i
def test_root_factors():
assert root_factors(Poly(1, x)) == [Poly(1, x)]
assert root_factors(Poly(x, x)) == [Poly(x, x)]
assert root_factors(x**2 - 1, x) == [x + 1, x - 1]
assert root_factors(x**2 - y, x) == [x - sqrt(y), x + sqrt(y)]
assert root_factors((x**4 - 1)**2) == \
[x + 1, x + 1, x - 1, x - 1, x - I, x - I, x + I, x + I]
assert root_factors(Poly(x**4 - 1, x), filter='Z') == \
[Poly(x + 1, x), Poly(x - 1, x), Poly(x**2 + 1, x)]
assert root_factors(8*x**2 + 12*x**4 + 6*x**6 + x**8, x, filter='Q') == \
[x, x, x**6 + 6*x**4 + 12*x**2 + 8]
@slow
def test_nroots1():
n = 64
p = legendre_poly(n, x, polys=True)
raises(mpmath.mp.NoConvergence, lambda: p.nroots(n=3, maxsteps=5))
roots = p.nroots(n=3)
# The order of roots matters. They are ordered from smallest to the
# largest.
assert [str(r) for r in roots] == \
['-0.999', '-0.996', '-0.991', '-0.983', '-0.973', '-0.961',
'-0.946', '-0.930', '-0.911', '-0.889', '-0.866', '-0.841',
'-0.813', '-0.784', '-0.753', '-0.720', '-0.685', '-0.649',
'-0.611', '-0.572', '-0.531', '-0.489', '-0.446', '-0.402',
'-0.357', '-0.311', '-0.265', '-0.217', '-0.170', '-0.121',
'-0.0730', '-0.0243', '0.0243', '0.0730', '0.121', '0.170',
'0.217', '0.265', '0.311', '0.357', '0.402', '0.446', '0.489',
'0.531', '0.572', '0.611', '0.649', '0.685', '0.720', '0.753',
'0.784', '0.813', '0.841', '0.866', '0.889', '0.911', '0.930',
'0.946', '0.961', '0.973', '0.983', '0.991', '0.996', '0.999']
def test_nroots2():
p = Poly(x**5 + 3*x + 1, x)
roots = p.nroots(n=3)
# The order of roots matters. The roots are ordered by their real
# components (if they agree, then by their imaginary components),
# with real roots appearing first.
assert [str(r) for r in roots] == \
['-0.332', '-0.839 - 0.944*I', '-0.839 + 0.944*I',
'1.01 - 0.937*I', '1.01 + 0.937*I']
roots = p.nroots(n=5)
assert [str(r) for r in roots] == \
['-0.33199', '-0.83907 - 0.94385*I', '-0.83907 + 0.94385*I',
'1.0051 - 0.93726*I', '1.0051 + 0.93726*I']
def test_roots_composite():
assert len(roots(Poly(y**3 + y**2*sqrt(x) + y + x, y, composite=True))) == 3
def test_issue_19113():
eq = cos(x)**3 - cos(x) + 1
raises(PolynomialError, lambda: roots(eq))
|
bf75662965d07b23eec0d1b28b37d1e9915be0691167def14441dac2e2d6aa98 | """Finite extensions of ring domains."""
from sympy.polys.domains.domain import Domain
from sympy.polys.domains.domainelement import DomainElement
from sympy.polys.polyerrors import (CoercionFailed, NotInvertible,
GeneratorsError)
from sympy.polys.polytools import Poly
from sympy.printing.defaults import DefaultPrinting
class ExtensionElement(DomainElement, DefaultPrinting):
"""
Element of a finite extension.
A class of univariate polynomials modulo the ``modulus``
of the extension ``ext``. It is represented by the
unique polynomial ``rep`` of lowest degree. Both
``rep`` and the representation ``mod`` of ``modulus``
are of class DMP.
"""
__slots__ = ('rep', 'ext')
def __init__(self, rep, ext):
self.rep = rep
self.ext = ext
def parent(f):
return f.ext
def __bool__(f):
return bool(f.rep)
def __pos__(f):
return f
def __neg__(f):
return ExtElem(-f.rep, f.ext)
def _get_rep(f, g):
if isinstance(g, ExtElem):
if g.ext == f.ext:
return g.rep
else:
return None
else:
try:
g = f.ext.convert(g)
return g.rep
except CoercionFailed:
return None
def __add__(f, g):
rep = f._get_rep(g)
if rep is not None:
return ExtElem(f.rep + rep, f.ext)
else:
return NotImplemented
__radd__ = __add__
def __sub__(f, g):
rep = f._get_rep(g)
if rep is not None:
return ExtElem(f.rep - rep, f.ext)
else:
return NotImplemented
def __rsub__(f, g):
rep = f._get_rep(g)
if rep is not None:
return ExtElem(rep - f.rep, f.ext)
else:
return NotImplemented
def __mul__(f, g):
rep = f._get_rep(g)
if rep is not None:
return ExtElem((f.rep * rep) % f.ext.mod, f.ext)
else:
return NotImplemented
__rmul__ = __mul__
def _divcheck(f):
"""Raise if division is not implemented for this divisor"""
if not f:
raise NotInvertible('Zero divisor')
elif f.ext.is_Field:
return True
elif f.rep.is_ground and f.ext.domain.is_unit(f.rep.rep[0]):
return True
else:
# Some cases like (2*x + 2)/2 over ZZ will fail here. It is
# unclear how to implement division in general if the ground
# domain is not a field so for now it was decided to restrict the
# implementation to division by invertible constants.
msg = (f"Can not invert {f} in {f.ext}. "
"Only division by invertible constants is implemented.")
raise NotImplementedError(msg)
def inverse(f):
"""Multiplicative inverse.
Raises
======
NotInvertible
If the element is a zero divisor.
"""
f._divcheck()
if f.ext.is_Field:
invrep = f.rep.invert(f.ext.mod)
else:
R = f.ext.ring
invrep = R.exquo(R.one, f.rep)
return ExtElem(invrep, f.ext)
def __truediv__(f, g):
rep = f._get_rep(g)
if rep is None:
return NotImplemented
g = ExtElem(rep, f.ext)
try:
ginv = g.inverse()
except NotInvertible:
raise ZeroDivisionError(f"{f} / {g}")
return f * ginv
__floordiv__ = __truediv__
def __rtruediv__(f, g):
try:
g = f.ext.convert(g)
except CoercionFailed:
return NotImplemented
return g / f
__rfloordiv__ = __rtruediv__
def __mod__(f, g):
rep = f._get_rep(g)
if rep is None:
return NotImplemented
g = ExtElem(rep, f.ext)
try:
g._divcheck()
except NotInvertible:
raise ZeroDivisionError(f"{f} % {g}")
# Division where defined is always exact so there is no remainder
return f.ext.zero
def __rmod__(f, g):
try:
g = f.ext.convert(g)
except CoercionFailed:
return NotImplemented
return g % f
def __pow__(f, n):
if not isinstance(n, int):
raise TypeError("exponent of type 'int' expected")
if n < 0:
try:
f, n = f.inverse(), -n
except NotImplementedError:
raise ValueError("negative powers are not defined")
b = f.rep
m = f.ext.mod
r = f.ext.one.rep
while n > 0:
if n % 2:
r = (r*b) % m
b = (b*b) % m
n //= 2
return ExtElem(r, f.ext)
def __eq__(f, g):
if isinstance(g, ExtElem):
return f.rep == g.rep and f.ext == g.ext
else:
return NotImplemented
def __ne__(f, g):
return not f == g
def __hash__(f):
return hash((f.rep, f.ext))
def __str__(f):
from sympy.printing.str import sstr
return sstr(f.rep)
__repr__ = __str__
@property
def is_ground(f):
return f.rep.is_ground
def to_ground(f):
[c] = f.rep.to_list()
return c
ExtElem = ExtensionElement
class MonogenicFiniteExtension(Domain):
r"""
Finite extension generated by an integral element.
The generator is defined by a monic univariate
polynomial derived from the argument ``mod``.
A shorter alias is ``FiniteExtension``.
Examples
========
Quadratic integer ring $\mathbb{Z}[\sqrt2]$:
>>> from sympy import Symbol, Poly
>>> from sympy.polys.agca.extensions import FiniteExtension
>>> x = Symbol('x')
>>> R = FiniteExtension(Poly(x**2 - 2)); R
ZZ[x]/(x**2 - 2)
>>> R.rank
2
>>> R(1 + x)*(3 - 2*x)
x - 1
Finite field $GF(5^3)$ defined by the primitive
polynomial $x^3 + x^2 + 2$ (over $\mathbb{Z}_5$).
>>> F = FiniteExtension(Poly(x**3 + x**2 + 2, modulus=5)); F
GF(5)[x]/(x**3 + x**2 + 2)
>>> F.basis
(1, x, x**2)
>>> F(x + 3)/(x**2 + 2)
-2*x**2 + x + 2
Function field of an elliptic curve:
>>> t = Symbol('t')
>>> FiniteExtension(Poly(t**2 - x**3 - x + 1, t, field=True))
ZZ(x)[t]/(t**2 - x**3 - x + 1)
"""
is_FiniteExtension = True
dtype = ExtensionElement
def __init__(self, mod):
if not (isinstance(mod, Poly) and mod.is_univariate):
raise TypeError("modulus must be a univariate Poly")
# Using auto=True (default) potentially changes the ground domain to a
# field whereas auto=False raises if division is not exact. We'll let
# the caller decide whether or not they want to put the ground domain
# over a field. In most uses mod is already monic.
mod = mod.monic(auto=False)
self.rank = mod.degree()
self.modulus = mod
self.mod = mod.rep # DMP representation
self.domain = dom = mod.domain
self.ring = mod.rep.ring or dom.old_poly_ring(*mod.gens)
self.zero = self.convert(self.ring.zero)
self.one = self.convert(self.ring.one)
gen = self.ring.gens[0]
self.symbol = self.ring.symbols[0]
self.generator = self.convert(gen)
self.basis = tuple(self.convert(gen**i) for i in range(self.rank))
# XXX: It might be necessary to check mod.is_irreducible here
self.is_Field = self.domain.is_Field
def new(self, arg):
rep = self.ring.convert(arg)
return ExtElem(rep % self.mod, self)
def __eq__(self, other):
if not isinstance(other, FiniteExtension):
return False
return self.modulus == other.modulus
def __hash__(self):
return hash((self.__class__.__name__, self.modulus))
def __str__(self):
return "%s/(%s)" % (self.ring, self.modulus.as_expr())
__repr__ = __str__
def convert(self, f, base=None):
rep = self.ring.convert(f, base)
return ExtElem(rep % self.mod, self)
def convert_from(self, f, base):
rep = self.ring.convert(f, base)
return ExtElem(rep % self.mod, self)
def to_sympy(self, f):
return self.ring.to_sympy(f.rep)
def from_sympy(self, f):
return self.convert(f)
def set_domain(self, K):
mod = self.modulus.set_domain(K)
return self.__class__(mod)
def drop(self, *symbols):
if self.symbol in symbols:
raise GeneratorsError('Can not drop generator from FiniteExtension')
K = self.domain.drop(*symbols)
return self.set_domain(K)
def quo(self, f, g):
return self.exquo(f, g)
def exquo(self, f, g):
rep = self.ring.exquo(f.rep, g.rep)
return ExtElem(rep % self.mod, self)
def is_negative(self, a):
return False
def is_unit(self, a):
if self.is_Field:
return bool(a)
elif a.is_ground:
return self.domain.is_unit(a.to_ground())
FiniteExtension = MonogenicFiniteExtension
|
076724f1ec36ec8e7c15508f2511dec861a11f96c869b9f240c80755d8df5339 | """
Module for the DomainMatrix class.
A DomainMatrix represents a matrix with elements that are in a particular
Domain. Each DomainMatrix internally wraps a DDM which is used for the
lower-level operations. The idea is that the DomainMatrix class provides the
convenience routines for converting between Expr and the poly domains as well
as unifying matrices with different domains.
"""
from sympy.core.sympify import _sympify
from ..constructor import construct_domain
from .exceptions import (NonSquareMatrixError, ShapeError, DDMShapeError,
DDMDomainError, DDMFormatError)
from .ddm import DDM
from .sdm import SDM
from .domainscalar import DomainScalar
from sympy.polys.domains import ZZ
class DomainMatrix:
r"""
Associate Matrix with :py:class:`~.Domain`
Explanation
===========
DomainMatrix uses :py:class:`~.Domain` for its internal representation
which makes it more faster for many common operations
than current sympy Matrix class, but this advantage makes it not
entirely compatible with Matrix.
DomainMatrix could be found analogous to numpy arrays with "dtype".
In the DomainMatrix, each matrix has a domain such as :ref:`ZZ`
or :ref:`QQ(a)`.
Examples
========
Creating a DomainMatrix from the existing Matrix class:
>>> from sympy import Matrix
>>> from sympy.polys.matrices import DomainMatrix
>>> Matrix1 = Matrix([
... [1, 2],
... [3, 4]])
>>> A = DomainMatrix.from_Matrix(Matrix1)
>>> A
DomainMatrix([[1, 2], [3, 4]], (2, 2), ZZ)
Driectly forming a DomainMatrix:
>>> from sympy import ZZ
>>> from sympy.polys.matrices import DomainMatrix
>>> A = DomainMatrix([
... [ZZ(1), ZZ(2)],
... [ZZ(3), ZZ(4)]], (2, 2), ZZ)
>>> A
DomainMatrix([[1, 2], [3, 4]], (2, 2), ZZ)
See Also
========
DDM
SDM
Domain
Poly
"""
def __new__(cls, rows, shape, domain, *, fmt=None):
"""
Creates a :py:class:`~.DomainMatrix`.
Parameters
==========
rows : Represents elements of DomainMatrix as list of lists
shape : Represents dimension of DomainMatrix
domain : Represents :py:class:`~.Domain` of DomainMatrix
Raises
======
TypeError
If any of rows, shape and domain are not provided
"""
if isinstance(rows, (DDM, SDM)):
raise TypeError("Use from_rep to initialise from SDM/DDM")
elif isinstance(rows, list):
rep = DDM(rows, shape, domain)
elif isinstance(rows, dict):
rep = SDM(rows, shape, domain)
else:
msg = "Input should be list-of-lists or dict-of-dicts"
raise TypeError(msg)
if fmt is not None:
if fmt == 'sparse':
rep = rep.to_sdm()
elif fmt == 'dense':
rep = rep.to_ddm()
else:
raise ValueError("fmt should be 'sparse' or 'dense'")
return cls.from_rep(rep)
@classmethod
def from_rep(cls, rep):
"""Create a new DomainMatrix efficiently from DDM/SDM.
Examples
========
Create a :py:class:`~.DomainMatrix` with an dense internal
representation as :py:class:`~.DDM`:
>>> from sympy.polys.domains import ZZ
>>> from sympy.polys.matrices import DomainMatrix
>>> from sympy.polys.matrices.ddm import DDM
>>> drep = DDM([[ZZ(1), ZZ(2)], [ZZ(3), ZZ(4)]], (2, 2), ZZ)
>>> dM = DomainMatrix.from_rep(drep)
>>> dM
DomainMatrix([[1, 2], [3, 4]], (2, 2), ZZ)
Create a :py:class:`~.DomainMatrix` with a sparse internal
representation as :py:class:`~.SDM`:
>>> from sympy.polys.matrices import DomainMatrix
>>> from sympy.polys.matrices.sdm import SDM
>>> from sympy import ZZ
>>> drep = SDM({0:{1:ZZ(1)},1:{0:ZZ(2)}}, (2, 2), ZZ)
>>> dM = DomainMatrix.from_rep(drep)
>>> dM
DomainMatrix({0: {1: 1}, 1: {0: 2}}, (2, 2), ZZ)
Parameters
==========
rep: SDM or DDM
The internal sparse or dense representation of the matrix.
Returns
=======
DomainMatrix
A :py:class:`~.DomainMatrix` wrapping *rep*.
Notes
=====
This takes ownership of rep as its internal representation. If rep is
being mutated elsewhere then a copy should be provided to
``from_rep``. Only minimal verification or checking is done on *rep*
as this is supposed to be an efficient internal routine.
"""
if not isinstance(rep, (DDM, SDM)):
raise TypeError("rep should be of type DDM or SDM")
self = super().__new__(cls)
self.rep = rep
self.shape = rep.shape
self.domain = rep.domain
return self
@classmethod
def from_list_sympy(cls, nrows, ncols, rows, **kwargs):
r"""
Convert a list of lists of Expr into a DomainMatrix using construct_domain
Parameters
==========
nrows: number of rows
ncols: number of columns
rows: list of lists
Returns
=======
DomainMatrix containing elements of rows
Examples
========
>>> from sympy.polys.matrices import DomainMatrix
>>> A = DomainMatrix.from_list_sympy(1, 2, [[1, 0]])
>>> A
DomainMatrix([[1, 0]], (1, 2), ZZ)
See Also
========
sympy.polys.constructor.construct_domain
"""
assert len(rows) == nrows
assert all(len(row) == ncols for row in rows)
items_sympy = [_sympify(item) for row in rows for item in row]
domain, items_domain = cls.get_domain(items_sympy, **kwargs)
domain_rows = [[items_domain[ncols*r + c] for c in range(ncols)] for r in range(nrows)]
return DomainMatrix(domain_rows, (nrows, ncols), domain)
@classmethod
def from_Matrix(cls, M, **kwargs):
r"""
Convert Matrix to DomainMatrix
Parameters
==========
M: Matrix
Returns
=======
Returns DomainMatrix with identical elements as M
Examples
========
>>> from sympy import Matrix
>>> from sympy.polys.matrices import DomainMatrix
>>> M = Matrix([
... [1.0, 3.4],
... [2.4, 1]])
>>> A = DomainMatrix.from_Matrix(M)
>>> A
DomainMatrix([[1.0, 3.4], [2.4, 1.0]], (2, 2), RR)
See Also
========
Matrix
"""
return cls.from_list_sympy(*M.shape, M.tolist(), **kwargs)
@classmethod
def get_domain(cls, items_sympy, **kwargs):
K, items_K = construct_domain(items_sympy, **kwargs)
return K, items_K
def convert_to(self, K):
r"""
Change the domain of DomainMatrix to desired domain or field
Parameters
==========
K : Represents the desired domain or field
Returns
=======
DomainMatrix
DomainMatrix with the desired domain or field
Examples
========
>>> from sympy import ZZ, ZZ_I
>>> from sympy.polys.matrices import DomainMatrix
>>> A = DomainMatrix([
... [ZZ(1), ZZ(2)],
... [ZZ(3), ZZ(4)]], (2, 2), ZZ)
>>> A.convert_to(ZZ_I)
DomainMatrix([[1, 2], [3, 4]], (2, 2), ZZ_I)
"""
return self.from_rep(self.rep.convert_to(K))
def to_field(self):
r"""
Returns a DomainMatrix with the appropriate field
Returns
=======
DomainMatrix
DomainMatrix with the appropriate field
Examples
========
>>> from sympy import ZZ
>>> from sympy.polys.matrices import DomainMatrix
>>> A = DomainMatrix([
... [ZZ(1), ZZ(2)],
... [ZZ(3), ZZ(4)]], (2, 2), ZZ)
>>> A.to_field()
DomainMatrix([[1, 2], [3, 4]], (2, 2), QQ)
"""
K = self.domain.get_field()
return self.convert_to(K)
def to_sparse(self):
"""
Return a sparse DomainMatrix representation of *self*.
Examples
========
>>> from sympy.polys.matrices import DomainMatrix
>>> from sympy import QQ
>>> A = DomainMatrix([[1, 0],[0, 2]], (2, 2), QQ)
>>> A.rep
[[1, 0], [0, 2]]
>>> B = A.to_sparse()
>>> B.rep
{0: {0: 1}, 1: {1: 2}}
"""
if self.rep.fmt == 'sparse':
return self
return self.from_rep(SDM.from_ddm(self.rep))
def to_dense(self):
"""
Return a dense DomainMatrix representation of *self*.
Examples
========
>>> from sympy.polys.matrices import DomainMatrix
>>> from sympy import QQ
>>> A = DomainMatrix({0: {0: 1}, 1: {1: 2}}, (2, 2), QQ)
>>> A.rep
{0: {0: 1}, 1: {1: 2}}
>>> B = A.to_dense()
>>> B.rep
[[1, 0], [0, 2]]
"""
if self.rep.fmt == 'dense':
return self
return self.from_rep(SDM.to_ddm(self.rep))
def _unify_domain(self, other):
"""Convert self and other to a common domain"""
K1 = self.domain
K2 = other.domain
if K1 == K2:
return self, other
K = K1.unify(K2)
if K1 != K:
self = self.convert_to(K)
if K2 != K:
other = other.convert_to(K)
return self, other
def _unify_fmt(self, other, fmt):
"""Convert self and other to the same format.
If both are sparse or both are dense then return both unmodified.
Otherwise convert both to the preferred format given as *fmt* which
should be 'dense' or 'sparse'.
"""
if self.rep.fmt == other.rep.fmt:
return self, other
elif fmt == 'sparse':
return self.to_sparse(), other.to_sparse()
elif fmt == 'dense':
return self.to_dense(), other.to_dense()
else:
raise ValueError("fmt should be 'sparse' or 'dense'")
def unify(self, other, *, fmt=None):
"""
Unifies the domains and the format of self and other
matrices.
Parameters
==========
other : another DomainMatrix
fmt: string 'dense', 'sparse' or `None` (default)
The preferred format to convert to if self and other are not
already in the same format. If `None` or not specified then no
conversion if performed.
Returns
=======
(dM1, dM2)
dM1, dM2 DomainMatrix matrices with unified Domain and format
Examples
========
Unify the domain of DomainMatrix that have different domains:
>>> from sympy import ZZ, QQ
>>> from sympy.polys.matrices import DomainMatrix
>>> A = DomainMatrix([[ZZ(1), ZZ(2)]], (1, 2), ZZ)
>>> B = DomainMatrix([[QQ(1, 2), QQ(2)]], (1, 2), QQ)
>>> Aq, Bq = A.unify(B)
>>> Aq
DomainMatrix([[1, 2]], (1, 2), QQ)
>>> Bq
DomainMatrix([[1/2, 2]], (1, 2), QQ)
Unify the format (dense or sparse):
>>> A = DomainMatrix([[ZZ(1), ZZ(2)]], (1, 2), ZZ)
>>> B = DomainMatrix({0:{0: ZZ(1)}}, (2, 2), ZZ)
>>> B.rep
{0: {0: 1}}
>>> A2, B2 = A.unify(B, fmt='dense')
>>> B2.rep
[[1, 0], [0, 0]]
See Also
========
convert_to, to_dense, to_sparse
"""
dM1, dM2 = self._unify_domain(other)
if fmt is not None:
dM1, dM2 = dM1._unify_fmt(dM2, fmt)
return dM1, dM2
def to_Matrix(self):
r"""
Convert DomainMatrix to Matrix
Returns
=======
Matrix
MutableDenseMatrix for the DomainMatrix
Examples
========
>>> from sympy import ZZ
>>> from sympy.polys.matrices import DomainMatrix
>>> A = DomainMatrix([
... [ZZ(1), ZZ(2)],
... [ZZ(3), ZZ(4)]], (2, 2), ZZ)
>>> A.to_Matrix()
Matrix([
[1, 2],
[3, 4]])
See Also
========
from_Matrix
"""
from sympy.matrices.dense import MutableDenseMatrix
elemlist = self.rep.to_list()
rows_sympy = [[self.domain.to_sympy(e) for e in row] for row in elemlist]
return MutableDenseMatrix(rows_sympy)
def __repr__(self):
return 'DomainMatrix(%s, %r, %r)' % (str(self.rep), self.shape, self.domain)
def hstack(A, B):
r"""
Horizontally stacks 2 Domain Matrices.
Parameters
==========
A, B: DomainMatrix
to stack the rows horizontally
Returns
=======
DomainMatrix
DomainMatrix by stacking the rows horizontally
Examples
========
>>> from sympy import ZZ, QQ
>>> from sympy.polys.matrices import DomainMatrix
>>> A = DomainMatrix([[ZZ(1), ZZ(2), ZZ(3)]], (1, 3), ZZ)
>>> B = DomainMatrix([[QQ(-1, 2), QQ(1, 2), QQ(1, 3)]],(1, 3), QQ)
>>> A.hstack(B)
DomainMatrix([[1, 2, 3, -1/2, 1/2, 1/3]], (1, 6), QQ)
See Also
========
unify
"""
A, B = A.unify(B, fmt='dense')
return A.from_rep(A.rep.hstack(B.rep))
def __add__(A, B):
if not isinstance(B, DomainMatrix):
return NotImplemented
A, B = A.unify(B, fmt='dense')
return A.add(B)
def __sub__(A, B):
if not isinstance(B, DomainMatrix):
return NotImplemented
A, B = A.unify(B, fmt='dense')
return A.sub(B)
def __neg__(A):
return A.neg()
def __mul__(A, B):
"""A * B"""
if isinstance(B, DomainMatrix):
A, B = A.unify(B, fmt='dense')
return A.matmul(B)
elif B in A.domain:
return A.from_rep(A.rep * B)
elif isinstance(B, DomainScalar):
A, B = A.unify(B)
return A.scalarmul(B)
else:
return NotImplemented
def __rmul__(A, B):
if B in A.domain:
return A.from_rep(A.rep * B)
elif isinstance(B, DomainScalar):
A, B = A.unify(B)
return A.scalarmul(B)
else:
return NotImplemented
def __pow__(A, n):
"""A ** n"""
if not isinstance(n, int):
return NotImplemented
return A.pow(n)
def _check(a, op, b, ashape, bshape):
if a.domain != b.domain:
msg = "Domain mismatch: %s %s %s" % (a.domain, op, b.domain)
raise DDMDomainError(msg)
if ashape != bshape:
msg = "Shape mismatch: %s %s %s" % (a.shape, op, b.shape)
raise DDMShapeError(msg)
if a.rep.fmt != b.rep.fmt:
msg = "Format mismatch: %s %s %s" % (a.rep.fmt, op, b.rep.fmt)
raise DDMFormatError(msg)
def add(A, B):
r"""
Adds two DomainMatrix matrices of the same Domain
Parameters
==========
A, B: DomainMatrix
matrices to add
Returns
=======
DomainMatrix
DomainMatrix after Addition
Raises
======
ShapeError
If the dimensions of the two DomainMatrix are not equal
ValueError
If the domain of the two DomainMatrix are not same
Examples
========
>>> from sympy import ZZ
>>> from sympy.polys.matrices import DomainMatrix
>>> A = DomainMatrix([
... [ZZ(1), ZZ(2)],
... [ZZ(3), ZZ(4)]], (2, 2), ZZ)
>>> B = DomainMatrix([
... [ZZ(4), ZZ(3)],
... [ZZ(2), ZZ(1)]], (2, 2), ZZ)
>>> A.add(B)
DomainMatrix([[5, 5], [5, 5]], (2, 2), ZZ)
See Also
========
sub, matmul
"""
A._check('+', B, A.shape, B.shape)
return A.from_rep(A.rep.add(B.rep))
def sub(A, B):
r"""
Subtracts two DomainMatrix matrices of the same Domain
Parameters
==========
A, B: DomainMatrix
matrices to substract
Returns
=======
DomainMatrix
DomainMatrix after Substraction
Raises
======
ShapeError
If the dimensions of the two DomainMatrix are not equal
ValueError
If the domain of the two DomainMatrix are not same
Examples
========
>>> from sympy import ZZ
>>> from sympy.polys.matrices import DomainMatrix
>>> A = DomainMatrix([
... [ZZ(1), ZZ(2)],
... [ZZ(3), ZZ(4)]], (2, 2), ZZ)
>>> B = DomainMatrix([
... [ZZ(4), ZZ(3)],
... [ZZ(2), ZZ(1)]], (2, 2), ZZ)
>>> A.sub(B)
DomainMatrix([[-3, -1], [1, 3]], (2, 2), ZZ)
See Also
========
add, matmul
"""
A._check('-', B, A.shape, B.shape)
return A.from_rep(A.rep.sub(B.rep))
def neg(A):
r"""
Returns the negative of DomainMatrix
Parameters
==========
A : Represents a DomainMatrix
Returns
=======
DomainMatrix
DomainMatrix after Negation
Examples
========
>>> from sympy import ZZ
>>> from sympy.polys.matrices import DomainMatrix
>>> A = DomainMatrix([
... [ZZ(1), ZZ(2)],
... [ZZ(3), ZZ(4)]], (2, 2), ZZ)
>>> A.neg()
DomainMatrix([[-1, -2], [-3, -4]], (2, 2), ZZ)
"""
return A.from_rep(A.rep.neg())
def mul(A, b):
r"""
Performs term by term multiplication for the second DomainMatrix
w.r.t first DomainMatrix. Returns a DomainMatrix whose rows are
list of DomainMatrix matrices created after term by term multiplication.
Parameters
==========
A, B: DomainMatrix
matrices to multiply term-wise
Returns
=======
DomainMatrix
DomainMatrix after term by term multiplication
Examples
========
>>> from sympy import ZZ
>>> from sympy.polys.matrices import DomainMatrix
>>> A = DomainMatrix([
... [ZZ(1), ZZ(2)],
... [ZZ(3), ZZ(4)]], (2, 2), ZZ)
>>> B = DomainMatrix([
... [ZZ(1), ZZ(1)],
... [ZZ(0), ZZ(1)]], (2, 2), ZZ)
>>> A.mul(B)
DomainMatrix([[DomainMatrix([[1, 1], [0, 1]], (2, 2), ZZ),
DomainMatrix([[2, 2], [0, 2]], (2, 2), ZZ)],
[DomainMatrix([[3, 3], [0, 3]], (2, 2), ZZ),
DomainMatrix([[4, 4], [0, 4]], (2, 2), ZZ)]], (2, 2), ZZ)
See Also
========
matmul
"""
return A.from_rep(A.rep.mul(b))
def matmul(A, B):
r"""
Performs matrix multiplication of two DomainMatrix matrices
Parameters
==========
A, B: DomainMatrix
to multiply
Returns
=======
DomainMatrix
DomainMatrix after multiplication
Examples
========
>>> from sympy import ZZ
>>> from sympy.polys.matrices import DomainMatrix
>>> A = DomainMatrix([
... [ZZ(1), ZZ(2)],
... [ZZ(3), ZZ(4)]], (2, 2), ZZ)
>>> B = DomainMatrix([
... [ZZ(1), ZZ(1)],
... [ZZ(0), ZZ(1)]], (2, 2), ZZ)
>>> A.matmul(B)
DomainMatrix([[1, 3], [3, 7]], (2, 2), ZZ)
See Also
========
mul, pow, add, sub
"""
A._check('*', B, A.shape[1], B.shape[0])
return A.from_rep(A.rep.matmul(B.rep))
def scalarmul(A, lamda):
if lamda.element == lamda.domain.zero:
m, n = A.shape
return DomainMatrix([[lamda.domain.zero]*n]*m, (m, n), A.domain)
if lamda.element == lamda.domain.one:
return A
return A.mul(lamda.element)
def __truediv__(A, lamda):
""" Method for Scalar Divison"""
if isinstance(lamda, int):
lamda = DomainScalar(ZZ(lamda), ZZ)
if not isinstance(lamda, DomainScalar):
return NotImplemented
A, lamda = A.to_field().unify(lamda)
if lamda.element == lamda.domain.zero:
raise ZeroDivisionError
if lamda.element == lamda.domain.one:
return A.to_field()
return A.mul(1 / lamda.element)
def pow(A, n):
r"""
Computes A**n
Parameters
==========
A : DomainMatrix
n : exponent for A
Returns
=======
DomainMatrix
DomainMatrix on computing A**n
Raises
======
NotImplementedError
if n is negative.
Examples
========
>>> from sympy import ZZ
>>> from sympy.polys.matrices import DomainMatrix
>>> A = DomainMatrix([
... [ZZ(1), ZZ(1)],
... [ZZ(0), ZZ(1)]], (2, 2), ZZ)
>>> A.pow(2)
DomainMatrix([[1, 2], [0, 1]], (2, 2), ZZ)
See Also
========
matmul
"""
nrows, ncols = A.shape
if nrows != ncols:
raise NonSquareMatrixError('Power of a nonsquare matrix')
if n < 0:
raise NotImplementedError('Negative powers')
elif n == 0:
return A.eye(nrows, A.domain)
elif n == 1:
return A
elif n % 2 == 1:
return A * A**(n - 1)
else:
sqrtAn = A ** (n // 2)
return sqrtAn * sqrtAn
def rref(self):
r"""
Returns reduced-row echelon form and list of pivots for the DomainMatrix
Returns
=======
(DomainMatrix, list)
reduced-row echelon form and list of pivots for the DomainMatrix
Raises
======
ValueError
If the domain of DomainMatrix not a Field
Examples
========
>>> from sympy import QQ
>>> from sympy.polys.matrices import DomainMatrix
>>> A = DomainMatrix([
... [QQ(2), QQ(-1), QQ(0)],
... [QQ(-1), QQ(2), QQ(-1)],
... [QQ(0), QQ(0), QQ(2)]], (3, 3), QQ)
>>> rref_matrix, rref_pivots = A.rref()
>>> rref_matrix
DomainMatrix([[1, 0, 0], [0, 1, 0], [0, 0, 1]], (3, 3), QQ)
>>> rref_pivots
(0, 1, 2)
See Also
========
convert_to, lu
"""
if not self.domain.is_Field:
raise ValueError('Not a field')
rref_ddm, pivots = self.rep.rref()
return self.from_rep(rref_ddm), tuple(pivots)
def nullspace(self):
r"""
Returns the Null Space for the DomainMatrix
Returns
=======
DomainMatrix
Null Space of the DomainMatrix
Examples
========
>>> from sympy import QQ
>>> from sympy.polys.matrices import DomainMatrix
>>> A = DomainMatrix([
... [QQ(1), QQ(-1)],
... [QQ(2), QQ(-2)]], (2, 2), QQ)
>>> A.nullspace()
DomainMatrix([[1, 1]], (1, 2), QQ)
"""
return self.from_rep(self.rep.nullspace()[0])
def inv(self):
r"""
Finds the inverse of the DomainMatrix if exists
Returns
=======
DomainMatrix
DomainMatrix after inverse
Raises
======
ValueError
If the domain of DomainMatrix not a Field
NonSquareMatrixError
If the DomainMatrix is not a not Square DomainMatrix
Examples
========
>>> from sympy import QQ
>>> from sympy.polys.matrices import DomainMatrix
>>> A = DomainMatrix([
... [QQ(2), QQ(-1), QQ(0)],
... [QQ(-1), QQ(2), QQ(-1)],
... [QQ(0), QQ(0), QQ(2)]], (3, 3), QQ)
>>> A.inv()
DomainMatrix([[2/3, 1/3, 1/6], [1/3, 2/3, 1/3], [0, 0, 1/2]], (3, 3), QQ)
See Also
========
neg
"""
if not self.domain.is_Field:
raise ValueError('Not a field')
m, n = self.shape
if m != n:
raise NonSquareMatrixError
inv = self.rep.inv()
return self.from_rep(inv)
def det(self):
r"""
Returns the determinant of a Square DomainMatrix
Returns
=======
S.Complexes
determinant of Square DomainMatrix
Raises
======
ValueError
If the domain of DomainMatrix not a Field
Examples
========
>>> from sympy import ZZ
>>> from sympy.polys.matrices import DomainMatrix
>>> A = DomainMatrix([
... [ZZ(1), ZZ(2)],
... [ZZ(3), ZZ(4)]], (2, 2), ZZ)
>>> A.det()
-2
"""
m, n = self.shape
if m != n:
raise NonSquareMatrixError
return self.rep.det()
def lu(self):
r"""
Returns Lower and Upper decomposition of the DomainMatrix
Returns
=======
(L, U, exchange)
L, U are Lower and Upper decomposition of the DomainMatrix,
exchange is the list of indices of rows exchanged in the decomposition.
Raises
======
ValueError
If the domain of DomainMatrix not a Field
Examples
========
>>> from sympy import QQ
>>> from sympy.polys.matrices import DomainMatrix
>>> A = DomainMatrix([
... [QQ(1), QQ(-1)],
... [QQ(2), QQ(-2)]], (2, 2), QQ)
>>> A.lu()
(DomainMatrix([[1, 0], [2, 1]], (2, 2), QQ), DomainMatrix([[1, -1], [0, 0]], (2, 2), QQ), [])
See Also
========
lu_solve
"""
if not self.domain.is_Field:
raise ValueError('Not a field')
L, U, swaps = self.rep.lu()
return self.from_rep(L), self.from_rep(U), swaps
def lu_solve(self, rhs):
r"""
Solver for DomainMatrix x in the A*x = B
Parameters
==========
rhs : DomainMatrix B
Returns
=======
DomainMatrix
x in A*x = B
Raises
======
ShapeError
If the DomainMatrix A and rhs have different number of rows
ValueError
If the domain of DomainMatrix A not a Field
Examples
========
>>> from sympy import QQ
>>> from sympy.polys.matrices import DomainMatrix
>>> A = DomainMatrix([
... [QQ(1), QQ(2)],
... [QQ(3), QQ(4)]], (2, 2), QQ)
>>> B = DomainMatrix([
... [QQ(1), QQ(1)],
... [QQ(0), QQ(1)]], (2, 2), QQ)
>>> A.lu_solve(B)
DomainMatrix([[-2, -1], [3/2, 1]], (2, 2), QQ)
See Also
========
lu
"""
if self.shape[0] != rhs.shape[0]:
raise ShapeError("Shape")
if not self.domain.is_Field:
raise ValueError('Not a field')
sol = self.rep.lu_solve(rhs.rep)
return self.from_rep(sol)
def charpoly(self):
r"""
Returns the coefficients of the characteristic polynomial
of the DomainMatrix. These elements will be domain elements.
The domain of the elements will be same as domain of the DomainMatrix.
Returns
=======
list
coefficients of the characteristic polynomial
Raises
======
NonSquareMatrixError
If the DomainMatrix is not a not Square DomainMatrix
Examples
========
>>> from sympy import ZZ
>>> from sympy.polys.matrices import DomainMatrix
>>> A = DomainMatrix([
... [ZZ(1), ZZ(2)],
... [ZZ(3), ZZ(4)]], (2, 2), ZZ)
>>> A.charpoly()
[1, -5, -2]
"""
m, n = self.shape
if m != n:
raise NonSquareMatrixError("not square")
return self.rep.charpoly()
@classmethod
def eye(cls, n, domain):
r"""
Return identity matrix of size n
Examples
========
>>> from sympy.polys.matrices import DomainMatrix
>>> from sympy import QQ
>>> DomainMatrix.eye(3, QQ)
DomainMatrix({0: {0: 1}, 1: {1: 1}, 2: {2: 1}}, (3, 3), QQ)
"""
return cls.from_rep(SDM.eye(n, domain))
@classmethod
def zeros(cls, shape, domain, *, fmt='sparse'):
"""Returns a zero DomainMatrix of size shape, belonging to the specified domain
Examples
========
>>> from sympy.polys.matrices import DomainMatrix
>>> from sympy import QQ
>>> DomainMatrix.zeros((2, 3), QQ)
DomainMatrix({}, (2, 3), QQ)
"""
return cls.from_rep(SDM.zeros(shape, domain))
@classmethod
def ones(cls, shape, domain):
"""Returns a zero DomainMatrix of size shape, belonging to the specified domain
Examples
========
>>> from sympy.polys.matrices import DomainMatrix
>>> from sympy import QQ
>>> DomainMatrix.ones((2,3), QQ)
DomainMatrix([[1, 1, 1], [1, 1, 1]], (2, 3), QQ)
"""
return cls.from_rep(DDM.ones(shape, domain))
def __eq__(A, B):
r"""
Checks for two DomainMatrix matrices to be equal or not
Parameters
==========
A, B: DomainMatrix
to check equality
Returns
=======
Boolean
True for equal, else False
Raises
======
NotImplementedError
If B is not a DomainMatrix
Examples
========
>>> from sympy import ZZ
>>> from sympy.polys.matrices import DomainMatrix
>>> A = DomainMatrix([
... [ZZ(1), ZZ(2)],
... [ZZ(3), ZZ(4)]], (2, 2), ZZ)
>>> B = DomainMatrix([
... [ZZ(1), ZZ(1)],
... [ZZ(0), ZZ(1)]], (2, 2), ZZ)
>>> A.__eq__(A)
True
>>> A.__eq__(B)
False
"""
if not isinstance(B, DomainMatrix):
return NotImplemented
return A.rep == B.rep
|
298d396d5ff9d44640d9daf41cb09f76c2b74ae47690f0e3a26de72c28c5afbc | """
Module to define exceptions to be used in sympy.polys.matrices modules and
classes.
Ideally all exceptions raised in these modules would be defined and documented
here and not e.g. imported from matrices. Also ideally generic exceptions like
ValueError/TypeError would not be raised anywhere.
"""
from sympy.matrices.common import (NonInvertibleMatrixError,
NonSquareMatrixError, ShapeError)
class DDMError(Exception):
"""Base class for errors raised by DDM"""
pass
class DDMBadInputError(DDMError):
"""list of lists is inconsistent with shape"""
pass
class DDMDomainError(DDMError):
"""domains do not match"""
pass
class DDMShapeError(DDMError):
"""shapes are inconsistent"""
pass
class DDMFormatError(DDMError):
"""mixed dense/sparse not supported"""
pass
__all__ = [
'DDMError', 'DDMShapeError', 'DDMDomainError', 'DDMFormatError',
'NonSquareMatrixError', 'NonInvertibleMatrixError', 'ShapeError',
]
|
2d576c0c6f0c3f59c001de18c2ea57fc26b83fbb604c7a9846178886953e0232 | #
# sympy.polys.matrices.linsolve module
#
# This module defines the _linsolve function which is the internal workhorse
# used by linsolve. This computes the solution of a system of linear equations
# using the SDM sparse matrix implementation in sympy.polys.matrices.sdm. This
# is a replacement for solve_lin_sys in sympy.polys.solvers which is
# inefficient for large sparse systems due to the use of a PolyRing with many
# generators:
#
# https://github.com/sympy/sympy/issues/20857
#
# The implementation of _linsolve here handles:
#
# - Extracting the coefficients from the Expr/Eq input equations.
# - Constructing a domain and converting the coefficients to
# that domain.
# - Using the SDM.rref, SDM.nullspace etc methods to generate the full
# solution working with arithmetic only in the domain of the coefficients.
#
# The routines here are particularly designed to be efficient for large sparse
# systems of linear equations although as well as dense systems. It is
# possible that for some small dense systems solve_lin_sys which uses the
# dense matrix implementation DDM will be more efficient. With smaller systems
# though the bulk of the time is spent just preprocessing the inputs and the
# relative time spent in rref is too small to be noticeable.
#
from collections import defaultdict
from sympy.core.add import Add
from sympy.core.mul import Mul
from sympy.core.singleton import S
from sympy.polys.constructor import construct_domain
from sympy.polys.solvers import PolyNonlinearError
from .sdm import (
SDM,
sdm_irref,
sdm_particular_from_rref,
sdm_nullspace_from_rref
)
def _linsolve(eqs, syms):
"""Solve a linear system of equations.
Examples
========
Solve a linear system with a unique solution:
>>> from sympy import symbols, Eq
>>> from sympy.polys.matrices.linsolve import _linsolve
>>> x, y = symbols('x, y')
>>> eqs = [Eq(x + y, 1), Eq(x - y, 2)]
>>> _linsolve(eqs, [x, y])
{x: 3/2, y: -1/2}
In the case of underdetermined systems the solution will be expressed in
terms of the unknown symbols that are unconstrained:
>>> _linsolve([Eq(x + y, 0)], [x, y])
{x: -y, y: y}
"""
# Number of unknowns (columns in the non-augmented matrix)
nsyms = len(syms)
# Convert to sparse augmented matrix (len(eqs) x (nsyms+1))
eqsdict, rhs = _linear_eq_to_dict(eqs, syms)
Aaug = sympy_dict_to_dm(eqsdict, rhs, syms)
K = Aaug.domain
# Compute reduced-row echelon form (RREF)
Arref, pivots, nzcols = sdm_irref(Aaug)
# No solution:
if pivots and pivots[-1] == nsyms:
return None
# Particular solution for non-homogeneous system:
P = sdm_particular_from_rref(Arref, nsyms+1, pivots)
# Nullspace - general solution to homogeneous system
# Note: using nsyms not nsyms+1 to ignore last column
V, nonpivots = sdm_nullspace_from_rref(Arref, K.one, nsyms, pivots, nzcols)
# Collect together terms from particular and nullspace:
sol = defaultdict(list)
for i, v in P.items():
sol[syms[i]].append(K.to_sympy(v))
for npi, Vi in zip(nonpivots, V):
sym = syms[npi]
for i, v in Vi.items():
sol[syms[i]].append(sym * K.to_sympy(v))
# Use a single call to Add for each term:
sol = {s: Add(*terms) for s, terms in sol.items()}
# Fill in the zeros:
zero = S.Zero
for s in set(syms) - set(sol):
sol[s] = zero
# All done!
return sol
def sympy_dict_to_dm(eqs_coeffs, eqs_rhs, syms):
"""Convert a system of dict equations to a sparse augmented matrix"""
elems = set(eqs_rhs).union(*(e.values() for e in eqs_coeffs))
K, elems_K = construct_domain(elems, field=True, extension=True)
elem_map = dict(zip(elems, elems_K))
neqs = len(eqs_coeffs)
nsyms = len(syms)
sym2index = dict(zip(syms, range(nsyms)))
eqsdict = []
for eq, rhs in zip(eqs_coeffs, eqs_rhs):
eqdict = {sym2index[s]: elem_map[c] for s, c in eq.items()}
if rhs:
eqdict[nsyms] = - elem_map[rhs]
if eqdict:
eqsdict.append(eqdict)
sdm_aug = SDM(enumerate(eqsdict), (neqs, nsyms+1), K)
return sdm_aug
def _expand_eqs_deprecated(eqs):
"""Use expand to cancel nonlinear terms.
This approach matches previous behaviour of linsolve but should be
deprecated.
"""
def expand_eq(eq):
if eq.is_Equality:
eq = eq.lhs - eq.rhs
return eq.expand()
return [expand_eq(eq) for eq in eqs]
def _linear_eq_to_dict(eqs, syms):
"""Convert a system Expr/Eq equations into dict form"""
try:
return _linear_eq_to_dict_inner(eqs, syms)
except PolyNonlinearError:
# XXX: This should be deprecated:
eqs = _expand_eqs_deprecated(eqs)
return _linear_eq_to_dict_inner(eqs, syms)
def _linear_eq_to_dict_inner(eqs, syms):
"""Convert a system Expr/Eq equations into dict form"""
syms = set(syms)
eqsdict, eqs_rhs = [], []
for eq in eqs:
rhs, eqdict = _lin_eq2dict(eq, syms)
eqsdict.append(eqdict)
eqs_rhs.append(rhs)
return eqsdict, eqs_rhs
def _lin_eq2dict(a, symset):
"""Efficiently convert a linear equation to a dict of coefficients"""
if a in symset:
return S.Zero, {a: S.One}
elif a.is_Add:
terms_list = defaultdict(list)
coeff_list = []
for ai in a.args:
ci, ti = _lin_eq2dict(ai, symset)
coeff_list.append(ci)
for mij, cij in ti.items():
terms_list[mij].append(cij)
coeff = Add(*coeff_list)
terms = {sym: Add(*coeffs) for sym, coeffs in terms_list.items()}
return coeff, terms
elif a.is_Mul:
terms = terms_coeff = None
coeff_list = []
for ai in a.args:
ci, ti = _lin_eq2dict(ai, symset)
if not ti:
coeff_list.append(ci)
elif terms is None:
terms = ti
terms_coeff = ci
else:
raise PolyNonlinearError
coeff = Mul(*coeff_list)
if terms is None:
return coeff, {}
else:
terms = {sym: coeff * c for sym, c in terms.items()}
return coeff * terms_coeff, terms
elif a.is_Equality:
return _lin_eq2dict(a.lhs - a.rhs, symset)
elif not a.free_symbols & symset:
return a, {}
else:
raise PolyNonlinearError
|
39257f813b0ea64e90e7e503b5398494b0256e72f20ec5e47427948946cee90b | """
Module for the SDM class.
"""
from operator import add, neg, pos, sub
from collections import defaultdict
from .exceptions import DDMBadInputError, DDMDomainError, DDMShapeError
from .ddm import DDM
class SDM(dict):
"""Sparse matrix based on polys domain elements
This is a dict subclass and is a wrapper for a dict of dicts that supports
basic matrix arithmetic +, -, *, **.
"""
fmt = 'sparse'
def __init__(self, elemsdict, shape, domain):
super().__init__(elemsdict)
self.shape = self.rows, self.cols = m, n = shape
self.domain = domain
if not all(0 <= r < m for r in self):
raise DDMBadInputError("Row out of range")
if not all(0 <= c < n for row in self.values() for c in row):
raise DDMBadInputError("Column out of range")
def __str__(self):
rowsstr = []
for i, row in self.items():
elemsstr = ', '.join('%s: %s' % (j, elem) for j, elem in row.items())
rowsstr.append('%s: {%s}' % (i, elemsstr))
return '{%s}' % ', '.join(rowsstr)
def __repr__(self):
cls = type(self).__name__
rows = dict.__repr__(self)
return '%s(%s, %s, %s)' % (cls, rows, self.shape, self.domain)
@classmethod
def new(cls, sdm, shape, domain):
return cls(sdm, shape, domain)
def copy(A):
Ac = {i: Ai.copy() for i, Ai in A.items()}
return A.new(Ac, A.shape, A.domain)
@classmethod
def from_list(cls, ddm, shape, domain):
m, n = shape
if not (len(ddm) == m and all(len(row) == n for row in ddm)):
raise DDMBadInputError("Inconsistent row-list/shape")
getrow = lambda i: {j:ddm[i][j] for j in range(n) if ddm[i][j]}
irows = ((i, getrow(i)) for i in range(m))
sdm = {i: row for i, row in irows if row}
return cls(sdm, shape, domain)
@classmethod
def from_ddm(cls, ddm):
return cls.from_list(ddm, ddm.shape, ddm.domain)
def to_list(M):
m, n = M.shape
zero = M.domain.zero
ddm = [[zero] * n for _ in range(m)]
for i, row in M.items():
for j, e in row.items():
ddm[i][j] = e
return ddm
def to_ddm(M):
return DDM(M.to_list(), M.shape, M.domain)
def to_sdm(M):
return M
@classmethod
def zeros(cls, shape, domain):
return cls({}, shape, domain)
@classmethod
def ones(cls, shape, domain):
one = domain.one
m, n = shape
row = dict(zip(range(n), [one]*n))
sdm = {i: row.copy() for i in range(m)}
return cls(sdm, shape, domain)
@classmethod
def eye(cls, size, domain):
one = domain.one
sdm = {i: {i: one} for i in range(size)}
return cls(sdm, (size, size), domain)
def transpose(M):
MT = sdm_transpose(M)
return M.new(MT, M.shape[::-1], M.domain)
def __mul__(a, b):
if b in a.domain:
return a.mul(b)
else:
return NotImplemented
def __rmul__(a, b):
if b in a.domain:
return a.mul(b)
else:
return NotImplemented
def matmul(A, B):
if A.domain != B.domain:
raise DDMDomainError
m, n = A.shape
n2, o = B.shape
if n != n2:
raise DDMShapeError
C = sdm_matmul(A, B)
return A.new(C, (m, o), A.domain)
def mul(A, b):
Csdm = unop_dict(A, lambda aij: aij*b)
return A.new(Csdm, A.shape, A.domain)
def add(A, B):
Csdm = binop_dict(A, B, add, pos, pos)
return A.new(Csdm, A.shape, A.domain)
def sub(A, B):
Csdm = binop_dict(A, B, sub, pos, neg)
return A.new(Csdm, A.shape, A.domain)
def neg(A):
Csdm = unop_dict(A, neg)
return A.new(Csdm, A.shape, A.domain)
def convert_to(A, K):
Kold = A.domain
if K == Kold:
return A.copy()
Ak = unop_dict(A, lambda e: K.convert_from(e, Kold))
return A.new(Ak, A.shape, K)
def rref(A):
B, pivots, _ = sdm_irref(A)
return A.new(B, A.shape, A.domain), pivots
def inv(A):
return A.from_ddm(A.to_ddm().inv())
def det(A):
return A.to_ddm().det()
def lu(A):
L, U, swaps = A.to_ddm().lu()
return A.from_ddm(L), A.from_ddm(U), swaps
def lu_solve(A, b):
return A.from_ddm(A.to_ddm().lu_solve(b.to_ddm()))
def nullspace(A):
ncols = A.shape[1]
one = A.domain.one
B, pivots, nzcols = sdm_irref(A)
K, nonpivots = sdm_nullspace_from_rref(B, one, ncols, pivots, nzcols)
K = dict(enumerate(K))
shape = (len(K), ncols)
return A.new(K, shape, A.domain), nonpivots
def particular(A):
ncols = A.shape[1]
B, pivots, nzcols = sdm_irref(A)
P = sdm_particular_from_rref(B, ncols, pivots)
rep = {0:P} if P else {}
return A.new(rep, (1, A.shape[1]), A.domain)
def hstack(A, *B):
Anew = dict(A.copy())
rows, cols = A.shape
domain = A.domain
for Bk in B:
Bkrows, Bkcols = Bk.shape
assert Bkrows == rows
assert Bk.domain == domain
for i, Bki in Bk.items():
Ai = Anew.get(i, None)
if Ai is None:
Anew[i] = Ai = {}
for j, Bkij in Bki.items():
Ai[j + cols] = Bkij
cols += Bkcols
return A.new(Anew, (rows, cols), A.domain)
def charpoly(A):
return A.to_ddm().charpoly()
def binop_dict(A, B, fab, fa, fb):
Anz, Bnz = set(A), set(B)
C = {}
for i in Anz & Bnz:
Ai, Bi = A[i], B[i]
Ci = {}
Anzi, Bnzi = set(Ai), set(Bi)
for j in Anzi & Bnzi:
elem = fab(Ai[j], Bi[j])
if elem:
Ci[j] = elem
for j in Anzi - Bnzi:
Ci[j] = fa(Ai[j])
for j in Bnzi - Anzi:
Ci[j] = fb(Bi[j])
if Ci:
C[i] = Ci
for i in Anz - Bnz:
Ai = A[i]
C[i] = {j: fa(Aij) for j, Aij in Ai.items()}
for i in Bnz - Anz:
Bi = B[i]
C[i] = {j: fb(Bij) for j, Bij in Bi.items()}
return C
def unop_dict(A, f):
B = {}
for i, Ai in A.items():
Bi = {}
for j, Aij in Ai.items():
Bij = f(Aij)
if Bij:
Bi[j] = Bij
if Bi:
B[i] = Bi
return B
def sdm_transpose(M):
MT = {}
for i, Mi in M.items():
for j, Mij in Mi.items():
try:
MT[j][i] = Mij
except KeyError:
MT[j] = {i: Mij}
return MT
def sdm_matmul(A, B):
#
# Should be fast if A and B are very sparse.
# Consider e.g. A = B = eye(1000).
#
# The idea here is that we compute C = A*B in terms of the rows of C and
# B since the dict of dicts representation naturally stores the matrix as
# rows. The ith row of C (Ci) is equal to the sum of Aik * Bk where Bk is
# the kth row of B. The algorithm below loops over each nonzero element
# Aik of A and if the corresponding row Bj is nonzero then we do
# Ci += Aik * Bk.
# To make this more efficient we don't need to loop over all elements Aik.
# Instead for each row Ai we compute the intersection of the nonzero
# columns in Ai with the nonzero rows in B. That gives the k such that
# Aik and Bk are both nonzero. In Python the intersection of two sets
# of int can be computed very efficiently.
#
C = {}
B_knz = set(B)
for i, Ai in A.items():
Ci = {}
Ai_knz = set(Ai)
for k in Ai_knz & B_knz:
Aik = Ai[k]
for j, Bkj in B[k].items():
Cij = Ci.get(j, None)
if Cij is not None:
Cij = Cij + Aik * Bkj
if Cij:
Ci[j] = Cij
else:
Ci.pop(j)
else:
Cij = Aik * Bkj
if Cij:
Ci[j] = Cij
if Ci:
C[i] = Ci
return C
def sdm_irref(A):
"""RREF and pivots of a sparse matrix *A*.
Compute the reduced row echelon form (RREF) of the matrix *A* and return a
list of the pivot columns. This routine does not work in place and leaves
the original matrix *A* unmodified.
Examples
========
This routine works with a dict of dicts sparse representation of a matrix:
>>> from sympy import QQ
>>> from sympy.polys.matrices.sdm import sdm_irref
>>> A = {0: {0: QQ(1), 1: QQ(2)}, 1: {0: QQ(3), 1: QQ(4)}}
>>> Arref, pivots, _ = sdm_irref(A)
>>> Arref
{0: {0: 1}, 1: {1: 1}}
>>> pivots
[0, 1]
The analogous calculation with :py:class:`~.Matrix` would be
>>> from sympy import Matrix
>>> M = Matrix([[1, 2], [3, 4]])
>>> Mrref, pivots = M.rref()
>>> Mrref
Matrix([
[1, 0],
[0, 1]])
>>> pivots
(0, 1)
Notes
=====
The cost of this algorithm is determined purely by the nonzero elements of
the matrix. No part of the cost of any step in this algorithm depends on
the number of rows or columns in the matrix. No step depends even on the
number of nonzero rows apart from the primary loop over those rows. The
implementation is much faster than ddm_rref for sparse matrices. In fact
at the time of writing it is also (slightly) faster than the dense
implementation even if the input is a fully dense matrix so it seems to be
faster in all cases.
The elements of the matrix should support exact division with ``/``. For
example elements of any domain that is a field (e.g. ``QQ``) should be
fine. No attempt is made to handle inexact arithmetic.
"""
#
# Any zeros in the matrix are not stored at all so an element is zero if
# its row dict has no index at that key. A row is entirely zero if its
# row index is not in the outer dict. Since rref reorders the rows and
# removes zero rows we can completely discard the row indices. The first
# step then copies the row dicts into a list sorted by the index of the
# first nonzero column in each row.
#
# The algorithm then processes each row Ai one at a time. Previously seen
# rows are used to cancel their pivot columns from Ai. Then a pivot from
# Ai is chosen and is cancelled from all previously seen rows. At this
# point Ai joins the previously seen rows. Once all rows are seen all
# elimination has occurred and the rows are sorted by pivot column index.
#
# The previously seen rows are stored in two separate groups. The reduced
# group consists of all rows that have been reduced to a single nonzero
# element (the pivot). There is no need to attempt any further reduction
# with these. Rows that still have other nonzeros need to be considered
# when Ai is cancelled from the previously seen rows.
#
# A dict nonzerocolumns is used to map from a column index to a set of
# previously seen rows that still have a nonzero element in that column.
# This means that we can cancel the pivot from Ai into the previously seen
# rows without needing to loop over each row that might have a zero in
# that column.
#
# Row dicts sorted by index of first nonzero column
# (Maybe sorting is not needed/useful.)
Arows = sorted((Ai.copy() for Ai in A.values()), key=min)
# Each processed row has an associated pivot column.
# pivot_row_map maps from the pivot column index to the row dict.
# This means that we can represent a set of rows purely as a set of their
# pivot indices.
pivot_row_map = {}
# Set of pivot indices for rows that are fully reduced to a single nonzero.
reduced_pivots = set()
# Set of pivot indices for rows not fully reduced
nonreduced_pivots = set()
# Map from column index to a set of pivot indices representing the rows
# that have a nonzero at that column.
nonzero_columns = defaultdict(set)
while Arows:
# Select pivot element and row
Ai = Arows.pop()
# Nonzero columns from fully reduced pivot rows can be removed
Ai = {j: Aij for j, Aij in Ai.items() if j not in reduced_pivots}
# Others require full row cancellation
for j in nonreduced_pivots & set(Ai):
Aj = pivot_row_map[j]
Aij = Ai[j]
Ainz = set(Ai)
Ajnz = set(Aj)
for k in Ajnz - Ainz:
Ai[k] = - Aij * Aj[k]
for k in Ajnz & Ainz:
Aik = Ai[k] - Aij * Aj[k]
if Aik:
Ai[k] = Aik
else:
Ai.pop(k)
# We have now cancelled previously seen pivots from Ai.
# If it is zero then discard it.
if not Ai:
continue
# Choose a pivot from Ai:
j = min(Ai)
Aij = Ai[j]
pivot_row_map[j] = Ai
Ainz = set(Ai)
# Normalise the pivot row to make the pivot 1.
#
# This approach is slow for some domains. Cross cancellation might be
# better for e.g. QQ(x) with division delayed to the final steps.
Aijinv = Aij**-1
for l in Ai:
Ai[l] *= Aijinv
# Use Aij to cancel column j from all previously seen rows
for k in nonzero_columns.pop(j, ()):
Ak = pivot_row_map[k]
Akj = Ak[j]
Aknz = set(Ak)
for l in Ainz - Aknz:
Ak[l] = - Akj * Ai[l]
nonzero_columns[l].add(k)
for l in Ainz & Aknz:
Akl = Ak[l] - Akj * Ai[l]
if Akl:
Ak[l] = Akl
else:
# Drop nonzero elements
Ak.pop(l)
if l != j:
nonzero_columns[l].remove(k)
if len(Ak) == 1:
reduced_pivots.add(k)
nonreduced_pivots.remove(k)
if len(Ai) == 1:
reduced_pivots.add(j)
else:
nonreduced_pivots.add(j)
for l in Ai:
if l != j:
nonzero_columns[l].add(j)
# All done!
pivots = sorted(reduced_pivots | nonreduced_pivots)
pivot2row = {p: n for n, p in enumerate(pivots)}
nonzero_columns = {c: set(pivot2row[p] for p in s) for c, s in nonzero_columns.items()}
rows = [pivot_row_map[i] for i in pivots]
rref = dict(enumerate(rows))
return rref, pivots, nonzero_columns
def sdm_nullspace_from_rref(A, one, ncols, pivots, nonzero_cols):
"""Get nullspace from A which is in RREF"""
nonpivots = sorted(set(range(ncols)) - set(pivots))
K = []
for j in nonpivots:
Kj = {j:one}
for i in nonzero_cols.get(j, ()):
Kj[pivots[i]] = -A[i][j]
K.append(Kj)
return K, nonpivots
def sdm_particular_from_rref(A, ncols, pivots):
"""Get a particular solution from A which is in RREF"""
P = {}
for i, j in enumerate(pivots):
Ain = A[i].get(ncols-1, None)
if Ain is not None:
P[j] = Ain / A[i][j]
return P
|
30411671308c12a6c96b142fa347f95d05636a57d3ba51ccca71d5176938f568 | """
Module for the DomainScalar class.
A DomainScalar represents an element which is in a particular
Domain. The idea is that the DomainScalar class provides the
convenience routines for unifying elements with different domains.
It assists in Scalar Multiplication and getitem for DomainMatrix.
"""
from ..constructor import construct_domain
from sympy.polys.domains import Domain, ZZ
class DomainScalar:
r"""
docstring
"""
def __new__(cls, element, domain):
if not isinstance(domain, Domain):
raise TypeError("domain should be of type Domain")
if not domain.of_type(element):
raise TypeError("element %s should be in domain %s" % (element, domain))
return cls.new(element, domain)
@classmethod
def new(cls, element, domain):
obj = super().__new__(cls)
obj.element = element
obj.domain = domain
return obj
def __repr__(self):
return repr(self.element)
@classmethod
def from_sympy(cls, expr):
[domain, [element]] = construct_domain([expr])
return cls.new(element, domain)
def to_domain(self, domain):
element = domain.convert_from(self.element, self.domain)
return self.new(element, domain)
def convert_to(self, domain):
return self.to_domain(domain)
def unify(self, other):
domain = self.domain.unify(other.domain)
return self.to_domain(domain), other.to_domain(domain)
def __add__(self, other):
if not isinstance(other, DomainScalar):
return NotImplemented
self, other = self.unify(other)
return self.new(self.element + other.element, self.domain)
def __sub__(self, other):
if not isinstance(other, DomainScalar):
return NotImplemented
self, other = self.unify(other)
return self.new(self.element - other.element, self.domain)
def __mul__(self, other):
if not isinstance(other, DomainScalar):
if isinstance(other, int):
other = DomainScalar(ZZ(other), ZZ)
else:
return NotImplemented
self, other = self.unify(other)
return self.new(self.element * other.element, self.domain)
def __floordiv__(self, other):
if not isinstance(other, DomainScalar):
return NotImplemented
self, other = self.unify(other)
return self.new(self.domain.quo(self.element, other.element), self.domain)
def __mod__(self, other):
if not isinstance(other, DomainScalar):
return NotImplemented
self, other = self.unify(other)
return self.new(self.domain.rem(self.element, other.element), self.domain)
def __divmod__(self, other):
if not isinstance(other, DomainScalar):
return NotImplemented
self, other = self.unify(other)
q, r = self.domain.div(self.element, other.element)
return (self.new(q, self.domain), self.new(r, self.domain))
def __pow__(self, n):
if not isinstance(n, int):
return NotImplemented
return self.new(self.element**n, self.domain)
def __pos__(self):
return self.new(+self.element, self.domain)
def __eq__(self, other):
if not isinstance(other, DomainScalar):
return NotImplemented
return self.element == other.element and self.domain == other.domain
def is_zero(self):
return self.element == self.domain.zero
def is_one(self):
return self.element == self.domain.one
|
a6f202588f3faffbac6b4105aa56f3ec8f004b328c2b68f6827fbae4299348f5 | """
Routines for computing eigenvectors with DomainMatrix.
"""
from sympy.core.symbol import Dummy
from ..agca.extensions import FiniteExtension
from ..factortools import dup_factor_list
from ..polyroots import roots
from ..polytools import Poly
from ..rootoftools import CRootOf
from .domainmatrix import DomainMatrix
def dom_eigenvects(A, l=Dummy('lambda')):
charpoly = A.charpoly()
rows, cols = A.shape
domain = A.domain
_, factors = dup_factor_list(charpoly, domain)
rational_eigenvects = []
algebraic_eigenvects = []
for base, exp in factors:
if len(base) == 2:
field = domain
eigenval = -base[1] / base[0]
EE_items = [
[eigenval if i == j else field.zero for j in range(cols)]
for i in range(rows)]
EE = DomainMatrix(EE_items, (rows, cols), field)
basis = (A - EE).nullspace()
rational_eigenvects.append((field, eigenval, exp, basis))
else:
minpoly = Poly.from_list(base, l, domain=domain)
field = FiniteExtension(minpoly)
eigenval = field(l)
AA_items = [
[Poly.from_list([item], l, domain=domain).rep for item in row]
for row in A.rep.to_ddm()]
AA_items = [[field(item) for item in row] for row in AA_items]
AA = DomainMatrix(AA_items, (rows, cols), field)
EE_items = [
[eigenval if i == j else field.zero for j in range(cols)]
for i in range(rows)]
EE = DomainMatrix(EE_items, (rows, cols), field)
basis = (AA - EE).nullspace()
algebraic_eigenvects.append((field, minpoly, exp, basis))
return rational_eigenvects, algebraic_eigenvects
def dom_eigenvects_to_sympy(
rational_eigenvects, algebraic_eigenvects,
Matrix, **kwargs
):
result = []
for field, eigenvalue, multiplicity, eigenvects in rational_eigenvects:
eigenvects = eigenvects.rep.to_ddm()
eigenvalue = field.to_sympy(eigenvalue)
new_eigenvects = [
Matrix([field.to_sympy(x) for x in vect])
for vect in eigenvects]
result.append((eigenvalue, multiplicity, new_eigenvects))
for field, minpoly, multiplicity, eigenvects in algebraic_eigenvects:
eigenvects = eigenvects.rep.to_ddm()
l = minpoly.gens[0]
eigenvects = [[field.to_sympy(x) for x in vect] for vect in eigenvects]
degree = minpoly.degree()
minpoly = minpoly.as_expr()
eigenvals = roots(minpoly, l, **kwargs)
if len(eigenvals) != degree:
eigenvals = [CRootOf(minpoly, l, idx) for idx in range(degree)]
for eigenvalue in eigenvals:
new_eigenvects = [
Matrix([x.subs(l, eigenvalue) for x in vect])
for vect in eigenvects]
result.append((eigenvalue, multiplicity, new_eigenvects))
return result
|
bfcc642b7982f7e43cfaa4ea2afc886e9479ca810e9c38ee2a3ef4df0b332ccc | """
Module for the DDM class.
The DDM class is an internal representation used by DomainMatrix. The letters
DDM stand for Dense Domain Matrix. A DDM instance represents a matrix using
elements from a polynomial Domain (e.g. ZZ, QQ, ...) in a dense-matrix
representation.
Basic usage:
>>> from sympy import ZZ, QQ
>>> from sympy.polys.matrices.ddm import DDM
>>> A = DDM([[ZZ(0), ZZ(1)], [ZZ(-1), ZZ(0)]], (2, 2), ZZ)
>>> A.shape
(2, 2)
>>> A
[[0, 1], [-1, 0]]
>>> type(A)
<class 'sympy.polys.matrices.ddm.DDM'>
>>> A @ A
[[-1, 0], [0, -1]]
The ddm_* functions are designed to operate on DDM as well as on an ordinary
list of lists:
>>> from sympy.polys.matrices.dense import ddm_idet
>>> ddm_idet(A, QQ)
1
>>> ddm_idet([[0, 1], [-1, 0]], QQ)
1
>>> A
[[-1, 0], [0, 1]]
Note that ddm_idet modifies the input matrix in-place. It is recommended to
use the DDM.det method as a friendlier interface to this instead which takes
care of copying the matrix:
>>> B = DDM([[ZZ(0), ZZ(1)], [ZZ(-1), ZZ(0)]], (2, 2), ZZ)
>>> B.det()
1
Normally DDM would not be used directly and is just part of the internal
representation of DomainMatrix which adds further functionality including e.g.
unifying domains.
The dense format used by DDM is a list of lists of elements e.g. the 2x2
identity matrix is like [[1, 0], [0, 1]]. The DDM class itself is a subclass
of list and its list items are plain lists. Elements are accessed as e.g.
ddm[i][j] where ddm[i] gives the ith row and ddm[i][j] gets the element in the
jth column of that row. Subclassing list makes e.g. iteration and indexing
very efficient. We do not override __getitem__ because it would lose that
benefit.
The core routines are implemented by the ddm_* functions defined in dense.py.
Those functions are intended to be able to operate on a raw list-of-lists
representation of matrices with most functions operating in-place. The DDM
class takes care of copying etc and also stores a Domain object associated
with its elements. This makes it possible to implement things like A + B with
domain checking and also shape checking so that the list of lists
representation is friendlier.
"""
from .exceptions import DDMBadInputError, DDMShapeError, DDMDomainError
from .dense import (
ddm_iadd,
ddm_isub,
ddm_ineg,
ddm_imul,
ddm_imatmul,
ddm_irref,
ddm_idet,
ddm_iinv,
ddm_ilu_split,
ddm_ilu_solve,
ddm_berk,
)
class DDM(list):
"""Dense matrix based on polys domain elements
This is a list subclass and is a wrapper for a list of lists that supports
basic matrix arithmetic +, -, *, **.
"""
fmt = 'dense'
def __init__(self, rowslist, shape, domain):
super().__init__(rowslist)
self.shape = self.rows, self.cols = m, n = shape
self.domain = domain
if not (len(self) == m and all(len(row) == n for row in self)):
raise DDMBadInputError("Inconsistent row-list/shape")
def to_list(self):
return list(self)
def to_ddm(self):
return self
def to_sdm(self):
return SDM.from_list(self, self.shape, self.domain)
def convert_to(self, K):
Kold = self.domain
if K == Kold:
return self.copy()
rows = ([K.convert_from(e, Kold) for e in row] for row in self)
return DDM(rows, self.shape, K)
def __str__(self):
rowsstr = ['[%s]' % ', '.join(map(str, row)) for row in self]
return '[%s]' % ', '.join(rowsstr)
def __repr__(self):
cls = type(self).__name__
rows = list.__repr__(self)
return '%s(%s, %s, %s)' % (cls, rows, self.shape, self.domain)
def __eq__(self, other):
if not isinstance(other, DDM):
return False
return (super().__eq__(other) and self.domain == other.domain)
def __ne__(self, other):
return not self.__eq__(other)
@classmethod
def zeros(cls, shape, domain):
z = domain.zero
m, n = shape
rowslist = ([z] * n for _ in range(m))
return DDM(rowslist, shape, domain)
@classmethod
def ones(cls, shape, domain):
one = domain.one
m, n = shape
rowlist = ([one] * n for _ in range(m))
return DDM(rowlist, shape, domain)
@classmethod
def eye(cls, size, domain):
one = domain.one
ddm = cls.zeros((size, size), domain)
for i in range(size):
ddm[i][i] = one
return ddm
def copy(self):
copyrows = (row[:] for row in self)
return DDM(copyrows, self.shape, self.domain)
def __add__(a, b):
if not isinstance(b, DDM):
return NotImplemented
return a.add(b)
def __sub__(a, b):
if not isinstance(b, DDM):
return NotImplemented
return a.sub(b)
def __neg__(a):
return a.neg()
def __mul__(a, b):
if b in a.domain:
return a.mul(b)
else:
return NotImplemented
def __rmul__(a, b):
if b in a.domain:
return a.mul(b)
else:
return NotImplemented
def __matmul__(a, b):
if isinstance(b, DDM):
return a.matmul(b)
else:
return NotImplemented
@classmethod
def _check(cls, a, op, b, ashape, bshape):
if a.domain != b.domain:
msg = "Domain mismatch: %s %s %s" % (a.domain, op, b.domain)
raise DDMDomainError(msg)
if ashape != bshape:
msg = "Shape mismatch: %s %s %s" % (a.shape, op, b.shape)
raise DDMShapeError(msg)
def add(a, b):
"""a + b"""
a._check(a, '+', b, a.shape, b.shape)
c = a.copy()
ddm_iadd(c, b)
return c
def sub(a, b):
"""a - b"""
a._check(a, '-', b, a.shape, b.shape)
c = a.copy()
ddm_isub(c, b)
return c
def neg(a):
"""-a"""
b = a.copy()
ddm_ineg(b)
return b
def mul(a, b):
c = a.copy()
ddm_imul(c, b)
return c
def matmul(a, b):
"""a @ b (matrix product)"""
m, o = a.shape
o2, n = b.shape
a._check(a, '*', b, o, o2)
c = a.zeros((m, n), a.domain)
ddm_imatmul(c, a, b)
return c
def hstack(A, B):
Anew = list(A.copy())
rows, cols = A.shape
domain = A.domain
Brows, Bcols = B.shape
assert Brows == rows
assert B.domain == domain
cols += Bcols
for i, Bi in enumerate(B):
Anew[i].extend(Bi)
return DDM(Anew, (rows, cols), A.domain)
def rref(a):
"""Reduced-row echelon form of a and list of pivots"""
b = a.copy()
pivots = ddm_irref(b)
return b, pivots
def nullspace(a):
rref, pivots = a.rref()
rows, cols = a.shape
domain = a.domain
basis = []
nonpivots = []
for i in range(cols):
if i in pivots:
continue
nonpivots.append(i)
vec = [domain.one if i == j else domain.zero for j in range(cols)]
for ii, jj in enumerate(pivots):
vec[jj] -= rref[ii][i]
basis.append(vec)
return DDM(basis, (len(basis), cols), domain), nonpivots
def det(a):
"""Determinant of a"""
m, n = a.shape
if m != n:
raise DDMShapeError("Determinant of non-square matrix")
b = a.copy()
K = b.domain
deta = ddm_idet(b, K)
return deta
def inv(a):
"""Inverse of a"""
m, n = a.shape
if m != n:
raise DDMShapeError("Determinant of non-square matrix")
ainv = a.copy()
K = a.domain
ddm_iinv(ainv, a, K)
return ainv
def lu(a):
"""L, U decomposition of a"""
m, n = a.shape
K = a.domain
U = a.copy()
L = a.eye(m, K)
swaps = ddm_ilu_split(L, U, K)
return L, U, swaps
def lu_solve(a, b):
"""x where a*x = b"""
m, n = a.shape
m2, o = b.shape
a._check(a, 'lu_solve', b, m, m2)
L, U, swaps = a.lu()
x = a.zeros((n, o), a.domain)
ddm_ilu_solve(x, L, U, swaps, b)
return x
def charpoly(a):
"""Coefficients of characteristic polynomial of a"""
K = a.domain
m, n = a.shape
if m != n:
raise DDMShapeError("Charpoly of non-square matrix")
vec = ddm_berk(a, K)
coeffs = [vec[i][0] for i in range(n+1)]
return coeffs
from .sdm import SDM
|
8ebbb84b84e4fee1b24795fab8198491d1b566be34b59d91c6a86b511ac1ffd6 | """Tests for classes defining properties of ground domains, e.g. ZZ, QQ, ZZ[x] ... """
from sympy import I, S, sqrt, sin, oo, Poly, Float, Integer, Rational, pi
from sympy.abc import x, y, z
from sympy.core.compatibility import HAS_GMPY
from sympy.polys.domains import (ZZ, QQ, RR, CC, FF, GF, EX, ZZ_gmpy,
ZZ_python, QQ_gmpy, QQ_python)
from sympy.polys.domains.algebraicfield import AlgebraicField
from sympy.polys.domains.gaussiandomains import ZZ_I, QQ_I
from sympy.polys.domains.polynomialring import PolynomialRing
from sympy.polys.domains.realfield import RealField
from sympy.polys.rings import ring
from sympy.polys.fields import field
from sympy.polys.agca.extensions import FiniteExtension
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(ZZ_I, F3) == ZZ_I
assert unify(ZZ_I, ZZ) == ZZ_I
assert unify(ZZ_I, ZZ_I) == ZZ_I
assert unify(ZZ_I, QQ) == QQ_I
assert unify(ZZ_I, ALG) == QQ.algebraic_field(I, sqrt(2), sqrt(3))
assert unify(ZZ_I, RR) == CC
assert unify(ZZ_I, CC) == CC
assert unify(ZZ_I, ZZ[x]) == ZZ_I[x]
assert unify(ZZ_I, ZZ_I[x]) == ZZ_I[x]
assert unify(ZZ_I, ZZ.frac_field(x)) == ZZ_I.frac_field(x)
assert unify(ZZ_I, ZZ_I.frac_field(x)) == ZZ_I.frac_field(x)
assert unify(ZZ_I, EX) == EX
assert unify(QQ_I, F3) == QQ_I
assert unify(QQ_I, ZZ) == QQ_I
assert unify(QQ_I, ZZ_I) == QQ_I
assert unify(QQ_I, QQ) == QQ_I
assert unify(QQ_I, ALG) == QQ.algebraic_field(I, sqrt(2), sqrt(3))
assert unify(QQ_I, RR) == CC
assert unify(QQ_I, CC) == CC
assert unify(QQ_I, ZZ[x]) == QQ_I[x]
assert unify(QQ_I, ZZ_I[x]) == QQ_I[x]
assert unify(QQ_I, QQ[x]) == QQ_I[x]
assert unify(QQ_I, QQ_I[x]) == QQ_I[x]
assert unify(QQ_I, ZZ.frac_field(x)) == QQ_I.frac_field(x)
assert unify(QQ_I, ZZ_I.frac_field(x)) == QQ_I.frac_field(x)
assert unify(QQ_I, QQ.frac_field(x)) == QQ_I.frac_field(x)
assert unify(QQ_I, QQ_I.frac_field(x)) == QQ_I.frac_field(x)
assert unify(QQ_I, 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_FiniteExtension():
KxZZ = FiniteExtension(Poly(x**2 - 2, x, domain=ZZ))
KxQQ = FiniteExtension(Poly(x**2 - 2, x, domain=QQ))
KxZZy = FiniteExtension(Poly(x**2 - 2, x, domain=ZZ[y]))
KxQQy = FiniteExtension(Poly(x**2 - 2, x, domain=QQ[y]))
assert KxZZ.unify(KxZZ) == KxZZ
assert KxQQ.unify(KxQQ) == KxQQ
assert KxZZy.unify(KxZZy) == KxZZy
assert KxQQy.unify(KxQQy) == KxQQy
assert KxZZ.unify(ZZ) == KxZZ
assert KxZZ.unify(QQ) == KxQQ
assert KxQQ.unify(ZZ) == KxQQ
assert KxQQ.unify(QQ) == KxQQ
assert KxZZ.unify(ZZ[y]) == KxZZy
assert KxZZ.unify(QQ[y]) == KxQQy
assert KxQQ.unify(ZZ[y]) == KxQQy
assert KxQQ.unify(QQ[y]) == KxQQy
assert KxZZy.unify(ZZ) == KxZZy
assert KxZZy.unify(QQ) == KxQQy
assert KxQQy.unify(ZZ) == KxQQy
assert KxQQy.unify(QQ) == KxQQy
assert KxZZy.unify(ZZ[y]) == KxZZy
assert KxZZy.unify(QQ[y]) == KxQQy
assert KxQQy.unify(ZZ[y]) == KxQQy
assert KxQQy.unify(QQ[y]) == KxQQy
K = FiniteExtension(Poly(x**2 - 2, x, domain=ZZ[y]))
assert K.unify(ZZ) == K
assert K.unify(ZZ[x]) == K
assert K.unify(ZZ[y]) == K
assert K.unify(ZZ[x, y]) == K
Kz = FiniteExtension(Poly(x**2 - 2, x, domain=ZZ[y, z]))
assert K.unify(ZZ[z]) == Kz
assert K.unify(ZZ[x, z]) == Kz
assert K.unify(ZZ[y, z]) == Kz
assert K.unify(ZZ[x, y, z]) == Kz
Kx = FiniteExtension(Poly(x**2 - 2, x, domain=ZZ))
Ky = FiniteExtension(Poly(y**2 - 2, y, domain=ZZ))
Kxy = FiniteExtension(Poly(y**2 - 2, y, domain=Kx))
assert Kx.unify(Kx) == Kx
assert Ky.unify(Ky) == Ky
assert Kx.unify(Ky) == Kxy
assert Ky.unify(Kx) == Kxy
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_is_unit():
nums = [-2, -1, 0, 1, 2]
invring = [False, True, False, True, False]
invfield = [True, True, False, True, True]
ZZx, QQx, QQxf = ZZ[x], QQ[x], QQ.frac_field(x)
assert [ZZ.is_unit(ZZ(n)) for n in nums] == invring
assert [QQ.is_unit(QQ(n)) for n in nums] == invfield
assert [ZZx.is_unit(ZZx(n)) for n in nums] == invring
assert [QQx.is_unit(QQx(n)) for n in nums] == invfield
assert [QQxf.is_unit(QQxf(n)) for n in nums] == invfield
assert ZZx.is_unit(ZZx(x)) is False
assert QQx.is_unit(QQx(x)) is False
assert QQxf.is_unit(QQxf(x)) is True
def test_Domain_convert():
def check_element(e1, e2, K1, K2, K3):
assert type(e1) is type(e2), '%s, %s: %s %s -> %s' % (e1, e2, K1, K2, K3)
assert e1 == e2, '%s, %s: %s %s -> %s' % (e1, e2, K1, K2, K3)
def check_domains(K1, K2):
K3 = K1.unify(K2)
check_element(K3.convert_from(K1.one, K1), K3.one , K1, K2, K3)
check_element(K3.convert_from(K2.one, K2), K3.one , K1, K2, K3)
check_element(K3.convert_from(K1.zero, K1), K3.zero, K1, K2, K3)
check_element(K3.convert_from(K2.zero, K2), K3.zero, K1, K2, K3)
def composite_domains(K):
return [K, K[y], K[z], K[y, z],
K.frac_field(y), K.frac_field(z), K.frac_field(y, z)]
QQ2 = QQ.algebraic_field(sqrt(2))
QQ3 = QQ.algebraic_field(sqrt(3))
doms = [ZZ, QQ, QQ2, QQ3, QQ_I, ZZ_I, RR, CC]
for i, K1 in enumerate(doms):
for K2 in doms[i:]:
for K3 in composite_domains(K1):
for K4 in composite_domains(K2):
check_domains(K3, K4)
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
assert CC.convert(ZZ_I(1, 2)) == CC(1, 2)
assert CC.convert(QQ_I(1, 2)) == CC(1, 2)
def test_GlobalPolynomialRing_convert():
K1 = QQ.old_poly_ring(x)
K2 = QQ[x]
assert K1.convert(x) == K1.convert(K2.convert(x), K2)
assert K2.convert(x) == K2.convert(K1.convert(x), K1)
K1 = QQ.old_poly_ring(x, y)
K2 = QQ[x]
assert K1.convert(x) == K1.convert(K2.convert(x), K2)
#assert K2.convert(x) == K2.convert(K1.convert(x), K1)
K1 = ZZ.old_poly_ring(x, y)
K2 = QQ[x]
assert K1.convert(x) == K1.convert(K2.convert(x), K2)
#assert K2.convert(x) == K2.convert(K1.convert(x), K1)
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_FractionField_convert():
K = QQ.frac_field(x)
assert K.convert(QQ(2, 3), QQ) == K.from_sympy(Rational(2, 3))
K = QQ.frac_field(x)
assert K.convert(ZZ(2), ZZ) == K.from_sympy(Integer(2))
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_drop():
assert ZZ.drop(x) == ZZ
assert ZZ[x].drop(x) == ZZ
assert ZZ[x, y].drop(x) == ZZ[y]
assert ZZ.frac_field(x).drop(x) == ZZ
assert ZZ.frac_field(x, y).drop(x) == ZZ.frac_field(y)
assert ZZ[x][y].drop(y) == ZZ[x]
assert ZZ[x][y].drop(x) == ZZ[y]
assert ZZ.frac_field(x)[y].drop(x) == ZZ[y]
assert ZZ.frac_field(x)[y].drop(y) == ZZ.frac_field(x)
Ky = FiniteExtension(Poly(x**2-1, x, domain=ZZ[y]))
K = FiniteExtension(Poly(x**2-1, x, domain=ZZ))
assert Ky.drop(y) == K
raises(GeneratorsError, lambda: Ky.drop(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, d = [ZZ_I.convert(x) for x in (5, 2 + I, 3 - I, 5 - 5)]
ZZ_I.gcd(a, b) == b
ZZ_I.gcd(a, c) == b
ZZ_I.lcm(a, b) == a
ZZ_I.lcm(a, c) == d
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
assert QQ_I.convert(QQ(3, 2)) == QQ_I(QQ(3, 2), QQ(0))
assert QQ_I.convert(QQ(3, 2), QQ) == QQ_I(QQ(3, 2), QQ(0))
for G in (QQ_I, ZZ_I):
q = G(3, 4)
assert str(q) == '3 + 4*I'
assert q.parent() == G
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 G(1, 2) == G(1, 2)
assert G(1, 2) != G(1, 3)
assert G(3, 0) == G(3)
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 q * q == G(-7, 24)
assert 3 * q == q * 3 == G(9, 12)
assert q ** 0 == G(1, 0)
assert q ** 1 == q
assert q ** 2 == q * q == G(-7, 24)
assert q ** 3 == q * q * q == G(-117, 44)
assert 1 / q == q ** -1 == QQ_I(S(3)/25, - S(4)/25)
assert q / 1 == QQ_I(3, 4)
assert q / 2 == QQ_I(S(3)/2, 2)
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 q*i + r == 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(TypeError, lambda: q + x)
raises(TypeError, lambda: q - x)
raises(TypeError, lambda: x + q)
raises(TypeError, lambda: x - q)
raises(TypeError, lambda: q * x)
raises(TypeError, lambda: x * q)
raises(TypeError, lambda: q / x)
raises(TypeError, lambda: x / q)
raises(TypeError, lambda: q // x)
raises(TypeError, lambda: x // q)
assert G.from_sympy(S(2)) == G(2, 0)
assert G.to_sympy(G(2, 0)) == S(2)
raises(CoercionFailed, lambda: G.from_sympy(pi))
PR = G.inject(x)
assert isinstance(PR, PolynomialRing)
assert PR.domain == G
assert len(PR.gens) == 1 and PR.gens[0].as_expr() == x
if G is QQ_I:
AF = G.as_AlgebraicField()
assert isinstance(AF, AlgebraicField)
assert AF.domain == QQ
assert AF.ext.args[0] == I
for qi in [G(-1, 0), G(1, 0), G(0, -1), G(0, 1)]:
assert G.is_negative(qi) is False
assert G.is_positive(qi) is False
assert G.is_nonnegative(qi) is False
assert G.is_nonpositive(qi) is False
domains = [ZZ_python(), QQ_python(), AlgebraicField(QQ, I)]
if HAS_GMPY:
domains += [ZZ_gmpy(), QQ_gmpy()]
for K in domains:
assert G.convert(K(2)) == G(2, 0)
assert G.convert(K(2), K) == G(2, 0)
for K in ZZ_I, QQ_I:
assert G.convert(K(1, 1)) == G(1, 1)
assert G.convert(K(1, 1), K) == G(1, 1)
if G == ZZ_I:
assert repr(q) == 'ZZ_I(3, 4)'
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 repr(q) == 'QQ_I(3, 4)'
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)
q2 = G(S(3)/2, S(5)/3)
assert G.numer(q2) == ZZ_I(9, 10)
assert G.denom(q2) == ZZ_I(6)
def test_canonical_unit():
for K in [ZZ, QQ, RR]: # CC?
assert K.canonical_unit(K(2)) == K(1)
assert K.canonical_unit(K(-2)) == K(-1)
for K in [ZZ_I, QQ_I]:
i = K.from_sympy(I)
assert K.canonical_unit(K(2)) == K(1)
assert K.canonical_unit(K(2)*i) == -i
assert K.canonical_unit(-K(2)) == K(-1)
assert K.canonical_unit(-K(2)*i) == i
K = ZZ[x]
assert K.canonical_unit(K(x + 1)) == K(1)
assert K.canonical_unit(K(-x + 1)) == K(-1)
K = ZZ_I[x]
assert K.canonical_unit(K.from_sympy(I*x)) == ZZ_I(0, -1)
K = ZZ_I.frac_field(x, y)
i = K.from_sympy(I)
assert i / i == K.one
assert (K.one + i)/(i - K.one) == -i
def test_issue_18278():
assert str(RR(2).parent()) == 'RR'
assert str(CC(2).parent()) == 'CC'
def test_Domain_is_negative():
I = S.ImaginaryUnit
a, b = [CC.convert(x) for x in (2 + I, 5)]
assert CC.is_negative(a) == False
assert CC.is_negative(b) == False
def test_Domain_is_positive():
I = S.ImaginaryUnit
a, b = [CC.convert(x) for x in (2 + I, 5)]
assert CC.is_positive(a) == False
assert CC.is_positive(b) == False
def test_Domain_is_nonnegative():
I = S.ImaginaryUnit
a, b = [CC.convert(x) for x in (2 + I, 5)]
assert CC.is_nonnegative(a) == False
assert CC.is_nonnegative(b) == False
def test_Domain_is_nonpositive():
I = S.ImaginaryUnit
a, b = [CC.convert(x) for x in (2 + I, 5)]
assert CC.is_nonpositive(a) == False
assert CC.is_nonpositive(b) == False
|
283b84b55bb47012b17b2fb0b4fc590e92d82c34b8e10375f11a922f07546711 | """
Tests for the sympy.polys.matrices.eigen module
"""
from sympy import S, Matrix, sqrt
from sympy.polys.agca.extensions import FiniteExtension
from sympy.polys.domains import QQ
from sympy.polys.polytools import Poly
from sympy.polys.rootoftools import CRootOf
from sympy.polys.matrices.domainmatrix import DomainMatrix
from sympy.polys.matrices.eigen import dom_eigenvects, dom_eigenvects_to_sympy
def test_dom_eigenvects_rational():
# Rational eigenvalues
A = DomainMatrix([[QQ(1), QQ(2)], [QQ(1), QQ(2)]], (2, 2), QQ)
rational_eigenvects = [
(QQ, QQ(3), 1, DomainMatrix([[QQ(1), QQ(1)]], (1, 2), QQ)),
(QQ, QQ(0), 1, DomainMatrix([[QQ(-2), QQ(1)]], (1, 2), QQ)),
]
assert dom_eigenvects(A) == (rational_eigenvects, [])
# Test converting to Expr:
sympy_eigenvects = [
(S(3), 1, [Matrix([1, 1])]),
(S(0), 1, [Matrix([-2, 1])]),
]
assert dom_eigenvects_to_sympy(rational_eigenvects, [], Matrix) == sympy_eigenvects
def test_dom_eigenvects_algebraic():
# Algebraic eigenvalues
A = DomainMatrix([[QQ(1), QQ(2)], [QQ(3), QQ(4)]], (2, 2), QQ)
Avects = dom_eigenvects(A)
# Extract the dummy to build the expected result:
lamda = Avects[1][0][1].gens[0]
irreducible = Poly(lamda**2 - 5*lamda - 2, lamda, domain=QQ)
K = FiniteExtension(irreducible)
KK = K.from_sympy
algebraic_eigenvects = [
(K, irreducible, 1, DomainMatrix([[KK((lamda-4)/3), KK(1)]], (1, 2), K)),
]
assert Avects == ([], algebraic_eigenvects)
# Test converting to Expr:
sympy_eigenvects = [
(S(5)/2 - sqrt(33)/2, 1, [Matrix([[-sqrt(33)/6 - S(1)/2], [1]])]),
(S(5)/2 + sqrt(33)/2, 1, [Matrix([[-S(1)/2 + sqrt(33)/6], [1]])]),
]
assert dom_eigenvects_to_sympy([], algebraic_eigenvects, Matrix) == sympy_eigenvects
def test_dom_eigenvects_rootof():
# Algebraic eigenvalues
A = DomainMatrix([
[0, 0, 0, 0, -1],
[1, 0, 0, 0, 1],
[0, 1, 0, 0, 0],
[0, 0, 1, 0, 0],
[0, 0, 0, 1, 0]], (5, 5), QQ)
Avects = dom_eigenvects(A)
# Extract the dummy to build the expected result:
lamda = Avects[1][0][1].gens[0]
irreducible = Poly(lamda**5 - lamda + 1, lamda, domain=QQ)
K = FiniteExtension(irreducible)
KK = K.from_sympy
algebraic_eigenvects = [
(K, irreducible, 1,
DomainMatrix([
[KK(lamda**4-1), KK(lamda**3), KK(lamda**2), KK(lamda), KK(1)]
], (1, 5), K)),
]
assert Avects == ([], algebraic_eigenvects)
# Test converting to Expr (slow):
l0, l1, l2, l3, l4 = [CRootOf(lamda**5 - lamda + 1, i) for i in range(5)]
sympy_eigenvects = [
(l0, 1, [Matrix([-1 + l0**4, l0**3, l0**2, l0, 1])]),
(l1, 1, [Matrix([-1 + l1**4, l1**3, l1**2, l1, 1])]),
(l2, 1, [Matrix([-1 + l2**4, l2**3, l2**2, l2, 1])]),
(l3, 1, [Matrix([-1 + l3**4, l3**3, l3**2, l3, 1])]),
(l4, 1, [Matrix([-1 + l4**4, l4**3, l4**2, l4, 1])]),
]
assert dom_eigenvects_to_sympy([], algebraic_eigenvects, Matrix) == sympy_eigenvects
|
05e5b55e50d9168d646768470bb66bbd47c0c4c34a953125e50f48a9ce9b7b01 | from sympy.testing.pytest import raises
from sympy.core.compatibility import HAS_GMPY
from sympy.polys import ZZ, QQ
from sympy.polys.matrices.ddm import DDM
from sympy.polys.matrices.exceptions import (
DDMShapeError, NonInvertibleMatrixError, DDMDomainError,
DDMBadInputError)
def test_DDM_init():
items = [[ZZ(0), ZZ(1), ZZ(2)], [ZZ(3), ZZ(4), ZZ(5)]]
shape = (2, 3)
ddm = DDM(items, shape, ZZ)
assert ddm.shape == shape
assert ddm.rows == 2
assert ddm.cols == 3
assert ddm.domain == ZZ
raises(DDMBadInputError, lambda: DDM([[ZZ(2), ZZ(3)]], (2, 2), ZZ))
raises(DDMBadInputError, lambda: DDM([[ZZ(1)], [ZZ(2), ZZ(3)]], (2, 2), ZZ))
def test_DDM_getsetitem():
ddm = DDM([[ZZ(2), ZZ(3)], [ZZ(4), ZZ(5)]], (2, 2), ZZ)
assert ddm[0][0] == ZZ(2)
assert ddm[0][1] == ZZ(3)
assert ddm[1][0] == ZZ(4)
assert ddm[1][1] == ZZ(5)
raises(IndexError, lambda: ddm[2][0])
raises(IndexError, lambda: ddm[0][2])
ddm[0][0] = ZZ(-1)
assert ddm[0][0] == ZZ(-1)
def test_DDM_str():
ddm = DDM([[ZZ(0), ZZ(1)], [ZZ(2), ZZ(3)]], (2, 2), ZZ)
if HAS_GMPY: # pragma: no cover
assert str(ddm) == '[[0, 1], [2, 3]]'
assert repr(ddm) == 'DDM([[mpz(0), mpz(1)], [mpz(2), mpz(3)]], (2, 2), ZZ)'
else: # pragma: no cover
assert repr(ddm) == 'DDM([[0, 1], [2, 3]], (2, 2), ZZ)'
assert str(ddm) == '[[0, 1], [2, 3]]'
def test_DDM_eq():
items = [[ZZ(0), ZZ(1)], [ZZ(2), ZZ(3)]]
ddm1 = DDM(items, (2, 2), ZZ)
ddm2 = DDM(items, (2, 2), ZZ)
assert (ddm1 == ddm1) is True
assert (ddm1 == items) is False
assert (items == ddm1) is False
assert (ddm1 == ddm2) is True
assert (ddm2 == ddm1) is True
assert (ddm1 != ddm1) is False
assert (ddm1 != items) is True
assert (items != ddm1) is True
assert (ddm1 != ddm2) is False
assert (ddm2 != ddm1) is False
ddm3 = DDM([[ZZ(0), ZZ(1)], [ZZ(3), ZZ(3)]], (2, 2), ZZ)
ddm3 = DDM(items, (2, 2), QQ)
assert (ddm1 == ddm3) is False
assert (ddm3 == ddm1) is False
assert (ddm1 != ddm3) is True
assert (ddm3 != ddm1) is True
def test_DDM_convert_to():
ddm = DDM([[ZZ(1), ZZ(2)]], (1, 2), ZZ)
assert ddm.convert_to(ZZ) == ddm
ddmq = ddm.convert_to(QQ)
assert ddmq.domain == QQ
def test_DDM_zeros():
ddmz = DDM.zeros((3, 4), QQ)
assert list(ddmz) == [[QQ(0)] * 4] * 3
assert ddmz.shape == (3, 4)
assert ddmz.domain == QQ
def test_DDM_ones():
ddmone = DDM.ones((2, 3), QQ)
assert list(ddmone) == [[QQ(1)] * 3] * 2
assert ddmone.shape == (2, 3)
assert ddmone.domain == QQ
def test_DDM_eye():
ddmz = DDM.eye(3, QQ)
f = lambda i, j: QQ(1) if i == j else QQ(0)
assert list(ddmz) == [[f(i, j) for i in range(3)] for j in range(3)]
assert ddmz.shape == (3, 3)
assert ddmz.domain == QQ
def test_DDM_copy():
ddm1 = DDM([[QQ(1)], [QQ(2)]], (2, 1), QQ)
ddm2 = ddm1.copy()
assert (ddm1 == ddm2) is True
ddm1[0][0] = QQ(-1)
assert (ddm1 == ddm2) is False
ddm2[0][0] = QQ(-1)
assert (ddm1 == ddm2) is True
def test_DDM_add():
A = DDM([[ZZ(1)], [ZZ(2)]], (2, 1), ZZ)
B = DDM([[ZZ(3)], [ZZ(4)]], (2, 1), ZZ)
C = DDM([[ZZ(4)], [ZZ(6)]], (2, 1), ZZ)
AQ = DDM([[QQ(1)], [QQ(2)]], (2, 1), QQ)
assert A + B == A.add(B) == C
raises(DDMShapeError, lambda: A + DDM([[ZZ(5)]], (1, 1), ZZ))
raises(TypeError, lambda: A + ZZ(1))
raises(TypeError, lambda: ZZ(1) + A)
raises(DDMDomainError, lambda: A + AQ)
raises(DDMDomainError, lambda: AQ + A)
def test_DDM_sub():
A = DDM([[ZZ(1)], [ZZ(2)]], (2, 1), ZZ)
B = DDM([[ZZ(3)], [ZZ(4)]], (2, 1), ZZ)
C = DDM([[ZZ(-2)], [ZZ(-2)]], (2, 1), ZZ)
AQ = DDM([[QQ(1)], [QQ(2)]], (2, 1), QQ)
D = DDM([[ZZ(5)]], (1, 1), ZZ)
assert A - B == A.sub(B) == C
raises(TypeError, lambda: A - ZZ(1))
raises(TypeError, lambda: ZZ(1) - A)
raises(DDMShapeError, lambda: A - D)
raises(DDMShapeError, lambda: D - A)
raises(DDMShapeError, lambda: A.sub(D))
raises(DDMShapeError, lambda: D.sub(A))
raises(DDMDomainError, lambda: A - AQ)
raises(DDMDomainError, lambda: AQ - A)
raises(DDMDomainError, lambda: A.sub(AQ))
raises(DDMDomainError, lambda: AQ.sub(A))
def test_DDM_neg():
A = DDM([[ZZ(1)], [ZZ(2)]], (2, 1), ZZ)
An = DDM([[ZZ(-1)], [ZZ(-2)]], (2, 1), ZZ)
assert -A == A.neg() == An
assert -An == An.neg() == A
def test_DDM_mul():
A = DDM([[ZZ(1)]], (1, 1), ZZ)
A2 = DDM([[ZZ(2)]], (1, 1), ZZ)
assert A * ZZ(2) == A2
assert ZZ(2) * A == A2
raises(TypeError, lambda: [[1]] * A)
raises(TypeError, lambda: A * [[1]])
def test_DDM_matmul():
A = DDM([[ZZ(1)], [ZZ(2)]], (2, 1), ZZ)
B = DDM([[ZZ(3), ZZ(4)]], (1, 2), ZZ)
AB = DDM([[ZZ(3), ZZ(4)], [ZZ(6), ZZ(8)]], (2, 2), ZZ)
BA = DDM([[ZZ(11)]], (1, 1), ZZ)
assert A @ B == A.matmul(B) == AB
assert B @ A == B.matmul(A) == BA
raises(TypeError, lambda: A @ 1)
raises(TypeError, lambda: A @ [[3, 4]])
Bq = DDM([[QQ(3), QQ(4)]], (1, 2), QQ)
raises(DDMDomainError, lambda: A @ Bq)
raises(DDMDomainError, lambda: Bq @ A)
C = DDM([[ZZ(1)]], (1, 1), ZZ)
assert A @ C == A.matmul(C) == A
raises(DDMShapeError, lambda: C @ A)
raises(DDMShapeError, lambda: C.matmul(A))
Z04 = DDM([], (0, 4), ZZ)
Z40 = DDM([[]]*4, (4, 0), ZZ)
Z50 = DDM([[]]*5, (5, 0), ZZ)
Z05 = DDM([], (0, 5), ZZ)
Z45 = DDM([[0] * 5] * 4, (4, 5), ZZ)
Z54 = DDM([[0] * 4] * 5, (5, 4), ZZ)
Z00 = DDM([], (0, 0), ZZ)
assert Z04 @ Z45 == Z04.matmul(Z45) == Z05
assert Z45 @ Z50 == Z45.matmul(Z50) == Z40
assert Z00 @ Z04 == Z00.matmul(Z04) == Z04
assert Z50 @ Z00 == Z50.matmul(Z00) == Z50
assert Z00 @ Z00 == Z00.matmul(Z00) == Z00
assert Z50 @ Z04 == Z50.matmul(Z04) == Z54
raises(DDMShapeError, lambda: Z05 @ Z40)
raises(DDMShapeError, lambda: Z05.matmul(Z40))
def test_DDM_hstack():
A = DDM([[ZZ(1), ZZ(2), ZZ(3)]], (1, 3), ZZ)
B = DDM([[ZZ(4), ZZ(5)]], (1, 2), ZZ)
Ah = A.hstack(B)
assert Ah.shape == (1, 5)
assert Ah.domain == ZZ
assert Ah == DDM([[ZZ(1), ZZ(2), ZZ(3), ZZ(4), ZZ(5)]], (1, 5), ZZ)
def test_DDM_rref():
A = DDM([], (0, 4), QQ)
assert A.rref() == (A, [])
A = DDM([[QQ(0), QQ(1)], [QQ(1), QQ(1)]], (2, 2), QQ)
Ar = DDM([[QQ(1), QQ(0)], [QQ(0), QQ(1)]], (2, 2), QQ)
pivots = [0, 1]
assert A.rref() == (Ar, pivots)
A = DDM([[QQ(1), QQ(2), QQ(1)], [QQ(3), QQ(4), QQ(1)]], (2, 3), QQ)
Ar = DDM([[QQ(1), QQ(0), QQ(-1)], [QQ(0), QQ(1), QQ(1)]], (2, 3), QQ)
pivots = [0, 1]
assert A.rref() == (Ar, pivots)
A = DDM([[QQ(3), QQ(4), QQ(1)], [QQ(1), QQ(2), QQ(1)]], (2, 3), QQ)
Ar = DDM([[QQ(1), QQ(0), QQ(-1)], [QQ(0), QQ(1), QQ(1)]], (2, 3), QQ)
pivots = [0, 1]
assert A.rref() == (Ar, pivots)
A = DDM([[QQ(1), QQ(0)], [QQ(1), QQ(3)], [QQ(0), QQ(1)]], (3, 2), QQ)
Ar = DDM([[QQ(1), QQ(0)], [QQ(0), QQ(1)], [QQ(0), QQ(0)]], (3, 2), QQ)
pivots = [0, 1]
assert A.rref() == (Ar, pivots)
A = DDM([[QQ(1), QQ(0), QQ(1)], [QQ(3), QQ(0), QQ(1)]], (2, 3), QQ)
Ar = DDM([[QQ(1), QQ(0), QQ(0)], [QQ(0), QQ(0), QQ(1)]], (2, 3), QQ)
pivots = [0, 2]
assert A.rref() == (Ar, pivots)
def test_DDM_nullspace():
A = DDM([[QQ(1), QQ(1)], [QQ(1), QQ(1)]], (2, 2), QQ)
Anull = DDM([[QQ(-1), QQ(1)]], (1, 2), QQ)
nonpivots = [1]
assert A.nullspace() == (Anull, nonpivots)
def test_DDM_det():
# 0x0 case
A = DDM([], (0, 0), ZZ)
assert A.det() == ZZ(1)
# 1x1 case
A = DDM([[ZZ(2)]], (1, 1), ZZ)
assert A.det() == ZZ(2)
# 2x2 case
A = DDM([[ZZ(1), ZZ(2)], [ZZ(3), ZZ(4)]], (2, 2), ZZ)
assert A.det() == ZZ(-2)
# 3x3 with swap
A = DDM([[ZZ(1), ZZ(2), ZZ(3)], [ZZ(1), ZZ(2), ZZ(4)], [ZZ(1), ZZ(2), ZZ(5)]], (3, 3), ZZ)
assert A.det() == ZZ(0)
# 2x2 QQ case
A = DDM([[QQ(1, 2), QQ(1, 2)], [QQ(1, 3), QQ(1, 4)]], (2, 2), QQ)
assert A.det() == QQ(-1, 24)
# Nonsquare error
A = DDM([[ZZ(1)], [ZZ(2)]], (2, 1), ZZ)
raises(DDMShapeError, lambda: A.det())
# Nonsquare error with empty matrix
A = DDM([], (0, 1), ZZ)
raises(DDMShapeError, lambda: A.det())
def test_DDM_inv():
A = DDM([[QQ(1, 1), QQ(2, 1)], [QQ(3, 1), QQ(4, 1)]], (2, 2), QQ)
Ainv = DDM([[QQ(-2, 1), QQ(1, 1)], [QQ(3, 2), QQ(-1, 2)]], (2, 2), QQ)
assert A.inv() == Ainv
A = DDM([[QQ(1), QQ(2)]], (1, 2), QQ)
raises(DDMShapeError, lambda: A.inv())
A = DDM([[ZZ(2)]], (1, 1), ZZ)
raises(ValueError, lambda: A.inv())
A = DDM([], (0, 0), QQ)
assert A.inv() == A
A = DDM([[QQ(1), QQ(2)], [QQ(2), QQ(4)]], (2, 2), QQ)
raises(NonInvertibleMatrixError, lambda: A.inv())
def test_DDM_lu():
A = DDM([[QQ(1), QQ(2)], [QQ(3), QQ(4)]], (2, 2), QQ)
L, U, swaps = A.lu()
assert L == DDM([[QQ(1), QQ(0)], [QQ(3), QQ(1)]], (2, 2), QQ)
assert U == DDM([[QQ(1), QQ(2)], [QQ(0), QQ(-2)]], (2, 2), QQ)
assert swaps == []
A = [[1, 0, 0, 0], [0, 0, 0, 0], [0, 0, 1, 1], [0, 0, 1, 2]]
Lexp = [[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 1, 1]]
Uexp = [[1, 0, 0, 0], [0, 0, 0, 0], [0, 0, 1, 1], [0, 0, 0, 1]]
to_dom = lambda rows, dom: [[dom(e) for e in row] for row in rows]
A = DDM(to_dom(A, QQ), (4, 4), QQ)
Lexp = DDM(to_dom(Lexp, QQ), (4, 4), QQ)
Uexp = DDM(to_dom(Uexp, QQ), (4, 4), QQ)
L, U, swaps = A.lu()
assert L == Lexp
assert U == Uexp
assert swaps == []
def test_DDM_lu_solve():
# Basic example
A = DDM([[QQ(1), QQ(2)], [QQ(3), QQ(4)]], (2, 2), QQ)
b = DDM([[QQ(1)], [QQ(2)]], (2, 1), QQ)
x = DDM([[QQ(0)], [QQ(1, 2)]], (2, 1), QQ)
assert A.lu_solve(b) == x
# Example with swaps
A = DDM([[QQ(0), QQ(2)], [QQ(3), QQ(4)]], (2, 2), QQ)
assert A.lu_solve(b) == x
# Overdetermined, consistent
A = DDM([[QQ(1), QQ(2)], [QQ(3), QQ(4)], [QQ(5), QQ(6)]], (3, 2), QQ)
b = DDM([[QQ(1)], [QQ(2)], [QQ(3)]], (3, 1), QQ)
assert A.lu_solve(b) == x
# Overdetermined, inconsistent
b = DDM([[QQ(1)], [QQ(2)], [QQ(4)]], (3, 1), QQ)
raises(NonInvertibleMatrixError, lambda: A.lu_solve(b))
# Square, noninvertible
A = DDM([[QQ(1), QQ(2)], [QQ(1), QQ(2)]], (2, 2), QQ)
b = DDM([[QQ(1)], [QQ(2)]], (2, 1), QQ)
raises(NonInvertibleMatrixError, lambda: A.lu_solve(b))
# Underdetermined
A = DDM([[QQ(1), QQ(2)]], (1, 2), QQ)
b = DDM([[QQ(3)]], (1, 1), QQ)
raises(NotImplementedError, lambda: A.lu_solve(b))
# Domain mismatch
bz = DDM([[ZZ(1)], [ZZ(2)]], (2, 1), ZZ)
raises(DDMDomainError, lambda: A.lu_solve(bz))
# Shape mismatch
b3 = DDM([[QQ(1)], [QQ(2)], [QQ(3)]], (3, 1), QQ)
raises(DDMShapeError, lambda: A.lu_solve(b3))
def test_DDM_charpoly():
A = DDM([], (0, 0), ZZ)
assert A.charpoly() == [ZZ(1)]
A = DDM([
[ZZ(1), ZZ(2), ZZ(3)],
[ZZ(4), ZZ(5), ZZ(6)],
[ZZ(7), ZZ(8), ZZ(9)]], (3, 3), ZZ)
Avec = [ZZ(1), ZZ(-15), ZZ(-18), ZZ(0)]
assert A.charpoly() == Avec
A = DDM([[ZZ(1), ZZ(2)]], (1, 2), ZZ)
raises(DDMShapeError, lambda: A.charpoly())
|
8f77f42cfb6f618ba5bfc5b5956a97bdd3771c915c43b9a91965a12773754da5 | from sympy.testing.pytest import raises
from sympy.core.symbol import S
from sympy.polys import ZZ, QQ
from sympy.polys.matrices.domainscalar import DomainScalar
from sympy.polys.matrices.domainmatrix import DomainMatrix
def test_DomainScalar___new__():
raises(TypeError, lambda: DomainScalar(ZZ(1), QQ))
raises(TypeError, lambda: DomainScalar(ZZ(1), 1))
def test_DomainScalar_new():
A = DomainScalar(ZZ(1), ZZ)
B = A.new(ZZ(4), ZZ)
assert B == DomainScalar(ZZ(4), ZZ)
def test_DomainScalar_repr():
A = DomainScalar(ZZ(1), ZZ)
assert repr(A) in {'1', 'mpz(1)'}
def test_DomainScalar_from_sympy():
expr = S(1)
B = DomainScalar.from_sympy(expr)
assert B == DomainScalar(ZZ(1), ZZ)
def test_DomainScalar_to_domain():
A = DomainScalar(ZZ(1), ZZ)
B = A.to_domain(QQ)
assert B == DomainScalar(QQ(1), QQ)
def test_DomainScalar_convert_to():
A = DomainScalar(ZZ(1), ZZ)
B = A.convert_to(QQ)
assert B == DomainScalar(QQ(1), QQ)
def test_DomainScalar_unify():
A = DomainScalar(ZZ(1), ZZ)
B = DomainScalar(QQ(2), QQ)
A, B = A.unify(B)
assert A.domain == B.domain == QQ
def test_DomainScalar_add():
A = DomainScalar(ZZ(1), ZZ)
B = DomainScalar(QQ(2), QQ)
assert A + B == DomainScalar(QQ(3), QQ)
raises(TypeError, lambda: A + 1.5)
def test_DomainScalar_sub():
A = DomainScalar(ZZ(1), ZZ)
B = DomainScalar(QQ(2), QQ)
assert A - B == DomainScalar(QQ(-1), QQ)
raises(TypeError, lambda: A - 1.5)
def test_DomainScalar_mul():
A = DomainScalar(ZZ(1), ZZ)
B = DomainScalar(QQ(2), QQ)
dm = DomainMatrix([[ZZ(1), ZZ(2)], [ZZ(3), ZZ(4)]], (2, 2), ZZ)
assert A * B == DomainScalar(QQ(2), QQ)
assert A * dm == dm
assert B * 2 == DomainScalar(QQ(4), QQ)
raises(TypeError, lambda: A * 1.5)
def test_DomainScalar_floordiv():
A = DomainScalar(ZZ(-5), ZZ)
B = DomainScalar(QQ(2), QQ)
assert A // B == DomainScalar(QQ(-5, 2), QQ)
C = DomainScalar(ZZ(2), ZZ)
assert A // C == DomainScalar(ZZ(-3), ZZ)
raises(TypeError, lambda: A // 1.5)
def test_DomainScalar_mod():
A = DomainScalar(ZZ(5), ZZ)
B = DomainScalar(QQ(2), QQ)
assert A % B == DomainScalar(QQ(0), QQ)
C = DomainScalar(ZZ(2), ZZ)
assert A % C == DomainScalar(ZZ(1), ZZ)
raises(TypeError, lambda: A % 1.5)
def test_DomainScalar_divmod():
A = DomainScalar(ZZ(5), ZZ)
B = DomainScalar(QQ(2), QQ)
assert divmod(A, B) == (DomainScalar(QQ(5, 2), QQ), DomainScalar(QQ(0), QQ))
C = DomainScalar(ZZ(2), ZZ)
assert divmod(A, C) == (DomainScalar(ZZ(2), ZZ), DomainScalar(ZZ(1), ZZ))
raises(TypeError, lambda: divmod(A, 1.5))
def test_DomainScalar_pow():
A = DomainScalar(ZZ(-5), ZZ)
B = A**(2)
assert B == DomainScalar(ZZ(25), ZZ)
raises(TypeError, lambda: A**(1.5))
def test_DomainScalar_pos():
A = DomainScalar(QQ(2), QQ)
B = DomainScalar(QQ(2), QQ)
assert +A == B
def test_DomainScalar_eq():
A = DomainScalar(QQ(2), QQ)
assert A == A
B = DomainScalar(ZZ(-5), ZZ)
assert A != B
C = DomainScalar(ZZ(2), ZZ)
assert A != C
D = [1]
assert A != D
def test_DomainScalar_isZero():
A = DomainScalar(ZZ(0), ZZ)
assert A.is_zero() == True
B = DomainScalar(ZZ(1), ZZ)
assert B.is_zero() == False
def test_DomainScalar_isOne():
A = DomainScalar(ZZ(1), ZZ)
assert A.is_one() == True
B = DomainScalar(ZZ(0), ZZ)
assert B.is_one() == False
|
646473dd030415d86398ef3f39b9dc2d02aa9b071bea10bbe459fae2478e2462 | from sympy.testing.pytest import raises
from sympy.core.numbers import Rational
from sympy.functions import sqrt
from sympy.matrices.common import (NonInvertibleMatrixError,
NonSquareMatrixError, ShapeError)
from sympy.matrices.dense import Matrix
from sympy.polys import ZZ, QQ
from sympy.polys.matrices.domainmatrix import DomainMatrix
from sympy.polys.matrices.exceptions import (DDMBadInputError, DDMDomainError,
DDMShapeError, DDMFormatError)
from sympy.polys.matrices.ddm import DDM
from sympy.polys.matrices.sdm import SDM
from sympy.polys.matrices.domainscalar import DomainScalar
def test_DomainMatrix_init():
lol = [[ZZ(1), ZZ(2)], [ZZ(3), ZZ(4)]]
dod = {0: {0: ZZ(1), 1:ZZ(2)}, 1: {0:ZZ(3), 1:ZZ(4)}}
ddm = DDM(lol, (2, 2), ZZ)
sdm = SDM(dod, (2, 2), ZZ)
A = DomainMatrix(lol, (2, 2), ZZ)
assert A.rep == ddm
assert A.shape == (2, 2)
assert A.domain == ZZ
A = DomainMatrix(dod, (2, 2), ZZ)
assert A.rep == sdm
assert A.shape == (2, 2)
assert A.domain == ZZ
raises(TypeError, lambda: DomainMatrix(ddm, (2, 2), ZZ))
raises(TypeError, lambda: DomainMatrix(sdm, (2, 2), ZZ))
raises(TypeError, lambda: DomainMatrix(Matrix([[1]]), (1, 1), ZZ))
for fmt, rep in [('sparse', sdm), ('dense', ddm)]:
A = DomainMatrix(lol, (2, 2), ZZ, fmt=fmt)
assert A.rep == rep
A = DomainMatrix(dod, (2, 2), ZZ, fmt=fmt)
assert A.rep == rep
raises(ValueError, lambda: DomainMatrix(lol, (2, 2), ZZ, fmt='invalid'))
raises(DDMBadInputError, lambda: DomainMatrix([[ZZ(1), ZZ(2)]], (2, 2), ZZ))
def test_DomainMatrix_from_rep():
ddm = DDM([[ZZ(1), ZZ(2)], [ZZ(3), ZZ(4)]], (2, 2), ZZ)
A = DomainMatrix.from_rep(ddm)
assert A.rep == ddm
assert A.shape == (2, 2)
assert A.domain == ZZ
sdm = SDM({0: {0: ZZ(1), 1:ZZ(2)}, 1: {0:ZZ(3), 1:ZZ(4)}}, (2, 2), ZZ)
A = DomainMatrix.from_rep(sdm)
assert A.rep == sdm
assert A.shape == (2, 2)
assert A.domain == ZZ
A = DomainMatrix([[ZZ(1)]], (1, 1), ZZ)
raises(TypeError, lambda: DomainMatrix.from_rep(A))
def test_DomainMatrix_from_list_sympy():
ddm = DDM([[ZZ(1), ZZ(2)], [ZZ(3), ZZ(4)]], (2, 2), ZZ)
A = DomainMatrix.from_list_sympy(2, 2, [[1, 2], [3, 4]])
assert A.rep == ddm
assert A.shape == (2, 2)
assert A.domain == ZZ
K = QQ.algebraic_field(sqrt(2))
ddm = DDM(
[[K.convert(1 + sqrt(2)), K.convert(2 + sqrt(2))],
[K.convert(3 + sqrt(2)), K.convert(4 + sqrt(2))]],
(2, 2),
K
)
A = DomainMatrix.from_list_sympy(
2, 2, [[1 + sqrt(2), 2 + sqrt(2)], [3 + sqrt(2), 4 + sqrt(2)]],
extension=True)
assert A.rep == ddm
assert A.shape == (2, 2)
assert A.domain == K
def test_DomainMatrix_from_Matrix():
ddm = DDM([[ZZ(1), ZZ(2)], [ZZ(3), ZZ(4)]], (2, 2), ZZ)
A = DomainMatrix.from_Matrix(Matrix([[1, 2], [3, 4]]))
assert A.rep == ddm
assert A.shape == (2, 2)
assert A.domain == ZZ
K = QQ.algebraic_field(sqrt(2))
ddm = DDM(
[[K.convert(1 + sqrt(2)), K.convert(2 + sqrt(2))],
[K.convert(3 + sqrt(2)), K.convert(4 + sqrt(2))]],
(2, 2),
K
)
A = DomainMatrix.from_Matrix(
Matrix([[1 + sqrt(2), 2 + sqrt(2)], [3 + sqrt(2), 4 + sqrt(2)]]),
extension=True)
assert A.rep == ddm
assert A.shape == (2, 2)
assert A.domain == K
def test_DomainMatrix_eq():
A = DomainMatrix([[ZZ(1), ZZ(2)], [ZZ(3), ZZ(4)]], (2, 2), ZZ)
assert A == A
B = DomainMatrix([[ZZ(1), ZZ(2)], [ZZ(3), ZZ(1)]], (2, 2), ZZ)
assert A != B
C = [[ZZ(1), ZZ(2)], [ZZ(3), ZZ(4)]]
assert A != C
def test_DomainMatrix_get_domain():
K, items = DomainMatrix.get_domain([1, 2, 3, 4])
assert items == [ZZ(1), ZZ(2), ZZ(3), ZZ(4)]
assert K == ZZ
K, items = DomainMatrix.get_domain([1, 2, 3, Rational(1, 2)])
assert items == [QQ(1), QQ(2), QQ(3), QQ(1, 2)]
assert K == QQ
def test_DomainMatrix_convert_to():
A = DomainMatrix([[ZZ(1), ZZ(2)], [ZZ(3), ZZ(4)]], (2, 2), ZZ)
Aq = A.convert_to(QQ)
assert Aq == DomainMatrix([[QQ(1), QQ(2)], [QQ(3), QQ(4)]], (2, 2), QQ)
def test_DomainMatrix_to_field():
A = DomainMatrix([[ZZ(1), ZZ(2)], [ZZ(3), ZZ(4)]], (2, 2), ZZ)
Aq = A.to_field()
assert Aq == DomainMatrix([[QQ(1), QQ(2)], [QQ(3), QQ(4)]], (2, 2), QQ)
def test_DomainMatrix_to_sparse():
A = DomainMatrix([[ZZ(1), ZZ(2)], [ZZ(3), ZZ(4)]], (2, 2), ZZ)
A_sparse = A.to_sparse()
assert A_sparse.rep == {0: {0: 1, 1: 2}, 1: {0: 3, 1: 4}}
def test_DomainMatrix_to_dense():
A = DomainMatrix({0: {0: 1, 1: 2}, 1: {0: 3, 1: 4}}, (2, 2), ZZ)
A_dense = A.to_dense()
assert A_dense.rep == DDM([[1, 2], [3, 4]], (2, 2), ZZ)
def test_DomainMatrix_unify():
Az = DomainMatrix([[ZZ(1), ZZ(2)], [ZZ(3), ZZ(4)]], (2, 2), ZZ)
Aq = DomainMatrix([[QQ(1), QQ(2)], [QQ(3), QQ(4)]], (2, 2), QQ)
assert Az.unify(Az) == (Az, Az)
assert Az.unify(Aq) == (Aq, Aq)
assert Aq.unify(Az) == (Aq, Aq)
assert Aq.unify(Aq) == (Aq, Aq)
As = DomainMatrix({0: {1: ZZ(1)}, 1:{0:ZZ(2)}}, (2, 2), ZZ)
Ad = DomainMatrix([[ZZ(1), ZZ(2)], [ZZ(3), ZZ(4)]], (2, 2), ZZ)
assert As.unify(As) == (As, As)
assert Ad.unify(Ad) == (Ad, Ad)
Bs, Bd = As.unify(Ad, fmt='dense')
assert Bs.rep == DDM([[0, 1], [2, 0]], (2, 2), ZZ)
assert Bd.rep == DDM([[1, 2],[3, 4]], (2, 2), ZZ)
Bs, Bd = As.unify(Ad, fmt='sparse')
assert Bs.rep == SDM({0: {1: 1}, 1: {0: 2}}, (2, 2), ZZ)
assert Bd.rep == SDM({0: {0: 1, 1: 2}, 1: {0: 3, 1: 4}}, (2, 2), ZZ)
raises(ValueError, lambda: As.unify(Ad, fmt='invalid'))
def test_DomainMatrix_to_Matrix():
A = DomainMatrix([[ZZ(1), ZZ(2)], [ZZ(3), ZZ(4)]], (2, 2), ZZ)
assert A.to_Matrix() == Matrix([[1, 2], [3, 4]])
def test_DomainMatrix_repr():
A = DomainMatrix([[ZZ(1), ZZ(2)], [ZZ(3), ZZ(4)]], (2, 2), ZZ)
assert repr(A) == 'DomainMatrix([[1, 2], [3, 4]], (2, 2), ZZ)'
def test_DomainMatrix_add():
A = DomainMatrix([[ZZ(1), ZZ(2)], [ZZ(3), ZZ(4)]], (2, 2), ZZ)
B = DomainMatrix([[ZZ(2), ZZ(4)], [ZZ(6), ZZ(8)]], (2, 2), ZZ)
assert A + A == A.add(A) == B
A = DomainMatrix([[ZZ(1), ZZ(2)], [ZZ(3), ZZ(4)]], (2, 2), ZZ)
L = [[2, 3], [3, 4]]
raises(TypeError, lambda: A + L)
raises(TypeError, lambda: L + A)
A1 = DomainMatrix([[ZZ(1), ZZ(2)], [ZZ(3), ZZ(4)]], (2, 2), ZZ)
A2 = DomainMatrix([[ZZ(1), ZZ(2)]], (1, 2), ZZ)
raises(DDMShapeError, lambda: A1 + A2)
raises(DDMShapeError, lambda: A2 + A1)
raises(DDMShapeError, lambda: A1.add(A2))
raises(DDMShapeError, lambda: A2.add(A1))
Az = DomainMatrix([[ZZ(1), ZZ(2)], [ZZ(3), ZZ(4)]], (2, 2), ZZ)
Aq = DomainMatrix([[QQ(1), QQ(2)], [QQ(3), QQ(4)]], (2, 2), QQ)
Asum = DomainMatrix([[QQ(2), QQ(4)], [QQ(6), QQ(8)]], (2, 2), QQ)
assert Az + Aq == Asum
assert Aq + Az == Asum
raises(DDMDomainError, lambda: Az.add(Aq))
raises(DDMDomainError, lambda: Aq.add(Az))
As = DomainMatrix({0: {1: ZZ(1)}, 1: {0: ZZ(2)}}, (2, 2), ZZ)
Ad = DomainMatrix([[ZZ(1), ZZ(2)], [ZZ(3), ZZ(4)]], (2, 2), ZZ)
Asd = As + Ad
Ads = Ad + As
assert Asd == DomainMatrix([[1, 3], [5, 4]], (2, 2), ZZ)
assert Asd.rep == DDM([[1, 3], [5, 4]], (2, 2), ZZ)
assert Ads == DomainMatrix([[1, 3], [5, 4]], (2, 2), ZZ)
assert Ads.rep == DDM([[1, 3], [5, 4]], (2, 2), ZZ)
raises(DDMFormatError, lambda: As.add(Ad))
def test_DomainMatrix_sub():
A = DomainMatrix([[ZZ(1), ZZ(2)], [ZZ(3), ZZ(4)]], (2, 2), ZZ)
B = DomainMatrix([[ZZ(0), ZZ(0)], [ZZ(0), ZZ(0)]], (2, 2), ZZ)
assert A - A == A.sub(A) == B
A = DomainMatrix([[ZZ(1), ZZ(2)], [ZZ(3), ZZ(4)]], (2, 2), ZZ)
L = [[2, 3], [3, 4]]
raises(TypeError, lambda: A - L)
raises(TypeError, lambda: L - A)
A1 = DomainMatrix([[ZZ(1), ZZ(2)], [ZZ(3), ZZ(4)]], (2, 2), ZZ)
A2 = DomainMatrix([[ZZ(1), ZZ(2)]], (1, 2), ZZ)
raises(DDMShapeError, lambda: A1 - A2)
raises(DDMShapeError, lambda: A2 - A1)
raises(DDMShapeError, lambda: A1.sub(A2))
raises(DDMShapeError, lambda: A2.sub(A1))
Az = DomainMatrix([[ZZ(1), ZZ(2)], [ZZ(3), ZZ(4)]], (2, 2), ZZ)
Aq = DomainMatrix([[QQ(1), QQ(2)], [QQ(3), QQ(4)]], (2, 2), QQ)
Adiff = DomainMatrix([[QQ(0), QQ(0)], [QQ(0), QQ(0)]], (2, 2), QQ)
assert Az - Aq == Adiff
assert Aq - Az == Adiff
raises(DDMDomainError, lambda: Az.sub(Aq))
raises(DDMDomainError, lambda: Aq.sub(Az))
As = DomainMatrix({0: {1: ZZ(1)}, 1: {0: ZZ(2)}}, (2, 2), ZZ)
Ad = DomainMatrix([[ZZ(1), ZZ(2)], [ZZ(3), ZZ(4)]], (2, 2), ZZ)
Asd = As - Ad
Ads = Ad - As
assert Asd == DomainMatrix([[-1, -1], [-1, -4]], (2, 2), ZZ)
assert Asd.rep == DDM([[-1, -1], [-1, -4]], (2, 2), ZZ)
assert Asd == -Ads
assert Asd.rep == -Ads.rep
def test_DomainMatrix_neg():
A = DomainMatrix([[ZZ(1), ZZ(2)], [ZZ(3), ZZ(4)]], (2, 2), ZZ)
Aneg = DomainMatrix([[ZZ(-1), ZZ(-2)], [ZZ(-3), ZZ(-4)]], (2, 2), ZZ)
assert -A == A.neg() == Aneg
def test_DomainMatrix_mul():
A = DomainMatrix([[ZZ(1), ZZ(2)], [ZZ(3), ZZ(4)]], (2, 2), ZZ)
A2 = DomainMatrix([[ZZ(7), ZZ(10)], [ZZ(15), ZZ(22)]], (2, 2), ZZ)
assert A*A == A.matmul(A) == A2
A = DomainMatrix([[ZZ(1), ZZ(2)], [ZZ(3), ZZ(4)]], (2, 2), ZZ)
L = [[1, 2], [3, 4]]
raises(TypeError, lambda: A * L)
raises(TypeError, lambda: L * A)
Az = DomainMatrix([[ZZ(1), ZZ(2)], [ZZ(3), ZZ(4)]], (2, 2), ZZ)
Aq = DomainMatrix([[QQ(1), QQ(2)], [QQ(3), QQ(4)]], (2, 2), QQ)
Aprod = DomainMatrix([[QQ(7), QQ(10)], [QQ(15), QQ(22)]], (2, 2), QQ)
assert Az * Aq == Aprod
assert Aq * Az == Aprod
raises(DDMDomainError, lambda: Az.matmul(Aq))
raises(DDMDomainError, lambda: Aq.matmul(Az))
A = DomainMatrix([[ZZ(1), ZZ(2)], [ZZ(3), ZZ(4)]], (2, 2), ZZ)
AA = DomainMatrix([[ZZ(2), ZZ(4)], [ZZ(6), ZZ(8)]], (2, 2), ZZ)
x = ZZ(2)
assert A * x == x * A == A.mul(x) == AA
A = DomainMatrix([[ZZ(1), ZZ(2)], [ZZ(3), ZZ(4)]], (2, 2), ZZ)
AA = DomainMatrix([[ZZ(0), ZZ(0)], [ZZ(0), ZZ(0)]], (2, 2), ZZ)
x = ZZ(0)
assert A * x == x * A == A.mul(x) == AA
As = DomainMatrix({0: {1: ZZ(1)}, 1: {0: ZZ(2)}}, (2, 2), ZZ)
Ad = DomainMatrix([[ZZ(1), ZZ(2)], [ZZ(3), ZZ(4)]], (2, 2), ZZ)
Asd = As * Ad
Ads = Ad * As
assert Asd == DomainMatrix([[3, 4], [2, 4]], (2, 2), ZZ)
assert Asd.rep == DDM([[3, 4], [2, 4]], (2, 2), ZZ)
assert Ads == DomainMatrix([[4, 1], [8, 3]], (2, 2), ZZ)
assert Ads.rep == DDM([[4, 1], [8, 3]], (2, 2), ZZ)
def test_DomainMatrix_pow():
eye = DomainMatrix.eye(2, ZZ)
A = DomainMatrix([[ZZ(1), ZZ(2)], [ZZ(3), ZZ(4)]], (2, 2), ZZ)
A2 = DomainMatrix([[ZZ(7), ZZ(10)], [ZZ(15), ZZ(22)]], (2, 2), ZZ)
A3 = DomainMatrix([[ZZ(37), ZZ(54)], [ZZ(81), ZZ(118)]], (2, 2), ZZ)
assert A**0 == A.pow(0) == eye
assert A**1 == A.pow(1) == A
assert A**2 == A.pow(2) == A2
assert A**3 == A.pow(3) == A3
raises(TypeError, lambda: A ** Rational(1, 2))
raises(NotImplementedError, lambda: A ** -1)
raises(NotImplementedError, lambda: A.pow(-1))
A = DomainMatrix.zeros((2, 1), ZZ)
raises(NonSquareMatrixError, lambda: A ** 1)
def test_DomainMatrix_rref():
A = DomainMatrix([], (0, 1), QQ)
assert A.rref() == (A, ())
A = DomainMatrix([[QQ(1)]], (1, 1), QQ)
assert A.rref() == (A, (0,))
A = DomainMatrix([[QQ(0)]], (1, 1), QQ)
assert A.rref() == (A, ())
A = DomainMatrix([[QQ(1), QQ(2)], [QQ(3), QQ(4)]], (2, 2), QQ)
Ar, pivots = A.rref()
assert Ar == DomainMatrix([[QQ(1), QQ(0)], [QQ(0), QQ(1)]], (2, 2), QQ)
assert pivots == (0, 1)
A = DomainMatrix([[QQ(0), QQ(2)], [QQ(3), QQ(4)]], (2, 2), QQ)
Ar, pivots = A.rref()
assert Ar == DomainMatrix([[QQ(1), QQ(0)], [QQ(0), QQ(1)]], (2, 2), QQ)
assert pivots == (0, 1)
A = DomainMatrix([[QQ(0), QQ(2)], [QQ(0), QQ(4)]], (2, 2), QQ)
Ar, pivots = A.rref()
assert Ar == DomainMatrix([[QQ(0), QQ(1)], [QQ(0), QQ(0)]], (2, 2), QQ)
assert pivots == (1,)
Az = DomainMatrix([[ZZ(1), ZZ(2)], [ZZ(3), ZZ(4)]], (2, 2), ZZ)
raises(ValueError, lambda: Az.rref())
def test_DomainMatrix_nullspace():
A = DomainMatrix([[QQ(1), QQ(1)], [QQ(1), QQ(1)]], (2, 2), ZZ)
Anull = DomainMatrix([[QQ(-1), QQ(1)]], (1, 2), ZZ)
assert A.nullspace() == Anull
def test_DomainMatrix_inv():
A = DomainMatrix([], (0, 0), QQ)
assert A.inv() == A
A = DomainMatrix([[QQ(1), QQ(2)], [QQ(3), QQ(4)]], (2, 2), QQ)
Ainv = DomainMatrix([[QQ(-2), QQ(1)], [QQ(3, 2), QQ(-1, 2)]], (2, 2), QQ)
assert A.inv() == Ainv
Az = DomainMatrix([[ZZ(1), ZZ(2)], [ZZ(3), ZZ(4)]], (2, 2), ZZ)
raises(ValueError, lambda: Az.inv())
Ans = DomainMatrix([[QQ(1), QQ(2)]], (1, 2), QQ)
raises(NonSquareMatrixError, lambda: Ans.inv())
Aninv = DomainMatrix([[QQ(1), QQ(2)], [QQ(3), QQ(6)]], (2, 2), QQ)
raises(NonInvertibleMatrixError, lambda: Aninv.inv())
def test_DomainMatrix_det():
A = DomainMatrix([], (0, 0), ZZ)
assert A.det() == 1
A = DomainMatrix([[1]], (1, 1), ZZ)
assert A.det() == 1
A = DomainMatrix([[ZZ(1), ZZ(2)], [ZZ(3), ZZ(4)]], (2, 2), ZZ)
assert A.det() == ZZ(-2)
A = DomainMatrix([[ZZ(1), ZZ(2), ZZ(3)], [ZZ(1), ZZ(2), ZZ(4)], [ZZ(1), ZZ(3), ZZ(5)]], (3, 3), ZZ)
assert A.det() == ZZ(-1)
A = DomainMatrix([[ZZ(1), ZZ(2), ZZ(3)], [ZZ(1), ZZ(2), ZZ(4)], [ZZ(1), ZZ(2), ZZ(5)]], (3, 3), ZZ)
assert A.det() == ZZ(0)
Ans = DomainMatrix([[QQ(1), QQ(2)]], (1, 2), QQ)
raises(NonSquareMatrixError, lambda: Ans.det())
A = DomainMatrix([[QQ(1), QQ(2)], [QQ(3), QQ(4)]], (2, 2), QQ)
assert A.det() == QQ(-2)
def test_DomainMatrix_lu():
A = DomainMatrix([], (0, 0), QQ)
assert A.lu() == (A, A, [])
A = DomainMatrix([[QQ(1), QQ(2)], [QQ(3), QQ(4)]], (2, 2), QQ)
L = DomainMatrix([[QQ(1), QQ(0)], [QQ(3), QQ(1)]], (2, 2), QQ)
U = DomainMatrix([[QQ(1), QQ(2)], [QQ(0), QQ(-2)]], (2, 2), QQ)
swaps = []
assert A.lu() == (L, U, swaps)
A = DomainMatrix([[QQ(0), QQ(2)], [QQ(3), QQ(4)]], (2, 2), QQ)
L = DomainMatrix([[QQ(1), QQ(0)], [QQ(0), QQ(1)]], (2, 2), QQ)
U = DomainMatrix([[QQ(3), QQ(4)], [QQ(0), QQ(2)]], (2, 2), QQ)
swaps = [(0, 1)]
assert A.lu() == (L, U, swaps)
A = DomainMatrix([[QQ(1), QQ(2)], [QQ(2), QQ(4)]], (2, 2), QQ)
L = DomainMatrix([[QQ(1), QQ(0)], [QQ(2), QQ(1)]], (2, 2), QQ)
U = DomainMatrix([[QQ(1), QQ(2)], [QQ(0), QQ(0)]], (2, 2), QQ)
swaps = []
assert A.lu() == (L, U, swaps)
A = DomainMatrix([[QQ(0), QQ(2)], [QQ(0), QQ(4)]], (2, 2), QQ)
L = DomainMatrix([[QQ(1), QQ(0)], [QQ(0), QQ(1)]], (2, 2), QQ)
U = DomainMatrix([[QQ(0), QQ(2)], [QQ(0), QQ(4)]], (2, 2), QQ)
swaps = []
assert A.lu() == (L, U, swaps)
A = DomainMatrix([[QQ(1), QQ(2), QQ(3)], [QQ(4), QQ(5), QQ(6)]], (2, 3), QQ)
L = DomainMatrix([[QQ(1), QQ(0)], [QQ(4), QQ(1)]], (2, 2), QQ)
U = DomainMatrix([[QQ(1), QQ(2), QQ(3)], [QQ(0), QQ(-3), QQ(-6)]], (2, 3), QQ)
swaps = []
assert A.lu() == (L, U, swaps)
A = DomainMatrix([[QQ(1), QQ(2)], [QQ(3), QQ(4)], [QQ(5), QQ(6)]], (3, 2), QQ)
L = DomainMatrix([
[QQ(1), QQ(0), QQ(0)],
[QQ(3), QQ(1), QQ(0)],
[QQ(5), QQ(2), QQ(1)]], (3, 3), QQ)
U = DomainMatrix([[QQ(1), QQ(2)], [QQ(0), QQ(-2)], [QQ(0), QQ(0)]], (3, 2), QQ)
swaps = []
assert A.lu() == (L, U, swaps)
A = [[1, 0, 0, 0], [0, 0, 0, 0], [0, 0, 1, 1], [0, 0, 1, 2]]
L = [[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 1, 1]]
U = [[1, 0, 0, 0], [0, 0, 0, 0], [0, 0, 1, 1], [0, 0, 0, 1]]
to_dom = lambda rows, dom: [[dom(e) for e in row] for row in rows]
A = DomainMatrix(to_dom(A, QQ), (4, 4), QQ)
L = DomainMatrix(to_dom(L, QQ), (4, 4), QQ)
U = DomainMatrix(to_dom(U, QQ), (4, 4), QQ)
assert A.lu() == (L, U, [])
A = DomainMatrix([[ZZ(1), ZZ(2)], [ZZ(3), ZZ(4)]], (2, 2), ZZ)
raises(ValueError, lambda: A.lu())
def test_DomainMatrix_lu_solve():
# Base case
A = b = x = DomainMatrix([], (0, 0), QQ)
assert A.lu_solve(b) == x
# Basic example
A = DomainMatrix([[QQ(1), QQ(2)], [QQ(3), QQ(4)]], (2, 2), QQ)
b = DomainMatrix([[QQ(1)], [QQ(2)]], (2, 1), QQ)
x = DomainMatrix([[QQ(0)], [QQ(1, 2)]], (2, 1), QQ)
assert A.lu_solve(b) == x
# Example with swaps
A = DomainMatrix([[QQ(0), QQ(2)], [QQ(3), QQ(4)]], (2, 2), QQ)
b = DomainMatrix([[QQ(1)], [QQ(2)]], (2, 1), QQ)
x = DomainMatrix([[QQ(0)], [QQ(1, 2)]], (2, 1), QQ)
assert A.lu_solve(b) == x
# Non-invertible
A = DomainMatrix([[QQ(1), QQ(2)], [QQ(2), QQ(4)]], (2, 2), QQ)
b = DomainMatrix([[QQ(1)], [QQ(2)]], (2, 1), QQ)
raises(NonInvertibleMatrixError, lambda: A.lu_solve(b))
# Overdetermined, consistent
A = DomainMatrix([[QQ(1), QQ(2)], [QQ(3), QQ(4)], [QQ(5), QQ(6)]], (3, 2), QQ)
b = DomainMatrix([[QQ(1)], [QQ(2)], [QQ(3)]], (3, 1), QQ)
x = DomainMatrix([[QQ(0)], [QQ(1, 2)]], (2, 1), QQ)
assert A.lu_solve(b) == x
# Overdetermined, inconsistent
A = DomainMatrix([[QQ(1), QQ(2)], [QQ(3), QQ(4)], [QQ(5), QQ(6)]], (3, 2), QQ)
b = DomainMatrix([[QQ(1)], [QQ(2)], [QQ(4)]], (3, 1), QQ)
raises(NonInvertibleMatrixError, lambda: A.lu_solve(b))
# Underdetermined
A = DomainMatrix([[QQ(1), QQ(2)]], (1, 2), QQ)
b = DomainMatrix([[QQ(1)]], (1, 1), QQ)
raises(NotImplementedError, lambda: A.lu_solve(b))
# Non-field
A = DomainMatrix([[ZZ(1), ZZ(2)], [ZZ(3), ZZ(4)]], (2, 2), ZZ)
b = DomainMatrix([[ZZ(1)], [ZZ(2)]], (2, 1), ZZ)
raises(ValueError, lambda: A.lu_solve(b))
# Shape mismatch
A = DomainMatrix([[QQ(1), QQ(2)], [QQ(3), QQ(4)]], (2, 2), QQ)
b = DomainMatrix([[QQ(1), QQ(2)]], (1, 2), QQ)
raises(ShapeError, lambda: A.lu_solve(b))
def test_DomainMatrix_charpoly():
A = DomainMatrix([], (0, 0), ZZ)
assert A.charpoly() == [ZZ(1)]
A = DomainMatrix([[1]], (1, 1), ZZ)
assert A.charpoly() == [ZZ(1), ZZ(-1)]
A = DomainMatrix([[ZZ(1), ZZ(2)], [ZZ(3), ZZ(4)]], (2, 2), ZZ)
assert A.charpoly() == [ZZ(1), ZZ(-5), ZZ(-2)]
A = DomainMatrix([[ZZ(1), ZZ(2), ZZ(3)], [ZZ(4), ZZ(5), ZZ(6)], [ZZ(7), ZZ(8), ZZ(9)]], (3, 3), ZZ)
assert A.charpoly() == [ZZ(1), ZZ(-15), ZZ(-18), ZZ(0)]
Ans = DomainMatrix([[QQ(1), QQ(2)]], (1, 2), QQ)
raises(NonSquareMatrixError, lambda: Ans.charpoly())
def test_DomainMatrix_eye():
A = DomainMatrix.eye(3, QQ)
assert A.rep == SDM.eye(3, QQ)
assert A.shape == (3, 3)
assert A.domain == QQ
def test_DomainMatrix_zeros():
A = DomainMatrix.zeros((1, 2), QQ)
assert A.rep == SDM.zeros((1, 2), QQ)
assert A.shape == (1, 2)
assert A.domain == QQ
def test_DomainMatrix_ones():
A = DomainMatrix.ones((2, 3), QQ)
assert A.rep == DDM.ones((2, 3), QQ)
assert A.shape == (2, 3)
assert A.domain == QQ
def test_DomainMatrix_hstack():
A = DomainMatrix([[ZZ(1)], [ZZ(2)]], (2, 1), ZZ)
B = DomainMatrix([[QQ(3), QQ(4)], [QQ(5), QQ(6)]], (2, 2), QQ)
AB = DomainMatrix([[QQ(1), QQ(3), QQ(4)], [QQ(2), QQ(5), QQ(6)]], (2, 3), QQ)
assert A.hstack(B) == AB
def test_DomainMatrix_scalarmul():
A = DomainMatrix([[ZZ(1), ZZ(2)], [ZZ(3), ZZ(4)]], (2, 2), ZZ)
lamda = DomainScalar(QQ(3)/QQ(2), QQ)
assert A * lamda == DomainMatrix([[QQ(3, 2), QQ(3)], [QQ(9, 2), QQ(6)]], (2, 2), QQ)
assert A * 2 == DomainMatrix([[ZZ(2), ZZ(4)], [ZZ(6), ZZ(8)]], (2, 2), ZZ)
assert A * DomainScalar(ZZ(0), ZZ) == DomainMatrix([[ZZ(0)]*2]*2, (2, 2), ZZ)
assert A * DomainScalar(ZZ(1), ZZ) == A
raises(TypeError, lambda: A * 1.5)
def test_DomainMatrix_truediv():
A = DomainMatrix.from_Matrix(Matrix([[1, 2], [3, 4]]))
lamda = DomainScalar(QQ(3)/QQ(2), QQ)
assert A / lamda == DomainMatrix([[QQ(2, 3), QQ(4, 3)], [QQ(2), QQ(8, 3)]], (2, 2), QQ)
b = DomainScalar(ZZ(1), ZZ)
assert A / b == DomainMatrix([[QQ(1), QQ(2)], [QQ(3), QQ(4)]], (2, 2), QQ)
assert A / 1 == DomainMatrix([[QQ(1), QQ(2)], [QQ(3), QQ(4)]], (2, 2), QQ)
assert A / 2 == DomainMatrix([[QQ(1, 2), QQ(1)], [QQ(3, 2), QQ(2)]], (2, 2), QQ)
raises(ZeroDivisionError, lambda: A / 0)
raises(TypeError, lambda: A / 1.5)
raises(ZeroDivisionError, lambda: A / DomainScalar(ZZ(0), ZZ))
|
ad62603a61c44c793c938020814c11eaf3387bcff91310c22a05430af1d93a07 | #
# test_linsolve.py
#
# Test the internal implementation of linsolve.
#
from sympy import S
from sympy.abc import x
from sympy.polys.matrices.linsolve import _linsolve
def test__linsolve():
assert _linsolve([], [x]) == {x:x}
assert _linsolve([S.Zero], [x]) == {x:x}
|
4af3be6557fae3a948f8cc5344c1b00d1c9722c9478a75adf4a27cbf47495e67 | """
Tests for the basic functionality of the SDM class.
"""
from sympy.core.compatibility import HAS_GMPY
from sympy.testing.pytest import raises
from sympy import QQ, ZZ
from sympy.polys.matrices.sdm import SDM
from sympy.polys.matrices.ddm import DDM
from sympy.polys.matrices.exceptions import (DDMBadInputError, DDMDomainError,
DDMShapeError)
def test_SDM():
A = SDM({0:{0:ZZ(1)}}, (2, 2), ZZ)
assert A.domain == ZZ
assert A.shape == (2, 2)
assert dict(A) == {0:{0:ZZ(1)}}
raises(DDMBadInputError, lambda: SDM({5:{1:ZZ(0)}}, (2, 2), ZZ))
raises(DDMBadInputError, lambda: SDM({0:{5:ZZ(0)}}, (2, 2), ZZ))
def test_DDM_str():
sdm = SDM({0:{0:ZZ(1)}, 1:{1:ZZ(1)}}, (2, 2), ZZ)
assert str(sdm) == '{0: {0: 1}, 1: {1: 1}}'
if HAS_GMPY: # pragma: no cover
assert repr(sdm) == 'SDM({0: {0: mpz(1)}, 1: {1: mpz(1)}}, (2, 2), ZZ)'
else: # pragma: no cover
assert repr(sdm) == 'SDM({0: {0: 1}, 1: {1: 1}}, (2, 2), ZZ)'
def test_SDM_new():
A = SDM({0:{0:ZZ(1)}}, (2, 2), ZZ)
B = A.new({}, (2, 2), ZZ)
assert B == SDM({}, (2, 2), ZZ)
def test_SDM_copy():
A = SDM({0:{0:ZZ(1)}}, (2, 2), ZZ)
B = A.copy()
assert A == B
A[0][0] = ZZ(2)
assert A != B
def test_SDM_from_list():
A = SDM.from_list([[ZZ(0), ZZ(1)], [ZZ(1), ZZ(0)]], (2, 2), ZZ)
assert A == SDM({0:{1:ZZ(1)}, 1:{0:ZZ(1)}}, (2, 2), ZZ)
raises(DDMBadInputError, lambda: SDM.from_list([[ZZ(0)], [ZZ(0), ZZ(1)]], (2, 2), ZZ))
raises(DDMBadInputError, lambda: SDM.from_list([[ZZ(0), ZZ(1)]], (2, 2), ZZ))
def test_SDM_to_list():
A = SDM({0:{1: ZZ(1)}}, (2, 2), ZZ)
assert A.to_list() == [[ZZ(0), ZZ(1)], [ZZ(0), ZZ(0)]]
A = SDM({}, (0, 2), ZZ)
assert A.to_list() == []
A = SDM({}, (2, 0), ZZ)
assert A.to_list() == [[], []]
def test_SDM_from_ddm():
A = DDM([[ZZ(1), ZZ(0)], [ZZ(1), ZZ(0)]], (2, 2), ZZ)
B = SDM.from_ddm(A)
assert B.domain == ZZ
assert B.shape == (2, 2)
assert dict(B) == {0:{0:ZZ(1)}, 1:{0:ZZ(1)}}
def test_SDM_to_ddm():
A = SDM({0:{1: ZZ(1)}}, (2, 2), ZZ)
B = DDM([[ZZ(0), ZZ(1)], [ZZ(0), ZZ(0)]], (2, 2), ZZ)
assert A.to_ddm() == B
def test_SDM_zeros():
A = SDM.zeros((2, 2), ZZ)
assert A.domain == ZZ
assert A.shape == (2, 2)
assert dict(A) == {}
def test_SDM_ones():
A = SDM.ones((1, 2), QQ)
assert A.domain == QQ
assert A.shape == (1, 2)
assert dict(A) == {0:{0:QQ(1), 1:QQ(1)}}
def test_SDM_eye():
A = SDM.eye(2, ZZ)
assert A.domain == ZZ
assert A.shape == (2, 2)
assert dict(A) == {0:{0:ZZ(1)}, 1:{1:ZZ(1)}}
def test_SDM_transpose():
A = SDM({0:{0:ZZ(1), 1:ZZ(2)}, 1:{0:ZZ(3), 1:ZZ(4)}}, (2, 2), ZZ)
B = SDM({0:{0:ZZ(1), 1:ZZ(3)}, 1:{0:ZZ(2), 1:ZZ(4)}}, (2, 2), ZZ)
assert A.transpose() == B
A = SDM({0:{1:ZZ(2)}}, (2, 2), ZZ)
B = SDM({1:{0:ZZ(2)}}, (2, 2), ZZ)
assert A.transpose() == B
A = SDM({0:{1:ZZ(2)}}, (1, 2), ZZ)
B = SDM({1:{0:ZZ(2)}}, (2, 1), ZZ)
assert A.transpose() == B
def test_SDM_mul():
A = SDM({0:{0:ZZ(2)}}, (2, 2), ZZ)
B = SDM({0:{0:ZZ(4)}}, (2, 2), ZZ)
assert A*ZZ(2) == B
assert ZZ(2)*A == B
raises(TypeError, lambda: A*QQ(1, 2))
raises(TypeError, lambda: QQ(1, 2)*A)
def test_SDM_matmul():
A = SDM({0:{0:ZZ(2)}}, (2, 2), ZZ)
B = SDM({0:{0:ZZ(4)}}, (2, 2), ZZ)
assert A.matmul(A) == B
C = SDM({0:{0:ZZ(2)}}, (2, 2), QQ)
raises(DDMDomainError, lambda: A.matmul(C))
A = SDM({0:{0:ZZ(1), 1:ZZ(2)}, 1:{0:ZZ(3), 1:ZZ(4)}}, (2, 2), ZZ)
B = SDM({0:{0:ZZ(7), 1:ZZ(10)}, 1:{0:ZZ(15), 1:ZZ(22)}}, (2, 2), ZZ)
assert A.matmul(A) == B
A22 = SDM({0:{0:ZZ(4)}}, (2, 2), ZZ)
A32 = SDM({0:{0:ZZ(2)}}, (3, 2), ZZ)
A23 = SDM({0:{0:ZZ(4)}}, (2, 3), ZZ)
A33 = SDM({0:{0:ZZ(8)}}, (3, 3), ZZ)
A22 = SDM({0:{0:ZZ(8)}}, (2, 2), ZZ)
assert A32.matmul(A23) == A33
assert A23.matmul(A32) == A22
# XXX: @ not supported by SDM...
#assert A32.matmul(A23) == A32 @ A23 == A33
#assert A23.matmul(A32) == A23 @ A32 == A22
#raises(DDMShapeError, lambda: A23 @ A22)
raises(DDMShapeError, lambda: A23.matmul(A22))
def test_SDM_add():
A = SDM({0:{1:ZZ(1)}, 1:{0:ZZ(2), 1:ZZ(3)}}, (2, 2), ZZ)
B = SDM({0:{0:ZZ(1)}, 1:{0:ZZ(-2), 1:ZZ(3)}}, (2, 2), ZZ)
C = SDM({0:{0:ZZ(1), 1:ZZ(1)}, 1:{1:ZZ(6)}}, (2, 2), ZZ)
assert A.add(B) == C
A = SDM({0:{1:ZZ(1)}}, (2, 2), ZZ)
B = SDM({0:{0:ZZ(1)}, 1:{0:ZZ(-2), 1:ZZ(3)}}, (2, 2), ZZ)
C = SDM({0:{0:ZZ(1), 1:ZZ(1)}, 1:{0:ZZ(-2), 1:ZZ(3)}}, (2, 2), ZZ)
assert A.add(B) == C
assert B.add(A) == C
def test_SDM_sub():
A = SDM({0:{1:ZZ(1)}, 1:{0:ZZ(2), 1:ZZ(3)}}, (2, 2), ZZ)
B = SDM({0:{0:ZZ(1)}, 1:{0:ZZ(-2), 1:ZZ(3)}}, (2, 2), ZZ)
C = SDM({0:{0:ZZ(-1), 1:ZZ(1)}, 1:{0:ZZ(4)}}, (2, 2), ZZ)
assert A.sub(B) == C
def test_SDM_neg():
A = SDM({0:{1:ZZ(1)}, 1:{0:ZZ(2), 1:ZZ(3)}}, (2, 2), ZZ)
B = SDM({0:{1:ZZ(-1)}, 1:{0:ZZ(-2), 1:ZZ(-3)}}, (2, 2), ZZ)
assert A.neg() == B
def test_SDM_convert_to():
A = SDM({0:{1:ZZ(1)}, 1:{0:ZZ(2), 1:ZZ(3)}}, (2, 2), ZZ)
B = SDM({0:{1:QQ(1)}, 1:{0:QQ(2), 1:QQ(3)}}, (2, 2), QQ)
C = A.convert_to(QQ)
assert C == B
assert C.domain == QQ
D = A.convert_to(ZZ)
assert D == A
assert D.domain == ZZ
def test_SDM_hstack():
A = SDM({0:{1:ZZ(1)}}, (2, 2), ZZ)
B = SDM({1:{1:ZZ(1)}}, (2, 2), ZZ)
AA = SDM({0:{1:ZZ(1), 3:ZZ(1)}}, (2, 4), ZZ)
AB = SDM({0:{1:ZZ(1)}, 1:{3:ZZ(1)}}, (2, 4), ZZ)
assert SDM.hstack(A) == A
assert SDM.hstack(A, A) == AA
assert SDM.hstack(A, B) == AB
def test_SDM_inv():
A = SDM({0:{0:QQ(1), 1:QQ(2)}, 1:{0:QQ(3), 1:QQ(4)}}, (2, 2), QQ)
B = SDM({0:{0:QQ(-2), 1:QQ(1)}, 1:{0:QQ(3, 2), 1:QQ(-1, 2)}}, (2, 2), QQ)
assert A.inv() == B
def test_SDM_det():
A = SDM({0:{0:QQ(1), 1:QQ(2)}, 1:{0:QQ(3), 1:QQ(4)}}, (2, 2), QQ)
assert A.det() == QQ(-2)
def test_SDM_lu():
A = SDM({0:{0:QQ(1), 1:QQ(2)}, 1:{0:QQ(3), 1:QQ(4)}}, (2, 2), QQ)
L = SDM({0:{0:QQ(1)}, 1:{0:QQ(3), 1:QQ(1)}}, (2, 2), QQ)
#U = SDM({0:{0:QQ(1), 1:QQ(2)}, 1:{0:QQ(3), 1:QQ(-2)}}, (2, 2), QQ)
#swaps = []
# This doesn't quite work. U has some nonzero elements in the lower part.
#assert A.lu() == (L, U, swaps)
assert A.lu()[0] == L
def test_SDM_lu_solve():
A = SDM({0:{0:QQ(1), 1:QQ(2)}, 1:{0:QQ(3), 1:QQ(4)}}, (2, 2), QQ)
b = SDM({0:{0:QQ(1)}, 1:{0:QQ(2)}}, (2, 1), QQ)
x = SDM({1:{0:QQ(1, 2)}}, (2, 1), QQ)
assert A.matmul(x) == b
assert A.lu_solve(b) == x
def test_SDM_charpoly():
A = SDM({0:{0:ZZ(1), 1:ZZ(2)}, 1:{0:ZZ(3), 1:ZZ(4)}}, (2, 2), ZZ)
assert A.charpoly() == [ZZ(1), ZZ(-5), ZZ(-2)]
def test_SDM_nullspace():
A = SDM({0:{0:QQ(1), 1:QQ(1)}}, (2, 2), QQ)
assert A.nullspace()[0] == SDM({0:{0:QQ(-1), 1:QQ(1)}}, (1, 2), QQ)
def test_SDM_rref():
eye2 = SDM({0:{0:QQ(1)}, 1:{1:QQ(1)}}, (2, 2), QQ)
A = SDM({0:{0:QQ(1), 1:QQ(2)}, 1:{0:QQ(3), 1:QQ(4)}}, (2, 2), QQ)
assert A.rref() == (eye2, [0, 1])
A = SDM({0:{0:QQ(1)}, 1:{0:QQ(3), 1:QQ(4)}}, (2, 2), QQ)
assert A.rref() == (eye2, [0, 1])
A = SDM({0:{1:QQ(2)}, 1:{0:QQ(3), 1:QQ(4)}}, (2, 2), QQ)
assert A.rref() == (eye2, [0, 1])
A = SDM({0:{0:QQ(1), 1:QQ(2), 2:QQ(3)},
1:{0:QQ(4), 1:QQ(5), 2:QQ(6)},
2:{0:QQ(7), 1:QQ(8), 2:QQ(9)} }, (3, 3), QQ)
Arref = SDM({0:{0:QQ(1), 2:QQ(-1)}, 1:{1:QQ(1), 2:QQ(2)}}, (3, 3), QQ)
assert A.rref() == (Arref, [0, 1])
A = SDM({0:{0:QQ(1), 1:QQ(2), 3:QQ(1)},
1:{0:QQ(1), 1:QQ(1), 2:QQ(9)}}, (2, 4), QQ)
Arref = SDM({0:{0:QQ(1), 2:QQ(18), 3:QQ(-1)},
1:{1:QQ(1), 2:QQ(-9), 3:QQ(1)}}, (2, 4), QQ)
assert A.rref() == (Arref, [0, 1])
A = SDM({0:{0:QQ(1), 1:QQ(1), 2:QQ(1)},
1:{0:QQ(1), 1:QQ(2), 2:QQ(2)}}, (2, 3), QQ)
Arref = SDM(
{0: {0: QQ(1,1)}, 1: {1: QQ(1,1), 2: QQ(1,1)}},
(2, 3), QQ)
assert A.rref() == (Arref, [0, 1])
|
df2432ae4f417eea0d22ded314695e2ee54ba0be65bc433e302ccc456cf8b054 | from sympy import Add, Basic, symbols, Symbol, And, S
from sympy.core.symbol import Str
from sympy.unify.core import Compound, Variable
from sympy.unify.usympy import (deconstruct, construct, unify, is_associative,
is_commutative)
from sympy.abc import x, y, z, n
def test_deconstruct():
expr = Basic(1, 2, 3)
expected = Compound(Basic, (1, 2, 3))
assert deconstruct(expr) == expected
assert deconstruct(1) == 1
assert deconstruct(x) == x
assert deconstruct(x, variables=(x,)) == Variable(x)
assert deconstruct(Add(1, x, evaluate=False)) == Compound(Add, (1, x))
assert deconstruct(Add(1, x, evaluate=False), variables=(x,)) == \
Compound(Add, (1, Variable(x)))
def test_construct():
expr = Compound(Basic, (1, 2, 3))
expected = Basic(1, 2, 3)
assert construct(expr) == expected
def test_nested():
expr = Basic(1, Basic(2), 3)
cmpd = Compound(Basic, (1, Compound(Basic, (2,)), 3))
assert deconstruct(expr) == cmpd
assert construct(cmpd) == expr
def test_unify():
expr = Basic(1, 2, 3)
a, b, c = map(Symbol, 'abc')
pattern = Basic(a, b, c)
assert list(unify(expr, pattern, {}, (a, b, c))) == [{a: 1, b: 2, c: 3}]
assert list(unify(expr, pattern, variables=(a, b, c))) == \
[{a: 1, b: 2, c: 3}]
def test_unify_variables():
assert list(unify(Basic(1, 2), Basic(1, x), {}, variables=(x,))) == [{x: 2}]
def test_s_input():
expr = Basic(1, 2)
a, b = map(Symbol, 'ab')
pattern = Basic(a, b)
assert list(unify(expr, pattern, {}, (a, b))) == [{a: 1, b: 2}]
assert list(unify(expr, pattern, {a: 5}, (a, b))) == []
def iterdicteq(a, b):
a = tuple(a)
b = tuple(b)
return len(a) == len(b) and all(x in b for x in a)
def test_unify_commutative():
expr = Add(1, 2, 3, evaluate=False)
a, b, c = map(Symbol, 'abc')
pattern = Add(a, b, c, evaluate=False)
result = tuple(unify(expr, pattern, {}, (a, b, c)))
expected = ({a: 1, b: 2, c: 3},
{a: 1, b: 3, c: 2},
{a: 2, b: 1, c: 3},
{a: 2, b: 3, c: 1},
{a: 3, b: 1, c: 2},
{a: 3, b: 2, c: 1})
assert iterdicteq(result, expected)
def test_unify_iter():
expr = Add(1, 2, 3, evaluate=False)
a, b, c = map(Symbol, 'abc')
pattern = Add(a, c, evaluate=False)
assert is_associative(deconstruct(pattern))
assert is_commutative(deconstruct(pattern))
result = list(unify(expr, pattern, {}, (a, c)))
expected = [{a: 1, c: Add(2, 3, evaluate=False)},
{a: 1, c: Add(3, 2, evaluate=False)},
{a: 2, c: Add(1, 3, evaluate=False)},
{a: 2, c: Add(3, 1, evaluate=False)},
{a: 3, c: Add(1, 2, evaluate=False)},
{a: 3, c: Add(2, 1, evaluate=False)},
{a: Add(1, 2, evaluate=False), c: 3},
{a: Add(2, 1, evaluate=False), c: 3},
{a: Add(1, 3, evaluate=False), c: 2},
{a: Add(3, 1, evaluate=False), c: 2},
{a: Add(2, 3, evaluate=False), c: 1},
{a: Add(3, 2, evaluate=False), c: 1}]
assert iterdicteq(result, expected)
def test_hard_match():
from sympy import sin, cos
expr = sin(x) + cos(x)**2
p, q = map(Symbol, 'pq')
pattern = sin(p) + cos(p)**2
assert list(unify(expr, pattern, {}, (p, q))) == [{p: x}]
def test_matrix():
from sympy import MatrixSymbol
X = MatrixSymbol('X', n, n)
Y = MatrixSymbol('Y', 2, 2)
Z = MatrixSymbol('Z', 2, 3)
assert list(unify(X, Y, {}, variables=[n, Str('X')])) == [{Str('X'): Str('Y'), n: 2}]
assert list(unify(X, Z, {}, variables=[n, Str('X')])) == []
def test_non_frankenAdds():
# the is_commutative property used to fail because of Basic.__new__
# This caused is_commutative and str calls to fail
expr = x+y*2
rebuilt = construct(deconstruct(expr))
# Ensure that we can run these commands without causing an error
str(rebuilt)
rebuilt.is_commutative
def test_FiniteSet_commutivity():
from sympy import FiniteSet
a, b, c, x, y = symbols('a,b,c,x,y')
s = FiniteSet(a, b, c)
t = FiniteSet(x, y)
variables = (x, y)
assert {x: FiniteSet(a, c), y: b} in tuple(unify(s, t, variables=variables))
def test_FiniteSet_complex():
from sympy import FiniteSet
a, b, c, x, y, z = symbols('a,b,c,x,y,z')
expr = FiniteSet(Basic(S(1), x), y, Basic(x, z))
pattern = FiniteSet(a, Basic(x, b))
variables = a, b
expected = tuple([{b: 1, a: FiniteSet(y, Basic(x, z))},
{b: z, a: FiniteSet(y, Basic(S(1), x))}])
assert iterdicteq(unify(expr, pattern, variables=variables), expected)
def test_and():
variables = x, y
expected = tuple([{x: z > 0, y: n < 3}])
assert iterdicteq(unify((z>0) & (n<3), And(x, y), variables=variables),
expected)
def test_Union():
from sympy import Interval
assert list(unify(Interval(0, 1) + Interval(10, 11),
Interval(0, 1) + Interval(12, 13),
variables=(Interval(12, 13),)))
def test_is_commutative():
assert is_commutative(deconstruct(x+y))
assert is_commutative(deconstruct(x*y))
assert not is_commutative(deconstruct(x**y))
def test_commutative_in_commutative():
from sympy.abc import a,b,c,d
from sympy import sin, cos
eq = sin(3)*sin(4)*sin(5) + 4*cos(3)*cos(4)
pat = a*cos(b)*cos(c) + d*sin(b)*sin(c)
assert next(unify(eq, pat, variables=(a,b,c,d)))
|
180d43ec78ee00c3137508dfcdf7dd3a853809e4455d5d0881d942bb89939901 | from sympy import sin, cos, pi, S, sqrt
from sympy.testing.pytest import raises
from sympy.vector.coordsysrect import CoordSys3D
from sympy.vector.integrals import ParametricIntegral, vector_integrate
from sympy.vector.parametricregion import ParametricRegion
from sympy.vector.implicitregion import ImplicitRegion
from sympy.abc import x, y, z, u, v, r, t, theta, phi
from sympy.geometry import Point, Segment, Curve, Circle, Polygon, Plane
C = CoordSys3D('C')
def test_parametric_lineintegrals():
halfcircle = ParametricRegion((4*cos(theta), 4*sin(theta)), (theta, -pi/2, pi/2))
assert ParametricIntegral(C.x*C.y**4, halfcircle) == S(8192)/5
curve = ParametricRegion((t, t**2, t**3), (t, 0, 1))
field1 = 8*C.x**2*C.y*C.z*C.i + 5*C.z*C.j - 4*C.x*C.y*C.k
assert ParametricIntegral(field1, curve) == 1
line = ParametricRegion((4*t - 1, 2 - 2*t, t), (t, 0, 1))
assert ParametricIntegral(C.x*C.z*C.i - C.y*C.z*C.k, line) == 3
assert ParametricIntegral(4*C.x**3, ParametricRegion((1, t), (t, 0, 2))) == 8
helix = ParametricRegion((cos(t), sin(t), 3*t), (t, 0, 4*pi))
assert ParametricIntegral(C.x*C.y*C.z, helix) == -3*sqrt(10)*pi
field2 = C.y*C.i + C.z*C.j + C.z*C.k
assert ParametricIntegral(field2, ParametricRegion((cos(t), sin(t), t**2), (t, 0, pi))) == -5*pi/2 + pi**4/2
def test_parametric_surfaceintegrals():
semisphere = ParametricRegion((2*sin(phi)*cos(theta), 2*sin(phi)*sin(theta), 2*cos(phi)),\
(theta, 0, 2*pi), (phi, 0, pi/2))
assert ParametricIntegral(C.z, semisphere) == 8*pi
cylinder = ParametricRegion((sqrt(3)*cos(theta), sqrt(3)*sin(theta), z), (z, 0, 6), (theta, 0, 2*pi))
assert ParametricIntegral(C.y, cylinder) == 0
cone = ParametricRegion((v*cos(u), v*sin(u), v), (u, 0, 2*pi), (v, 0, 1))
assert ParametricIntegral(C.x*C.i + C.y*C.j + C.z**4*C.k, cone) == pi/3
triangle1 = ParametricRegion((x, y), (x, 0, 2), (y, 0, 10 - 5*x))
triangle2 = ParametricRegion((x, y), (y, 0, 10 - 5*x), (x, 0, 2))
assert ParametricIntegral(-15.6*C.y*C.k, triangle1) == ParametricIntegral(-15.6*C.y*C.k, triangle2)
assert ParametricIntegral(C.z, triangle1) == 10*C.z
def test_parametric_volumeintegrals():
cube = ParametricRegion((x, y, z), (x, 0, 1), (y, 0, 1), (z, 0, 1))
assert ParametricIntegral(1, cube) == 1
solidsphere1 = ParametricRegion((r*sin(phi)*cos(theta), r*sin(phi)*sin(theta), r*cos(phi)),\
(r, 0, 2), (theta, 0, 2*pi), (phi, 0, pi))
solidsphere2 = ParametricRegion((r*sin(phi)*cos(theta), r*sin(phi)*sin(theta), r*cos(phi)),\
(r, 0, 2), (phi, 0, pi), (theta, 0, 2*pi))
assert ParametricIntegral(C.x**2 + C.y**2, solidsphere1) == -256*pi/15
assert ParametricIntegral(C.x**2 + C.y**2, solidsphere2) == 256*pi/15
region_under_plane1 = ParametricRegion((x, y, z), (x, 0, 3), (y, 0, -2*x/3 + 2),\
(z, 0, 6 - 2*x - 3*y))
region_under_plane2 = ParametricRegion((x, y, z), (x, 0, 3), (z, 0, 6 - 2*x - 3*y),\
(y, 0, -2*x/3 + 2))
assert ParametricIntegral(C.x*C.i + C.j - 100*C.k, region_under_plane1) == \
ParametricIntegral(C.x*C.i + C.j - 100*C.k, region_under_plane2)
assert ParametricIntegral(2*C.x, region_under_plane2) == -9
def test_vector_integrate():
halfdisc = ParametricRegion((r*cos(theta), r* sin(theta)), (r, -2, 2), (theta, 0, pi))
assert vector_integrate(C.x**2, halfdisc) == 4*pi
vector_integrate(C.x, ParametricRegion((t, t**2), (t, 2, 3))) == -17*sqrt(17)/12 + 37*sqrt(37)/12
assert vector_integrate(C.y**3*C.z, (C.x, 0, 3), (C.y, -1, 4)) == 765*C.z/4
s1 = Segment(Point(0, 0), Point(0, 1))
assert vector_integrate(-15*C.y, s1) == S(-15)/2
s2 = Segment(Point(4, 3, 9), Point(1, 1, 7))
assert vector_integrate(C.y*C.i, s2) == -6
curve = Curve((sin(t), cos(t)), (t, 0, 2))
assert vector_integrate(5*C.z, curve) == 10*C.z
c1 = Circle(Point(2, 3), 6)
assert vector_integrate(C.x*C.y, c1) == 72*pi
c2 = Circle(Point(0, 0), Point(1, 1), Point(1, 0))
assert vector_integrate(1, c2) == c2.circumference
triangle = Polygon((0, 0), (1, 0), (1, 1))
assert vector_integrate(C.x*C.i - 14*C.y*C.j, triangle) == 0
p1, p2, p3, p4 = [(0, 0), (1, 0), (5, 1), (0, 1)]
poly = Polygon(p1, p2, p3, p4)
assert vector_integrate(-23*C.z, poly) == -161*C.z - 23*sqrt(17)*C.z
point = Point(2, 3)
assert vector_integrate(C.i*C.y - C.z, point) == ParametricIntegral(C.y*C.i, ParametricRegion((2, 3)))
c3 = ImplicitRegion((x, y), x**2 + y**2 - 4)
assert vector_integrate(45, c3) == 180*pi
c4 = ImplicitRegion((x, y), (x - 3)**2 + (y - 4)**2 - 9)
assert vector_integrate(1, c4) == 6*pi
pl = Plane(Point(1, 1, 1), Point(2, 3, 4), Point(2, 2, 2))
raises(ValueError, lambda: vector_integrate(C.x*C.z*C.i + C.k, pl))
|
d338aab6aa5fce70f1a5cf31ed79a6a88da2732a8f973678aba5e4fcedf47df1 | from sympy import Eq, Rational, S, Symbol, symbols, pi, sqrt, oo, Point2D, Segment2D, Abs, sec
from sympy.geometry import (Circle, Ellipse, GeometryError, Line, Point,
Polygon, Ray, RegularPolygon, Segment,
Triangle, intersection)
from sympy.testing.pytest import raises, slow
from sympy import integrate
from sympy.functions.special.elliptic_integrals import elliptic_e
from sympy.functions.elementary.miscellaneous import Max
def test_ellipse_equation_using_slope():
from sympy.abc import x, y
e1 = Ellipse(Point(1, 0), 3, 2)
assert str(e1.equation(_slope=1)) == str((-x + y + 1)**2/8 + (x + y - 1)**2/18 - 1)
e2 = Ellipse(Point(0, 0), 4, 1)
assert str(e2.equation(_slope=1)) == str((-x + y)**2/2 + (x + y)**2/32 - 1)
e3 = Ellipse(Point(1, 5), 6, 2)
assert str(e3.equation(_slope=2)) == str((-2*x + y - 3)**2/20 + (x + 2*y - 11)**2/180 - 1)
def test_object_from_equation():
from sympy.abc import x, y, a, b
assert Circle(x**2 + y**2 + 3*x + 4*y - 8) == Circle(Point2D(S(-3) / 2, -2),
sqrt(57) / 2)
assert Circle(x**2 + y**2 + 6*x + 8*y + 25) == Circle(Point2D(-3, -4), 0)
assert Circle(a**2 + b**2 + 6*a + 8*b + 25, x='a', y='b') == Circle(Point2D(-3, -4), 0)
assert Circle(x**2 + y**2 - 25) == Circle(Point2D(0, 0), 5)
assert Circle(x**2 + y**2) == Circle(Point2D(0, 0), 0)
assert Circle(a**2 + b**2, x='a', y='b') == Circle(Point2D(0, 0), 0)
assert Circle(x**2 + y**2 + 6*x + 8) == Circle(Point2D(-3, 0), 1)
assert Circle(x**2 + y**2 + 6*y + 8) == Circle(Point2D(0, -3), 1)
assert Circle(6*(x**2) + 6*(y**2) + 6*x + 8*y - 25) == Circle(Point2D(Rational(-1, 2), Rational(-2, 3)), 5*sqrt(37)/6)
assert Circle(Eq(a**2 + b**2, 25), x='a', y=b) == Circle(Point2D(0, 0), 5)
raises(GeometryError, lambda: Circle(x**2 + y**2 + 3*x + 4*y + 26))
raises(GeometryError, lambda: Circle(x**2 + y**2 + 25))
raises(GeometryError, lambda: Circle(a**2 + b**2 + 25, x='a', y='b'))
raises(GeometryError, lambda: Circle(x**2 + 6*y + 8))
raises(GeometryError, lambda: Circle(6*(x ** 2) + 4*(y**2) + 6*x + 8*y + 25))
raises(ValueError, lambda: Circle(a**2 + b**2 + 3*a + 4*b - 8))
@slow
def test_ellipse_geom():
x = Symbol('x', real=True)
y = Symbol('y', real=True)
t = Symbol('t', real=True)
y1 = Symbol('y1', real=True)
half = S.Half
p1 = Point(0, 0)
p2 = Point(1, 1)
p4 = Point(0, 1)
e1 = Ellipse(p1, 1, 1)
e2 = Ellipse(p2, half, 1)
e3 = Ellipse(p1, y1, y1)
c1 = Circle(p1, 1)
c2 = Circle(p2, 1)
c3 = Circle(Point(sqrt(2), sqrt(2)), 1)
l1 = Line(p1, p2)
# Test creation with three points
cen, rad = Point(3*half, 2), 5*half
assert Circle(Point(0, 0), Point(3, 0), Point(0, 4)) == Circle(cen, rad)
assert Circle(Point(0, 0), Point(1, 1), Point(2, 2)) == Segment2D(Point2D(0, 0), Point2D(2, 2))
raises(ValueError, lambda: Ellipse(None, None, None, 1))
raises(ValueError, lambda: Ellipse())
raises(GeometryError, lambda: Circle(Point(0, 0)))
raises(GeometryError, lambda: Circle(Symbol('x')*Symbol('y')))
# Basic Stuff
assert Ellipse(None, 1, 1).center == Point(0, 0)
assert e1 == c1
assert e1 != e2
assert e1 != l1
assert p4 in e1
assert e1 in e1
assert e2 in e2
assert 1 not in e2
assert p2 not in e2
assert e1.area == pi
assert e2.area == pi/2
assert e3.area == pi*y1*abs(y1)
assert c1.area == e1.area
assert c1.circumference == e1.circumference
assert e3.circumference == 2*pi*y1
assert e1.plot_interval() == e2.plot_interval() == [t, -pi, pi]
assert e1.plot_interval(x) == e2.plot_interval(x) == [x, -pi, pi]
assert c1.minor == 1
assert c1.major == 1
assert c1.hradius == 1
assert c1.vradius == 1
assert Ellipse((1, 1), 0, 0) == Point(1, 1)
assert Ellipse((1, 1), 1, 0) == Segment(Point(0, 1), Point(2, 1))
assert Ellipse((1, 1), 0, 1) == Segment(Point(1, 0), Point(1, 2))
# Private Functions
assert hash(c1) == hash(Circle(Point(1, 0), Point(0, 1), Point(0, -1)))
assert c1 in e1
assert (Line(p1, p2) in e1) is False
assert e1.__cmp__(e1) == 0
assert e1.__cmp__(Point(0, 0)) > 0
# Encloses
assert e1.encloses(Segment(Point(-0.5, -0.5), Point(0.5, 0.5))) is True
assert e1.encloses(Line(p1, p2)) is False
assert e1.encloses(Ray(p1, p2)) is False
assert e1.encloses(e1) is False
assert e1.encloses(
Polygon(Point(-0.5, -0.5), Point(-0.5, 0.5), Point(0.5, 0.5))) is True
assert e1.encloses(RegularPolygon(p1, 0.5, 3)) is True
assert e1.encloses(RegularPolygon(p1, 5, 3)) is False
assert e1.encloses(RegularPolygon(p2, 5, 3)) is False
assert e2.arbitrary_point() in e2
raises(ValueError, lambda: Ellipse(Point(x, y), 1, 1).arbitrary_point(parameter='x'))
# Foci
f1, f2 = Point(sqrt(12), 0), Point(-sqrt(12), 0)
ef = Ellipse(Point(0, 0), 4, 2)
assert ef.foci in [(f1, f2), (f2, f1)]
# Tangents
v = sqrt(2) / 2
p1_1 = Point(v, v)
p1_2 = p2 + Point(half, 0)
p1_3 = p2 + Point(0, 1)
assert e1.tangent_lines(p4) == c1.tangent_lines(p4)
assert e2.tangent_lines(p1_2) == [Line(Point(Rational(3, 2), 1), Point(Rational(3, 2), S.Half))]
assert e2.tangent_lines(p1_3) == [Line(Point(1, 2), Point(Rational(5, 4), 2))]
assert c1.tangent_lines(p1_1) != [Line(p1_1, Point(0, sqrt(2)))]
assert c1.tangent_lines(p1) == []
assert e2.is_tangent(Line(p1_2, p2 + Point(half, 1)))
assert e2.is_tangent(Line(p1_3, p2 + Point(half, 1)))
assert c1.is_tangent(Line(p1_1, Point(0, sqrt(2))))
assert e1.is_tangent(Line(Point(0, 0), Point(1, 1))) is False
assert c1.is_tangent(e1) is True
assert c1.is_tangent(Ellipse(Point(2, 0), 1, 1)) is True
assert c1.is_tangent(
Polygon(Point(1, 1), Point(1, -1), Point(2, 0))) is True
assert c1.is_tangent(
Polygon(Point(1, 1), Point(1, 0), Point(2, 0))) is False
assert Circle(Point(5, 5), 3).is_tangent(Circle(Point(0, 5), 1)) is False
assert Ellipse(Point(5, 5), 2, 1).tangent_lines(Point(0, 0)) == \
[Line(Point(0, 0), Point(Rational(77, 25), Rational(132, 25))),
Line(Point(0, 0), Point(Rational(33, 5), Rational(22, 5)))]
assert Ellipse(Point(5, 5), 2, 1).tangent_lines(Point(3, 4)) == \
[Line(Point(3, 4), Point(4, 4)), Line(Point(3, 4), Point(3, 5))]
assert Circle(Point(5, 5), 2).tangent_lines(Point(3, 3)) == \
[Line(Point(3, 3), Point(4, 3)), Line(Point(3, 3), Point(3, 4))]
assert Circle(Point(5, 5), 2).tangent_lines(Point(5 - 2*sqrt(2), 5)) == \
[Line(Point(5 - 2*sqrt(2), 5), Point(5 - sqrt(2), 5 - sqrt(2))),
Line(Point(5 - 2*sqrt(2), 5), Point(5 - sqrt(2), 5 + sqrt(2))), ]
assert Circle(Point(5, 5), 5).tangent_lines(Point(4, 0)) == \
[Line(Point(4, 0), Point(Rational(40, 13), Rational(5, 13))),
Line(Point(4, 0), Point(5, 0))]
assert Circle(Point(5, 5), 5).tangent_lines(Point(0, 6)) == \
[Line(Point(0, 6), Point(0, 7)),
Line(Point(0, 6), Point(Rational(5, 13), Rational(90, 13)))]
# for numerical calculations, we shouldn't demand exact equality,
# so only test up to the desired precision
def lines_close(l1, l2, prec):
""" tests whether l1 and 12 are within 10**(-prec)
of each other """
return abs(l1.p1 - l2.p1) < 10**(-prec) and abs(l1.p2 - l2.p2) < 10**(-prec)
def line_list_close(ll1, ll2, prec):
return all(lines_close(l1, l2, prec) for l1, l2 in zip(ll1, ll2))
e = Ellipse(Point(0, 0), 2, 1)
assert e.normal_lines(Point(0, 0)) == \
[Line(Point(0, 0), Point(0, 1)), Line(Point(0, 0), Point(1, 0))]
assert e.normal_lines(Point(1, 0)) == \
[Line(Point(0, 0), Point(1, 0))]
assert e.normal_lines((0, 1)) == \
[Line(Point(0, 0), Point(0, 1))]
assert line_list_close(e.normal_lines(Point(1, 1), 2), [
Line(Point(Rational(-51, 26), Rational(-1, 5)), Point(Rational(-25, 26), Rational(17, 83))),
Line(Point(Rational(28, 29), Rational(-7, 8)), Point(Rational(57, 29), Rational(-9, 2)))], 2)
# test the failure of Poly.intervals and checks a point on the boundary
p = Point(sqrt(3), S.Half)
assert p in e
assert line_list_close(e.normal_lines(p, 2), [
Line(Point(Rational(-341, 171), Rational(-1, 13)), Point(Rational(-170, 171), Rational(5, 64))),
Line(Point(Rational(26, 15), Rational(-1, 2)), Point(Rational(41, 15), Rational(-43, 26)))], 2)
# be sure to use the slope that isn't undefined on boundary
e = Ellipse((0, 0), 2, 2*sqrt(3)/3)
assert line_list_close(e.normal_lines((1, 1), 2), [
Line(Point(Rational(-64, 33), Rational(-20, 71)), Point(Rational(-31, 33), Rational(2, 13))),
Line(Point(1, -1), Point(2, -4))], 2)
# general ellipse fails except under certain conditions
e = Ellipse((0, 0), x, 1)
assert e.normal_lines((x + 1, 0)) == [Line(Point(0, 0), Point(1, 0))]
raises(NotImplementedError, lambda: e.normal_lines((x + 1, 1)))
# Properties
major = 3
minor = 1
e4 = Ellipse(p2, minor, major)
assert e4.focus_distance == sqrt(major**2 - minor**2)
ecc = e4.focus_distance / major
assert e4.eccentricity == ecc
assert e4.periapsis == major*(1 - ecc)
assert e4.apoapsis == major*(1 + ecc)
assert e4.semilatus_rectum == major*(1 - ecc ** 2)
# independent of orientation
e4 = Ellipse(p2, major, minor)
assert e4.focus_distance == sqrt(major**2 - minor**2)
ecc = e4.focus_distance / major
assert e4.eccentricity == ecc
assert e4.periapsis == major*(1 - ecc)
assert e4.apoapsis == major*(1 + ecc)
# Intersection
l1 = Line(Point(1, -5), Point(1, 5))
l2 = Line(Point(-5, -1), Point(5, -1))
l3 = Line(Point(-1, -1), Point(1, 1))
l4 = Line(Point(-10, 0), Point(0, 10))
pts_c1_l3 = [Point(sqrt(2)/2, sqrt(2)/2), Point(-sqrt(2)/2, -sqrt(2)/2)]
assert intersection(e2, l4) == []
assert intersection(c1, Point(1, 0)) == [Point(1, 0)]
assert intersection(c1, l1) == [Point(1, 0)]
assert intersection(c1, l2) == [Point(0, -1)]
assert intersection(c1, l3) in [pts_c1_l3, [pts_c1_l3[1], pts_c1_l3[0]]]
assert intersection(c1, c2) == [Point(0, 1), Point(1, 0)]
assert intersection(c1, c3) == [Point(sqrt(2)/2, sqrt(2)/2)]
assert e1.intersection(l1) == [Point(1, 0)]
assert e2.intersection(l4) == []
assert e1.intersection(Circle(Point(0, 2), 1)) == [Point(0, 1)]
assert e1.intersection(Circle(Point(5, 0), 1)) == []
assert e1.intersection(Ellipse(Point(2, 0), 1, 1)) == [Point(1, 0)]
assert e1.intersection(Ellipse(Point(5, 0), 1, 1)) == []
assert e1.intersection(Point(2, 0)) == []
assert e1.intersection(e1) == e1
assert intersection(Ellipse(Point(0, 0), 2, 1), Ellipse(Point(3, 0), 1, 2)) == [Point(2, 0)]
assert intersection(Circle(Point(0, 0), 2), Circle(Point(3, 0), 1)) == [Point(2, 0)]
assert intersection(Circle(Point(0, 0), 2), Circle(Point(7, 0), 1)) == []
assert intersection(Ellipse(Point(0, 0), 5, 17), Ellipse(Point(4, 0), 1, 0.2)) == [Point(5, 0)]
assert intersection(Ellipse(Point(0, 0), 5, 17), Ellipse(Point(4, 0), 0.999, 0.2)) == []
assert Circle((0, 0), S.Half).intersection(
Triangle((-1, 0), (1, 0), (0, 1))) == [
Point(Rational(-1, 2), 0), Point(S.Half, 0)]
raises(TypeError, lambda: intersection(e2, Line((0, 0, 0), (0, 0, 1))))
raises(TypeError, lambda: intersection(e2, Rational(12)))
raises(TypeError, lambda: Ellipse.intersection(e2, 1))
# some special case intersections
csmall = Circle(p1, 3)
cbig = Circle(p1, 5)
cout = Circle(Point(5, 5), 1)
# one circle inside of another
assert csmall.intersection(cbig) == []
# separate circles
assert csmall.intersection(cout) == []
# coincident circles
assert csmall.intersection(csmall) == csmall
v = sqrt(2)
t1 = Triangle(Point(0, v), Point(0, -v), Point(v, 0))
points = intersection(t1, c1)
assert len(points) == 4
assert Point(0, 1) in points
assert Point(0, -1) in points
assert Point(v/2, v/2) in points
assert Point(v/2, -v/2) in points
circ = Circle(Point(0, 0), 5)
elip = Ellipse(Point(0, 0), 5, 20)
assert intersection(circ, elip) in \
[[Point(5, 0), Point(-5, 0)], [Point(-5, 0), Point(5, 0)]]
assert elip.tangent_lines(Point(0, 0)) == []
elip = Ellipse(Point(0, 0), 3, 2)
assert elip.tangent_lines(Point(3, 0)) == \
[Line(Point(3, 0), Point(3, -12))]
e1 = Ellipse(Point(0, 0), 5, 10)
e2 = Ellipse(Point(2, 1), 4, 8)
a = Rational(53, 17)
c = 2*sqrt(3991)/17
ans = [Point(a - c/8, a/2 + c), Point(a + c/8, a/2 - c)]
assert e1.intersection(e2) == ans
e2 = Ellipse(Point(x, y), 4, 8)
c = sqrt(3991)
ans = [Point(-c/68 + a, c*Rational(2, 17) + a/2), Point(c/68 + a, c*Rational(-2, 17) + a/2)]
assert [p.subs({x: 2, y:1}) for p in e1.intersection(e2)] == ans
# Combinations of above
assert e3.is_tangent(e3.tangent_lines(p1 + Point(y1, 0))[0])
e = Ellipse((1, 2), 3, 2)
assert e.tangent_lines(Point(10, 0)) == \
[Line(Point(10, 0), Point(1, 0)),
Line(Point(10, 0), Point(Rational(14, 5), Rational(18, 5)))]
# encloses_point
e = Ellipse((0, 0), 1, 2)
assert e.encloses_point(e.center)
assert e.encloses_point(e.center + Point(0, e.vradius - Rational(1, 10)))
assert e.encloses_point(e.center + Point(e.hradius - Rational(1, 10), 0))
assert e.encloses_point(e.center + Point(e.hradius, 0)) is False
assert e.encloses_point(
e.center + Point(e.hradius + Rational(1, 10), 0)) is False
e = Ellipse((0, 0), 2, 1)
assert e.encloses_point(e.center)
assert e.encloses_point(e.center + Point(0, e.vradius - Rational(1, 10)))
assert e.encloses_point(e.center + Point(e.hradius - Rational(1, 10), 0))
assert e.encloses_point(e.center + Point(e.hradius, 0)) is False
assert e.encloses_point(
e.center + Point(e.hradius + Rational(1, 10), 0)) is False
assert c1.encloses_point(Point(1, 0)) is False
assert c1.encloses_point(Point(0.3, 0.4)) is True
assert e.scale(2, 3) == Ellipse((0, 0), 4, 3)
assert e.scale(3, 6) == Ellipse((0, 0), 6, 6)
assert e.rotate(pi) == e
assert e.rotate(pi, (1, 2)) == Ellipse(Point(2, 4), 2, 1)
raises(NotImplementedError, lambda: e.rotate(pi/3))
# Circle rotation tests (Issue #11743)
# Link - https://github.com/sympy/sympy/issues/11743
cir = Circle(Point(1, 0), 1)
assert cir.rotate(pi/2) == Circle(Point(0, 1), 1)
assert cir.rotate(pi/3) == Circle(Point(S.Half, sqrt(3)/2), 1)
assert cir.rotate(pi/3, Point(1, 0)) == Circle(Point(1, 0), 1)
assert cir.rotate(pi/3, Point(0, 1)) == Circle(Point(S.Half + sqrt(3)/2, S.Half + sqrt(3)/2), 1)
def test_construction():
e1 = Ellipse(hradius=2, vradius=1, eccentricity=None)
assert e1.eccentricity == sqrt(3)/2
e2 = Ellipse(hradius=2, vradius=None, eccentricity=sqrt(3)/2)
assert e2.vradius == 1
e3 = Ellipse(hradius=None, vradius=1, eccentricity=sqrt(3)/2)
assert e3.hradius == 2
# filter(None, iterator) filters out anything falsey, including 0
# eccentricity would be filtered out in this case and the constructor would throw an error
e4 = Ellipse(Point(0, 0), hradius=1, eccentricity=0)
assert e4.vradius == 1
#tests for eccentricity > 1
raises(GeometryError, lambda: Ellipse(Point(3, 1), hradius=3, eccentricity = S(3)/2))
raises(GeometryError, lambda: Ellipse(Point(3, 1), hradius=3, eccentricity=sec(5)))
raises(GeometryError, lambda: Ellipse(Point(3, 1), hradius=3, eccentricity=S.Pi-S(2)))
#tests for eccentricity = 1
#if vradius is not defined
assert Ellipse(None, 1, None, 1).length == 2
#if hradius is not defined
raises(GeometryError, lambda: Ellipse(None, None, 1, eccentricity = 1))
#tests for eccentricity < 0
raises(GeometryError, lambda: Ellipse(Point(3, 1), hradius=3, eccentricity = -3))
raises(GeometryError, lambda: Ellipse(Point(3, 1), hradius=3, eccentricity = -0.5))
def test_ellipse_random_point():
y1 = Symbol('y1', real=True)
e3 = Ellipse(Point(0, 0), y1, y1)
rx, ry = Symbol('rx'), Symbol('ry')
for ind in range(0, 5):
r = e3.random_point()
# substitution should give zero*y1**2
assert e3.equation(rx, ry).subs(zip((rx, ry), r.args)).equals(0)
# test for the case with seed
r = e3.random_point(seed=1)
assert e3.equation(rx, ry).subs(zip((rx, ry), r.args)).equals(0)
def test_repr():
assert repr(Circle((0, 1), 2)) == 'Circle(Point2D(0, 1), 2)'
def test_transform():
c = Circle((1, 1), 2)
assert c.scale(-1) == Circle((-1, 1), 2)
assert c.scale(y=-1) == Circle((1, -1), 2)
assert c.scale(2) == Ellipse((2, 1), 4, 2)
assert Ellipse((0, 0), 2, 3).scale(2, 3, (4, 5)) == \
Ellipse(Point(-4, -10), 4, 9)
assert Circle((0, 0), 2).scale(2, 3, (4, 5)) == \
Ellipse(Point(-4, -10), 4, 6)
assert Ellipse((0, 0), 2, 3).scale(3, 3, (4, 5)) == \
Ellipse(Point(-8, -10), 6, 9)
assert Circle((0, 0), 2).scale(3, 3, (4, 5)) == \
Circle(Point(-8, -10), 6)
assert Circle(Point(-8, -10), 6).scale(Rational(1, 3), Rational(1, 3), (4, 5)) == \
Circle((0, 0), 2)
assert Circle((0, 0), 2).translate(4, 5) == \
Circle((4, 5), 2)
assert Circle((0, 0), 2).scale(3, 3) == \
Circle((0, 0), 6)
def test_bounds():
e1 = Ellipse(Point(0, 0), 3, 5)
e2 = Ellipse(Point(2, -2), 7, 7)
c1 = Circle(Point(2, -2), 7)
c2 = Circle(Point(-2, 0), Point(0, 2), Point(2, 0))
assert e1.bounds == (-3, -5, 3, 5)
assert e2.bounds == (-5, -9, 9, 5)
assert c1.bounds == (-5, -9, 9, 5)
assert c2.bounds == (-2, -2, 2, 2)
def test_reflect():
b = Symbol('b')
m = Symbol('m')
l = Line((0, b), slope=m)
t1 = Triangle((0, 0), (1, 0), (2, 3))
assert t1.area == -t1.reflect(l).area
e = Ellipse((1, 0), 1, 2)
assert e.area == -e.reflect(Line((1, 0), slope=0)).area
assert e.area == -e.reflect(Line((1, 0), slope=oo)).area
raises(NotImplementedError, lambda: e.reflect(Line((1, 0), slope=m)))
assert Circle((0, 1), 1).reflect(Line((0, 0), (1, 1))) == Circle(Point2D(1, 0), -1)
def test_is_tangent():
e1 = Ellipse(Point(0, 0), 3, 5)
c1 = Circle(Point(2, -2), 7)
assert e1.is_tangent(Point(0, 0)) is False
assert e1.is_tangent(Point(3, 0)) is False
assert e1.is_tangent(e1) is True
assert e1.is_tangent(Ellipse((0, 0), 1, 2)) is False
assert e1.is_tangent(Ellipse((0, 0), 3, 2)) is True
assert c1.is_tangent(Ellipse((2, -2), 7, 1)) is True
assert c1.is_tangent(Circle((11, -2), 2)) is True
assert c1.is_tangent(Circle((7, -2), 2)) is True
assert c1.is_tangent(Ray((-5, -2), (-15, -20))) is False
assert c1.is_tangent(Ray((-3, -2), (-15, -20))) is False
assert c1.is_tangent(Ray((-3, -22), (15, 20))) is False
assert c1.is_tangent(Ray((9, 20), (9, -20))) is True
assert e1.is_tangent(Segment((2, 2), (-7, 7))) is False
assert e1.is_tangent(Segment((0, 0), (1, 2))) is False
assert c1.is_tangent(Segment((0, 0), (-5, -2))) is False
assert e1.is_tangent(Segment((3, 0), (12, 12))) is False
assert e1.is_tangent(Segment((12, 12), (3, 0))) is False
assert e1.is_tangent(Segment((-3, 0), (3, 0))) is False
assert e1.is_tangent(Segment((-3, 5), (3, 5))) is True
assert e1.is_tangent(Line((10, 0), (10, 10))) is False
assert e1.is_tangent(Line((0, 0), (1, 1))) is False
assert e1.is_tangent(Line((-3, 0), (-2.99, -0.001))) is False
assert e1.is_tangent(Line((-3, 0), (-3, 1))) is True
assert e1.is_tangent(Polygon((0, 0), (5, 5), (5, -5))) is False
assert e1.is_tangent(Polygon((-100, -50), (-40, -334), (-70, -52))) is False
assert e1.is_tangent(Polygon((-3, 0), (3, 0), (0, 1))) is False
assert e1.is_tangent(Polygon((-3, 0), (3, 0), (0, 5))) is False
assert e1.is_tangent(Polygon((-3, 0), (0, -5), (3, 0), (0, 5))) is False
assert e1.is_tangent(Polygon((-3, -5), (-3, 5), (3, 5), (3, -5))) is True
assert c1.is_tangent(Polygon((-3, -5), (-3, 5), (3, 5), (3, -5))) is False
assert e1.is_tangent(Polygon((0, 0), (3, 0), (7, 7), (0, 5))) is False
assert e1.is_tangent(Polygon((3, 12), (3, -12), (6, 5))) is True
assert e1.is_tangent(Polygon((3, 12), (3, -12), (0, -5), (0, 5))) is False
assert e1.is_tangent(Polygon((3, 0), (5, 7), (6, -5))) is False
raises(TypeError, lambda: e1.is_tangent(Point(0, 0, 0)))
raises(TypeError, lambda: e1.is_tangent(Rational(5)))
def test_parameter_value():
t = Symbol('t')
e = Ellipse(Point(0, 0), 3, 5)
assert e.parameter_value((3, 0), t) == {t: 0}
raises(ValueError, lambda: e.parameter_value((4, 0), t))
@slow
def test_second_moment_of_area():
x, y = symbols('x, y')
e = Ellipse(Point(0, 0), 5, 4)
I_yy = 2*4*integrate(sqrt(25 - x**2)*x**2, (x, -5, 5))/5
I_xx = 2*5*integrate(sqrt(16 - y**2)*y**2, (y, -4, 4))/4
Y = 3*sqrt(1 - x**2/5**2)
I_xy = integrate(integrate(y, (y, -Y, Y))*x, (x, -5, 5))
assert I_yy == e.second_moment_of_area()[1]
assert I_xx == e.second_moment_of_area()[0]
assert I_xy == e.second_moment_of_area()[2]
#checking for other point
t1 = e.second_moment_of_area(Point(6,5))
t2 = (580*pi, 845*pi, 600*pi)
assert t1==t2
def test_section_modulus_and_polar_second_moment_of_area():
d = Symbol('d', positive=True)
c = Circle((3, 7), 8)
assert c.polar_second_moment_of_area() == 2048*pi
assert c.section_modulus() == (128*pi, 128*pi)
c = Circle((2, 9), d/2)
assert c.polar_second_moment_of_area() == pi*d**3*Abs(d)/64 + pi*d*Abs(d)**3/64
assert c.section_modulus() == (pi*d**3/S(32), pi*d**3/S(32))
a, b = symbols('a, b', positive=True)
e = Ellipse((4, 6), a, b)
assert e.section_modulus() == (pi*a*b**2/S(4), pi*a**2*b/S(4))
assert e.polar_second_moment_of_area() == pi*a**3*b/S(4) + pi*a*b**3/S(4)
e = e.rotate(pi/2) # no change in polar and section modulus
assert e.section_modulus() == (pi*a**2*b/S(4), pi*a*b**2/S(4))
assert e.polar_second_moment_of_area() == pi*a**3*b/S(4) + pi*a*b**3/S(4)
e = Ellipse((a, b), 2, 6)
assert e.section_modulus() == (18*pi, 6*pi)
assert e.polar_second_moment_of_area() == 120*pi
e = Ellipse(Point(0, 0), 2, 2)
assert e.section_modulus() == (2*pi, 2*pi)
assert e.section_modulus(Point(2, 2)) == (2*pi, 2*pi)
assert e.section_modulus((2, 2)) == (2*pi, 2*pi)
def test_circumference():
M = Symbol('M')
m = Symbol('m')
assert Ellipse(Point(0, 0), M, m).circumference == 4 * M * elliptic_e((M ** 2 - m ** 2) / M**2)
assert Ellipse(Point(0, 0), 5, 4).circumference == 20 * elliptic_e(S(9) / 25)
# circle
assert Ellipse(None, 1, None, 0).circumference == 2*pi
# test numerically
assert abs(Ellipse(None, hradius=5, vradius=3).circumference.evalf(16) - 25.52699886339813) < 1e-10
def test_issue_15259():
assert Circle((1, 2), 0) == Point(1, 2)
def test_issue_15797_equals():
Ri = 0.024127189424130748
Ci = (0.0864931002830291, 0.0819863295239654)
A = Point(0, 0.0578591400998346)
c = Circle(Ci, Ri) # evaluated
assert c.is_tangent(c.tangent_lines(A)[0]) == True
assert c.center.x.is_Rational
assert c.center.y.is_Rational
assert c.radius.is_Rational
u = Circle(Ci, Ri, evaluate=False) # unevaluated
assert u.center.x.is_Float
assert u.center.y.is_Float
assert u.radius.is_Float
def test_auxiliary_circle():
x, y, a, b = symbols('x y a b')
e = Ellipse((x, y), a, b)
# the general result
assert e.auxiliary_circle() == Circle((x, y), Max(a, b))
# a special case where Ellipse is a Circle
assert Circle((3, 4), 8).auxiliary_circle() == Circle((3, 4), 8)
def test_director_circle():
x, y, a, b = symbols('x y a b')
e = Ellipse((x, y), a, b)
# the general result
assert e.director_circle() == Circle((x, y), sqrt(a**2 + b**2))
# a special case where Ellipse is a Circle
assert Circle((3, 4), 8).director_circle() == Circle((3, 4), 8*sqrt(2))
def test_evolute():
#ellipse centered at h,k
x, y, h, k = symbols('x y h k',real = True)
a, b = symbols('a b')
e = Ellipse(Point(h, k), a, b)
t1 = (e.hradius*(x - e.center.x))**Rational(2, 3)
t2 = (e.vradius*(y - e.center.y))**Rational(2, 3)
E = t1 + t2 - (e.hradius**2 - e.vradius**2)**Rational(2, 3)
assert e.evolute() == E
#Numerical Example
e = Ellipse(Point(1, 1), 6, 3)
t1 = (6*(x - 1))**Rational(2, 3)
t2 = (3*(y - 1))**Rational(2, 3)
E = t1 + t2 - (27)**Rational(2, 3)
assert e.evolute() == E
def test_svg():
e1 = Ellipse(Point(1, 0), 3, 2)
assert e1._svg(2, "#FFAAFF") == '<ellipse fill="#FFAAFF" stroke="#555555" stroke-width="4.0" opacity="0.6" cx="1.00000000000000" cy="0" rx="3.00000000000000" ry="2.00000000000000"/>'
|
dd4e38acea08620865aebea176369f0577859ff08f2fca81d315aad90ee644bc | import sympy
from sympy.parsing.sympy_parser import (
parse_expr,
standard_transformations,
convert_xor,
implicit_multiplication_application,
implicit_multiplication,
implicit_application,
function_exponentiation,
split_symbols,
split_symbols_custom,
_token_splittable
)
from sympy.testing.pytest import raises
def test_implicit_multiplication():
cases = {
'5x': '5*x',
'abc': 'a*b*c',
'3sin(x)': '3*sin(x)',
'(x+1)(x+2)': '(x+1)*(x+2)',
'(5 x**2)sin(x)': '(5*x**2)*sin(x)',
'2 sin(x) cos(x)': '2*sin(x)*cos(x)',
'pi x': 'pi*x',
'x pi': 'x*pi',
'E x': 'E*x',
'EulerGamma y': 'EulerGamma*y',
'E pi': 'E*pi',
'pi (x + 2)': 'pi*(x+2)',
'(x + 2) pi': '(x+2)*pi',
'pi sin(x)': 'pi*sin(x)',
}
transformations = standard_transformations + (convert_xor,)
transformations2 = transformations + (split_symbols,
implicit_multiplication)
for case in cases:
implicit = parse_expr(case, transformations=transformations2)
normal = parse_expr(cases[case], transformations=transformations)
assert(implicit == normal)
application = ['sin x', 'cos 2*x', 'sin cos x']
for case in application:
raises(SyntaxError,
lambda: parse_expr(case, transformations=transformations2))
raises(TypeError,
lambda: parse_expr('sin**2(x)', transformations=transformations2))
def test_implicit_application():
cases = {
'factorial': 'factorial',
'sin x': 'sin(x)',
'tan y**3': 'tan(y**3)',
'cos 2*x': 'cos(2*x)',
'(cot)': 'cot',
'sin cos tan x': 'sin(cos(tan(x)))'
}
transformations = standard_transformations + (convert_xor,)
transformations2 = transformations + (implicit_application,)
for case in cases:
implicit = parse_expr(case, transformations=transformations2)
normal = parse_expr(cases[case], transformations=transformations)
assert(implicit == normal), (implicit, normal)
multiplication = ['x y', 'x sin x', '2x']
for case in multiplication:
raises(SyntaxError,
lambda: parse_expr(case, transformations=transformations2))
raises(TypeError,
lambda: parse_expr('sin**2(x)', transformations=transformations2))
def test_function_exponentiation():
cases = {
'sin**2(x)': 'sin(x)**2',
'exp^y(z)': 'exp(z)^y',
'sin**2(E^(x))': 'sin(E^(x))**2'
}
transformations = standard_transformations + (convert_xor,)
transformations2 = transformations + (function_exponentiation,)
for case in cases:
implicit = parse_expr(case, transformations=transformations2)
normal = parse_expr(cases[case], transformations=transformations)
assert(implicit == normal)
other_implicit = ['x y', 'x sin x', '2x', 'sin x',
'cos 2*x', 'sin cos x']
for case in other_implicit:
raises(SyntaxError,
lambda: parse_expr(case, transformations=transformations2))
assert parse_expr('x**2', local_dict={ 'x': sympy.Symbol('x') },
transformations=transformations2) == parse_expr('x**2')
def test_symbol_splitting():
# By default Greek letter names should not be split (lambda is a keyword
# so skip it)
transformations = standard_transformations + (split_symbols,)
greek_letters = ('alpha', 'beta', 'gamma', 'delta', 'epsilon', 'zeta',
'eta', 'theta', 'iota', 'kappa', 'mu', 'nu', 'xi',
'omicron', 'pi', 'rho', 'sigma', 'tau', 'upsilon',
'phi', 'chi', 'psi', 'omega')
for letter in greek_letters:
assert(parse_expr(letter, transformations=transformations) ==
parse_expr(letter))
# Make sure symbol splitting resolves names
transformations += (implicit_multiplication,)
local_dict = { 'e': sympy.E }
cases = {
'xe': 'E*x',
'Iy': 'I*y',
'ee': 'E*E',
}
for case, expected in cases.items():
assert(parse_expr(case, local_dict=local_dict,
transformations=transformations) ==
parse_expr(expected))
# Make sure custom splitting works
def can_split(symbol):
if symbol not in ('unsplittable', 'names'):
return _token_splittable(symbol)
return False
transformations = standard_transformations
transformations += (split_symbols_custom(can_split),
implicit_multiplication)
assert(parse_expr('unsplittable', transformations=transformations) ==
parse_expr('unsplittable'))
assert(parse_expr('names', transformations=transformations) ==
parse_expr('names'))
assert(parse_expr('xy', transformations=transformations) ==
parse_expr('x*y'))
for letter in greek_letters:
assert(parse_expr(letter, transformations=transformations) ==
parse_expr(letter))
def test_all_implicit_steps():
cases = {
'2x': '2*x', # implicit multiplication
'x y': 'x*y',
'xy': 'x*y',
'sin x': 'sin(x)', # add parentheses
'2sin x': '2*sin(x)',
'x y z': 'x*y*z',
'sin(2 * 3x)': 'sin(2 * 3 * x)',
'sin(x) (1 + cos(x))': 'sin(x) * (1 + cos(x))',
'(x + 2) sin(x)': '(x + 2) * sin(x)',
'(x + 2) sin x': '(x + 2) * sin(x)',
'sin(sin x)': 'sin(sin(x))',
'sin x!': 'sin(factorial(x))',
'sin x!!': 'sin(factorial2(x))',
'factorial': 'factorial', # don't apply a bare function
'x sin x': 'x * sin(x)', # both application and multiplication
'xy sin x': 'x * y * sin(x)',
'(x+2)(x+3)': '(x + 2) * (x+3)',
'x**2 + 2xy + y**2': 'x**2 + 2 * x * y + y**2', # split the xy
'pi': 'pi', # don't mess with constants
'None': 'None',
'ln sin x': 'ln(sin(x))', # multiple implicit function applications
'factorial': 'factorial', # don't add parentheses
'sin x**2': 'sin(x**2)', # implicit application to an exponential
'alpha': 'Symbol("alpha")', # don't split Greek letters/subscripts
'x_2': 'Symbol("x_2")',
'sin^2 x**2': 'sin(x**2)**2', # function raised to a power
'sin**3(x)': 'sin(x)**3',
'(factorial)': 'factorial',
'tan 3x': 'tan(3*x)',
'sin^2(3*E^(x))': 'sin(3*E**(x))**2',
'sin**2(E^(3x))': 'sin(E**(3*x))**2',
'sin^2 (3x*E^(x))': 'sin(3*x*E^x)**2',
'pi sin x': 'pi*sin(x)',
}
transformations = standard_transformations + (convert_xor,)
transformations2 = transformations + (implicit_multiplication_application,)
for case in cases:
implicit = parse_expr(case, transformations=transformations2)
normal = parse_expr(cases[case], transformations=transformations)
assert(implicit == normal)
def test_no_methods_implicit_multiplication():
# Issue 21020
u = sympy.Symbol('u')
transformations = standard_transformations + \
(implicit_multiplication,)
expr = parse_expr('x.is_polynomial(x)', transformations=transformations)
assert expr == True
expr = parse_expr('(exp(x) / (1 + exp(2x))).subs(exp(x), u)',
transformations=transformations)
assert expr == u/(u**2 + 1)
|
5ec745f0191c5337aca7c123e2b1802075a3132ad9f91bb3f3f6f60b1b90caef | # -*- coding: utf-8 -*-
import sys
from sympy.assumptions import Q
from sympy.core import Symbol, Function, Float, Rational, Integer, I, Mul, Pow, Eq
from sympy.functions import exp, factorial, factorial2, sin
from sympy.logic import And
from sympy.series import Limit
from sympy.testing.pytest import raises, skip
from sympy.parsing.sympy_parser import (
parse_expr, standard_transformations, rationalize, TokenError,
split_symbols, implicit_multiplication, convert_equals_signs,
convert_xor, function_exponentiation,
implicit_multiplication_application,
)
def test_sympy_parser():
x = Symbol('x')
inputs = {
'2*x': 2 * x,
'3.00': Float(3),
'22/7': Rational(22, 7),
'2+3j': 2 + 3*I,
'exp(x)': exp(x),
'x!': factorial(x),
'x!!': factorial2(x),
'(x + 1)! - 1': factorial(x + 1) - 1,
'3.[3]': Rational(10, 3),
'.0[3]': Rational(1, 30),
'3.2[3]': Rational(97, 30),
'1.3[12]': Rational(433, 330),
'1 + 3.[3]': Rational(13, 3),
'1 + .0[3]': Rational(31, 30),
'1 + 3.2[3]': Rational(127, 30),
'.[0011]': Rational(1, 909),
'0.1[00102] + 1': Rational(366697, 333330),
'1.[0191]': Rational(10190, 9999),
'10!': 3628800,
'-(2)': -Integer(2),
'[-1, -2, 3]': [Integer(-1), Integer(-2), Integer(3)],
'Symbol("x").free_symbols': x.free_symbols,
"S('S(3).n(n=3)')": 3.00,
'factorint(12, visual=True)': Mul(
Pow(2, 2, evaluate=False),
Pow(3, 1, evaluate=False),
evaluate=False),
'Limit(sin(x), x, 0, dir="-")': Limit(sin(x), x, 0, dir='-'),
'Q.even(x)': Q.even(x),
}
for text, result in inputs.items():
assert parse_expr(text) == result
raises(TypeError, lambda:
parse_expr('x', standard_transformations))
raises(TypeError, lambda:
parse_expr('x', transformations=lambda x,y: 1))
raises(TypeError, lambda:
parse_expr('x', transformations=(lambda x,y: 1,)))
raises(TypeError, lambda: parse_expr('x', transformations=((),)))
raises(TypeError, lambda: parse_expr('x', {}, [], []))
raises(TypeError, lambda: parse_expr('x', [], [], {}))
raises(TypeError, lambda: parse_expr('x', [], [], {}))
def test_rationalize():
inputs = {
'0.123': Rational(123, 1000)
}
transformations = standard_transformations + (rationalize,)
for text, result in inputs.items():
assert parse_expr(text, transformations=transformations) == result
def test_factorial_fail():
inputs = ['x!!!', 'x!!!!', '(!)']
for text in inputs:
try:
parse_expr(text)
assert False
except TokenError:
assert True
def test_repeated_fail():
inputs = ['1[1]', '.1e1[1]', '0x1[1]', '1.1j[1]', '1.1[1 + 1]',
'0.1[[1]]', '0x1.1[1]']
# All are valid Python, so only raise TypeError for invalid indexing
for text in inputs:
raises(TypeError, lambda: parse_expr(text))
inputs = ['0.1[', '0.1[1', '0.1[]']
for text in inputs:
raises((TokenError, SyntaxError), lambda: parse_expr(text))
def test_repeated_dot_only():
assert parse_expr('.[1]') == Rational(1, 9)
assert parse_expr('1 + .[1]') == Rational(10, 9)
def test_local_dict():
local_dict = {
'my_function': lambda x: x + 2
}
inputs = {
'my_function(2)': Integer(4)
}
for text, result in inputs.items():
assert parse_expr(text, local_dict=local_dict) == result
def test_local_dict_split_implmult():
t = standard_transformations + (split_symbols, implicit_multiplication,)
w = Symbol('w', real=True)
y = Symbol('y')
assert parse_expr('yx', local_dict={'x':w}, transformations=t) == y*w
def test_local_dict_symbol_to_fcn():
x = Symbol('x')
d = {'foo': Function('bar')}
assert parse_expr('foo(x)', local_dict=d) == d['foo'](x)
# XXX: bit odd, but would be error if parser left the Symbol
d = {'foo': Symbol('baz')}
assert parse_expr('foo(x)', local_dict=d) == Function('baz')(x)
def test_global_dict():
global_dict = {
'Symbol': Symbol
}
inputs = {
'Q & S': And(Symbol('Q'), Symbol('S'))
}
for text, result in inputs.items():
assert parse_expr(text, global_dict=global_dict) == result
def test_issue_2515():
raises(TokenError, lambda: parse_expr('(()'))
raises(TokenError, lambda: parse_expr('"""'))
def test_issue_7663():
x = Symbol('x')
e = '2*(x+1)'
assert parse_expr(e, evaluate=0) == parse_expr(e, evaluate=False)
assert parse_expr(e, evaluate=0).equals(2*(x+1))
def test_issue_10560():
inputs = {
'4*-3' : '(-3)*4',
'-4*3' : '(-4)*3',
}
for text, result in inputs.items():
assert parse_expr(text, evaluate=False) == parse_expr(result, evaluate=False)
def test_issue_10773():
inputs = {
'-10/5': '(-10)/5',
'-10/-5' : '(-10)/(-5)',
}
for text, result in inputs.items():
assert parse_expr(text, evaluate=False) == parse_expr(result, evaluate=False)
def test_split_symbols():
transformations = standard_transformations + \
(split_symbols, implicit_multiplication,)
x = Symbol('x')
y = Symbol('y')
xy = Symbol('xy')
assert parse_expr("xy") == xy
assert parse_expr("xy", transformations=transformations) == x*y
def test_split_symbols_function():
transformations = standard_transformations + \
(split_symbols, implicit_multiplication,)
x = Symbol('x')
y = Symbol('y')
a = Symbol('a')
f = Function('f')
assert parse_expr("ay(x+1)", transformations=transformations) == a*y*(x+1)
assert parse_expr("af(x+1)", transformations=transformations,
local_dict={'f':f}) == a*f(x+1)
def test_functional_exponent():
t = standard_transformations + (convert_xor, function_exponentiation)
x = Symbol('x')
y = Symbol('y')
a = Symbol('a')
yfcn = Function('y')
assert parse_expr("sin^2(x)", transformations=t) == (sin(x))**2
assert parse_expr("sin^y(x)", transformations=t) == (sin(x))**y
assert parse_expr("exp^y(x)", transformations=t) == (exp(x))**y
assert parse_expr("E^y(x)", transformations=t) == exp(yfcn(x))
assert parse_expr("a^y(x)", transformations=t) == a**(yfcn(x))
def test_match_parentheses_implicit_multiplication():
transformations = standard_transformations + \
(implicit_multiplication,)
raises(TokenError, lambda: parse_expr('(1,2),(3,4]',transformations=transformations))
def test_convert_equals_signs():
transformations = standard_transformations + \
(convert_equals_signs, )
x = Symbol('x')
y = Symbol('y')
assert parse_expr("1*2=x", transformations=transformations) == Eq(2, x)
assert parse_expr("y = x", transformations=transformations) == Eq(y, x)
assert parse_expr("(2*y = x) = False",
transformations=transformations) == Eq(Eq(2*y, x), False)
def test_parse_function_issue_3539():
x = Symbol('x')
f = Function('f')
assert parse_expr('f(x)') == f(x)
def test_split_symbols_numeric():
transformations = (
standard_transformations +
(implicit_multiplication_application,))
n = Symbol('n')
expr1 = parse_expr('2**n * 3**n')
expr2 = parse_expr('2**n3**n', transformations=transformations)
assert expr1 == expr2 == 2**n*3**n
expr1 = parse_expr('n12n34', transformations=transformations)
assert expr1 == n*12*n*34
def test_unicode_names():
assert parse_expr('α') == Symbol('α')
def test_python3_features():
# Make sure the tokenizer can handle Python 3-only features
if sys.version_info < (3, 6):
skip("test_python3_features requires Python 3.6 or newer")
assert parse_expr("123_456") == 123456
assert parse_expr("1.2[3_4]") == parse_expr("1.2[34]") == Rational(611, 495)
assert parse_expr("1.2[012_012]") == parse_expr("1.2[012012]") == Rational(400, 333)
assert parse_expr('.[3_4]') == parse_expr('.[34]') == Rational(34, 99)
assert parse_expr('.1[3_4]') == parse_expr('.1[34]') == Rational(133, 990)
assert parse_expr('123_123.123_123[3_4]') == parse_expr('123123.123123[34]') == Rational(12189189189211, 99000000)
|
b41f15d579b2669bac0aef45a8b71ab19d520eb14c3c6501f6b3dd7934e6e5eb | from sympy.testing.pytest import raises, XFAIL
from sympy.external import import_module
from sympy import (
Symbol, Mul, Add, Abs, sin, asin, cos, Pow, csc, sec,
Limit, oo, Derivative, Integral, factorial, sqrt, root,
conjugate, StrictLessThan, LessThan, StrictGreaterThan,
GreaterThan, Sum, Product, E, log, tan, Function, binomial,
exp, floor, ceiling, Unequality
)
from sympy.core.relational import Eq, Ne, Lt, Le, Gt, Ge
from sympy.physics.quantum.state import Bra, Ket
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 _Sqrt(a):
return sqrt(a, evaluate=False)
def _Conjugate(a):
return conjugate(a, 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 = [
(r"0", 0),
(r"1", 1),
(r"-3.14", _Mul(-1, 3.14)),
(r"(-7.13)(1.5)", _Mul(_Mul(-1, 7.13), 1.5)),
(r"x", x),
(r"2x", 2*x),
(r"x^2", x**2),
(r"x^{3 + 1}", x**_Add(3, 1)),
(r"-c", -c),
(r"a \cdot b", a * b),
(r"a / b", a / b),
(r"a \div b", a / b),
(r"a + b", a + b),
(r"a + b - a", _Add(a+b, -a)),
(r"a^2 + b^2 = c^2", Eq(a**2 + b**2, c**2)),
(r"(x + y) z", _Mul(_Add(x, y), z)),
(r"\left(x + y\right) z", _Mul(_Add(x, y), z)),
(r"\left( x + y\right ) z", _Mul(_Add(x, y), z)),
(r"\left( x + y\right ) z", _Mul(_Add(x, y), z)),
(r"\left[x + y\right] z", _Mul(_Add(x, y), z)),
(r"\left\{x + y\right\} z", _Mul(_Add(x, y), z)),
(r"1+1", _Add(1, 1)),
(r"0+1", _Add(0, 1)),
(r"1*2", _Mul(1, 2)),
(r"0*1", _Mul(0, 1)),
(r"x = y", Eq(x, y)),
(r"x \neq y", Ne(x, y)),
(r"x < y", Lt(x, y)),
(r"x > y", Gt(x, y)),
(r"x \leq y", Le(x, y)),
(r"x \geq y", Ge(x, y)),
(r"x \le y", Le(x, y)),
(r"x \ge y", Ge(x, y)),
(r"\lfloor x \rfloor", floor(x)),
(r"\lceil x \rceil", ceiling(x)),
(r"\langle x |", Bra('x')),
(r"| x \rangle", Ket('x')),
(r"\sin \theta", sin(theta)),
(r"\sin(\theta)", sin(theta)),
(r"\sin^{-1} a", asin(a)),
(r"\sin a \cos b", _Mul(sin(a), cos(b))),
(r"\sin \cos \theta", sin(cos(theta))),
(r"\sin(\cos \theta)", sin(cos(theta))),
(r"\frac{a}{b}", a / b),
(r"\frac{a + b}{c}", _Mul(a + b, _Pow(c, -1))),
(r"\frac{7}{3}", _Mul(7, _Pow(3, -1))),
(r"(\csc x)(\sec y)", csc(x)*sec(y)),
(r"\lim_{x \to 3} a", Limit(a, x, 3)),
(r"\lim_{x \rightarrow 3} a", Limit(a, x, 3)),
(r"\lim_{x \Rightarrow 3} a", Limit(a, x, 3)),
(r"\lim_{x \longrightarrow 3} a", Limit(a, x, 3)),
(r"\lim_{x \Longrightarrow 3} a", Limit(a, x, 3)),
(r"\lim_{x \to 3^{+}} a", Limit(a, x, 3, dir='+')),
(r"\lim_{x \to 3^{-}} a", Limit(a, x, 3, dir='-')),
(r"\infty", oo),
(r"\lim_{x \to \infty} \frac{1}{x}", Limit(_Pow(x, -1), x, oo)),
(r"\frac{d}{dx} x", Derivative(x, x)),
(r"\frac{d}{dt} x", Derivative(x, t)),
(r"f(x)", f(x)),
(r"f(x, y)", f(x, y)),
(r"f(x, y, z)", f(x, y, z)),
(r"\frac{d f(x)}{dx}", Derivative(f(x), x)),
(r"\frac{d\theta(x)}{dx}", Derivative(Function('theta')(x), x)),
(r"x \neq y", Unequality(x, y)),
(r"|x|", _Abs(x)),
(r"||x||", _Abs(Abs(x))),
(r"|x||y|", _Abs(x)*_Abs(y)),
(r"||x||y||", _Abs(_Abs(x)*_Abs(y))),
(r"\pi^{|xy|}", Symbol('pi')**_Abs(x*y)),
(r"\int x dx", Integral(x, x)),
(r"\int x d\theta", Integral(x, theta)),
(r"\int (x^2 - y)dx", Integral(x**2 - y, x)),
(r"\int x + a dx", Integral(_Add(x, a), x)),
(r"\int da", Integral(1, a)),
(r"\int_0^7 dx", Integral(1, (x, 0, 7))),
(r"\int_a^b x dx", Integral(x, (x, a, b))),
(r"\int^b_a x dx", Integral(x, (x, a, b))),
(r"\int_{a}^b x dx", Integral(x, (x, a, b))),
(r"\int^{b}_a x dx", Integral(x, (x, a, b))),
(r"\int_{a}^{b} x dx", Integral(x, (x, a, b))),
(r"\int^{b}_{a} x dx", Integral(x, (x, a, b))),
(r"\int_{f(a)}^{f(b)} f(z) dz", Integral(f(z), (z, f(a), f(b)))),
(r"\int (x+a)", Integral(_Add(x, a), x)),
(r"\int a + b + c dx", Integral(_Add(_Add(a, b), c), x)),
(r"\int \frac{dz}{z}", Integral(Pow(z, -1), z)),
(r"\int \frac{3 dz}{z}", Integral(3*Pow(z, -1), z)),
(r"\int \frac{1}{x} dx", Integral(Pow(x, -1), x)),
(r"\int \frac{1}{a} + \frac{1}{b} dx",
Integral(_Add(_Pow(a, -1), Pow(b, -1)), x)),
(r"\int \frac{3 \cdot d\theta}{\theta}",
Integral(3*_Pow(theta, -1), theta)),
(r"\int \frac{1}{x} + 1 dx", Integral(_Add(_Pow(x, -1), 1), x)),
(r"x_0", Symbol('x_{0}')),
(r"x_{1}", Symbol('x_{1}')),
(r"x_a", Symbol('x_{a}')),
(r"x_{b}", Symbol('x_{b}')),
(r"h_\theta", Symbol('h_{theta}')),
(r"h_{\theta}", Symbol('h_{theta}')),
(r"h_{\theta}(x_0, x_1)",
Function('h_{theta}')(Symbol('x_{0}'), Symbol('x_{1}'))),
(r"x!", _factorial(x)),
(r"100!", _factorial(100)),
(r"\theta!", _factorial(theta)),
(r"(x + 1)!", _factorial(_Add(x, 1))),
(r"(x!)!", _factorial(_factorial(x))),
(r"x!!!", _factorial(_factorial(_factorial(x)))),
(r"5!7!", _Mul(_factorial(5), _factorial(7))),
(r"\sqrt{x}", sqrt(x)),
(r"\sqrt{x + b}", sqrt(_Add(x, b))),
(r"\sqrt[3]{\sin x}", root(sin(x), 3)),
(r"\sqrt[y]{\sin x}", root(sin(x), y)),
(r"\sqrt[\theta]{\sin x}", root(sin(x), theta)),
(r"\sqrt{\frac{12}{6}}", _Sqrt(_Mul(12, _Pow(6, -1)))),
(r"\overline{z}", _Conjugate(z)),
(r"\overline{\overline{z}}", _Conjugate(_Conjugate(z))),
(r"\overline{x + y}", _Conjugate(_Add(x, y))),
(r"\overline{x} + \overline{y}", _Conjugate(x) + _Conjugate(y)),
(r"x < y", StrictLessThan(x, y)),
(r"x \leq y", LessThan(x, y)),
(r"x > y", StrictGreaterThan(x, y)),
(r"x \geq y", GreaterThan(x, y)),
(r"\mathit{x}", Symbol('x')),
(r"\mathit{test}", Symbol('test')),
(r"\mathit{TEST}", Symbol('TEST')),
(r"\mathit{HELLO world}", Symbol('HELLO world')),
(r"\sum_{k = 1}^{3} c", Sum(c, (k, 1, 3))),
(r"\sum_{k = 1}^3 c", Sum(c, (k, 1, 3))),
(r"\sum^{3}_{k = 1} c", Sum(c, (k, 1, 3))),
(r"\sum^3_{k = 1} c", Sum(c, (k, 1, 3))),
(r"\sum_{k = 1}^{10} k^2", Sum(k**2, (k, 1, 10))),
(r"\sum_{n = 0}^{\infty} \frac{1}{n!}",
Sum(_Pow(_factorial(n), -1), (n, 0, oo))),
(r"\prod_{a = b}^{c} x", Product(x, (a, b, c))),
(r"\prod_{a = b}^c x", Product(x, (a, b, c))),
(r"\prod^{c}_{a = b} x", Product(x, (a, b, c))),
(r"\prod^c_{a = b} x", Product(x, (a, b, c))),
(r"\exp x", _exp(x)),
(r"\exp(x)", _exp(x)),
(r"\ln x", _log(x, E)),
(r"\ln xy", _log(x*y, E)),
(r"\log x", _log(x, 10)),
(r"\log xy", _log(x*y, 10)),
(r"\log_{2} x", _log(x, 2)),
(r"\log_{a} x", _log(x, a)),
(r"\log_{11} x", _log(x, 11)),
(r"\log_{a^2} x", _log(x, _Pow(a, 2))),
(r"[x]", x),
(r"[a + b]", _Add(a, b)),
(r"\frac{d}{dx} [ \tan x ]", Derivative(tan(x), x)),
(r"\binom{n}{k}", _binomial(n, k)),
(r"\tbinom{n}{k}", _binomial(n, k)),
(r"\dbinom{n}{k}", _binomial(n, k)),
(r"\binom{n}{0}", _binomial(n, 0)),
(r"a \, b", _Mul(a, b)),
(r"a \thinspace b", _Mul(a, b)),
(r"a \: b", _Mul(a, b)),
(r"a \medspace b", _Mul(a, b)),
(r"a \; b", _Mul(a, b)),
(r"a \thickspace b", _Mul(a, b)),
(r"a \quad b", _Mul(a, b)),
(r"a \qquad b", _Mul(a, b)),
(r"a \! b", _Mul(a, b)),
(r"a \negthinspace b", _Mul(a, b)),
(r"a \negmedspace b", _Mul(a, b)),
(r"a \negthickspace b", _Mul(a, b)),
(r"\int x \, dx", Integral(x, x)),
(r"\log_2 x", _log(x, 2)),
(r"\log_a x", _log(x, a)),
]
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
# These bad LaTeX strings should raise a LaTeXParsingError when parsed
BAD_STRINGS = [
r"(",
r")",
r"\frac{d}{dx}",
r"(\frac{d}{dx})",
r"\sqrt{}",
r"\sqrt",
r"\overline{}",
r"\overline",
r"{",
r"}",
r"\mathit{x + y}",
r"\mathit{21}",
r"\frac{2}{}",
r"\frac{}{2}",
r"\int",
r"!",
r"!0",
r"_",
r"^",
r"|",
r"||x|",
r"()",
r"((((((((((((((((()))))))))))))))))",
r"-",
r"\frac{d}{dx} + \frac{d}{dt}",
r"f(x,,y)",
r"f(x,y,",
r"\sin^x",
r"\cos^2",
r"@",
r"#",
r"$",
r"%",
r"&",
r"*",
r"" "\\",
r"~",
r"\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 = [
r"\cos 1 \cos",
r"f(,",
r"f()",
r"a \div \div b",
r"a \cdot \cdot b",
r"a // b",
r"a +",
r"1.1.1",
r"1 +",
r"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)
|
3f95d3a365239320f2c7877d929996226e81a025f23a2103f6cf13af7c37026e | import collections
import warnings
from sympy.external import import_module
autolevparser = import_module('sympy.parsing.autolev._antlr.autolevparser',
import_kwargs={'fromlist': ['AutolevParser']})
autolevlexer = import_module('sympy.parsing.autolev._antlr.autolevlexer',
import_kwargs={'fromlist': ['AutolevLexer']})
autolevlistener = import_module('sympy.parsing.autolev._antlr.autolevlistener',
import_kwargs={'fromlist': ['AutolevListener']})
AutolevParser = getattr(autolevparser, 'AutolevParser', None)
AutolevLexer = getattr(autolevlexer, 'AutolevLexer', None)
AutolevListener = getattr(autolevlistener, 'AutolevListener', None)
def strfunc(z):
if z == 0:
return ""
elif z == 1:
return "_d"
else:
return "_" + "d" * z
def declare_phy_entities(self, ctx, phy_type, i, j=None):
if phy_type in ("frame", "newtonian"):
declare_frames(self, ctx, i, j)
elif phy_type == "particle":
declare_particles(self, ctx, i, j)
elif phy_type == "point":
declare_points(self, ctx, i, j)
elif phy_type == "bodies":
declare_bodies(self, ctx, i, j)
def declare_frames(self, ctx, i, j=None):
if "{" in ctx.getText():
if j:
name1 = ctx.ID().getText().lower() + str(i) + str(j)
else:
name1 = ctx.ID().getText().lower() + str(i)
else:
name1 = ctx.ID().getText().lower()
name2 = "frame_" + name1
if self.getValue(ctx.parentCtx.varType()) == "newtonian":
self.newtonian = name2
self.symbol_table2.update({name1: name2})
self.symbol_table.update({name1 + "1>": name2 + ".x"})
self.symbol_table.update({name1 + "2>": name2 + ".y"})
self.symbol_table.update({name1 + "3>": name2 + ".z"})
self.type2.update({name1: "frame"})
self.write(name2 + " = " + "_me.ReferenceFrame('" + name1 + "')\n")
def declare_points(self, ctx, i, j=None):
if "{" in ctx.getText():
if j:
name1 = ctx.ID().getText().lower() + str(i) + str(j)
else:
name1 = ctx.ID().getText().lower() + str(i)
else:
name1 = ctx.ID().getText().lower()
name2 = "point_" + name1
self.symbol_table2.update({name1: name2})
self.type2.update({name1: "point"})
self.write(name2 + " = " + "_me.Point('" + name1 + "')\n")
def declare_particles(self, ctx, i, j=None):
if "{" in ctx.getText():
if j:
name1 = ctx.ID().getText().lower() + str(i) + str(j)
else:
name1 = ctx.ID().getText().lower() + str(i)
else:
name1 = ctx.ID().getText().lower()
name2 = "particle_" + name1
self.symbol_table2.update({name1: name2})
self.type2.update({name1: "particle"})
self.bodies.update({name1: name2})
self.write(name2 + " = " + "_me.Particle('" + name1 + "', " + "_me.Point('" +
name1 + "_pt" + "'), " + "_sm.Symbol('m'))\n")
def declare_bodies(self, ctx, i, j=None):
if "{" in ctx.getText():
if j:
name1 = ctx.ID().getText().lower() + str(i) + str(j)
else:
name1 = ctx.ID().getText().lower() + str(i)
else:
name1 = ctx.ID().getText().lower()
name2 = "body_" + name1
self.bodies.update({name1: name2})
masscenter = name2 + "_cm"
refFrame = name2 + "_f"
self.symbol_table2.update({name1: name2})
self.symbol_table2.update({name1 + "o": masscenter})
self.symbol_table.update({name1 + "1>": refFrame+".x"})
self.symbol_table.update({name1 + "2>": refFrame+".y"})
self.symbol_table.update({name1 + "3>": refFrame+".z"})
self.type2.update({name1: "bodies"})
self.type2.update({name1+"o": "point"})
self.write(masscenter + " = " + "_me.Point('" + name1 + "_cm" + "')\n")
if self.newtonian:
self.write(masscenter + ".set_vel(" + self.newtonian + ", " + "0)\n")
self.write(refFrame + " = " + "_me.ReferenceFrame('" + name1 + "_f" + "')\n")
# We set a dummy mass and inertia here.
# They will be reset using the setters later in the code anyway.
self.write(name2 + " = " + "_me.RigidBody('" + name1 + "', " + masscenter + ", " +
refFrame + ", " + "_sm.symbols('m'), (_me.outer(" + refFrame +
".x," + refFrame + ".x)," + masscenter + "))\n")
def inertia_func(self, v1, v2, l, frame):
if self.type2[v1] == "particle":
l.append("_me.inertia_of_point_mass(" + self.bodies[v1] + ".mass, " + self.bodies[v1] +
".point.pos_from(" + self.symbol_table2[v2] + "), " + frame + ")")
elif self.type2[v1] == "bodies":
# Inertia has been defined about center of mass.
if self.inertia_point[v1] == v1 + "o":
# Asking point is cm as well
if v2 == self.inertia_point[v1]:
l.append(self.symbol_table2[v1] + ".inertia[0]")
# Asking point is not cm
else:
l.append(self.bodies[v1] + ".inertia[0]" + " + " +
"_me.inertia_of_point_mass(" + self.bodies[v1] +
".mass, " + self.bodies[v1] + ".masscenter" +
".pos_from(" + self.symbol_table2[v2] +
"), " + frame + ")")
# Inertia has been defined about another point
else:
# Asking point is the defined point
if v2 == self.inertia_point[v1]:
l.append(self.symbol_table2[v1] + ".inertia[0]")
# Asking point is cm
elif v2 == v1 + "o":
l.append(self.bodies[v1] + ".inertia[0]" + " - " +
"_me.inertia_of_point_mass(" + self.bodies[v1] +
".mass, " + self.bodies[v1] + ".masscenter" +
".pos_from(" + self.symbol_table2[self.inertia_point[v1]] +
"), " + frame + ")")
# Asking point is some other point
else:
l.append(self.bodies[v1] + ".inertia[0]" + " - " +
"_me.inertia_of_point_mass(" + self.bodies[v1] +
".mass, " + self.bodies[v1] + ".masscenter" +
".pos_from(" + self.symbol_table2[self.inertia_point[v1]] +
"), " + frame + ")" + " + " +
"_me.inertia_of_point_mass(" + self.bodies[v1] +
".mass, " + self.bodies[v1] + ".masscenter" +
".pos_from(" + self.symbol_table2[v2] +
"), " + frame + ")")
def processConstants(self, ctx):
# Process constant declarations of the type: Constants F = 3, g = 9.81
name = ctx.ID().getText().lower()
if "=" in ctx.getText():
self.symbol_table.update({name: name})
# self.inputs.update({self.symbol_table[name]: self.getValue(ctx.getChild(2))})
self.write(self.symbol_table[name] + " = " + "_sm.S(" + self.getValue(ctx.getChild(2)) + ")\n")
self.type.update({name: "constants"})
return
# Constants declarations of the type: Constants A, B
else:
if "{" not in ctx.getText():
self.symbol_table[name] = name
self.type[name] = "constants"
# Process constant declarations of the type: Constants C+, D-
if ctx.getChildCount() == 2:
# This is set for declaring nonpositive=True and nonnegative=True
if ctx.getChild(1).getText() == "+":
self.sign[name] = "+"
elif ctx.getChild(1).getText() == "-":
self.sign[name] = "-"
else:
if "{" not in ctx.getText():
self.sign[name] = "o"
# Process constant declarations of the type: Constants K{4}, a{1:2, 1:2}, b{1:2}
if "{" in ctx.getText():
if ":" in ctx.getText():
num1 = int(ctx.INT(0).getText())
num2 = int(ctx.INT(1).getText()) + 1
else:
num1 = 1
num2 = int(ctx.INT(0).getText()) + 1
if ":" in ctx.getText():
if "," in ctx.getText():
num3 = int(ctx.INT(2).getText())
num4 = int(ctx.INT(3).getText()) + 1
for i in range(num1, num2):
for j in range(num3, num4):
self.symbol_table[name + str(i) + str(j)] = name + str(i) + str(j)
self.type[name + str(i) + str(j)] = "constants"
self.var_list.append(name + str(i) + str(j))
self.sign[name + str(i) + str(j)] = "o"
else:
for i in range(num1, num2):
self.symbol_table[name + str(i)] = name + str(i)
self.type[name + str(i)] = "constants"
self.var_list.append(name + str(i))
self.sign[name + str(i)] = "o"
elif "," in ctx.getText():
for i in range(1, int(ctx.INT(0).getText()) + 1):
for j in range(1, int(ctx.INT(1).getText()) + 1):
self.symbol_table[name] = name + str(i) + str(j)
self.type[name + str(i) + str(j)] = "constants"
self.var_list.append(name + str(i) + str(j))
self.sign[name + str(i) + str(j)] = "o"
else:
for i in range(num1, num2):
self.symbol_table[name + str(i)] = name + str(i)
self.type[name + str(i)] = "constants"
self.var_list.append(name + str(i))
self.sign[name + str(i)] = "o"
if "{" not in ctx.getText():
self.var_list.append(name)
def writeConstants(self, ctx):
l1 = list(filter(lambda x: self.sign[x] == "o", self.var_list))
l2 = list(filter(lambda x: self.sign[x] == "+", self.var_list))
l3 = list(filter(lambda x: self.sign[x] == "-", self.var_list))
try:
if self.settings["complex"] == "on":
real = ", real=True"
elif self.settings["complex"] == "off":
real = ""
except Exception:
real = ", real=True"
if l1:
a = ", ".join(l1) + " = " + "_sm.symbols(" + "'" +\
" ".join(l1) + "'" + real + ")\n"
self.write(a)
if l2:
a = ", ".join(l2) + " = " + "_sm.symbols(" + "'" +\
" ".join(l2) + "'" + real + ", nonnegative=True)\n"
self.write(a)
if l3:
a = ", ".join(l3) + " = " + "_sm.symbols(" + "'" + \
" ".join(l3) + "'" + real + ", nonpositive=True)\n"
self.write(a)
self.var_list = []
def processVariables(self, ctx):
# Specified F = x*N1> + y*N2>
name = ctx.ID().getText().lower()
if "=" in ctx.getText():
text = name + "'"*(ctx.getChildCount()-3)
self.write(text + " = " + self.getValue(ctx.expr()) + "\n")
return
# Process variables of the type: Variables qA, qB
if ctx.getChildCount() == 1:
self.symbol_table[name] = name
if self.getValue(ctx.parentCtx.getChild(0)) in ("variable", "specified", "motionvariable", "motionvariable'"):
self.type.update({name: self.getValue(ctx.parentCtx.getChild(0))})
self.var_list.append(name)
self.sign[name] = 0
# Process variables of the type: Variables x', y''
elif "'" in ctx.getText() and "{" not in ctx.getText():
if ctx.getText().count("'") > self.maxDegree:
self.maxDegree = ctx.getText().count("'")
for i in range(ctx.getChildCount()):
self.sign[name + strfunc(i)] = i
self.symbol_table[name + "'"*i] = name + strfunc(i)
if self.getValue(ctx.parentCtx.getChild(0)) in ("variable", "specified", "motionvariable", "motionvariable'"):
self.type.update({name + "'"*i: self.getValue(ctx.parentCtx.getChild(0))})
self.var_list.append(name + strfunc(i))
elif "{" in ctx.getText():
# Process variables of the type: Variales x{3}, y{2}
if "'" in ctx.getText():
dash_count = ctx.getText().count("'")
if dash_count > self.maxDegree:
self.maxDegree = dash_count
if ":" in ctx.getText():
# Variables C{1:2, 1:2}
if "," in ctx.getText():
num1 = int(ctx.INT(0).getText())
num2 = int(ctx.INT(1).getText()) + 1
num3 = int(ctx.INT(2).getText())
num4 = int(ctx.INT(3).getText()) + 1
# Variables C{1:2}
else:
num1 = int(ctx.INT(0).getText())
num2 = int(ctx.INT(1).getText()) + 1
# Variables C{1,3}
elif "," in ctx.getText():
num1 = 1
num2 = int(ctx.INT(0).getText()) + 1
num3 = 1
num4 = int(ctx.INT(1).getText()) + 1
else:
num1 = 1
num2 = int(ctx.INT(0).getText()) + 1
for i in range(num1, num2):
try:
for j in range(num3, num4):
try:
for z in range(dash_count+1):
self.symbol_table.update({name + str(i) + str(j) + "'"*z: name + str(i) + str(j) + strfunc(z)})
if self.getValue(ctx.parentCtx.getChild(0)) in ("variable", "specified", "motionvariable", "motionvariable'"):
self.type.update({name + str(i) + str(j) + "'"*z: self.getValue(ctx.parentCtx.getChild(0))})
self.var_list.append(name + str(i) + str(j) + strfunc(z))
self.sign.update({name + str(i) + str(j) + strfunc(z): z})
if dash_count > self.maxDegree:
self.maxDegree = dash_count
except Exception:
self.symbol_table.update({name + str(i) + str(j): name + str(i) + str(j)})
if self.getValue(ctx.parentCtx.getChild(0)) in ("variable", "specified", "motionvariable", "motionvariable'"):
self.type.update({name + str(i) + str(j): self.getValue(ctx.parentCtx.getChild(0))})
self.var_list.append(name + str(i) + str(j))
self.sign.update({name + str(i) + str(j): 0})
except Exception:
try:
for z in range(dash_count+1):
self.symbol_table.update({name + str(i) + "'"*z: name + str(i) + strfunc(z)})
if self.getValue(ctx.parentCtx.getChild(0)) in ("variable", "specified", "motionvariable", "motionvariable'"):
self.type.update({name + str(i) + "'"*z: self.getValue(ctx.parentCtx.getChild(0))})
self.var_list.append(name + str(i) + strfunc(z))
self.sign.update({name + str(i) + strfunc(z): z})
if dash_count > self.maxDegree:
self.maxDegree = dash_count
except Exception:
self.symbol_table.update({name + str(i): name + str(i)})
if self.getValue(ctx.parentCtx.getChild(0)) in ("variable", "specified", "motionvariable", "motionvariable'"):
self.type.update({name + str(i): self.getValue(ctx.parentCtx.getChild(0))})
self.var_list.append(name + str(i))
self.sign.update({name + str(i): 0})
def writeVariables(self, ctx):
#print(self.sign)
#print(self.symbol_table)
if self.var_list:
for i in range(self.maxDegree+1):
if i == 0:
j = ""
t = ""
else:
j = str(i)
t = ", "
l = []
for k in list(filter(lambda x: self.sign[x] == i, self.var_list)):
if i == 0:
l.append(k)
if i == 1:
l.append(k[:-1])
if i > 1:
l.append(k[:-2])
a = ", ".join(list(filter(lambda x: self.sign[x] == i, self.var_list))) + " = " +\
"_me.dynamicsymbols(" + "'" + " ".join(l) + "'" + t + j + ")\n"
l = []
self.write(a)
self.maxDegree = 0
self.var_list = []
def processImaginary(self, ctx):
name = ctx.ID().getText().lower()
self.symbol_table[name] = name
self.type[name] = "imaginary"
self.var_list.append(name)
def writeImaginary(self, ctx):
a = ", ".join(self.var_list) + " = " + "_sm.symbols(" + "'" + \
" ".join(self.var_list) + "')\n"
b = ", ".join(self.var_list) + " = " + "_sm.I\n"
self.write(a)
self.write(b)
self.var_list = []
if AutolevListener:
class MyListener(AutolevListener): # type: ignore
def __init__(self, include_numeric=False):
# Stores data in tree nodes(tree annotation). Especially useful for expr reconstruction.
self.tree_property = {}
# Stores the declared variables, constants etc as they are declared in Autolev and SymPy
# {"<Autolev symbol>": "<SymPy symbol>"}.
self.symbol_table = collections.OrderedDict()
# Similar to symbol_table. Used for storing Physical entities like Frames, Points,
# Particles, Bodies etc
self.symbol_table2 = collections.OrderedDict()
# Used to store nonpositive, nonnegative etc for constants and number of "'"s (order of diff)
# in variables.
self.sign = {}
# Simple list used as a store to pass around variables between the 'process' and 'write'
# methods.
self.var_list = []
# Stores the type of a declared variable (constants, variables, specifieds etc)
self.type = collections.OrderedDict()
# Similar to self.type. Used for storing the type of Physical entities like Frames, Points,
# Particles, Bodies etc
self.type2 = collections.OrderedDict()
# These lists are used to distinguish matrix, numeric and vector expressions.
self.matrix_expr = []
self.numeric_expr = []
self.vector_expr = []
self.fr_expr = []
self.output_code = []
# Stores the variables and their rhs for substituting upon the Autolev command EXPLICIT.
self.explicit = collections.OrderedDict()
# Write code to import common dependencies.
self.output_code.append("import sympy.physics.mechanics as _me\n")
self.output_code.append("import sympy as _sm\n")
self.output_code.append("import math as m\n")
self.output_code.append("import numpy as _np\n")
self.output_code.append("\n")
# Just a store for the max degree variable in a line.
self.maxDegree = 0
# Stores the input parameters which are then used for codegen and numerical analysis.
self.inputs = collections.OrderedDict()
# Stores the variables which appear in Output Autolev commands.
self.outputs = []
# Stores the settings specified by the user. Ex: Complex on/off, Degrees on/off
self.settings = {}
# Boolean which changes the behaviour of some expression reconstruction
# when parsing Input Autolev commands.
self.in_inputs = False
self.in_outputs = False
# Stores for the physical entities.
self.newtonian = None
self.bodies = collections.OrderedDict()
self.constants = []
self.forces = collections.OrderedDict()
self.q_ind = []
self.q_dep = []
self.u_ind = []
self.u_dep = []
self.kd_eqs = []
self.dependent_variables = []
self.kd_equivalents = collections.OrderedDict()
self.kd_equivalents2 = collections.OrderedDict()
self.kd_eqs_supplied = None
self.kane_type = "no_args"
self.inertia_point = collections.OrderedDict()
self.kane_parsed = False
self.t = False
# PyDy ode code will be included only if this flag is set to True.
self.include_numeric = include_numeric
def write(self, string):
self.output_code.append(string)
def getValue(self, node):
return self.tree_property[node]
def setValue(self, node, value):
self.tree_property[node] = value
def getSymbolTable(self):
return self.symbol_table
def getType(self):
return self.type
def exitVarDecl(self, ctx):
# This event method handles variable declarations. The parse tree node varDecl contains
# one or more varDecl2 nodes. Eg varDecl for 'Constants a{1:2, 1:2}, b{1:2}' has two varDecl2
# nodes(one for a{1:2, 1:2} and one for b{1:2}).
# Variable declarations are processed and stored in the event method exitVarDecl2.
# This stored information is used to write the final SymPy output code in the exitVarDecl event method.
# determine the type of declaration
if self.getValue(ctx.varType()) == "constant":
writeConstants(self, ctx)
elif self.getValue(ctx.varType()) in\
("variable", "motionvariable", "motionvariable'", "specified"):
writeVariables(self, ctx)
elif self.getValue(ctx.varType()) == "imaginary":
writeImaginary(self, ctx)
def exitVarType(self, ctx):
# Annotate the varType tree node with the type of the variable declaration.
name = ctx.getChild(0).getText().lower()
if name[-1] == "s" and name != "bodies":
self.setValue(ctx, name[:-1])
else:
self.setValue(ctx, name)
def exitVarDecl2(self, ctx):
# Variable declarations are processed and stored in the event method exitVarDecl2.
# This stored information is used to write the final SymPy output code in the exitVarDecl event method.
# This is the case for constants, variables, specifieds etc.
# This isn't the case for all types of declarations though. For instance
# Frames A, B, C, N cannot be defined on one line in SymPy. So we do not append A, B, C, N
# to a var_list or use exitVarDecl. exitVarDecl2 directly writes out to the file.
# determine the type of declaration
if self.getValue(ctx.parentCtx.varType()) == "constant":
processConstants(self, ctx)
elif self.getValue(ctx.parentCtx.varType()) in \
("variable", "motionvariable", "motionvariable'", "specified"):
processVariables(self, ctx)
elif self.getValue(ctx.parentCtx.varType()) == "imaginary":
processImaginary(self, ctx)
elif self.getValue(ctx.parentCtx.varType()) in ("frame", "newtonian", "point", "particle", "bodies"):
if "{" in ctx.getText():
if ":" in ctx.getText() and "," not in ctx.getText():
num1 = int(ctx.INT(0).getText())
num2 = int(ctx.INT(1).getText()) + 1
elif ":" not in ctx.getText() and "," in ctx.getText():
num1 = 1
num2 = int(ctx.INT(0).getText()) + 1
num3 = 1
num4 = int(ctx.INT(1).getText()) + 1
elif ":" in ctx.getText() and "," in ctx.getText():
num1 = int(ctx.INT(0).getText())
num2 = int(ctx.INT(1).getText()) + 1
num3 = int(ctx.INT(2).getText())
num4 = int(ctx.INT(3).getText()) + 1
else:
num1 = 1
num2 = int(ctx.INT(0).getText()) + 1
else:
num1 = 1
num2 = 2
for i in range(num1, num2):
try:
for j in range(num3, num4):
declare_phy_entities(self, ctx, self.getValue(ctx.parentCtx.varType()), i, j)
except Exception:
declare_phy_entities(self, ctx, self.getValue(ctx.parentCtx.varType()), i)
# ================== Subrules of parser rule expr (Start) ====================== #
def exitId(self, ctx):
# Tree annotation for ID which is a labeled subrule of the parser rule expr.
# A_C
python_keywords = ["and", "as", "assert", "break", "class", "continue", "def", "del", "elif", "else", "except",\
"exec", "finally", "for", "from", "global", "if", "import", "in", "is", "lambda", "not", "or", "pass", "print",\
"raise", "return", "try", "while", "with", "yield"]
if ctx.ID().getText().lower() in python_keywords:
warnings.warn("Python keywords must not be used as identifiers. Please refer to the list of keywords at https://docs.python.org/2.5/ref/keywords.html",
SyntaxWarning)
if "_" in ctx.ID().getText() and ctx.ID().getText().count('_') == 1:
e1, e2 = ctx.ID().getText().lower().split('_')
try:
if self.type2[e1] == "frame":
e1 = self.symbol_table2[e1]
elif self.type2[e1] == "bodies":
e1 = self.symbol_table2[e1] + "_f"
if self.type2[e2] == "frame":
e2 = self.symbol_table2[e2]
elif self.type2[e2] == "bodies":
e2 = self.symbol_table2[e2] + "_f"
self.setValue(ctx, e1 + ".dcm(" + e2 + ")")
except Exception:
self.setValue(ctx, ctx.ID().getText().lower())
else:
# Reserved constant Pi
if ctx.ID().getText().lower() == "pi":
self.setValue(ctx, "_sm.pi")
self.numeric_expr.append(ctx)
# Reserved variable T (for time)
elif ctx.ID().getText().lower() == "t":
self.setValue(ctx, "_me.dynamicsymbols._t")
if not self.in_inputs and not self.in_outputs:
self.t = True
else:
idText = ctx.ID().getText().lower() + "'"*(ctx.getChildCount() - 1)
if idText in self.type.keys() and self.type[idText] == "matrix":
self.matrix_expr.append(ctx)
if self.in_inputs:
try:
self.setValue(ctx, self.symbol_table[idText])
except Exception:
self.setValue(ctx, idText.lower())
else:
try:
self.setValue(ctx, self.symbol_table[idText])
except Exception:
pass
def exitInt(self, ctx):
# Tree annotation for int which is a labeled subrule of the parser rule expr.
int_text = ctx.INT().getText()
self.setValue(ctx, int_text)
self.numeric_expr.append(ctx)
def exitFloat(self, ctx):
# Tree annotation for float which is a labeled subrule of the parser rule expr.
floatText = ctx.FLOAT().getText()
self.setValue(ctx, floatText)
self.numeric_expr.append(ctx)
def exitAddSub(self, ctx):
# Tree annotation for AddSub which is a labeled subrule of the parser rule expr.
# The subrule is expr = expr (+|-) expr
if ctx.expr(0) in self.matrix_expr or ctx.expr(1) in self.matrix_expr:
self.matrix_expr.append(ctx)
if ctx.expr(0) in self.vector_expr or ctx.expr(1) in self.vector_expr:
self.vector_expr.append(ctx)
if ctx.expr(0) in self.numeric_expr and ctx.expr(1) in self.numeric_expr:
self.numeric_expr.append(ctx)
self.setValue(ctx, self.getValue(ctx.expr(0)) + ctx.getChild(1).getText() +
self.getValue(ctx.expr(1)))
def exitMulDiv(self, ctx):
# Tree annotation for MulDiv which is a labeled subrule of the parser rule expr.
# The subrule is expr = expr (*|/) expr
try:
if ctx.expr(0) in self.vector_expr and ctx.expr(1) in self.vector_expr:
self.setValue(ctx, "_me.outer(" + self.getValue(ctx.expr(0)) + ", " +
self.getValue(ctx.expr(1)) + ")")
else:
if ctx.expr(0) in self.matrix_expr or ctx.expr(1) in self.matrix_expr:
self.matrix_expr.append(ctx)
if ctx.expr(0) in self.vector_expr or ctx.expr(1) in self.vector_expr:
self.vector_expr.append(ctx)
if ctx.expr(0) in self.numeric_expr and ctx.expr(1) in self.numeric_expr:
self.numeric_expr.append(ctx)
self.setValue(ctx, self.getValue(ctx.expr(0)) + ctx.getChild(1).getText() +
self.getValue(ctx.expr(1)))
except Exception:
pass
def exitNegativeOne(self, ctx):
# Tree annotation for negativeOne which is a labeled subrule of the parser rule expr.
self.setValue(ctx, "-1*" + self.getValue(ctx.getChild(1)))
if ctx.getChild(1) in self.matrix_expr:
self.matrix_expr.append(ctx)
if ctx.getChild(1) in self.numeric_expr:
self.numeric_expr.append(ctx)
def exitParens(self, ctx):
# Tree annotation for parens which is a labeled subrule of the parser rule expr.
# The subrule is expr = '(' expr ')'
if ctx.expr() in self.matrix_expr:
self.matrix_expr.append(ctx)
if ctx.expr() in self.vector_expr:
self.vector_expr.append(ctx)
if ctx.expr() in self.numeric_expr:
self.numeric_expr.append(ctx)
self.setValue(ctx, "(" + self.getValue(ctx.expr()) + ")")
def exitExponent(self, ctx):
# Tree annotation for Exponent which is a labeled subrule of the parser rule expr.
# The subrule is expr = expr ^ expr
if ctx.expr(0) in self.matrix_expr or ctx.expr(1) in self.matrix_expr:
self.matrix_expr.append(ctx)
if ctx.expr(0) in self.vector_expr or ctx.expr(1) in self.vector_expr:
self.vector_expr.append(ctx)
if ctx.expr(0) in self.numeric_expr and ctx.expr(1) in self.numeric_expr:
self.numeric_expr.append(ctx)
self.setValue(ctx, self.getValue(ctx.expr(0)) + "**" + self.getValue(ctx.expr(1)))
def exitExp(self, ctx):
s = ctx.EXP().getText()[ctx.EXP().getText().index('E')+1:]
if "-" in s:
s = s[0] + s[1:].lstrip("0")
else:
s = s.lstrip("0")
self.setValue(ctx, ctx.EXP().getText()[:ctx.EXP().getText().index('E')] +
"*10**(" + s + ")")
def exitFunction(self, ctx):
# Tree annotation for function which is a labeled subrule of the parser rule expr.
# The difference between this and FunctionCall is that this is used for non standalone functions
# appearing in expressions and assignments.
# Eg:
# When we come across a standalone function say Expand(E, n:m) then it is categorized as FunctionCall
# which is a parser rule in itself under rule stat. exitFunctionCall() takes care of it and writes to the file.
#
# On the other hand, while we come across E_diff = D(E, y), we annotate the tree node
# of the function D(E, y) with the SymPy equivalent in exitFunction().
# In this case it is the method exitAssignment() that writes the code to the file and not exitFunction().
ch = ctx.getChild(0)
func_name = ch.getChild(0).getText().lower()
# Expand(y, n:m) *
if func_name == "expand":
expr = self.getValue(ch.expr(0))
if ch.expr(0) in self.matrix_expr or (expr in self.type.keys() and self.type[expr] == "matrix"):
self.matrix_expr.append(ctx)
# _sm.Matrix([i.expand() for i in z]).reshape(z.shape[0], z.shape[1])
self.setValue(ctx, "_sm.Matrix([i.expand() for i in " + expr + "])" +
".reshape((" + expr + ").shape[0], " + "(" + expr + ").shape[1])")
else:
self.setValue(ctx, "(" + expr + ")" + "." + "expand()")
# Factor(y, x) *
elif func_name == "factor":
expr = self.getValue(ch.expr(0))
if ch.expr(0) in self.matrix_expr or (expr in self.type.keys() and self.type[expr] == "matrix"):
self.matrix_expr.append(ctx)
self.setValue(ctx, "_sm.Matrix([_sm.factor(i, " + self.getValue(ch.expr(1)) + ") for i in " +
expr + "])" + ".reshape((" + expr + ").shape[0], " + "(" + expr + ").shape[1])")
else:
self.setValue(ctx, "_sm.factor(" + "(" + expr + ")" +
", " + self.getValue(ch.expr(1)) + ")")
# D(y, x)
elif func_name == "d":
expr = self.getValue(ch.expr(0))
if ch.expr(0) in self.matrix_expr or (expr in self.type.keys() and self.type[expr] == "matrix"):
self.matrix_expr.append(ctx)
self.setValue(ctx, "_sm.Matrix([i.diff(" + self.getValue(ch.expr(1)) + ") for i in " +
expr + "])" + ".reshape((" + expr + ").shape[0], " + "(" + expr + ").shape[1])")
else:
if ch.getChildCount() == 8:
frame = self.symbol_table2[ch.expr(2).getText().lower()]
self.setValue(ctx, "(" + expr + ")" + "." + "diff(" + self.getValue(ch.expr(1)) +
", " + frame + ")")
else:
self.setValue(ctx, "(" + expr + ")" + "." + "diff(" +
self.getValue(ch.expr(1)) + ")")
# Dt(y)
elif func_name == "dt":
expr = self.getValue(ch.expr(0))
if ch.expr(0) in self.vector_expr:
text = "dt("
else:
text = "diff(_sm.Symbol('t')"
if ch.expr(0) in self.matrix_expr or (expr in self.type.keys() and self.type[expr] == "matrix"):
self.matrix_expr.append(ctx)
self.setValue(ctx, "_sm.Matrix([i." + text +
") for i in " + expr + "])" +
".reshape((" + expr + ").shape[0], " + "(" + expr + ").shape[1])")
else:
if ch.getChildCount() == 6:
frame = self.symbol_table2[ch.expr(1).getText().lower()]
self.setValue(ctx, "(" + expr + ")" + "." + "dt(" +
frame + ")")
else:
self.setValue(ctx, "(" + expr + ")" + "." + text + ")")
# Explicit(EXPRESS(IMPLICIT>,C))
elif func_name == "explicit":
if ch.expr(0) in self.vector_expr:
self.vector_expr.append(ctx)
expr = self.getValue(ch.expr(0))
if self.explicit.keys():
explicit_list = []
for i in self.explicit.keys():
explicit_list.append(i + ":" + self.explicit[i])
self.setValue(ctx, "(" + expr + ")" + ".subs({" + ", ".join(explicit_list) + "})")
else:
self.setValue(ctx, expr)
# Taylor(y, 0:2, w=a, x=0)
# TODO: Currently only works with symbols. Make it work for dynamicsymbols.
elif func_name == "taylor":
exp = self.getValue(ch.expr(0))
order = self.getValue(ch.expr(1).expr(1))
x = (ch.getChildCount()-6)//2
l = []
for i in range(x):
index = 2 + i
child = ch.expr(index)
l.append(".series(" + self.getValue(child.getChild(0)) +
", " + self.getValue(child.getChild(2)) +
", " + order + ").removeO()")
self.setValue(ctx, "(" + exp + ")" + "".join(l))
# Evaluate(y, a=x, b=2)
elif func_name == "evaluate":
expr = self.getValue(ch.expr(0))
l = []
x = (ch.getChildCount()-4)//2
for i in range(x):
index = 1 + i
child = ch.expr(index)
l.append(self.getValue(child.getChild(0)) + ":" +
self.getValue(child.getChild(2)))
if ch.expr(0) in self.matrix_expr or (expr in self.type.keys() and self.type[expr] == "matrix"):
self.matrix_expr.append(ctx)
self.setValue(ctx, "_sm.Matrix([i.subs({" + ",".join(l) + "}) for i in " +
expr + "])" +
".reshape((" + expr + ").shape[0], " + "(" + expr + ").shape[1])")
else:
if self.explicit:
explicit_list = []
for i in self.explicit.keys():
explicit_list.append(i + ":" + self.explicit[i])
self.setValue(ctx, "(" + expr + ")" + ".subs({" + ",".join(explicit_list) +
"}).subs({" + ",".join(l) + "})")
else:
self.setValue(ctx, "(" + expr + ")" + ".subs({" + ",".join(l) + "})")
# Polynomial([a, b, c], x)
elif func_name == "polynomial":
self.setValue(ctx, "_sm.Poly(" + self.getValue(ch.expr(0)) + ", " +
self.getValue(ch.expr(1)) + ")")
# Roots(Poly, x, 2)
# Roots([1; 2; 3; 4])
elif func_name == "roots":
self.matrix_expr.append(ctx)
expr = self.getValue(ch.expr(0))
if ch.expr(0) in self.matrix_expr or (expr in self.type.keys() and self.type[expr] == "matrix"):
self.setValue(ctx, "[i.evalf() for i in " + "_sm.solve(" +
"_sm.Poly(" + expr + ", " + "x),x)]")
else:
self.setValue(ctx, "[i.evalf() for i in " + "_sm.solve(" +
expr + ", " + self.getValue(ch.expr(1)) + ")]")
# Transpose(A), Inv(A)
elif func_name in ("transpose", "inv", "inverse"):
self.matrix_expr.append(ctx)
if func_name == "transpose":
e = ".T"
elif func_name in ("inv", "inverse"):
e = "**(-1)"
self.setValue(ctx, "(" + self.getValue(ch.expr(0)) + ")" + e)
# Eig(A)
elif func_name == "eig":
# "_sm.Matrix([i.evalf() for i in " +
self.setValue(ctx, "_sm.Matrix([i.evalf() for i in (" +
self.getValue(ch.expr(0)) + ").eigenvals().keys()])")
# Diagmat(n, m, x)
# Diagmat(3, 1)
elif func_name == "diagmat":
self.matrix_expr.append(ctx)
if ch.getChildCount() == 6:
l = []
for i in range(int(self.getValue(ch.expr(0)))):
l.append(self.getValue(ch.expr(1)) + ",")
self.setValue(ctx, "_sm.diag(" + ("".join(l))[:-1] + ")")
elif ch.getChildCount() == 8:
# _sm.Matrix([x if i==j else 0 for i in range(n) for j in range(m)]).reshape(n, m)
n = self.getValue(ch.expr(0))
m = self.getValue(ch.expr(1))
x = self.getValue(ch.expr(2))
self.setValue(ctx, "_sm.Matrix([" + x + " if i==j else 0 for i in range(" +
n + ") for j in range(" + m + ")]).reshape(" + n + ", " + m + ")")
# Cols(A)
# Cols(A, 1)
# Cols(A, 1, 2:4, 3)
elif func_name in ("cols", "rows"):
self.matrix_expr.append(ctx)
if func_name == "cols":
e1 = ".cols"
e2 = ".T."
else:
e1 = ".rows"
e2 = "."
if ch.getChildCount() == 4:
self.setValue(ctx, "(" + self.getValue(ch.expr(0)) + ")" + e1)
elif ch.getChildCount() == 6:
self.setValue(ctx, "(" + self.getValue(ch.expr(0)) + ")" +
e1[:-1] + "(" + str(int(self.getValue(ch.expr(1))) - 1) + ")")
else:
l = []
for i in range(4, ch.getChildCount()):
try:
if ch.getChild(i).getChildCount() > 1 and ch.getChild(i).getChild(1).getText() == ":":
for j in range(int(ch.getChild(i).getChild(0).getText()),
int(ch.getChild(i).getChild(2).getText())+1):
l.append("(" + self.getValue(ch.getChild(2)) + ")" + e2 +
"row(" + str(j-1) + ")")
else:
l.append("(" + self.getValue(ch.getChild(2)) + ")" + e2 +
"row(" + str(int(ch.getChild(i).getText())-1) + ")")
except Exception:
pass
self.setValue(ctx, "_sm.Matrix([" + ",".join(l) + "])")
# Det(A) Trace(A)
elif func_name in ["det", "trace"]:
self.setValue(ctx, "(" + self.getValue(ch.expr(0)) + ")" + "." +
func_name + "()")
# Element(A, 2, 3)
elif func_name == "element":
self.setValue(ctx, "(" + self.getValue(ch.expr(0)) + ")" + "[" +
str(int(self.getValue(ch.expr(1)))-1) + "," +
str(int(self.getValue(ch.expr(2)))-1) + "]")
elif func_name in \
["cos", "sin", "tan", "cosh", "sinh", "tanh", "acos", "asin", "atan",
"log", "exp", "sqrt", "factorial", "floor", "sign"]:
self.setValue(ctx, "_sm." + func_name + "(" + self.getValue(ch.expr(0)) + ")")
elif func_name == "ceil":
self.setValue(ctx, "_sm.ceiling" + "(" + self.getValue(ch.expr(0)) + ")")
elif func_name == "sqr":
self.setValue(ctx, "(" + self.getValue(ch.expr(0)) +
")" + "**2")
elif func_name == "log10":
self.setValue(ctx, "_sm.log" +
"(" + self.getValue(ch.expr(0)) + ", 10)")
elif func_name == "atan2":
self.setValue(ctx, "_sm.atan2" + "(" + self.getValue(ch.expr(0)) + ", " +
self.getValue(ch.expr(1)) + ")")
elif func_name in ["int", "round"]:
self.setValue(ctx, func_name +
"(" + self.getValue(ch.expr(0)) + ")")
elif func_name == "abs":
self.setValue(ctx, "_sm.Abs(" + self.getValue(ch.expr(0)) + ")")
elif func_name in ["max", "min"]:
# max(x, y, z)
l = []
for i in range(1, ch.getChildCount()):
if ch.getChild(i) in self.tree_property.keys():
l.append(self.getValue(ch.getChild(i)))
elif ch.getChild(i).getText() in [",", "(", ")"]:
l.append(ch.getChild(i).getText())
self.setValue(ctx, "_sm." + ch.getChild(0).getText().capitalize() + "".join(l))
# Coef(y, x)
elif func_name == "coef":
#A41_A53=COEF([RHS(U4);RHS(U5)],[U1,U2,U3])
if ch.expr(0) in self.matrix_expr and ch.expr(1) in self.matrix_expr:
icount = jcount = 0
for i in range(ch.expr(0).getChild(0).getChildCount()):
try:
ch.expr(0).getChild(0).getChild(i).getRuleIndex()
icount+=1
except Exception:
pass
for j in range(ch.expr(1).getChild(0).getChildCount()):
try:
ch.expr(1).getChild(0).getChild(j).getRuleIndex()
jcount+=1
except Exception:
pass
l = []
for i in range(icount):
for j in range(jcount):
# a41_a53[i,j] = u4.expand().coeff(u1)
l.append(self.getValue(ch.expr(0).getChild(0).expr(i)) + ".expand().coeff("
+ self.getValue(ch.expr(1).getChild(0).expr(j)) + ")")
self.setValue(ctx, "_sm.Matrix([" + ", ".join(l) + "]).reshape(" + str(icount) + ", " + str(jcount) + ")")
else:
self.setValue(ctx, "(" + self.getValue(ch.expr(0)) +
")" + ".expand().coeff(" + self.getValue(ch.expr(1)) + ")")
# Exclude(y, x) Include(y, x)
elif func_name in ("exclude", "include"):
if func_name == "exclude":
e = "0"
else:
e = "1"
expr = self.getValue(ch.expr(0))
if ch.expr(0) in self.matrix_expr or (expr in self.type.keys() and self.type[expr] == "matrix"):
self.matrix_expr.append(ctx)
self.setValue(ctx, "_sm.Matrix([i.collect(" + self.getValue(ch.expr(1)) + "])" +
".coeff(" + self.getValue(ch.expr(1)) + "," + e + ")" + "for i in " + expr + ")" +
".reshape((" + expr + ").shape[0], " + "(" + expr + ").shape[1])")
else:
self.setValue(ctx, "(" + expr +
")" + ".collect(" + self.getValue(ch.expr(1)) + ")" +
".coeff(" + self.getValue(ch.expr(1)) + "," + e + ")")
# RHS(y)
elif func_name == "rhs":
self.setValue(ctx, self.explicit[self.getValue(ch.expr(0))])
# Arrange(y, n, x) *
elif func_name == "arrange":
expr = self.getValue(ch.expr(0))
if ch.expr(0) in self.matrix_expr or (expr in self.type.keys() and self.type[expr] == "matrix"):
self.matrix_expr.append(ctx)
self.setValue(ctx, "_sm.Matrix([i.collect(" + self.getValue(ch.expr(2)) +
")" + "for i in " + expr + "])"+
".reshape((" + expr + ").shape[0], " + "(" + expr + ").shape[1])")
else:
self.setValue(ctx, "(" + expr +
")" + ".collect(" + self.getValue(ch.expr(2)) + ")")
# Replace(y, sin(x)=3)
elif func_name == "replace":
l = []
for i in range(1, ch.getChildCount()):
try:
if ch.getChild(i).getChild(1).getText() == "=":
l.append(self.getValue(ch.getChild(i).getChild(0)) +
":" + self.getValue(ch.getChild(i).getChild(2)))
except Exception:
pass
expr = self.getValue(ch.expr(0))
if ch.expr(0) in self.matrix_expr or (expr in self.type.keys() and self.type[expr] == "matrix"):
self.matrix_expr.append(ctx)
self.setValue(ctx, "_sm.Matrix([i.subs({" + ",".join(l) + "}) for i in " +
expr + "])" +
".reshape((" + expr + ").shape[0], " + "(" + expr + ").shape[1])")
else:
self.setValue(ctx, "(" + self.getValue(ch.expr(0)) + ")" +
".subs({" + ",".join(l) + "})")
# Dot(Loop>, N1>)
elif func_name == "dot":
l = []
num = (ch.expr(1).getChild(0).getChildCount()-1)//2
if ch.expr(1) in self.matrix_expr:
for i in range(num):
l.append("_me.dot(" + self.getValue(ch.expr(0)) + ", " + self.getValue(ch.expr(1).getChild(0).expr(i)) + ")")
self.setValue(ctx, "_sm.Matrix([" + ",".join(l) + "]).reshape(" + str(num) + ", " + "1)")
else:
self.setValue(ctx, "_me.dot(" + self.getValue(ch.expr(0)) + ", " + self.getValue(ch.expr(1)) + ")")
# Cross(w_A_N>, P_NA_AB>)
elif func_name == "cross":
self.vector_expr.append(ctx)
self.setValue(ctx, "_me.cross(" + self.getValue(ch.expr(0)) + ", " + self.getValue(ch.expr(1)) + ")")
# Mag(P_O_Q>)
elif func_name == "mag":
self.setValue(ctx, self.getValue(ch.expr(0)) + "." + "magnitude()")
# MATRIX(A, I_R>>)
elif func_name == "matrix":
if self.type2[ch.expr(0).getText().lower()] == "frame":
text = ""
elif self.type2[ch.expr(0).getText().lower()] == "bodies":
text = "_f"
self.setValue(ctx, "(" + self.getValue(ch.expr(1)) + ")" + ".to_matrix(" +
self.symbol_table2[ch.expr(0).getText().lower()] + text + ")")
# VECTOR(A, ROWS(EIGVECS,1))
elif func_name == "vector":
if self.type2[ch.expr(0).getText().lower()] == "frame":
text = ""
elif self.type2[ch.expr(0).getText().lower()] == "bodies":
text = "_f"
v = self.getValue(ch.expr(1))
f = self.symbol_table2[ch.expr(0).getText().lower()] + text
self.setValue(ctx, v + "[0]*" + f + ".x +" + v + "[1]*" + f + ".y +" +
v + "[2]*" + f + ".z")
# Express(A2>, B)
# Here I am dealing with all the Inertia commands as I expect the users to use Inertia
# commands only with Express because SymPy needs the Reference frame to be specified unlike Autolev.
elif func_name == "express":
self.vector_expr.append(ctx)
if self.type2[ch.expr(1).getText().lower()] == "frame":
frame = self.symbol_table2[ch.expr(1).getText().lower()]
else:
frame = self.symbol_table2[ch.expr(1).getText().lower()] + "_f"
if ch.expr(0).getText().lower() == "1>>":
self.setValue(ctx, "_me.inertia(" + frame + ", 1, 1, 1)")
elif '_' in ch.expr(0).getText().lower() and ch.expr(0).getText().lower().count('_') == 2\
and ch.expr(0).getText().lower()[0] == "i" and ch.expr(0).getText().lower()[-2:] == ">>":
v1 = ch.expr(0).getText().lower()[:-2].split('_')[1]
v2 = ch.expr(0).getText().lower()[:-2].split('_')[2]
l = []
inertia_func(self, v1, v2, l, frame)
self.setValue(ctx, " + ".join(l))
elif ch.expr(0).getChild(0).getChild(0).getText().lower() == "inertia":
if ch.expr(0).getChild(0).getChildCount() == 4:
l = []
v2 = ch.expr(0).getChild(0).ID(0).getText().lower()
for v1 in self.bodies:
inertia_func(self, v1, v2, l, frame)
self.setValue(ctx, " + ".join(l))
else:
l = []
l2 = []
v2 = ch.expr(0).getChild(0).ID(0).getText().lower()
for i in range(1, (ch.expr(0).getChild(0).getChildCount()-2)//2):
l2.append(ch.expr(0).getChild(0).ID(i).getText().lower())
for v1 in l2:
inertia_func(self, v1, v2, l, frame)
self.setValue(ctx, " + ".join(l))
else:
self.setValue(ctx, "(" + self.getValue(ch.expr(0)) + ")" + ".express(" +
self.symbol_table2[ch.expr(1).getText().lower()] + ")")
# CM(P)
elif func_name == "cm":
if self.type2[ch.expr(0).getText().lower()] == "point":
text = ""
else:
text = ".point"
if ch.getChildCount() == 4:
self.setValue(ctx, "_me.functions.center_of_mass(" + self.symbol_table2[ch.expr(0).getText().lower()] +
text + "," + ", ".join(self.bodies.values()) + ")")
else:
bodies = []
for i in range(1, (ch.getChildCount()-1)//2):
bodies.append(self.symbol_table2[ch.expr(i).getText().lower()])
self.setValue(ctx, "_me.functions.center_of_mass(" + self.symbol_table2[ch.expr(0).getText().lower()] +
text + "," + ", ".join(bodies) + ")")
# PARTIALS(V_P1_E>,U1)
elif func_name == "partials":
speeds = []
for i in range(1, (ch.getChildCount()-1)//2):
if self.kd_equivalents2:
speeds.append(self.kd_equivalents2[self.symbol_table[ch.expr(i).getText().lower()]])
else:
speeds.append(self.symbol_table[ch.expr(i).getText().lower()])
v1, v2, v3 = ch.expr(0).getText().lower().replace(">","").split('_')
if self.type2[v2] == "point":
point = self.symbol_table2[v2]
elif self.type2[v2] == "particle":
point = self.symbol_table2[v2] + ".point"
frame = self.symbol_table2[v3]
self.setValue(ctx, point + ".partial_velocity(" + frame + ", " + ",".join(speeds) + ")")
# UnitVec(A1>+A2>+A3>)
elif func_name == "unitvec":
self.setValue(ctx, "(" + self.getValue(ch.expr(0)) + ")" + ".normalize()")
# Units(deg, rad)
elif func_name == "units":
if ch.expr(0).getText().lower() == "deg" and ch.expr(1).getText().lower() == "rad":
factor = 0.0174533
elif ch.expr(0).getText().lower() == "rad" and ch.expr(1).getText().lower() == "deg":
factor = 57.2958
self.setValue(ctx, str(factor))
# Mass(A)
elif func_name == "mass":
l = []
try:
ch.ID(0).getText().lower()
for i in range((ch.getChildCount()-1)//2):
l.append(self.symbol_table2[ch.ID(i).getText().lower()] + ".mass")
self.setValue(ctx, "+".join(l))
except Exception:
for i in self.bodies.keys():
l.append(self.bodies[i] + ".mass")
self.setValue(ctx, "+".join(l))
# Fr() FrStar()
# _me.KanesMethod(n, q_ind, u_ind, kd, velocity_constraints).kanes_equations(pl, fl)[0]
elif func_name in ["fr", "frstar"]:
if not self.kane_parsed:
if self.kd_eqs:
for i in self.kd_eqs:
self.q_ind.append(self.symbol_table[i.strip().split('-')[0].replace("'","")])
self.u_ind.append(self.symbol_table[i.strip().split('-')[1].replace("'","")])
for i in range(len(self.kd_eqs)):
self.kd_eqs[i] = self.symbol_table[self.kd_eqs[i].strip().split('-')[0]] + " - " +\
self.symbol_table[self.kd_eqs[i].strip().split('-')[1]]
# Do all of this if kd_eqs are not specified
if not self.kd_eqs:
self.kd_eqs_supplied = False
self.matrix_expr.append(ctx)
for i in self.type.keys():
if self.type[i] == "motionvariable":
if self.sign[self.symbol_table[i.lower()]] == 0:
self.q_ind.append(self.symbol_table[i.lower()])
elif self.sign[self.symbol_table[i.lower()]] == 1:
name = "u_" + self.symbol_table[i.lower()]
self.symbol_table.update({name: name})
self.write(name + " = " + "_me.dynamicsymbols('" + name + "')\n")
if self.symbol_table[i.lower()] not in self.dependent_variables:
self.u_ind.append(name)
self.kd_equivalents.update({name: self.symbol_table[i.lower()]})
else:
self.u_dep.append(name)
self.kd_equivalents.update({name: self.symbol_table[i.lower()]})
for i in self.kd_equivalents.keys():
self.kd_eqs.append(self.kd_equivalents[i] + "-" + i)
if not self.u_ind and not self.kd_eqs:
self.u_ind = self.q_ind.copy()
self.q_ind = []
# deal with velocity constraints
if self.dependent_variables:
for i in self.dependent_variables:
self.u_dep.append(i)
if i in self.u_ind:
self.u_ind.remove(i)
self.u_dep[:] = [i for i in self.u_dep if i not in self.kd_equivalents.values()]
force_list = []
for i in self.forces.keys():
force_list.append("(" + i + "," + self.forces[i] + ")")
if self.u_dep:
u_dep_text = ", u_dependent=[" + ", ".join(self.u_dep) + "]"
else:
u_dep_text = ""
if self.dependent_variables:
velocity_constraints_text = ", velocity_constraints = velocity_constraints"
else:
velocity_constraints_text = ""
if ctx.parentCtx not in self.fr_expr:
self.write("kd_eqs = [" + ", ".join(self.kd_eqs) + "]\n")
self.write("forceList = " + "[" + ", ".join(force_list) + "]\n")
self.write("kane = _me.KanesMethod(" + self.newtonian + ", " + "q_ind=[" +
",".join(self.q_ind) + "], " + "u_ind=[" +
", ".join(self.u_ind) + "]" + u_dep_text + ", " +
"kd_eqs = kd_eqs" + velocity_constraints_text + ")\n")
self.write("fr, frstar = kane." + "kanes_equations([" +
", ".join(self.bodies.values()) + "], forceList)\n")
self.fr_expr.append(ctx.parentCtx)
self.kane_parsed = True
self.setValue(ctx, func_name)
def exitMatrices(self, ctx):
# Tree annotation for Matrices which is a labeled subrule of the parser rule expr.
# MO = [a, b; c, d]
# we generate _sm.Matrix([a, b, c, d]).reshape(2, 2)
# The reshape values are determined by counting the "," and ";" in the Autolev matrix
# Eg:
# [1, 2, 3; 4, 5, 6; 7, 8, 9; 10, 11, 12]
# semicolon_count = 3 and rows = 3+1 = 4
# comma_count = 8 and cols = 8/rows + 1 = 8/4 + 1 = 3
# TODO** Parse block matrices
self.matrix_expr.append(ctx)
l = []
semicolon_count = 0
comma_count = 0
for i in range(ctx.matrix().getChildCount()):
child = ctx.matrix().getChild(i)
if child == AutolevParser.ExprContext:
l.append(self.getValue(child))
elif child.getText() == ";":
semicolon_count += 1
l.append(",")
elif child.getText() == ",":
comma_count += 1
l.append(",")
else:
try:
try:
l.append(self.getValue(child))
except Exception:
l.append(self.symbol_table[child.getText().lower()])
except Exception:
l.append(child.getText().lower())
num_of_rows = semicolon_count + 1
num_of_cols = (comma_count//num_of_rows) + 1
self.setValue(ctx, "_sm.Matrix(" + "".join(l) + ")" + ".reshape(" +
str(num_of_rows) + ", " + str(num_of_cols) + ")")
def exitVectorOrDyadic(self, ctx):
self.vector_expr.append(ctx)
ch = ctx.vec()
if ch.getChild(0).getText() == "0>":
self.setValue(ctx, "0")
elif ch.getChild(0).getText() == "1>>":
self.setValue(ctx, "1>>")
elif "_" in ch.ID().getText() and ch.ID().getText().count('_') == 2:
vec_text = ch.getText().lower()
v1, v2, v3 = ch.ID().getText().lower().split('_')
if v1 == "p":
if self.type2[v2] == "point":
e2 = self.symbol_table2[v2]
elif self.type2[v2] == "particle":
e2 = self.symbol_table2[v2] + ".point"
if self.type2[v3] == "point":
e3 = self.symbol_table2[v3]
elif self.type2[v3] == "particle":
e3 = self.symbol_table2[v3] + ".point"
get_vec = e3 + ".pos_from(" + e2 + ")"
self.setValue(ctx, get_vec)
elif v1 in ("w", "alf"):
if v1 == "w":
text = ".ang_vel_in("
elif v1 == "alf":
text = ".ang_acc_in("
if self.type2[v2] == "bodies":
e2 = self.symbol_table2[v2] + "_f"
elif self.type2[v2] == "frame":
e2 = self.symbol_table2[v2]
if self.type2[v3] == "bodies":
e3 = self.symbol_table2[v3] + "_f"
elif self.type2[v3] == "frame":
e3 = self.symbol_table2[v3]
get_vec = e2 + text + e3 + ")"
self.setValue(ctx, get_vec)
elif v1 in ("v", "a"):
if v1 == "v":
text = ".vel("
elif v1 == "a":
text = ".acc("
if self.type2[v2] == "point":
e2 = self.symbol_table2[v2]
elif self.type2[v2] == "particle":
e2 = self.symbol_table2[v2] + ".point"
get_vec = e2 + text + self.symbol_table2[v3] + ")"
self.setValue(ctx, get_vec)
else:
self.setValue(ctx, vec_text.replace(">", ""))
else:
vec_text = ch.getText().lower()
name = self.symbol_table[vec_text]
self.setValue(ctx, name)
def exitIndexing(self, ctx):
if ctx.getChildCount() == 4:
try:
int_text = str(int(self.getValue(ctx.getChild(2))) - 1)
except Exception:
int_text = self.getValue(ctx.getChild(2)) + " - 1"
self.setValue(ctx, ctx.ID().getText().lower() + "[" + int_text + "]")
elif ctx.getChildCount() == 6:
try:
int_text1 = str(int(self.getValue(ctx.getChild(2))) - 1)
except Exception:
int_text1 = self.getValue(ctx.getChild(2)) + " - 1"
try:
int_text2 = str(int(self.getValue(ctx.getChild(4))) - 1)
except Exception:
int_text2 = self.getValue(ctx.getChild(2)) + " - 1"
self.setValue(ctx, ctx.ID().getText().lower() + "[" + int_text1 + ", " + int_text2 + "]")
# ================== Subrules of parser rule expr (End) ====================== #
def exitRegularAssign(self, ctx):
# Handle assignments of type ID = expr
if ctx.equals().getText() in ["=", "+=", "-=", "*=", "/="]:
equals = ctx.equals().getText()
elif ctx.equals().getText() == ":=":
equals = " = "
elif ctx.equals().getText() == "^=":
equals = "**="
try:
a = ctx.ID().getText().lower() + "'"*ctx.diff().getText().count("'")
except Exception:
a = ctx.ID().getText().lower()
if a in self.type.keys() and self.type[a] in ("motionvariable", "motionvariable'") and\
self.type[ctx.expr().getText().lower()] in ("motionvariable", "motionvariable'"):
b = ctx.expr().getText().lower()
if "'" in b and "'" not in a:
a, b = b, a
if not self.kane_parsed:
self.kd_eqs.append(a + "-" + b)
self.kd_equivalents.update({self.symbol_table[a]:
self.symbol_table[b]})
self.kd_equivalents2.update({self.symbol_table[b]:
self.symbol_table[a]})
if a in self.symbol_table.keys() and a in self.type.keys() and self.type[a] in ("variable", "motionvariable"):
self.explicit.update({self.symbol_table[a]: self.getValue(ctx.expr())})
else:
if ctx.expr() in self.matrix_expr:
self.type.update({a: "matrix"})
try:
b = self.symbol_table[a]
except KeyError:
self.symbol_table[a] = a
if "_" in a and a.count("_") == 1:
e1, e2 = a.split('_')
if e1 in self.type2.keys() and self.type2[e1] in ("frame", "bodies")\
and e2 in self.type2.keys() and self.type2[e2] in ("frame", "bodies"):
if self.type2[e1] == "bodies":
t1 = "_f"
else:
t1 = ""
if self.type2[e2] == "bodies":
t2 = "_f"
else:
t2 = ""
self.write(self.symbol_table2[e2] + t2 + ".orient(" + self.symbol_table2[e1] +
t1 + ", 'DCM', " + self.getValue(ctx.expr()) + ")\n")
else:
self.write(self.symbol_table[a] + " " + equals + " " +
self.getValue(ctx.expr()) + "\n")
else:
self.write(self.symbol_table[a] + " " + equals + " " +
self.getValue(ctx.expr()) + "\n")
def exitIndexAssign(self, ctx):
# Handle assignments of type ID[index] = expr
if ctx.equals().getText() in ["=", "+=", "-=", "*=", "/="]:
equals = ctx.equals().getText()
elif ctx.equals().getText() == ":=":
equals = " = "
elif ctx.equals().getText() == "^=":
equals = "**="
text = ctx.ID().getText().lower()
self.type.update({text: "matrix"})
# Handle assignments of type ID[2] = expr
if ctx.index().getChildCount() == 1:
if ctx.index().getChild(0).getText() == "1":
self.type.update({text: "matrix"})
self.symbol_table.update({text: text})
self.write(text + " = " + "_sm.Matrix([[0]])\n")
self.write(text + "[0] = " + self.getValue(ctx.expr()) + "\n")
else:
# m = m.row_insert(m.shape[0], _sm.Matrix([[0]]))
self.write(text + " = " + text +
".row_insert(" + text + ".shape[0]" + ", " + "_sm.Matrix([[0]])" + ")\n")
self.write(text + "[" + text + ".shape[0]-1" + "] = " + self.getValue(ctx.expr()) + "\n")
# Handle assignments of type ID[2, 2] = expr
elif ctx.index().getChildCount() == 3:
l = []
try:
l.append(str(int(self.getValue(ctx.index().getChild(0)))-1))
except Exception:
l.append(self.getValue(ctx.index().getChild(0)) + "-1")
l.append(",")
try:
l.append(str(int(self.getValue(ctx.index().getChild(2)))-1))
except Exception:
l.append(self.getValue(ctx.index().getChild(2)) + "-1")
self.write(self.symbol_table[ctx.ID().getText().lower()] +
"[" + "".join(l) + "]" + " " + equals + " " + self.getValue(ctx.expr()) + "\n")
def exitVecAssign(self, ctx):
# Handle assignments of the type vec = expr
ch = ctx.vec()
vec_text = ch.getText().lower()
if "_" in ch.ID().getText():
num = ch.ID().getText().count('_')
if num == 2:
v1, v2, v3 = ch.ID().getText().lower().split('_')
if v1 == "p":
if self.type2[v2] == "point":
e2 = self.symbol_table2[v2]
elif self.type2[v2] == "particle":
e2 = self.symbol_table2[v2] + ".point"
if self.type2[v3] == "point":
e3 = self.symbol_table2[v3]
elif self.type2[v3] == "particle":
e3 = self.symbol_table2[v3] + ".point"
# ab.set_pos(na, la*a.x)
self.write(e3 + ".set_pos(" + e2 + ", " + self.getValue(ctx.expr()) + ")\n")
elif v1 in ("w", "alf"):
if v1 == "w":
text = ".set_ang_vel("
elif v1 == "alf":
text = ".set_ang_acc("
# a.set_ang_vel(n, qad*a.z)
if self.type2[v2] == "bodies":
e2 = self.symbol_table2[v2] + "_f"
else:
e2 = self.symbol_table2[v2]
if self.type2[v3] == "bodies":
e3 = self.symbol_table2[v3] + "_f"
else:
e3 = self.symbol_table2[v3]
self.write(e2 + text + e3 + ", " + self.getValue(ctx.expr()) + ")\n")
elif v1 in ("v", "a"):
if v1 == "v":
text = ".set_vel("
elif v1 == "a":
text = ".set_acc("
if self.type2[v2] == "point":
e2 = self.symbol_table2[v2]
elif self.type2[v2] == "particle":
e2 = self.symbol_table2[v2] + ".point"
self.write(e2 + text + self.symbol_table2[v3] +
", " + self.getValue(ctx.expr()) + ")\n")
elif v1 == "i":
if v2 in self.type2.keys() and self.type2[v2] == "bodies":
self.write(self.symbol_table2[v2] + ".inertia = (" + self.getValue(ctx.expr()) +
", " + self.symbol_table2[v3] + ")\n")
self.inertia_point.update({v2: v3})
elif v2 in self.type2.keys() and self.type2[v2] == "particle":
self.write(ch.ID().getText().lower() + " = " + self.getValue(ctx.expr()) + "\n")
else:
self.write(ch.ID().getText().lower() + " = " + self.getValue(ctx.expr()) + "\n")
else:
self.write(ch.ID().getText().lower() + " = " + self.getValue(ctx.expr()) + "\n")
elif num == 1:
v1, v2 = ch.ID().getText().lower().split('_')
if v1 in ("force", "torque"):
if self.type2[v2] in ("point", "frame"):
e2 = self.symbol_table2[v2]
elif self.type2[v2] == "particle":
e2 = self.symbol_table2[v2] + ".point"
self.symbol_table.update({vec_text: ch.ID().getText().lower()})
if e2 in self.forces.keys():
self.forces[e2] = self.forces[e2] + " + " + self.getValue(ctx.expr())
else:
self.forces.update({e2: self.getValue(ctx.expr())})
self.write(ch.ID().getText().lower() + " = " + self.forces[e2] + "\n")
else:
name = ch.ID().getText().lower()
self.symbol_table.update({vec_text: name})
self.write(ch.ID().getText().lower() + " = " + self.getValue(ctx.expr()) + "\n")
else:
name = ch.ID().getText().lower()
self.symbol_table.update({vec_text: name})
self.write(name + " " + ctx.getChild(1).getText() + " " + self.getValue(ctx.expr()) + "\n")
else:
name = ch.ID().getText().lower()
self.symbol_table.update({vec_text: name})
self.write(name + " " + ctx.getChild(1).getText() + " " + self.getValue(ctx.expr()) + "\n")
def enterInputs2(self, ctx):
self.in_inputs = True
# Inputs
def exitInputs2(self, ctx):
# Stores numerical values given by the input command which
# are used for codegen and numerical analysis.
if ctx.getChildCount() == 3:
try:
self.inputs.update({self.symbol_table[ctx.id_diff().getText().lower()]: self.getValue(ctx.expr(0))})
except Exception:
self.inputs.update({ctx.id_diff().getText().lower(): self.getValue(ctx.expr(0))})
elif ctx.getChildCount() == 4:
try:
self.inputs.update({self.symbol_table[ctx.id_diff().getText().lower()]:
(self.getValue(ctx.expr(0)), self.getValue(ctx.expr(1)))})
except Exception:
self.inputs.update({ctx.id_diff().getText().lower():
(self.getValue(ctx.expr(0)), self.getValue(ctx.expr(1)))})
self.in_inputs = False
def enterOutputs(self, ctx):
self.in_outputs = True
def exitOutputs(self, ctx):
self.in_outputs = False
def exitOutputs2(self, ctx):
try:
if "[" in ctx.expr(1).getText():
self.outputs.append(self.symbol_table[ctx.expr(0).getText().lower()] +
ctx.expr(1).getText().lower())
else:
self.outputs.append(self.symbol_table[ctx.expr(0).getText().lower()])
except Exception:
pass
# Code commands
def exitCodegen(self, ctx):
# Handles the CODE() command ie the solvers and the codgen part.
# Uses linsolve for the algebraic solvers and nsolve for non linear solvers.
if ctx.functionCall().getChild(0).getText().lower() == "algebraic":
matrix_name = self.getValue(ctx.functionCall().expr(0))
e = []
d = []
for i in range(1, (ctx.functionCall().getChildCount()-2)//2):
a = self.getValue(ctx.functionCall().expr(i))
e.append(a)
for i in self.inputs.keys():
d.append(i + ":" + self.inputs[i])
self.write(matrix_name + "_list" + " = " + "[]\n")
self.write("for i in " + matrix_name + ": " + matrix_name +
"_list" + ".append(i.subs({" + ", ".join(d) + "}))\n")
self.write("print(_sm.linsolve(" + matrix_name + "_list" + ", " + ",".join(e) + "))\n")
elif ctx.functionCall().getChild(0).getText().lower() == "nonlinear":
e = []
d = []
guess = []
for i in range(1, (ctx.functionCall().getChildCount()-2)//2):
a = self.getValue(ctx.functionCall().expr(i))
e.append(a)
#print(self.inputs)
for i in self.inputs.keys():
if i in self.symbol_table.keys():
if type(self.inputs[i]) is tuple:
j, z = self.inputs[i]
else:
j = self.inputs[i]
z = ""
if i not in e:
if z == "deg":
d.append(i + ":" + "_np.deg2rad(" + j + ")")
else:
d.append(i + ":" + j)
else:
if z == "deg":
guess.append("_np.deg2rad(" + j + ")")
else:
guess.append(j)
self.write("matrix_list" + " = " + "[]\n")
self.write("for i in " + self.getValue(ctx.functionCall().expr(0)) + ":")
self.write("matrix_list" + ".append(i.subs({" + ", ".join(d) + "}))\n")
self.write("print(_sm.nsolve(matrix_list," + "(" + ",".join(e) + ")" +
",(" + ",".join(guess) + ")" + "))\n")
elif ctx.functionCall().getChild(0).getText().lower() in ["ode", "dynamics"] and self.include_numeric:
if self.kane_type == "no_args":
for i in self.symbol_table.keys():
try:
if self.type[i] == "constants" or self.type[self.symbol_table[i]] == "constants":
self.constants.append(self.symbol_table[i])
except Exception:
pass
q_add_u = self.q_ind + self.q_dep + self.u_ind + self.u_dep
x0 = []
for i in q_add_u:
try:
if i in self.inputs.keys():
if type(self.inputs[i]) is tuple:
if self.inputs[i][1] == "deg":
x0.append(i + ":" + "_np.deg2rad(" + self.inputs[i][0] + ")")
else:
x0.append(i + ":" + self.inputs[i][0])
else:
x0.append(i + ":" + self.inputs[i])
elif self.kd_equivalents[i] in self.inputs.keys():
if type(self.inputs[self.kd_equivalents[i]]) is tuple:
x0.append(i + ":" + self.inputs[self.kd_equivalents[i]][0])
else:
x0.append(i + ":" + self.inputs[self.kd_equivalents[i]])
except Exception:
pass
# numerical constants
numerical_constants = []
for i in self.constants:
if i in self.inputs.keys():
if type(self.inputs[i]) is tuple:
numerical_constants.append(self.inputs[i][0])
else:
numerical_constants.append(self.inputs[i])
# t = linspace
t_final = self.inputs["tfinal"]
integ_stp = self.inputs["integstp"]
self.write("from pydy.system import System\n")
const_list = []
if numerical_constants:
for i in range(len(self.constants)):
const_list.append(self.constants[i] + ":" + numerical_constants[i])
specifieds = []
if self.t:
specifieds.append("_me.dynamicsymbols('t')" + ":" + "lambda x, t: t")
for i in self.inputs:
if i in self.symbol_table.keys() and self.symbol_table[i] not in\
self.constants + self.q_ind + self.q_dep + self.u_ind + self.u_dep:
specifieds.append(self.symbol_table[i] + ":" + self.inputs[i])
self.write("sys = System(kane, constants = {" + ", ".join(const_list) + "},\n" +
"specifieds={" + ", ".join(specifieds) + "},\n" +
"initial_conditions={" + ", ".join(x0) + "},\n" +
"times = _np.linspace(0.0, " + str(t_final) + ", " + str(t_final) +
"/" + str(integ_stp) + "))\n\ny=sys.integrate()\n")
# For outputs other than qs and us.
other_outputs = []
for i in self.outputs:
if i not in q_add_u:
if "[" in i:
other_outputs.append((i[:-3] + i[-2], i[:-3] + "[" + str(int(i[-2])-1) + "]"))
else:
other_outputs.append((i, i))
for i in other_outputs:
self.write(i[0] + "_out" + " = " + "[]\n")
if other_outputs:
self.write("for i in y:\n")
self.write(" q_u_dict = dict(zip(sys.coordinates+sys.speeds, i))\n")
for i in other_outputs:
self.write(" "*4 + i[0] + "_out" + ".append(" + i[1] + ".subs(q_u_dict)" +
".subs(sys.constants).evalf())\n")
# Standalone function calls (used for dual functions)
def exitFunctionCall(self, ctx):
# Basically deals with standalone function calls ie functions which are not a part of
# expressions and assignments. Autolev Dual functions can both appear in standalone
# function calls and also on the right hand side as part of expr or assignment.
# Dual functions are indicated by a * in the comments below
# Checks if the function is a statement on its own
if ctx.parentCtx.getRuleIndex() == AutolevParser.RULE_stat:
func_name = ctx.getChild(0).getText().lower()
# Expand(E, n:m) *
if func_name == "expand":
# If the first argument is a pre declared variable.
expr = self.getValue(ctx.expr(0))
symbol = self.symbol_table[ctx.expr(0).getText().lower()]
if ctx.expr(0) in self.matrix_expr or (expr in self.type.keys() and self.type[expr] == "matrix"):
self.write(symbol + " = " + "_sm.Matrix([i.expand() for i in " + expr + "])" +
".reshape((" + expr + ").shape[0], " + "(" + expr + ").shape[1])\n")
else:
self.write(symbol + " = " + symbol + "." + "expand()\n")
# Factor(E, x) *
elif func_name == "factor":
expr = self.getValue(ctx.expr(0))
symbol = self.symbol_table[ctx.expr(0).getText().lower()]
if ctx.expr(0) in self.matrix_expr or (expr in self.type.keys() and self.type[expr] == "matrix"):
self.write(symbol + " = " + "_sm.Matrix([_sm.factor(i," + self.getValue(ctx.expr(1)) +
") for i in " + expr + "])" +
".reshape((" + expr + ").shape[0], " + "(" + expr + ").shape[1])\n")
else:
self.write(expr + " = " + "_sm.factor(" + expr + ", " +
self.getValue(ctx.expr(1)) + ")\n")
# Solve(Zero, x, y)
elif func_name == "solve":
l = []
l2 = []
num = 0
for i in range(1, ctx.getChildCount()):
if ctx.getChild(i).getText() == ",":
num+=1
try:
l.append(self.getValue(ctx.getChild(i)))
except Exception:
l.append(ctx.getChild(i).getText())
if i != 2:
try:
l2.append(self.getValue(ctx.getChild(i)))
except Exception:
pass
for i in l2:
self.explicit.update({i: "_sm.solve" + "".join(l) + "[" + i + "]"})
self.write("print(_sm.solve" + "".join(l) + ")\n")
# Arrange(y, n, x) *
elif func_name == "arrange":
expr = self.getValue(ctx.expr(0))
symbol = self.symbol_table[ctx.expr(0).getText().lower()]
if ctx.expr(0) in self.matrix_expr or (expr in self.type.keys() and self.type[expr] == "matrix"):
self.write(symbol + " = " + "_sm.Matrix([i.collect(" + self.getValue(ctx.expr(2)) +
")" + "for i in " + expr + "])" +
".reshape((" + expr + ").shape[0], " + "(" + expr + ").shape[1])\n")
else:
self.write(self.getValue(ctx.expr(0)) + ".collect(" +
self.getValue(ctx.expr(2)) + ")\n")
# Eig(M, EigenValue, EigenVec)
elif func_name == "eig":
self.symbol_table.update({ctx.expr(1).getText().lower(): ctx.expr(1).getText().lower()})
self.symbol_table.update({ctx.expr(2).getText().lower(): ctx.expr(2).getText().lower()})
# _sm.Matrix([i.evalf() for i in (i_s_so).eigenvals().keys()])
self.write(ctx.expr(1).getText().lower() + " = " +
"_sm.Matrix([i.evalf() for i in " +
"(" + self.getValue(ctx.expr(0)) + ")" + ".eigenvals().keys()])\n")
# _sm.Matrix([i[2][0].evalf() for i in (i_s_o).eigenvects()]).reshape(i_s_o.shape[0], i_s_o.shape[1])
self.write(ctx.expr(2).getText().lower() + " = " +
"_sm.Matrix([i[2][0].evalf() for i in " + "(" + self.getValue(ctx.expr(0)) + ")" +
".eigenvects()]).reshape(" + self.getValue(ctx.expr(0)) + ".shape[0], " +
self.getValue(ctx.expr(0)) + ".shape[1])\n")
# Simprot(N, A, 3, qA)
elif func_name == "simprot":
# A.orient(N, 'Axis', qA, N.z)
if self.type2[ctx.expr(0).getText().lower()] == "frame":
frame1 = self.symbol_table2[ctx.expr(0).getText().lower()]
elif self.type2[ctx.expr(0).getText().lower()] == "bodies":
frame1 = self.symbol_table2[ctx.expr(0).getText().lower()] + "_f"
if self.type2[ctx.expr(1).getText().lower()] == "frame":
frame2 = self.symbol_table2[ctx.expr(1).getText().lower()]
elif self.type2[ctx.expr(1).getText().lower()] == "bodies":
frame2 = self.symbol_table2[ctx.expr(1).getText().lower()] + "_f"
e2 = ""
if ctx.expr(2).getText()[0] == "-":
e2 = "-1*"
if ctx.expr(2).getText() in ("1", "-1"):
e = frame1 + ".x"
elif ctx.expr(2).getText() in ("2", "-2"):
e = frame1 + ".y"
elif ctx.expr(2).getText() in ("3", "-3"):
e = frame1 + ".z"
else:
e = self.getValue(ctx.expr(2))
e2 = ""
if "degrees" in self.settings.keys() and self.settings["degrees"] == "off":
value = self.getValue(ctx.expr(3))
else:
if ctx.expr(3) in self.numeric_expr:
value = "_np.deg2rad(" + self.getValue(ctx.expr(3)) + ")"
else:
value = self.getValue(ctx.expr(3))
self.write(frame2 + ".orient(" + frame1 +
", " + "'Axis'" + ", " + "[" + value +
", " + e2 + e + "]" + ")\n")
# Express(A2>, B) *
elif func_name == "express":
if self.type2[ctx.expr(1).getText().lower()] == "bodies":
f = "_f"
else:
f = ""
if '_' in ctx.expr(0).getText().lower() and ctx.expr(0).getText().count('_') == 2:
vec = ctx.expr(0).getText().lower().replace(">", "").split('_')
v1 = self.symbol_table2[vec[1]]
v2 = self.symbol_table2[vec[2]]
if vec[0] == "p":
self.write(v2 + ".set_pos(" + v1 + ", " + "(" + self.getValue(ctx.expr(0)) +
")" + ".express(" + self.symbol_table2[ctx.expr(1).getText().lower()] + f + "))\n")
elif vec[0] == "v":
self.write(v1 + ".set_vel(" + v2 + ", " + "(" + self.getValue(ctx.expr(0)) +
")" + ".express(" + self.symbol_table2[ctx.expr(1).getText().lower()] + f + "))\n")
elif vec[0] == "a":
self.write(v1 + ".set_acc(" + v2 + ", " + "(" + self.getValue(ctx.expr(0)) +
")" + ".express(" + self.symbol_table2[ctx.expr(1).getText().lower()] + f + "))\n")
else:
self.write(self.getValue(ctx.expr(0)) + " = " + "(" + self.getValue(ctx.expr(0)) + ")" + ".express(" +
self.symbol_table2[ctx.expr(1).getText().lower()] + f + ")\n")
else:
self.write(self.getValue(ctx.expr(0)) + " = " + "(" + self.getValue(ctx.expr(0)) + ")" + ".express(" +
self.symbol_table2[ctx.expr(1).getText().lower()] + f + ")\n")
# Angvel(A, B)
elif func_name == "angvel":
self.write("print(" + self.symbol_table2[ctx.expr(1).getText().lower()] +
".ang_vel_in(" + self.symbol_table2[ctx.expr(0).getText().lower()] + "))\n")
# v2pts(N, A, O, P)
elif func_name in ("v2pts", "a2pts", "v2pt", "a1pt"):
if func_name == "v2pts":
text = ".v2pt_theory("
elif func_name == "a2pts":
text = ".a2pt_theory("
elif func_name == "v1pt":
text = ".v1pt_theory("
elif func_name == "a1pt":
text = ".a1pt_theory("
if self.type2[ctx.expr(1).getText().lower()] == "frame":
frame = self.symbol_table2[ctx.expr(1).getText().lower()]
elif self.type2[ctx.expr(1).getText().lower()] == "bodies":
frame = self.symbol_table2[ctx.expr(1).getText().lower()] + "_f"
expr_list = []
for i in range(2, 4):
if self.type2[ctx.expr(i).getText().lower()] == "point":
expr_list.append(self.symbol_table2[ctx.expr(i).getText().lower()])
elif self.type2[ctx.expr(i).getText().lower()] == "particle":
expr_list.append(self.symbol_table2[ctx.expr(i).getText().lower()] + ".point")
self.write(expr_list[1] + text + expr_list[0] +
"," + self.symbol_table2[ctx.expr(0).getText().lower()] + "," +
frame + ")\n")
# Gravity(g*N1>)
elif func_name == "gravity":
for i in self.bodies.keys():
if self.type2[i] == "bodies":
e = self.symbol_table2[i] + ".masscenter"
elif self.type2[i] == "particle":
e = self.symbol_table2[i] + ".point"
if e in self.forces.keys():
self.forces[e] = self.forces[e] + self.symbol_table2[i] +\
".mass*(" + self.getValue(ctx.expr(0)) + ")"
else:
self.forces.update({e: self.symbol_table2[i] +
".mass*(" + self.getValue(ctx.expr(0)) + ")"})
self.write("force_" + i + " = " + self.forces[e] + "\n")
# Explicit(EXPRESS(IMPLICIT>,C))
elif func_name == "explicit":
if ctx.expr(0) in self.vector_expr:
self.vector_expr.append(ctx)
expr = self.getValue(ctx.expr(0))
if self.explicit.keys():
explicit_list = []
for i in self.explicit.keys():
explicit_list.append(i + ":" + self.explicit[i])
if '_' in ctx.expr(0).getText().lower() and ctx.expr(0).getText().count('_') == 2:
vec = ctx.expr(0).getText().lower().replace(">", "").split('_')
v1 = self.symbol_table2[vec[1]]
v2 = self.symbol_table2[vec[2]]
if vec[0] == "p":
self.write(v2 + ".set_pos(" + v1 + ", " + "(" + expr +
")" + ".subs({" + ", ".join(explicit_list) + "}))\n")
elif vec[0] == "v":
self.write(v2 + ".set_vel(" + v1 + ", " + "(" + expr +
")" + ".subs({" + ", ".join(explicit_list) + "}))\n")
elif vec[0] == "a":
self.write(v2 + ".set_acc(" + v1 + ", " + "(" + expr +
")" + ".subs({" + ", ".join(explicit_list) + "}))\n")
else:
self.write(expr + " = " + "(" + expr + ")" + ".subs({" + ", ".join(explicit_list) + "})\n")
else:
self.write(expr + " = " + "(" + expr + ")" + ".subs({" + ", ".join(explicit_list) + "})\n")
# Force(O/Q, -k*Stretch*Uvec>)
elif func_name in ("force", "torque"):
if "/" in ctx.expr(0).getText().lower():
p1 = ctx.expr(0).getText().lower().split('/')[0]
p2 = ctx.expr(0).getText().lower().split('/')[1]
if self.type2[p1] in ("point", "frame"):
pt1 = self.symbol_table2[p1]
elif self.type2[p1] == "particle":
pt1 = self.symbol_table2[p1] + ".point"
if self.type2[p2] in ("point", "frame"):
pt2 = self.symbol_table2[p2]
elif self.type2[p2] == "particle":
pt2 = self.symbol_table2[p2] + ".point"
if pt1 in self.forces.keys():
self.forces[pt1] = self.forces[pt1] + " + -1*("+self.getValue(ctx.expr(1)) + ")"
self.write("force_" + p1 + " = " + self.forces[pt1] + "\n")
else:
self.forces.update({pt1: "-1*("+self.getValue(ctx.expr(1)) + ")"})
self.write("force_" + p1 + " = " + self.forces[pt1] + "\n")
if pt2 in self.forces.keys():
self.forces[pt2] = self.forces[pt2] + "+ " + self.getValue(ctx.expr(1))
self.write("force_" + p2 + " = " + self.forces[pt2] + "\n")
else:
self.forces.update({pt2: self.getValue(ctx.expr(1))})
self.write("force_" + p2 + " = " + self.forces[pt2] + "\n")
elif ctx.expr(0).getChildCount() == 1:
p1 = ctx.expr(0).getText().lower()
if self.type2[p1] in ("point", "frame"):
pt1 = self.symbol_table2[p1]
elif self.type2[p1] == "particle":
pt1 = self.symbol_table2[p1] + ".point"
if pt1 in self.forces.keys():
self.forces[pt1] = self.forces[pt1] + "+ -1*(" + self.getValue(ctx.expr(1)) + ")"
else:
self.forces.update({pt1: "-1*(" + self.getValue(ctx.expr(1)) + ")"})
# Constrain(Dependent[qB])
elif func_name == "constrain":
if ctx.getChild(2).getChild(0).getText().lower() == "dependent":
self.write("velocity_constraints = [i for i in dependent]\n")
x = (ctx.expr(0).getChildCount()-2)//2
for i in range(x):
self.dependent_variables.append(self.getValue(ctx.expr(0).expr(i)))
# Kane()
elif func_name == "kane":
if ctx.getChildCount() == 3:
self.kane_type = "no_args"
# Settings
def exitSettings(self, ctx):
# Stores settings like Complex on/off, Degrees on/off etc in self.settings.
try:
self.settings.update({ctx.getChild(0).getText().lower():
ctx.getChild(1).getText().lower()})
except Exception:
pass
def exitMassDecl2(self, ctx):
# Used for declaring the masses of particles and rigidbodies.
particle = self.symbol_table2[ctx.getChild(0).getText().lower()]
if ctx.getText().count("=") == 2:
if ctx.expr().expr(1) in self.numeric_expr:
e = "_sm.S(" + self.getValue(ctx.expr().expr(1)) + ")"
else:
e = self.getValue(ctx.expr().expr(1))
self.symbol_table.update({ctx.expr().expr(0).getText().lower(): ctx.expr().expr(0).getText().lower()})
self.write(ctx.expr().expr(0).getText().lower() + " = " + e + "\n")
mass = ctx.expr().expr(0).getText().lower()
else:
try:
if ctx.expr() in self.numeric_expr:
mass = "_sm.S(" + self.getValue(ctx.expr()) + ")"
else:
mass = self.getValue(ctx.expr())
except Exception:
a_text = ctx.expr().getText().lower()
self.symbol_table.update({a_text: a_text})
self.type.update({a_text: "constants"})
self.write(a_text + " = " + "_sm.symbols('" + a_text + "')\n")
mass = a_text
self.write(particle + ".mass = " + mass + "\n")
def exitInertiaDecl(self, ctx):
inertia_list = []
try:
ctx.ID(1).getText()
num = 5
except Exception:
num = 2
for i in range((ctx.getChildCount()-num)//2):
try:
if ctx.expr(i) in self.numeric_expr:
inertia_list.append("_sm.S(" + self.getValue(ctx.expr(i)) + ")")
else:
inertia_list.append(self.getValue(ctx.expr(i)))
except Exception:
a_text = ctx.expr(i).getText().lower()
self.symbol_table.update({a_text: a_text})
self.type.update({a_text: "constants"})
self.write(a_text + " = " + "_sm.symbols('" + a_text + "')\n")
inertia_list.append(a_text)
if len(inertia_list) < 6:
for i in range(6-len(inertia_list)):
inertia_list.append("0")
# body_a.inertia = (_me.inertia(body_a, I1, I2, I3, 0, 0, 0), body_a_cm)
try:
frame = self.symbol_table2[ctx.ID(1).getText().lower()]
point = self.symbol_table2[ctx.ID(0).getText().lower().split('_')[1]]
body = self.symbol_table2[ctx.ID(0).getText().lower().split('_')[0]]
self.inertia_point.update({ctx.ID(0).getText().lower().split('_')[0]
: ctx.ID(0).getText().lower().split('_')[1]})
self.write(body + ".inertia" + " = " + "(_me.inertia(" + frame + ", " +
", ".join(inertia_list) + "), " + point + ")\n")
except Exception:
body_name = self.symbol_table2[ctx.ID(0).getText().lower()]
body_name_cm = body_name + "_cm"
self.inertia_point.update({ctx.ID(0).getText().lower(): ctx.ID(0).getText().lower() + "o"})
self.write(body_name + ".inertia" + " = " + "(_me.inertia(" + body_name + "_f" + ", " +
", ".join(inertia_list) + "), " + body_name_cm + ")\n")
|
27353aa5c529c6f31b1aa66ac506c9a63140dcea76a4d7f983de801b18304710 | from sympy.external import import_module
from sympy.utilities.decorator import doctest_depends_on
@doctest_depends_on(modules=('antlr4',))
def parse_autolev(autolev_code, include_numeric=False):
"""Parses Autolev code (version 4.1) to SymPy code.
Parameters
=========
autolev_code : Can be an str or any object with a readlines() method (such as a file handle or StringIO).
include_numeric : boolean, optional
If True NumPy, PyDy, or other numeric code is included for numeric evaluation lines in the Autolev code.
Returns
=======
sympy_code : str
Equivalent sympy and/or numpy/pydy code as the input code.
Example (Double Pendulum)
=========================
>>> my_al_text = ("MOTIONVARIABLES' Q{2}', U{2}'",
... "CONSTANTS L,M,G",
... "NEWTONIAN N",
... "FRAMES A,B",
... "SIMPROT(N, A, 3, Q1)",
... "SIMPROT(N, B, 3, Q2)",
... "W_A_N>=U1*N3>",
... "W_B_N>=U2*N3>",
... "POINT O",
... "PARTICLES P,R",
... "P_O_P> = L*A1>",
... "P_P_R> = L*B1>",
... "V_O_N> = 0>",
... "V2PTS(N, A, O, P)",
... "V2PTS(N, B, P, R)",
... "MASS P=M, R=M",
... "Q1' = U1",
... "Q2' = U2",
... "GRAVITY(G*N1>)",
... "ZERO = FR() + FRSTAR()",
... "KANE()",
... "INPUT M=1,G=9.81,L=1",
... "INPUT Q1=.1,Q2=.2,U1=0,U2=0",
... "INPUT TFINAL=10, INTEGSTP=.01",
... "CODE DYNAMICS() some_filename.c")
>>> my_al_text = '\\n'.join(my_al_text)
>>> from sympy.parsing.autolev import parse_autolev
>>> print(parse_autolev(my_al_text, include_numeric=True))
import sympy.physics.mechanics as _me
import sympy as _sm
import math as m
import numpy as _np
<BLANKLINE>
q1, q2, u1, u2 = _me.dynamicsymbols('q1 q2 u1 u2')
q1_d, q2_d, u1_d, u2_d = _me.dynamicsymbols('q1_ q2_ u1_ u2_', 1)
l, m, g = _sm.symbols('l m g', real=True)
frame_n = _me.ReferenceFrame('n')
frame_a = _me.ReferenceFrame('a')
frame_b = _me.ReferenceFrame('b')
frame_a.orient(frame_n, 'Axis', [q1, frame_n.z])
frame_b.orient(frame_n, 'Axis', [q2, frame_n.z])
frame_a.set_ang_vel(frame_n, u1*frame_n.z)
frame_b.set_ang_vel(frame_n, u2*frame_n.z)
point_o = _me.Point('o')
particle_p = _me.Particle('p', _me.Point('p_pt'), _sm.Symbol('m'))
particle_r = _me.Particle('r', _me.Point('r_pt'), _sm.Symbol('m'))
particle_p.point.set_pos(point_o, l*frame_a.x)
particle_r.point.set_pos(particle_p.point, l*frame_b.x)
point_o.set_vel(frame_n, 0)
particle_p.point.v2pt_theory(point_o,frame_n,frame_a)
particle_r.point.v2pt_theory(particle_p.point,frame_n,frame_b)
particle_p.mass = m
particle_r.mass = m
force_p = particle_p.mass*(g*frame_n.x)
force_r = particle_r.mass*(g*frame_n.x)
kd_eqs = [q1_d - u1, q2_d - u2]
forceList = [(particle_p.point,particle_p.mass*(g*frame_n.x)), (particle_r.point,particle_r.mass*(g*frame_n.x))]
kane = _me.KanesMethod(frame_n, q_ind=[q1,q2], u_ind=[u1, u2], kd_eqs = kd_eqs)
fr, frstar = kane.kanes_equations([particle_p, particle_r], forceList)
zero = fr+frstar
from pydy.system import System
sys = System(kane, constants = {l:1, m:1, g:9.81},
specifieds={},
initial_conditions={q1:.1, q2:.2, u1:0, u2:0},
times = _np.linspace(0.0, 10, 10/.01))
<BLANKLINE>
y=sys.integrate()
<BLANKLINE>
"""
_autolev = import_module(
'sympy.parsing.autolev._parse_autolev_antlr',
import_kwargs={'fromlist': ['X']})
if _autolev is not None:
return _autolev.parse_autolev(autolev_code, include_numeric)
|
f76dc4aaa21de29b811ba56c1f6cf7177e51d3ea318a5bf72fba5dc13ee3ffb7 | # 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 sympy.physics.quantum.state import Bra, Ket
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)
elif rel.NEQ():
return sympy.Ne(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.floor():
return convert_floor(comp.floor())
elif comp.ceil():
return convert_ceil(comp.ceil())
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)
elif atom.bra():
val = convert_expr(atom.bra().expr())
return Bra(val)
elif atom.ket():
val = convert_expr(atom.ket().expr())
return Ket(val)
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_floor(floor):
val = convert_expr(floor.val)
return sympy.floor(val, evaluate=False)
def convert_ceil(ceil):
val = convert_expr(ceil.val)
return sympy.ceiling(val, 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():
if func.subexpr().expr():
base = convert_expr(func.subexpr().expr())
else:
base = convert_atom(func.subexpr().atom())
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, evaluate=False)
else:
return sympy.sqrt(expr, evaluate=False)
elif func.FUNC_OVERLINE():
expr = convert_expr(func.base)
return sympy.conjugate(expr, evaluate=False)
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
|
7882841731309021613c5d064e67fa48bbb15d3dfce06a9d0899e188bd94d014 | import sympy.physics.mechanics as _me
import sympy as _sm
import math as m
import numpy as _np
f = _sm.S(3)
g = _sm.S(9.81)
a, b = _sm.symbols('a b', real=True)
s, s1 = _sm.symbols('s s1', real=True)
s2, s3 = _sm.symbols('s2 s3', real=True, nonnegative=True)
s4 = _sm.symbols('s4', real=True, nonpositive=True)
k1, k2, k3, k4, l1, l2, l3, p11, p12, p13, p21, p22, p23 = _sm.symbols('k1 k2 k3 k4 l1 l2 l3 p11 p12 p13 p21 p22 p23', real=True)
c11, c12, c13, c21, c22, c23 = _sm.symbols('c11 c12 c13 c21 c22 c23', real=True)
e1 = a*f+s2-g
e2 = f**2+k3*k2*g
|
235b6c41c4805bac2a220b85fbb970963f43e1867c90266a3ea4ca3d41d1f682 | import sympy.physics.mechanics as _me
import sympy as _sm
import math as m
import numpy as _np
x, y = _me.dynamicsymbols('x y')
a, b = _sm.symbols('a b', real=True)
e = a*(b*x+y)**2
m = _sm.Matrix([e,e]).reshape(2, 1)
e = e.expand()
m = _sm.Matrix([i.expand() for i in m]).reshape((m).shape[0], (m).shape[1])
e = _sm.factor(e, x)
m = _sm.Matrix([_sm.factor(i,x) for i in m]).reshape((m).shape[0], (m).shape[1])
eqn = _sm.Matrix([[0]])
eqn[0] = a*x+b*y
eqn = eqn.row_insert(eqn.shape[0], _sm.Matrix([[0]]))
eqn[eqn.shape[0]-1] = 2*a*x-3*b*y
print(_sm.solve(eqn,x,y))
rhs_y = _sm.solve(eqn,x,y)[y]
e = (x+y)**2+2*x**2
e.collect(x)
a, b, c = _sm.symbols('a b c', real=True)
m = _sm.Matrix([a,b,c,0]).reshape(2, 2)
m2 = _sm.Matrix([i.subs({a:1,b:2,c:3}) for i in m]).reshape((m).shape[0], (m).shape[1])
eigvalue = _sm.Matrix([i.evalf() for i in (m2).eigenvals().keys()])
eigvec = _sm.Matrix([i[2][0].evalf() for i in (m2).eigenvects()]).reshape(m2.shape[0], m2.shape[1])
frame_n = _me.ReferenceFrame('n')
frame_a = _me.ReferenceFrame('a')
frame_a.orient(frame_n, 'Axis', [x, frame_n.x])
frame_a.orient(frame_n, 'Axis', [_sm.pi/2, frame_n.x])
c1, c2, c3 = _sm.symbols('c1 c2 c3', real=True)
v = c1*frame_a.x+c2*frame_a.y+c3*frame_a.z
point_o = _me.Point('o')
point_p = _me.Point('p')
point_o.set_pos(point_p, c1*frame_a.x)
v = (v).express(frame_n)
point_o.set_pos(point_p, (point_o.pos_from(point_p)).express(frame_n))
frame_a.set_ang_vel(frame_n, c3*frame_a.z)
print(frame_n.ang_vel_in(frame_a))
point_p.v2pt_theory(point_o,frame_n,frame_a)
particle_p1 = _me.Particle('p1', _me.Point('p1_pt'), _sm.Symbol('m'))
particle_p2 = _me.Particle('p2', _me.Point('p2_pt'), _sm.Symbol('m'))
particle_p2.point.v2pt_theory(particle_p1.point,frame_n,frame_a)
point_p.a2pt_theory(particle_p1.point,frame_n,frame_a)
body_b1_cm = _me.Point('b1_cm')
body_b1_cm.set_vel(frame_n, 0)
body_b1_f = _me.ReferenceFrame('b1_f')
body_b1 = _me.RigidBody('b1', body_b1_cm, body_b1_f, _sm.symbols('m'), (_me.outer(body_b1_f.x,body_b1_f.x),body_b1_cm))
body_b2_cm = _me.Point('b2_cm')
body_b2_cm.set_vel(frame_n, 0)
body_b2_f = _me.ReferenceFrame('b2_f')
body_b2 = _me.RigidBody('b2', body_b2_cm, body_b2_f, _sm.symbols('m'), (_me.outer(body_b2_f.x,body_b2_f.x),body_b2_cm))
g = _sm.symbols('g', real=True)
force_p1 = particle_p1.mass*(g*frame_n.x)
force_p2 = particle_p2.mass*(g*frame_n.x)
force_b1 = body_b1.mass*(g*frame_n.x)
force_b2 = body_b2.mass*(g*frame_n.x)
z = _me.dynamicsymbols('z')
v = x*frame_a.x+y*frame_a.z
point_o.set_pos(point_p, x*frame_a.x+y*frame_a.y)
v = (v).subs({x:2*z, y:z})
point_o.set_pos(point_p, (point_o.pos_from(point_p)).subs({x:2*z, y:z}))
force_o = -1*(x*y*frame_a.x)
force_p1 = particle_p1.mass*(g*frame_n.x)+ x*y*frame_a.x
|
3b72b7210a3e1c28c020e4a47f3ddb0e5b2dedd554891c21399d2aff78dbe4b5 | import sympy.physics.mechanics as _me
import sympy as _sm
import math as m
import numpy as _np
frame_a = _me.ReferenceFrame('a')
frame_b = _me.ReferenceFrame('b')
frame_n = _me.ReferenceFrame('n')
x1, x2, x3 = _me.dynamicsymbols('x1 x2 x3')
l = _sm.symbols('l', real=True)
v1 = x1*frame_a.x+x2*frame_a.y+x3*frame_a.z
v2 = x1*frame_b.x+x2*frame_b.y+x3*frame_b.z
v3 = x1*frame_n.x+x2*frame_n.y+x3*frame_n.z
v = v1+v2+v3
point_c = _me.Point('c')
point_d = _me.Point('d')
point_po1 = _me.Point('po1')
point_po2 = _me.Point('po2')
point_po3 = _me.Point('po3')
particle_l = _me.Particle('l', _me.Point('l_pt'), _sm.Symbol('m'))
particle_p1 = _me.Particle('p1', _me.Point('p1_pt'), _sm.Symbol('m'))
particle_p2 = _me.Particle('p2', _me.Point('p2_pt'), _sm.Symbol('m'))
particle_p3 = _me.Particle('p3', _me.Point('p3_pt'), _sm.Symbol('m'))
body_s_cm = _me.Point('s_cm')
body_s_cm.set_vel(frame_n, 0)
body_s_f = _me.ReferenceFrame('s_f')
body_s = _me.RigidBody('s', body_s_cm, body_s_f, _sm.symbols('m'), (_me.outer(body_s_f.x,body_s_f.x),body_s_cm))
body_r1_cm = _me.Point('r1_cm')
body_r1_cm.set_vel(frame_n, 0)
body_r1_f = _me.ReferenceFrame('r1_f')
body_r1 = _me.RigidBody('r1', body_r1_cm, body_r1_f, _sm.symbols('m'), (_me.outer(body_r1_f.x,body_r1_f.x),body_r1_cm))
body_r2_cm = _me.Point('r2_cm')
body_r2_cm.set_vel(frame_n, 0)
body_r2_f = _me.ReferenceFrame('r2_f')
body_r2 = _me.RigidBody('r2', body_r2_cm, body_r2_f, _sm.symbols('m'), (_me.outer(body_r2_f.x,body_r2_f.x),body_r2_cm))
v4 = x1*body_s_f.x+x2*body_s_f.y+x3*body_s_f.z
body_s_cm.set_pos(point_c, l*frame_n.x)
|
bc43b48cc383f8a21ebc07157b19a9bda7343068cdecefdd362a4e04924dcc5d | import sympy.physics.mechanics as _me
import sympy as _sm
import math as m
import numpy as _np
q1, q2 = _me.dynamicsymbols('q1 q2')
x, y, z = _me.dynamicsymbols('x y z')
e = q1+q2
a = (e).subs({q1:x**2+y**2, q2:x-y})
e2 = _sm.cos(x)
e3 = _sm.cos(x*y)
a = (e2).series(x, 0, 2).removeO()
b = (e3).series(x, 0, 2).removeO().series(y, 0, 2).removeO()
e = ((x+y)**2).expand()
a = (e).subs({q1:x**2+y**2,q2:x-y}).subs({x:1,y:z})
bm = _sm.Matrix([i.subs({x:1,y:z}) for i in _sm.Matrix([e,2*e]).reshape(2, 1)]).reshape((_sm.Matrix([e,2*e]).reshape(2, 1)).shape[0], (_sm.Matrix([e,2*e]).reshape(2, 1)).shape[1])
e = q1+q2
a = (e).subs({q1:x**2+y**2,q2:x-y}).subs({x:2,y:z**2})
j, k, l = _sm.symbols('j k l', real=True)
p1 = _sm.Poly(_sm.Matrix([j,k,l]).reshape(1, 3), x)
p2 = _sm.Poly(j*x+k, x)
root1 = [i.evalf() for i in _sm.solve(p1, x)]
root2 = [i.evalf() for i in _sm.solve(_sm.Poly(_sm.Matrix([1,2,3]).reshape(3, 1), x),x)]
m = _sm.Matrix([1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16]).reshape(4, 4)
am = (m).T+m
bm = _sm.Matrix([i.evalf() for i in (m).eigenvals().keys()])
c1 = _sm.diag(1,1,1,1)
c2 = _sm.Matrix([2 if i==j else 0 for i in range(3) for j in range(4)]).reshape(3, 4)
dm = (m+c1)**(-1)
e = (m+c1).det()+(_sm.Matrix([1,0,0,1]).reshape(2, 2)).trace()
f = (m)[1,2]
a = (m).cols
bm = (m).col(0)
cm = _sm.Matrix([(m).T.row(0),(m).T.row(1),(m).T.row(2),(m).T.row(3),(m).T.row(2)])
dm = (m).row(0)
em = _sm.Matrix([(m).row(0),(m).row(1),(m).row(2),(m).row(3),(m).row(2)])
|
f2d81bc09fa2af4c22382b20205080bb8b83f1289e611acba0bcc47da7b96106 | import sympy.physics.mechanics as _me
import sympy as _sm
import math as m
import numpy as _np
frame_a = _me.ReferenceFrame('a')
c1, c2, c3 = _sm.symbols('c1 c2 c3', real=True)
a = _me.inertia(frame_a, 1, 1, 1)
particle_p1 = _me.Particle('p1', _me.Point('p1_pt'), _sm.Symbol('m'))
particle_p2 = _me.Particle('p2', _me.Point('p2_pt'), _sm.Symbol('m'))
body_r_cm = _me.Point('r_cm')
body_r_f = _me.ReferenceFrame('r_f')
body_r = _me.RigidBody('r', body_r_cm, body_r_f, _sm.symbols('m'), (_me.outer(body_r_f.x,body_r_f.x),body_r_cm))
frame_a.orient(body_r_f, 'DCM', _sm.Matrix([1,1,1,1,1,0,0,0,1]).reshape(3, 3))
point_o = _me.Point('o')
m1 = _sm.symbols('m1')
particle_p1.mass = m1
m2 = _sm.symbols('m2')
particle_p2.mass = m2
mr = _sm.symbols('mr')
body_r.mass = mr
i1 = _sm.symbols('i1')
i2 = _sm.symbols('i2')
i3 = _sm.symbols('i3')
body_r.inertia = (_me.inertia(body_r_f, i1, i2, i3, 0, 0, 0), body_r_cm)
point_o.set_pos(particle_p1.point, c1*frame_a.x)
point_o.set_pos(particle_p2.point, c2*frame_a.y)
point_o.set_pos(body_r_cm, c3*frame_a.z)
a = _me.inertia_of_point_mass(particle_p1.mass, particle_p1.point.pos_from(point_o), frame_a)
a = _me.inertia_of_point_mass(particle_p2.mass, particle_p2.point.pos_from(point_o), frame_a)
a = body_r.inertia[0] + _me.inertia_of_point_mass(body_r.mass, body_r.masscenter.pos_from(point_o), frame_a)
a = _me.inertia_of_point_mass(particle_p1.mass, particle_p1.point.pos_from(point_o), frame_a) + _me.inertia_of_point_mass(particle_p2.mass, particle_p2.point.pos_from(point_o), frame_a) + body_r.inertia[0] + _me.inertia_of_point_mass(body_r.mass, body_r.masscenter.pos_from(point_o), frame_a)
a = _me.inertia_of_point_mass(particle_p1.mass, particle_p1.point.pos_from(point_o), frame_a) + body_r.inertia[0] + _me.inertia_of_point_mass(body_r.mass, body_r.masscenter.pos_from(point_o), frame_a)
a = body_r.inertia[0] + _me.inertia_of_point_mass(body_r.mass, body_r.masscenter.pos_from(point_o), frame_a)
a = body_r.inertia[0]
particle_p2.point.set_pos(particle_p1.point, c1*frame_a.x+c2*frame_a.y)
body_r_cm.set_pos(particle_p1.point, c3*frame_a.x)
body_r_cm.set_pos(particle_p2.point, c3*frame_a.y)
b = _me.functions.center_of_mass(point_o,particle_p1, particle_p2, body_r)
b = _me.functions.center_of_mass(point_o,particle_p1, body_r)
b = _me.functions.center_of_mass(particle_p1.point,particle_p1, particle_p2, body_r)
u1, u2, u3 = _me.dynamicsymbols('u1 u2 u3')
v = u1*frame_a.x+u2*frame_a.y+u3*frame_a.z
u = (v+c1*frame_a.x).normalize()
particle_p1.point.set_vel(frame_a, u1*frame_a.x)
a = particle_p1.point.partial_velocity(frame_a, u1)
m = particle_p1.mass+body_r.mass
m = particle_p2.mass
m = particle_p1.mass+particle_p2.mass+body_r.mass
|
1ada95f96ab61862737db94cb1c033f8a5c2cecd0fff85eabc0dc521d94f7b4e | import sympy.physics.mechanics as _me
import sympy as _sm
import math as m
import numpy as _np
frame_n = _me.ReferenceFrame('n')
frame_a = _me.ReferenceFrame('a')
a = 0
d = _me.inertia(frame_a, 1, 1, 1)
point_po1 = _me.Point('po1')
point_po2 = _me.Point('po2')
particle_p1 = _me.Particle('p1', _me.Point('p1_pt'), _sm.Symbol('m'))
particle_p2 = _me.Particle('p2', _me.Point('p2_pt'), _sm.Symbol('m'))
c1, c2, c3 = _me.dynamicsymbols('c1 c2 c3')
c1_d, c2_d, c3_d = _me.dynamicsymbols('c1_ c2_ c3_', 1)
body_r_cm = _me.Point('r_cm')
body_r_cm.set_vel(frame_n, 0)
body_r_f = _me.ReferenceFrame('r_f')
body_r = _me.RigidBody('r', body_r_cm, body_r_f, _sm.symbols('m'), (_me.outer(body_r_f.x,body_r_f.x),body_r_cm))
point_po2.set_pos(particle_p1.point, c1*frame_a.x)
v = 2*point_po2.pos_from(particle_p1.point)+c2*frame_a.y
frame_a.set_ang_vel(frame_n, c3*frame_a.z)
v = 2*frame_a.ang_vel_in(frame_n)+c2*frame_a.y
body_r_f.set_ang_vel(frame_n, c3*frame_a.z)
v = 2*body_r_f.ang_vel_in(frame_n)+c2*frame_a.y
frame_a.set_ang_acc(frame_n, (frame_a.ang_vel_in(frame_n)).dt(frame_a))
v = 2*frame_a.ang_acc_in(frame_n)+c2*frame_a.y
particle_p1.point.set_vel(frame_a, c1*frame_a.x+c3*frame_a.y)
body_r_cm.set_acc(frame_n, c2*frame_a.y)
v_a = _me.cross(body_r_cm.acc(frame_n), particle_p1.point.vel(frame_a))
x_b_c = v_a
x_b_d = 2*x_b_c
a_b_c_d_e = x_b_d*2
a_b_c = 2*c1*c2*c3
a_b_c += 2*c1
a_b_c = 3*c1
q1, q2, u1, u2 = _me.dynamicsymbols('q1 q2 u1 u2')
q1_d, q2_d, u1_d, u2_d = _me.dynamicsymbols('q1_ q2_ u1_ u2_', 1)
x, y = _me.dynamicsymbols('x y')
x_d, y_d = _me.dynamicsymbols('x_ y_', 1)
x_dd, y_dd = _me.dynamicsymbols('x_ y_', 2)
yy = _me.dynamicsymbols('yy')
yy = x*x_d**2+1
m = _sm.Matrix([[0]])
m[0] = 2*x
m = m.row_insert(m.shape[0], _sm.Matrix([[0]]))
m[m.shape[0]-1] = 2*y
a = 2*m[0]
m = _sm.Matrix([1,2,3,4,5,6,7,8,9]).reshape(3, 3)
m[0,1] = 5
a = m[0, 1]*2
force_ro = q1*frame_n.x
torque_a = q2*frame_n.z
force_ro = q1*frame_n.x + q2*frame_n.y
f = force_ro*2
|
fd7bccacc7b9afd44ba294eb22a3062e1698bc72f5aa3b5e5b3f4229ca2d08bf | import sympy.physics.mechanics as _me
import sympy as _sm
import math as m
import numpy as _np
x, y = _me.dynamicsymbols('x y')
x_d, y_d = _me.dynamicsymbols('x_ y_', 1)
e = _sm.cos(x)+_sm.sin(x)+_sm.tan(x)+_sm.cosh(x)+_sm.sinh(x)+_sm.tanh(x)+_sm.acos(x)+_sm.asin(x)+_sm.atan(x)+_sm.log(x)+_sm.exp(x)+_sm.sqrt(x)+_sm.factorial(x)+_sm.ceiling(x)+_sm.floor(x)+_sm.sign(x)
e = (x)**2+_sm.log(x, 10)
a = _sm.Abs(-1*1)+int(1.5)+round(1.9)
e1 = 2*x+3*y
e2 = x+y
am = _sm.Matrix([e1.expand().coeff(x), e1.expand().coeff(y), e2.expand().coeff(x), e2.expand().coeff(y)]).reshape(2, 2)
b = (e1).expand().coeff(x)
c = (e2).expand().coeff(y)
d1 = (e1).collect(x).coeff(x,0)
d2 = (e1).collect(x).coeff(x,1)
fm = _sm.Matrix([i.collect(x)for i in _sm.Matrix([e1,e2]).reshape(1, 2)]).reshape((_sm.Matrix([e1,e2]).reshape(1, 2)).shape[0], (_sm.Matrix([e1,e2]).reshape(1, 2)).shape[1])
f = (e1).collect(y)
g = (e1).subs({x:2*x})
gm = _sm.Matrix([i.subs({x:3}) for i in _sm.Matrix([e1,e2]).reshape(2, 1)]).reshape((_sm.Matrix([e1,e2]).reshape(2, 1)).shape[0], (_sm.Matrix([e1,e2]).reshape(2, 1)).shape[1])
frame_a = _me.ReferenceFrame('a')
frame_b = _me.ReferenceFrame('b')
theta = _me.dynamicsymbols('theta')
frame_b.orient(frame_a, 'Axis', [theta, frame_a.z])
v1 = 2*frame_a.x-3*frame_a.y+frame_a.z
v2 = frame_b.x+frame_b.y+frame_b.z
a = _me.dot(v1, v2)
bm = _sm.Matrix([_me.dot(v1, v2),_me.dot(v1, 2*v2)]).reshape(2, 1)
c = _me.cross(v1, v2)
d = 2*v1.magnitude()+3*v1.magnitude()
dyadic = _me.outer(3*frame_a.x, frame_a.x)+_me.outer(frame_a.y, frame_a.y)+_me.outer(2*frame_a.z, frame_a.z)
am = (dyadic).to_matrix(frame_b)
m = _sm.Matrix([1,2,3]).reshape(3, 1)
v = m[0]*frame_a.x +m[1]*frame_a.y +m[2]*frame_a.z
|
646dfab373e7913d1a28133d371e87d2c749aeda0bc1a79b53ddfce98494aa5f | import sympy.physics.mechanics as _me
import sympy as _sm
import math as m
import numpy as _np
x, y = _me.dynamicsymbols('x y')
a, b, r = _sm.symbols('a b r', real=True)
eqn = _sm.Matrix([[0]])
eqn[0] = a*x**3+b*y**2-r
eqn = eqn.row_insert(eqn.shape[0], _sm.Matrix([[0]]))
eqn[eqn.shape[0]-1] = a*_sm.sin(x)**2+b*_sm.cos(2*y)-r**2
matrix_list = []
for i in eqn:matrix_list.append(i.subs({a:2.0, b:3.0, r:1.0}))
print(_sm.nsolve(matrix_list,(x,y),(_np.deg2rad(30),3.14)))
|
5af52d9e8d43dc1ae614d3d860804a47f80e03e3da1e8c4b9524cd0d7ebb75c4 | import sympy.physics.mechanics as _me
import sympy as _sm
import math as m
import numpy as _np
x, y = _me.dynamicsymbols('x y')
x_d, y_d = _me.dynamicsymbols('x_ y_', 1)
e1 = (x+y)**2+(x-y)**3
e2 = (x-y)**2
e3 = x**2+y**2+2*x*y
m1 = _sm.Matrix([e1,e2]).reshape(2, 1)
m2 = _sm.Matrix([(x+y)**2,(x-y)**2]).reshape(1, 2)
m3 = m1+_sm.Matrix([x,y]).reshape(2, 1)
am = _sm.Matrix([i.expand() for i in m1]).reshape((m1).shape[0], (m1).shape[1])
cm = _sm.Matrix([i.expand() for i in _sm.Matrix([(x+y)**2,(x-y)**2]).reshape(1, 2)]).reshape((_sm.Matrix([(x+y)**2,(x-y)**2]).reshape(1, 2)).shape[0], (_sm.Matrix([(x+y)**2,(x-y)**2]).reshape(1, 2)).shape[1])
em = _sm.Matrix([i.expand() for i in m1+_sm.Matrix([x,y]).reshape(2, 1)]).reshape((m1+_sm.Matrix([x,y]).reshape(2, 1)).shape[0], (m1+_sm.Matrix([x,y]).reshape(2, 1)).shape[1])
f = (e1).expand()
g = (e2).expand()
a = _sm.factor((e3), x)
bm = _sm.Matrix([_sm.factor(i, x) for i in m1]).reshape((m1).shape[0], (m1).shape[1])
cm = _sm.Matrix([_sm.factor(i, x) for i in m1+_sm.Matrix([x,y]).reshape(2, 1)]).reshape((m1+_sm.Matrix([x,y]).reshape(2, 1)).shape[0], (m1+_sm.Matrix([x,y]).reshape(2, 1)).shape[1])
a = (e3).diff(x)
b = (e3).diff(y)
cm = _sm.Matrix([i.diff(x) for i in m2]).reshape((m2).shape[0], (m2).shape[1])
dm = _sm.Matrix([i.diff(x) for i in m1+_sm.Matrix([x,y]).reshape(2, 1)]).reshape((m1+_sm.Matrix([x,y]).reshape(2, 1)).shape[0], (m1+_sm.Matrix([x,y]).reshape(2, 1)).shape[1])
frame_a = _me.ReferenceFrame('a')
frame_b = _me.ReferenceFrame('b')
frame_b.orient(frame_a, 'DCM', _sm.Matrix([1,0,0,1,0,0,1,0,0]).reshape(3, 3))
v1 = x*frame_a.x+y*frame_a.y+x*y*frame_a.z
e = (v1).diff(x, frame_b)
fm = _sm.Matrix([i.diff(_sm.Symbol('t')) for i in m1]).reshape((m1).shape[0], (m1).shape[1])
gm = _sm.Matrix([i.diff(_sm.Symbol('t')) for i in _sm.Matrix([(x+y)**2,(x-y)**2]).reshape(1, 2)]).reshape((_sm.Matrix([(x+y)**2,(x-y)**2]).reshape(1, 2)).shape[0], (_sm.Matrix([(x+y)**2,(x-y)**2]).reshape(1, 2)).shape[1])
h = (v1).dt(frame_b)
|
587a1de72cca1784cd6c47f861fc66c7d5a78a603b34e5eab538d95d1f05b0fd | import sympy.physics.mechanics as _me
import sympy as _sm
import math as m
import numpy as _np
frame_a = _me.ReferenceFrame('a')
frame_b = _me.ReferenceFrame('b')
q1, q2, q3 = _me.dynamicsymbols('q1 q2 q3')
frame_b.orient(frame_a, 'Axis', [q3, frame_a.x])
dcm = frame_a.dcm(frame_b)
m = dcm*3-frame_a.dcm(frame_b)
r = _me.dynamicsymbols('r')
circle_area = _sm.pi*r**2
u, a = _me.dynamicsymbols('u a')
x, y = _me.dynamicsymbols('x y')
s = u*_me.dynamicsymbols._t-1/2*a*_me.dynamicsymbols._t**2
expr1 = 2*a*0.5-1.25+0.25
expr2 = -1*x**2+y**2+0.25*(x+y)**2
expr3 = 0.5*10**(-10)
dyadic = _me.outer(frame_a.x, frame_a.x)+_me.outer(frame_a.y, frame_a.y)+_me.outer(frame_a.z, frame_a.z)
|
7584d1b57bcc0d71e22b35c70f64a1d1f704ba4a1bdf0aff19b49ea9a66bacef | import sympy.physics.mechanics as _me
import sympy as _sm
import math as m
import numpy as _np
x, y = _me.dynamicsymbols('x y')
a11, a12, a21, a22, b1, b2 = _sm.symbols('a11 a12 a21 a22 b1 b2', real=True)
eqn = _sm.Matrix([[0]])
eqn[0] = a11*x+a12*y-b1
eqn = eqn.row_insert(eqn.shape[0], _sm.Matrix([[0]]))
eqn[eqn.shape[0]-1] = a21*x+a22*y-b2
eqn_list = []
for i in eqn: eqn_list.append(i.subs({a11:2, a12:5, a21:3, a22:4, b1:7, b2:6}))
print(_sm.linsolve(eqn_list, x,y))
|
8eb25f6f4264d853f8192e690dad14079a61d228c455dd57f2678a799ad35678 | import sympy.physics.mechanics as _me
import sympy as _sm
import math as m
import numpy as _np
x1, x2 = _me.dynamicsymbols('x1 x2')
f1 = x1*x2+3*x1**2
f2 = x1*_me.dynamicsymbols._t+x2*_me.dynamicsymbols._t**2
x, y = _me.dynamicsymbols('x y')
x_d, y_d = _me.dynamicsymbols('x_ y_', 1)
y_dd = _me.dynamicsymbols('y_', 2)
q1, q2, q3, u1, u2 = _me.dynamicsymbols('q1 q2 q3 u1 u2')
p1, p2 = _me.dynamicsymbols('p1 p2')
p1_d, p2_d = _me.dynamicsymbols('p1_ p2_', 1)
w1, w2, w3, r1, r2 = _me.dynamicsymbols('w1 w2 w3 r1 r2')
w1_d, w2_d, w3_d, r1_d, r2_d = _me.dynamicsymbols('w1_ w2_ w3_ r1_ r2_', 1)
r1_dd, r2_dd = _me.dynamicsymbols('r1_ r2_', 2)
c11, c12, c21, c22 = _me.dynamicsymbols('c11 c12 c21 c22')
d11, d12, d13 = _me.dynamicsymbols('d11 d12 d13')
j1, j2 = _me.dynamicsymbols('j1 j2')
n = _sm.symbols('n')
n = _sm.I
|
152bb84cfd810d3423cd884c91ca518576dbde4003dbb5c2c8efc4a0be7901c9 | import sympy.physics.mechanics as _me
import sympy as _sm
import math as m
import numpy as _np
g, lb, w, h = _sm.symbols('g lb w h', real=True)
theta, phi, omega, alpha = _me.dynamicsymbols('theta phi omega alpha')
theta_d, phi_d, omega_d, alpha_d = _me.dynamicsymbols('theta_ phi_ omega_ alpha_', 1)
theta_dd, phi_dd = _me.dynamicsymbols('theta_ phi_', 2)
frame_n = _me.ReferenceFrame('n')
body_a_cm = _me.Point('a_cm')
body_a_cm.set_vel(frame_n, 0)
body_a_f = _me.ReferenceFrame('a_f')
body_a = _me.RigidBody('a', body_a_cm, body_a_f, _sm.symbols('m'), (_me.outer(body_a_f.x,body_a_f.x),body_a_cm))
body_b_cm = _me.Point('b_cm')
body_b_cm.set_vel(frame_n, 0)
body_b_f = _me.ReferenceFrame('b_f')
body_b = _me.RigidBody('b', body_b_cm, body_b_f, _sm.symbols('m'), (_me.outer(body_b_f.x,body_b_f.x),body_b_cm))
body_a_f.orient(frame_n, 'Axis', [theta, frame_n.y])
body_b_f.orient(body_a_f, 'Axis', [phi, body_a_f.z])
point_o = _me.Point('o')
la = (lb-h/2)/2
body_a_cm.set_pos(point_o, la*body_a_f.z)
body_b_cm.set_pos(point_o, lb*body_a_f.z)
body_a_f.set_ang_vel(frame_n, omega*frame_n.y)
body_b_f.set_ang_vel(body_a_f, alpha*body_a_f.z)
point_o.set_vel(frame_n, 0)
body_a_cm.v2pt_theory(point_o,frame_n,body_a_f)
body_b_cm.v2pt_theory(point_o,frame_n,body_a_f)
ma = _sm.symbols('ma')
body_a.mass = ma
mb = _sm.symbols('mb')
body_b.mass = mb
iaxx = 1/12*ma*(2*la)**2
iayy = iaxx
iazz = 0
ibxx = 1/12*mb*h**2
ibyy = 1/12*mb*(w**2+h**2)
ibzz = 1/12*mb*w**2
body_a.inertia = (_me.inertia(body_a_f, iaxx, iayy, iazz, 0, 0, 0), body_a_cm)
body_b.inertia = (_me.inertia(body_b_f, ibxx, ibyy, ibzz, 0, 0, 0), body_b_cm)
force_a = body_a.mass*(g*frame_n.z)
force_b = body_b.mass*(g*frame_n.z)
kd_eqs = [theta_d - omega, phi_d - alpha]
forceList = [(body_a.masscenter,body_a.mass*(g*frame_n.z)), (body_b.masscenter,body_b.mass*(g*frame_n.z))]
kane = _me.KanesMethod(frame_n, q_ind=[theta,phi], u_ind=[omega, alpha], kd_eqs = kd_eqs)
fr, frstar = kane.kanes_equations([body_a, body_b], forceList)
zero = fr+frstar
from pydy.system import System
sys = System(kane, constants = {g:9.81, lb:0.2, w:0.2, h:0.1, ma:0.01, mb:0.1},
specifieds={},
initial_conditions={theta:_np.deg2rad(90), phi:_np.deg2rad(0.5), omega:0, alpha:0},
times = _np.linspace(0.0, 10, 10/0.02))
y=sys.integrate()
|
0ddc5c5ab9b71cc42ecb26373e4eac2876f845785010cfc4298dc761908ff1e7 | import sympy.physics.mechanics as _me
import sympy as _sm
import math as m
import numpy as _np
q1, q2 = _me.dynamicsymbols('q1 q2')
q1_d, q2_d = _me.dynamicsymbols('q1_ q2_', 1)
q1_dd, q2_dd = _me.dynamicsymbols('q1_ q2_', 2)
l, m, g = _sm.symbols('l m g', real=True)
frame_n = _me.ReferenceFrame('n')
point_pn = _me.Point('pn')
point_pn.set_vel(frame_n, 0)
theta1 = _sm.atan(q2/q1)
frame_a = _me.ReferenceFrame('a')
frame_a.orient(frame_n, 'Axis', [theta1, frame_n.z])
particle_p = _me.Particle('p', _me.Point('p_pt'), _sm.Symbol('m'))
particle_p.point.set_pos(point_pn, q1*frame_n.x+q2*frame_n.y)
particle_p.mass = m
particle_p.point.set_vel(frame_n, (point_pn.pos_from(particle_p.point)).dt(frame_n))
f_v = _me.dot((particle_p.point.vel(frame_n)).express(frame_a), frame_a.x)
force_p = particle_p.mass*(g*frame_n.x)
dependent = _sm.Matrix([[0]])
dependent[0] = f_v
velocity_constraints = [i for i in dependent]
u_q1_d = _me.dynamicsymbols('u_q1_d')
u_q2_d = _me.dynamicsymbols('u_q2_d')
kd_eqs = [q1_d-u_q1_d, q2_d-u_q2_d]
forceList = [(particle_p.point,particle_p.mass*(g*frame_n.x))]
kane = _me.KanesMethod(frame_n, q_ind=[q1,q2], u_ind=[u_q2_d], u_dependent=[u_q1_d], kd_eqs = kd_eqs, velocity_constraints = velocity_constraints)
fr, frstar = kane.kanes_equations([particle_p], forceList)
zero = fr+frstar
f_c = point_pn.pos_from(particle_p.point).magnitude()-l
config = _sm.Matrix([[0]])
config[0] = f_c
zero = zero.row_insert(zero.shape[0], _sm.Matrix([[0]]))
zero[zero.shape[0]-1] = config[0]
|
b94f5acd3506ad8d790520edc394ff57eee09a3ca11db5ec973926c01bc58c69 | import sympy.physics.mechanics as _me
import sympy as _sm
import math as m
import numpy as _np
q1, q2, u1, u2 = _me.dynamicsymbols('q1 q2 u1 u2')
q1_d, q2_d, u1_d, u2_d = _me.dynamicsymbols('q1_ q2_ u1_ u2_', 1)
l, m, g = _sm.symbols('l m g', real=True)
frame_n = _me.ReferenceFrame('n')
frame_a = _me.ReferenceFrame('a')
frame_b = _me.ReferenceFrame('b')
frame_a.orient(frame_n, 'Axis', [q1, frame_n.z])
frame_b.orient(frame_n, 'Axis', [q2, frame_n.z])
frame_a.set_ang_vel(frame_n, u1*frame_n.z)
frame_b.set_ang_vel(frame_n, u2*frame_n.z)
point_o = _me.Point('o')
particle_p = _me.Particle('p', _me.Point('p_pt'), _sm.Symbol('m'))
particle_r = _me.Particle('r', _me.Point('r_pt'), _sm.Symbol('m'))
particle_p.point.set_pos(point_o, l*frame_a.x)
particle_r.point.set_pos(particle_p.point, l*frame_b.x)
point_o.set_vel(frame_n, 0)
particle_p.point.v2pt_theory(point_o,frame_n,frame_a)
particle_r.point.v2pt_theory(particle_p.point,frame_n,frame_b)
particle_p.mass = m
particle_r.mass = m
force_p = particle_p.mass*(g*frame_n.x)
force_r = particle_r.mass*(g*frame_n.x)
kd_eqs = [q1_d - u1, q2_d - u2]
forceList = [(particle_p.point,particle_p.mass*(g*frame_n.x)), (particle_r.point,particle_r.mass*(g*frame_n.x))]
kane = _me.KanesMethod(frame_n, q_ind=[q1,q2], u_ind=[u1, u2], kd_eqs = kd_eqs)
fr, frstar = kane.kanes_equations([particle_p, particle_r], forceList)
zero = fr+frstar
from pydy.system import System
sys = System(kane, constants = {l:1, m:1, g:9.81},
specifieds={},
initial_conditions={q1:.1, q2:.2, u1:0, u2:0},
times = _np.linspace(0.0, 10, 10/.01))
y=sys.integrate()
|
f5178202a7141f91d80601e6a29f61e59c79e267d271664de4be45eab087938c | import sympy.physics.mechanics as _me
import sympy as _sm
import math as m
import numpy as _np
m, k, b, g = _sm.symbols('m k b g', real=True)
position, speed = _me.dynamicsymbols('position speed')
position_d, speed_d = _me.dynamicsymbols('position_ speed_', 1)
o = _me.dynamicsymbols('o')
force = o*_sm.sin(_me.dynamicsymbols._t)
frame_ceiling = _me.ReferenceFrame('ceiling')
point_origin = _me.Point('origin')
point_origin.set_vel(frame_ceiling, 0)
particle_block = _me.Particle('block', _me.Point('block_pt'), _sm.Symbol('m'))
particle_block.point.set_pos(point_origin, position*frame_ceiling.x)
particle_block.mass = m
particle_block.point.set_vel(frame_ceiling, speed*frame_ceiling.x)
force_magnitude = m*g-k*position-b*speed+force
force_block = (force_magnitude*frame_ceiling.x).subs({position_d:speed})
kd_eqs = [position_d - speed]
forceList = [(particle_block.point,(force_magnitude*frame_ceiling.x).subs({position_d:speed}))]
kane = _me.KanesMethod(frame_ceiling, q_ind=[position], u_ind=[speed], kd_eqs = kd_eqs)
fr, frstar = kane.kanes_equations([particle_block], forceList)
zero = fr+frstar
from pydy.system import System
sys = System(kane, constants = {m:1.0, k:1.0, b:0.2, g:9.8},
specifieds={_me.dynamicsymbols('t'):lambda x, t: t, o:2},
initial_conditions={position:0.1, speed:-1*1.0},
times = _np.linspace(0.0, 10.0, 10.0/0.01))
y=sys.integrate()
|
e1fbafbfbe870792072d3d45b65f55008fb748b943d3a9ff3c7e619e17ac96ea | 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 * 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
|
394d305ac5d216f86b1738d6b09e9c97737c0747c775caf8c2ec0035282ed151 | """
Definition of physical dimensions.
Unit systems will be constructed on top of these dimensions.
Most of the examples in the doc use MKS system and are presented from the
computer point of view: from a human point, adding length to time is not legal
in MKS but it is in natural system; for a computer in natural system there is
no time dimension (but a velocity dimension instead) - in the basis - so the
question of adding time to length has no meaning.
"""
from typing import Dict as tDict
import collections
from functools import reduce
from sympy import (Integer, Matrix, S, Symbol, sympify, Basic, Tuple, Dict,
default_sort_key)
from sympy.functions.elementary.trigonometric import TrigonometricFunction
from sympy.core.expr import Expr
from sympy.core.power import Pow
from sympy.utilities.exceptions import SymPyDeprecationWarning
class _QuantityMapper:
_quantity_scale_factors_global = {} # type: tDict[Expr, Expr]
_quantity_dimensional_equivalence_map_global = {} # type: tDict[Expr, Expr]
_quantity_dimension_global = {} # type: tDict[Expr, Expr]
def __init__(self, *args, **kwargs):
self._quantity_dimension_map = {}
self._quantity_scale_factors = {}
def set_quantity_dimension(self, unit, dimension):
from sympy.physics.units import Quantity
dimension = sympify(dimension)
if not isinstance(dimension, Dimension):
if dimension == 1:
dimension = Dimension(1)
else:
raise ValueError("expected dimension or 1")
elif isinstance(dimension, Quantity):
dimension = self.get_quantity_dimension(dimension)
self._quantity_dimension_map[unit] = dimension
def set_quantity_scale_factor(self, unit, scale_factor):
from sympy.physics.units import Quantity
from sympy.physics.units.prefixes import Prefix
scale_factor = sympify(scale_factor)
# replace all prefixes by their ratio to canonical units:
scale_factor = scale_factor.replace(
lambda x: isinstance(x, Prefix),
lambda x: x.scale_factor
)
# replace all quantities by their ratio to canonical units:
scale_factor = scale_factor.replace(
lambda x: isinstance(x, Quantity),
lambda x: self.get_quantity_scale_factor(x)
)
self._quantity_scale_factors[unit] = scale_factor
def get_quantity_dimension(self, unit):
from sympy.physics.units import Quantity
# First look-up the local dimension map, then the global one:
if unit in self._quantity_dimension_map:
return self._quantity_dimension_map[unit]
if unit in self._quantity_dimension_global:
return self._quantity_dimension_global[unit]
if unit in self._quantity_dimensional_equivalence_map_global:
dep_unit = self._quantity_dimensional_equivalence_map_global[unit]
if isinstance(dep_unit, Quantity):
return self.get_quantity_dimension(dep_unit)
else:
return Dimension(self.get_dimensional_expr(dep_unit))
if isinstance(unit, Quantity):
return Dimension(unit.name)
else:
return Dimension(1)
def get_quantity_scale_factor(self, unit):
if unit in self._quantity_scale_factors:
return self._quantity_scale_factors[unit]
if unit in self._quantity_scale_factors_global:
mul_factor, other_unit = self._quantity_scale_factors_global[unit]
return mul_factor*self.get_quantity_scale_factor(other_unit)
return S.One
class Dimension(Expr):
"""
This class represent the dimension of a physical quantities.
The ``Dimension`` constructor takes as parameters a name and an optional
symbol.
For example, in classical mechanics we know that time is different from
temperature and dimensions make this difference (but they do not provide
any measure of these quantites.
>>> from sympy.physics.units import Dimension
>>> length = Dimension('length')
>>> length
Dimension(length)
>>> time = Dimension('time')
>>> time
Dimension(time)
Dimensions can be composed using multiplication, division and
exponentiation (by a number) to give new dimensions. Addition and
subtraction is defined only when the two objects are the same dimension.
>>> velocity = length / time
>>> velocity
Dimension(length/time)
It is possible to use a dimension system object to get the dimensionsal
dependencies of a dimension, for example the dimension system used by the
SI units convention can be used:
>>> from sympy.physics.units.systems.si import dimsys_SI
>>> dimsys_SI.get_dimensional_dependencies(velocity)
{'length': 1, 'time': -1}
>>> length + length
Dimension(length)
>>> l2 = length**2
>>> l2
Dimension(length**2)
>>> dimsys_SI.get_dimensional_dependencies(l2)
{'length': 2}
"""
_op_priority = 13.0
# XXX: This doesn't seem to be used anywhere...
_dimensional_dependencies = dict() # type: ignore
is_commutative = True
is_number = False
# make sqrt(M**2) --> M
is_positive = True
is_real = True
def __new__(cls, name, symbol=None):
if isinstance(name, str):
name = Symbol(name)
else:
name = sympify(name)
if not isinstance(name, Expr):
raise TypeError("Dimension name needs to be a valid math expression")
if isinstance(symbol, str):
symbol = Symbol(symbol)
elif symbol is not None:
assert isinstance(symbol, Symbol)
if symbol is not None:
obj = Expr.__new__(cls, name, symbol)
else:
obj = Expr.__new__(cls, name)
obj._name = name
obj._symbol = symbol
return obj
@property
def name(self):
return self._name
@property
def symbol(self):
return self._symbol
def __hash__(self):
return Expr.__hash__(self)
def __eq__(self, other):
if isinstance(other, Dimension):
return self.name == other.name
return False
def __str__(self):
"""
Display the string representation of the dimension.
"""
if self.symbol is None:
return "Dimension(%s)" % (self.name)
else:
return "Dimension(%s, %s)" % (self.name, self.symbol)
def __repr__(self):
return self.__str__()
def __neg__(self):
return self
def __add__(self, other):
from sympy.physics.units.quantities import Quantity
other = sympify(other)
if isinstance(other, Basic):
if other.has(Quantity):
raise TypeError("cannot sum dimension and quantity")
if isinstance(other, Dimension) and self == other:
return self
return super().__add__(other)
return self
def __radd__(self, other):
return self.__add__(other)
def __sub__(self, other):
# there is no notion of ordering (or magnitude) among dimension,
# subtraction is equivalent to addition when the operation is legal
return self + other
def __rsub__(self, other):
# there is no notion of ordering (or magnitude) among dimension,
# subtraction is equivalent to addition when the operation is legal
return self + other
def __pow__(self, other):
return self._eval_power(other)
def _eval_power(self, other):
other = sympify(other)
return Dimension(self.name**other)
def __mul__(self, other):
from sympy.physics.units.quantities import Quantity
if isinstance(other, Basic):
if other.has(Quantity):
raise TypeError("cannot sum dimension and quantity")
if isinstance(other, Dimension):
return Dimension(self.name*other.name)
if not other.free_symbols: # other.is_number cannot be used
return self
return super().__mul__(other)
return self
def __rmul__(self, other):
return self.__mul__(other)
def __truediv__(self, other):
return self*Pow(other, -1)
def __rtruediv__(self, other):
return other * pow(self, -1)
@classmethod
def _from_dimensional_dependencies(cls, dependencies):
return reduce(lambda x, y: x * y, (
Dimension(d)**e for d, e in dependencies.items()
), 1)
@classmethod
def _get_dimensional_dependencies_for_name(cls, name):
from sympy.physics.units.systems.si import dimsys_default
SymPyDeprecationWarning(
deprecated_since_version="1.2",
issue=13336,
feature="do not call from `Dimension` objects.",
useinstead="DimensionSystem"
).warn()
return dimsys_default.get_dimensional_dependencies(name)
@property
def is_dimensionless(self):
"""
Check if the dimension object really has a dimension.
A dimension should have at least one component with non-zero power.
"""
if self.name == 1:
return True
from sympy.physics.units.systems.si import dimsys_default
SymPyDeprecationWarning(
deprecated_since_version="1.2",
issue=13336,
feature="wrong class",
).warn()
dimensional_dependencies=dimsys_default
return dimensional_dependencies.get_dimensional_dependencies(self) == {}
def has_integer_powers(self, dim_sys):
"""
Check if the dimension object has only integer powers.
All the dimension powers should be integers, but rational powers may
appear in intermediate steps. This method may be used to check that the
final result is well-defined.
"""
for dpow in dim_sys.get_dimensional_dependencies(self).values():
if not isinstance(dpow, (int, Integer)):
return False
return True
# Create dimensions according the the base units in MKSA.
# For other unit systems, they can be derived by transforming the base
# dimensional dependency dictionary.
class DimensionSystem(Basic, _QuantityMapper):
r"""
DimensionSystem represents a coherent set of dimensions.
The constructor takes three parameters:
- base dimensions;
- derived dimensions: these are defined in terms of the base dimensions
(for example velocity is defined from the division of length by time);
- dependency of dimensions: how the derived dimensions depend
on the base dimensions.
Optionally either the ``derived_dims`` or the ``dimensional_dependencies``
may be omitted.
"""
def __new__(cls, base_dims, derived_dims=[], dimensional_dependencies={}, name=None, descr=None):
dimensional_dependencies = dict(dimensional_dependencies)
if (name is not None) or (descr is not None):
SymPyDeprecationWarning(
deprecated_since_version="1.2",
issue=13336,
useinstead="do not define a `name` or `descr`",
).warn()
def parse_dim(dim):
if isinstance(dim, str):
dim = Dimension(Symbol(dim))
elif isinstance(dim, Dimension):
pass
elif isinstance(dim, Symbol):
dim = Dimension(dim)
else:
raise TypeError("%s wrong type" % dim)
return dim
base_dims = [parse_dim(i) for i in base_dims]
derived_dims = [parse_dim(i) for i in derived_dims]
for dim in base_dims:
dim = dim.name
if (dim in dimensional_dependencies
and (len(dimensional_dependencies[dim]) != 1 or
dimensional_dependencies[dim].get(dim, None) != 1)):
raise IndexError("Repeated value in base dimensions")
dimensional_dependencies[dim] = Dict({dim: 1})
def parse_dim_name(dim):
if isinstance(dim, Dimension):
return dim.name
elif isinstance(dim, str):
return Symbol(dim)
elif isinstance(dim, Symbol):
return dim
else:
raise TypeError("unrecognized type %s for %s" % (type(dim), dim))
for dim in dimensional_dependencies.keys():
dim = parse_dim(dim)
if (dim not in derived_dims) and (dim not in base_dims):
derived_dims.append(dim)
def parse_dict(d):
return Dict({parse_dim_name(i): j for i, j in d.items()})
# Make sure everything is a SymPy type:
dimensional_dependencies = {parse_dim_name(i): parse_dict(j) for i, j in
dimensional_dependencies.items()}
for dim in derived_dims:
if dim in base_dims:
raise ValueError("Dimension %s both in base and derived" % dim)
if dim.name not in dimensional_dependencies:
# TODO: should this raise a warning?
dimensional_dependencies[dim.name] = Dict({dim.name: 1})
base_dims.sort(key=default_sort_key)
derived_dims.sort(key=default_sort_key)
base_dims = Tuple(*base_dims)
derived_dims = Tuple(*derived_dims)
dimensional_dependencies = Dict({i: Dict(j) for i, j in dimensional_dependencies.items()})
obj = Basic.__new__(cls, base_dims, derived_dims, dimensional_dependencies)
return obj
@property
def base_dims(self):
return self.args[0]
@property
def derived_dims(self):
return self.args[1]
@property
def dimensional_dependencies(self):
return self.args[2]
def _get_dimensional_dependencies_for_name(self, name):
if isinstance(name, Dimension):
name = name.name
if isinstance(name, str):
name = Symbol(name)
if name.is_Symbol:
# Dimensions not included in the dependencies are considered
# as base dimensions:
return dict(self.dimensional_dependencies.get(name, {name: 1}))
if name.is_number or name.is_NumberSymbol:
return {}
get_for_name = self._get_dimensional_dependencies_for_name
if name.is_Mul:
ret = collections.defaultdict(int)
dicts = [get_for_name(i) for i in name.args]
for d in dicts:
for k, v in d.items():
ret[k] += v
return {k: v for (k, v) in ret.items() if v != 0}
if name.is_Add:
dicts = [get_for_name(i) for i in name.args]
if all([d == dicts[0] for d in dicts[1:]]):
return dicts[0]
raise TypeError("Only equivalent dimensions can be added or subtracted.")
if name.is_Pow:
dim_base = get_for_name(name.base)
dim_exp = get_for_name(name.exp)
if dim_exp == {} or name.exp.is_Symbol:
return {k: v*name.exp for (k, v) in dim_base.items()}
else:
raise TypeError("The exponent for the power operator must be a Symbol or dimensionless.")
if name.is_Function:
args = (Dimension._from_dimensional_dependencies(
get_for_name(arg)) for arg in name.args)
result = name.func(*args)
dicts = [get_for_name(i) for i in name.args]
if isinstance(result, Dimension):
return self.get_dimensional_dependencies(result)
elif result.func == name.func:
if isinstance(name, TrigonometricFunction):
if dicts[0] == {} or dicts[0] == {Symbol('angle'): 1}:
return {}
else:
raise TypeError("The input argument for the function {} must be dimensionless or have dimensions of angle.".format(name.func))
else:
if all( (item == {} for item in dicts) ):
return {}
else:
raise TypeError("The input arguments for the function {} must be dimensionless.".format(name.func))
else:
return get_for_name(result)
raise TypeError("Type {} not implemented for get_dimensional_dependencies".format(type(name)))
def get_dimensional_dependencies(self, name, mark_dimensionless=False):
dimdep = self._get_dimensional_dependencies_for_name(name)
if mark_dimensionless and dimdep == {}:
return {'dimensionless': 1}
return {str(i): j for i, j in dimdep.items()}
def equivalent_dims(self, dim1, dim2):
deps1 = self.get_dimensional_dependencies(dim1)
deps2 = self.get_dimensional_dependencies(dim2)
return deps1 == deps2
def extend(self, new_base_dims, new_derived_dims=[], new_dim_deps={}, name=None, description=None):
if (name is not None) or (description is not None):
SymPyDeprecationWarning(
deprecated_since_version="1.2",
issue=13336,
feature="name and descriptions of DimensionSystem",
useinstead="do not specify `name` or `description`",
).warn()
deps = dict(self.dimensional_dependencies)
deps.update(new_dim_deps)
new_dim_sys = DimensionSystem(
tuple(self.base_dims) + tuple(new_base_dims),
tuple(self.derived_dims) + tuple(new_derived_dims),
deps
)
new_dim_sys._quantity_dimension_map.update(self._quantity_dimension_map)
new_dim_sys._quantity_scale_factors.update(self._quantity_scale_factors)
return new_dim_sys
@staticmethod
def sort_dims(dims):
"""
Useless method, kept for compatibility with previous versions.
DO NOT USE.
Sort dimensions given in argument using their str function.
This function will ensure that we get always the same tuple for a given
set of dimensions.
"""
SymPyDeprecationWarning(
deprecated_since_version="1.2",
issue=13336,
feature="sort_dims",
useinstead="sorted(..., key=default_sort_key)",
).warn()
return tuple(sorted(dims, key=str))
def __getitem__(self, key):
"""
Useless method, kept for compatibility with previous versions.
DO NOT USE.
Shortcut to the get_dim method, using key access.
"""
SymPyDeprecationWarning(
deprecated_since_version="1.2",
issue=13336,
feature="the get [ ] operator",
useinstead="the dimension definition",
).warn()
d = self.get_dim(key)
#TODO: really want to raise an error?
if d is None:
raise KeyError(key)
return d
def __call__(self, unit):
"""
Useless method, kept for compatibility with previous versions.
DO NOT USE.
Wrapper to the method print_dim_base
"""
SymPyDeprecationWarning(
deprecated_since_version="1.2",
issue=13336,
feature="call DimensionSystem",
useinstead="the dimension definition",
).warn()
return self.print_dim_base(unit)
def is_dimensionless(self, dimension):
"""
Check if the dimension object really has a dimension.
A dimension should have at least one component with non-zero power.
"""
if dimension.name == 1:
return True
return self.get_dimensional_dependencies(dimension) == {}
@property
def list_can_dims(self):
"""
Useless method, kept for compatibility with previous versions.
DO NOT USE.
List all canonical dimension names.
"""
dimset = set()
for i in self.base_dims:
dimset.update(set(self.get_dimensional_dependencies(i).keys()))
return tuple(sorted(dimset, key=str))
@property
def inv_can_transf_matrix(self):
"""
Useless method, kept for compatibility with previous versions.
DO NOT USE.
Compute the inverse transformation matrix from the base to the
canonical dimension basis.
It corresponds to the matrix where columns are the vector of base
dimensions in canonical basis.
This matrix will almost never be used because dimensions are always
defined with respect to the canonical basis, so no work has to be done
to get them in this basis. Nonetheless if this matrix is not square
(or not invertible) it means that we have chosen a bad basis.
"""
matrix = reduce(lambda x, y: x.row_join(y),
[self.dim_can_vector(d) for d in self.base_dims])
return matrix
@property
def can_transf_matrix(self):
"""
Useless method, kept for compatibility with previous versions.
DO NOT USE.
Return the canonical transformation matrix from the canonical to the
base dimension basis.
It is the inverse of the matrix computed with inv_can_transf_matrix().
"""
#TODO: the inversion will fail if the system is inconsistent, for
# example if the matrix is not a square
return reduce(lambda x, y: x.row_join(y),
[self.dim_can_vector(d) for d in sorted(self.base_dims, key=str)]
).inv()
def dim_can_vector(self, dim):
"""
Useless method, kept for compatibility with previous versions.
DO NOT USE.
Dimensional representation in terms of the canonical base dimensions.
"""
vec = []
for d in self.list_can_dims:
vec.append(self.get_dimensional_dependencies(dim).get(d, 0))
return Matrix(vec)
def dim_vector(self, dim):
"""
Useless method, kept for compatibility with previous versions.
DO NOT USE.
Vector representation in terms of the base dimensions.
"""
return self.can_transf_matrix * Matrix(self.dim_can_vector(dim))
def print_dim_base(self, dim):
"""
Give the string expression of a dimension in term of the basis symbols.
"""
dims = self.dim_vector(dim)
symbols = [i.symbol if i.symbol is not None else i.name for i in self.base_dims]
res = S.One
for (s, p) in zip(symbols, dims):
res *= s**p
return res
@property
def dim(self):
"""
Useless method, kept for compatibility with previous versions.
DO NOT USE.
Give the dimension of the system.
That is return the number of dimensions forming the basis.
"""
return len(self.base_dims)
@property
def is_consistent(self):
"""
Useless method, kept for compatibility with previous versions.
DO NOT USE.
Check if the system is well defined.
"""
# not enough or too many base dimensions compared to independent
# dimensions
# in vector language: the set of vectors do not form a basis
return self.inv_can_transf_matrix.is_square
|
4fc2c0d8144d141dae7af740debbeca855e197dcafe929c8f89f33644b02a838 | from .vector import Vector, _check_vector
from .frame import _check_frame
from warnings import warn
__all__ = ['Point']
class Point:
"""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
>>> from sympy.physics.vector import init_vprinting
>>> init_vprinting(pretty_print=False)
>>> 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.physics.vector import init_vprinting
>>> init_vprinting(pretty_print=False)
>>> 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):
"""Returns a list of points that gives the shortest path with respect
to position, velocity, or acceleration from this point to the provided
point.
Parameters
==========
other : Point
A point that may be related to this point by position, velocity, or
acceleration.
num : integer
0 for searching the position tree, 1 for searching the velocity
tree, and 2 for searching the acceleration tree.
Returns
=======
list of Points
A sequence of points from self to other.
Notes
=====
It isn't clear if num = 1 or num = 2 actually works because the keys to
``_vel_dict`` and ``_acc_dict`` are :class:`ReferenceFrame` objects which
do not have the ``_pdlist`` attribute.
"""
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
>>> from sympy.physics.vector import init_vprinting
>>> init_vprinting(pretty_print=False)
>>> 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
>>> from sympy.physics.vector import init_vprinting
>>> init_vprinting(pretty_print=False)
>>> 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 1-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
>>> from sympy.physics.vector import init_vprinting
>>> init_vprinting(pretty_print=False)
>>> 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
>>> from sympy.physics.vector import init_vprinting
>>> init_vprinting(pretty_print=False)
>>> 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, dynamicsymbols
>>> N = ReferenceFrame('N')
>>> p1 = Point('p1')
>>> p1.set_vel(N, 10 * N.x)
>>> p1.vel(N)
10*N.x
Velocities will be automatically calculated if possible, otherwise a ``ValueError`` will be returned. If it is possible to calculate multiple different velocities from the relative points, the points defined most directly relative to this point will be used. In the case of inconsistent relative positions of points, incorrect velocities may be returned. It is up to the user to define prior relative positions and velocities of points in a self-consistent way.
>>> p = Point('p')
>>> q = dynamicsymbols('q')
>>> p.set_vel(N, 10 * N.x)
>>> p2 = Point('p2')
>>> p2.set_pos(p, q*N.x)
>>> p2.vel(N)
(Derivative(q(t), t) + 10)*N.x
"""
_check_frame(frame)
if not (frame in self._vel_dict):
valid_neighbor_found = False
is_cyclic = False
visited = []
queue = [self]
candidate_neighbor = []
while queue: #BFS to find nearest point
node = queue.pop(0)
if node not in visited:
visited.append(node)
for neighbor, neighbor_pos in node._pos_dict.items():
if neighbor in visited:
continue
try:
neighbor_pos.express(frame) #Checks if pos vector is valid
except ValueError:
continue
if neighbor in queue:
is_cyclic = True
try :
neighbor_velocity = neighbor._vel_dict[frame] #Checks if point has its vel defined in req frame
except KeyError:
queue.append(neighbor)
continue
candidate_neighbor.append(neighbor)
if not valid_neighbor_found:
self.set_vel(frame, self.pos_from(neighbor).dt(frame) + neighbor_velocity)
valid_neighbor_found = True
if is_cyclic:
warn('Kinematic loops are defined among the positions of points. This is likely not desired and may cause errors in your calculations.')
if len(candidate_neighbor) > 1:
warn('Velocity automatically calculated based on point ' +
candidate_neighbor[0].name + ' but it is also possible from points(s):' +
str(candidate_neighbor[1:]) + '. Velocities from these points are not necessarily the same. This may cause errors in your calculations.')
if valid_neighbor_found:
return self._vel_dict[frame]
else:
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)
|
fb7b2fd593bb85a17c0a88a8c88d0d90ffc251a13283b495d245b92e9fd55c52 | 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()._sanitize(assumptions, cls)
obj = super().__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:
"""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() + "_" + indices[0]),
(name.lower() + "_" + indices[1]),
(name.lower() + "_" + 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() + "_x",
name.lower() + "_y",
name.lower() + "_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 adjacent parent-child
#relationships. The _dcm_cache dictionary will store calculated dcm along with
#all content of _dcm_dict for faster retrieval of dcms.
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):
"""Returns an inclusive list of reference frames that connect this
reference frame to the provided reference frame.
Parameters
==========
other : ReferenceFrame
The other reference frame to look for a connecting relationship to.
num : integer
``0``, ``1``, and ``2`` will look for orientation, angular
velocity, and angular acceleration relationships between the two
frames, respectively.
Returns
=======
list
Inclusive list of reference frames that connect this reference
frame to the other reference frame.
Examples
========
>>> from sympy.physics.vector import ReferenceFrame
>>> A = ReferenceFrame('A')
>>> B = ReferenceFrame('B')
>>> C = ReferenceFrame('C')
>>> D = ReferenceFrame('D')
>>> B.orient_axis(A, A.x, 1.0)
>>> C.orient_axis(B, B.x, 1.0)
>>> D.orient_axis(C, C.x, 1.0)
>>> D._dict_list(A, 0)
[D, C, B, A]
Raises
======
ValueError
When no path is found between the two reference frames or ``num``
is an incorrect value.
"""
connect_type = {0: 'orientation',
1: 'angular velocity',
2: 'angular acceleration'}
if num not in connect_type.keys():
raise ValueError('Valid values for num are 0, 1, or 2.')
possible_connecting_paths = [[self]]
oldlist = [[]]
while possible_connecting_paths != oldlist:
oldlist = possible_connecting_paths[:] # make a copy
for frame_list in possible_connecting_paths:
frames_adjacent_to_last = frame_list[-1]._dlist[num].keys()
for adjacent_frame in frames_adjacent_to_last:
if adjacent_frame not in frame_list:
connecting_path = frame_list + [adjacent_frame]
if connecting_path not in possible_connecting_paths:
possible_connecting_paths.append(connecting_path)
for connecting_path in oldlist:
if connecting_path[-1] != other:
possible_connecting_paths.remove(connecting_path)
possible_connecting_paths.sort(key=len)
if len(possible_connecting_paths) != 0:
return possible_connecting_paths[0] # selects the shortest path
msg = 'No connecting {} path found between {} and {}.'
raise ValueError(msg.format(connect_type[num], self.name, 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:`^{\mathbf{A}} \mathbf{R} ^{\mathbf{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 _dcm(self, parent, parent_orient):
# If parent.oreint(self) is already defined,then
# update the _dcm_dict of parent while over write
# all content of self._dcm_dict and self._dcm_cache
# with new dcm relation.
# Else update _dcm_cache and _dcm_dict of both
# self and parent.
frames = self._dcm_cache.keys()
dcm_dict_del = []
dcm_cache_del = []
if parent in frames:
for frame in frames:
if frame in self._dcm_dict:
dcm_dict_del += [frame]
dcm_cache_del += [frame]
# 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
for frame in dcm_dict_del:
del frame._dcm_dict[self]
for frame in dcm_cache_del:
del frame._dcm_cache[self]
# Reset the _dcm_dict
self._dcm_dict = self._dlist[0] = {}
# Reset the _dcm_cache
self._dcm_cache = {}
# Add the dcm relationship to _dcm_dict
self._dcm_dict.update({parent: parent_orient.T})
parent._dcm_dict.update({self: parent_orient})
# Update the dcm cache
self._dcm_cache.update({parent: parent_orient.T})
parent._dcm_cache.update({self: parent_orient})
def orient_axis(self, parent, axis, angle):
"""Sets the orientation of this reference frame with respect to a
parent reference frame by rotating through an angle about an axis fixed
in the parent reference frame.
Parameters
==========
parent : ReferenceFrame
Reference frame that this reference frame will be rotated relative
to.
axis : Vector
Vector fixed in the parent frame about about which this frame is
rotated. It need not be a unit vector and the rotation follows the
right hand rule.
angle : sympifiable
Angle in radians by which it the frame is to be rotated.
Examples
========
Setup variables for the examples:
>>> from sympy import symbols
>>> from sympy.physics.vector import ReferenceFrame
>>> q1 = symbols('q1')
>>> N = ReferenceFrame('N')
>>> B = ReferenceFrame('B')
>>> B.orient_axis(N, N.x, q1)
The ``orient_axis()`` 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)]])
>>> N.dcm(B)
Matrix([
[1, 0, 0],
[0, cos(q1), -sin(q1)],
[0, sin(q1), cos(q1)]])
The following two lines show that the sense of the rotation can be
defined by negating the vector direction or the angle. Both lines
produce the same result.
>>> B.orient_axis(N, -N.x, q1)
>>> B.orient_axis(N, N.x, -q1)
"""
from sympy.physics.vector.functions import dynamicsymbols
_check_frame(parent)
amount = sympify(angle)
theta = amount
axis = _check_vector(axis)
parent_orient_axis = []
if not axis.dt(parent) == 0:
raise ValueError('Axis cannot be time-varying.')
unit_axis = axis.express(parent).normalize()
unit_col = unit_axis.args[0][0]
parent_orient_axis = (
(eye(3) - unit_col * unit_col.T) * cos(theta) +
Matrix([[0, -unit_col[2], unit_col[1]],
[unit_col[2], 0, -unit_col[0]],
[-unit_col[1], unit_col[0], 0]]) *
sin(theta) + unit_col * unit_col.T)
self._dcm(parent, parent_orient_axis)
thetad = (amount).diff(dynamicsymbols._t)
wvec = thetad*axis.express(parent).normalize()
self._ang_vel_dict.update({parent: wvec})
parent._ang_vel_dict.update({self: -wvec})
self._var_dict = {}
def orient_explicit(self, parent, dcm):
"""Sets the orientation of this reference frame relative to a parent
reference frame by explicitly setting the direction cosine matrix.
Parameters
==========
parent : ReferenceFrame
Reference frame that this reference frame will be rotated relative
to.
dcm : Matrix, shape(3, 3)
Direction cosine matrix that specifies the relative rotation
between the two reference frames.
Examples
========
Setup variables for the examples:
>>> from sympy import symbols, Matrix, sin, cos
>>> from sympy.physics.vector import ReferenceFrame
>>> q1 = symbols('q1')
>>> A = ReferenceFrame('A')
>>> B = ReferenceFrame('B')
>>> N = ReferenceFrame('N')
A simple rotation of ``A`` relative to ``N`` about ``N.x`` is defined
by the following direction cosine matrix:
>>> dcm = Matrix([[1, 0, 0],
... [0, cos(q1), -sin(q1)],
... [0, sin(q1), cos(q1)]])
>>> A.orient_explicit(N, dcm)
>>> A.dcm(N)
Matrix([
[1, 0, 0],
[0, cos(q1), sin(q1)],
[0, -sin(q1), cos(q1)]])
This is equivalent to using ``orient_axis()``:
>>> B.orient_axis(N, N.x, q1)
>>> B.dcm(N)
Matrix([
[1, 0, 0],
[0, cos(q1), sin(q1)],
[0, -sin(q1), cos(q1)]])
**Note carefully that** ``N.dcm(B)`` **(the transpose) would be passed
into** ``orient_explicit()`` **for** ``A.dcm(N)`` **to match**
``B.dcm(N)``:
>>> A.orient_explicit(N, N.dcm(B))
>>> A.dcm(N)
Matrix([
[1, 0, 0],
[0, cos(q1), sin(q1)],
[0, -sin(q1), cos(q1)]])
"""
_check_frame(parent)
# amounts must be a Matrix type object
# (e.g. sympy.matrices.dense.MutableDenseMatrix).
if not isinstance(dcm, MatrixBase):
raise TypeError("Amounts must be a sympy Matrix type object.")
parent_orient_dcm = []
parent_orient_dcm = dcm
self._dcm(parent, parent_orient_dcm)
wvec = self._w_diff_dcm(parent)
self._ang_vel_dict.update({parent: wvec})
parent._ang_vel_dict.update({self: -wvec})
self._var_dict = {}
def _rot(self, 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]])
def orient_body_fixed(self, parent, angles, rotation_order):
"""Rotates this reference frame relative to the parent reference frame
by right hand rotating through three successive body fixed simple axis
rotations. Each subsequent axis of rotation is about the "body fixed"
unit vectors of a new intermediate reference frame. This type of
rotation is also referred to rotating through the `Euler and Tait-Bryan
Angles`_.
.. _Euler and Tait-Bryan Angles: https://en.wikipedia.org/wiki/Euler_angles
Parameters
==========
parent : ReferenceFrame
Reference frame that this reference frame will be rotated relative
to.
angles : 3-tuple of sympifiable
Three angles in radians used for the successive rotations.
rotation_order : 3 character string or 3 digit integer
Order of the rotations about each intermediate reference frames'
unit vectors. The Euler rotation about the X, Z', X'' axes can be
specified by the strings ``'XZX'``, ``'131'``, or the integer
``131``. There are 12 unique valid rotation orders (6 Euler and 6
Tait-Bryan): zxz, xyx, yzy, zyz, xzx, yxy, xyz, yzx, zxy, xzy, zyx,
and yxz.
Examples
========
Setup variables for the examples:
>>> from sympy import symbols
>>> from sympy.physics.vector import ReferenceFrame
>>> q1, q2, q3 = symbols('q1, q2, q3')
>>> N = ReferenceFrame('N')
>>> B = ReferenceFrame('B')
>>> B1 = ReferenceFrame('B1')
>>> B2 = ReferenceFrame('B2')
For example, a classic Euler Angle rotation can be done by:
>>> B.orient_body_fixed(N, (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 reference frame B relative to reference frame 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
three successive ``orient_axis()`` calls:
>>> B1.orient_axis(N, N.x, q1)
>>> B2.orient_axis(B1, B1.y, q2)
>>> B.orient_axis(B2, B2.x, q3)
>>> 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_body_fixed(N, (q1, q2, 0), 'ZXZ')
>>> B.orient_body_fixed(N, (q1, q2, 0), '121')
>>> B.orient_body_fixed(N, (q1, q2, q3), 123)
"""
_check_frame(parent)
amounts = list(angles)
for i, v in enumerate(amounts):
if not isinstance(v, Vector):
amounts[i] = sympify(v)
approved_orders = ('123', '231', '312', '132', '213', '321', '121',
'131', '212', '232', '313', '323', '')
# make sure XYZ => 123
rot_order = translate(str(rotation_order), 'XYZxyz', '123123')
if rot_order not in approved_orders:
raise TypeError('The rotation order is not a valid order.')
parent_orient_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_body = (self._rot(a1, amounts[0]) *
self._rot(a2, amounts[1]) *
self._rot(a3, amounts[2]))
self._dcm(parent, parent_orient_body)
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],
'body', 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 orient_space_fixed(self, parent, angles, rotation_order):
"""Rotates this reference frame relative to the parent reference frame
by right hand rotating through three successive space fixed simple axis
rotations. Each subsequent axis of rotation is about the "space fixed"
unit vectors of the parent reference frame.
Parameters
==========
parent : ReferenceFrame
Reference frame that this reference frame will be rotated relative
to.
angles : 3-tuple of sympifiable
Three angles in radians used for the successive rotations.
rotation_order : 3 character string or 3 digit integer
Order of the rotations about the parent reference frame's unit
vectors. The order can be specified by the strings ``'XZX'``,
``'131'``, or the integer ``131``. There are 12 unique valid
rotation orders.
Examples
========
Setup variables for the examples:
>>> from sympy import symbols
>>> from sympy.physics.vector import ReferenceFrame
>>> q1, q2, q3 = symbols('q1, q2, q3')
>>> N = ReferenceFrame('N')
>>> B = ReferenceFrame('B')
>>> B1 = ReferenceFrame('B1')
>>> B2 = ReferenceFrame('B2')
>>> B.orient_space_fixed(N, (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_axis(N, N.z, q1)
>>> B2.orient_axis(B1, N.x, q2)
>>> B.orient_axis(B2, N.y, q3)
>>> B.dcm(N).simplify()
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_space_fixed(N, (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_body_fixed(N, (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)]])
"""
_check_frame(parent)
amounts = list(angles)
for i, v in enumerate(amounts):
if not isinstance(v, Vector):
amounts[i] = sympify(v)
approved_orders = ('123', '231', '312', '132', '213', '321', '121',
'131', '212', '232', '313', '323', '')
# make sure XYZ => 123
rot_order = translate(str(rotation_order), 'XYZxyz', '123123')
if rot_order not in approved_orders:
raise TypeError('The supplied order is not an approved type')
parent_orient_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_space = (self._rot(a3, amounts[2]) *
self._rot(a2, amounts[1]) *
self._rot(a1, amounts[0]))
self._dcm(parent, parent_orient_space)
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],
'space', 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 orient_quaternion(self, parent, numbers):
"""Sets the orientation of this reference frame relative to a parent
reference frame via an orientation quaternion. An orientation
quaternion is defined as a finite rotation a unit vector, ``(lambda_x,
lambda_y, lambda_z)``, by an angle ``theta``. The orientation
quaternion 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)``
See `Quaternions and Spatial Rotation
<https://en.wikipedia.org/wiki/Quaternions_and_spatial_rotation>`_ on
Wikipedia for more information.
Parameters
==========
parent : ReferenceFrame
Reference frame that this reference frame will be rotated relative
to.
numbers : 4-tuple of sympifiable
The four quaternion scalar numbers as defined above: ``q0``,
``q1``, ``q2``, ``q3``.
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')
Set the orientation:
>>> B.orient_quaternion(N, (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)
numbers = list(numbers)
for i, v in enumerate(numbers):
if not isinstance(v, Vector):
numbers[i] = sympify(v)
parent_orient_quaternion = []
if not (isinstance(numbers, (list, tuple)) & (len(numbers) == 4)):
raise TypeError('Amounts are a list or tuple of length 4')
q0, q1, q2, q3 = numbers
parent_orient_quaternion = (
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]]))
self._dcm(parent, parent_orient_quaternion)
t = dynamicsymbols._t
q0, q1, q2, q3 = numbers
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)])
self._ang_vel_dict.update({parent: wvec})
parent._ang_vel_dict.update({self: -wvec})
self._var_dict = {}
def orient(self, parent, rot_type, amounts, rot_order=''):
"""Sets the orientation of this reference frame relative to another
(parent) reference frame.
.. note:: It is now recommended to use the ``.orient_axis,
.orient_body_fixed, .orient_space_fixed, .orient_quaternion``
methods for the different rotation types.
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'``.
"""
_check_frame(parent)
approved_orders = ('123', '231', '312', '132', '213', '321', '121',
'131', '212', '232', '313', '323', '')
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')
if rot_type == 'AXIS':
self.orient_axis(parent, amounts[1], amounts[0])
elif rot_type == 'DCM':
self.orient_explicit(parent, amounts)
elif rot_type == 'BODY':
self.orient_body_fixed(parent, amounts, rot_order)
elif rot_type == 'SPACE':
self.orient_space_fixed(parent, amounts, rot_order)
elif rot_type == 'QUATERNION':
self.orient_quaternion(parent, amounts)
else:
raise NotImplementedError('That is not an implemented rotation')
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)
approved_orders = ('123', '231', '312', '132', '213', '321', '121',
'131', '212', '232', '313', '323', '')
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')
if rot_type == 'AXIS':
newframe.orient_axis(self, amounts[1], amounts[0])
elif rot_type == 'DCM':
newframe.orient_explicit(self, amounts)
elif rot_type == 'BODY':
newframe.orient_body_fixed(self, amounts, rot_order)
elif rot_type == 'SPACE':
newframe.orient_space_fixed(self, amounts, rot_order)
elif rot_type == 'QUATERNION':
newframe.orient_quaternion(self, amounts)
else:
raise NotImplementedError('That is not an implemented rotation')
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'))
|
e884211e08c8fb90338bf6ff7afd9d11fe3695a8bce9dd7c8c7328cf48157632 | """
This module can be used to solve 2D beam bending problems with
singularity functions in mechanics.
"""
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:
"""
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. However, the
chosen sign convention must respect the rule that, on the positive
side of beam's axis (in respect to current section), a loading force
giving positive shear yields a negative moment, as below (the
curved arrow shows the positive moment and rotation):
.. image:: allowed-sign-conventions.png
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:
# load has an end point within the length of the beam.
self._handle_end(x, value, start, order, end, type="apply")
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:
# load has an end point within the length of the beam.
self._handle_end(x, value, start, order, end, type="remove")
def _handle_end(self, x, value, start, order, end, type):
"""
This functions handles the optional `end` value in the
`apply_load` and `remove_load` functions. When the value
of end is not NULL, this function will be executed.
"""
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
if type == "apply":
# iterating for "apply_load" method
for i in range(0, order + 1):
self._load -= (f.diff(x, i).subs(x, end - start) *
SingularityFunction(x, end, i)/factorial(i))
elif type == "remove":
# iterating for "remove_load" method
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")
>>> p = b.draw()
>>> p
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)
>>> p.show()
"""
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, factor
>>> 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()
>>> factor(b.slope())
[0, 0, x*(-l + x)*(-A*G*l**3*q + 2*A*G*l**2*q*x - 12*E*I*l*q
- 72*E*I*m + 24*E*I*q*x)/(12*E*I*(A*G*l**2 + 12*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().__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
def _plot_shear_force(self, dir, subs=None):
shear_force = self.shear_force()
if dir == 'x':
dir_num = 0
color = 'r'
elif dir == 'y':
dir_num = 1
color = 'g'
elif dir == 'z':
dir_num = 2
color = 'b'
if subs is None:
subs = {}
for sym in shear_force[dir_num].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[dir_num].subs(subs), (self.variable, 0, length), show = False, title='Shear Force along %c direction'%dir,
xlabel=r'$\mathrm{%c}$'%dir, ylabel=r'$\mathrm{V[%c]}$'%dir, line_color=color)
def plot_shear_force(self, dir="all", subs=None):
"""
Returns a plot for Shear force along all three directions
present in the Beam object.
Parameters
==========
dir : string (default : "all")
Direction along which shear force plot is required.
If no direction is specified, all plots are displayed.
subs : dictionary
Python dictionary containing Symbols as key and their
corresponding values.
Examples
========
There is a beam of length 20 meters. It it supported by rollers
at of its end. A linear load having slope equal to 12 is applied
along y-axis. A constant distributed load of magnitude 15 N is
applied from start till its end along z-axis.
.. plot::
:context: close-figs
:format: doctest
:include-source: True
>>> 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(20, E, G, I, A, x)
>>> b.apply_load(15, start=0, order=0, dir="z")
>>> b.apply_load(12*x, start=0, order=0, dir="y")
>>> b.bc_deflection = [(0, [0, 0, 0]), (20, [0, 0, 0])]
>>> R1, R2, R3, R4 = symbols('R1, R2, R3, R4')
>>> b.apply_load(R1, start=0, order=-1, dir="z")
>>> b.apply_load(R2, start=20, order=-1, dir="z")
>>> b.apply_load(R3, start=0, order=-1, dir="y")
>>> b.apply_load(R4, start=20, order=-1, dir="y")
>>> b.solve_for_reaction_loads(R1, R2, R3, R4)
>>> b.plot_shear_force()
PlotGrid object containing:
Plot[0]:Plot object containing:
[0]: cartesian line: 0 for x over (0.0, 20.0)
Plot[1]:Plot object containing:
[0]: cartesian line: -6*x**2 for x over (0.0, 20.0)
Plot[2]:Plot object containing:
[0]: cartesian line: -15*x for x over (0.0, 20.0)
"""
Px = self._plot_shear_force('x')
Py = self._plot_shear_force('y')
Pz = self._plot_shear_force('z')
# For shear force along x direction
if dir == "x":
return Px.show()
# For shear force along y direction
elif dir == "y":
return Py.show()
# For shear force along z direction
elif dir == "z":
return Pz.show()
# For shear force along all direction
else:
return PlotGrid(3, 1, Px, Py, Pz)
|
d149aa22ddbebbc5b036402ab27395b7d565901f4ad024482588906a5b49ef7c | __all__ = [
'TWave',
'RayTransferMatrix', 'FreeSpace', 'FlatRefraction', 'CurvedRefraction',
'FlatMirror', 'CurvedMirror', 'ThinLens', 'GeometricRay', 'BeamParameter',
'waist2rayleigh', 'rayleigh2waist', 'geometric_conj_ab',
'geometric_conj_af', 'geometric_conj_bf', 'gaussian_conj',
'conjugate_gauss_beams',
'Medium',
'refraction_angle', 'deviation', 'fresnel_coefficients', 'brewster_angle',
'critical_angle', 'lens_makers_formula', 'mirror_formula', 'lens_formula',
'hyperfocal_distance', 'transverse_magnification',
'jones_vector', 'stokes_vector', 'jones_2_stokes', 'linear_polarizer',
'phase_retarder', 'half_wave_retarder', 'quarter_wave_retarder',
'transmissive_filter', 'reflective_filter', 'mueller_matrix',
'polarizing_beam_splitter',
]
from .waves import TWave
from .gaussopt import (RayTransferMatrix, FreeSpace, FlatRefraction,
CurvedRefraction, FlatMirror, CurvedMirror, ThinLens, GeometricRay,
BeamParameter, waist2rayleigh, rayleigh2waist, geometric_conj_ab,
geometric_conj_af, geometric_conj_bf, gaussian_conj,
conjugate_gauss_beams)
from .medium import Medium
from .utils import (refraction_angle, deviation, fresnel_coefficients,
brewster_angle, critical_angle, lens_makers_formula, mirror_formula,
lens_formula, hyperfocal_distance, transverse_magnification)
from .polarization import (jones_vector, stokes_vector, jones_2_stokes,
linear_polarizer, phase_retarder, half_wave_retarder,
quarter_wave_retarder, transmissive_filter, reflective_filter,
mueller_matrix, polarizing_beam_splitter)
|
c46c16ad746bce2c297370e86c6f72b3c0d486f1422c4839f67141a0f2b291f3 | """
This module has all the classes and functions related to waves in optics.
**Contains**
* TWave
"""
__all__ = ['TWave']
from sympy import (sympify, pi, sin, cos, sqrt, Number, Symbol, S,
symbols, Derivative, atan2)
from sympy.core.expr import Expr
from sympy.physics.units import speed_of_light, meter, second
c = speed_of_light.convert_to(meter/second)
class TWave(Expr):
r"""
This is a simple transverse sine wave travelling in a one-dimensional space.
Basic properties are required at the time of creation of the object,
but they can be changed later with respective methods provided.
Explanation
===========
It is represented as :math:`A \times cos(k*x - \omega \times t + \phi )`,
where :math:`A` is the amplitude, :math:`\omega` is the angular frequency,
:math:`k` is the wavenumber (spatial frequency), :math:`x` is a spatial variable
to represent the position on the dimension on which the wave propagates,
and :math:`\phi` is the phase angle of the wave.
Arguments
=========
amplitude : Sympifyable
Amplitude of the wave.
frequency : Sympifyable
Frequency of the wave.
phase : Sympifyable
Phase angle of the wave.
time_period : Sympifyable
Time period of the wave.
n : Sympifyable
Refractive index of the medium.
Raises
=======
ValueError : When neither frequency nor time period is provided
or they are not consistent.
TypeError : When anything other than TWave objects is added.
Examples
========
>>> from sympy import symbols
>>> from sympy.physics.optics import TWave
>>> A1, phi1, A2, phi2, f = symbols('A1, phi1, A2, phi2, f')
>>> w1 = TWave(A1, f, phi1)
>>> w2 = TWave(A2, f, phi2)
>>> w3 = w1 + w2 # Superposition of two waves
>>> w3
TWave(sqrt(A1**2 + 2*A1*A2*cos(phi1 - phi2) + A2**2), f,
atan2(A1*sin(phi1) + A2*sin(phi2), A1*cos(phi1) + A2*cos(phi2)))
>>> w3.amplitude
sqrt(A1**2 + 2*A1*A2*cos(phi1 - phi2) + A2**2)
>>> w3.phase
atan2(A1*sin(phi1) + A2*sin(phi2), A1*cos(phi1) + A2*cos(phi2))
>>> w3.speed
299792458*meter/(second*n)
>>> w3.angular_velocity
2*pi*f
"""
def __init__(
self,
amplitude,
frequency=None,
phase=S.Zero,
time_period=None,
n=Symbol('n')):
frequency = sympify(frequency)
amplitude = sympify(amplitude)
phase = sympify(phase)
time_period = sympify(time_period)
n = sympify(n)
self._frequency = frequency
self._amplitude = amplitude
self._phase = phase
self._time_period = time_period
self._n = n
if time_period is not None:
self._frequency = 1/self._time_period
if frequency is not None:
self._time_period = 1/self._frequency
if time_period is not None:
if frequency != 1/time_period:
raise ValueError("frequency and time_period should be consistent.")
if frequency is None and time_period is None:
raise ValueError("Either frequency or time period is needed.")
@property
def frequency(self):
"""
Returns the frequency of the wave,
in cycles per second.
Examples
========
>>> from sympy import symbols
>>> from sympy.physics.optics import TWave
>>> A, phi, f = symbols('A, phi, f')
>>> w = TWave(A, f, phi)
>>> w.frequency
f
"""
return self._frequency
@property
def time_period(self):
"""
Returns the temporal period of the wave,
in seconds per cycle.
Examples
========
>>> from sympy import symbols
>>> from sympy.physics.optics import TWave
>>> A, phi, f = symbols('A, phi, f')
>>> w = TWave(A, f, phi)
>>> w.time_period
1/f
"""
return self._time_period
@property
def wavelength(self):
"""
Returns the wavelength (spatial period) of the wave,
in meters per cycle.
It depends on the medium of the wave.
Examples
========
>>> from sympy import symbols
>>> from sympy.physics.optics import TWave
>>> A, phi, f = symbols('A, phi, f')
>>> w = TWave(A, f, phi)
>>> w.wavelength
299792458*meter/(second*f*n)
"""
return c/(self._frequency*self._n)
@property
def amplitude(self):
"""
Returns the amplitude of the wave.
Examples
========
>>> from sympy import symbols
>>> from sympy.physics.optics import TWave
>>> A, phi, f = symbols('A, phi, f')
>>> w = TWave(A, f, phi)
>>> w.amplitude
A
"""
return self._amplitude
@property
def phase(self):
"""
Returns the phase angle of the wave,
in radians.
Examples
========
>>> from sympy import symbols
>>> from sympy.physics.optics import TWave
>>> A, phi, f = symbols('A, phi, f')
>>> w = TWave(A, f, phi)
>>> w.phase
phi
"""
return self._phase
@property
def speed(self):
"""
Returns the propagation speed of the wave,
in meters per second.
It is dependent on the propagation medium.
Examples
========
>>> from sympy import symbols
>>> from sympy.physics.optics import TWave
>>> A, phi, f = symbols('A, phi, f')
>>> w = TWave(A, f, phi)
>>> w.speed
299792458*meter/(second*n)
"""
return self.wavelength*self._frequency
@property
def angular_velocity(self):
"""
Returns the angular velocity of the wave,
in radians per second.
Examples
========
>>> from sympy import symbols
>>> from sympy.physics.optics import TWave
>>> A, phi, f = symbols('A, phi, f')
>>> w = TWave(A, f, phi)
>>> w.angular_velocity
2*pi*f
"""
return 2*pi*self._frequency
@property
def wavenumber(self):
"""
Returns the wavenumber of the wave,
in radians per meter.
Examples
========
>>> from sympy import symbols
>>> from sympy.physics.optics import TWave
>>> A, phi, f = symbols('A, phi, f')
>>> w = TWave(A, f, phi)
>>> w.wavenumber
pi*second*f*n/(149896229*meter)
"""
return 2*pi/self.wavelength
def __str__(self):
"""String representation of a TWave."""
from sympy.printing import sstr
return type(self).__name__ + sstr(self.args)
__repr__ = __str__
def __add__(self, other):
"""
Addition of two waves will result in their superposition.
The type of interference will depend on their phase angles.
"""
if isinstance(other, TWave):
if self._frequency == other._frequency and self.wavelength == other.wavelength:
return TWave(sqrt(self._amplitude**2 + other._amplitude**2 + 2 *
self._amplitude*other._amplitude*cos(
self._phase - other.phase)),
self._frequency,
atan2(self._amplitude*sin(self._phase)
+ other._amplitude*sin(other._phase),
self._amplitude*cos(self._phase)
+ other._amplitude*cos(other._phase))
)
else:
raise NotImplementedError("Interference of waves with different frequencies"
" has not been implemented.")
else:
raise TypeError(type(other).__name__ + " and TWave objects can't be added.")
def __mul__(self, other):
"""
Multiplying a wave by a scalar rescales the amplitude of the wave.
"""
other = sympify(other)
if isinstance(other, Number):
return TWave(self._amplitude*other, *self.args[1:])
else:
raise TypeError(type(other).__name__ + " and TWave objects can't be multiplied.")
def __sub__(self, other):
return self.__add__(-1*other)
def __neg__(self):
return self.__mul__(-1)
def __radd__(self, other):
return self.__add__(other)
def __rmul__(self, other):
return self.__mul__(other)
def __rsub__(self, other):
return (-self).__radd__(other)
def _eval_rewrite_as_sin(self, *args, **kwargs):
return self._amplitude*sin(self.wavenumber*Symbol('x')
- self.angular_velocity*Symbol('t') + self._phase + pi/2, evaluate=False)
def _eval_rewrite_as_cos(self, *args, **kwargs):
return self._amplitude*cos(self.wavenumber*Symbol('x')
- self.angular_velocity*Symbol('t') + self._phase)
def _eval_rewrite_as_pde(self, *args, **kwargs):
from sympy import Function
mu, epsilon, x, t = symbols('mu, epsilon, x, t')
E = Function('E')
return Derivative(E(x, t), x, 2) + mu*epsilon*Derivative(E(x, t), t, 2)
def _eval_rewrite_as_exp(self, *args, **kwargs):
from sympy import exp, I
return self._amplitude*exp(I*(self.wavenumber*Symbol('x')
- self.angular_velocity*Symbol('t') + self._phase))
|
7e051180d522bb483346382e5847414ae10172fca94ae8f440d83b68f9e83398 | from sympy.physics.units.systems.si import dimsys_SI
from sympy import S, Symbol, sqrt, cos, acos, log, atan2, pi, Abs
from sympy.physics.units.dimensions import Dimension
from sympy.physics.units.definitions.dimension_definitions import (
length, time, mass, force, pressure, angle
)
from sympy.physics.units import foot
from sympy.testing.pytest import raises
def test_Dimension_definition():
assert dimsys_SI.get_dimensional_dependencies(length) == {"length": 1}
assert length.name == Symbol("length")
assert length.symbol == Symbol("L")
halflength = sqrt(length)
assert dimsys_SI.get_dimensional_dependencies(halflength) == {"length": S.Half}
def test_Dimension_error_definition():
# tuple with more or less than two entries
raises(TypeError, lambda: Dimension(("length", 1, 2)))
raises(TypeError, lambda: Dimension(["length"]))
# non-number power
raises(TypeError, lambda: Dimension({"length": "a"}))
# non-number with named argument
raises(TypeError, lambda: Dimension({"length": (1, 2)}))
# symbol should by Symbol or str
raises(AssertionError, lambda: Dimension("length", symbol=1))
def test_str():
assert str(Dimension("length")) == "Dimension(length)"
assert str(Dimension("length", "L")) == "Dimension(length, L)"
def test_Dimension_properties():
assert dimsys_SI.is_dimensionless(length) is False
assert dimsys_SI.is_dimensionless(length/length) is True
assert dimsys_SI.is_dimensionless(Dimension("undefined")) is False
assert length.has_integer_powers(dimsys_SI) is True
assert (length**(-1)).has_integer_powers(dimsys_SI) is True
assert (length**1.5).has_integer_powers(dimsys_SI) is False
def test_Dimension_add_sub():
assert length + length == length
assert length - length == length
assert -length == length
raises(TypeError, lambda: length + foot)
raises(TypeError, lambda: foot + length)
raises(TypeError, lambda: length - foot)
raises(TypeError, lambda: foot - length)
# issue 14547 - only raise error for dimensional args; allow
# others to pass
x = Symbol('x')
e = length + x
assert e == x + length and e.is_Add and set(e.args) == {length, x}
e = length + 1
assert e == 1 + length == 1 - length and e.is_Add and set(e.args) == {length, 1}
assert dimsys_SI.get_dimensional_dependencies(mass * length / time**2 + force) == \
{'length': 1, 'mass': 1, 'time': -2}
assert dimsys_SI.get_dimensional_dependencies(mass * length / time**2 + force -
pressure * length**2) == \
{'length': 1, 'mass': 1, 'time': -2}
raises(TypeError, lambda: dimsys_SI.get_dimensional_dependencies(mass * length / time**2 + pressure))
def test_Dimension_mul_div_exp():
assert 2*length == length*2 == length/2 == length
assert 2/length == 1/length
x = Symbol('x')
m = x*length
assert m == length*x and m.is_Mul and set(m.args) == {x, length}
d = x/length
assert d == x*length**-1 and d.is_Mul and set(d.args) == {x, 1/length}
d = length/x
assert d == length*x**-1 and d.is_Mul and set(d.args) == {1/x, length}
velo = length / time
assert (length * length) == length ** 2
assert dimsys_SI.get_dimensional_dependencies(length * length) == {"length": 2}
assert dimsys_SI.get_dimensional_dependencies(length ** 2) == {"length": 2}
assert dimsys_SI.get_dimensional_dependencies(length * time) == { "length": 1, "time": 1}
assert dimsys_SI.get_dimensional_dependencies(velo) == { "length": 1, "time": -1}
assert dimsys_SI.get_dimensional_dependencies(velo ** 2) == {"length": 2, "time": -2}
assert dimsys_SI.get_dimensional_dependencies(length / length) == {}
assert dimsys_SI.get_dimensional_dependencies(velo / length * time) == {}
assert dimsys_SI.get_dimensional_dependencies(length ** -1) == {"length": -1}
assert dimsys_SI.get_dimensional_dependencies(velo ** -1.5) == {"length": -1.5, "time": 1.5}
length_a = length**"a"
assert dimsys_SI.get_dimensional_dependencies(length_a) == {"length": Symbol("a")}
assert dimsys_SI.get_dimensional_dependencies(length**pi) == {"length": pi}
assert dimsys_SI.get_dimensional_dependencies(length**(length/length)) == {"length": Dimension(1)}
raises(TypeError, lambda: dimsys_SI.get_dimensional_dependencies(length**length))
assert length != 1
assert length / length != 1
length_0 = length ** 0
assert dimsys_SI.get_dimensional_dependencies(length_0) == {}
# issue 18738
a = Symbol('a')
b = Symbol('b')
c = sqrt(a**2 + b**2)
c_dim = c.subs({a: length, b: length})
assert dimsys_SI.equivalent_dims(c_dim, length)
def test_Dimension_functions():
raises(TypeError, lambda: dimsys_SI.get_dimensional_dependencies(cos(length)))
raises(TypeError, lambda: dimsys_SI.get_dimensional_dependencies(acos(angle)))
raises(TypeError, lambda: dimsys_SI.get_dimensional_dependencies(atan2(length, time)))
raises(TypeError, lambda: dimsys_SI.get_dimensional_dependencies(log(length)))
raises(TypeError, lambda: dimsys_SI.get_dimensional_dependencies(log(100, length)))
raises(TypeError, lambda: dimsys_SI.get_dimensional_dependencies(log(length, 10)))
assert dimsys_SI.get_dimensional_dependencies(pi) == {}
assert dimsys_SI.get_dimensional_dependencies(cos(1)) == {}
assert dimsys_SI.get_dimensional_dependencies(cos(angle)) == {}
assert dimsys_SI.get_dimensional_dependencies(atan2(length, length)) == {}
assert dimsys_SI.get_dimensional_dependencies(log(length / length, length / length)) == {}
assert dimsys_SI.get_dimensional_dependencies(Abs(length)) == {"length": 1}
assert dimsys_SI.get_dimensional_dependencies(Abs(length / length)) == {}
assert dimsys_SI.get_dimensional_dependencies(sqrt(-1)) == {}
|
42bbe6f56c6eb4ee6df93e035e015ee1cd035d3f429b06e29e368cab10200d7e | from sympy import (symbols, sin, cos, pi, zeros, eye, simplify, ImmutableMatrix
as Matrix)
from sympy.physics.vector import (ReferenceFrame, Vector, CoordinateSym,
dynamicsymbols, time_derivative, express,
dot)
from sympy.physics.vector.frame import _check_frame
from sympy.physics.vector.vector import VectorTypeError
from sympy.testing.pytest import raises
Vector.simp = True
def test_dict_list():
A = ReferenceFrame('A')
B = ReferenceFrame('B')
C = ReferenceFrame('C')
D = ReferenceFrame('D')
E = ReferenceFrame('E')
F = ReferenceFrame('F')
B.orient_axis(A, A.x, 1.0)
C.orient_axis(B, B.x, 1.0)
D.orient_axis(C, C.x, 1.0)
assert D._dict_list(A, 0) == [D, C, B, A]
E.orient_axis(D, D.x, 1.0)
assert C._dict_list(A, 0) == [C, B, A]
assert C._dict_list(E, 0) == [C, D, E]
# only 0, 1, 2 permitted for second argument
raises(ValueError, lambda: C._dict_list(E, 5))
# no connecting path
raises(ValueError, lambda: F._dict_list(A, 0))
def test_coordinate_vars():
"""Tests the coordinate variables functionality"""
A = ReferenceFrame('A')
assert CoordinateSym('Ax', A, 0) == A[0]
assert CoordinateSym('Ax', A, 1) == A[1]
assert CoordinateSym('Ax', A, 2) == A[2]
raises(ValueError, lambda: CoordinateSym('Ax', A, 3))
q = dynamicsymbols('q')
qd = dynamicsymbols('q', 1)
assert isinstance(A[0], CoordinateSym) and \
isinstance(A[0], CoordinateSym) and \
isinstance(A[0], CoordinateSym)
assert A.variable_map(A) == {A[0]:A[0], A[1]:A[1], A[2]:A[2]}
assert A[0].frame == A
B = A.orientnew('B', 'Axis', [q, A.z])
assert B.variable_map(A) == {B[2]: A[2], B[1]: -A[0]*sin(q) + A[1]*cos(q),
B[0]: A[0]*cos(q) + A[1]*sin(q)}
assert A.variable_map(B) == {A[0]: B[0]*cos(q) - B[1]*sin(q),
A[1]: B[0]*sin(q) + B[1]*cos(q), A[2]: B[2]}
assert time_derivative(B[0], A) == -A[0]*sin(q)*qd + A[1]*cos(q)*qd
assert time_derivative(B[1], A) == -A[0]*cos(q)*qd - A[1]*sin(q)*qd
assert time_derivative(B[2], A) == 0
assert express(B[0], A, variables=True) == A[0]*cos(q) + A[1]*sin(q)
assert express(B[1], A, variables=True) == -A[0]*sin(q) + A[1]*cos(q)
assert express(B[2], A, variables=True) == A[2]
assert time_derivative(A[0]*A.x + A[1]*A.y + A[2]*A.z, B) == A[1]*qd*A.x - A[0]*qd*A.y
assert time_derivative(B[0]*B.x + B[1]*B.y + B[2]*B.z, A) == - B[1]*qd*B.x + B[0]*qd*B.y
assert express(B[0]*B[1]*B[2], A, variables=True) == \
A[2]*(-A[0]*sin(q) + A[1]*cos(q))*(A[0]*cos(q) + A[1]*sin(q))
assert (time_derivative(B[0]*B[1]*B[2], A) -
(A[2]*(-A[0]**2*cos(2*q) -
2*A[0]*A[1]*sin(2*q) +
A[1]**2*cos(2*q))*qd)).trigsimp() == 0
assert express(B[0]*B.x + B[1]*B.y + B[2]*B.z, A) == \
(B[0]*cos(q) - B[1]*sin(q))*A.x + (B[0]*sin(q) + \
B[1]*cos(q))*A.y + B[2]*A.z
assert express(B[0]*B.x + B[1]*B.y + B[2]*B.z, A, variables=True) == \
A[0]*A.x + A[1]*A.y + A[2]*A.z
assert express(A[0]*A.x + A[1]*A.y + A[2]*A.z, B) == \
(A[0]*cos(q) + A[1]*sin(q))*B.x + \
(-A[0]*sin(q) + A[1]*cos(q))*B.y + A[2]*B.z
assert express(A[0]*A.x + A[1]*A.y + A[2]*A.z, B, variables=True) == \
B[0]*B.x + B[1]*B.y + B[2]*B.z
N = B.orientnew('N', 'Axis', [-q, B.z])
assert N.variable_map(A) == {N[0]: A[0], N[2]: A[2], N[1]: A[1]}
C = A.orientnew('C', 'Axis', [q, A.x + A.y + A.z])
mapping = A.variable_map(C)
assert mapping[A[0]] == 2*C[0]*cos(q)/3 + C[0]/3 - 2*C[1]*sin(q + pi/6)/3 +\
C[1]/3 - 2*C[2]*cos(q + pi/3)/3 + C[2]/3
assert mapping[A[1]] == -2*C[0]*cos(q + pi/3)/3 + \
C[0]/3 + 2*C[1]*cos(q)/3 + C[1]/3 - 2*C[2]*sin(q + pi/6)/3 + C[2]/3
assert mapping[A[2]] == -2*C[0]*sin(q + pi/6)/3 + C[0]/3 - \
2*C[1]*cos(q + pi/3)/3 + C[1]/3 + 2*C[2]*cos(q)/3 + C[2]/3
def test_ang_vel():
q1, q2, q3, q4 = dynamicsymbols('q1 q2 q3 q4')
q1d, q2d, q3d, q4d = dynamicsymbols('q1 q2 q3 q4', 1)
N = ReferenceFrame('N')
A = N.orientnew('A', 'Axis', [q1, N.z])
B = A.orientnew('B', 'Axis', [q2, A.x])
C = B.orientnew('C', 'Axis', [q3, B.y])
D = N.orientnew('D', 'Axis', [q4, N.y])
u1, u2, u3 = dynamicsymbols('u1 u2 u3')
assert A.ang_vel_in(N) == (q1d)*A.z
assert B.ang_vel_in(N) == (q2d)*B.x + (q1d)*A.z
assert C.ang_vel_in(N) == (q3d)*C.y + (q2d)*B.x + (q1d)*A.z
A2 = N.orientnew('A2', 'Axis', [q4, N.y])
assert N.ang_vel_in(N) == 0
assert N.ang_vel_in(A) == -q1d*N.z
assert N.ang_vel_in(B) == -q1d*A.z - q2d*B.x
assert N.ang_vel_in(C) == -q1d*A.z - q2d*B.x - q3d*B.y
assert N.ang_vel_in(A2) == -q4d*N.y
assert A.ang_vel_in(N) == q1d*N.z
assert A.ang_vel_in(A) == 0
assert A.ang_vel_in(B) == - q2d*B.x
assert A.ang_vel_in(C) == - q2d*B.x - q3d*B.y
assert A.ang_vel_in(A2) == q1d*N.z - q4d*N.y
assert B.ang_vel_in(N) == q1d*A.z + q2d*A.x
assert B.ang_vel_in(A) == q2d*A.x
assert B.ang_vel_in(B) == 0
assert B.ang_vel_in(C) == -q3d*B.y
assert B.ang_vel_in(A2) == q1d*A.z + q2d*A.x - q4d*N.y
assert C.ang_vel_in(N) == q1d*A.z + q2d*A.x + q3d*B.y
assert C.ang_vel_in(A) == q2d*A.x + q3d*C.y
assert C.ang_vel_in(B) == q3d*B.y
assert C.ang_vel_in(C) == 0
assert C.ang_vel_in(A2) == q1d*A.z + q2d*A.x + q3d*B.y - q4d*N.y
assert A2.ang_vel_in(N) == q4d*A2.y
assert A2.ang_vel_in(A) == q4d*A2.y - q1d*N.z
assert A2.ang_vel_in(B) == q4d*N.y - q1d*A.z - q2d*A.x
assert A2.ang_vel_in(C) == q4d*N.y - q1d*A.z - q2d*A.x - q3d*B.y
assert A2.ang_vel_in(A2) == 0
C.set_ang_vel(N, u1*C.x + u2*C.y + u3*C.z)
assert C.ang_vel_in(N) == (u1)*C.x + (u2)*C.y + (u3)*C.z
assert N.ang_vel_in(C) == (-u1)*C.x + (-u2)*C.y + (-u3)*C.z
assert C.ang_vel_in(D) == (u1)*C.x + (u2)*C.y + (u3)*C.z + (-q4d)*D.y
assert D.ang_vel_in(C) == (-u1)*C.x + (-u2)*C.y + (-u3)*C.z + (q4d)*D.y
q0 = dynamicsymbols('q0')
q0d = dynamicsymbols('q0', 1)
E = N.orientnew('E', 'Quaternion', (q0, q1, q2, q3))
assert E.ang_vel_in(N) == (
2 * (q1d * q0 + q2d * q3 - q3d * q2 - q0d * q1) * E.x +
2 * (q2d * q0 + q3d * q1 - q1d * q3 - q0d * q2) * E.y +
2 * (q3d * q0 + q1d * q2 - q2d * q1 - q0d * q3) * E.z)
F = N.orientnew('F', 'Body', (q1, q2, q3), 313)
assert F.ang_vel_in(N) == ((sin(q2)*sin(q3)*q1d + cos(q3)*q2d)*F.x +
(sin(q2)*cos(q3)*q1d - sin(q3)*q2d)*F.y + (cos(q2)*q1d + q3d)*F.z)
G = N.orientnew('G', 'Axis', (q1, N.x + N.y))
assert G.ang_vel_in(N) == q1d * (N.x + N.y).normalize()
assert N.ang_vel_in(G) == -q1d * (N.x + N.y).normalize()
def test_dcm():
q1, q2, q3, q4 = dynamicsymbols('q1 q2 q3 q4')
N = ReferenceFrame('N')
A = N.orientnew('A', 'Axis', [q1, N.z])
B = A.orientnew('B', 'Axis', [q2, A.x])
C = B.orientnew('C', 'Axis', [q3, B.y])
D = N.orientnew('D', 'Axis', [q4, N.y])
E = N.orientnew('E', 'Space', [q1, q2, q3], '123')
assert N.dcm(C) == 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)]])
# This is a little touchy. Is it ok to use simplify in assert?
test_mat = D.dcm(C) - Matrix(
[[cos(q1) * cos(q3) * cos(q4) - sin(q3) * (- sin(q4) * cos(q2) +
sin(q1) * sin(q2) * cos(q4)), - sin(q2) * sin(q4) - sin(q1) *
cos(q2) * cos(q4), sin(q3) * cos(q1) * cos(q4) + cos(q3) * (- sin(q4) *
cos(q2) + sin(q1) * sin(q2) * cos(q4))], [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(q4) * cos(q1) * cos(q3) -
sin(q3) * (cos(q2) * cos(q4) + sin(q1) * sin(q2) * sin(q4)), sin(q2) *
cos(q4) - sin(q1) * sin(q4) * cos(q2), sin(q3) * sin(q4) * cos(q1) +
cos(q3) * (cos(q2) * cos(q4) + sin(q1) * sin(q2) * sin(q4))]])
assert test_mat.expand() == zeros(3, 3)
assert E.dcm(N) == Matrix(
[[cos(q2)*cos(q3), sin(q3)*cos(q2), -sin(q2)],
[sin(q1)*sin(q2)*cos(q3) - sin(q3)*cos(q1), sin(q1)*sin(q2)*sin(q3) +
cos(q1)*cos(q3), sin(q1)*cos(q2)], [sin(q1)*sin(q3) +
sin(q2)*cos(q1)*cos(q3), - sin(q1)*cos(q3) + sin(q2)*sin(q3)*cos(q1),
cos(q1)*cos(q2)]])
def test_w_diff_dcm1():
# Ref:
# Dynamics Theory and Applications, Kane 1985
# Sec. 2.1 ANGULAR VELOCITY
A = ReferenceFrame('A')
B = ReferenceFrame('B')
c11, c12, c13 = dynamicsymbols('C11 C12 C13')
c21, c22, c23 = dynamicsymbols('C21 C22 C23')
c31, c32, c33 = dynamicsymbols('C31 C32 C33')
c11d, c12d, c13d = dynamicsymbols('C11 C12 C13', level=1)
c21d, c22d, c23d = dynamicsymbols('C21 C22 C23', level=1)
c31d, c32d, c33d = dynamicsymbols('C31 C32 C33', level=1)
DCM = Matrix([
[c11, c12, c13],
[c21, c22, c23],
[c31, c32, c33]
])
B.orient(A, 'DCM', DCM)
b1a = (B.x).express(A)
b2a = (B.y).express(A)
b3a = (B.z).express(A)
# Equation (2.1.1)
B.set_ang_vel(A, B.x*(dot((b3a).dt(A), B.y))
+ B.y*(dot((b1a).dt(A), B.z))
+ B.z*(dot((b2a).dt(A), B.x)))
# Equation (2.1.21)
expr = ( (c12*c13d + c22*c23d + c32*c33d)*B.x
+ (c13*c11d + c23*c21d + c33*c31d)*B.y
+ (c11*c12d + c21*c22d + c31*c32d)*B.z)
assert B.ang_vel_in(A) - expr == 0
def test_w_diff_dcm2():
q1, q2, q3 = dynamicsymbols('q1:4')
N = ReferenceFrame('N')
A = N.orientnew('A', 'axis', [q1, N.x])
B = A.orientnew('B', 'axis', [q2, A.y])
C = B.orientnew('C', 'axis', [q3, B.z])
DCM = C.dcm(N).T
D = N.orientnew('D', 'DCM', DCM)
# Frames D and C are the same ReferenceFrame,
# since they have equal DCM respect to frame N.
# Therefore, D and C should have same angle velocity in N.
assert D.dcm(N) == C.dcm(N) == Matrix([
[cos(q2)*cos(q3), sin(q1)*sin(q2)*cos(q3) +
sin(q3)*cos(q1), sin(q1)*sin(q3) -
sin(q2)*cos(q1)*cos(q3)], [-sin(q3)*cos(q2),
-sin(q1)*sin(q2)*sin(q3) + cos(q1)*cos(q3),
sin(q1)*cos(q3) + sin(q2)*sin(q3)*cos(q1)],
[sin(q2), -sin(q1)*cos(q2), cos(q1)*cos(q2)]])
assert (D.ang_vel_in(N) - C.ang_vel_in(N)).express(N).simplify() == 0
def test_orientnew_respects_parent_class():
class MyReferenceFrame(ReferenceFrame):
pass
B = MyReferenceFrame('B')
C = B.orientnew('C', 'Axis', [0, B.x])
assert isinstance(C, MyReferenceFrame)
def test_orientnew_respects_input_indices():
N = ReferenceFrame('N')
q1 = dynamicsymbols('q1')
A = N.orientnew('a', 'Axis', [q1, N.z])
#modify default indices:
minds = [x+'1' for x in N.indices]
B = N.orientnew('b', 'Axis', [q1, N.z], indices=minds)
assert N.indices == A.indices
assert B.indices == minds
def test_orientnew_respects_input_latexs():
N = ReferenceFrame('N')
q1 = dynamicsymbols('q1')
A = N.orientnew('a', 'Axis', [q1, N.z])
#build default and alternate latex_vecs:
def_latex_vecs = [(r"\mathbf{\hat{%s}_%s}" % (A.name.lower(),
A.indices[0])), (r"\mathbf{\hat{%s}_%s}" %
(A.name.lower(), A.indices[1])),
(r"\mathbf{\hat{%s}_%s}" % (A.name.lower(),
A.indices[2]))]
name = 'b'
indices = [x+'1' for x in N.indices]
new_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]))]
B = N.orientnew(name, 'Axis', [q1, N.z], latexs=new_latex_vecs)
assert A.latex_vecs == def_latex_vecs
assert B.latex_vecs == new_latex_vecs
assert B.indices != indices
def test_orientnew_respects_input_variables():
N = ReferenceFrame('N')
q1 = dynamicsymbols('q1')
A = N.orientnew('a', 'Axis', [q1, N.z])
#build non-standard variable names
name = 'b'
new_variables = ['notb_'+x+'1' for x in N.indices]
B = N.orientnew(name, 'Axis', [q1, N.z], variables=new_variables)
for j,var in enumerate(A.varlist):
assert var.name == A.name + '_' + A.indices[j]
for j,var in enumerate(B.varlist):
assert var.name == new_variables[j]
def test_issue_10348():
u = dynamicsymbols('u:3')
I = ReferenceFrame('I')
I.orientnew('A', 'space', u, 'XYZ')
def test_issue_11503():
A = ReferenceFrame("A")
A.orientnew("B", "Axis", [35, A.y])
C = ReferenceFrame("C")
A.orient(C, "Axis", [70, C.z])
def test_partial_velocity():
N = ReferenceFrame('N')
A = ReferenceFrame('A')
u1, u2 = dynamicsymbols('u1, u2')
A.set_ang_vel(N, u1 * A.x + u2 * N.y)
assert N.partial_velocity(A, u1) == -A.x
assert N.partial_velocity(A, u1, u2) == (-A.x, -N.y)
assert A.partial_velocity(N, u1) == A.x
assert A.partial_velocity(N, u1, u2) == (A.x, N.y)
assert N.partial_velocity(N, u1) == 0
assert A.partial_velocity(A, u1) == 0
def test_issue_11498():
A = ReferenceFrame('A')
B = ReferenceFrame('B')
# Identity transformation
A.orient(B, 'DCM', eye(3))
assert A.dcm(B) == Matrix([[1, 0, 0], [0, 1, 0], [0, 0, 1]])
assert B.dcm(A) == Matrix([[1, 0, 0], [0, 1, 0], [0, 0, 1]])
# x -> y
# y -> -z
# z -> -x
A.orient(B, 'DCM', Matrix([[0, 1, 0], [0, 0, -1], [-1, 0, 0]]))
assert B.dcm(A) == Matrix([[0, 1, 0], [0, 0, -1], [-1, 0, 0]])
assert A.dcm(B) == Matrix([[0, 0, -1], [1, 0, 0], [0, -1, 0]])
assert B.dcm(A).T == A.dcm(B)
def test_reference_frame():
raises(TypeError, lambda: ReferenceFrame(0))
raises(TypeError, lambda: ReferenceFrame('N', 0))
raises(ValueError, lambda: ReferenceFrame('N', [0, 1]))
raises(TypeError, lambda: ReferenceFrame('N', [0, 1, 2]))
raises(TypeError, lambda: ReferenceFrame('N', ['a', 'b', 'c'], 0))
raises(ValueError, lambda: ReferenceFrame('N', ['a', 'b', 'c'], [0, 1]))
raises(TypeError, lambda: ReferenceFrame('N', ['a', 'b', 'c'], [0, 1, 2]))
raises(TypeError, lambda: ReferenceFrame('N', ['a', 'b', 'c'],
['a', 'b', 'c'], 0))
raises(ValueError, lambda: ReferenceFrame('N', ['a', 'b', 'c'],
['a', 'b', 'c'], [0, 1]))
raises(TypeError, lambda: ReferenceFrame('N', ['a', 'b', 'c'],
['a', 'b', 'c'], [0, 1, 2]))
N = ReferenceFrame('N')
assert N[0] == CoordinateSym('N_x', N, 0)
assert N[1] == CoordinateSym('N_y', N, 1)
assert N[2] == CoordinateSym('N_z', N, 2)
raises(ValueError, lambda: N[3])
N = ReferenceFrame('N', ['a', 'b', 'c'])
assert N['a'] == N.x
assert N['b'] == N.y
assert N['c'] == N.z
raises(ValueError, lambda: N['d'])
assert str(N) == 'N'
A = ReferenceFrame('A')
B = ReferenceFrame('B')
q0, q1, q2, q3 = symbols('q0 q1 q2 q3')
raises(TypeError, lambda: A.orient(B, 'DCM', 0))
raises(TypeError, lambda: B.orient(N, 'Space', [q1, q2, q3], '222'))
raises(TypeError, lambda: B.orient(N, 'Axis', [q1, N.x + 2 * N.y], '222'))
raises(TypeError, lambda: B.orient(N, 'Axis', q1))
raises(IndexError, lambda: B.orient(N, 'Axis', [q1]))
raises(TypeError, lambda: B.orient(N, 'Quaternion', [q0, q1, q2, q3], '222'))
raises(TypeError, lambda: B.orient(N, 'Quaternion', q0))
raises(TypeError, lambda: B.orient(N, 'Quaternion', [q0, q1, q2]))
raises(NotImplementedError, lambda: B.orient(N, 'Foo', [q0, q1, q2]))
raises(TypeError, lambda: B.orient(N, 'Body', [q1, q2], '232'))
raises(TypeError, lambda: B.orient(N, 'Space', [q1, q2], '232'))
N.set_ang_acc(B, 0)
assert N.ang_acc_in(B) == Vector(0)
N.set_ang_vel(B, 0)
assert N.ang_vel_in(B) == Vector(0)
def test_check_frame():
raises(VectorTypeError, lambda: _check_frame(0))
def test_dcm_diff_16824():
# NOTE : This is a regression test for the bug introduced in PR 14758,
# identified in 16824, and solved by PR 16828.
# This is the solution to Problem 2.2 on page 264 in Kane & Lenvinson's
# 1985 book.
q1, q2, q3 = dynamicsymbols('q1:4')
s1 = sin(q1)
c1 = cos(q1)
s2 = sin(q2)
c2 = cos(q2)
s3 = sin(q3)
c3 = cos(q3)
dcm = Matrix([[c2*c3, s1*s2*c3 - s3*c1, c1*s2*c3 + s3*s1],
[c2*s3, s1*s2*s3 + c3*c1, c1*s2*s3 - c3*s1],
[-s2, s1*c2, c1*c2]])
A = ReferenceFrame('A')
B = ReferenceFrame('B')
B.orient(A, 'DCM', dcm)
AwB = B.ang_vel_in(A)
alpha2 = s3*c2*q1.diff() + c3*q2.diff()
beta2 = s1*c2*q3.diff() + c1*q2.diff()
assert simplify(AwB.dot(A.y) - alpha2) == 0
assert simplify(AwB.dot(B.y) - beta2) == 0
def test_orient_explicit():
A = ReferenceFrame('A')
B = ReferenceFrame('B')
A.orient_explicit(B, eye(3))
assert A.dcm(B) == Matrix([[1, 0, 0], [0, 1, 0], [0, 0, 1]])
def test_orient_axis():
A = ReferenceFrame('A')
B = ReferenceFrame('B')
assert A.orient_axis(B,-B.x, 1) == A.orient_axis(B, B.x, -1)
def test_orient_body():
A = ReferenceFrame('A')
B = ReferenceFrame('B')
B.orient_body_fixed(A, (1,1,0), 'XYX')
assert B.dcm(A) == Matrix([[cos(1), sin(1)**2, -sin(1)*cos(1)], [0, cos(1), sin(1)], [sin(1), -sin(1)*cos(1), cos(1)**2]])
def test_orient_space():
A = ReferenceFrame('A')
B = ReferenceFrame('B')
B.orient_space_fixed(A, (0,0,0), '123')
assert B.dcm(A) == Matrix([[1, 0, 0], [0, 1, 0], [0, 0, 1]])
def test_orient_quaternion():
A = ReferenceFrame('A')
B = ReferenceFrame('B')
B.orient_quaternion(A, (0,0,0,0))
assert B.dcm(A) == Matrix([[0, 0, 0], [0, 0, 0], [0, 0, 0]])
def test_frame_dict():
A = ReferenceFrame('A')
B = ReferenceFrame('B')
C = ReferenceFrame('C')
a, b, c = symbols('a b c')
B.orient_axis(A, A.x, a)
assert A._dcm_dict == {B: Matrix([[1, 0, 0],[0, cos(a), -sin(a)],[0, sin(a), cos(a)]])}
assert B._dcm_dict == {A: Matrix([[1, 0, 0],[0, cos(a), sin(a)],[0, -sin(a), cos(a)]])}
assert C._dcm_dict == {}
B.orient_axis(C, C.x, b)
# Previous relation is not wiped
assert A._dcm_dict == {B: Matrix([[1, 0, 0],[0, cos(a), -sin(a)],[0, sin(a), cos(a)]])}
assert B._dcm_dict == {A: Matrix([[1, 0, 0],[0, cos(a), sin(a)],[0, -sin(a), cos(a)]]), \
C: Matrix([[1, 0, 0],[0, cos(b), sin(b)],[0, -sin(b), cos(b)]])}
assert C._dcm_dict == {B: Matrix([[1, 0, 0],[0, cos(b), -sin(b)],[0, sin(b), cos(b)]])}
A.orient_axis(B, B.x, c)
# Previous relation is updated
assert B._dcm_dict == {C: Matrix([[1, 0, 0],[0, cos(b), sin(b)],[0, -sin(b), cos(b)]]),\
A: Matrix([[1, 0, 0],[0, cos(c), -sin(c)],[0, sin(c), cos(c)]])}
assert A._dcm_dict == {B: Matrix([[1, 0, 0],[0, cos(c), sin(c)],[0, -sin(c), cos(c)]])}
assert C._dcm_dict == {B: Matrix([[1, 0, 0],[0, cos(b), -sin(b)],[0, sin(b), cos(b)]])}
def test_dcm_cache_dict():
A = ReferenceFrame('A')
B = ReferenceFrame('B')
C = ReferenceFrame('C')
D = ReferenceFrame('D')
a, b, c = symbols('a b c')
B.orient_axis(A, A.x, a)
C.orient_axis(B, B.x, b)
D.orient_axis(C, C.x, c)
assert D._dcm_dict == {C: Matrix([[1, 0, 0],[0, cos(c), sin(c)],[0, -sin(c), cos(c)]])}
assert C._dcm_dict == {B: Matrix([[1, 0, 0],[0, cos(b), sin(b)],[0, -sin(b), cos(b)]]), \
D: Matrix([[1, 0, 0],[0, cos(c), -sin(c)],[0, sin(c), cos(c)]])}
assert B._dcm_dict == {A: Matrix([[1, 0, 0],[0, cos(a), sin(a)],[0, -sin(a), cos(a)]]), \
C: Matrix([[1, 0, 0],[0, cos(b), -sin(b)],[0, sin(b), cos(b)]])}
assert A._dcm_dict == {B: Matrix([[1, 0, 0],[0, cos(a), -sin(a)],[0, sin(a), cos(a)]])}
assert D._dcm_dict == D._dcm_cache
D.dcm(A) # Check calculated dcm relation is stored in _dcm_cache and not in _dcm_dict
assert list(A._dcm_cache.keys()) == [A, B, D]
assert list(D._dcm_cache.keys()) == [C, A]
assert list(A._dcm_dict.keys()) == [B]
assert list(D._dcm_dict.keys()) == [C]
assert A._dcm_dict != A._dcm_cache
A.orient_axis(B, B.x, b) # _dcm_cache of A is wiped out and new relation is stored.
assert A._dcm_dict == {B: Matrix([[1, 0, 0],[0, cos(b), sin(b)],[0, -sin(b), cos(b)]])}
assert A._dcm_dict == A._dcm_cache
assert B._dcm_dict == {C: Matrix([[1, 0, 0],[0, cos(b), -sin(b)],[0, sin(b), cos(b)]]), \
A: Matrix([[1, 0, 0],[0, cos(b), -sin(b)],[0, sin(b), cos(b)]])}
|
24c60ca9b04cdf262ea908896f9c9ed0c73b0ce108951e66c8c19d99a676fcb1 | from sympy import (symbols, Symbol, pi, sqrt, cos, sin, Derivative,
Function, simplify, I, atan2)
from sympy.abc import epsilon, mu
from sympy.functions.elementary.exponential import exp
from sympy.physics.units import speed_of_light, m, s
from sympy.physics.optics import TWave
from sympy.testing.pytest import raises
c = speed_of_light.convert_to(m/s)
def test_twave():
A1, phi1, A2, phi2, f = symbols('A1, phi1, A2, phi2, f')
n = Symbol('n') # Refractive index
t = Symbol('t') # Time
x = Symbol('x') # Spatial variable
E = Function('E')
w1 = TWave(A1, f, phi1)
w2 = TWave(A2, f, phi2)
assert w1.amplitude == A1
assert w1.frequency == f
assert w1.phase == phi1
assert w1.wavelength == c/(f*n)
assert w1.time_period == 1/f
assert w1.angular_velocity == 2*pi*f
assert w1.wavenumber == 2*pi*f*n/c
assert w1.speed == c/n
w3 = w1 + w2
assert w3.amplitude == sqrt(A1**2 + 2*A1*A2*cos(phi1 - phi2) + A2**2)
assert w3.frequency == f
assert w3.phase == atan2(A1*sin(phi1) + A2*sin(phi2), A1*cos(phi1) + A2*cos(phi2))
assert w3.wavelength == c/(f*n)
assert w3.time_period == 1/f
assert w3.angular_velocity == 2*pi*f
assert w3.wavenumber == 2*pi*f*n/c
assert w3.speed == c/n
assert simplify(w3.rewrite(sin) - w2.rewrite(sin) - w1.rewrite(sin)) == 0
assert w3.rewrite('pde') == epsilon*mu*Derivative(E(x, t), t, t) + Derivative(E(x, t), x, x)
assert w3.rewrite(cos) == sqrt(A1**2 + 2*A1*A2*cos(phi1 - phi2)
+ A2**2)*cos(pi*f*n*x*s/(149896229*m) - 2*pi*f*t + atan2(A1*sin(phi1)
+ A2*sin(phi2), A1*cos(phi1) + A2*cos(phi2)))
assert w3.rewrite(exp) == sqrt(A1**2 + 2*A1*A2*cos(phi1 - phi2)
+ A2**2)*exp(I*(-2*pi*f*t + atan2(A1*sin(phi1) + A2*sin(phi2), A1*cos(phi1)
+ A2*cos(phi2)) + pi*s*f*n*x/(149896229*m)))
w4 = TWave(A1, None, 0, 1/f)
assert w4.frequency == f
w5 = w1 - w2
assert w5.amplitude == sqrt(A1**2 - 2*A1*A2*cos(phi1 - phi2) + A2**2)
assert w5.frequency == f
assert w5.phase == atan2(A1*sin(phi1) - A2*sin(phi2), A1*cos(phi1) - A2*cos(phi2))
assert w5.wavelength == c/(f*n)
assert w5.time_period == 1/f
assert w5.angular_velocity == 2*pi*f
assert w5.wavenumber == 2*pi*f*n/c
assert w5.speed == c/n
assert simplify(w5.rewrite(sin) - w1.rewrite(sin) + w2.rewrite(sin)) == 0
assert w5.rewrite('pde') == epsilon*mu*Derivative(E(x, t), t, t) + Derivative(E(x, t), x, x)
assert w5.rewrite(cos) == sqrt(A1**2 - 2*A1*A2*cos(phi1 - phi2)
+ A2**2)*cos(-2*pi*f*t + atan2(A1*sin(phi1) - A2*sin(phi2), A1*cos(phi1)
- A2*cos(phi2)) + pi*s*f*n*x/(149896229*m))
assert w5.rewrite(exp) == sqrt(A1**2 - 2*A1*A2*cos(phi1 - phi2)
+ A2**2)*exp(I*(-2*pi*f*t + atan2(A1*sin(phi1) - A2*sin(phi2), A1*cos(phi1)
- A2*cos(phi2)) + pi*s*f*n*x/(149896229*m)))
w6 = 2*w1
assert w6.amplitude == 2*A1
assert w6.frequency == f
assert w6.phase == phi1
w7 = -w6
assert w7.amplitude == -2*A1
assert w7.frequency == f
assert w7.phase == phi1
raises(ValueError, lambda:TWave(A1))
raises(ValueError, lambda:TWave(A1, f, phi1, t))
|
08b734e38bd0e2193696c33f5e7a1200916a51ac205f43145a3c105a48745eab | import itertools
from collections.abc import Iterable
from sympy import S, Tuple, diff, Basic
from sympy.core.sympify import _sympify
from sympy.tensor.array.ndim_array import NDimArray
from sympy.tensor.array.dense_ndim_array import DenseNDimArray, ImmutableDenseNDimArray
from sympy.tensor.array.sparse_ndim_array import SparseNDimArray
def _arrayfy(a):
from sympy.matrices import MatrixBase
if isinstance(a, NDimArray):
return a
if isinstance(a, (MatrixBase, list, tuple, Tuple)):
return ImmutableDenseNDimArray(a)
return a
def tensorproduct(*args):
"""
Tensor product among scalars or array-like objects.
Examples
========
>>> from sympy.tensor.array import tensorproduct, Array
>>> from sympy.abc import x, y, z, t
>>> A = Array([[1, 2], [3, 4]])
>>> B = Array([x, y])
>>> tensorproduct(A, B)
[[[x, y], [2*x, 2*y]], [[3*x, 3*y], [4*x, 4*y]]]
>>> tensorproduct(A, x)
[[x, 2*x], [3*x, 4*x]]
>>> tensorproduct(A, B, B)
[[[[x**2, x*y], [x*y, y**2]], [[2*x**2, 2*x*y], [2*x*y, 2*y**2]]], [[[3*x**2, 3*x*y], [3*x*y, 3*y**2]], [[4*x**2, 4*x*y], [4*x*y, 4*y**2]]]]
Applying this function on two matrices will result in a rank 4 array.
>>> from sympy import Matrix, eye
>>> m = Matrix([[x, y], [z, t]])
>>> p = tensorproduct(eye(3), m)
>>> p
[[[[x, y], [z, t]], [[0, 0], [0, 0]], [[0, 0], [0, 0]]], [[[0, 0], [0, 0]], [[x, y], [z, t]], [[0, 0], [0, 0]]], [[[0, 0], [0, 0]], [[0, 0], [0, 0]], [[x, y], [z, t]]]]
"""
from sympy.tensor.array import SparseNDimArray, ImmutableSparseNDimArray
if len(args) == 0:
return S.One
if len(args) == 1:
return _arrayfy(args[0])
from sympy.tensor.array.expressions.array_expressions import _CodegenArrayAbstract
from sympy.tensor.array.expressions.array_expressions import ArrayTensorProduct
from sympy.tensor.array.expressions.array_expressions import _ArrayExpr
from sympy import MatrixSymbol
if any(isinstance(arg, (_ArrayExpr, _CodegenArrayAbstract, MatrixSymbol)) for arg in args):
return ArrayTensorProduct(*args)
if len(args) > 2:
return tensorproduct(tensorproduct(args[0], args[1]), *args[2:])
# length of args is 2:
a, b = map(_arrayfy, args)
if not isinstance(a, NDimArray) or not isinstance(b, NDimArray):
return a*b
if isinstance(a, SparseNDimArray) and isinstance(b, SparseNDimArray):
lp = len(b)
new_array = {k1*lp + k2: v1*v2 for k1, v1 in a._sparse_array.items() for k2, v2 in b._sparse_array.items()}
return ImmutableSparseNDimArray(new_array, a.shape + b.shape)
product_list = [i*j for i in Flatten(a) for j in Flatten(b)]
return ImmutableDenseNDimArray(product_list, a.shape + b.shape)
def _util_contraction_diagonal(array, *contraction_or_diagonal_axes):
array = _arrayfy(array)
# Verify contraction_axes:
taken_dims = set()
for axes_group in contraction_or_diagonal_axes:
if not isinstance(axes_group, Iterable):
raise ValueError("collections of contraction/diagonal axes expected")
dim = array.shape[axes_group[0]]
for d in axes_group:
if d in taken_dims:
raise ValueError("dimension specified more than once")
if dim != array.shape[d]:
raise ValueError("cannot contract or diagonalize between axes of different dimension")
taken_dims.add(d)
rank = array.rank()
remaining_shape = [dim for i, dim in enumerate(array.shape) if i not in taken_dims]
cum_shape = [0]*rank
_cumul = 1
for i in range(rank):
cum_shape[rank - i - 1] = _cumul
_cumul *= int(array.shape[rank - i - 1])
# DEFINITION: by absolute position it is meant the position along the one
# dimensional array containing all the tensor components.
# Possible future work on this module: move computation of absolute
# positions to a class method.
# Determine absolute positions of the uncontracted indices:
remaining_indices = [[cum_shape[i]*j for j in range(array.shape[i])]
for i in range(rank) if i not in taken_dims]
# Determine absolute positions of the contracted indices:
summed_deltas = []
for axes_group in contraction_or_diagonal_axes:
lidx = []
for js in range(array.shape[axes_group[0]]):
lidx.append(sum([cum_shape[ig] * js for ig in axes_group]))
summed_deltas.append(lidx)
return array, remaining_indices, remaining_shape, summed_deltas
def tensorcontraction(array, *contraction_axes):
"""
Contraction of an array-like object on the specified axes.
Examples
========
>>> from sympy import Array, tensorcontraction
>>> from sympy import Matrix, eye
>>> tensorcontraction(eye(3), (0, 1))
3
>>> A = Array(range(18), (3, 2, 3))
>>> A
[[[0, 1, 2], [3, 4, 5]], [[6, 7, 8], [9, 10, 11]], [[12, 13, 14], [15, 16, 17]]]
>>> tensorcontraction(A, (0, 2))
[21, 30]
Matrix multiplication may be emulated with a proper combination of
``tensorcontraction`` and ``tensorproduct``
>>> from sympy import tensorproduct
>>> from sympy.abc import a,b,c,d,e,f,g,h
>>> m1 = Matrix([[a, b], [c, d]])
>>> m2 = Matrix([[e, f], [g, h]])
>>> p = tensorproduct(m1, m2)
>>> p
[[[[a*e, a*f], [a*g, a*h]], [[b*e, b*f], [b*g, b*h]]], [[[c*e, c*f], [c*g, c*h]], [[d*e, d*f], [d*g, d*h]]]]
>>> tensorcontraction(p, (1, 2))
[[a*e + b*g, a*f + b*h], [c*e + d*g, c*f + d*h]]
>>> m1*m2
Matrix([
[a*e + b*g, a*f + b*h],
[c*e + d*g, c*f + d*h]])
"""
from sympy.tensor.array.expressions.array_expressions import ArrayContraction
from sympy.tensor.array.expressions.array_expressions import _CodegenArrayAbstract
from sympy.tensor.array.expressions.array_expressions import _ArrayExpr
from sympy import MatrixSymbol
if isinstance(array, (_ArrayExpr, _CodegenArrayAbstract, MatrixSymbol)):
return ArrayContraction(array, *contraction_axes)
array, remaining_indices, remaining_shape, summed_deltas = _util_contraction_diagonal(array, *contraction_axes)
# Compute the contracted array:
#
# 1. external for loops on all uncontracted indices.
# Uncontracted indices are determined by the combinatorial product of
# the absolute positions of the remaining indices.
# 2. internal loop on all contracted indices.
# It sums the values of the absolute contracted index and the absolute
# uncontracted index for the external loop.
contracted_array = []
for icontrib in itertools.product(*remaining_indices):
index_base_position = sum(icontrib)
isum = S.Zero
for sum_to_index in itertools.product(*summed_deltas):
idx = array._get_tuple_index(index_base_position + sum(sum_to_index))
isum += array[idx]
contracted_array.append(isum)
if len(remaining_indices) == 0:
assert len(contracted_array) == 1
return contracted_array[0]
return type(array)(contracted_array, remaining_shape)
def tensordiagonal(array, *diagonal_axes):
"""
Diagonalization of an array-like object on the specified axes.
This is equivalent to multiplying the expression by Kronecker deltas
uniting the axes.
The diagonal indices are put at the end of the axes.
Examples
========
``tensordiagonal`` acting on a 2-dimensional array by axes 0 and 1 is
equivalent to the diagonal of the matrix:
>>> from sympy import Array, tensordiagonal
>>> from sympy import Matrix, eye
>>> tensordiagonal(eye(3), (0, 1))
[1, 1, 1]
>>> from sympy.abc import a,b,c,d
>>> m1 = Matrix([[a, b], [c, d]])
>>> tensordiagonal(m1, [0, 1])
[a, d]
In case of higher dimensional arrays, the diagonalized out dimensions
are appended removed and appended as a single dimension at the end:
>>> A = Array(range(18), (3, 2, 3))
>>> A
[[[0, 1, 2], [3, 4, 5]], [[6, 7, 8], [9, 10, 11]], [[12, 13, 14], [15, 16, 17]]]
>>> tensordiagonal(A, (0, 2))
[[0, 7, 14], [3, 10, 17]]
>>> from sympy import permutedims
>>> tensordiagonal(A, (0, 2)) == permutedims(Array([A[0, :, 0], A[1, :, 1], A[2, :, 2]]), [1, 0])
True
"""
if any([len(i) <= 1 for i in diagonal_axes]):
raise ValueError("need at least two axes to diagonalize")
from sympy.tensor.array.expressions.array_expressions import _ArrayExpr
from sympy.tensor.array.expressions.array_expressions import _CodegenArrayAbstract
from sympy.tensor.array.expressions.array_expressions import ArrayDiagonal
from sympy import MatrixSymbol
if isinstance(array, (_ArrayExpr, _CodegenArrayAbstract, MatrixSymbol)):
return ArrayDiagonal(array, *diagonal_axes)
array, remaining_indices, remaining_shape, diagonal_deltas = _util_contraction_diagonal(array, *diagonal_axes)
# Compute the diagonalized array:
#
# 1. external for loops on all undiagonalized indices.
# Undiagonalized indices are determined by the combinatorial product of
# the absolute positions of the remaining indices.
# 2. internal loop on all diagonal indices.
# It appends the values of the absolute diagonalized index and the absolute
# undiagonalized index for the external loop.
diagonalized_array = []
diagonal_shape = [len(i) for i in diagonal_deltas]
for icontrib in itertools.product(*remaining_indices):
index_base_position = sum(icontrib)
isum = []
for sum_to_index in itertools.product(*diagonal_deltas):
idx = array._get_tuple_index(index_base_position + sum(sum_to_index))
isum.append(array[idx])
isum = type(array)(isum).reshape(*diagonal_shape)
diagonalized_array.append(isum)
return type(array)(diagonalized_array, remaining_shape + diagonal_shape)
def derive_by_array(expr, dx):
r"""
Derivative by arrays. Supports both arrays and scalars.
Explanation
===========
Given the array `A_{i_1, \ldots, i_N}` and the array `X_{j_1, \ldots, j_M}`
this function will return a new array `B` defined by
`B_{j_1,\ldots,j_M,i_1,\ldots,i_N} := \frac{\partial A_{i_1,\ldots,i_N}}{\partial X_{j_1,\ldots,j_M}}`
Examples
========
>>> from sympy import derive_by_array
>>> from sympy.abc import x, y, z, t
>>> from sympy import cos
>>> derive_by_array(cos(x*t), x)
-t*sin(t*x)
>>> derive_by_array(cos(x*t), [x, y, z, t])
[-t*sin(t*x), 0, 0, -x*sin(t*x)]
>>> derive_by_array([x, y**2*z], [[x, y], [z, t]])
[[[1, 0], [0, 2*y*z]], [[0, y**2], [0, 0]]]
"""
from sympy.matrices import MatrixBase
from sympy.tensor.array import SparseNDimArray
array_types = (Iterable, MatrixBase, NDimArray)
if isinstance(dx, array_types):
dx = ImmutableDenseNDimArray(dx)
for i in dx:
if not i._diff_wrt:
raise ValueError("cannot derive by this array")
if isinstance(expr, array_types):
if isinstance(expr, NDimArray):
expr = expr.as_immutable()
else:
expr = ImmutableDenseNDimArray(expr)
if isinstance(dx, array_types):
if isinstance(expr, SparseNDimArray):
lp = len(expr)
new_array = {k + i*lp: v
for i, x in enumerate(Flatten(dx))
for k, v in expr.diff(x)._sparse_array.items()}
else:
new_array = [[y.diff(x) for y in Flatten(expr)] for x in Flatten(dx)]
return type(expr)(new_array, dx.shape + expr.shape)
else:
return expr.diff(dx)
else:
expr = _sympify(expr)
if isinstance(dx, array_types):
return ImmutableDenseNDimArray([expr.diff(i) for i in Flatten(dx)], dx.shape)
else:
dx = _sympify(dx)
return diff(expr, dx)
def permutedims(expr, perm):
"""
Permutes the indices of an array.
Parameter specifies the permutation of the indices.
Examples
========
>>> from sympy.abc import x, y, z, t
>>> from sympy import sin
>>> from sympy import Array, permutedims
>>> a = Array([[x, y, z], [t, sin(x), 0]])
>>> a
[[x, y, z], [t, sin(x), 0]]
>>> permutedims(a, (1, 0))
[[x, t], [y, sin(x)], [z, 0]]
If the array is of second order, ``transpose`` can be used:
>>> from sympy import transpose
>>> transpose(a)
[[x, t], [y, sin(x)], [z, 0]]
Examples on higher dimensions:
>>> b = Array([[[1, 2], [3, 4]], [[5, 6], [7, 8]]])
>>> permutedims(b, (2, 1, 0))
[[[1, 5], [3, 7]], [[2, 6], [4, 8]]]
>>> permutedims(b, (1, 2, 0))
[[[1, 5], [2, 6]], [[3, 7], [4, 8]]]
``Permutation`` objects are also allowed:
>>> from sympy.combinatorics import Permutation
>>> permutedims(b, Permutation([1, 2, 0]))
[[[1, 5], [2, 6]], [[3, 7], [4, 8]]]
"""
from sympy.tensor.array import SparseNDimArray
from sympy.tensor.array.expressions.array_expressions import _ArrayExpr
from sympy.tensor.array.expressions.array_expressions import _CodegenArrayAbstract
from sympy.tensor.array.expressions.array_expressions import PermuteDims
from sympy import MatrixSymbol
if isinstance(expr, (_ArrayExpr, _CodegenArrayAbstract, MatrixSymbol)):
return PermuteDims(expr, perm)
if not isinstance(expr, NDimArray):
expr = ImmutableDenseNDimArray(expr)
from sympy.combinatorics import Permutation
if not isinstance(perm, Permutation):
perm = Permutation(list(perm))
if perm.size != expr.rank():
raise ValueError("wrong permutation size")
# Get the inverse permutation:
iperm = ~perm
new_shape = perm(expr.shape)
if isinstance(expr, SparseNDimArray):
return type(expr)({tuple(perm(expr._get_tuple_index(k))): v
for k, v in expr._sparse_array.items()}, new_shape)
indices_span = perm([range(i) for i in expr.shape])
new_array = [None]*len(expr)
for i, idx in enumerate(itertools.product(*indices_span)):
t = iperm(idx)
new_array[i] = expr[t]
return type(expr)(new_array, new_shape)
class Flatten(Basic):
'''
Flatten an iterable object to a list in a lazy-evaluation way.
Notes
=====
This class is an iterator with which the memory cost can be economised.
Optimisation has been considered to ameliorate the performance for some
specific data types like DenseNDimArray and SparseNDimArray.
Examples
========
>>> from sympy.tensor.array.arrayop import Flatten
>>> from sympy.tensor.array import Array
>>> A = Array(range(6)).reshape(2, 3)
>>> Flatten(A)
Flatten([[0, 1, 2], [3, 4, 5]])
>>> [i for i in Flatten(A)]
[0, 1, 2, 3, 4, 5]
'''
def __init__(self, iterable):
from sympy.matrices.matrices import MatrixBase
from sympy.tensor.array import NDimArray
if not isinstance(iterable, (Iterable, MatrixBase)):
raise NotImplementedError("Data type not yet supported")
if isinstance(iterable, list):
iterable = NDimArray(iterable)
self._iter = iterable
self._idx = 0
def __iter__(self):
return self
def __next__(self):
from sympy.matrices.matrices import MatrixBase
if len(self._iter) > self._idx:
if isinstance(self._iter, DenseNDimArray):
result = self._iter._array[self._idx]
elif isinstance(self._iter, SparseNDimArray):
if self._idx in self._iter._sparse_array:
result = self._iter._sparse_array[self._idx]
else:
result = 0
elif isinstance(self._iter, MatrixBase):
result = self._iter[self._idx]
elif hasattr(self._iter, '__next__'):
result = next(self._iter)
else:
result = self._iter[self._idx]
else:
raise StopIteration
self._idx += 1
return result
def next(self):
return self.__next__()
|
3923d4d76063552658500bd97bd3525f6d7429f02a0024c0ee44ee7dd2fa3d62 | from collections import defaultdict
from sympy import Sum, Mul, KroneckerDelta, Indexed, IndexedBase, Add
from sympy.combinatorics import Permutation
from sympy.matrices.expressions.matexpr import MatrixElement
from sympy.tensor.array.expressions.array_expressions import PermuteDims, ArrayDiagonal, \
ArrayContraction, ArrayTensorProduct, ArrayAdd
from sympy.tensor.array.expressions.utils import _get_argindex, _get_diagonal_indices
def convert_indexed_to_array(expr, first_indices=None):
r"""
Parse indexed expression into a form useful for code generation.
Examples
========
>>> from sympy.tensor.array.expressions.conv_indexed_to_array import convert_indexed_to_array
>>> from sympy import MatrixSymbol, Sum, symbols
>>> i, j, k, d = symbols("i j k d")
>>> M = MatrixSymbol("M", d, d)
>>> N = MatrixSymbol("N", d, d)
Recognize the trace in summation form:
>>> expr = Sum(M[i, i], (i, 0, d-1))
>>> convert_indexed_to_array(expr)
ArrayContraction(M, (0, 1))
Recognize the extraction of the diagonal by using the same index `i` on
both axes of the matrix:
>>> expr = M[i, i]
>>> convert_indexed_to_array(expr)
ArrayDiagonal(M, (0, 1))
This function can help perform the transformation expressed in two
different mathematical notations as:
`\sum_{j=0}^{N-1} A_{i,j} B_{j,k} \Longrightarrow \mathbf{A}\cdot \mathbf{B}`
Recognize the matrix multiplication in summation form:
>>> expr = Sum(M[i, j]*N[j, k], (j, 0, d-1))
>>> convert_indexed_to_array(expr)
ArrayContraction(ArrayTensorProduct(M, N), (1, 2))
Specify that ``k`` has to be the starting index:
>>> convert_indexed_to_array(expr, first_indices=[k])
ArrayContraction(ArrayTensorProduct(N, M), (0, 3))
"""
result, indices = _convert_indexed_to_array(expr)
if not first_indices:
return result
for i in first_indices:
if i not in indices:
first_indices.remove(i)
first_indices.extend([i for i in indices if i not in first_indices])
permutation = [first_indices.index(i) for i in indices]
return PermuteDims(result, permutation)
def _convert_indexed_to_array(expr):
if isinstance(expr, Sum):
function = expr.function
summation_indices = expr.variables
subexpr, subindices = _convert_indexed_to_array(function)
# Check dimensional consistency:
shape = subexpr.shape
if shape:
for ind, istart, iend in expr.limits:
i = _get_argindex(subindices, ind)
if istart != 0 or iend+1 != shape[i]:
raise ValueError("summation index and array dimension mismatch: %s" % ind)
contraction_indices = []
subindices = list(subindices)
if isinstance(subexpr, ArrayDiagonal):
diagonal_indices = list(subexpr.diagonal_indices)
dindices = subindices[-len(diagonal_indices):]
subindices = subindices[:-len(diagonal_indices)]
for index in summation_indices:
if index in dindices:
position = dindices.index(index)
contraction_indices.append(diagonal_indices[position])
diagonal_indices[position] = None
diagonal_indices = [i for i in diagonal_indices if i is not None]
for i, ind in enumerate(subindices):
if ind in summation_indices:
pass
if diagonal_indices:
subexpr = ArrayDiagonal(subexpr.expr, *diagonal_indices)
else:
subexpr = subexpr.expr
axes_contraction = defaultdict(list)
for i, ind in enumerate(subindices):
if ind in summation_indices:
axes_contraction[ind].append(i)
subindices[i] = None
for k, v in axes_contraction.items():
contraction_indices.append(tuple(v))
free_indices = [i for i in subindices if i is not None]
indices_ret = list(free_indices)
indices_ret.sort(key=lambda x: free_indices.index(x))
return ArrayContraction(
subexpr,
*contraction_indices,
free_indices=free_indices
), tuple(indices_ret)
if isinstance(expr, Mul):
args, indices = zip(*[_convert_indexed_to_array(arg) for arg in expr.args])
# Check if there are KroneckerDelta objects:
kronecker_delta_repl = {}
for arg in args:
if not isinstance(arg, KroneckerDelta):
continue
# Diagonalize two indices:
i, j = arg.indices
kindices = set(arg.indices)
if i in kronecker_delta_repl:
kindices.update(kronecker_delta_repl[i])
if j in kronecker_delta_repl:
kindices.update(kronecker_delta_repl[j])
kindices = frozenset(kindices)
for index in kindices:
kronecker_delta_repl[index] = kindices
# Remove KroneckerDelta objects, their relations should be handled by
# ArrayDiagonal:
newargs = []
newindices = []
for arg, loc_indices in zip(args, indices):
if isinstance(arg, KroneckerDelta):
continue
newargs.append(arg)
newindices.append(loc_indices)
flattened_indices = [kronecker_delta_repl.get(j, j) for i in newindices for j in i]
diagonal_indices, ret_indices = _get_diagonal_indices(flattened_indices)
tp = ArrayTensorProduct(*newargs)
if diagonal_indices:
return (ArrayDiagonal(tp, *diagonal_indices), ret_indices)
else:
return tp, ret_indices
if isinstance(expr, MatrixElement):
indices = expr.args[1:]
diagonal_indices, ret_indices = _get_diagonal_indices(indices)
if diagonal_indices:
return (ArrayDiagonal(expr.args[0], *diagonal_indices), ret_indices)
else:
return expr.args[0], ret_indices
if isinstance(expr, Indexed):
indices = expr.indices
diagonal_indices, ret_indices = _get_diagonal_indices(indices)
if diagonal_indices:
return (ArrayDiagonal(expr.base, *diagonal_indices), ret_indices)
else:
return expr.args[0], ret_indices
if isinstance(expr, IndexedBase):
raise NotImplementedError
if isinstance(expr, KroneckerDelta):
return expr, expr.indices
if isinstance(expr, Add):
args, indices = zip(*[_convert_indexed_to_array(arg) for arg in expr.args])
args = list(args)
# Check if all indices are compatible. Otherwise expand the dimensions:
index0set = set(indices[0])
index0 = indices[0]
for i in range(1, len(args)):
if set(indices[i]) != index0set:
raise NotImplementedError("indices must be the same")
permutation = Permutation([index0.index(j) for j in indices[i]])
# Perform index permutations:
args[i] = PermuteDims(args[i], permutation)
return ArrayAdd(*args), index0
return expr, ()
|
ac3f3e25aede3afc79729b8754e69651f712d8c689ce1d06e07d8c90bf31d477 | import itertools
from typing import Tuple
from functools import reduce, singledispatch
from itertools import accumulate
from sympy import S, Trace, MatrixExpr, Transpose, DiagMatrix, Mul, ZeroMatrix
from sympy.combinatorics.permutations import _af_invert, Permutation
from sympy.matrices.common import MatrixCommon
from sympy.matrices.expressions.applyfunc import ElementwiseApplyFunction
from sympy.tensor.array.expressions.array_expressions import PermuteDims, ArrayDiagonal, \
ArrayTensorProduct, OneArray, get_rank, _get_subrank, ZeroArray, ArrayContraction, \
ArrayAdd, _CodegenArrayAbstract, get_shape, ArrayElementwiseApplyFunc, _ArrayExpr
from sympy.tensor.array.expressions.utils import _get_mapping_from_subranks
def _support_function_tp1_recognize(contraction_indices, args):
subranks = [get_rank(i) for i in args]
coeff = reduce(lambda x, y: x*y, [arg for arg, srank in zip(args, subranks) if srank == 0], S.One)
mapping = _get_mapping_from_subranks(subranks)
new_contraction_indices = list(contraction_indices)
newargs = args[:] # make a copy of the list
removed = [None for i in newargs]
cumul = list(accumulate([0] + [get_rank(arg) for arg in args]))
new_perms = [list(range(cumul[i], cumul[i+1])) for i, arg in enumerate(args)]
for pi, contraction_pair in enumerate(contraction_indices):
if len(contraction_pair) != 2:
continue
i1, i2 = contraction_pair
a1, e1 = mapping[i1]
a2, e2 = mapping[i2]
while removed[a1] is not None:
a1, e1 = removed[a1]
while removed[a2] is not None:
a2, e2 = removed[a2]
if a1 == a2:
trace_arg = newargs[a1]
newargs[a1] = Trace(trace_arg)._normalize()
new_contraction_indices[pi] = None
continue
if not isinstance(newargs[a1], MatrixExpr) or not isinstance(newargs[a2], MatrixExpr):
continue
arg1 = newargs[a1]
arg2 = newargs[a2]
if (e1 == 1 and e2 == 1) or (e1 == 0 and e2 == 0):
arg2 = Transpose(arg2)
if e1 == 1:
argnew = arg1*arg2
else:
argnew = arg2*arg1
removed[a2] = a1, e1
new_perms[a1][e1] = new_perms[a2][1 - e2]
new_perms[a2] = None
newargs[a1] = argnew
newargs[a2] = None
new_contraction_indices[pi] = None
new_contraction_indices = [i for i in new_contraction_indices if i is not None]
newargs2 = [arg for arg in newargs if arg is not None]
if len(newargs2) == 0:
return coeff
tp = _a2m_tensor_product(*newargs2)
tc = ArrayContraction(tp, *new_contraction_indices)
new_perms2 = ArrayContraction._push_indices_up(contraction_indices, [i for i in new_perms if i is not None])
permutation = _af_invert([j for i in new_perms2 for j in i if j is not None])
if permutation == [1, 0] and len(newargs2) == 1:
return Transpose(newargs2[0]).doit()
tperm = PermuteDims(tc, permutation)
return tperm
@singledispatch
def _array2matrix(expr):
return expr
@_array2matrix.register(ZeroArray)
def _(expr: ZeroArray):
if get_rank(expr) == 2:
return ZeroMatrix(*expr.shape)
else:
return expr
@_array2matrix.register(ArrayTensorProduct)
def _(expr: ArrayTensorProduct):
return _a2m_tensor_product(*[_array2matrix(arg) for arg in expr.args])
@_array2matrix.register(ArrayContraction)
def _(expr: ArrayContraction):
expr = expr.flatten_contraction_of_diagonal()
expr = expr.split_multiple_contractions()
subexpr = expr.expr
contraction_indices: Tuple[Tuple[int]] = expr.contraction_indices
if isinstance(subexpr, ArrayTensorProduct):
newexpr = ArrayContraction(_array2matrix(subexpr), *contraction_indices)
contraction_indices = newexpr.contraction_indices
if any(i > 2 for i in newexpr.subranks):
addends = ArrayAdd(*[_a2m_tensor_product(*j) for j in itertools.product(*[i.args if isinstance(i,
ArrayAdd) else [i] for i in expr.expr.args])])
newexpr = ArrayContraction(addends, *contraction_indices)
if isinstance(newexpr, ArrayAdd):
ret = _array2matrix(newexpr)
return ret
assert isinstance(newexpr, ArrayContraction)
ret = _support_function_tp1_recognize(contraction_indices, list(newexpr.expr.args))
return ret
elif not isinstance(subexpr, _CodegenArrayAbstract):
ret = _array2matrix(subexpr)
if isinstance(ret, MatrixExpr):
assert expr.contraction_indices == ((0, 1),)
return _a2m_trace(ret)
else:
return ArrayContraction(ret, *expr.contraction_indices)
@_array2matrix.register(ArrayDiagonal)
def _(expr: ArrayDiagonal):
expr2 = _array2matrix(expr.expr)
pexpr = _array_diag2contr_diagmatrix(ArrayDiagonal(expr2, *expr.diagonal_indices))
if expr == pexpr:
return expr
return _array2matrix(pexpr)
@_array2matrix.register(PermuteDims)
def _(expr: PermuteDims):
if expr.permutation.array_form == [1, 0]:
return _a2m_transpose(_array2matrix(expr.expr))
elif isinstance(expr.expr, ArrayTensorProduct):
ranks = expr.expr.subranks
inv_permutation = expr.permutation**(-1)
newrange = [inv_permutation(i) for i in range(sum(ranks))]
newpos = []
counter = 0
for rank in ranks:
newpos.append(newrange[counter:counter+rank])
counter += rank
newargs = []
newperm = []
scalars = []
for pos, arg in zip(newpos, expr.expr.args):
if len(pos) == 0:
scalars.append(_array2matrix(arg))
elif pos == sorted(pos):
newargs.append((_array2matrix(arg), pos[0]))
newperm.extend(pos)
elif len(pos) == 2:
newargs.append((_a2m_transpose(_array2matrix(arg)), pos[0]))
newperm.extend(reversed(pos))
else:
raise NotImplementedError()
newargs = [i[0] for i in newargs]
return PermuteDims(_a2m_tensor_product(*scalars, *newargs), _af_invert(newperm))
elif isinstance(expr.expr, ArrayContraction):
mat_mul_lines = _array2matrix(expr.expr)
if not isinstance(mat_mul_lines, ArrayTensorProduct):
flat_cyclic_form = [j for i in expr.permutation.cyclic_form for j in i]
expr_shape = get_shape(expr)
if all(expr_shape[i] == 1 for i in flat_cyclic_form):
return mat_mul_lines
return mat_mul_lines
permutation = Permutation(2*len(mat_mul_lines.args)-1)*expr.permutation
permuted = [permutation(i) for i in range(2*len(mat_mul_lines.args))]
args_array = [None for i in mat_mul_lines.args]
for i in range(len(mat_mul_lines.args)):
p1 = permuted[2*i]
p2 = permuted[2*i+1]
if p1 // 2 != p2 // 2:
return PermuteDims(mat_mul_lines, permutation)
pos = p1 // 2
if p1 > p2:
args_array[i] = _a2m_transpose(mat_mul_lines.args[pos])
else:
args_array[i] = mat_mul_lines.args[pos]
return _a2m_tensor_product(*args_array)
else:
raise NotImplementedError()
@_array2matrix.register(ArrayAdd)
def _(expr: ArrayAdd):
addends = [_array2matrix(arg) for arg in expr.args]
return _a2m_add(*addends)
@_array2matrix.register(ArrayElementwiseApplyFunc)
def _(expr: ArrayElementwiseApplyFunc):
subexpr = _array2matrix(expr.expr)
if isinstance(subexpr, MatrixExpr):
return ElementwiseApplyFunction(expr.function, subexpr)
else:
return ArrayElementwiseApplyFunc(expr.function, subexpr)
@singledispatch
def _remove_trivial_dims(expr):
return expr, []
@_remove_trivial_dims.register(ArrayTensorProduct)
def _(expr: ArrayTensorProduct):
# Recognize expressions like [x, y] with shape (k, 1, k, 1) as `x*y.T`.
# The matrix expression has to be equivalent to the tensor product of the
# matrices, with trivial dimensions (i.e. dim=1) dropped.
# That is, add contractions over trivial dimensions:
removed = []
newargs = []
cumul = list(accumulate([0] + [get_rank(arg) for arg in expr.args]))
pending = None
prev_i = None
for i, arg in enumerate(expr.args):
current_range = list(range(cumul[i], cumul[i+1]))
if isinstance(arg, OneArray):
removed.extend(current_range)
continue
if not isinstance(arg, (MatrixExpr, MatrixCommon)):
rarg, rem = _remove_trivial_dims(arg)
removed.extend(rem)
newargs.append(rarg)
continue
elif getattr(arg, "is_Identity", False):
if arg.shape == (1, 1):
# Ignore identity matrices of shape (1, 1) - they are equivalent to scalar 1.
removed.extend(current_range)
continue
k = arg.shape[0]
if pending == k:
# OK, there is already
removed.extend(current_range)
continue
elif pending is None:
newargs.append(arg)
pending = k
prev_i = i
elif pending != k:
pending = k
prev_i = i
newargs.append(arg)
elif arg.shape == (1, 1):
arg, _ = _remove_trivial_dims(arg)
# Matrix is equivalent to scalar:
if len(newargs) == 0:
newargs.append(arg)
elif 1 in get_shape(newargs[-1]):
if newargs[-1].shape[1] == 1:
newargs[-1] = newargs[-1]*arg
else:
newargs[-1] = arg*newargs[-1]
removed.extend(current_range)
else:
newargs.append(arg)
elif 1 in arg.shape:
k = [i for i in arg.shape if i != 1][0]
if pending is None:
pending = k
prev_i = i
newargs.append(arg)
elif pending == k:
prev = newargs[-1]
if prev.is_Identity:
removed.extend([cumul[prev_i], cumul[prev_i]+1])
newargs[-1] = arg
prev_i = i
continue
if prev.shape[0] == 1:
d1 = cumul[prev_i]
prev = _a2m_transpose(prev)
else:
d1 = cumul[prev_i] + 1
if arg.shape[1] == 1:
d2 = cumul[i] + 1
arg = _a2m_transpose(arg)
else:
d2 = cumul[i]
newargs[-1] = prev*arg
pending = None
removed.extend([d1, d2])
else:
newargs.append(arg)
pending = k
prev_i = i
else:
newargs.append(arg)
pending = None
return _a2m_tensor_product(*newargs), sorted(removed)
@_remove_trivial_dims.register(ArrayAdd)
def _(expr: ArrayAdd):
rec = [_remove_trivial_dims(arg) for arg in expr.args]
newargs, removed = zip(*rec)
if len(set(map(tuple, removed))) != 1:
return expr, []
return _a2m_add(*newargs), removed[0]
@_remove_trivial_dims.register(PermuteDims)
def _(expr: PermuteDims):
subexpr, subremoved = _remove_trivial_dims(expr.expr)
p = expr.permutation.array_form
pinv = _af_invert(expr.permutation.array_form)
shift = list(accumulate([1 if i in subremoved else 0 for i in range(len(p))]))
premoved = [pinv[i] for i in subremoved]
p2 = [e - shift[e] for i, e in enumerate(p) if e not in subremoved]
# TODO: check if subremoved should be permuted as well...
newexpr = PermuteDims(subexpr, p2)
if newexpr != expr:
newexpr = _array2matrix(newexpr)
return newexpr, sorted(premoved)
@_remove_trivial_dims.register(ArrayContraction)
def _(expr: ArrayContraction):
newexpr, removed = _remove_trivial_dims(expr.expr)
new_contraction_indices = [tuple(j for j in i if j not in removed) for i in expr.contraction_indices]
# Remove possible empty tuples "()":
new_contraction_indices = [i for i in new_contraction_indices if i]
return ArrayContraction(newexpr, *new_contraction_indices), removed
@_remove_trivial_dims.register(ArrayDiagonal)
def _(expr: ArrayDiagonal):
newexpr, removed = _remove_trivial_dims(expr.expr)
new_diag_indices = [tuple(j for j in i if j not in removed) for i in expr.diagonal_indices]
return ArrayDiagonal(newexpr, *new_diag_indices), removed
@_remove_trivial_dims.register(ElementwiseApplyFunction)
def _(expr: ElementwiseApplyFunction):
subexpr, removed = _remove_trivial_dims(expr.expr)
if subexpr.shape == (1, 1):
# TODO: move this to ElementwiseApplyFunction
return expr.function(subexpr), removed + [0, 1]
return ElementwiseApplyFunction(expr.function, subexpr)
@_remove_trivial_dims.register(ArrayElementwiseApplyFunc)
def _(expr: ArrayElementwiseApplyFunc):
subexpr, removed = _remove_trivial_dims(expr.expr)
return ArrayElementwiseApplyFunc(expr.function, subexpr), removed
def convert_array_to_matrix(expr):
r"""
Recognize matrix expressions in codegen objects.
If more than one matrix multiplication line have been detected, return a
list with the matrix expressions.
Examples
========
>>> from sympy.tensor.array.expressions.conv_indexed_to_array import convert_indexed_to_array
>>> from sympy.tensor.array.expressions.array_expressions import ArrayTensorProduct
>>> from sympy import MatrixSymbol, Sum
>>> from sympy.abc import i, j, k, l, N
>>> from sympy.tensor.array.expressions.array_expressions import ArrayContraction
>>> from sympy.tensor.array.expressions.conv_matrix_to_array import convert_matrix_to_array
>>> from sympy.tensor.array.expressions.conv_array_to_matrix import convert_array_to_matrix
>>> A = MatrixSymbol("A", N, N)
>>> B = MatrixSymbol("B", N, N)
>>> C = MatrixSymbol("C", N, N)
>>> D = MatrixSymbol("D", N, N)
>>> expr = Sum(A[i, j]*B[j, k], (j, 0, N-1))
>>> cg = convert_indexed_to_array(expr)
>>> convert_array_to_matrix(cg)
A*B
>>> cg = convert_indexed_to_array(expr, first_indices=[k])
>>> convert_array_to_matrix(cg)
B.T*A.T
Transposition is detected:
>>> expr = Sum(A[j, i]*B[j, k], (j, 0, N-1))
>>> cg = convert_indexed_to_array(expr)
>>> convert_array_to_matrix(cg)
A.T*B
>>> cg = convert_indexed_to_array(expr, first_indices=[k])
>>> convert_array_to_matrix(cg)
B.T*A
Detect the trace:
>>> expr = Sum(A[i, i], (i, 0, N-1))
>>> cg = convert_indexed_to_array(expr)
>>> convert_array_to_matrix(cg)
Trace(A)
Recognize some more complex traces:
>>> expr = Sum(A[i, j]*B[j, i], (i, 0, N-1), (j, 0, N-1))
>>> cg = convert_indexed_to_array(expr)
>>> convert_array_to_matrix(cg)
Trace(A*B)
More complicated expressions:
>>> expr = Sum(A[i, j]*B[k, j]*A[l, k], (j, 0, N-1), (k, 0, N-1))
>>> cg = convert_indexed_to_array(expr)
>>> convert_array_to_matrix(cg)
A*B.T*A.T
Expressions constructed from matrix expressions do not contain literal
indices, the positions of free indices are returned instead:
>>> expr = A*B
>>> cg = convert_matrix_to_array(expr)
>>> convert_array_to_matrix(cg)
A*B
If more than one line of matrix multiplications is detected, return
separate matrix multiplication factors embedded in a tensor product object:
>>> cg = ArrayContraction(ArrayTensorProduct(A, B, C, D), (1, 2), (5, 6))
>>> convert_array_to_matrix(cg)
ArrayTensorProduct(A*B, C*D)
The two lines have free indices at axes 0, 3 and 4, 7, respectively.
"""
rec = _array2matrix(expr)
rec, removed = _remove_trivial_dims(rec)
return rec
def _array_diag2contr_diagmatrix(expr: ArrayDiagonal):
if isinstance(expr.expr, ArrayTensorProduct):
args = list(expr.expr.args)
diag_indices = list(expr.diagonal_indices)
mapping = _get_mapping_from_subranks([_get_subrank(arg) for arg in args])
tuple_links = [[mapping[j] for j in i] for i in diag_indices]
contr_indices = []
total_rank = get_rank(expr)
replaced = [False for arg in args]
for i, (abs_pos, rel_pos) in enumerate(zip(diag_indices, tuple_links)):
if len(abs_pos) != 2:
continue
(pos1_outer, pos1_inner), (pos2_outer, pos2_inner) = rel_pos
arg1 = args[pos1_outer]
arg2 = args[pos2_outer]
if get_rank(arg1) != 2 or get_rank(arg2) != 2:
if replaced[pos1_outer]:
diag_indices[i] = None
if replaced[pos2_outer]:
diag_indices[i] = None
continue
pos1_in2 = 1 - pos1_inner
pos2_in2 = 1 - pos2_inner
if arg1.shape[pos1_in2] == 1:
darg1 = DiagMatrix(arg1)
args.append(darg1)
contr_indices.append(((pos2_outer, pos2_inner), (len(args)-1, pos1_inner)))
total_rank += 1
diag_indices[i] = None
args[pos1_outer] = OneArray(arg1.shape[pos1_in2])
replaced[pos1_outer] = True
elif arg2.shape[pos2_in2] == 1:
darg2 = DiagMatrix(arg2)
args.append(darg2)
contr_indices.append(((pos1_outer, pos1_inner), (len(args)-1, pos2_inner)))
total_rank += 1
diag_indices[i] = None
args[pos2_outer] = OneArray(arg2.shape[pos2_in2])
replaced[pos2_outer] = True
diag_indices_new = [i for i in diag_indices if i is not None]
cumul = list(accumulate([0] + [get_rank(arg) for arg in args]))
contr_indices2 = [tuple(cumul[a] + b for a, b in i) for i in contr_indices]
tc = ArrayContraction(
ArrayTensorProduct(*args), *contr_indices2
)
td = ArrayDiagonal(tc, *diag_indices_new)
return td
return expr
def _a2m_mul(*args):
if all(not isinstance(i, _CodegenArrayAbstract) for i in args):
from sympy import MatMul
return MatMul(*args).doit()
else:
return ArrayContraction(
ArrayTensorProduct(*args),
*[(2*i-1, 2*i) for i in range(1, len(args))]
)
def _a2m_tensor_product(*args):
scalars = []
arrays = []
for arg in args:
if isinstance(arg, (MatrixExpr, _ArrayExpr, _CodegenArrayAbstract)):
arrays.append(arg)
else:
scalars.append(arg)
scalar = Mul.fromiter(scalars)
if len(arrays) == 0:
return scalar
if scalar != 1:
if isinstance(arrays[0], _CodegenArrayAbstract):
arrays = [scalar] + arrays
else:
arrays[0] *= scalar
return ArrayTensorProduct(*arrays)
def _a2m_add(*args):
if all(not isinstance(i, _CodegenArrayAbstract) for i in args):
from sympy import MatAdd
return MatAdd(*args).doit()
else:
return ArrayAdd(*args)
def _a2m_trace(arg):
if isinstance(arg, _CodegenArrayAbstract):
return ArrayContraction(arg, (0, 1))
else:
from sympy import Trace
return Trace(arg)
def _a2m_transpose(arg):
if isinstance(arg, _CodegenArrayAbstract):
return PermuteDims(arg, [1, 0])
else:
from sympy import Transpose
return Transpose(arg).doit()
|
8aa8d435687a227720b01e5ed7e23a6fff579c8878bbb9f67bae6b94cbd311da | from sympy import Mul, Basic, MatMul, MatAdd, Transpose, Trace, Pow, \
MatPow, symbols, Dummy, Lambda, HadamardProduct, HadamardPower, S
from sympy.matrices.expressions.matexpr import MatrixExpr
from sympy.tensor.array.expressions.array_expressions import ArrayDiagonal, ArrayTensorProduct, \
PermuteDims, ArrayAdd, ArrayContraction, ArrayElementwiseApplyFunc
def convert_matrix_to_array(expr: MatrixExpr) -> Basic:
if isinstance(expr, MatMul):
args_nonmat = []
args = []
for arg in expr.args:
if isinstance(arg, MatrixExpr):
args.append(arg)
else:
args_nonmat.append(convert_matrix_to_array(arg))
contractions = [(2*i+1, 2*i+2) for i in range(len(args)-1)]
scalar = ArrayTensorProduct.fromiter(args_nonmat) if args_nonmat else S.One
if scalar == 1:
tprod = ArrayTensorProduct(
*[convert_matrix_to_array(arg) for arg in args])
else:
tprod = ArrayTensorProduct(
scalar,
*[convert_matrix_to_array(arg) for arg in args])
return ArrayContraction(
tprod,
*contractions
)
elif isinstance(expr, MatAdd):
return ArrayAdd(
*[convert_matrix_to_array(arg) for arg in expr.args]
)
elif isinstance(expr, Transpose):
return PermuteDims(
convert_matrix_to_array(expr.args[0]), [1, 0]
)
elif isinstance(expr, Trace):
inner_expr = convert_matrix_to_array(expr.arg)
return ArrayContraction(inner_expr, (0, len(inner_expr.shape) - 1))
elif isinstance(expr, Mul):
return ArrayTensorProduct.fromiter(convert_matrix_to_array(i) for i in expr.args)
elif isinstance(expr, Pow):
base = convert_matrix_to_array(expr.base)
if (expr.exp > 0) == True:
return ArrayTensorProduct.fromiter(base for i in range(expr.exp))
else:
return expr
elif isinstance(expr, MatPow):
base = convert_matrix_to_array(expr.base)
if expr.exp.is_Integer != True:
b = symbols("b", cls=Dummy)
return ArrayElementwiseApplyFunc(Lambda(b, b**expr.exp), convert_matrix_to_array(base))
elif (expr.exp > 0) == True:
return convert_matrix_to_array(MatMul.fromiter(base for i in range(expr.exp)))
else:
return expr
elif isinstance(expr, HadamardProduct):
tp = ArrayTensorProduct.fromiter([convert_matrix_to_array(arg) for arg in expr.args])
diag = [[2*i for i in range(len(expr.args))], [2*i+1 for i in range(len(expr.args))]]
return ArrayDiagonal(tp, *diag)
elif isinstance(expr, HadamardPower):
base, exp = expr.args
if exp.is_Integer and exp > 0:
return convert_matrix_to_array(HadamardProduct.fromiter(base for i in range(exp)))
else:
raise NotImplementedError("conversion of Hadamard symbolic power is currently not supported")
else:
return expr
|
24ca4df9ba9c8646768c8fc6b018c3cce2f7a3c492391ed19220db84eb5869d2 | import bisect
from collections import defaultdict
from sympy import Tuple, Integer
def _get_mapping_from_subranks(subranks):
mapping = {}
counter = 0
for i, rank in enumerate(subranks):
for j in range(rank):
mapping[counter] = (i, j)
counter += 1
return mapping
def _get_contraction_links(args, subranks, *contraction_indices):
mapping = _get_mapping_from_subranks(subranks)
contraction_tuples = [[mapping[j] for j in i] for i in contraction_indices]
dlinks = defaultdict(dict)
for links in contraction_tuples:
if len(links) == 2:
(arg1, pos1), (arg2, pos2) = links
dlinks[arg1][pos1] = (arg2, pos2)
dlinks[arg2][pos2] = (arg1, pos1)
continue
return args, dict(dlinks)
def _sort_contraction_indices(pairing_indices):
pairing_indices = [Tuple(*sorted(i)) for i in pairing_indices]
pairing_indices.sort(key=lambda x: min(x))
return pairing_indices
def _get_diagonal_indices(flattened_indices):
axes_contraction = defaultdict(list)
for i, ind in enumerate(flattened_indices):
if isinstance(ind, (int, Integer)):
# If the indices is a number, there can be no diagonal operation:
continue
axes_contraction[ind].append(i)
axes_contraction = {k: v for k, v in axes_contraction.items() if len(v) > 1}
# Put the diagonalized indices at the end:
ret_indices = [i for i in flattened_indices if i not in axes_contraction]
diag_indices = list(axes_contraction)
diag_indices.sort(key=lambda x: flattened_indices.index(x))
diagonal_indices = [tuple(axes_contraction[i]) for i in diag_indices]
ret_indices += diag_indices
ret_indices = tuple(ret_indices)
return diagonal_indices, ret_indices
def _get_argindex(subindices, ind):
for i, sind in enumerate(subindices):
if ind == sind:
return i
if isinstance(sind, (set, frozenset)) and ind in sind:
return i
raise IndexError("%s not found in %s" % (ind, subindices))
def _apply_recursively_over_nested_lists(func, arr):
if isinstance(arr, (tuple, list, Tuple)):
return tuple(_apply_recursively_over_nested_lists(func, i) for i in arr)
elif isinstance(arr, Tuple):
return Tuple.fromiter(_apply_recursively_over_nested_lists(func, i) for i in arr)
else:
return func(arr)
def _build_push_indices_up_func_transformation(flattened_contraction_indices):
shifts = {0: 0}
i = 0
cumulative = 0
while i < len(flattened_contraction_indices):
j = 1
while i+j < len(flattened_contraction_indices):
if flattened_contraction_indices[i] + j != flattened_contraction_indices[i+j]:
break
j += 1
cumulative += j
shifts[flattened_contraction_indices[i]] = cumulative
i += j
shift_keys = sorted(shifts.keys())
def func(idx):
return shifts[shift_keys[bisect.bisect_right(shift_keys, idx)-1]]
def transform(j):
if j in flattened_contraction_indices:
return None
else:
return j - func(j)
return transform
def _build_push_indices_down_func_transformation(flattened_contraction_indices):
N = flattened_contraction_indices[-1]+2
shifts = [i for i in range(N) if i not in flattened_contraction_indices]
def transform(j):
if j < len(shifts):
return shifts[j]
else:
return j + shifts[-1] - len(shifts) + 1
return transform
|
54c4c1da12f2d637692a7d1d3988af4f300dcea2ab73c02da9e3052aac049ab8 | import operator
from functools import reduce
import itertools
from itertools import accumulate
from sympy import Expr, ImmutableDenseNDimArray, S, Symbol, Integer, ZeroMatrix, Basic, tensorproduct, Add, permutedims, \
Tuple, tensordiagonal, Lambda, Dummy, Function, MatrixExpr, NDimArray, Indexed, IndexedBase, default_sort_key, \
tensorcontraction, diagonalize_vector
from sympy.matrices.expressions.matexpr import MatrixElement
from sympy.tensor.array.expressions.utils import _apply_recursively_over_nested_lists, _sort_contraction_indices, \
_get_mapping_from_subranks, _build_push_indices_up_func_transformation, _get_contraction_links, \
_build_push_indices_down_func_transformation
from sympy.combinatorics import Permutation
from sympy.combinatorics.permutations import _af_invert
from sympy.core.sympify import _sympify
class _ArrayExpr(Expr):
pass
class ArraySymbol(_ArrayExpr):
"""
Symbol representing an array expression
"""
def __new__(cls, symbol, *shape):
if isinstance(symbol, str):
symbol = Symbol(symbol)
# symbol = _sympify(symbol)
shape = map(_sympify, shape)
obj = Expr.__new__(cls, symbol, *shape)
return obj
@property
def name(self):
return self._args[0]
@property
def shape(self):
return self._args[1:]
def __getitem__(self, item):
return ArrayElement(self, item)
def as_explicit(self):
if any(not isinstance(i, (int, Integer)) for i in self.shape):
raise ValueError("cannot express explicit array with symbolic shape")
data = [self[i] for i in itertools.product(*[range(j) for j in self.shape])]
return ImmutableDenseNDimArray(data).reshape(*self.shape)
class ArrayElement(_ArrayExpr):
"""
An element of an array.
"""
def __new__(cls, name, indices):
if isinstance(name, str):
name = Symbol(name)
name = _sympify(name)
indices = _sympify(indices)
if hasattr(name, "shape"):
if any([(i >= s) == True for i, s in zip(indices, name.shape)]):
raise ValueError("shape is out of bounds")
if any([(i < 0) == True for i in indices]):
raise ValueError("shape contains negative values")
obj = Expr.__new__(cls, name, indices)
return obj
@property
def name(self):
return self._args[0]
@property
def indices(self):
return self._args[1]
class ZeroArray(_ArrayExpr):
"""
Symbolic array of zeros. Equivalent to ``ZeroMatrix`` for matrices.
"""
def __new__(cls, *shape):
if len(shape) == 0:
return S.Zero
shape = map(_sympify, shape)
obj = Expr.__new__(cls, *shape)
return obj
@property
def shape(self):
return self._args
def as_explicit(self):
if any(not i.is_Integer for i in self.shape):
raise ValueError("Cannot return explicit form for symbolic shape.")
return ImmutableDenseNDimArray.zeros(*self.shape)
class OneArray(_ArrayExpr):
"""
Symbolic array of ones.
"""
def __new__(cls, *shape):
if len(shape) == 0:
return S.One
shape = map(_sympify, shape)
obj = Expr.__new__(cls, *shape)
return obj
@property
def shape(self):
return self._args
def as_explicit(self):
if any(not i.is_Integer for i in self.shape):
raise ValueError("Cannot return explicit form for symbolic shape.")
return ImmutableDenseNDimArray([S.One for i in range(reduce(operator.mul, self.shape))]).reshape(*self.shape)
class _CodegenArrayAbstract(Basic):
@property
def subranks(self):
"""
Returns the ranks of the objects in the uppermost tensor product inside
the current object. In case no tensor products are contained, return
the atomic ranks.
Examples
========
>>> from sympy.tensor.array.expressions.array_expressions import ArrayTensorProduct, ArrayContraction
>>> from sympy import MatrixSymbol
>>> M = MatrixSymbol("M", 3, 3)
>>> N = MatrixSymbol("N", 3, 3)
>>> P = MatrixSymbol("P", 3, 3)
Important: do not confuse the rank of the matrix with the rank of an array.
>>> tp = ArrayTensorProduct(M, N, P)
>>> tp.subranks
[2, 2, 2]
>>> co = ArrayContraction(tp, (1, 2), (3, 4))
>>> co.subranks
[2, 2, 2]
"""
return self._subranks[:]
def subrank(self):
"""
The sum of ``subranks``.
"""
return sum(self.subranks)
@property
def shape(self):
return self._shape
class ArrayTensorProduct(_CodegenArrayAbstract):
r"""
Class to represent the tensor product of array-like objects.
"""
def __new__(cls, *args):
args = [_sympify(arg) for arg in args]
args = cls._flatten(args)
ranks = [get_rank(arg) for arg in args]
# Check if there are nested permutation and lift them up:
permutation_cycles = []
for i, arg in enumerate(args):
if not isinstance(arg, PermuteDims):
continue
permutation_cycles.extend([[k + sum(ranks[:i]) for k in j] for j in arg.permutation.cyclic_form])
args[i] = arg.expr
if permutation_cycles:
return PermuteDims(ArrayTensorProduct(*args), Permutation(sum(ranks)-1)*Permutation(permutation_cycles))
if len(args) == 1:
return args[0]
# If any object is a ZeroArray, return a ZeroArray:
if any(isinstance(arg, (ZeroArray, ZeroMatrix)) for arg in args):
shapes = reduce(operator.add, [get_shape(i) for i in args], ())
return ZeroArray(*shapes)
# If there are contraction objects inside, transform the whole
# expression into `ArrayContraction`:
contractions = {i: arg for i, arg in enumerate(args) if isinstance(arg, ArrayContraction)}
if contractions:
ranks = [_get_subrank(arg) if isinstance(arg, ArrayContraction) else get_rank(arg) for arg in args]
cumulative_ranks = list(accumulate([0] + ranks))[:-1]
tp = cls(*[arg.expr if isinstance(arg, ArrayContraction) else arg for arg in args])
contraction_indices = [tuple(cumulative_ranks[i] + k for k in j) for i, arg in contractions.items() for j in arg.contraction_indices]
return ArrayContraction(tp, *contraction_indices)
diagonals = {i: arg for i, arg in enumerate(args) if isinstance(arg, ArrayDiagonal)}
if diagonals:
permutation = []
last_perm = []
ranks = [get_rank(arg) for arg in args]
cumulative_ranks = list(accumulate([0] + ranks))[:-1]
for i, arg in enumerate(args):
if isinstance(arg, ArrayDiagonal):
i1 = get_rank(arg) - len(arg.diagonal_indices)
i2 = len(arg.diagonal_indices)
permutation.extend([cumulative_ranks[i] + j for j in range(i1)])
last_perm.extend([cumulative_ranks[i] + j for j in range(i1, i1 + i2)])
else:
permutation.extend([cumulative_ranks[i] + j for j in range(get_rank(arg))])
permutation.extend(last_perm)
tp = cls(*[arg.expr if isinstance(arg, ArrayDiagonal) else arg for arg in args])
ranks2 = [_get_subrank(arg) if isinstance(arg, ArrayDiagonal) else get_rank(arg) for arg in args]
cumulative_ranks2 = list(accumulate([0] + ranks2))[:-1]
diagonal_indices = [tuple(cumulative_ranks2[i] + k for k in j) for i, arg in diagonals.items() for j in arg.diagonal_indices]
return PermuteDims(ArrayDiagonal(tp, *diagonal_indices), permutation)
obj = Basic.__new__(cls, *args)
obj._subranks = ranks
shapes = [get_shape(i) for i in args]
if any(i is None for i in shapes):
obj._shape = None
else:
obj._shape = tuple(j for i in shapes for j in i)
return obj
@classmethod
def _flatten(cls, args):
args = [i for arg in args for i in (arg.args if isinstance(arg, cls) else [arg])]
return args
def as_explicit(self):
return tensorproduct(*[arg.as_explicit() if hasattr(arg, "as_explicit") else arg for arg in self.args])
class ArrayAdd(_CodegenArrayAbstract):
r"""
Class for elementwise array additions.
"""
def __new__(cls, *args):
args = [_sympify(arg) for arg in args]
ranks = [get_rank(arg) for arg in args]
ranks = list(set(ranks))
if len(ranks) != 1:
raise ValueError("summing arrays of different ranks")
shapes = [arg.shape for arg in args]
if len({i for i in shapes if i is not None}) > 1:
raise ValueError("mismatching shapes in addition")
# Flatten:
args = cls._flatten_args(args)
args = [arg for arg in args if not isinstance(arg, (ZeroArray, ZeroMatrix))]
if len(args) == 0:
if any(i for i in shapes if i is None):
raise NotImplementedError("cannot handle addition of ZeroMatrix/ZeroArray and undefined shape object")
return ZeroArray(*shapes[0])
elif len(args) == 1:
return args[0]
obj = Basic.__new__(cls, *args)
obj._subranks = ranks
if any(i is None for i in shapes):
obj._shape = None
else:
obj._shape = shapes[0]
return obj
@classmethod
def _flatten_args(cls, args):
new_args = []
for arg in args:
if isinstance(arg, ArrayAdd):
new_args.extend(arg.args)
else:
new_args.append(arg)
return new_args
def as_explicit(self):
return Add.fromiter([arg.as_explicit() for arg in self.args])
class PermuteDims(_CodegenArrayAbstract):
r"""
Class to represent permutation of axes of arrays.
Examples
========
>>> from sympy.tensor.array.expressions.array_expressions import PermuteDims
>>> from sympy import MatrixSymbol
>>> M = MatrixSymbol("M", 3, 3)
>>> cg = PermuteDims(M, [1, 0])
The object ``cg`` represents the transposition of ``M``, as the permutation
``[1, 0]`` will act on its indices by switching them:
`M_{ij} \Rightarrow M_{ji}`
This is evident when transforming back to matrix form:
>>> from sympy.tensor.array.expressions.conv_array_to_matrix import convert_array_to_matrix
>>> convert_array_to_matrix(cg)
M.T
>>> N = MatrixSymbol("N", 3, 2)
>>> cg = PermuteDims(N, [1, 0])
>>> cg.shape
(2, 3)
Permutations of tensor products are simplified in order to achieve a
standard form:
>>> from sympy.tensor.array.expressions.array_expressions import ArrayTensorProduct
>>> M = MatrixSymbol("M", 4, 5)
>>> tp = ArrayTensorProduct(M, N)
>>> tp.shape
(4, 5, 3, 2)
>>> perm1 = PermuteDims(tp, [2, 3, 1, 0])
The args ``(M, N)`` have been sorted and the permutation has been
simplified, the expression is equivalent:
>>> perm1.expr.args
(N, M)
>>> perm1.shape
(3, 2, 5, 4)
>>> perm1.permutation
(2 3)
The permutation in its array form has been simplified from
``[2, 3, 1, 0]`` to ``[0, 1, 3, 2]``, as the arguments of the tensor
product `M` and `N` have been switched:
>>> perm1.permutation.array_form
[0, 1, 3, 2]
We can nest a second permutation:
>>> perm2 = PermuteDims(perm1, [1, 0, 2, 3])
>>> perm2.shape
(2, 3, 5, 4)
>>> perm2.permutation.array_form
[1, 0, 3, 2]
"""
def __new__(cls, expr, permutation, nest_permutation=True):
from sympy.combinatorics import Permutation
expr = _sympify(expr)
permutation = Permutation(permutation)
permutation_size = permutation.size
expr_rank = get_rank(expr)
if permutation_size != expr_rank:
raise ValueError("Permutation size must be the length of the shape of expr")
if isinstance(expr, PermuteDims):
subexpr = expr.expr
subperm = expr.permutation
permutation = permutation * subperm
expr = subexpr
if isinstance(expr, ArrayContraction):
expr, permutation = cls._handle_nested_contraction(expr, permutation)
if isinstance(expr, ArrayTensorProduct):
expr, permutation = cls._sort_components(expr, permutation)
if isinstance(expr, (ZeroArray, ZeroMatrix)):
return ZeroArray(*[expr.shape[i] for i in permutation.array_form])
plist = permutation.array_form
if plist == sorted(plist):
return expr
obj = Basic.__new__(cls, expr, permutation)
obj._subranks = [get_rank(expr)]
shape = expr.shape
if shape is None:
obj._shape = None
else:
obj._shape = tuple(shape[permutation(i)] for i in range(len(shape)))
return obj
@property
def expr(self):
return self.args[0]
@property
def permutation(self):
return self.args[1]
@classmethod
def _sort_components(cls, expr, permutation):
# Get the permutation in its image-form:
perm_image_form = _af_invert(permutation.array_form)
args = list(expr.args)
# Starting index global position for every arg:
cumul = list(accumulate([0] + expr.subranks))
# Split `perm_image_form` into a list of list corresponding to the indices
# of every argument:
perm_image_form_in_components = [perm_image_form[cumul[i]:cumul[i+1]] for i in range(len(args))]
# Create an index, target-position-key array:
ps = [(i, sorted(comp)) for i, comp in enumerate(perm_image_form_in_components)]
# Sort the array according to the target-position-key:
# In this way, we define a canonical way to sort the arguments according
# to the permutation.
ps.sort(key=lambda x: x[1])
# Read the inverse-permutation (i.e. image-form) of the args:
perm_args_image_form = [i[0] for i in ps]
# Apply the args-permutation to the `args`:
args_sorted = [args[i] for i in perm_args_image_form]
# Apply the args-permutation to the array-form of the permutation of the axes (of `expr`):
perm_image_form_sorted_args = [perm_image_form_in_components[i] for i in perm_args_image_form]
new_permutation = Permutation(_af_invert([j for i in perm_image_form_sorted_args for j in i]))
return ArrayTensorProduct(*args_sorted), new_permutation
@classmethod
def _handle_nested_contraction(cls, expr, permutation):
if not isinstance(expr, ArrayContraction):
return expr, permutation
if not isinstance(expr.expr, ArrayTensorProduct):
return expr, permutation
args = expr.expr.args
subranks = [get_rank(arg) for arg in expr.expr.args]
contraction_indices = expr.contraction_indices
contraction_indices_flat = [j for i in contraction_indices for j in i]
cumul = list(accumulate([0] + subranks))
# Spread the permutation in its array form across the args in the corresponding
# tensor-product arguments with free indices:
permutation_array_blocks_up = []
image_form = _af_invert(permutation.array_form)
counter = 0
for i, e in enumerate(subranks):
current = []
for j in range(cumul[i], cumul[i+1]):
if j in contraction_indices_flat:
continue
current.append(image_form[counter])
counter += 1
permutation_array_blocks_up.append(current)
# Get the map of axis repositioning for every argument of tensor-product:
index_blocks = [[j for j in range(cumul[i], cumul[i+1])] for i, e in enumerate(expr.subranks)]
index_blocks_up = expr._push_indices_up(expr.contraction_indices, index_blocks)
inverse_permutation = permutation**(-1)
index_blocks_up_permuted = [[inverse_permutation(j) for j in i if j is not None] for i in index_blocks_up]
# Sorting key is a list of tuple, first element is the index of `args`, second element of
# the tuple is the sorting key to sort `args` of the tensor product:
sorting_keys = list(enumerate(index_blocks_up_permuted))
sorting_keys.sort(key=lambda x: x[1])
# Now we can get the permutation acting on the args in its image-form:
new_perm_image_form = [i[0] for i in sorting_keys]
# Apply the args-level permutation to various elements:
new_index_blocks = [index_blocks[i] for i in new_perm_image_form]
new_index_perm_array_form = _af_invert([j for i in new_index_blocks for j in i])
new_args = [args[i] for i in new_perm_image_form]
new_contraction_indices = [tuple(new_index_perm_array_form[j] for j in i) for i in contraction_indices]
new_expr = ArrayContraction(ArrayTensorProduct(*new_args), *new_contraction_indices)
new_permutation = Permutation(_af_invert([j for i in [permutation_array_blocks_up[k] for k in new_perm_image_form] for j in i]))
return new_expr, new_permutation
@classmethod
def _check_permutation_mapping(cls, expr, permutation):
subranks = expr.subranks
index2arg = [i for i, arg in enumerate(expr.args) for j in range(expr.subranks[i])]
permuted_indices = [permutation(i) for i in range(expr.subrank())]
new_args = list(expr.args)
arg_candidate_index = index2arg[permuted_indices[0]]
current_indices = []
new_permutation = []
inserted_arg_cand_indices = set([])
for i, idx in enumerate(permuted_indices):
if index2arg[idx] != arg_candidate_index:
new_permutation.extend(current_indices)
current_indices = []
arg_candidate_index = index2arg[idx]
current_indices.append(idx)
arg_candidate_rank = subranks[arg_candidate_index]
if len(current_indices) == arg_candidate_rank:
new_permutation.extend(sorted(current_indices))
local_current_indices = [j - min(current_indices) for j in current_indices]
i1 = index2arg[i]
new_args[i1] = PermuteDims(new_args[i1], Permutation(local_current_indices))
inserted_arg_cand_indices.add(arg_candidate_index)
current_indices = []
new_permutation.extend(current_indices)
# TODO: swap args positions in order to simplify the expression:
# TODO: this should be in a function
args_positions = list(range(len(new_args)))
# Get possible shifts:
maps = {}
cumulative_subranks = [0] + list(accumulate(subranks))
for i in range(0, len(subranks)):
s = set([index2arg[new_permutation[j]] for j in range(cumulative_subranks[i], cumulative_subranks[i+1])])
if len(s) != 1:
continue
elem = next(iter(s))
if i != elem:
maps[i] = elem
# Find cycles in the map:
lines = []
current_line = []
while maps:
if len(current_line) == 0:
k, v = maps.popitem()
current_line.append(k)
else:
k = current_line[-1]
if k not in maps:
current_line = []
continue
v = maps.pop(k)
if v in current_line:
lines.append(current_line)
current_line = []
continue
current_line.append(v)
for line in lines:
for i, e in enumerate(line):
args_positions[line[(i + 1) % len(line)]] = e
# TODO: function in order to permute the args:
permutation_blocks = [[new_permutation[cumulative_subranks[i] + j] for j in range(e)] for i, e in enumerate(subranks)]
new_args = [new_args[i] for i in args_positions]
new_permutation_blocks = [permutation_blocks[i] for i in args_positions]
new_permutation2 = [j for i in new_permutation_blocks for j in i]
return ArrayTensorProduct(*new_args), Permutation(new_permutation2) # **(-1)
@classmethod
def _check_if_there_are_closed_cycles(cls, expr, permutation):
args = list(expr.args)
subranks = expr.subranks
cyclic_form = permutation.cyclic_form
cumulative_subranks = [0] + list(accumulate(subranks))
cyclic_min = [min(i) for i in cyclic_form]
cyclic_max = [max(i) for i in cyclic_form]
cyclic_keep = []
for i, cycle in enumerate(cyclic_form):
flag = True
for j in range(0, len(cumulative_subranks) - 1):
if cyclic_min[i] >= cumulative_subranks[j] and cyclic_max[i] < cumulative_subranks[j+1]:
# Found a sinkable cycle.
args[j] = PermuteDims(args[j], Permutation([[k - cumulative_subranks[j] for k in cyclic_form[i]]]))
flag = False
break
if flag:
cyclic_keep.append(cyclic_form[i])
return ArrayTensorProduct(*args), Permutation(cyclic_keep, size=permutation.size)
def nest_permutation(self):
r"""
DEPRECATED.
"""
ret = self._nest_permutation(self.expr, self.permutation)
if ret is None:
return self
return ret
@classmethod
def _nest_permutation(cls, expr, permutation):
if isinstance(expr, ArrayTensorProduct):
return PermuteDims(*cls._check_if_there_are_closed_cycles(expr, permutation))
elif isinstance(expr, ArrayContraction):
# Invert tree hierarchy: put the contraction above.
cycles = permutation.cyclic_form
newcycles = ArrayContraction._convert_outer_indices_to_inner_indices(expr, *cycles)
newpermutation = Permutation(newcycles)
new_contr_indices = [tuple(newpermutation(j) for j in i) for i in expr.contraction_indices]
return ArrayContraction(PermuteDims(expr.expr, newpermutation), *new_contr_indices)
elif isinstance(expr, ArrayAdd):
return ArrayAdd(*[PermuteDims(arg, permutation) for arg in expr.args])
return None
def as_explicit(self):
return permutedims(self.expr.as_explicit(), self.permutation)
class ArrayDiagonal(_CodegenArrayAbstract):
r"""
Class to represent the diagonal operator.
Explanation
===========
In a 2-dimensional array it returns the diagonal, this looks like the
operation:
`A_{ij} \rightarrow A_{ii}`
The diagonal over axes 1 and 2 (the second and third) of the tensor product
of two 2-dimensional arrays `A \otimes B` is
`\Big[ A_{ab} B_{cd} \Big]_{abcd} \rightarrow \Big[ A_{ai} B_{id} \Big]_{adi}`
In this last example the array expression has been reduced from
4-dimensional to 3-dimensional. Notice that no contraction has occurred,
rather there is a new index `i` for the diagonal, contraction would have
reduced the array to 2 dimensions.
Notice that the diagonalized out dimensions are added as new dimensions at
the end of the indices.
"""
def __new__(cls, expr, *diagonal_indices):
expr = _sympify(expr)
diagonal_indices = [Tuple(*sorted(i)) for i in diagonal_indices]
if isinstance(expr, ArrayAdd):
return ArrayAdd(*[ArrayDiagonal(arg, *diagonal_indices) for arg in expr.args])
if isinstance(expr, ArrayDiagonal):
return cls._flatten(expr, *diagonal_indices)
if isinstance(expr, PermuteDims):
return cls._handle_nested_permutedims_in_diag(expr, *diagonal_indices)
shape = expr.shape
if shape is not None:
cls._validate(expr, *diagonal_indices)
# Get new shape:
positions, shape = cls._get_positions_shape(shape, diagonal_indices)
else:
positions = None
if len(diagonal_indices) == 0:
return expr
if isinstance(expr, (ZeroArray, ZeroMatrix)):
return ZeroArray(*shape)
obj = Basic.__new__(cls, expr, *diagonal_indices)
obj._positions = positions
obj._subranks = _get_subranks(expr)
obj._shape = shape
return obj
@staticmethod
def _validate(expr, *diagonal_indices):
# Check that no diagonalization happens on indices with mismatched
# dimensions:
shape = expr.shape
for i in diagonal_indices:
if len({shape[j] for j in i}) != 1:
raise ValueError("diagonalizing indices of different dimensions")
if len(i) <= 1:
raise ValueError("need at least two axes to diagonalize")
@staticmethod
def _remove_trivial_dimensions(shape, *diagonal_indices):
return [tuple(j for j in i) for i in diagonal_indices if shape[i[0]] != 1]
@property
def expr(self):
return self.args[0]
@property
def diagonal_indices(self):
return self.args[1:]
@staticmethod
def _flatten(expr, *outer_diagonal_indices):
inner_diagonal_indices = expr.diagonal_indices
all_inner = [j for i in inner_diagonal_indices for j in i]
all_inner.sort()
# TODO: add API for total rank and cumulative rank:
total_rank = _get_subrank(expr)
inner_rank = len(all_inner)
outer_rank = total_rank - inner_rank
shifts = [0 for i in range(outer_rank)]
counter = 0
pointer = 0
for i in range(outer_rank):
while pointer < inner_rank and counter >= all_inner[pointer]:
counter += 1
pointer += 1
shifts[i] += pointer
counter += 1
outer_diagonal_indices = tuple(tuple(shifts[j] + j for j in i) for i in outer_diagonal_indices)
diagonal_indices = inner_diagonal_indices + outer_diagonal_indices
return ArrayDiagonal(expr.expr, *diagonal_indices)
@classmethod
def _handle_nested_permutedims_in_diag(cls, expr: PermuteDims, *diagonal_indices):
back_diagonal_indices = [[expr.permutation(j) for j in i] for i in diagonal_indices]
nondiag = [i for i in range(get_rank(expr)) if not any(i in j for j in diagonal_indices)]
back_nondiag = [expr.permutation(i) for i in nondiag]
remap = {e: i for i, e in enumerate(sorted(back_nondiag))}
new_permutation1 = [remap[i] for i in back_nondiag]
shift = len(new_permutation1)
diag_block_perm = [i + shift for i in range(len(back_diagonal_indices))]
new_permutation = new_permutation1 + diag_block_perm
return PermuteDims(
ArrayDiagonal(
expr.expr,
*back_diagonal_indices
),
new_permutation
)
def _push_indices_down_nonstatic(self, indices):
transform = lambda x: self._positions[x] if x < len(self._positions) else None
return _apply_recursively_over_nested_lists(transform, indices)
def _push_indices_up_nonstatic(self, indices):
def transform(x):
for i, e in enumerate(self._positions):
if (isinstance(e, int) and x == e) or (isinstance(e, tuple) and x in e):
return i
return _apply_recursively_over_nested_lists(transform, indices)
@classmethod
def _push_indices_down(cls, diagonal_indices, indices, rank):
positions, shape = cls._get_positions_shape(range(rank), diagonal_indices)
transform = lambda x: positions[x] if x < len(positions) else None
return _apply_recursively_over_nested_lists(transform, indices)
@classmethod
def _push_indices_up(cls, diagonal_indices, indices, rank):
positions, shape = cls._get_positions_shape(range(rank), diagonal_indices)
def transform(x):
for i, e in enumerate(positions):
if (isinstance(e, int) and x == e) or (isinstance(e, tuple) and x in e):
return i
return _apply_recursively_over_nested_lists(transform, indices)
@classmethod
def _get_positions_shape(cls, shape, diagonal_indices):
data1 = tuple((i, shp) for i, shp in enumerate(shape) if not any(i in j for j in diagonal_indices))
pos1, shp1 = zip(*data1) if data1 else ((), ())
data2 = tuple((i, shape[i[0]]) for i in diagonal_indices)
pos2, shp2 = zip(*data2) if data2 else ((), ())
positions = pos1 + pos2
shape = shp1 + shp2
return positions, shape
def as_explicit(self):
return tensordiagonal(self.expr.as_explicit(), *self.diagonal_indices)
class ArrayElementwiseApplyFunc(_CodegenArrayAbstract):
def __new__(cls, function, element):
if not isinstance(function, Lambda):
d = Dummy('d')
function = Lambda(d, function(d))
obj = _CodegenArrayAbstract.__new__(cls, function, element)
obj._subranks = _get_subranks(element)
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 _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
class ArrayContraction(_CodegenArrayAbstract):
r"""
This class is meant to represent contractions of arrays in a form easily
processable by the code printers.
"""
def __new__(cls, expr, *contraction_indices, **kwargs):
contraction_indices = _sort_contraction_indices(contraction_indices)
expr = _sympify(expr)
if len(contraction_indices) == 0:
return expr
if isinstance(expr, ArrayContraction):
return cls._flatten(expr, *contraction_indices)
if isinstance(expr, (ZeroArray, ZeroMatrix)):
contraction_indices_flat = [j for i in contraction_indices for j in i]
shape = [e for i, e in enumerate(expr.shape) if i not in contraction_indices_flat]
return ZeroArray(*shape)
if isinstance(expr, PermuteDims):
return cls._handle_nested_permute_dims(expr, *contraction_indices)
if isinstance(expr, ArrayTensorProduct):
expr, contraction_indices = cls._sort_fully_contracted_args(expr, contraction_indices)
expr, contraction_indices = cls._lower_contraction_to_addends(expr, contraction_indices)
if len(contraction_indices) == 0:
return expr
if isinstance(expr, ArrayDiagonal):
return cls._handle_nested_diagonal(expr, *contraction_indices)
if isinstance(expr, ArrayAdd):
return ArrayAdd(*[ArrayContraction(i, *contraction_indices) for i in expr.args])
obj = Basic.__new__(cls, expr, *contraction_indices)
obj._subranks = _get_subranks(expr)
obj._mapping = _get_mapping_from_subranks(obj._subranks)
free_indices_to_position = {i: i for i in range(sum(obj._subranks)) if all([i not in cind for cind in contraction_indices])}
obj._free_indices_to_position = free_indices_to_position
shape = expr.shape
cls._validate(expr, *contraction_indices)
if shape:
shape = tuple(shp for i, shp in enumerate(shape) if not any(i in j for j in contraction_indices))
obj._shape = shape
return obj
def __mul__(self, other):
if other == 1:
return self
else:
raise NotImplementedError("Product of N-dim arrays is not uniquely defined. Use another method.")
def __rmul__(self, other):
if other == 1:
return self
else:
raise NotImplementedError("Product of N-dim arrays is not uniquely defined. Use another method.")
@staticmethod
def _validate(expr, *contraction_indices):
shape = expr.shape
if shape is None:
return
# Check that no contraction happens when the shape is mismatched:
for i in contraction_indices:
if len({shape[j] for j in i if shape[j] != -1}) != 1:
raise ValueError("contracting indices of different dimensions")
@classmethod
def _push_indices_down(cls, contraction_indices, indices):
flattened_contraction_indices = [j for i in contraction_indices for j in i]
flattened_contraction_indices.sort()
transform = _build_push_indices_down_func_transformation(flattened_contraction_indices)
return _apply_recursively_over_nested_lists(transform, indices)
@classmethod
def _push_indices_up(cls, contraction_indices, indices):
flattened_contraction_indices = [j for i in contraction_indices for j in i]
flattened_contraction_indices.sort()
transform = _build_push_indices_up_func_transformation(flattened_contraction_indices)
return _apply_recursively_over_nested_lists(transform, indices)
@classmethod
def _lower_contraction_to_addends(cls, expr, contraction_indices):
if isinstance(expr, ArrayAdd):
raise NotImplementedError()
if not isinstance(expr, ArrayTensorProduct):
return expr, contraction_indices
subranks = expr.subranks
cumranks = list(accumulate([0] + subranks))
contraction_indices_remaining = []
contraction_indices_args = [[] for i in expr.args]
backshift = set([])
for i, contraction_group in enumerate(contraction_indices):
for j in range(len(expr.args)):
if not isinstance(expr.args[j], ArrayAdd):
continue
if all(cumranks[j] <= k < cumranks[j+1] for k in contraction_group):
contraction_indices_args[j].append([k - cumranks[j] for k in contraction_group])
backshift.update(contraction_group)
break
else:
contraction_indices_remaining.append(contraction_group)
if len(contraction_indices_remaining) == len(contraction_indices):
return expr, contraction_indices
total_rank = get_rank(expr)
shifts = list(accumulate([1 if i in backshift else 0 for i in range(total_rank)]))
contraction_indices_remaining = [Tuple.fromiter(j - shifts[j] for j in i) for i in contraction_indices_remaining]
ret = ArrayTensorProduct(*[
ArrayContraction(arg, *contr) for arg, contr in zip(expr.args, contraction_indices_args)
])
return ret, contraction_indices_remaining
def split_multiple_contractions(self):
"""
Recognize multiple contractions and attempt at rewriting them as paired-contractions.
"""
from sympy import ask, Q
contraction_indices = self.contraction_indices
if isinstance(self.expr, ArrayTensorProduct):
args = list(self.expr.args)
else:
args = [self.expr]
# TODO: unify API, best location in ArrayTensorProduct
subranks = [get_rank(i) for i in args]
# TODO: unify API
mapping = _get_mapping_from_subranks(subranks)
reverse_mapping = {v:k for k, v in mapping.items()}
new_contraction_indices = []
for indl, links in enumerate(contraction_indices):
if len(links) <= 2:
new_contraction_indices.append(links)
continue
# Check multiple contractions:
#
# Examples:
#
# * `A_ij b_j0 C_jk` ===> `A*DiagMatrix(b)*C`
#
# Care for:
# - matrix being diagonalized (i.e. `A_ii`)
# - vectors being diagonalized (i.e. `a_i0`)
# Also consider the case of diagonal matrices being contracted:
current_dimension = self.expr.shape[links[0]]
tuple_links = [mapping[i] for i in links]
arg_indices, arg_positions = zip(*tuple_links)
args_updates = {}
if len(arg_indices) != len(set(arg_indices)):
# Maybe trace should be supported?
raise NotImplementedError()
not_vectors = []
vectors = []
for arg_ind, arg_pos in tuple_links:
mat = args[arg_ind]
other_arg_pos = 1-arg_pos
other_arg_abs = reverse_mapping[arg_ind, other_arg_pos]
if (((1 not in mat.shape) and (not ask(Q.diagonal(mat)))) or
((current_dimension == 1) is True and mat.shape != (1, 1)) or
any([other_arg_abs in l for li, l in enumerate(contraction_indices) if li != indl])
):
not_vectors.append((arg_ind, arg_pos))
continue
args_updates[arg_ind] = diagonalize_vector(mat)
vectors.append((arg_ind, arg_pos))
vectors.append((arg_ind, 1-arg_pos))
if len(not_vectors) > 2:
new_contraction_indices.append(links)
continue
if len(not_vectors) == 0:
new_sequence = vectors[:1] + vectors[2:]
elif len(not_vectors) == 1:
new_sequence = not_vectors[:1] + vectors[:-1]
else:
new_sequence = not_vectors[:1] + vectors + not_vectors[1:]
for i in range(0, len(new_sequence) - 1, 2):
arg1, pos1 = new_sequence[i]
arg2, pos2 = new_sequence[i+1]
if arg1 == arg2:
raise NotImplementedError
continue
abspos1 = reverse_mapping[arg1, pos1]
abspos2 = reverse_mapping[arg2, pos2]
new_contraction_indices.append((abspos1, abspos2))
for ind, newarg in args_updates.items():
args[ind] = newarg
return ArrayContraction(
ArrayTensorProduct(*args),
*new_contraction_indices
)
def flatten_contraction_of_diagonal(self):
if not isinstance(self.expr, ArrayDiagonal):
return self
contraction_down = self.expr._push_indices_down(self.expr.diagonal_indices, self.contraction_indices)
new_contraction_indices = []
diagonal_indices = self.expr.diagonal_indices[:]
for i in contraction_down:
contraction_group = list(i)
for j in i:
diagonal_with = [k for k in diagonal_indices if j in k]
contraction_group.extend([l for k in diagonal_with for l in k])
diagonal_indices = [k for k in diagonal_indices if k not in diagonal_with]
new_contraction_indices.append(sorted(set(contraction_group)))
new_contraction_indices = ArrayDiagonal._push_indices_up(diagonal_indices, new_contraction_indices)
return ArrayContraction(
ArrayDiagonal(
self.expr.expr,
*diagonal_indices
),
*new_contraction_indices
)
@staticmethod
def _get_free_indices_to_position_map(free_indices, contraction_indices):
free_indices_to_position = {}
flattened_contraction_indices = [j for i in contraction_indices for j in i]
counter = 0
for ind in free_indices:
while counter in flattened_contraction_indices:
counter += 1
free_indices_to_position[ind] = counter
counter += 1
return free_indices_to_position
@staticmethod
def _get_index_shifts(expr):
"""
Get the mapping of indices at the positions before the contraction
occurs.
Examples
========
>>> from sympy.tensor.array.expressions.array_expressions import ArrayTensorProduct
>>> from sympy.tensor.array.expressions.array_expressions import ArrayContraction
>>> from sympy import MatrixSymbol
>>> M = MatrixSymbol("M", 3, 3)
>>> N = MatrixSymbol("N", 3, 3)
>>> cg = ArrayContraction(ArrayTensorProduct(M, N), [1, 2])
>>> cg._get_index_shifts(cg)
[0, 2]
Indeed, ``cg`` after the contraction has two dimensions, 0 and 1. They
need to be shifted by 0 and 2 to get the corresponding positions before
the contraction (that is, 0 and 3).
"""
inner_contraction_indices = expr.contraction_indices
all_inner = [j for i in inner_contraction_indices for j in i]
all_inner.sort()
# TODO: add API for total rank and cumulative rank:
total_rank = _get_subrank(expr)
inner_rank = len(all_inner)
outer_rank = total_rank - inner_rank
shifts = [0 for i in range(outer_rank)]
counter = 0
pointer = 0
for i in range(outer_rank):
while pointer < inner_rank and counter >= all_inner[pointer]:
counter += 1
pointer += 1
shifts[i] += pointer
counter += 1
return shifts
@staticmethod
def _convert_outer_indices_to_inner_indices(expr, *outer_contraction_indices):
shifts = ArrayContraction._get_index_shifts(expr)
outer_contraction_indices = tuple(tuple(shifts[j] + j for j in i) for i in outer_contraction_indices)
return outer_contraction_indices
@staticmethod
def _flatten(expr, *outer_contraction_indices):
inner_contraction_indices = expr.contraction_indices
outer_contraction_indices = ArrayContraction._convert_outer_indices_to_inner_indices(expr, *outer_contraction_indices)
contraction_indices = inner_contraction_indices + outer_contraction_indices
return ArrayContraction(expr.expr, *contraction_indices)
@classmethod
def _handle_nested_permute_dims(cls, expr, *contraction_indices):
permutation = expr.permutation
plist = permutation.array_form
new_contraction_indices = [tuple(permutation(j) for j in i) for i in contraction_indices]
new_plist = [i for i in plist if all(i not in j for j in new_contraction_indices)]
new_plist = cls._push_indices_up(new_contraction_indices, new_plist)
return PermuteDims(
ArrayContraction(expr.expr, *new_contraction_indices),
Permutation(new_plist)
)
@classmethod
def _handle_nested_diagonal(cls, expr: 'ArrayDiagonal', *contraction_indices):
diagonal_indices = list(expr.diagonal_indices)
down_contraction_indices = expr._push_indices_down(expr.diagonal_indices, contraction_indices, get_rank(expr.expr))
# Flatten diagonally contracted indices:
down_contraction_indices = [[k for j in i for k in (j if isinstance(j, (tuple, Tuple)) else [j])] for i in down_contraction_indices]
new_contraction_indices = []
for contr_indgrp in down_contraction_indices:
ind = contr_indgrp[:]
for j, diag_indgrp in enumerate(diagonal_indices):
if diag_indgrp is None:
continue
if any(i in diag_indgrp for i in contr_indgrp):
ind.extend(diag_indgrp)
diagonal_indices[j] = None
new_contraction_indices.append(sorted(set(ind)))
new_diagonal_indices_down = [i for i in diagonal_indices if i is not None]
new_diagonal_indices = ArrayContraction._push_indices_up(new_contraction_indices, new_diagonal_indices_down)
return ArrayDiagonal(
ArrayContraction(expr.expr, *new_contraction_indices),
*new_diagonal_indices
)
@classmethod
def _sort_fully_contracted_args(cls, expr, contraction_indices):
if expr.shape is None:
return expr, contraction_indices
cumul = list(accumulate([0] + expr.subranks))
index_blocks = [list(range(cumul[i], cumul[i+1])) for i in range(len(expr.args))]
contraction_indices_flat = {j for i in contraction_indices for j in i}
fully_contracted = [all(j in contraction_indices_flat for j in range(cumul[i], cumul[i+1])) for i, arg in enumerate(expr.args)]
new_pos = sorted(range(len(expr.args)), key=lambda x: (0, default_sort_key(expr.args[x])) if fully_contracted[x] else (1,))
new_args = [expr.args[i] for i in new_pos]
new_index_blocks_flat = [j for i in new_pos for j in index_blocks[i]]
index_permutation_array_form = _af_invert(new_index_blocks_flat)
new_contraction_indices = [tuple(index_permutation_array_form[j] for j in i) for i in contraction_indices]
new_contraction_indices = _sort_contraction_indices(new_contraction_indices)
return ArrayTensorProduct(*new_args), new_contraction_indices
def _get_contraction_tuples(self):
r"""
Return tuples containing the argument index and position within the
argument of the index position.
Examples
========
>>> from sympy.tensor.array.expressions.array_expressions import ArrayTensorProduct
>>> from sympy import MatrixSymbol
>>> from sympy.abc import N
>>> from sympy.tensor.array.expressions.array_expressions import ArrayContraction
>>> A = MatrixSymbol("A", N, N)
>>> B = MatrixSymbol("B", N, N)
>>> cg = ArrayContraction(ArrayTensorProduct(A, B), (1, 2))
>>> cg._get_contraction_tuples()
[[(0, 1), (1, 0)]]
Notes
=====
Here the contraction pair `(1, 2)` meaning that the 2nd and 3rd indices
of the tensor product `A\otimes B` are contracted, has been transformed
into `(0, 1)` and `(1, 0)`, identifying the same indices in a different
notation. `(0, 1)` is the second index (1) of the first argument (i.e.
0 or `A`). `(1, 0)` is the first index (i.e. 0) of the second
argument (i.e. 1 or `B`).
"""
mapping = self._mapping
return [[mapping[j] for j in i] for i in self.contraction_indices]
@staticmethod
def _contraction_tuples_to_contraction_indices(expr, contraction_tuples):
# TODO: check that `expr` has `.subranks`:
ranks = expr.subranks
cumulative_ranks = [0] + list(accumulate(ranks))
return [tuple(cumulative_ranks[j]+k for j, k in i) for i in contraction_tuples]
@property
def free_indices(self):
return self._free_indices[:]
@property
def free_indices_to_position(self):
return dict(self._free_indices_to_position)
@property
def expr(self):
return self.args[0]
@property
def contraction_indices(self):
return self.args[1:]
def _contraction_indices_to_components(self):
expr = self.expr
if not isinstance(expr, ArrayTensorProduct):
raise NotImplementedError("only for contractions of tensor products")
ranks = expr.subranks
mapping = {}
counter = 0
for i, rank in enumerate(ranks):
for j in range(rank):
mapping[counter] = (i, j)
counter += 1
return mapping
def sort_args_by_name(self):
"""
Sort arguments in the tensor product so that their order is lexicographical.
Examples
========
>>> from sympy.tensor.array.expressions.conv_matrix_to_array import convert_matrix_to_array
>>> from sympy import MatrixSymbol
>>> from sympy.abc import N
>>> A = MatrixSymbol("A", N, N)
>>> B = MatrixSymbol("B", N, N)
>>> C = MatrixSymbol("C", N, N)
>>> D = MatrixSymbol("D", N, N)
>>> cg = convert_matrix_to_array(C*D*A*B)
>>> cg
ArrayContraction(ArrayTensorProduct(A, D, C, B), (0, 3), (1, 6), (2, 5))
>>> cg.sort_args_by_name()
ArrayContraction(ArrayTensorProduct(A, D, B, C), (0, 3), (1, 4), (2, 7))
"""
expr = self.expr
if not isinstance(expr, ArrayTensorProduct):
return self
args = expr.args
sorted_data = sorted(enumerate(args), key=lambda x: default_sort_key(x[1]))
pos_sorted, args_sorted = zip(*sorted_data)
reordering_map = {i: pos_sorted.index(i) for i, arg in enumerate(args)}
contraction_tuples = self._get_contraction_tuples()
contraction_tuples = [[(reordering_map[j], k) for j, k in i] for i in contraction_tuples]
c_tp = ArrayTensorProduct(*args_sorted)
new_contr_indices = self._contraction_tuples_to_contraction_indices(
c_tp,
contraction_tuples
)
return ArrayContraction(c_tp, *new_contr_indices)
def _get_contraction_links(self):
r"""
Returns a dictionary of links between arguments in the tensor product
being contracted.
See the example for an explanation of the values.
Examples
========
>>> from sympy import MatrixSymbol
>>> from sympy.abc import N
>>> from sympy.tensor.array.expressions.conv_matrix_to_array import convert_matrix_to_array
>>> A = MatrixSymbol("A", N, N)
>>> B = MatrixSymbol("B", N, N)
>>> C = MatrixSymbol("C", N, N)
>>> D = MatrixSymbol("D", N, N)
Matrix multiplications are pairwise contractions between neighboring
matrices:
`A_{ij} B_{jk} C_{kl} D_{lm}`
>>> cg = convert_matrix_to_array(A*B*C*D)
>>> cg
ArrayContraction(ArrayTensorProduct(B, C, A, D), (0, 5), (1, 2), (3, 6))
>>> cg._get_contraction_links()
{0: {0: (2, 1), 1: (1, 0)}, 1: {0: (0, 1), 1: (3, 0)}, 2: {1: (0, 0)}, 3: {0: (1, 1)}}
This dictionary is interpreted as follows: argument in position 0 (i.e.
matrix `A`) has its second index (i.e. 1) contracted to `(1, 0)`, that
is argument in position 1 (matrix `B`) on the first index slot of `B`,
this is the contraction provided by the index `j` from `A`.
The argument in position 1 (that is, matrix `B`) has two contractions,
the ones provided by the indices `j` and `k`, respectively the first
and second indices (0 and 1 in the sub-dict). The link `(0, 1)` and
`(2, 0)` respectively. `(0, 1)` is the index slot 1 (the 2nd) of
argument in position 0 (that is, `A_{\ldot j}`), and so on.
"""
args, dlinks = _get_contraction_links([self], self.subranks, *self.contraction_indices)
return dlinks
def as_explicit(self):
return tensorcontraction(self.expr.as_explicit(), *self.contraction_indices)
def get_rank(expr):
if isinstance(expr, (MatrixExpr, MatrixElement)):
return 2
if isinstance(expr, _CodegenArrayAbstract):
return len(expr.shape)
if isinstance(expr, NDimArray):
return expr.rank()
if isinstance(expr, Indexed):
return expr.rank
if isinstance(expr, IndexedBase):
shape = expr.shape
if shape is None:
return -1
else:
return len(shape)
if hasattr(expr, "shape"):
return len(expr.shape)
return 0
def _get_subrank(expr):
if isinstance(expr, _CodegenArrayAbstract):
return expr.subrank()
return get_rank(expr)
def _get_subranks(expr):
if isinstance(expr, _CodegenArrayAbstract):
return expr.subranks
else:
return [get_rank(expr)]
def get_shape(expr):
if hasattr(expr, "shape"):
return expr.shape
return ()
def nest_permutation(expr):
if isinstance(expr, PermuteDims):
return expr.nest_permutation()
else:
return expr
|
e4fa93730e54beaed8274f49af2b642b84ff4fcac152a952411d92da5b99c038 | import operator
from functools import reduce, singledispatch
from sympy import Expr, Transpose, Identity, MatrixSymbol, S, Inverse, MatrixExpr, HadamardProduct
from sympy.combinatorics.permutations import _af_invert
from sympy.matrices.expressions.applyfunc import ElementwiseApplyFunction
from sympy.tensor.array.expressions.array_expressions import ZeroArray, ArraySymbol, ArrayTensorProduct, \
ArrayAdd, PermuteDims, ArrayDiagonal, ArrayElementwiseApplyFunc, get_rank, \
get_shape, ArrayContraction
from sympy.tensor.array.expressions.conv_matrix_to_array import convert_matrix_to_array
@singledispatch
def array_derive(expr, x):
raise NotImplementedError(f"not implemented for type {type(expr)}")
@array_derive.register(Expr)
def _(expr: Expr, x: Expr):
return ZeroArray(*x.shape)
@array_derive.register(ArrayTensorProduct)
def _(expr: ArrayTensorProduct, x: Expr):
args = expr.args
addend_list = []
for i, arg in enumerate(expr.args):
darg = array_derive(arg, x)
if darg == 0:
continue
args_prev = args[:i]
args_succ = args[i+1:]
shape_prev = reduce(operator.add, map(get_shape, args_prev), ())
shape_succ = reduce(operator.add, map(get_shape, args_succ), ())
addend = ArrayTensorProduct(*args_prev, darg, *args_succ)
tot1 = len(get_shape(x))
tot2 = tot1 + len(shape_prev)
tot3 = tot2 + len(get_shape(arg))
tot4 = tot3 + len(shape_succ)
perm = [i for i in range(tot1, tot2)] + \
[i for i in range(tot1)] + [i for i in range(tot2, tot3)] + \
[i for i in range(tot3, tot4)]
addend = PermuteDims(addend, _af_invert(perm))
addend_list.append(addend)
if len(addend_list) == 1:
return addend_list[0]
elif len(addend_list) == 0:
return S.Zero
else:
return ArrayAdd(*addend_list)
@array_derive.register(ArraySymbol)
def _(expr: ArraySymbol, x: Expr):
if expr == x:
return PermuteDims(
ArrayTensorProduct.fromiter(Identity(i) for i in expr.shape),
[2*i for i in range(len(expr.shape))] + [2*i+1 for i in range(len(expr.shape))]
)
return ZeroArray(*(x.shape + expr.shape))
@array_derive.register(MatrixSymbol)
def _(expr: MatrixSymbol, x: Expr):
m, n = expr.shape
if expr == x:
return PermuteDims(
ArrayTensorProduct(Identity(m), Identity(n)),
[0, 2, 1, 3]
)
return ZeroArray(*(x.shape + expr.shape))
@array_derive.register(Identity)
def _(expr: Identity, x: Expr):
return ZeroArray(*(x.shape + expr.shape))
@array_derive.register(Transpose)
def _(expr: Transpose, x: Expr):
# D(A.T, A) ==> (m,n,i,j) ==> D(A_ji, A_mn) = d_mj d_ni
# D(B.T, A) ==> (m,n,i,j) ==> D(B_ji, A_mn)
fd = array_derive(expr.arg, x)
return PermuteDims(fd, [0, 1, 3, 2])
@array_derive.register(Inverse)
def _(expr: Inverse, x: Expr):
mat = expr.I
dexpr = array_derive(mat, x)
tp = ArrayTensorProduct(-expr, dexpr, expr)
mp = ArrayContraction(tp, (1, 4), (5, 6))
pp = PermuteDims(mp, [1, 2, 0, 3])
return pp
@array_derive.register(ElementwiseApplyFunction)
def _(expr: ElementwiseApplyFunction, x: Expr):
assert get_rank(expr) == 2
assert get_rank(x) == 2
fdiff = expr._get_function_fdiff()
dexpr = array_derive(expr.expr, x)
tp = ArrayTensorProduct(
ElementwiseApplyFunction(fdiff, expr.expr),
dexpr
)
td = ArrayDiagonal(
tp, (0, 4), (1, 5)
)
return td
@array_derive.register(ArrayElementwiseApplyFunc)
def _(expr: ArrayElementwiseApplyFunc, x: Expr):
fdiff = expr._get_function_fdiff()
subexpr = expr.expr
dsubexpr = array_derive(subexpr, x)
tp = ArrayTensorProduct(
dsubexpr,
ArrayElementwiseApplyFunc(fdiff, subexpr)
)
b = get_rank(x)
c = get_rank(expr)
diag_indices = [(b + i, b + c + i) for i in range(c)]
return ArrayDiagonal(tp, *diag_indices)
@array_derive.register(MatrixExpr)
def _(expr: MatrixExpr, x: Expr):
cg = convert_matrix_to_array(expr)
return array_derive(cg, x)
@array_derive.register(HadamardProduct)
def _(expr: HadamardProduct, x: Expr):
raise NotImplementedError()
@array_derive.register(ArrayContraction)
def _(expr: ArrayContraction, x: Expr):
fd = array_derive(expr.expr, x)
rank_x = len(get_shape(x))
contraction_indices = expr.contraction_indices
new_contraction_indices = [tuple(j + rank_x for j in i) for i in contraction_indices]
return ArrayContraction(fd, *new_contraction_indices)
@array_derive.register(ArrayDiagonal)
def _(expr: ArrayDiagonal, x: Expr):
dsubexpr = array_derive(expr.expr, x)
rank_x = len(get_shape(x))
diag_indices = [[j + rank_x for j in i] for i in expr.diagonal_indices]
return ArrayDiagonal(dsubexpr, *diag_indices)
@array_derive.register(ArrayAdd)
def _(expr: ArrayAdd, x: Expr):
return ArrayAdd(*[array_derive(arg, x) for arg in expr.args])
@array_derive.register(PermuteDims)
def _(expr: PermuteDims, x: Expr):
de = array_derive(expr.expr, x)
perm = [0, 1] + [i + 2 for i in expr.permutation.array_form]
return PermuteDims(de, perm)
def matrix_derive(expr, x):
from sympy.tensor.array.expressions.conv_array_to_matrix import convert_array_to_matrix
ce = convert_matrix_to_array(expr)
dce = array_derive(ce, x)
return convert_array_to_matrix(dce).doit()
|
dae68711529a978d7b2c060348923515bd3a0d2e76a663ef77e49f3d0e5caed0 | from sympy import (
symbols, Identity, cos, ZeroMatrix)
from sympy.tensor.array.expressions.conv_matrix_to_array import convert_matrix_to_array
from sympy.tensor.array.expressions.conv_array_to_matrix import _support_function_tp1_recognize, \
_array_diag2contr_diagmatrix, convert_array_to_matrix, _remove_trivial_dims, _array2matrix
from sympy import MatrixSymbol
from sympy.combinatorics import Permutation
from sympy.matrices.expressions.diagonal import DiagMatrix
from sympy.matrices import Trace, MatMul, Transpose
from sympy.tensor.array.expressions.array_expressions import ZeroArray, OneArray, \
ArrayTensorProduct, ArrayAdd, PermuteDims, ArrayDiagonal, \
ArrayContraction
from sympy.testing.pytest import raises
i, j, k, l, m, n = symbols("i j k l m n")
I = Identity(k)
I1 = Identity(1)
M = MatrixSymbol("M", k, k)
N = MatrixSymbol("N", k, k)
P = MatrixSymbol("P", k, k)
Q = MatrixSymbol("Q", k, k)
A = MatrixSymbol("A", k, k)
B = MatrixSymbol("B", k, k)
C = MatrixSymbol("C", k, k)
D = MatrixSymbol("D", k, k)
X = MatrixSymbol("X", k, k)
Y = MatrixSymbol("Y", k, k)
a = MatrixSymbol("a", k, 1)
b = MatrixSymbol("b", k, 1)
c = MatrixSymbol("c", k, 1)
d = MatrixSymbol("d", k, 1)
x = MatrixSymbol("x", k, 1)
def test_arrayexpr_convert_array_to_matrix():
cg = ArrayContraction(ArrayTensorProduct(M), (0, 1))
assert convert_array_to_matrix(cg) == Trace(M)
cg = ArrayContraction(ArrayTensorProduct(M, N), (0, 1), (2, 3))
assert convert_array_to_matrix(cg) == Trace(M) * Trace(N)
cg = ArrayContraction(ArrayTensorProduct(M, N), (0, 3), (1, 2))
assert convert_array_to_matrix(cg) == Trace(M * N)
cg = ArrayContraction(ArrayTensorProduct(M, N), (0, 2), (1, 3))
assert convert_array_to_matrix(cg) == Trace(M * N.T)
cg = convert_matrix_to_array(M * N * P)
assert convert_array_to_matrix(cg) == M * N * P
cg = convert_matrix_to_array(M * N.T * P)
assert convert_array_to_matrix(cg) == M * N.T * P
cg = ArrayContraction(ArrayTensorProduct(M,N,P,Q), (1, 2), (5, 6))
assert convert_array_to_matrix(cg) == ArrayTensorProduct(M * N, P * Q)
cg = ArrayContraction(ArrayTensorProduct(-2, M, N), (1, 2))
assert convert_array_to_matrix(cg) == -2 * M * N
a = MatrixSymbol("a", k, 1)
b = MatrixSymbol("b", k, 1)
c = MatrixSymbol("c", k, 1)
cg = PermuteDims(
ArrayContraction(
ArrayTensorProduct(
a,
ArrayAdd(
ArrayTensorProduct(b, c),
ArrayTensorProduct(c, b),
)
), (2, 4)), [0, 1, 3, 2])
assert convert_array_to_matrix(cg) == a * (b.T * c + c.T * b)
za = ZeroArray(m, n)
assert convert_array_to_matrix(za) == ZeroMatrix(m, n)
cg = ArrayTensorProduct(3, M)
assert convert_array_to_matrix(cg) == 3 * M
# TODO: not yet supported:
# cg = ArrayDiagonal(ArrayTensorProduct(M, N, P), (0, 2, 4), (1, 3, 5))
# assert recognize_matrix_expression(cg) == HadamardProduct(M, N, P)
# cg = ArrayDiagonal(ArrayTensorProduct(M, N, P), (0, 3, 4), (1, 2, 5))
# assert recognize_matrix_expression(cg) == HadamardProduct(M, N.T, P)
x = MatrixSymbol("x", k, 1)
cg = PermuteDims(
ArrayContraction(ArrayTensorProduct(OneArray(1), x, OneArray(1), DiagMatrix(Identity(1))),
(0, 5)), Permutation(1, 2, 3))
assert convert_array_to_matrix(cg) == x
expr = ArrayAdd(M, PermuteDims(M, [1, 0]))
assert convert_array_to_matrix(expr) == M + Transpose(M)
def test_arrayexpr_convert_array_to_matrix2():
cg = ArrayContraction(ArrayTensorProduct(M, N), (1, 3))
assert convert_array_to_matrix(cg) == M * N.T
cg = PermuteDims(ArrayTensorProduct(M, N), Permutation([0, 1, 3, 2]))
assert convert_array_to_matrix(cg) == ArrayTensorProduct(M, N.T)
cg = ArrayTensorProduct(M, PermuteDims(N, Permutation([1, 0])))
assert convert_array_to_matrix(cg) == ArrayTensorProduct(M, N.T)
cg = ArrayContraction(
PermuteDims(
ArrayTensorProduct(M, N, P, Q), Permutation([0, 2, 3, 1, 4, 5, 7, 6])),
(1, 2), (3, 5)
)
assert convert_array_to_matrix(cg) == ArrayTensorProduct(M * P.T * Trace(N), Q.T)
cg = ArrayContraction(
ArrayTensorProduct(M, N, P, PermuteDims(Q, Permutation([1, 0]))),
(1, 5), (2, 3)
)
assert convert_array_to_matrix(cg) == ArrayTensorProduct(M * P.T * Trace(N), Q.T)
cg = ArrayTensorProduct(M, PermuteDims(N, [1, 0]))
assert convert_array_to_matrix(cg) == ArrayTensorProduct(M, N.T)
cg = ArrayTensorProduct(PermuteDims(M, [1, 0]), PermuteDims(N, [1, 0]))
assert convert_array_to_matrix(cg) == ArrayTensorProduct(M.T, N.T)
cg = ArrayTensorProduct(PermuteDims(N, [1, 0]), PermuteDims(M, [1, 0]))
assert convert_array_to_matrix(cg) == ArrayTensorProduct(N.T, M.T)
def test_arrayexpr_convert_array_to_diagonalized_vector():
# Check matrix recognition over trivial dimensions:
cg = ArrayTensorProduct(a, b)
assert convert_array_to_matrix(cg) == a * b.T
cg = ArrayTensorProduct(I1, a, b)
assert convert_array_to_matrix(cg) == a * b.T
# Recognize trace inside a tensor product:
cg = ArrayContraction(ArrayTensorProduct(A, B, C), (0, 3), (1, 2))
assert convert_array_to_matrix(cg) == Trace(A * B) * C
# Transform diagonal operator to contraction:
cg = ArrayDiagonal(ArrayTensorProduct(A, a), (1, 2))
assert _array_diag2contr_diagmatrix(cg) == ArrayContraction(ArrayTensorProduct(A, OneArray(1), DiagMatrix(a)), (1, 3))
assert convert_array_to_matrix(cg) == A * DiagMatrix(a)
cg = ArrayDiagonal(ArrayTensorProduct(a, b), (0, 2))
assert _array_diag2contr_diagmatrix(cg) == PermuteDims(
ArrayContraction(ArrayTensorProduct(DiagMatrix(a), OneArray(1), b), (0, 3)), [1, 2, 0]
)
assert convert_array_to_matrix(cg) == b.T * DiagMatrix(a)
cg = ArrayDiagonal(ArrayTensorProduct(A, a), (0, 2))
assert _array_diag2contr_diagmatrix(cg) == ArrayContraction(ArrayTensorProduct(A, OneArray(1), DiagMatrix(a)), (0, 3))
assert convert_array_to_matrix(cg) == A.T * DiagMatrix(a)
cg = ArrayDiagonal(ArrayTensorProduct(I, x, I1), (0, 2), (3, 5))
assert _array_diag2contr_diagmatrix(cg) == ArrayContraction(ArrayTensorProduct(I, OneArray(1), I1, DiagMatrix(x)), (0, 5))
assert convert_array_to_matrix(cg) == DiagMatrix(x)
cg = ArrayDiagonal(ArrayTensorProduct(I, x, A, B), (1, 2), (5, 6))
assert _array_diag2contr_diagmatrix(cg) == ArrayDiagonal(ArrayContraction(ArrayTensorProduct(I, OneArray(1), A, B, DiagMatrix(x)), (1, 7)), (5, 6))
# TODO: not yet working
# assert recognize_matrix_expression(cg)
cg = ArrayDiagonal(ArrayTensorProduct(x, I1), (1, 2))
assert isinstance(cg, ArrayDiagonal)
assert cg.diagonal_indices == ((1, 2),)
assert convert_array_to_matrix(cg) == x
cg = ArrayDiagonal(ArrayTensorProduct(x, I), (0, 2))
assert _array_diag2contr_diagmatrix(cg) == ArrayContraction(ArrayTensorProduct(OneArray(1), I, DiagMatrix(x)), (1, 3))
assert convert_array_to_matrix(cg).doit() == DiagMatrix(x)
raises(ValueError, lambda: ArrayDiagonal(x, (1,)))
# Ignore identity matrices with contractions:
cg = ArrayContraction(ArrayTensorProduct(I, A, I, I), (0, 2), (1, 3), (5, 7))
assert cg.split_multiple_contractions() == cg
assert convert_array_to_matrix(cg) == Trace(A) * I
cg = ArrayContraction(ArrayTensorProduct(Trace(A) * I, I, I), (1, 5), (3, 4))
assert cg.split_multiple_contractions() == cg
assert convert_array_to_matrix(cg).doit() == Trace(A) * I
# Add DiagMatrix when required:
cg = ArrayContraction(ArrayTensorProduct(A, a), (1, 2))
assert cg.split_multiple_contractions() == cg
assert convert_array_to_matrix(cg) == A * a
cg = ArrayContraction(ArrayTensorProduct(A, a, B), (1, 2, 4))
assert cg.split_multiple_contractions() == ArrayContraction(ArrayTensorProduct(A, DiagMatrix(a), B), (1, 2), (3, 4))
assert convert_array_to_matrix(cg) == A * DiagMatrix(a) * B
cg = ArrayContraction(ArrayTensorProduct(A, a, B), (0, 2, 4))
assert cg.split_multiple_contractions() == ArrayContraction(ArrayTensorProduct(A, DiagMatrix(a), B), (0, 2), (3, 4))
assert convert_array_to_matrix(cg) == A.T * DiagMatrix(a) * B
cg = ArrayContraction(ArrayTensorProduct(A, a, b, a.T, B), (0, 2, 4, 7, 9))
assert cg.split_multiple_contractions() == ArrayContraction(ArrayTensorProduct(A, DiagMatrix(a), DiagMatrix(b),
DiagMatrix(a), B),
(0, 2), (3, 4), (5, 7), (6, 9))
assert convert_array_to_matrix(cg).doit() == A.T * DiagMatrix(a) * DiagMatrix(b) * DiagMatrix(a) * B.T
cg = ArrayContraction(ArrayTensorProduct(I1, I1, I1), (1, 2, 4))
assert cg.split_multiple_contractions() == ArrayContraction(ArrayTensorProduct(I1, I1, I1), (1, 2), (3, 4))
assert convert_array_to_matrix(cg).doit() == Identity(1)
cg = ArrayContraction(ArrayTensorProduct(I, I, I, I, A), (1, 2, 8), (5, 6, 9))
assert convert_array_to_matrix(cg.split_multiple_contractions()).doit() == A
cg = ArrayContraction(ArrayTensorProduct(A, a, C, a, B), (1, 2, 4), (5, 6, 8))
expected = ArrayContraction(ArrayTensorProduct(DiagMatrix(a), DiagMatrix(a), C, A, B), (0, 4), (1, 7), (2, 5), (3, 8))
assert cg.split_multiple_contractions() == expected
assert convert_array_to_matrix(cg) == A * DiagMatrix(a) * C * DiagMatrix(a) * B
cg = ArrayContraction(ArrayTensorProduct(a, I1, b, I1, (a.T*b).applyfunc(cos)), (1, 2, 8), (5, 6, 9))
assert cg.split_multiple_contractions().dummy_eq(ArrayContraction(ArrayTensorProduct((a.T * b).applyfunc(cos), I1, I1, a, b), (0, 2), (1, 4), (3, 7), (5, 9)))
assert convert_array_to_matrix(cg).doit().dummy_eq(MatMul(a, (a.T * b).applyfunc(cos), b.T))
def test_arrayexpr_convert_array_contraction_tp_additions():
a = ArrayAdd(
ArrayTensorProduct(M, N),
ArrayTensorProduct(N, M)
)
tp = ArrayTensorProduct(P, a, Q)
expr = ArrayContraction(tp, (3, 4))
expected = ArrayTensorProduct(
P,
ArrayAdd(
ArrayContraction(ArrayTensorProduct(M, N), (1, 2)),
ArrayContraction(ArrayTensorProduct(N, M), (1, 2)),
),
Q
)
assert expr == expected
assert convert_array_to_matrix(expr) == ArrayTensorProduct(P, M * N + N * M, Q)
expr = ArrayContraction(tp, (1, 2), (3, 4), (5, 6))
result = ArrayContraction(
ArrayTensorProduct(
P,
ArrayAdd(
ArrayContraction(ArrayTensorProduct(M, N), (1, 2)),
ArrayContraction(ArrayTensorProduct(N, M), (1, 2)),
),
Q
), (1, 2), (3, 4))
assert expr == result
assert convert_array_to_matrix(expr) == P * (M * N + N * M) * Q
def test_arrayexpr_convert_array_to_implicit_matmul():
# Trivial dimensions are suppressed, so the result can be expressed in matrix form:
cg = ArrayTensorProduct(a, b)
assert convert_array_to_matrix(cg) == a * b.T
cg = ArrayTensorProduct(a, I, b)
assert convert_array_to_matrix(cg) == a * b.T
cg = ArrayContraction(ArrayTensorProduct(I, I), (1, 2))
assert convert_array_to_matrix(cg) == I
cg = PermuteDims(ArrayTensorProduct(I, Identity(1)), [0, 2, 1, 3])
assert convert_array_to_matrix(cg) == I
def test_arrayexpr_convert_array_to_matrix_remove_trivial_dims():
# Tensor Product:
assert _remove_trivial_dims(ArrayTensorProduct(a, b)) == (a * b.T, [1, 3])
assert _remove_trivial_dims(ArrayTensorProduct(a.T, b)) == (a * b.T, [0, 3])
assert _remove_trivial_dims(ArrayTensorProduct(a, b.T)) == (a * b.T, [1, 2])
assert _remove_trivial_dims(ArrayTensorProduct(a.T, b.T)) == (a * b.T, [0, 2])
assert _remove_trivial_dims(ArrayTensorProduct(I, a.T, b.T)) == (a * b.T, [0, 1, 2, 4])
assert _remove_trivial_dims(ArrayTensorProduct(a.T, I, b.T)) == (a * b.T, [0, 2, 3, 4])
assert _remove_trivial_dims(ArrayTensorProduct(a, I)) == (a, [2, 3])
assert _remove_trivial_dims(ArrayTensorProduct(I, a)) == (a, [0, 1])
assert _remove_trivial_dims(ArrayTensorProduct(a.T, b.T, c, d)) == (
ArrayTensorProduct(a * b.T, c * d.T), [0, 2, 5, 7])
assert _remove_trivial_dims(ArrayTensorProduct(a.T, I, b.T, c, d, I)) == (
ArrayTensorProduct(a * b.T, c * d.T, I), [0, 2, 3, 4, 7, 9])
# Addition:
cg = ArrayAdd(ArrayTensorProduct(a, b), ArrayTensorProduct(c, d))
assert _remove_trivial_dims(cg) == (a * b.T + c * d.T, [1, 3])
# Permute Dims:
cg = PermuteDims(ArrayTensorProduct(a, b), Permutation(3)(1, 2))
assert _remove_trivial_dims(cg) == (a * b.T, [2, 3])
cg = PermuteDims(ArrayTensorProduct(a, I, b), Permutation(5)(1, 2, 3, 4))
assert _remove_trivial_dims(cg) == (a * b.T, [1, 2, 4, 5])
cg = PermuteDims(ArrayTensorProduct(I, b, a), Permutation(5)(1, 2, 4, 5, 3))
assert _remove_trivial_dims(cg) == (b * a.T, [0, 3, 4, 5])
# Diagonal:
cg = ArrayDiagonal(ArrayTensorProduct(M, a), (1, 2))
assert _remove_trivial_dims(cg) == (cg, [])
# Contraction:
cg = ArrayContraction(ArrayTensorProduct(M, a), (1, 2))
assert _remove_trivial_dims(cg) == (cg, [])
def test_arrayexpr_convert_array_to_matrix_diag2contraction_diagmatrix():
cg = ArrayDiagonal(ArrayTensorProduct(M, a), (1, 2))
res = _array_diag2contr_diagmatrix(cg)
assert res.shape == cg.shape
assert res == ArrayContraction(ArrayTensorProduct(M, OneArray(1), DiagMatrix(a)), (1, 3))
raises(ValueError, lambda: ArrayDiagonal(ArrayTensorProduct(a, M), (1, 2)))
cg = ArrayDiagonal(ArrayTensorProduct(a.T, M), (1, 2))
res = _array_diag2contr_diagmatrix(cg)
assert res.shape == cg.shape
assert res == ArrayContraction(ArrayTensorProduct(OneArray(1), M, DiagMatrix(a.T)), (1, 4))
cg = ArrayDiagonal(ArrayTensorProduct(a.T, M, N, b.T), (1, 2), (4, 7))
res = _array_diag2contr_diagmatrix(cg)
assert res.shape == cg.shape
assert res == ArrayContraction(
ArrayTensorProduct(OneArray(1), M, N, OneArray(1), DiagMatrix(a.T), DiagMatrix(b.T)), (1, 7), (3, 9))
cg = ArrayDiagonal(ArrayTensorProduct(a, M, N, b.T), (0, 2), (4, 7))
res = _array_diag2contr_diagmatrix(cg)
assert res.shape == cg.shape
assert res == ArrayContraction(
ArrayTensorProduct(OneArray(1), M, N, OneArray(1), DiagMatrix(a), DiagMatrix(b.T)), (1, 6), (3, 9))
cg = ArrayDiagonal(ArrayTensorProduct(a, M, N, b.T), (0, 4), (3, 7))
res = _array_diag2contr_diagmatrix(cg)
assert res.shape == cg.shape
assert res == ArrayContraction(
ArrayTensorProduct(OneArray(1), M, N, OneArray(1), DiagMatrix(a), DiagMatrix(b.T)), (3, 6), (2, 9))
I1 = Identity(1)
x = MatrixSymbol("x", k, 1)
A = MatrixSymbol("A", k, k)
cg = ArrayDiagonal(ArrayTensorProduct(x, A.T, I1), (0, 2))
assert _array_diag2contr_diagmatrix(cg).shape == cg.shape
assert _array2matrix(cg).shape == cg.shape
def test_arrayexpr_convert_array_to_matrix_support_function():
assert _support_function_tp1_recognize([], [2 * k]) == 2 * k
assert _support_function_tp1_recognize([(1, 2)], [A, 2 * k, B, 3]) == 6 * k * A * B
assert _support_function_tp1_recognize([(0, 3), (1, 2)], [A, B]) == Trace(A * B)
assert _support_function_tp1_recognize([(1, 2)], [A, B]) == A * B
assert _support_function_tp1_recognize([(0, 2)], [A, B]) == A.T * B
assert _support_function_tp1_recognize([(1, 3)], [A, B]) == A * B.T
assert _support_function_tp1_recognize([(0, 3)], [A, B]) == A.T * B.T
assert _support_function_tp1_recognize([(1, 2), (5, 6)], [A, B, C, D]) == ArrayTensorProduct(A * B, C * D)
assert _support_function_tp1_recognize([(1, 4), (3, 6)], [A, B, C, D]) == PermuteDims(
ArrayTensorProduct(A * C, B * D), [0, 2, 1, 3])
assert _support_function_tp1_recognize([(0, 3), (1, 4)], [A, B, C]) == B * A * C
assert _support_function_tp1_recognize([(9, 10), (1, 2), (5, 6), (3, 4), (7, 8)],
[X, Y, A, B, C, D]) == X * Y * A * B * C * D
assert _support_function_tp1_recognize([(9, 10), (1, 2), (5, 6), (3, 4)],
[X, Y, A, B, C, D]) == ArrayTensorProduct(X * Y * A * B, C * D)
assert _support_function_tp1_recognize([(1, 7), (3, 8), (4, 11)], [X, Y, A, B, C, D]) == PermuteDims(
ArrayTensorProduct(X * B.T, Y * C, D * A), [0, 2, 5, 1, 3, 4]
)
assert _support_function_tp1_recognize([(0, 1), (3, 6), (5, 8)], [X, A, B, C, D]) == PermuteDims(
ArrayTensorProduct(Trace(X) * A * C, B * D), [0, 2, 1, 3])
assert _support_function_tp1_recognize([(1, 2), (3, 4), (5, 6), (7, 8)], [A, A, B, C, D]) == A ** 2 * B * C * D
assert _support_function_tp1_recognize([(1, 2), (3, 4), (5, 6), (7, 8)], [X, A, B, C, D]) == X * A * B * C * D
assert _support_function_tp1_recognize([(1, 6), (3, 8), (5, 10)], [X, Y, A, B, C, D]) == PermuteDims(
ArrayTensorProduct(X * B, Y * C, A * D), [0, 2, 4, 1, 3, 5]
)
assert _support_function_tp1_recognize([(1, 4), (3, 6)], [A, B, C, D]) == PermuteDims(
ArrayTensorProduct(A * C, B * D), [0, 2, 1, 3])
assert _support_function_tp1_recognize([(0, 4), (1, 7), (2, 5), (3, 8)], [X, A, B, C, D]) == C*X.T*B*A*D
assert _support_function_tp1_recognize([(0, 4), (1, 7), (2, 5), (3, 8)], [X, A, B, C, D]) == C*X.T*B*A*D
|
74496453b9a25e65fa19d278a5fe796f0b2ac948866f3e6466a435c572e4a5ac | from sympy import ImmutableDenseNDimArray, tensorproduct, MatrixSymbol, tensorcontraction, tensordiagonal, permutedims, \
Symbol
from sympy.tensor.array.expressions.array_expressions import ZeroArray, OneArray, ArraySymbol, \
ArrayTensorProduct, PermuteDims, ArrayDiagonal, ArrayContraction
from sympy.testing.pytest import raises
def test_array_as_explicit_call():
assert ZeroArray(3, 2, 4).as_explicit() == ImmutableDenseNDimArray.zeros(3, 2, 4)
assert OneArray(3, 2, 4).as_explicit() == ImmutableDenseNDimArray([1 for i in range(3*2*4)]).reshape(3, 2, 4)
k = Symbol("k")
X = ArraySymbol("X", k, 3, 2)
raises(ValueError, lambda: X.as_explicit())
raises(ValueError, lambda: ZeroArray(k, 2, 3).as_explicit())
raises(ValueError, lambda: OneArray(2, k, 2).as_explicit())
A = ArraySymbol("A", 3, 3)
B = ArraySymbol("B", 3, 3)
texpr = tensorproduct(A, B)
assert isinstance(texpr, ArrayTensorProduct)
assert texpr.as_explicit() == tensorproduct(A.as_explicit(), B.as_explicit())
texpr = tensorcontraction(A, (0, 1))
assert isinstance(texpr, ArrayContraction)
assert texpr.as_explicit() == A[0, 0] + A[1, 1] + A[2, 2]
texpr = tensordiagonal(A, (0, 1))
assert isinstance(texpr, ArrayDiagonal)
assert texpr.as_explicit() == ImmutableDenseNDimArray([A[0, 0], A[1, 1], A[2, 2]])
texpr = permutedims(A, [1, 0])
assert isinstance(texpr, PermuteDims)
assert texpr.as_explicit() == permutedims(A.as_explicit(), [1, 0])
def test_array_as_explicit_matrix_symbol():
A = MatrixSymbol("A", 3, 3)
B = MatrixSymbol("B", 3, 3)
texpr = tensorproduct(A, B)
assert isinstance(texpr, ArrayTensorProduct)
assert texpr.as_explicit() == tensorproduct(A.as_explicit(), B.as_explicit())
texpr = tensorcontraction(A, (0, 1))
assert isinstance(texpr, ArrayContraction)
assert texpr.as_explicit() == A[0, 0] + A[1, 1] + A[2, 2]
texpr = tensordiagonal(A, (0, 1))
assert isinstance(texpr, ArrayDiagonal)
assert texpr.as_explicit() == ImmutableDenseNDimArray([A[0, 0], A[1, 1], A[2, 2]])
texpr = permutedims(A, [1, 0])
assert isinstance(texpr, PermuteDims)
assert texpr.as_explicit() == permutedims(A.as_explicit(), [1, 0])
|
983786d5091893814288a971135dc76211a040983f23e6118bd681b0a50c07bd | from sympy import Sum, MatrixSymbol, Identity, symbols, IndexedBase, KroneckerDelta
from sympy.combinatorics import Permutation
from sympy.tensor.array.expressions.array_expressions import ArrayContraction, ArrayTensorProduct, \
ArrayDiagonal, ArrayAdd, PermuteDims
from sympy.tensor.array.expressions.conv_array_to_matrix import convert_array_to_matrix
from sympy.tensor.array.expressions.conv_indexed_to_array import convert_indexed_to_array, _convert_indexed_to_array
from sympy.testing.pytest import raises
A, B = symbols("A B", cls=IndexedBase)
i, j, k, l, m, n = symbols("i j k l m n")
I = Identity(k)
M = MatrixSymbol("M", k, k)
N = MatrixSymbol("N", k, k)
P = MatrixSymbol("P", k, k)
Q = MatrixSymbol("Q", k, k)
a = MatrixSymbol("a", k, 1)
b = MatrixSymbol("b", k, 1)
c = MatrixSymbol("c", k, 1)
d = MatrixSymbol("d", k, 1)
def test_arrayexpr_convert_index_to_array_support_function():
expr = M[i, j]
assert _convert_indexed_to_array(expr) == (M, (i, j))
expr = M[i, j]*N[k, l]
assert _convert_indexed_to_array(expr) == (ArrayTensorProduct(M, N), (i, j, k, l))
expr = M[i, j]*N[j, k]
assert _convert_indexed_to_array(expr) == (ArrayDiagonal(ArrayTensorProduct(M, N), (1, 2)), (i, k, j))
expr = Sum(M[i, j]*N[j, k], (j, 0, k-1))
assert _convert_indexed_to_array(expr) == (ArrayContraction(ArrayTensorProduct(M, N), (1, 2)), (i, k))
expr = M[i, j] + N[i, j]
assert _convert_indexed_to_array(expr) == (ArrayAdd(M, N), (i, j))
expr = M[i, j] + N[j, i]
assert _convert_indexed_to_array(expr) == (ArrayAdd(M, PermuteDims(N, Permutation([1, 0]))), (i, j))
expr = M[i, j] + M[j, i]
assert _convert_indexed_to_array(expr) == (ArrayAdd(M, PermuteDims(M, Permutation([1, 0]))), (i, j))
expr = (M*N*P)[i, j]
assert _convert_indexed_to_array(expr) == (ArrayContraction(ArrayTensorProduct(M, N, P), (1, 2), (3, 4)), (i, j))
expr = expr.function # Disregard summation in previous expression
ret1, ret2 = _convert_indexed_to_array(expr)
assert ret1 == ArrayDiagonal(ArrayTensorProduct(M, N, P), (1, 2), (3, 4))
assert str(ret2) == "(i, j, _i_1, _i_2)"
expr = KroneckerDelta(i, j)*M[i, k]
assert _convert_indexed_to_array(expr) == (M, ({i, j}, k))
expr = KroneckerDelta(i, j)*KroneckerDelta(j, k)*M[i, l]
assert _convert_indexed_to_array(expr) == (M, ({i, j, k}, l))
expr = KroneckerDelta(j, k)*(M[i, j]*N[k, l] + N[i, j]*M[k, l])
assert _convert_indexed_to_array(expr) == (ArrayDiagonal(ArrayAdd(
ArrayTensorProduct(M, N),
PermuteDims(ArrayTensorProduct(M, N), Permutation(0, 2)(1, 3))
), (1, 2)), (i, l, frozenset({j, k})))
expr = KroneckerDelta(j, m)*KroneckerDelta(m, k)*(M[i, j]*N[k, l] + N[i, j]*M[k, l])
assert _convert_indexed_to_array(expr) == (ArrayDiagonal(ArrayAdd(
ArrayTensorProduct(M, N),
PermuteDims(ArrayTensorProduct(M, N), Permutation(0, 2)(1, 3))
), (1, 2)), (i, l, frozenset({j, m, k})))
expr = KroneckerDelta(i, j)*KroneckerDelta(j, k)*KroneckerDelta(k,m)*M[i, 0]*KroneckerDelta(m, n)
assert _convert_indexed_to_array(expr) == (M, ({i, j, k, m, n}, 0))
expr = M[i, i]
assert _convert_indexed_to_array(expr) == (ArrayDiagonal(M, (0, 1)), (i,))
def test_arrayexpr_convert_indexed_to_array_expression():
s = Sum(A[i]*B[i], (i, 0, 3))
cg = convert_indexed_to_array(s)
assert cg == ArrayContraction(ArrayTensorProduct(A, B), (0, 1))
expr = M*N
result = ArrayContraction(ArrayTensorProduct(M, N), (1, 2))
elem = expr[i, j]
assert convert_indexed_to_array(elem) == result
expr = M*N*M
elem = expr[i, j]
result = ArrayContraction(ArrayTensorProduct(M, M, N), (1, 4), (2, 5))
cg = convert_indexed_to_array(elem)
assert cg == result
cg = convert_indexed_to_array((M * N * P)[i, j])
assert cg == ArrayContraction(ArrayTensorProduct(M, N, P), (1, 2), (3, 4))
cg = convert_indexed_to_array((M * N.T * P)[i, j])
assert cg == ArrayContraction(ArrayTensorProduct(M, N, P), (1, 3), (2, 4))
expr = -2*M*N
elem = expr[i, j]
cg = convert_indexed_to_array(elem)
assert cg == ArrayContraction(ArrayTensorProduct(-2, M, N), (1, 2))
def test_arrayexpr_convert_indexed_to_array_and_back_to_matrix():
expr = a.T*b
elem = expr[0, 0]
cg = convert_indexed_to_array(elem)
assert cg == ArrayContraction(ArrayTensorProduct(a, b), (0, 2))
expr = M[i,j] + N[i,j]
p1, p2 = _convert_indexed_to_array(expr)
assert convert_array_to_matrix(p1) == M + N
expr = M[i,j] + N[j,i]
p1, p2 = _convert_indexed_to_array(expr)
assert convert_array_to_matrix(p1) == M + N.T
expr = M[i,j]*N[k,l] + N[i,j]*M[k,l]
p1, p2 = _convert_indexed_to_array(expr)
assert convert_array_to_matrix(p1) == ArrayAdd(
ArrayTensorProduct(M, N),
ArrayTensorProduct(N, M))
expr = (M*N*P)[i, j]
p1, p2 = _convert_indexed_to_array(expr)
assert convert_array_to_matrix(p1) == M * N * P
expr = Sum(M[i,j]*(N*P)[j,m], (j, 0, k-1))
p1, p2 = _convert_indexed_to_array(expr)
assert convert_array_to_matrix(p1) == M * N * P
expr = Sum((P[j, m] + P[m, j])*(M[i,j]*N[m,n] + N[i,j]*M[m,n]), (j, 0, k-1), (m, 0, k-1))
p1, p2 = _convert_indexed_to_array(expr)
assert convert_array_to_matrix(p1) == M * P * N + M * P.T * N + N * P * M + N * P.T * M
def test_arrayexpr_convert_indexed_to_array_out_of_bounds():
expr = Sum(M[i, i], (i, 0, 4))
raises(ValueError, lambda: convert_indexed_to_array(expr))
expr = Sum(M[i, i], (i, 0, k))
raises(ValueError, lambda: convert_indexed_to_array(expr))
expr = Sum(M[i, i], (i, 1, k-1))
raises(ValueError, lambda: convert_indexed_to_array(expr))
expr = Sum(M[i, j]*N[j,m], (j, 0, 4))
raises(ValueError, lambda: convert_indexed_to_array(expr))
expr = Sum(M[i, j]*N[j,m], (j, 0, k))
raises(ValueError, lambda: convert_indexed_to_array(expr))
expr = Sum(M[i, j]*N[j,m], (j, 1, k-1))
raises(ValueError, lambda: convert_indexed_to_array(expr))
|
364e2373f49b93951523eaa74513c4d3ef5473692c40d426ce3b4f1b4d0008a6 | from sympy import MatrixSymbol, Transpose, Inverse, Trace, HadamardProduct, HadamardPower, MatPow, symbols, Identity
from sympy.tensor.array.expressions.array_expressions import ArrayTensorProduct, ArrayContraction, \
PermuteDims, ArrayDiagonal
from sympy.tensor.array.expressions.conv_array_to_matrix import convert_array_to_matrix
from sympy.tensor.array.expressions.conv_matrix_to_array import convert_matrix_to_array
i, j, k, l, m, n = symbols("i j k l m n")
I = Identity(k)
M = MatrixSymbol("M", k, k)
N = MatrixSymbol("N", k, k)
P = MatrixSymbol("P", k, k)
Q = MatrixSymbol("Q", k, k)
A = MatrixSymbol("A", k, k)
B = MatrixSymbol("B", k, k)
C = MatrixSymbol("C", k, k)
D = MatrixSymbol("D", k, k)
X = MatrixSymbol("X", k, k)
Y = MatrixSymbol("Y", k, k)
a = MatrixSymbol("a", k, 1)
b = MatrixSymbol("b", k, 1)
c = MatrixSymbol("c", k, 1)
d = MatrixSymbol("d", k, 1)
def test_arrayexpr_convert_matrix_to_array():
expr = M*N
result = ArrayContraction(ArrayTensorProduct(M, N), (1, 2))
assert convert_matrix_to_array(expr) == result
expr = M*N*M
result = ArrayContraction(ArrayTensorProduct(M, N, M), (1, 2), (3, 4))
assert convert_matrix_to_array(expr) == result
expr = Transpose(M)
assert convert_matrix_to_array(expr) == PermuteDims(M, [1, 0])
expr = M*Transpose(N)
assert convert_matrix_to_array(expr) == ArrayContraction(ArrayTensorProduct(M, PermuteDims(N, [1, 0])), (1, 2))
expr = 3*M*N
res = convert_matrix_to_array(expr)
rexpr = convert_array_to_matrix(res)
assert expr == rexpr
expr = 3*M + N*M.T*M + 4*k*N
res = convert_matrix_to_array(expr)
rexpr = convert_array_to_matrix(res)
assert expr == rexpr
expr = Inverse(M)*N
rexpr = convert_array_to_matrix(convert_matrix_to_array(expr))
assert expr == rexpr
expr = M**2
rexpr = convert_array_to_matrix(convert_matrix_to_array(expr))
assert expr == rexpr
expr = M*(2*N + 3*M)
res = convert_matrix_to_array(expr)
rexpr = convert_array_to_matrix(res)
assert expr == rexpr
expr = Trace(M)
result = ArrayContraction(M, (0, 1))
assert convert_matrix_to_array(expr) == result
expr = 3*Trace(M)
result = ArrayContraction(ArrayTensorProduct(3, M), (0, 1))
assert convert_matrix_to_array(expr) == result
expr = 3*Trace(Trace(M) * M)
result = ArrayContraction(ArrayTensorProduct(3, M, M), (0, 1), (2, 3))
assert convert_matrix_to_array(expr) == result
expr = 3*Trace(M)**2
result = ArrayContraction(ArrayTensorProduct(3, M, M), (0, 1), (2, 3))
assert convert_matrix_to_array(expr) == result
expr = HadamardProduct(M, N)
result = ArrayDiagonal(ArrayTensorProduct(M, N), (0, 2), (1, 3))
assert convert_matrix_to_array(expr) == result
expr = HadamardProduct(M*N, N*M)
result = ArrayDiagonal(ArrayContraction(ArrayTensorProduct(M, N, N, M), (1, 2), (5, 6)), (0, 2), (1, 3))
assert convert_matrix_to_array(expr) == result
expr = HadamardPower(M, 2)
result = ArrayDiagonal(ArrayTensorProduct(M, M), (0, 2), (1, 3))
assert convert_matrix_to_array(expr) == result
expr = HadamardPower(M*N, 2)
result = ArrayDiagonal(ArrayContraction(ArrayTensorProduct(M, N, M, N), (1, 2), (5, 6)), (0, 2), (1, 3))
assert convert_matrix_to_array(expr) == result
expr = M**2
assert isinstance(expr, MatPow)
assert convert_matrix_to_array(expr) == ArrayContraction(ArrayTensorProduct(M, M), (1, 2))
expr = a.T*b
cg = convert_matrix_to_array(expr)
assert cg == ArrayContraction(ArrayTensorProduct(a, b), (0, 2))
|
efadfcef79b93fe1375d53f9798ba63372e8e1eb1bba5401fd664e9b7c363c82 | import random
from sympy import symbols, ImmutableDenseNDimArray, tensorproduct, tensorcontraction, permutedims, MatrixSymbol, \
ZeroMatrix, sin, cos, DiagMatrix
from sympy.combinatorics import Permutation
from sympy.tensor.array.expressions.array_expressions import ZeroArray, OneArray, ArraySymbol, ArrayElement, \
PermuteDims, ArrayContraction, ArrayTensorProduct, ArrayDiagonal, \
ArrayAdd, nest_permutation, ArrayElementwiseApplyFunc
from sympy.testing.pytest import raises
i, j, k, l, m, n = symbols("i j k l m n")
M = ArraySymbol("M", k, k)
N = ArraySymbol("N", k, k)
P = ArraySymbol("P", k, k)
Q = ArraySymbol("Q", k, k)
A = ArraySymbol("A", k, k)
B = ArraySymbol("B", k, k)
C = ArraySymbol("C", k, k)
D = ArraySymbol("D", k, k)
X = ArraySymbol("X", k, k)
Y = ArraySymbol("Y", k, k)
a = ArraySymbol("a", k, 1)
b = ArraySymbol("b", k, 1)
c = ArraySymbol("c", k, 1)
d = ArraySymbol("d", k, 1)
def test_array_symbol_and_element():
A = ArraySymbol("A", 2)
A0 = ArrayElement(A, (0,))
A1 = ArrayElement(A, (1,))
assert A.as_explicit() == ImmutableDenseNDimArray([A0, A1])
A2 = tensorproduct(A, A)
assert A2.shape == (2, 2)
# TODO: not yet supported:
# assert A2.as_explicit() == Array([[A[0]*A[0], A[1]*A[0]], [A[0]*A[1], A[1]*A[1]]])
A3 = tensorcontraction(A2, (0, 1))
assert A3.shape == ()
# TODO: not yet supported:
# assert A3.as_explicit() == Array([])
A = ArraySymbol("A", 2, 3, 4)
Ae = A.as_explicit()
assert Ae == ImmutableDenseNDimArray(
[[[ArrayElement(A, (i, j, k)) for k in range(4)] for j in range(3)] for i in range(2)])
p = permutedims(A, Permutation(0, 2, 1))
assert isinstance(p, PermuteDims)
def test_zero_array():
assert ZeroArray() == 0
assert ZeroArray().is_Integer
za = ZeroArray(3, 2, 4)
assert za.shape == (3, 2, 4)
za_e = za.as_explicit()
assert za_e.shape == (3, 2, 4)
m, n, k = symbols("m n k")
za = ZeroArray(m, n, k, 2)
assert za.shape == (m, n, k, 2)
raises(ValueError, lambda: za.as_explicit())
def test_one_array():
assert OneArray() == 1
assert OneArray().is_Integer
oa = OneArray(3, 2, 4)
assert oa.shape == (3, 2, 4)
oa_e = oa.as_explicit()
assert oa_e.shape == (3, 2, 4)
m, n, k = symbols("m n k")
oa = OneArray(m, n, k, 2)
assert oa.shape == (m, n, k, 2)
raises(ValueError, lambda: oa.as_explicit())
def test_arrayexpr_contraction_construction():
cg = ArrayContraction(A)
assert cg == A
cg = ArrayContraction(ArrayTensorProduct(A, B), (1, 0))
assert cg == ArrayContraction(ArrayTensorProduct(A, B), (0, 1))
cg = ArrayContraction(ArrayTensorProduct(M, N), (0, 1))
indtup = cg._get_contraction_tuples()
assert indtup == [[(0, 0), (0, 1)]]
assert cg._contraction_tuples_to_contraction_indices(cg.expr, indtup) == [(0, 1)]
cg = ArrayContraction(ArrayTensorProduct(M, N), (1, 2))
indtup = cg._get_contraction_tuples()
assert indtup == [[(0, 1), (1, 0)]]
assert cg._contraction_tuples_to_contraction_indices(cg.expr, indtup) == [(1, 2)]
cg = ArrayContraction(ArrayTensorProduct(M, M, N), (1, 4), (2, 5))
indtup = cg._get_contraction_tuples()
assert indtup == [[(0, 0), (1, 1)], [(0, 1), (2, 0)]]
assert cg._contraction_tuples_to_contraction_indices(cg.expr, indtup) == [(0, 3), (1, 4)]
def test_arrayexpr_array_flatten():
# Flatten nested ArrayTensorProduct objects:
expr1 = ArrayTensorProduct(M, N)
expr2 = ArrayTensorProduct(P, Q)
expr = ArrayTensorProduct(expr1, expr2)
assert expr == ArrayTensorProduct(M, N, P, Q)
assert expr.args == (M, N, P, Q)
# Flatten mixed ArrayTensorProduct and ArrayContraction objects:
cg1 = ArrayContraction(expr1, (1, 2))
cg2 = ArrayContraction(expr2, (0, 3))
expr = ArrayTensorProduct(cg1, cg2)
assert expr == ArrayContraction(ArrayTensorProduct(M, N, P, Q), (1, 2), (4, 7))
expr = ArrayTensorProduct(M, cg1)
assert expr == ArrayContraction(ArrayTensorProduct(M, M, N), (3, 4))
# Flatten nested ArrayContraction objects:
cgnested = ArrayContraction(cg1, (0, 1))
assert cgnested == ArrayContraction(ArrayTensorProduct(M, N), (0, 3), (1, 2))
cgnested = ArrayContraction(ArrayTensorProduct(cg1, cg2), (0, 3))
assert cgnested == ArrayContraction(ArrayTensorProduct(M, N, P, Q), (0, 6), (1, 2), (4, 7))
cg3 = ArrayContraction(ArrayTensorProduct(M, N, P, Q), (1, 3), (2, 4))
cgnested = ArrayContraction(cg3, (0, 1))
assert cgnested == ArrayContraction(ArrayTensorProduct(M, N, P, Q), (0, 5), (1, 3), (2, 4))
cgnested = ArrayContraction(cg3, (0, 3), (1, 2))
assert cgnested == ArrayContraction(ArrayTensorProduct(M, N, P, Q), (0, 7), (1, 3), (2, 4), (5, 6))
cg4 = ArrayContraction(ArrayTensorProduct(M, N, P, Q), (1, 5), (3, 7))
cgnested = ArrayContraction(cg4, (0, 1))
assert cgnested == ArrayContraction(ArrayTensorProduct(M, N, P, Q), (0, 2), (1, 5), (3, 7))
cgnested = ArrayContraction(cg4, (0, 1), (2, 3))
assert cgnested == ArrayContraction(ArrayTensorProduct(M, N, P, Q), (0, 2), (1, 5), (3, 7), (4, 6))
cg = ArrayDiagonal(cg4)
assert cg == cg4
assert isinstance(cg, type(cg4))
# Flatten nested ArrayDiagonal objects:
cg1 = ArrayDiagonal(expr1, (1, 2))
cg2 = ArrayDiagonal(expr2, (0, 3))
cg3 = ArrayDiagonal(ArrayTensorProduct(M, N, P, Q), (1, 3), (2, 4))
cg4 = ArrayDiagonal(ArrayTensorProduct(M, N, P, Q), (1, 5), (3, 7))
cgnested = ArrayDiagonal(cg1, (0, 1))
assert cgnested == ArrayDiagonal(ArrayTensorProduct(M, N), (1, 2), (0, 3))
cgnested = ArrayDiagonal(cg3, (1, 2))
assert cgnested == ArrayDiagonal(ArrayTensorProduct(M, N, P, Q), (1, 3), (2, 4), (5, 6))
cgnested = ArrayDiagonal(cg4, (1, 2))
assert cgnested == ArrayDiagonal(ArrayTensorProduct(M, N, P, Q), (1, 5), (3, 7), (2, 4))
cg = ArrayAdd(M, N)
cg2 = ArrayAdd(cg, P)
assert isinstance(cg2, ArrayAdd)
assert cg2.args == (M, N, P)
assert cg2.shape == (k, k)
expr = ArrayTensorProduct(ArrayDiagonal(X, (0, 1)), ArrayDiagonal(A, (0, 1)))
assert expr == ArrayDiagonal(ArrayTensorProduct(X, A), (0, 1), (2, 3))
expr1 = ArrayDiagonal(ArrayTensorProduct(X, A), (1, 2))
expr2 = ArrayTensorProduct(expr1, a)
assert expr2 == PermuteDims(ArrayDiagonal(ArrayTensorProduct(X, A, a), (1, 2)), [0, 1, 3, 4, 2])
expr1 = ArrayContraction(ArrayTensorProduct(X, A), (1, 2))
expr2 = ArrayTensorProduct(expr1, a)
assert isinstance(expr2, ArrayContraction)
assert isinstance(expr2.expr, ArrayTensorProduct)
def test_arrayexpr_array_diagonal():
cg = ArrayDiagonal(M, (1, 0))
assert cg == ArrayDiagonal(M, (0, 1))
cg = ArrayDiagonal(ArrayTensorProduct(M, N, P), (4, 1), (2, 0))
assert cg == ArrayDiagonal(ArrayTensorProduct(M, N, P), (1, 4), (0, 2))
def test_arrayexpr_array_shape():
expr = ArrayTensorProduct(M, N, P, Q)
assert expr.shape == (k, k, k, k, k, k, k, k)
Z = MatrixSymbol("Z", m, n)
expr = ArrayTensorProduct(M, Z)
assert expr.shape == (k, k, m, n)
expr2 = ArrayContraction(expr, (0, 1))
assert expr2.shape == (m, n)
expr2 = ArrayDiagonal(expr, (0, 1))
assert expr2.shape == (m, n, k)
exprp = PermuteDims(expr, [2, 1, 3, 0])
assert exprp.shape == (m, k, n, k)
expr3 = ArrayTensorProduct(N, Z)
expr2 = ArrayAdd(expr, expr3)
assert expr2.shape == (k, k, m, n)
# Contraction along axes with discordant dimensions:
raises(ValueError, lambda: ArrayContraction(expr, (1, 2)))
# Also diagonal needs the same dimensions:
raises(ValueError, lambda: ArrayDiagonal(expr, (1, 2)))
# Diagonal requires at least to axes to compute the diagonal:
raises(ValueError, lambda: ArrayDiagonal(expr, (1,)))
def test_arrayexpr_permutedims_sink():
cg = PermuteDims(ArrayTensorProduct(M, N), [0, 1, 3, 2], nest_permutation=False)
sunk = nest_permutation(cg)
assert sunk == ArrayTensorProduct(M, PermuteDims(N, [1, 0]))
cg = PermuteDims(ArrayTensorProduct(M, N), [1, 0, 3, 2], nest_permutation=False)
sunk = nest_permutation(cg)
assert sunk == ArrayTensorProduct(PermuteDims(M, [1, 0]), PermuteDims(N, [1, 0]))
cg = PermuteDims(ArrayTensorProduct(M, N), [3, 2, 1, 0], nest_permutation=False)
sunk = nest_permutation(cg)
assert sunk == ArrayTensorProduct(PermuteDims(N, [1, 0]), PermuteDims(M, [1, 0]))
cg = PermuteDims(ArrayContraction(ArrayTensorProduct(M, N), (1, 2)), [1, 0], nest_permutation=False)
sunk = nest_permutation(cg)
assert sunk == ArrayContraction(PermuteDims(ArrayTensorProduct(M, N), [[0, 3]]), (1, 2))
cg = PermuteDims(ArrayTensorProduct(M, N), [1, 0, 3, 2], nest_permutation=False)
sunk = nest_permutation(cg)
assert sunk == ArrayTensorProduct(PermuteDims(M, [1, 0]), PermuteDims(N, [1, 0]))
cg = PermuteDims(ArrayContraction(ArrayTensorProduct(M, N, P), (1, 2), (3, 4)), [1, 0], nest_permutation=False)
sunk = nest_permutation(cg)
assert sunk == ArrayContraction(PermuteDims(ArrayTensorProduct(M, N, P), [[0, 5]]), (1, 2), (3, 4))
def test_arrayexpr_push_indices_up_and_down():
indices = list(range(12))
contr_diag_indices = [(0, 6), (2, 8)]
assert ArrayContraction._push_indices_down(contr_diag_indices, indices) == (1, 3, 4, 5, 7, 9, 10, 11, 12, 13, 14, 15)
assert ArrayContraction._push_indices_up(contr_diag_indices, indices) == (None, 0, None, 1, 2, 3, None, 4, None, 5, 6, 7)
assert ArrayDiagonal._push_indices_down(contr_diag_indices, indices, 10) == (1, 3, 4, 5, 7, 9, (0, 6), (2, 8), None, None, None, None)
assert ArrayDiagonal._push_indices_up(contr_diag_indices, indices, 10) == (6, 0, 7, 1, 2, 3, 6, 4, 7, 5, None, None)
contr_diag_indices = [(1, 2), (7, 8)]
assert ArrayContraction._push_indices_down(contr_diag_indices, indices) == (0, 3, 4, 5, 6, 9, 10, 11, 12, 13, 14, 15)
assert ArrayContraction._push_indices_up(contr_diag_indices, indices) == (0, None, None, 1, 2, 3, 4, None, None, 5, 6, 7)
assert ArrayDiagonal._push_indices_down(contr_diag_indices, indices, 10) == (0, 3, 4, 5, 6, 9, (1, 2), (7, 8), None, None, None, None)
assert ArrayDiagonal._push_indices_up(contr_diag_indices, indices, 10) == (0, 6, 6, 1, 2, 3, 4, 7, 7, 5, None, None)
def test_arrayexpr_split_multiple_contractions():
a = MatrixSymbol("a", k, 1)
b = MatrixSymbol("b", k, 1)
A = MatrixSymbol("A", k, k)
B = MatrixSymbol("B", k, k)
C = MatrixSymbol("C", k, k)
X = MatrixSymbol("X", k, k)
cg = ArrayContraction(ArrayTensorProduct(A.T, a, b, b.T, (A*X*b).applyfunc(cos)), (1, 2, 8), (5, 6, 9))
assert cg.split_multiple_contractions().dummy_eq(ArrayContraction(ArrayTensorProduct(DiagMatrix(a), (A*X*b).applyfunc(cos), A.T, b, b.T), (0, 2), (1, 5), (3, 7, 8)))
# assert recognize_matrix_expression(cg)
# Check no overlap of lines:
cg = ArrayContraction(ArrayTensorProduct(A, a, C, a, B), (1, 2, 4), (5, 6, 8), (3, 7))
assert cg.split_multiple_contractions() == cg
cg = ArrayContraction(ArrayTensorProduct(a, b, A), (0, 2, 4), (1, 3))
assert cg.split_multiple_contractions() == cg
def test_arrayexpr_nested_permutations():
cg = PermuteDims(PermuteDims(M, (1, 0)), (1, 0))
assert cg == M
times = 3
plist1 = [list(range(6)) for i in range(times)]
plist2 = [list(range(6)) for i in range(times)]
for i in range(times):
random.shuffle(plist1[i])
random.shuffle(plist2[i])
plist1.append([2, 5, 4, 1, 0, 3])
plist2.append([3, 5, 0, 4, 1, 2])
plist1.append([2, 5, 4, 0, 3, 1])
plist2.append([3, 0, 5, 1, 2, 4])
plist1.append([5, 4, 2, 0, 3, 1])
plist2.append([4, 5, 0, 2, 3, 1])
Me = M.subs(k, 3).as_explicit()
Ne = N.subs(k, 3).as_explicit()
Pe = P.subs(k, 3).as_explicit()
cge = tensorproduct(Me, Ne, Pe)
for permutation_array1, permutation_array2 in zip(plist1, plist2):
p1 = Permutation(permutation_array1)
p2 = Permutation(permutation_array2)
cg = PermuteDims(
PermuteDims(
ArrayTensorProduct(M, N, P),
p1),
p2
)
result = PermuteDims(
ArrayTensorProduct(M, N, P),
p2*p1
)
assert cg == result
# Check that `permutedims` behaves the same way with explicit-component arrays:
result1 = permutedims(permutedims(cge, p1), p2)
result2 = permutedims(cge, p2*p1)
assert result1 == result2
def test_arrayexpr_contraction_permutation_mix():
Me = M.subs(k, 3).as_explicit()
Ne = N.subs(k, 3).as_explicit()
cg1 = ArrayContraction(PermuteDims(ArrayTensorProduct(M, N), Permutation([0, 2, 1, 3])), (2, 3))
cg2 = ArrayContraction(ArrayTensorProduct(M, N), (1, 3))
assert cg1 == cg2
cge1 = tensorcontraction(permutedims(tensorproduct(Me, Ne), Permutation([0, 2, 1, 3])), (2, 3))
cge2 = tensorcontraction(tensorproduct(Me, Ne), (1, 3))
assert cge1 == cge2
cg1 = PermuteDims(ArrayTensorProduct(M, N), Permutation([0, 1, 3, 2]))
cg2 = ArrayTensorProduct(M, PermuteDims(N, Permutation([1, 0])))
assert cg1 == cg2
cg1 = ArrayContraction(
PermuteDims(
ArrayTensorProduct(M, N, P, Q), Permutation([0, 2, 3, 1, 4, 5, 7, 6])),
(1, 2), (3, 5)
)
cg2 = ArrayContraction(
ArrayTensorProduct(M, N, P, PermuteDims(Q, Permutation([1, 0]))),
(1, 5), (2, 3)
)
assert cg1 == cg2
cg1 = ArrayContraction(
PermuteDims(
ArrayTensorProduct(M, N, P, Q), Permutation([1, 0, 4, 6, 2, 7, 5, 3])),
(0, 1), (2, 6), (3, 7)
)
cg2 = PermuteDims(
ArrayContraction(
ArrayTensorProduct(M, P, Q, N),
(0, 1), (2, 3), (4, 7)),
[1, 0]
)
assert cg1 == cg2
cg1 = ArrayContraction(
PermuteDims(
ArrayTensorProduct(M, N, P, Q), Permutation([1, 0, 4, 6, 7, 2, 5, 3])),
(0, 1), (2, 6), (3, 7)
)
cg2 = PermuteDims(
ArrayContraction(
ArrayTensorProduct(PermuteDims(M, [1, 0]), N, P, Q),
(0, 1), (3, 6), (4, 5)
),
Permutation([1, 0])
)
assert cg1 == cg2
def test_arrayexpr_permute_tensor_product():
cg1 = PermuteDims(ArrayTensorProduct(M, N, P, Q), Permutation([2, 3, 1, 0, 5, 4, 6, 7]))
cg2 = ArrayTensorProduct(N, PermuteDims(M, [1, 0]),
PermuteDims(P, [1, 0]), Q)
assert cg1 == cg2
# TODO: reverse operation starting with `PermuteDims` and getting down to `bb`...
cg1 = PermuteDims(ArrayTensorProduct(M, N, P, Q), Permutation([2, 3, 4, 5, 0, 1, 6, 7]))
cg2 = ArrayTensorProduct(N, P, M, Q)
assert cg1 == cg2
cg1 = PermuteDims(ArrayTensorProduct(M, N, P, Q), Permutation([2, 3, 4, 6, 5, 7, 0, 1]))
assert cg1.expr == ArrayTensorProduct(N, P, Q, M)
assert cg1.permutation == Permutation([0, 1, 2, 4, 3, 5, 6, 7])
cg1 = ArrayContraction(
PermuteDims(
ArrayTensorProduct(N, Q, Q, M),
[2, 1, 5, 4, 0, 3, 6, 7]),
[1, 2, 6])
cg2 = PermuteDims(ArrayContraction(ArrayTensorProduct(Q, Q, N, M), (3, 5, 6)), [0, 2, 3, 1, 4])
assert cg1 == cg2
cg1 = ArrayContraction(
ArrayContraction(
ArrayContraction(
ArrayContraction(
PermuteDims(
ArrayTensorProduct(N, Q, Q, M),
[2, 1, 5, 4, 0, 3, 6, 7]),
[1, 2, 6]),
[1, 3, 4]),
[1]),
[0])
cg2 = ArrayContraction(ArrayTensorProduct(M, N, Q, Q), (0, 3, 5), (1, 4, 7), (2,), (6,))
assert cg1 == cg2
def test_arrayexpr_normalize_diagonal_permutedims():
tp = ArrayTensorProduct(M, Q, N, P)
expr = ArrayDiagonal(
PermuteDims(tp, [0, 1, 2, 4, 7, 6, 3, 5]), (2, 4, 5), (6, 7),
(0, 3))
result = ArrayDiagonal(tp, (2, 6, 7), (3, 5), (0, 4))
assert expr == result
tp = ArrayTensorProduct(M, N, P, Q)
expr = ArrayDiagonal(PermuteDims(tp, [0, 5, 2, 4, 1, 6, 3, 7]), (1, 2, 6), (3, 4))
result = ArrayDiagonal(ArrayTensorProduct(M, P, N, Q), (3, 4, 5), (1, 2))
assert expr == result
def test_arrayexpr_normalize_diagonal_contraction():
tp = ArrayTensorProduct(M, N, P, Q)
expr = ArrayContraction(ArrayDiagonal(tp, (1, 3, 4)), (0, 3))
result = ArrayDiagonal(ArrayContraction(ArrayTensorProduct(M, N, P, Q), (0, 6)), (0, 2, 3))
assert expr == result
expr = ArrayContraction(ArrayDiagonal(tp, (0, 1, 2, 3, 7)), (1, 2, 3))
result = ArrayContraction(ArrayTensorProduct(M, N, P, Q), (0, 1, 2, 3, 5, 6, 7))
assert expr == result
expr = ArrayContraction(ArrayDiagonal(tp, (0, 2, 6, 7)), (1, 2, 3))
result = ArrayDiagonal(ArrayContraction(tp, (3, 4, 5)), (0, 2, 3, 4))
assert expr == result
td = ArrayDiagonal(ArrayTensorProduct(M, N, P, Q), (0, 3))
expr = ArrayContraction(td, (2, 1), (0, 4, 6, 5, 3))
result = ArrayContraction(ArrayTensorProduct(M, N, P, Q), (0, 1, 3, 5, 6, 7), (2, 4))
assert expr == result
def test_arrayexpr_array_wrong_permutation_size():
cg = ArrayTensorProduct(M, N)
raises(ValueError, lambda: PermuteDims(cg, [1, 0]))
raises(ValueError, lambda: PermuteDims(cg, [1, 0, 2, 3, 5, 4]))
def test_arrayexpr_nested_array_elementwise_add():
cg = ArrayContraction(ArrayAdd(
ArrayTensorProduct(M, N),
ArrayTensorProduct(N, M)
), (1, 2))
result = ArrayAdd(
ArrayContraction(ArrayTensorProduct(M, N), (1, 2)),
ArrayContraction(ArrayTensorProduct(N, M), (1, 2))
)
assert cg == result
cg = ArrayDiagonal(ArrayAdd(
ArrayTensorProduct(M, N),
ArrayTensorProduct(N, M)
), (1, 2))
result = ArrayAdd(
ArrayDiagonal(ArrayTensorProduct(M, N), (1, 2)),
ArrayDiagonal(ArrayTensorProduct(N, M), (1, 2))
)
assert cg == result
def test_arrayexpr_array_expr_zero_array():
za1 = ZeroArray(k, l, m, n)
zm1 = ZeroMatrix(m, n)
za2 = ZeroArray(k, m, m, n)
zm2 = ZeroMatrix(m, m)
zm3 = ZeroMatrix(k, k)
assert ArrayTensorProduct(M, N, za1) == ZeroArray(k, k, k, k, k, l, m, n)
assert ArrayTensorProduct(M, N, zm1) == ZeroArray(k, k, k, k, m, n)
assert ArrayContraction(za1, (3,)) == ZeroArray(k, l, m)
assert ArrayContraction(zm1, (1,)) == ZeroArray(m)
assert ArrayContraction(za2, (1, 2)) == ZeroArray(k, n)
assert ArrayContraction(zm2, (0, 1)) == 0
assert ArrayDiagonal(za2, (1, 2)) == ZeroArray(k, n, m)
assert ArrayDiagonal(zm2, (0, 1)) == ZeroArray(m)
assert PermuteDims(za1, [2, 1, 3, 0]) == ZeroArray(m, l, n, k)
assert PermuteDims(zm1, [1, 0]) == ZeroArray(n, m)
assert ArrayAdd(za1) == za1
assert ArrayAdd(zm1) == ZeroArray(m, n)
tp1 = ArrayTensorProduct(MatrixSymbol("A", k, l), MatrixSymbol("B", m, n))
assert ArrayAdd(tp1, za1) == tp1
tp2 = ArrayTensorProduct(MatrixSymbol("C", k, l), MatrixSymbol("D", m, n))
assert ArrayAdd(tp1, za1, tp2) == ArrayAdd(tp1, tp2)
assert ArrayAdd(M, zm3) == M
assert ArrayAdd(M, N, zm3) == ArrayAdd(M, N)
def test_arrayexpr_array_expr_applyfunc():
A = ArraySymbol("A", 3, k, 2)
aaf = ArrayElementwiseApplyFunc(sin, A)
assert aaf.shape == (3, k, 2)
|
3dfe0ca9efd8f01376d0e84182b10d045aea61f8d70a4b19b183ad8fed115c94 | from sympy import MatrixSymbol, symbols, Identity, sin, cos
from sympy.matrices.expressions.applyfunc import ElementwiseApplyFunction
from sympy.tensor.array.expressions.array_expressions import ArraySymbol, ArrayTensorProduct, \
PermuteDims, ArrayDiagonal, ArrayElementwiseApplyFunc, ArrayContraction
from sympy.tensor.array.expressions.arrayexpr_derivatives import array_derive
k = symbols("k")
I = Identity(k)
X = MatrixSymbol("X", k, k)
x = MatrixSymbol("x", k, 1)
A = MatrixSymbol("A", k, k)
B = MatrixSymbol("B", k, k)
C = MatrixSymbol("C", k, k)
D = MatrixSymbol("D", k, k)
A1 = ArraySymbol("A", 3, 2, k)
def test_arrayexpr_derivatives1():
res = array_derive(X, X)
assert res == PermuteDims(ArrayTensorProduct(I, I), [0, 2, 1, 3])
cg = ArrayTensorProduct(A, X, B)
res = array_derive(cg, X)
assert res == PermuteDims(
ArrayTensorProduct(I, A, I, B),
[0, 4, 2, 3, 1, 5, 6, 7])
cg = ArrayContraction(X, (0, 1))
res = array_derive(cg, X)
assert res == ArrayContraction(ArrayTensorProduct(I, I), (1, 3))
cg = ArrayDiagonal(X, (0, 1))
res = array_derive(cg, X)
assert res == ArrayDiagonal(ArrayTensorProduct(I, I), (1, 3))
cg = ElementwiseApplyFunction(sin, X)
res = array_derive(cg, X)
assert res.dummy_eq(ArrayDiagonal(
ArrayTensorProduct(
ElementwiseApplyFunction(cos, X),
I,
I
), (0, 3), (1, 5)))
cg = ArrayElementwiseApplyFunc(sin, X)
res = array_derive(cg, X)
assert res.dummy_eq(ArrayDiagonal(
ArrayTensorProduct(
I,
I,
ArrayElementwiseApplyFunc(cos, X)
), (1, 4), (3, 5)))
res = array_derive(A1, A1)
assert res == PermuteDims(
ArrayTensorProduct(Identity(3), Identity(2), Identity(k)),
[0, 2, 4, 1, 3, 5]
)
cg = ArrayElementwiseApplyFunc(sin, A1)
res = array_derive(cg, A1)
assert res.dummy_eq(ArrayDiagonal(
ArrayTensorProduct(
Identity(3), Identity(2), Identity(k),
ArrayElementwiseApplyFunc(cos, A1)
), (1, 6), (3, 7), (5, 8)
))
|
3b8bf20b61137bd7686645138f3f181f155e77c453b03af77704d638bb83f3ea | from sympy.assumptions.ask import Q
from sympy.assumptions.refine import refine
from sympy.core.numbers import oo
from sympy.core.relational import Equality, Eq, Ne
from sympy.core.singleton import S
from sympy.core.symbol import (Dummy, symbols)
from sympy.functions import Piecewise
from sympy.functions.elementary.miscellaneous import Max, Min
from sympy.functions.elementary.trigonometric import sin
from sympy.sets.sets import (EmptySet, Interval, Union)
from sympy.simplify.simplify import simplify
from sympy.logic.boolalg import (
And, Boolean, Equivalent, ITE, Implies, Nand, Nor, Not, Or,
POSform, SOPform, Xor, Xnor, conjuncts, disjuncts,
distribute_or_over_and, distribute_and_over_or,
eliminate_implications, is_nnf, is_cnf, is_dnf, simplify_logic,
to_nnf, to_cnf, to_dnf, to_int_repr, bool_map, true, false,
BooleanAtom, is_literal, term_to_integer, integer_to_term,
truth_table, as_Boolean, to_anf, is_anf, distribute_xor_over_and,
anf_coeffs, ANFform, bool_minterm, bool_maxterm, bool_monomial,
_check_pair, _convert_to_varsSOP, _convert_to_varsPOS)
from sympy.assumptions.cnf import CNF
from sympy.testing.pytest import raises, XFAIL, slow
from sympy.utilities.iterables import cartes
from itertools import combinations, permutations
A, B, C, D = symbols('A:D')
a, b, c, d, e, w, x, y, z = symbols('a:e w:z')
def test_overloading():
"""Test that |, & are overloaded as expected"""
assert A & B == And(A, B)
assert A | B == Or(A, B)
assert (A & B) | C == Or(And(A, B), C)
assert A >> B == Implies(A, B)
assert A << B == Implies(B, A)
assert ~A == Not(A)
assert A ^ B == Xor(A, B)
def test_And():
assert And() is true
assert And(A) == A
assert And(True) is true
assert And(False) is false
assert And(True, True) is true
assert And(True, False) is false
assert And(False, False) is false
assert And(True, A) == A
assert And(False, A) is false
assert And(True, True, True) is true
assert And(True, True, A) == A
assert And(True, False, A) is false
assert And(1, A) == A
raises(TypeError, lambda: And(2, A))
raises(TypeError, lambda: And(A < 2, A))
assert And(A < 1, A >= 1) is false
e = A > 1
assert And(e, e.canonical) == e.canonical
g, l, ge, le = A > B, B < A, A >= B, B <= A
assert And(g, l, ge, le) == And(ge, g)
assert {And(*i) for i in permutations((l,g,le,ge))} == {And(ge, g)}
assert And(And(Eq(a, 0), Eq(b, 0)), And(Ne(a, 0), Eq(c, 0))) is false
def test_Or():
assert Or() is false
assert Or(A) == A
assert Or(True) is true
assert Or(False) is false
assert Or(True, True) is true
assert Or(True, False) is true
assert Or(False, False) is false
assert Or(True, A) is true
assert Or(False, A) == A
assert Or(True, False, False) is true
assert Or(True, False, A) is true
assert Or(False, False, A) == A
assert Or(1, A) is true
raises(TypeError, lambda: Or(2, A))
raises(TypeError, lambda: Or(A < 2, A))
assert Or(A < 1, A >= 1) is true
e = A > 1
assert Or(e, e.canonical) == e
g, l, ge, le = A > B, B < A, A >= B, B <= A
assert Or(g, l, ge, le) == Or(g, ge)
def test_Xor():
assert Xor() is false
assert Xor(A) == A
assert Xor(A, A) is false
assert Xor(True, A, A) is true
assert Xor(A, A, A, A, A) == A
assert Xor(True, False, False, A, B) == ~Xor(A, B)
assert Xor(True) is true
assert Xor(False) is false
assert Xor(True, True) is false
assert Xor(True, False) is true
assert Xor(False, False) is false
assert Xor(True, A) == ~A
assert Xor(False, A) == A
assert Xor(True, False, False) is true
assert Xor(True, False, A) == ~A
assert Xor(False, False, A) == A
assert isinstance(Xor(A, B), Xor)
assert Xor(A, B, Xor(C, D)) == Xor(A, B, C, D)
assert Xor(A, B, Xor(B, C)) == Xor(A, C)
assert Xor(A < 1, A >= 1, B) == Xor(0, 1, B) == Xor(1, 0, B)
e = A > 1
assert Xor(e, e.canonical) == Xor(0, 0) == Xor(1, 1)
def test_rewrite_as_And():
expr = x ^ y
assert expr.rewrite(And) == (x | y) & (~x | ~y)
def test_rewrite_as_Or():
expr = x ^ y
assert expr.rewrite(Or) == (x & ~y) | (y & ~x)
def test_rewrite_as_Nand():
expr = (y & z) | (z & ~w)
assert expr.rewrite(Nand) == ~(~(y & z) & ~(z & ~w))
def test_rewrite_as_Nor():
expr = z & (y | ~w)
assert expr.rewrite(Nor) == ~(~z | ~(y | ~w))
def test_Not():
raises(TypeError, lambda: Not(True, False))
assert Not(True) is false
assert Not(False) is true
assert Not(0) is true
assert Not(1) is false
assert Not(2) is false
def test_Nand():
assert Nand() is false
assert Nand(A) == ~A
assert Nand(True) is false
assert Nand(False) is true
assert Nand(True, True) is false
assert Nand(True, False) is true
assert Nand(False, False) is true
assert Nand(True, A) == ~A
assert Nand(False, A) is true
assert Nand(True, True, True) is false
assert Nand(True, True, A) == ~A
assert Nand(True, False, A) is true
def test_Nor():
assert Nor() is true
assert Nor(A) == ~A
assert Nor(True) is false
assert Nor(False) is true
assert Nor(True, True) is false
assert Nor(True, False) is false
assert Nor(False, False) is true
assert Nor(True, A) is false
assert Nor(False, A) == ~A
assert Nor(True, True, True) is false
assert Nor(True, True, A) is false
assert Nor(True, False, A) is false
def test_Xnor():
assert Xnor() is true
assert Xnor(A) == ~A
assert Xnor(A, A) is true
assert Xnor(True, A, A) is false
assert Xnor(A, A, A, A, A) == ~A
assert Xnor(True) is false
assert Xnor(False) is true
assert Xnor(True, True) is true
assert Xnor(True, False) is false
assert Xnor(False, False) is true
assert Xnor(True, A) == A
assert Xnor(False, A) == ~A
assert Xnor(True, False, False) is false
assert Xnor(True, False, A) == A
assert Xnor(False, False, A) == ~A
def test_Implies():
raises(ValueError, lambda: Implies(A, B, C))
assert Implies(True, True) is true
assert Implies(True, False) is false
assert Implies(False, True) is true
assert Implies(False, False) is true
assert Implies(0, A) is true
assert Implies(1, 1) is true
assert Implies(1, 0) is false
assert A >> B == B << A
assert (A < 1) >> (A >= 1) == (A >= 1)
assert (A < 1) >> (S.One > A) is true
assert A >> A is true
def test_Equivalent():
assert Equivalent(A, B) == Equivalent(B, A) == Equivalent(A, B, A)
assert Equivalent() is true
assert Equivalent(A, A) == Equivalent(A) is true
assert Equivalent(True, True) == Equivalent(False, False) is true
assert Equivalent(True, False) == Equivalent(False, True) is false
assert Equivalent(A, True) == A
assert Equivalent(A, False) == Not(A)
assert Equivalent(A, B, True) == A & B
assert Equivalent(A, B, False) == ~A & ~B
assert Equivalent(1, A) == A
assert Equivalent(0, A) == Not(A)
assert Equivalent(A, Equivalent(B, C)) != Equivalent(Equivalent(A, B), C)
assert Equivalent(A < 1, A >= 1) is false
assert Equivalent(A < 1, A >= 1, 0) is false
assert Equivalent(A < 1, A >= 1, 1) is false
assert Equivalent(A < 1, S.One > A) == Equivalent(1, 1) == Equivalent(0, 0)
assert Equivalent(Equality(A, B), Equality(B, A)) is true
def test_equals():
assert Not(Or(A, B)).equals(And(Not(A), Not(B))) is True
assert Equivalent(A, B).equals((A >> B) & (B >> A)) is True
assert ((A | ~B) & (~A | B)).equals((~A & ~B) | (A & B)) is True
assert (A >> B).equals(~A >> ~B) is False
assert (A >> (B >> A)).equals(A >> (C >> A)) is False
raises(NotImplementedError, lambda: (A & B).equals(A > B))
def test_simplification():
"""
Test working of simplification methods.
"""
set1 = [[0, 0, 1], [0, 1, 1], [1, 0, 0], [1, 1, 0]]
set2 = [[0, 0, 0], [0, 1, 0], [1, 0, 1], [1, 1, 1]]
assert SOPform([x, y, z], set1) == Or(And(Not(x), z), And(Not(z), x))
assert Not(SOPform([x, y, z], set2)) == \
Not(Or(And(Not(x), Not(z)), And(x, z)))
assert POSform([x, y, z], set1 + set2) is true
assert SOPform([x, y, z], set1 + set2) is true
assert SOPform([Dummy(), Dummy(), Dummy()], set1 + set2) is true
minterms = [[0, 0, 0, 1], [0, 0, 1, 1], [0, 1, 1, 1], [1, 0, 1, 1],
[1, 1, 1, 1]]
dontcares = [[0, 0, 0, 0], [0, 0, 1, 0], [0, 1, 0, 1]]
assert (
SOPform([w, x, y, z], minterms, dontcares) ==
Or(And(y, z), And(Not(w), Not(x))))
assert POSform([w, x, y, z], minterms, dontcares) == And(Or(Not(w), y), z)
minterms = [1, 3, 7, 11, 15]
dontcares = [0, 2, 5]
assert (
SOPform([w, x, y, z], minterms, dontcares) ==
Or(And(y, z), And(Not(w), Not(x))))
assert POSform([w, x, y, z], minterms, dontcares) == And(Or(Not(w), y), z)
minterms = [1, [0, 0, 1, 1], 7, [1, 0, 1, 1],
[1, 1, 1, 1]]
dontcares = [0, [0, 0, 1, 0], 5]
assert (
SOPform([w, x, y, z], minterms, dontcares) ==
Or(And(y, z), And(Not(w), Not(x))))
assert POSform([w, x, y, z], minterms, dontcares) == And(Or(Not(w), y), z)
minterms = [1, {y: 1, z: 1}]
dontcares = [0, [0, 0, 1, 0], 5]
assert (
SOPform([w, x, y, z], minterms, dontcares) ==
Or(And(y, z), And(Not(w), Not(x))))
assert POSform([w, x, y, z], minterms, dontcares) == And(Or(Not(w), y), z)
minterms = [{y: 1, z: 1}, 1]
dontcares = [[0, 0, 0, 0]]
minterms = [[0, 0, 0]]
raises(ValueError, lambda: SOPform([w, x, y, z], minterms))
raises(ValueError, lambda: POSform([w, x, y, z], minterms))
raises(TypeError, lambda: POSform([w, x, y, z], ["abcdefg"]))
# test simplification
ans = And(A, Or(B, C))
assert simplify_logic(A & (B | C)) == ans
assert simplify_logic((A & B) | (A & C)) == ans
assert simplify_logic(Implies(A, B)) == Or(Not(A), B)
assert simplify_logic(Equivalent(A, B)) == \
Or(And(A, B), And(Not(A), Not(B)))
assert simplify_logic(And(Equality(A, 2), C)) == And(Equality(A, 2), C)
assert simplify_logic(And(Equality(A, 2), A)) is S.false
assert simplify_logic(And(Equality(A, 2), A)) == And(Equality(A, 2), A)
assert simplify_logic(And(Equality(A, B), C)) == And(Equality(A, B), C)
assert simplify_logic(Or(And(Equality(A, 3), B), And(Equality(A, 3), C))) \
== And(Equality(A, 3), Or(B, C))
b = (~x & ~y & ~z) | (~x & ~y & z)
e = And(A, b)
assert simplify_logic(e) == A & ~x & ~y
raises(ValueError, lambda: simplify_logic(A & (B | C), form='blabla'))
# Check that expressions with nine variables or more are not simplified
# (without the force-flag)
a, b, c, d, e, f, g, h, j = symbols('a b c d e f g h j')
expr = a & b & c & d & e & f & g & h & j | \
a & b & c & d & e & f & g & h & ~j
# This expression can be simplified to get rid of the j variables
assert simplify_logic(expr) == expr
# check input
ans = SOPform([x, y], [[1, 0]])
assert SOPform([x, y], [[1, 0]]) == ans
assert POSform([x, y], [[1, 0]]) == ans
raises(ValueError, lambda: SOPform([x], [[1]], [[1]]))
assert SOPform([x], [[1]], [[0]]) is true
assert SOPform([x], [[0]], [[1]]) is true
assert SOPform([x], [], []) is false
raises(ValueError, lambda: POSform([x], [[1]], [[1]]))
assert POSform([x], [[1]], [[0]]) is true
assert POSform([x], [[0]], [[1]]) is true
assert POSform([x], [], []) is false
# check working of simplify
assert simplify((A & B) | (A & C)) == And(A, Or(B, C))
assert simplify(And(x, Not(x))) == False
assert simplify(Or(x, Not(x))) == True
assert simplify(And(Eq(x, 0), Eq(x, y))) == And(Eq(x, 0), Eq(y, 0))
assert And(Eq(x - 1, 0), Eq(x, y)).simplify() == And(Eq(x, 1), Eq(y, 1))
assert And(Ne(x - 1, 0), Ne(x, y)).simplify() == And(Ne(x, 1), Ne(x, y))
assert And(Eq(x - 1, 0), Ne(x, y)).simplify() == And(Eq(x, 1), Ne(y, 1))
assert And(Eq(x - 1, 0), Eq(x, z + y), Eq(y + x, 0)).simplify(
) == And(Eq(x, 1), Eq(y, -1), Eq(z, 2))
assert And(Eq(x - 1, 0), Eq(x + 2, 3)).simplify() == Eq(x, 1)
assert And(Ne(x - 1, 0), Ne(x + 2, 3)).simplify() == Ne(x, 1)
assert And(Eq(x - 1, 0), Eq(x + 2, 2)).simplify() == False
assert And(Ne(x - 1, 0), Ne(x + 2, 2)).simplify(
) == And(Ne(x, 1), Ne(x, 0))
def test_bool_map():
"""
Test working of bool_map function.
"""
minterms = [[0, 0, 0, 1], [0, 0, 1, 1], [0, 1, 1, 1], [1, 0, 1, 1],
[1, 1, 1, 1]]
assert bool_map(Not(Not(a)), a) == (a, {a: a})
assert bool_map(SOPform([w, x, y, z], minterms),
POSform([w, x, y, z], minterms)) == \
(And(Or(Not(w), y), Or(Not(x), y), z), {x: x, w: w, z: z, y: y})
assert bool_map(SOPform([x, z, y], [[1, 0, 1]]),
SOPform([a, b, c], [[1, 0, 1]])) != False
function1 = SOPform([x, z, y], [[1, 0, 1], [0, 0, 1]])
function2 = SOPform([a, b, c], [[1, 0, 1], [1, 0, 0]])
assert bool_map(function1, function2) == \
(function1, {y: a, z: b})
assert bool_map(Xor(x, y), ~Xor(x, y)) == False
assert bool_map(And(x, y), Or(x, y)) is None
assert bool_map(And(x, y), And(x, y, z)) is None
# issue 16179
assert bool_map(Xor(x, y, z), ~Xor(x, y, z)) == False
assert bool_map(Xor(a, x, y, z), ~Xor(a, x, y, z)) == False
def test_bool_symbol():
"""Test that mixing symbols with boolean values
works as expected"""
assert And(A, True) == A
assert And(A, True, True) == A
assert And(A, False) is false
assert And(A, True, False) is false
assert Or(A, True) is true
assert Or(A, False) == A
def test_is_boolean():
assert isinstance(True, Boolean) is False
assert isinstance(true, Boolean) is True
assert 1 == True
assert 1 != true
assert (1 == true) is False
assert 0 == False
assert 0 != false
assert (0 == false) is False
assert true.is_Boolean is True
assert (A & B).is_Boolean
assert (A | B).is_Boolean
assert (~A).is_Boolean
assert (A ^ B).is_Boolean
assert A.is_Boolean != isinstance(A, Boolean)
assert isinstance(A, Boolean)
def test_subs():
assert (A & B).subs(A, True) == B
assert (A & B).subs(A, False) is false
assert (A & B).subs(B, True) == A
assert (A & B).subs(B, False) is false
assert (A & B).subs({A: True, B: True}) is true
assert (A | B).subs(A, True) is true
assert (A | B).subs(A, False) == B
assert (A | B).subs(B, True) is true
assert (A | B).subs(B, False) == A
assert (A | B).subs({A: True, B: True}) is true
"""
we test for axioms of boolean algebra
see https://en.wikipedia.org/wiki/Boolean_algebra_(structure)
"""
def test_commutative():
"""Test for commutativity of And and Or"""
A, B = map(Boolean, symbols('A,B'))
assert A & B == B & A
assert A | B == B | A
def test_and_associativity():
"""Test for associativity of And"""
assert (A & B) & C == A & (B & C)
def test_or_assicativity():
assert ((A | B) | C) == (A | (B | C))
def test_double_negation():
a = Boolean()
assert ~(~a) == a
# test methods
def test_eliminate_implications():
assert eliminate_implications(Implies(A, B, evaluate=False)) == (~A) | B
assert eliminate_implications(
A >> (C >> Not(B))) == Or(Or(Not(B), Not(C)), Not(A))
assert eliminate_implications(Equivalent(A, B, C, D)) == \
(~A | B) & (~B | C) & (~C | D) & (~D | A)
def test_conjuncts():
assert conjuncts(A & B & C) == {A, B, C}
assert conjuncts((A | B) & C) == {A | B, C}
assert conjuncts(A) == {A}
assert conjuncts(True) == {True}
assert conjuncts(False) == {False}
def test_disjuncts():
assert disjuncts(A | B | C) == {A, B, C}
assert disjuncts((A | B) & C) == {(A | B) & C}
assert disjuncts(A) == {A}
assert disjuncts(True) == {True}
assert disjuncts(False) == {False}
def test_distribute():
assert distribute_and_over_or(Or(And(A, B), C)) == And(Or(A, C), Or(B, C))
assert distribute_or_over_and(And(A, Or(B, C))) == Or(And(A, B), And(A, C))
assert distribute_xor_over_and(And(A, Xor(B, C))) == Xor(And(A, B), And(A, C))
def test_to_anf():
x, y, z = symbols('x,y,z')
assert to_anf(And(x, y)) == And(x, y)
assert to_anf(Or(x, y)) == Xor(x, y, And(x, y))
assert to_anf(Or(Implies(x, y), And(x, y), y)) == \
Xor(x, True, x & y, remove_true=False)
assert to_anf(Or(Nand(x, y), Nor(x, y), Xnor(x, y), Implies(x, y))) == True
assert to_anf(Or(x, Not(y), Nor(x,z), And(x, y), Nand(y, z))) == \
Xor(True, And(y, z), And(x, y, z), remove_true=False)
assert to_anf(Xor(x, y)) == Xor(x, y)
assert to_anf(Not(x)) == Xor(x, True, remove_true=False)
assert to_anf(Nand(x, y)) == Xor(True, And(x, y), remove_true=False)
assert to_anf(Nor(x, y)) == Xor(x, y, True, And(x, y), remove_true=False)
assert to_anf(Implies(x, y)) == Xor(x, True, And(x, y), remove_true=False)
assert to_anf(Equivalent(x, y)) == Xor(x, y, True, remove_true=False)
assert to_anf(Nand(x | y, x >> y), deep=False) == \
Xor(True, And(Or(x, y), Implies(x, y)), remove_true=False)
assert to_anf(Nor(x ^ y, x & y), deep=False) == \
Xor(True, Or(Xor(x, y), And(x, y)), remove_true=False)
def test_to_nnf():
assert to_nnf(true) is true
assert to_nnf(false) is false
assert to_nnf(A) == A
assert to_nnf(A | ~A | B) is true
assert to_nnf(A & ~A & B) is false
assert to_nnf(A >> B) == ~A | B
assert to_nnf(Equivalent(A, B, C)) == (~A | B) & (~B | C) & (~C | A)
assert to_nnf(A ^ B ^ C) == \
(A | B | C) & (~A | ~B | C) & (A | ~B | ~C) & (~A | B | ~C)
assert to_nnf(ITE(A, B, C)) == (~A | B) & (A | C)
assert to_nnf(Not(A | B | C)) == ~A & ~B & ~C
assert to_nnf(Not(A & B & C)) == ~A | ~B | ~C
assert to_nnf(Not(A >> B)) == A & ~B
assert to_nnf(Not(Equivalent(A, B, C))) == And(Or(A, B, C), Or(~A, ~B, ~C))
assert to_nnf(Not(A ^ B ^ C)) == \
(~A | B | C) & (A | ~B | C) & (A | B | ~C) & (~A | ~B | ~C)
assert to_nnf(Not(ITE(A, B, C))) == (~A | ~B) & (A | ~C)
assert to_nnf((A >> B) ^ (B >> A)) == (A & ~B) | (~A & B)
assert to_nnf((A >> B) ^ (B >> A), False) == \
(~A | ~B | A | B) & ((A & ~B) | (~A & B))
assert ITE(A, 1, 0).to_nnf() == A
assert ITE(A, 0, 1).to_nnf() == ~A
# although ITE can hold non-Boolean, it will complain if
# an attempt is made to convert the ITE to Boolean nnf
raises(TypeError, lambda: ITE(A < 1, [1], B).to_nnf())
def test_to_cnf():
assert to_cnf(~(B | C)) == And(Not(B), Not(C))
assert to_cnf((A & B) | C) == And(Or(A, C), Or(B, C))
assert to_cnf(A >> B) == (~A) | B
assert to_cnf(A >> (B & C)) == (~A | B) & (~A | C)
assert to_cnf(A & (B | C) | ~A & (B | C), True) == B | C
assert to_cnf(A & B) == And(A, B)
assert to_cnf(Equivalent(A, B)) == And(Or(A, Not(B)), Or(B, Not(A)))
assert to_cnf(Equivalent(A, B & C)) == \
(~A | B) & (~A | C) & (~B | ~C | A)
assert to_cnf(Equivalent(A, B | C), True) == \
And(Or(Not(B), A), Or(Not(C), A), Or(B, C, Not(A)))
assert to_cnf(A + 1) == A + 1
def test_issue_18904():
x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15 = symbols('x1:16')
eq = (( x1 & x2 & x3 & x4 & x5 & x6 & x7 & x8 & x9 ) |
( x1 & x2 & x3 & x4 & x5 & x6 & x7 & x10 & x9 ) |
( x1 & x11 & x3 & x12 & x5 & x13 & x14 & x15 & x9 ))
assert is_cnf(to_cnf(eq))
raises(ValueError, lambda: to_cnf(eq, simplify=True))
for f, t in zip((And, Or), (to_cnf, to_dnf)):
eq = f(x1, x2, x3, x4, x5, x6, x7, x8, x9)
raises(ValueError, lambda: to_cnf(eq, simplify=True))
assert t(eq, simplify=True, force=True) == eq
def test_issue_9949():
assert is_cnf(to_cnf((b > -5) | (a > 2) & (a < 4)))
def test_to_CNF():
assert CNF.CNF_to_cnf(CNF.to_CNF(~(B | C))) == to_cnf(~(B | C))
assert CNF.CNF_to_cnf(CNF.to_CNF((A & B) | C)) == to_cnf((A & B) | C)
assert CNF.CNF_to_cnf(CNF.to_CNF(A >> B)) == to_cnf(A >> B)
assert CNF.CNF_to_cnf(CNF.to_CNF(A >> (B & C))) == to_cnf(A >> (B & C))
assert CNF.CNF_to_cnf(CNF.to_CNF(A & (B | C) | ~A & (B | C))) == to_cnf(A & (B | C) | ~A & (B | C))
assert CNF.CNF_to_cnf(CNF.to_CNF(A & B)) == to_cnf(A & B)
def test_to_dnf():
assert to_dnf(~(B | C)) == And(Not(B), Not(C))
assert to_dnf(A & (B | C)) == Or(And(A, B), And(A, C))
assert to_dnf(A >> B) == (~A) | B
assert to_dnf(A >> (B & C)) == (~A) | (B & C)
assert to_dnf(A | B) == A | B
assert to_dnf(Equivalent(A, B), True) == \
Or(And(A, B), And(Not(A), Not(B)))
assert to_dnf(Equivalent(A, B & C), True) == \
Or(And(A, B, C), And(Not(A), Not(B)), And(Not(A), Not(C)))
assert to_dnf(A + 1) == A + 1
def test_to_int_repr():
x, y, z = map(Boolean, symbols('x,y,z'))
def sorted_recursive(arg):
try:
return sorted(sorted_recursive(x) for x in arg)
except TypeError: # arg is not a sequence
return arg
assert sorted_recursive(to_int_repr([x | y, z | x], [x, y, z])) == \
sorted_recursive([[1, 2], [1, 3]])
assert sorted_recursive(to_int_repr([x | y, z | ~x], [x, y, z])) == \
sorted_recursive([[1, 2], [3, -1]])
def test_is_anf():
x, y = symbols('x,y')
assert is_anf(true) is True
assert is_anf(false) is True
assert is_anf(x) is True
assert is_anf(And(x, y)) is True
assert is_anf(Xor(x, y, And(x, y))) is True
assert is_anf(Xor(x, y, Or(x, y))) is False
assert is_anf(Xor(Not(x), y)) is False
def test_is_nnf():
assert is_nnf(true) is True
assert is_nnf(A) is True
assert is_nnf(~A) is True
assert is_nnf(A & B) is True
assert is_nnf((A & B) | (~A & A) | (~B & B) | (~A & ~B), False) is True
assert is_nnf((A | B) & (~A | ~B)) is True
assert is_nnf(Not(Or(A, B))) is False
assert is_nnf(A ^ B) is False
assert is_nnf((A & B) | (~A & A) | (~B & B) | (~A & ~B), True) is False
def test_is_cnf():
assert is_cnf(x) is True
assert is_cnf(x | y | z) is True
assert is_cnf(x & y & z) is True
assert is_cnf((x | y) & z) is True
assert is_cnf((x & y) | z) is False
assert is_cnf(~(x & y) | z) is False
def test_is_dnf():
assert is_dnf(x) is True
assert is_dnf(x | y | z) is True
assert is_dnf(x & y & z) is True
assert is_dnf((x & y) | z) is True
assert is_dnf((x | y) & z) is False
assert is_dnf(~(x | y) & z) is False
def test_ITE():
A, B, C = symbols('A:C')
assert ITE(True, False, True) is false
assert ITE(True, True, False) is true
assert ITE(False, True, False) is false
assert ITE(False, False, True) is true
assert isinstance(ITE(A, B, C), ITE)
A = True
assert ITE(A, B, C) == B
A = False
assert ITE(A, B, C) == C
B = True
assert ITE(And(A, B), B, C) == C
assert ITE(Or(A, False), And(B, True), False) is false
assert ITE(x, A, B) == Not(x)
assert ITE(x, B, A) == x
assert ITE(1, x, y) == x
assert ITE(0, x, y) == y
raises(TypeError, lambda: ITE(2, x, y))
raises(TypeError, lambda: ITE(1, [], y))
raises(TypeError, lambda: ITE(1, (), y))
raises(TypeError, lambda: ITE(1, y, []))
assert ITE(1, 1, 1) is S.true
assert isinstance(ITE(1, 1, 1, evaluate=False), ITE)
raises(TypeError, lambda: ITE(x > 1, y, x))
assert ITE(Eq(x, True), y, x) == ITE(x, y, x)
assert ITE(Eq(x, False), y, x) == ITE(~x, y, x)
assert ITE(Ne(x, True), y, x) == ITE(~x, y, x)
assert ITE(Ne(x, False), y, x) == ITE(x, y, x)
assert ITE(Eq(S. true, x), y, x) == ITE(x, y, x)
assert ITE(Eq(S.false, x), y, x) == ITE(~x, y, x)
assert ITE(Ne(S.true, x), y, x) == ITE(~x, y, x)
assert ITE(Ne(S.false, x), y, x) == ITE(x, y, x)
# 0 and 1 in the context are not treated as True/False
# so the equality must always be False since dissimilar
# objects cannot be equal
assert ITE(Eq(x, 0), y, x) == x
assert ITE(Eq(x, 1), y, x) == x
assert ITE(Ne(x, 0), y, x) == y
assert ITE(Ne(x, 1), y, x) == y
assert ITE(Eq(x, 0), y, z).subs(x, 0) == y
assert ITE(Eq(x, 0), y, z).subs(x, 1) == z
raises(ValueError, lambda: ITE(x > 1, y, x, z))
def test_is_literal():
assert is_literal(True) is True
assert is_literal(False) is True
assert is_literal(A) is True
assert is_literal(~A) is True
assert is_literal(Or(A, B)) is False
assert is_literal(Q.zero(A)) is True
assert is_literal(Not(Q.zero(A))) is True
assert is_literal(Or(A, B)) is False
assert is_literal(And(Q.zero(A), Q.zero(B))) is False
assert is_literal(x < 3)
assert not is_literal(x + y < 3)
def test_operators():
# Mostly test __and__, __rand__, and so on
assert True & A == A & True == A
assert False & A == A & False == False
assert A & B == And(A, B)
assert True | A == A | True == True
assert False | A == A | False == A
assert A | B == Or(A, B)
assert ~A == Not(A)
assert True >> A == A << True == A
assert False >> A == A << False == True
assert A >> True == True << A == True
assert A >> False == False << A == ~A
assert A >> B == B << A == Implies(A, B)
assert True ^ A == A ^ True == ~A
assert False ^ A == A ^ False == A
assert A ^ B == Xor(A, B)
def test_true_false():
assert true is S.true
assert false is S.false
assert true is not True
assert false is not False
assert true
assert not false
assert true == True
assert false == False
assert not (true == False)
assert not (false == True)
assert not (true == false)
assert hash(true) == hash(True)
assert hash(false) == hash(False)
assert len({true, True}) == len({false, False}) == 1
assert isinstance(true, BooleanAtom)
assert isinstance(false, BooleanAtom)
# We don't want to subclass from bool, because bool subclasses from
# int. But operators like &, |, ^, <<, >>, and ~ act differently on 0 and
# 1 then we want them to on true and false. See the docstrings of the
# various And, Or, etc. functions for examples.
assert not isinstance(true, bool)
assert not isinstance(false, bool)
# Note: using 'is' comparison is important here. We want these to return
# true and false, not True and False
assert Not(true) is false
assert Not(True) is false
assert Not(false) is true
assert Not(False) is true
assert ~true is false
assert ~false is true
for T, F in cartes([True, true], [False, false]):
assert And(T, F) is false
assert And(F, T) is false
assert And(F, F) is false
assert And(T, T) is true
assert And(T, x) == x
assert And(F, x) is false
if not (T is True and F is False):
assert T & F is false
assert F & T is false
if F is not False:
assert F & F is false
if T is not True:
assert T & T is true
assert Or(T, F) is true
assert Or(F, T) is true
assert Or(F, F) is false
assert Or(T, T) is true
assert Or(T, x) is true
assert Or(F, x) == x
if not (T is True and F is False):
assert T | F is true
assert F | T is true
if F is not False:
assert F | F is false
if T is not True:
assert T | T is true
assert Xor(T, F) is true
assert Xor(F, T) is true
assert Xor(F, F) is false
assert Xor(T, T) is false
assert Xor(T, x) == ~x
assert Xor(F, x) == x
if not (T is True and F is False):
assert T ^ F is true
assert F ^ T is true
if F is not False:
assert F ^ F is false
if T is not True:
assert T ^ T is false
assert Nand(T, F) is true
assert Nand(F, T) is true
assert Nand(F, F) is true
assert Nand(T, T) is false
assert Nand(T, x) == ~x
assert Nand(F, x) is true
assert Nor(T, F) is false
assert Nor(F, T) is false
assert Nor(F, F) is true
assert Nor(T, T) is false
assert Nor(T, x) is false
assert Nor(F, x) == ~x
assert Implies(T, F) is false
assert Implies(F, T) is true
assert Implies(F, F) is true
assert Implies(T, T) is true
assert Implies(T, x) == x
assert Implies(F, x) is true
assert Implies(x, T) is true
assert Implies(x, F) == ~x
if not (T is True and F is False):
assert T >> F is false
assert F << T is false
assert F >> T is true
assert T << F is true
if F is not False:
assert F >> F is true
assert F << F is true
if T is not True:
assert T >> T is true
assert T << T is true
assert Equivalent(T, F) is false
assert Equivalent(F, T) is false
assert Equivalent(F, F) is true
assert Equivalent(T, T) is true
assert Equivalent(T, x) == x
assert Equivalent(F, x) == ~x
assert Equivalent(x, T) == x
assert Equivalent(x, F) == ~x
assert ITE(T, T, T) is true
assert ITE(T, T, F) is true
assert ITE(T, F, T) is false
assert ITE(T, F, F) is false
assert ITE(F, T, T) is true
assert ITE(F, T, F) is false
assert ITE(F, F, T) is true
assert ITE(F, F, F) is false
assert all(i.simplify(1, 2) is i for i in (S.true, S.false))
def test_bool_as_set():
assert ITE(y <= 0, False, y >= 1).as_set() == Interval(1, oo)
assert And(x <= 2, x >= -2).as_set() == Interval(-2, 2)
assert Or(x >= 2, x <= -2).as_set() == Interval(-oo, -2) + Interval(2, oo)
assert Not(x > 2).as_set() == Interval(-oo, 2)
# issue 10240
assert Not(And(x > 2, x < 3)).as_set() == \
Union(Interval(-oo, 2), Interval(3, oo))
assert true.as_set() == S.UniversalSet
assert false.as_set() == EmptySet()
assert x.as_set() == S.UniversalSet
assert And(Or(x < 1, x > 3), x < 2).as_set() == Interval.open(-oo, 1)
assert And(x < 1, sin(x) < 3).as_set() == (x < 1).as_set()
raises(NotImplementedError, lambda: (sin(x) < 1).as_set())
@XFAIL
def test_multivariate_bool_as_set():
x, y = symbols('x,y')
assert And(x >= 0, y >= 0).as_set() == Interval(0, oo)*Interval(0, oo)
assert Or(x >= 0, y >= 0).as_set() == S.Reals*S.Reals - \
Interval(-oo, 0, True, True)*Interval(-oo, 0, True, True)
def test_all_or_nothing():
x = symbols('x', extended_real=True)
args = x >= -oo, x <= oo
v = And(*args)
if v.func is And:
assert len(v.args) == len(args) - args.count(S.true)
else:
assert v == True
v = Or(*args)
if v.func is Or:
assert len(v.args) == 2
else:
assert v == True
def test_canonical_atoms():
assert true.canonical == true
assert false.canonical == false
def test_negated_atoms():
assert true.negated == false
assert false.negated == true
def test_issue_8777():
assert And(x > 2, x < oo).as_set() == Interval(2, oo, left_open=True)
assert And(x >= 1, x < oo).as_set() == Interval(1, oo)
assert (x < oo).as_set() == Interval(-oo, oo)
assert (x > -oo).as_set() == Interval(-oo, oo)
def test_issue_8975():
assert Or(And(-oo < x, x <= -2), And(2 <= x, x < oo)).as_set() == \
Interval(-oo, -2) + Interval(2, oo)
def test_term_to_integer():
assert term_to_integer([1, 0, 1, 0, 0, 1, 0]) == 82
assert term_to_integer('0010101000111001') == 10809
def test_integer_to_term():
assert integer_to_term(777) == [1, 1, 0, 0, 0, 0, 1, 0, 0, 1]
assert integer_to_term(123, 3) == [1, 1, 1, 1, 0, 1, 1]
assert integer_to_term(456, 16) == [0, 0, 0, 0, 0, 0, 0, 1,
1, 1, 0, 0, 1, 0, 0, 0]
def test_truth_table():
assert list(truth_table(And(x, y), [x, y], input=False)) == \
[False, False, False, True]
assert list(truth_table(x | y, [x, y], input=False)) == \
[False, True, True, True]
assert list(truth_table(x >> y, [x, y], input=False)) == \
[True, True, False, True]
assert list(truth_table(And(x, y), [x, y])) == \
[([0, 0], False), ([0, 1], False), ([1, 0], False), ([1, 1], True)]
def test_issue_8571():
for t in (S.true, S.false):
raises(TypeError, lambda: +t)
raises(TypeError, lambda: -t)
raises(TypeError, lambda: abs(t))
# use int(bool(t)) to get 0 or 1
raises(TypeError, lambda: int(t))
for o in [S.Zero, S.One, x]:
for _ in range(2):
raises(TypeError, lambda: o + t)
raises(TypeError, lambda: o - t)
raises(TypeError, lambda: o % t)
raises(TypeError, lambda: o*t)
raises(TypeError, lambda: o/t)
raises(TypeError, lambda: o**t)
o, t = t, o # do again in reversed order
def test_expand_relational():
n = symbols('n', negative=True)
p, q = symbols('p q', positive=True)
r = ((n + q*(-n/q + 1))/(q*(-n/q + 1)) < 0)
assert r is not S.false
assert r.expand() is S.false
assert (q > 0).expand() is S.true
def test_issue_12717():
assert S.true.is_Atom == True
assert S.false.is_Atom == True
def test_as_Boolean():
nz = symbols('nz', nonzero=True)
assert all(as_Boolean(i) is S.true for i in (True, S.true, 1, nz))
z = symbols('z', zero=True)
assert all(as_Boolean(i) is S.false for i in (False, S.false, 0, z))
assert all(as_Boolean(i) == i for i in (x, x < 0))
for i in (2, S(2), x + 1, []):
raises(TypeError, lambda: as_Boolean(i))
def test_binary_symbols():
assert ITE(x < 1, y, z).binary_symbols == {y, z}
for f in (Eq, Ne):
assert f(x, 1).binary_symbols == set()
assert f(x, True).binary_symbols == {x}
assert f(x, False).binary_symbols == {x}
assert S.true.binary_symbols == set()
assert S.false.binary_symbols == set()
assert x.binary_symbols == {x}
assert And(x, Eq(y, False), Eq(z, 1)).binary_symbols == {x, y}
assert Q.prime(x).binary_symbols == set()
assert Q.lt(x, 1).binary_symbols == set()
assert Q.is_true(x).binary_symbols == {x}
assert Q.eq(x, True).binary_symbols == {x}
assert Q.prime(x).binary_symbols == set()
def test_BooleanFunction_diff():
assert And(x, y).diff(x) == Piecewise((0, Eq(y, False)), (1, True))
def test_issue_14700():
A, B, C, D, E, F, G, H = symbols('A B C D E F G H')
q = ((B & D & H & ~F) | (B & H & ~C & ~D) | (B & H & ~C & ~F) |
(B & H & ~D & ~G) | (B & H & ~F & ~G) | (C & G & ~B & ~D) |
(C & G & ~D & ~H) | (C & G & ~F & ~H) | (D & F & H & ~B) |
(D & F & ~G & ~H) | (B & D & F & ~C & ~H) | (D & E & F & ~B & ~C) |
(D & F & ~A & ~B & ~C) | (D & F & ~A & ~C & ~H) |
(A & B & D & F & ~E & ~H))
soldnf = ((B & D & H & ~F) | (D & F & H & ~B) | (B & H & ~C & ~D) |
(B & H & ~D & ~G) | (C & G & ~B & ~D) | (C & G & ~D & ~H) |
(C & G & ~F & ~H) | (D & F & ~G & ~H) | (D & E & F & ~C & ~H) |
(D & F & ~A & ~C & ~H) | (A & B & D & F & ~E & ~H))
solcnf = ((B | C | D) & (B | D | G) & (C | D | H) & (C | F | H) &
(D | G | H) & (F | G | H) & (B | F | ~D | ~H) &
(~B | ~D | ~F | ~H) & (D | ~B | ~C | ~G | ~H) &
(A | H | ~C | ~D | ~F | ~G) & (H | ~C | ~D | ~E | ~F | ~G) &
(B | E | H | ~A | ~D | ~F | ~G))
assert simplify_logic(q, "dnf") == soldnf
assert simplify_logic(q, "cnf") == solcnf
minterms = [[0, 1, 0, 0], [0, 1, 0, 1], [0, 1, 1, 0], [0, 1, 1, 1],
[0, 0, 1, 1], [1, 0, 1, 1]]
dontcares = [[1, 0, 0, 0], [1, 0, 0, 1], [1, 1, 0, 0], [1, 1, 0, 1]]
assert SOPform([w, x, y, z], minterms) == (x & ~w) | (y & z & ~x)
# Should not be more complicated with don't cares
assert SOPform([w, x, y, z], minterms, dontcares) == \
(x & ~w) | (y & z & ~x)
def test_relational_simplification():
w, x, y, z = symbols('w x y z', real=True)
d, e = symbols('d e', real=False)
# Test all combinations or sign and order
assert Or(x >= y, x < y).simplify() == S.true
assert Or(x >= y, y > x).simplify() == S.true
assert Or(x >= y, -x > -y).simplify() == S.true
assert Or(x >= y, -y < -x).simplify() == S.true
assert Or(-x <= -y, x < y).simplify() == S.true
assert Or(-x <= -y, -x > -y).simplify() == S.true
assert Or(-x <= -y, y > x).simplify() == S.true
assert Or(-x <= -y, -y < -x).simplify() == S.true
assert Or(y <= x, x < y).simplify() == S.true
assert Or(y <= x, y > x).simplify() == S.true
assert Or(y <= x, -x > -y).simplify() == S.true
assert Or(y <= x, -y < -x).simplify() == S.true
assert Or(-y >= -x, x < y).simplify() == S.true
assert Or(-y >= -x, y > x).simplify() == S.true
assert Or(-y >= -x, -x > -y).simplify() == S.true
assert Or(-y >= -x, -y < -x).simplify() == S.true
assert Or(x < y, x >= y).simplify() == S.true
assert Or(y > x, x >= y).simplify() == S.true
assert Or(-x > -y, x >= y).simplify() == S.true
assert Or(-y < -x, x >= y).simplify() == S.true
assert Or(x < y, -x <= -y).simplify() == S.true
assert Or(-x > -y, -x <= -y).simplify() == S.true
assert Or(y > x, -x <= -y).simplify() == S.true
assert Or(-y < -x, -x <= -y).simplify() == S.true
assert Or(x < y, y <= x).simplify() == S.true
assert Or(y > x, y <= x).simplify() == S.true
assert Or(-x > -y, y <= x).simplify() == S.true
assert Or(-y < -x, y <= x).simplify() == S.true
assert Or(x < y, -y >= -x).simplify() == S.true
assert Or(y > x, -y >= -x).simplify() == S.true
assert Or(-x > -y, -y >= -x).simplify() == S.true
assert Or(-y < -x, -y >= -x).simplify() == S.true
# Some other tests
assert Or(x >= y, w < z, x <= y).simplify() == S.true
assert And(x >= y, x < y).simplify() == S.false
assert Or(x >= y, Eq(y, x)).simplify() == (x >= y)
assert And(x >= y, Eq(y, x)).simplify() == Eq(x, y)
assert Or(Eq(x, y), x >= y, w < y, z < y).simplify() == \
Or(x >= y, y > Min(w, z))
assert And(Eq(x, y), x >= y, w < y, y >= z, z < y).simplify() == \
And(Eq(x, y), y > Max(w, z))
assert Or(Eq(x, y), x >= 1, 2 < y, y >= 5, z < y).simplify() == \
(Eq(x, y) | (x >= 1) | (y > Min(2, z)))
assert And(Eq(x, y), x >= 1, 2 < y, y >= 5, z < y).simplify() == \
(Eq(x, y) & (x >= 1) & (y >= 5) & (y > z))
assert (Eq(x, y) & Eq(d, e) & (x >= y) & (d >= e)).simplify() == \
(Eq(x, y) & Eq(d, e) & (d >= e))
assert And(Eq(x, y), Eq(x, -y)).simplify() == And(Eq(x, 0), Eq(y, 0))
assert Xor(x >= y, x <= y).simplify() == Ne(x, y)
@slow
def test_relational_simplification_numerically():
def test_simplification_numerically_function(original, simplified):
symb = original.free_symbols
n = len(symb)
valuelist = list(set(list(combinations(list(range(-(n-1), n))*n, n))))
for values in valuelist:
sublist = dict(zip(symb, values))
originalvalue = original.subs(sublist)
simplifiedvalue = simplified.subs(sublist)
assert originalvalue == simplifiedvalue, "Original: {}\nand"\
" simplified: {}\ndo not evaluate to the same value for {}"\
"".format(original, simplified, sublist)
w, x, y, z = symbols('w x y z', real=True)
d, e = symbols('d e', real=False)
expressions = (And(Eq(x, y), x >= y, w < y, y >= z, z < y),
And(Eq(x, y), x >= 1, 2 < y, y >= 5, z < y),
Or(Eq(x, y), x >= 1, 2 < y, y >= 5, z < y),
And(x >= y, Eq(y, x)),
Or(And(Eq(x, y), x >= y, w < y, Or(y >= z, z < y)),
And(Eq(x, y), x >= 1, 2 < y, y >= -1, z < y)),
(Eq(x, y) & Eq(d, e) & (x >= y) & (d >= e)),
)
for expression in expressions:
test_simplification_numerically_function(expression,
expression.simplify())
def test_relational_simplification_patterns_numerically():
from sympy.core import Wild
from sympy.logic.boolalg import simplify_patterns_and, \
simplify_patterns_or, simplify_patterns_xor
a = Wild('a')
b = Wild('b')
c = Wild('c')
symb = [a, b, c]
patternlists = [simplify_patterns_and(), simplify_patterns_or(),
simplify_patterns_xor()]
for patternlist in patternlists:
for pattern in patternlist:
original = pattern[0]
simplified = pattern[1]
valuelist = list(set(list(combinations(list(range(-2, 2))*3, 3))))
for values in valuelist:
sublist = dict(zip(symb, values))
originalvalue = original.subs(sublist)
simplifiedvalue = simplified.subs(sublist)
assert originalvalue == simplifiedvalue, "Original: {}\nand"\
" simplified: {}\ndo not evaluate to the same value for"\
"{}".format(original, simplified, sublist)
def test_issue_16803():
n = symbols('n')
# No simplification done, but should not raise an exception
assert ((n > 3) | (n < 0) | ((n > 0) & (n < 3))).simplify() == \
((n > 3) | (n < 0) | ((n > 0) & (n < 3)))
def test_issue_17530():
r = {x: oo, y: oo}
assert Or(x + y > 0, x - y < 0).subs(r)
assert not And(x + y < 0, x - y < 0).subs(r)
raises(TypeError, lambda: Or(x + y < 0, x - y < 0).subs(r))
raises(TypeError, lambda: And(x + y > 0, x - y < 0).subs(r))
raises(TypeError, lambda: And(x + y > 0, x - y < 0).subs(r))
def test_anf_coeffs():
assert anf_coeffs([1, 0]) == [1, 1]
assert anf_coeffs([0, 0, 0, 1]) == [0, 0, 0, 1]
assert anf_coeffs([0, 1, 1, 1]) == [0, 1, 1, 1]
assert anf_coeffs([1, 1, 1, 0]) == [1, 0, 0, 1]
assert anf_coeffs([1, 0, 0, 0]) == [1, 1, 1, 1]
assert anf_coeffs([1, 0, 0, 1]) == [1, 1, 1, 0]
assert anf_coeffs([1, 1, 0, 1]) == [1, 0, 1, 1]
def test_ANFform():
x, y = symbols('x,y')
assert ANFform([x], [1, 1]) == True
assert ANFform([x], [0, 0]) == False
assert ANFform([x], [1, 0]) == Xor(x, True, remove_true=False)
assert ANFform([x, y], [1, 1, 1, 0]) == \
Xor(True, And(x, y), remove_true=False)
def test_bool_minterm():
x, y = symbols('x,y')
assert bool_minterm(3, [x, y]) == And(x, y)
assert bool_minterm([1, 0], [x, y]) == And(Not(y), x)
def test_bool_maxterm():
x, y = symbols('x,y')
assert bool_maxterm(2, [x, y]) == Or(Not(x), y)
assert bool_maxterm([0, 1], [x, y]) == Or(Not(y), x)
def test_bool_monomial():
x, y = symbols('x,y')
assert bool_monomial(1, [x, y]) == y
assert bool_monomial([1, 1], [x, y]) == And(x, y)
def test_check_pair():
assert _check_pair([0, 1, 0], [0, 1, 1]) == 2
assert _check_pair([0, 1, 0], [1, 1, 1]) == -1
def test_convert_to_varsSOP():
assert _convert_to_varsSOP([0, 1, 0], [x, y, z]) == And(Not(x), y, Not(z))
assert _convert_to_varsSOP([3, 1, 0], [x, y, z]) == And(y, Not(z))
def test_convert_to_varsPOS():
assert _convert_to_varsPOS([0, 1, 0], [x, y, z]) == Or(x, Not(y), z)
assert _convert_to_varsPOS([3, 1, 0], [x, y, z]) == Or(Not(y), z)
def test_refine():
# relational
assert not refine(x < 0, ~(x < 0))
assert refine(x < 0, (x < 0))
assert refine(x < 0, (0 > x)) is S.true
assert refine(x < 0, (y < 0)) == (x < 0)
assert not refine(x <= 0, ~(x <= 0))
assert refine(x <= 0, (x <= 0))
assert refine(x <= 0, (0 >= x)) is S.true
assert refine(x <= 0, (y <= 0)) == (x <= 0)
assert not refine(x > 0, ~(x > 0))
assert refine(x > 0, (x > 0))
assert refine(x > 0, (0 < x)) is S.true
assert refine(x > 0, (y > 0)) == (x > 0)
assert not refine(x >= 0, ~(x >= 0))
assert refine(x >= 0, (x >= 0))
assert refine(x >= 0, (0 <= x)) is S.true
assert refine(x >= 0, (y >= 0)) == (x >= 0)
assert not refine(Eq(x, 0), ~(Eq(x, 0)))
assert refine(Eq(x, 0), (Eq(x, 0)))
assert refine(Eq(x, 0), (Eq(0, x))) is S.true
assert refine(Eq(x, 0), (Eq(y, 0))) == Eq(x, 0)
assert not refine(Ne(x, 0), ~(Ne(x, 0)))
assert refine(Ne(x, 0), (Ne(0, x))) is S.true
assert refine(Ne(x, 0), (Ne(x, 0)))
assert refine(Ne(x, 0), (Ne(y, 0))) == (Ne(x, 0))
# boolean functions
assert refine(And(x > 0, y > 0), (x > 0)) == (y > 0)
assert refine(And(x > 0, y > 0), (x > 0) & (y > 0)) is S.true
# predicates
assert refine(Q.positive(x), Q.positive(x)) is S.true
assert refine(Q.positive(x), Q.negative(x)) is S.false
assert refine(Q.positive(x), Q.real(x)) == Q.positive(x)
|
722e58875b9ad2cb6959d1b7e77aa19a877b18a90f4e486ac6f140949892e789 | from sympy.assumptions import Q
from sympy.core.expr import Expr
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 import Array
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(IndexError, lambda: e.row_del(5))
raises(IndexError, lambda: e.row_del(-5))
raises(IndexError, lambda: e.col_del(5))
raises(IndexError, 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]])
def test_upper_triangular():
A = OperationsOnlyMatrix([
[1, 1, 1, 1],
[1, 1, 1, 1],
[1, 1, 1, 1],
[1, 1, 1, 1]
])
R = A.upper_triangular(2)
assert R == OperationsOnlyMatrix([
[0, 0, 1, 1],
[0, 0, 0, 1],
[0, 0, 0, 0],
[0, 0, 0, 0]
])
R = A.upper_triangular(-2)
assert R == OperationsOnlyMatrix([
[1, 1, 1, 1],
[1, 1, 1, 1],
[1, 1, 1, 1],
[0, 1, 1, 1]
])
R = A.upper_triangular()
assert R == OperationsOnlyMatrix([
[1, 1, 1, 1],
[0, 1, 1, 1],
[0, 0, 1, 1],
[0, 0, 0, 1]
])
def test_lower_triangular():
A = OperationsOnlyMatrix([
[1, 1, 1, 1],
[1, 1, 1, 1],
[1, 1, 1, 1],
[1, 1, 1, 1]
])
L = A.lower_triangular()
assert L == ArithmeticOnlyMatrix([
[1, 0, 0, 0],
[1, 1, 0, 0],
[1, 1, 1, 0],
[1, 1, 1, 1]])
L = A.lower_triangular(2)
assert L == ArithmeticOnlyMatrix([
[1, 1, 1, 0],
[1, 1, 1, 1],
[1, 1, 1, 1],
[1, 1, 1, 1]
])
L = A.lower_triangular(-2)
assert L == ArithmeticOnlyMatrix([
[0, 0, 0, 0],
[0, 0, 0, 0],
[1, 0, 0, 0],
[1, 1, 0, 0]
])
# 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_non_matmul():
"""
Test that if explicitly specified as non-matrix, mul reverts
to scalar multiplication.
"""
class foo(Expr):
is_Matrix=False
is_MatrixLike=False
shape = (1, 1)
A = Matrix([[1, 2], [3, 4]])
b = foo()
assert b*A == Matrix([[b, 2*b], [3*b, 4*b]])
assert A*b == Matrix([[b, 2*b], [3*b, 4*b]])
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))
def test_wilkinson():
wminus, wplus = Matrix.wilkinson(1)
assert wminus == Matrix([
[-1, 1, 0],
[1, 0, 1],
[0, 1, 1]])
assert wplus == Matrix([
[1, 1, 0],
[1, 0, 1],
[0, 1, 1]])
wminus, wplus = Matrix.wilkinson(3)
assert wminus == Matrix([
[-3, 1, 0, 0, 0, 0, 0],
[1, -2, 1, 0, 0, 0, 0],
[0, 1, -1, 1, 0, 0, 0],
[0, 0, 1, 0, 1, 0, 0],
[0, 0, 0, 1, 1, 1, 0],
[0, 0, 0, 0, 1, 2, 1],
[0, 0, 0, 0, 0, 1, 3]])
assert wplus == Matrix([
[3, 1, 0, 0, 0, 0, 0],
[1, 2, 1, 0, 0, 0, 0],
[0, 1, 1, 1, 0, 0, 0],
[0, 0, 1, 0, 1, 0, 0],
[0, 0, 0, 1, 1, 1, 0],
[0, 0, 0, 0, 1, 2, 1],
[0, 0, 0, 0, 0, 1, 3]])
# 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]])
def test_issue_10589():
x, y, z = symbols("x, y z")
M1 = Matrix([x, y, z])
M1 = M1.subs(zip([x, y, z], [1, 2, 3]))
assert M1 == Matrix([[1], [2], [3]])
M2 = Matrix([[x, x, x, x, x], [x, x, x, x, x], [x, x, x, x, x]])
M2 = M2.subs(zip([x], [1]))
assert M2 == Matrix([[1, 1, 1, 1, 1], [1, 1, 1, 1, 1], [1, 1, 1, 1, 1]])
def test_rmul_pr19860():
class Foo(ImmutableDenseMatrix):
_op_priority = MutableDenseMatrix._op_priority + 0.01
a = Matrix(2, 2, [1, 2, 3, 4])
b = Foo(2, 2, [1, 2, 3, 4])
# This would throw a RecursionError: maximum recursion depth
# since b always has higher priority even after a.as_mutable()
c = a*b
assert isinstance(c, Foo)
assert c == Matrix([[7, 10], [15, 22]])
def test_issue_18956():
A = Array([[1, 2], [3, 4]])
B = Matrix([[1,2],[3,4]])
raises(TypeError, lambda: B + A)
raises(TypeError, lambda: A + B)
|
4d55a7d056158e79b05f4839b3b119f2aa1d129213c9d19f8ee7296975e831f8 | import random
from sympy.core.numbers import I
from sympy import symbols, Symbol, Rational, sqrt, Poly
from sympy.matrices import Matrix, eye, ones
from sympy.abc import x, y, z
from sympy.testing.pytest import raises
from sympy.matrices.matrices import MatrixDeterminant
from sympy.matrices.common import NonSquareMatrixError, _MinimalMatrix, _CastableMatrix
from sympy.functions.combinatorial.factorials import factorial, subfactorial
class DeterminantOnlyMatrix(_MinimalMatrix, _CastableMatrix, MatrixDeterminant):
pass
def test_determinant():
for M in [Matrix(), Matrix([[1]])]:
assert (
M.det() ==
M._eval_det_bareiss() ==
M._eval_det_berkowitz() ==
M._eval_det_lu() ==
1)
M = Matrix(( (-3, 2),
( 8, -5) ))
assert M.det(method="domain-ge") == -1
assert M.det(method="bareiss") == -1
assert M.det(method="berkowitz") == -1
assert M.det(method="lu") == -1
M = Matrix(( (x, 1),
(y, 2*y) ))
assert M.det(method="domain-ge") == 2*x*y - y
assert M.det(method="bareiss") == 2*x*y - y
assert M.det(method="berkowitz") == 2*x*y - y
assert M.det(method="lu") == 2*x*y - y
M = Matrix(( (1, 1, 1),
(1, 2, 3),
(1, 3, 6) ))
assert M.det(method="domain-ge") == 1
assert M.det(method="bareiss") == 1
assert M.det(method="berkowitz") == 1
assert M.det(method="lu") == 1
M = Matrix(( ( 3, -2, 0, 5),
(-2, 1, -2, 2),
( 0, -2, 5, 0),
( 5, 0, 3, 4) ))
assert M.det(method="domain-ge") == -289
assert M.det(method="bareiss") == -289
assert M.det(method="berkowitz") == -289
assert M.det(method="lu") == -289
M = Matrix(( ( 1, 2, 3, 4),
( 5, 6, 7, 8),
( 9, 10, 11, 12),
(13, 14, 15, 16) ))
assert M.det(method="domain-ge") == 0
assert M.det(method="bareiss") == 0
assert M.det(method="berkowitz") == 0
assert M.det(method="lu") == 0
M = Matrix(( (3, 2, 0, 0, 0),
(0, 3, 2, 0, 0),
(0, 0, 3, 2, 0),
(0, 0, 0, 3, 2),
(2, 0, 0, 0, 3) ))
assert M.det(method="domain-ge") == 275
assert M.det(method="bareiss") == 275
assert M.det(method="berkowitz") == 275
assert M.det(method="lu") == 275
M = Matrix(( ( 3, 0, 0, 0),
(-2, 1, 0, 0),
( 0, -2, 5, 0),
( 5, 0, 3, 4) ))
assert M.det(method="domain-ge") == 60
assert M.det(method="bareiss") == 60
assert M.det(method="berkowitz") == 60
assert M.det(method="lu") == 60
M = Matrix(( ( 1, 0, 0, 0),
( 5, 0, 0, 0),
( 9, 10, 11, 0),
(13, 14, 15, 16) ))
assert M.det(method="domain-ge") == 0
assert M.det(method="bareiss") == 0
assert M.det(method="berkowitz") == 0
assert M.det(method="lu") == 0
M = Matrix(( (3, 2, 0, 0, 0),
(0, 3, 2, 0, 0),
(0, 0, 3, 2, 0),
(0, 0, 0, 3, 2),
(0, 0, 0, 0, 3) ))
assert M.det(method="domain-ge") == 243
assert M.det(method="bareiss") == 243
assert M.det(method="berkowitz") == 243
assert M.det(method="lu") == 243
M = Matrix(( (1, 0, 1, 2, 12),
(2, 0, 1, 1, 4),
(2, 1, 1, -1, 3),
(3, 2, -1, 1, 8),
(1, 1, 1, 0, 6) ))
assert M.det(method="domain-ge") == -55
assert M.det(method="bareiss") == -55
assert M.det(method="berkowitz") == -55
assert M.det(method="lu") == -55
M = Matrix(( (-5, 2, 3, 4, 5),
( 1, -4, 3, 4, 5),
( 1, 2, -3, 4, 5),
( 1, 2, 3, -2, 5),
( 1, 2, 3, 4, -1) ))
assert M.det(method="domain-ge") == 11664
assert M.det(method="bareiss") == 11664
assert M.det(method="berkowitz") == 11664
assert M.det(method="lu") == 11664
M = Matrix(( ( 2, 7, -1, 3, 2),
( 0, 0, 1, 0, 1),
(-2, 0, 7, 0, 2),
(-3, -2, 4, 5, 3),
( 1, 0, 0, 0, 1) ))
assert M.det(method="domain-ge") == 123
assert M.det(method="bareiss") == 123
assert M.det(method="berkowitz") == 123
assert M.det(method="lu") == 123
M = Matrix(( (x, y, z),
(1, 0, 0),
(y, z, x) ))
assert M.det(method="domain-ge") == z**2 - x*y
assert M.det(method="bareiss") == z**2 - x*y
assert M.det(method="berkowitz") == z**2 - x*y
assert M.det(method="lu") == z**2 - x*y
# issue 13835
a = symbols('a')
M = lambda n: Matrix([[i + a*j for i in range(n)]
for j in range(n)])
assert M(5).det() == 0
assert M(6).det() == 0
assert M(7).det() == 0
def test_issue_14517():
M = Matrix([
[ 0, 10*I, 10*I, 0],
[10*I, 0, 0, 10*I],
[10*I, 0, 5 + 2*I, 10*I],
[ 0, 10*I, 10*I, 5 + 2*I]])
ev = M.eigenvals()
# test one random eigenvalue, the computation is a little slow
test_ev = random.choice(list(ev.keys()))
assert (M - test_ev*eye(4)).det() == 0
def test_legacy_det():
# Minimal support for legacy keys for 'method' in det()
# Partially copied from test_determinant()
M = Matrix(( ( 3, -2, 0, 5),
(-2, 1, -2, 2),
( 0, -2, 5, 0),
( 5, 0, 3, 4) ))
assert M.det(method="bareis") == -289
assert M.det(method="det_lu") == -289
assert M.det(method="det_LU") == -289
M = Matrix(( (3, 2, 0, 0, 0),
(0, 3, 2, 0, 0),
(0, 0, 3, 2, 0),
(0, 0, 0, 3, 2),
(2, 0, 0, 0, 3) ))
assert M.det(method="bareis") == 275
assert M.det(method="det_lu") == 275
assert M.det(method="Bareis") == 275
M = Matrix(( (1, 0, 1, 2, 12),
(2, 0, 1, 1, 4),
(2, 1, 1, -1, 3),
(3, 2, -1, 1, 8),
(1, 1, 1, 0, 6) ))
assert M.det(method="bareis") == -55
assert M.det(method="det_lu") == -55
assert M.det(method="BAREISS") == -55
M = Matrix(( ( 3, 0, 0, 0),
(-2, 1, 0, 0),
( 0, -2, 5, 0),
( 5, 0, 3, 4) ))
assert M.det(method="bareiss") == 60
assert M.det(method="berkowitz") == 60
assert M.det(method="lu") == 60
M = Matrix(( ( 1, 0, 0, 0),
( 5, 0, 0, 0),
( 9, 10, 11, 0),
(13, 14, 15, 16) ))
assert M.det(method="bareiss") == 0
assert M.det(method="berkowitz") == 0
assert M.det(method="lu") == 0
M = Matrix(( (3, 2, 0, 0, 0),
(0, 3, 2, 0, 0),
(0, 0, 3, 2, 0),
(0, 0, 0, 3, 2),
(0, 0, 0, 0, 3) ))
assert M.det(method="bareiss") == 243
assert M.det(method="berkowitz") == 243
assert M.det(method="lu") == 243
M = Matrix(( (-5, 2, 3, 4, 5),
( 1, -4, 3, 4, 5),
( 1, 2, -3, 4, 5),
( 1, 2, 3, -2, 5),
( 1, 2, 3, 4, -1) ))
assert M.det(method="bareis") == 11664
assert M.det(method="det_lu") == 11664
assert M.det(method="BERKOWITZ") == 11664
M = Matrix(( ( 2, 7, -1, 3, 2),
( 0, 0, 1, 0, 1),
(-2, 0, 7, 0, 2),
(-3, -2, 4, 5, 3),
( 1, 0, 0, 0, 1) ))
assert M.det(method="bareis") == 123
assert M.det(method="det_lu") == 123
assert M.det(method="LU") == 123
def eye_Determinant(n):
return DeterminantOnlyMatrix(n, n, lambda i, j: int(i == j))
def zeros_Determinant(n):
return DeterminantOnlyMatrix(n, n, lambda i, j: 0)
def test_det():
a = DeterminantOnlyMatrix(2, 3, [1, 2, 3, 4, 5, 6])
raises(NonSquareMatrixError, lambda: a.det())
z = zeros_Determinant(2)
ey = eye_Determinant(2)
assert z.det() == 0
assert ey.det() == 1
x = Symbol('x')
a = DeterminantOnlyMatrix(0, 0, [])
b = DeterminantOnlyMatrix(1, 1, [5])
c = DeterminantOnlyMatrix(2, 2, [1, 2, 3, 4])
d = DeterminantOnlyMatrix(3, 3, [1, 2, 3, 4, 5, 6, 7, 8, 8])
e = DeterminantOnlyMatrix(4, 4,
[x, 1, 2, 3, 4, 5, 6, 7, 2, 9, 10, 11, 12, 13, 14, 14])
from sympy.abc import i, j, k, l, m, n
f = DeterminantOnlyMatrix(3, 3, [i, l, m, 0, j, n, 0, 0, k])
g = DeterminantOnlyMatrix(3, 3, [i, 0, 0, l, j, 0, m, n, k])
h = DeterminantOnlyMatrix(3, 3, [x**3, 0, 0, i, x**-1, 0, j, k, x**-2])
# the method keyword for `det` doesn't kick in until 4x4 matrices,
# so there is no need to test all methods on smaller ones
assert a.det() == 1
assert b.det() == 5
assert c.det() == -2
assert d.det() == 3
assert e.det() == 4*x - 24
assert e.det(method="domain-ge") == 4*x - 24
assert e.det(method='bareiss') == 4*x - 24
assert e.det(method='berkowitz') == 4*x - 24
assert f.det() == i*j*k
assert g.det() == i*j*k
assert h.det() == 1
raises(ValueError, lambda: e.det(iszerofunc="test"))
def test_permanent():
M = Matrix([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
assert M.per() == 450
for i in range(1, 12):
assert ones(i, i).per() == ones(i, i).T.per() == factorial(i)
assert (ones(i, i)-eye(i)).per() == (ones(i, i)-eye(i)).T.per() == subfactorial(i)
a1, a2, a3, a4, a5 = symbols('a_1 a_2 a_3 a_4 a_5')
M = Matrix([a1, a2, a3, a4, a5])
assert M.per() == M.T.per() == a1 + a2 + a3 + a4 + a5
def test_adjugate():
x = Symbol('x')
e = DeterminantOnlyMatrix(4, 4,
[x, 1, 2, 3, 4, 5, 6, 7, 2, 9, 10, 11, 12, 13, 14, 14])
adj = Matrix([
[ 4, -8, 4, 0],
[ 76, -14*x - 68, 14*x - 8, -4*x + 24],
[-122, 17*x + 142, -21*x + 4, 8*x - 48],
[ 48, -4*x - 72, 8*x, -4*x + 24]])
assert e.adjugate() == adj
assert e.adjugate(method='bareiss') == adj
assert e.adjugate(method='berkowitz') == adj
a = DeterminantOnlyMatrix(2, 3, [1, 2, 3, 4, 5, 6])
raises(NonSquareMatrixError, lambda: a.adjugate())
def test_util():
R = Rational
v1 = Matrix(1, 3, [1, 2, 3])
v2 = Matrix(1, 3, [3, 4, 5])
assert v1.norm() == sqrt(14)
assert v1.project(v2) == Matrix(1, 3, [R(39)/25, R(52)/25, R(13)/5])
assert Matrix.zeros(1, 2) == Matrix(1, 2, [0, 0])
assert ones(1, 2) == Matrix(1, 2, [1, 1])
assert v1.copy() == v1
# cofactor
assert eye(3) == eye(3).cofactor_matrix()
test = Matrix([[1, 3, 2], [2, 6, 3], [2, 3, 6]])
assert test.cofactor_matrix() == \
Matrix([[27, -6, -6], [-12, 2, 3], [-3, 1, 0]])
test = Matrix([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
assert test.cofactor_matrix() == \
Matrix([[-3, 6, -3], [6, -12, 6], [-3, 6, -3]])
def test_cofactor_and_minors():
x = Symbol('x')
e = DeterminantOnlyMatrix(4, 4,
[x, 1, 2, 3, 4, 5, 6, 7, 2, 9, 10, 11, 12, 13, 14, 14])
m = Matrix([
[ x, 1, 3],
[ 2, 9, 11],
[12, 13, 14]])
cm = Matrix([
[ 4, 76, -122, 48],
[-8, -14*x - 68, 17*x + 142, -4*x - 72],
[ 4, 14*x - 8, -21*x + 4, 8*x],
[ 0, -4*x + 24, 8*x - 48, -4*x + 24]])
sub = Matrix([
[x, 1, 2],
[4, 5, 6],
[2, 9, 10]])
assert e.minor_submatrix(1, 2) == m
assert e.minor_submatrix(-1, -1) == sub
assert e.minor(1, 2) == -17*x - 142
assert e.cofactor(1, 2) == 17*x + 142
assert e.cofactor_matrix() == cm
assert e.cofactor_matrix(method="bareiss") == cm
assert e.cofactor_matrix(method="berkowitz") == cm
raises(ValueError, lambda: e.cofactor(4, 5))
raises(ValueError, lambda: e.minor(4, 5))
raises(ValueError, lambda: e.minor_submatrix(4, 5))
a = DeterminantOnlyMatrix(2, 3, [1, 2, 3, 4, 5, 6])
assert a.minor_submatrix(0, 0) == Matrix([[5, 6]])
raises(ValueError, lambda:
DeterminantOnlyMatrix(0, 0, []).minor_submatrix(0, 0))
raises(NonSquareMatrixError, lambda: a.cofactor(0, 0))
raises(NonSquareMatrixError, lambda: a.minor(0, 0))
raises(NonSquareMatrixError, lambda: a.cofactor_matrix())
def test_charpoly():
x, y = Symbol('x'), Symbol('y')
z, t = Symbol('z'), Symbol('t')
from sympy.abc import a,b,c
m = DeterminantOnlyMatrix(3, 3, [1, 2, 3, 4, 5, 6, 7, 8, 9])
assert eye_Determinant(3).charpoly(x) == Poly((x - 1)**3, x)
assert eye_Determinant(3).charpoly(y) == Poly((y - 1)**3, y)
assert m.charpoly() == Poly(x**3 - 15*x**2 - 18*x, x)
raises(NonSquareMatrixError, lambda: Matrix([[1], [2]]).charpoly())
n = DeterminantOnlyMatrix(4, 4, [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0])
assert n.charpoly() == Poly(x**4, x)
n = DeterminantOnlyMatrix(4, 4, [45, 0, 0, 0, 0, 23, 0, 0, 0, 0, 87, 0, 0, 0, 0, 12])
assert n.charpoly() == Poly(x**4 - 167*x**3 + 8811*x**2 - 173457*x + 1080540, x)
n = DeterminantOnlyMatrix(3, 3, [x, 0, 0, a, y, 0, b, c, z])
assert n.charpoly() == Poly(t**3 - (x+y+z)*t**2 + t*(x*y+y*z+x*z) - x*y*z , t)
|
b5cc6cbc2ddfaedf49a0d374b80ddf9d3e683b5a8929b50bb9619831fc4f1c4f | 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.tensor.array.expressions.array_expressions import ArrayContraction
from sympy.tensor.array.expressions.array_expressions import ArrayDiagonal
from sympy.tensor.array.expressions.array_expressions import ArrayTensorProduct
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(
ArrayDiagonal,
[
ExprBuilder(
ArrayTensorProduct,
[
ewdiff,
ptr1,
ptr2,
]
),
(0, 2) if iscolumn else (1, 4)
],
validator=ArrayDiagonal._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(
ArrayContraction,
[
ExprBuilder(
ArrayTensorProduct,
[ptr1, newptr1, ewdiff, ptr2, newptr2]
),
(1, 2, 4),
(5, 7, 8),
],
validator=ArrayContraction._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
def _eval_transpose(self):
from sympy import Transpose
return self.func(self.function, Transpose(self.expr).doit())
|
fcc6710454cd42e9f42633cf2e81912c095d63fd3a1f2c5f1b5da9edcb0376ce | from sympy import Number
from sympy.core import Mul, Basic, sympify, S
from sympy.core.mul import mul
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
from .matpow import MatPow
from .transpose import transpose
from .permutation import PermutationMatrix
from .special import ZeroMatrix, Identity, GenericIdentity, OneMatrix
# 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, check=True, _sympify=True):
if not args:
return cls.identity
# This must be removed aggressively in the constructor to avoid
# TypeErrors from GenericIdentity().shape
args = list(filter(lambda i: cls.identity != i, args))
if _sympify:
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
mul.register_handlerclass((Mul, MatMul), MatMul)
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)
def distribute_monom(mul):
"""
Simplify MatMul expressions but distributing
rational term to MatMul.
e.g. 2*(A+B) -> 2*A + 2*B
"""
args = mul.args
if len(args) == 2:
from .matadd import MatAdd
if args[0].is_MatAdd and args[1].is_Rational:
return MatAdd(*[MatMul(mat, args[1]).doit() for mat in args[0].args])
if args[1].is_MatAdd and args[0].is_Rational:
return MatAdd(*[MatMul(args[0], mat).doit() for mat in args[1].args])
return mul
rules = (
distribute_monom, 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
|
e39ea8ef53498da10e0975603f10da642aa8dd2d3429d1424c1a070ed8a08f70 | from sympy.matrices.common import NonSquareMatrixError
from .matexpr import MatrixExpr
from .special import Identity
from sympy.core import S
from sympy.core.sympify import _sympify
from sympy.matrices import MatrixBase
class MatPow(MatrixExpr):
def __new__(cls, base, exp, evaluate=False, **options):
base = _sympify(base)
if not base.is_Matrix:
raise TypeError("MatPow base should be a matrix")
if not base.is_square:
raise NonSquareMatrixError("Power of non-square matrix %s" % base)
exp = _sympify(exp)
obj = super().__new__(cls, base, exp)
if evaluate:
obj = obj.doit(deep=False)
return obj
@property
def base(self):
return self.args[0]
@property
def exp(self):
return self.args[1]
@property
def shape(self):
return self.base.shape
def _entry(self, i, j, **kwargs):
from sympy.matrices.expressions import MatMul
A = self.doit()
if isinstance(A, MatPow):
# We still have a MatPow, make an explicit MatMul out of it.
if A.exp.is_Integer and A.exp.is_positive:
A = MatMul(*[A.base for k in range(A.exp)])
#elif A.exp.is_Integer and self.exp.is_negative:
# Note: possible future improvement: in principle we can take
# positive powers of the inverse, but carefully avoid recursion,
# perhaps by adding `_entry` to Inverse (as it is our subclass).
# T = A.base.as_explicit().inverse()
# A = MatMul(*[T for k in range(-A.exp)])
else:
# Leave the expression unevaluated:
from sympy.matrices.expressions.matexpr import MatrixElement
return MatrixElement(self, i, j)
return A[i, j]
def doit(self, **kwargs):
if kwargs.get('deep', True):
base, exp = [arg.doit(**kwargs) for arg in self.args]
else:
base, exp = self.args
# combine all powers, e.g. (A ** 2) ** 3 -> A ** 6
while isinstance(base, MatPow):
exp *= base.args[1]
base = base.args[0]
if isinstance(base, MatrixBase):
# Delegate
return base ** exp
# Handle simple cases so that _eval_power() in MatrixExpr sub-classes can ignore them
if exp == S.One:
return base
if exp == S.Zero:
return Identity(base.rows)
if exp == S.NegativeOne:
from sympy.matrices.expressions import Inverse
return Inverse(base).doit(**kwargs)
eval_power = getattr(base, '_eval_power', None)
if eval_power is not None:
return eval_power(exp)
return MatPow(base, exp)
def _eval_transpose(self):
base, exp = self.args
return MatPow(base.T, exp)
def _eval_derivative(self, x):
from sympy import Pow
return Pow._eval_derivative(self, x)
def _eval_derivative_matrix_lines(self, x):
from sympy.core.expr import ExprBuilder
from sympy.tensor.array.expressions.array_expressions import ArrayContraction
from ...tensor.array.expressions.array_expressions import ArrayTensorProduct
from .matmul import MatMul
from .inverse import Inverse
exp = self.exp
if self.base.shape == (1, 1) and not exp.has(x):
lr = self.base._eval_derivative_matrix_lines(x)
for i in lr:
subexpr = ExprBuilder(
ArrayContraction,
[
ExprBuilder(
ArrayTensorProduct,
[
Identity(1),
i._lines[0],
exp*self.base**(exp-1),
i._lines[1],
Identity(1),
]
),
(0, 3, 4), (5, 7, 8)
],
validator=ArrayContraction._validate
)
i._first_pointer_parent = subexpr.args[0].args
i._first_pointer_index = 0
i._second_pointer_parent = subexpr.args[0].args
i._second_pointer_index = 4
i._lines = [subexpr]
return lr
if (exp > 0) == True:
newexpr = MatMul.fromiter([self.base for i in range(exp)])
elif (exp == -1) == True:
return Inverse(self.base)._eval_derivative_matrix_lines(x)
elif (exp < 0) == True:
newexpr = MatMul.fromiter([Inverse(self.base) for i in range(-exp)])
elif (exp == 0) == True:
return self.doit()._eval_derivative_matrix_lines(x)
else:
raise NotImplementedError("cannot evaluate %s derived by %s" % (self, x))
return newexpr._eval_derivative_matrix_lines(x)
def _eval_inverse(self):
return MatPow(self.base, -self.exp)
|
806e2e4d7e889b105f502c654efba789564b5f24b64640c21b8f6bc006066ba9 | from typing import Tuple as tTuple
from sympy.core.logic import FuzzyBool
from functools import wraps, reduce
import collections
from sympy.core import S, Symbol, Integer, Basic, Expr, Mul, Add
from sympy.core.decorators import call_highest_priority
from sympy.core.compatibility import SYMPY_INTS, default_sort_key
from sympy.core.symbol import Str
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
from sympy.simplify import simplify
from sympy.matrices.matrices import MatrixKind
from sympy.utilities.misc import filldedent
from sympy.multipledispatch import dispatch
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
kind = MatrixKind()
def __new__(cls, *args, **kwargs):
args = map(_sympify, args)
return Basic.__new__(cls, *args, **kwargs)
# The following is adapted from the core Expr object
@property
def shape(self) -> tTuple[Expr, Expr]:
raise NotImplementedError
@property
def _add_handler(self):
return MatAdd
@property
def _mul_handler(self):
return MatMul
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('__rtruediv__')
def __truediv__(self, other):
return self * other**S.NegativeOne
@_sympifyit('other', NotImplemented)
@call_highest_priority('__truediv__')
def __rtruediv__(self, other):
raise NotImplementedError()
#return MatMul(other, Pow(self, S.NegativeOne))
@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_n_times(self, x, n):
return Basic._eval_derivative_n_times(self, x, n)
def _eval_derivative(self, x):
# `x` is a scalar:
if self.has(x):
# See if there are other methods using it:
return super()._eval_derivative(x)
else:
return ZeroMatrix(*self.shape)
@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)
@dispatch(MatrixExpr, Expr)
def _eval_is_eq(lhs, rhs): # noqa:F811
return False
@dispatch(MatrixExpr, MatrixExpr) # type: ignore
def _eval_is_eq(lhs, rhs): # noqa:F811
if lhs.shape != rhs.shape:
return False
if (lhs - rhs).is_ZeroMatrix:
return True
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.tensor.array.array_derivatives import ArrayDerivative
lines = expr._eval_derivative_matrix_lines(x)
parts = [i.build() for i in lines]
from sympy.tensor.array.expressions.conv_array_to_matrix import convert_array_to_matrix
parts = [[convert_array_to_matrix(j) 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 ArrayDerivative(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 = Str(name)
obj = Basic.__new__(cls, name, n, m)
return obj
@property
def shape(self):
return self.args[1], self.args[2]
@property
def name(self):
return self.args[0].name
def _entry(self, i, j, **kwargs):
return MatrixElement(self, i, j)
@property
def free_symbols(self):
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],
)]
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 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 ...tensor.array.expressions.array_expressions import ArrayTensorProduct
from ...tensor.array.expressions.array_expressions import ArrayContraction
subexpr = ExprBuilder(
ArrayContraction,
[
ExprBuilder(
ArrayTensorProduct,
[
pointer,
other
]
),
(1, 2)
],
validator=ArrayContraction._validate
)
return subexpr
def append_first(self, other):
self.first_pointer *= other
def append_second(self, other):
self.second_pointer *= other
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
from .special import ZeroMatrix, Identity
|
6a3603845e8e2523bc7c956de6d3ac15d8d8b97140d2cb217116746d76076afb | 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.tensor.array.expressions.array_expressions import ArrayTensorProduct, ArrayContraction
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(
ArrayContraction,
[
ExprBuilder(
ArrayTensorProduct,
[
lr._lines[0],
lr._lines[1],
]
),
(1, 3),
],
validator=ArrayContraction._validate
)
else:
# This is not a matrix line:
lr.higher = ExprBuilder(
ArrayContraction,
[
ExprBuilder(
ArrayTensorProduct,
[
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 _normalize(self):
# Normalization of trace of matrix products. Use transposition and
# cyclic properties of traces to make sure the arguments of the matrix
# product are sorted and the first argument is not a trasposition.
from sympy import MatMul, Transpose, default_sort_key
trace_arg = self.arg
if isinstance(trace_arg, MatMul):
indmin = min(range(len(trace_arg.args)), key=lambda x: default_sort_key(trace_arg.args[x]))
if isinstance(trace_arg.args[indmin], Transpose):
trace_arg = Transpose(trace_arg).doit()
indmin = min(range(len(trace_arg.args)), key=lambda x: default_sort_key(trace_arg.args[x]))
trace_arg = MatMul.fromiter(trace_arg.args[indmin:] + trace_arg.args[:indmin])
return Trace(trace_arg)
return self
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()
|
31f8bd45ee55154f8c5abf7e8947bd6da756b09e4df325c8e1b6a0fc53963750 | from sympy.core import Mul, sympify
from sympy.matrices.common import ShapeError
from sympy.matrices.expressions.matexpr import MatrixExpr
from sympy.matrices.expressions.special import ZeroMatrix, OneMatrix
from sympy.strategies import (
unpack, flatten, condition, exhaust, rm_id, sort
)
def hadamard_product(*matrices):
"""
Return the elementwise (aka Hadamard) product of matrices.
Examples
========
>>> from sympy.matrices import hadamard_product, MatrixSymbol
>>> A = MatrixSymbol('A', 2, 3)
>>> B = MatrixSymbol('B', 2, 3)
>>> hadamard_product(A)
A
>>> hadamard_product(A, B)
HadamardProduct(A, B)
>>> hadamard_product(A, B)[0, 1]
A[0, 1]*B[0, 1]
"""
if not matrices:
raise TypeError("Empty Hadamard product is undefined")
validate(*matrices)
if len(matrices) == 1:
return matrices[0]
else:
matrices = [i for i in matrices if not i.is_Identity]
return HadamardProduct(*matrices).doit()
class HadamardProduct(MatrixExpr):
"""
Elementwise product of matrix expressions
Examples
========
Hadamard product for matrix symbols:
>>> from sympy.matrices import hadamard_product, HadamardProduct, MatrixSymbol
>>> A = MatrixSymbol('A', 5, 5)
>>> B = MatrixSymbol('B', 5, 5)
>>> isinstance(hadamard_product(A, B), HadamardProduct)
True
Notes
=====
This is a symbolic object that simply stores its argument without
evaluating it. To actually compute the product, use the function
``hadamard_product()`` or ``HadamardProduct.doit``
"""
is_HadamardProduct = True
def __new__(cls, *args, evaluate=False, check=True):
args = list(map(sympify, args))
if check:
validate(*args)
obj = super().__new__(cls, *args)
if evaluate:
obj = obj.doit(deep=False)
return obj
@property
def shape(self):
return self.args[0].shape
def _entry(self, i, j, **kwargs):
return Mul(*[arg._entry(i, j, **kwargs) for arg in self.args])
def _eval_transpose(self):
from sympy.matrices.expressions.transpose import transpose
return HadamardProduct(*list(map(transpose, self.args)))
def doit(self, **ignored):
expr = self.func(*[i.doit(**ignored) for i in self.args])
# Check for explicit matrices:
from sympy import MatrixBase
from sympy.matrices.immutable import ImmutableMatrix
explicit = [i for i in expr.args if isinstance(i, MatrixBase)]
if explicit:
remainder = [i for i in expr.args if i not in explicit]
expl_mat = ImmutableMatrix([
Mul.fromiter(i) for i in zip(*explicit)
]).reshape(*self.shape)
expr = HadamardProduct(*([expl_mat] + remainder))
return canonicalize(expr)
def _eval_derivative(self, x):
from sympy import Add
terms = []
args = list(self.args)
for i in range(len(args)):
factors = args[:i] + [args[i].diff(x)] + args[i+1:]
terms.append(hadamard_product(*factors))
return Add.fromiter(terms)
def _eval_derivative_matrix_lines(self, x):
from sympy.core.expr import ExprBuilder
from sympy.tensor.array.expressions.array_expressions import ArrayDiagonal
from sympy.tensor.array.expressions.array_expressions import ArrayTensorProduct
from sympy.matrices.expressions.matexpr import _make_matrix
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:]
d = self.args[ind]._eval_derivative_matrix_lines(x)
hadam = hadamard_product(*(right_args + left_args))
diagonal = [(0, 2), (3, 4)]
diagonal = [e for j, e in enumerate(diagonal) if self.shape[j] != 1]
for i in d:
l1 = i._lines[i._first_line_index]
l2 = i._lines[i._second_line_index]
subexpr = ExprBuilder(
ArrayDiagonal,
[
ExprBuilder(
ArrayTensorProduct,
[
ExprBuilder(_make_matrix, [l1]),
hadam,
ExprBuilder(_make_matrix, [l2]),
]
),
*diagonal],
)
i._first_pointer_parent = subexpr.args[0].args[0].args
i._first_pointer_index = 0
i._second_pointer_parent = subexpr.args[0].args[2].args
i._second_pointer_index = 0
i._lines = [subexpr]
lines.append(i)
return lines
def validate(*args):
if not all(arg.is_Matrix for arg in args):
raise TypeError("Mix of Matrix and Scalar symbols")
A = args[0]
for B in args[1:]:
if A.shape != B.shape:
raise ShapeError("Matrices %s and %s are not aligned" % (A, B))
# TODO Implement algorithm for rewriting Hadamard product as diagonal matrix
# if matmul identy matrix is multiplied.
def canonicalize(x):
"""Canonicalize the Hadamard product ``x`` with mathematical properties.
Examples
========
>>> from sympy.matrices.expressions import MatrixSymbol, HadamardProduct
>>> from sympy.matrices.expressions import OneMatrix, ZeroMatrix
>>> from sympy.matrices.expressions.hadamard import canonicalize
>>> from sympy import init_printing
>>> init_printing(use_unicode=False)
>>> A = MatrixSymbol('A', 2, 2)
>>> B = MatrixSymbol('B', 2, 2)
>>> C = MatrixSymbol('C', 2, 2)
Hadamard product associativity:
>>> X = HadamardProduct(A, HadamardProduct(B, C))
>>> X
A.*(B.*C)
>>> canonicalize(X)
A.*B.*C
Hadamard product commutativity:
>>> X = HadamardProduct(A, B)
>>> Y = HadamardProduct(B, A)
>>> X
A.*B
>>> Y
B.*A
>>> canonicalize(X)
A.*B
>>> canonicalize(Y)
A.*B
Hadamard product identity:
>>> X = HadamardProduct(A, OneMatrix(2, 2))
>>> X
A.*1
>>> canonicalize(X)
A
Absorbing element of Hadamard product:
>>> X = HadamardProduct(A, ZeroMatrix(2, 2))
>>> X
A.*0
>>> canonicalize(X)
0
Rewriting to Hadamard Power
>>> X = HadamardProduct(A, A, A)
>>> X
A.*A.*A
>>> canonicalize(X)
.3
A
Notes
=====
As the Hadamard product is associative, nested products can be flattened.
The Hadamard product is commutative so that factors can be sorted for
canonical form.
A matrix of only ones is an identity for Hadamard product,
so every matrices of only ones can be removed.
Any zero matrix will make the whole product a zero matrix.
Duplicate elements can be collected and rewritten as HadamardPower
References
==========
.. [1] https://en.wikipedia.org/wiki/Hadamard_product_(matrices)
"""
from sympy.core.compatibility import default_sort_key
# Associativity
rule = condition(
lambda x: isinstance(x, HadamardProduct),
flatten
)
fun = exhaust(rule)
x = fun(x)
# Identity
fun = condition(
lambda x: isinstance(x, HadamardProduct),
rm_id(lambda x: isinstance(x, OneMatrix))
)
x = fun(x)
# Absorbing by Zero Matrix
def absorb(x):
if any(isinstance(c, ZeroMatrix) for c in x.args):
return ZeroMatrix(*x.shape)
else:
return x
fun = condition(
lambda x: isinstance(x, HadamardProduct),
absorb
)
x = fun(x)
# Rewriting with HadamardPower
if isinstance(x, HadamardProduct):
from collections import Counter
tally = Counter(x.args)
new_arg = []
for base, exp in tally.items():
if exp == 1:
new_arg.append(base)
else:
new_arg.append(HadamardPower(base, exp))
x = HadamardProduct(*new_arg)
# Commutativity
fun = condition(
lambda x: isinstance(x, HadamardProduct),
sort(default_sort_key)
)
x = fun(x)
# Unpacking
x = unpack(x)
return x
def hadamard_power(base, exp):
base = sympify(base)
exp = sympify(exp)
if exp == 1:
return base
if not base.is_Matrix:
return base**exp
if exp.is_Matrix:
raise ValueError("cannot raise expression to a matrix")
return HadamardPower(base, exp)
class HadamardPower(MatrixExpr):
r"""
Elementwise power of matrix expressions
Parameters
==========
base : scalar or matrix
exp : scalar or matrix
Notes
=====
There are four definitions for the hadamard power which can be used.
Let's consider `A, B` as `(m, n)` matrices, and `a, b` as scalars.
Matrix raised to a scalar exponent:
.. math::
A^{\circ b} = \begin{bmatrix}
A_{0, 0}^b & A_{0, 1}^b & \cdots & A_{0, n-1}^b \\
A_{1, 0}^b & A_{1, 1}^b & \cdots & A_{1, n-1}^b \\
\vdots & \vdots & \ddots & \vdots \\
A_{m-1, 0}^b & A_{m-1, 1}^b & \cdots & A_{m-1, n-1}^b
\end{bmatrix}
Scalar raised to a matrix exponent:
.. math::
a^{\circ B} = \begin{bmatrix}
a^{B_{0, 0}} & a^{B_{0, 1}} & \cdots & a^{B_{0, n-1}} \\
a^{B_{1, 0}} & a^{B_{1, 1}} & \cdots & a^{B_{1, n-1}} \\
\vdots & \vdots & \ddots & \vdots \\
a^{B_{m-1, 0}} & a^{B_{m-1, 1}} & \cdots & a^{B_{m-1, n-1}}
\end{bmatrix}
Matrix raised to a matrix exponent:
.. math::
A^{\circ B} = \begin{bmatrix}
A_{0, 0}^{B_{0, 0}} & A_{0, 1}^{B_{0, 1}} &
\cdots & A_{0, n-1}^{B_{0, n-1}} \\
A_{1, 0}^{B_{1, 0}} & A_{1, 1}^{B_{1, 1}} &
\cdots & A_{1, n-1}^{B_{1, n-1}} \\
\vdots & \vdots &
\ddots & \vdots \\
A_{m-1, 0}^{B_{m-1, 0}} & A_{m-1, 1}^{B_{m-1, 1}} &
\cdots & A_{m-1, n-1}^{B_{m-1, n-1}}
\end{bmatrix}
Scalar raised to a scalar exponent:
.. math::
a^{\circ b} = a^b
"""
def __new__(cls, base, exp):
base = sympify(base)
exp = sympify(exp)
if base.is_scalar and exp.is_scalar:
return base ** exp
if base.is_Matrix and exp.is_Matrix and base.shape != exp.shape:
raise ValueError(
'The shape of the base {} and '
'the shape of the exponent {} do not match.'
.format(base.shape, exp.shape)
)
obj = super().__new__(cls, base, exp)
return obj
@property
def base(self):
return self._args[0]
@property
def exp(self):
return self._args[1]
@property
def shape(self):
if self.base.is_Matrix:
return self.base.shape
return self.exp.shape
def _entry(self, i, j, **kwargs):
base = self.base
exp = self.exp
if base.is_Matrix:
a = base._entry(i, j, **kwargs)
elif base.is_scalar:
a = base
else:
raise ValueError(
'The base {} must be a scalar or a matrix.'.format(base))
if exp.is_Matrix:
b = exp._entry(i, j, **kwargs)
elif exp.is_scalar:
b = exp
else:
raise ValueError(
'The exponent {} must be a scalar or a matrix.'.format(exp))
return a ** b
def _eval_transpose(self):
from sympy.matrices.expressions.transpose import transpose
return HadamardPower(transpose(self.base), self.exp)
def _eval_derivative(self, x):
from sympy import log
dexp = self.exp.diff(x)
logbase = self.base.applyfunc(log)
dlbase = logbase.diff(x)
return hadamard_product(
dexp*logbase + self.exp*dlbase,
self
)
def _eval_derivative_matrix_lines(self, x):
from sympy.tensor.array.expressions.array_expressions import ArrayTensorProduct
from sympy.tensor.array.expressions.array_expressions import ArrayDiagonal
from sympy.core.expr import ExprBuilder
from sympy.matrices.expressions.matexpr import _make_matrix
lr = self.base._eval_derivative_matrix_lines(x)
for i in lr:
diagonal = [(1, 2), (3, 4)]
diagonal = [e for j, e in enumerate(diagonal) if self.base.shape[j] != 1]
l1 = i._lines[i._first_line_index]
l2 = i._lines[i._second_line_index]
subexpr = ExprBuilder(
ArrayDiagonal,
[
ExprBuilder(
ArrayTensorProduct,
[
ExprBuilder(_make_matrix, [l1]),
self.exp*hadamard_power(self.base, self.exp-1),
ExprBuilder(_make_matrix, [l2]),
]
),
*diagonal],
validator=ArrayDiagonal._validate
)
i._first_pointer_parent = subexpr.args[0].args[0].args
i._first_pointer_index = 0
i._first_line_index = 0
i._second_pointer_parent = subexpr.args[0].args[2].args
i._second_pointer_index = 0
i._second_line_index = 0
i._lines = [subexpr]
return lr
|
d86cd8344899f36c2e160f8333075a188d26288006b3b755e23ed578be8fc8e3 | """
Some examples have been taken from:
http://www.math.uwaterloo.ca/~hwolkowi//matrixcookbook.pdf
"""
from sympy import (MatrixSymbol, Inverse, symbols, Determinant, Trace,
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.tensor.array.array_derivatives import ArrayDerivative
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) == ArrayDerivative(A, A)
assert A.T.diff(A) == ArrayDerivative(A.T, A)
assert (2*A).diff(A) == ArrayDerivative(2*A, A)
assert MatAdd(A, A).diff(A) == ArrayDerivative(MatAdd(A, A), A)
assert (A + B).diff(A) == ArrayDerivative(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) == ArrayDerivative(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) == ArrayDerivative(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) == ArrayDerivative(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*d.T*x/(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)
# TODO: restore this result (currently returning the transpose):
# 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
# TODO: restore (currently returning the traspose):
# 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
|
1fc3db4b298a7770d6dabffdfeebb965fa1b52002b165be6be1fd7bef9554510 | from sympy import (KroneckerDelta, diff, Sum, Dummy, factor,
expand, zeros, gcd_terms, Eq, Symbol)
from sympy.core import (S, symbols, Add, Mul, SympifyError, Rational,
Function)
from sympy.functions import sin, cos, tan, sqrt, cbrt, exp
from sympy.simplify import simplify
from sympy.matrices import (ImmutableMatrix, Inverse, MatAdd, MatMul,
MatPow, Matrix, MatrixExpr, MatrixSymbol, ShapeError,
SparseMatrix, Transpose, Adjoint, NonSquareMatrixError, MatrixSet)
from sympy.matrices.expressions.matexpr import MatrixElement
from sympy.matrices.expressions.special import ZeroMatrix, Identity
from sympy.testing.pytest import raises, XFAIL
n, m, l, k, p = symbols('n m l k p', integer=True)
x = symbols('x')
A = MatrixSymbol('A', n, m)
B = MatrixSymbol('B', m, l)
C = MatrixSymbol('C', n, n)
D = MatrixSymbol('D', n, n)
E = MatrixSymbol('E', m, n)
w = MatrixSymbol('w', n, 1)
def test_matrix_symbol_creation():
assert MatrixSymbol('A', 2, 2)
assert MatrixSymbol('A', 0, 0)
raises(ValueError, lambda: MatrixSymbol('A', -1, 2))
raises(ValueError, lambda: MatrixSymbol('A', 2.0, 2))
raises(ValueError, lambda: MatrixSymbol('A', 2j, 2))
raises(ValueError, lambda: MatrixSymbol('A', 2, -1))
raises(ValueError, lambda: MatrixSymbol('A', 2, 2.0))
raises(ValueError, lambda: MatrixSymbol('A', 2, 2j))
n = symbols('n')
assert MatrixSymbol('A', n, n)
n = symbols('n', integer=False)
raises(ValueError, lambda: MatrixSymbol('A', n, n))
n = symbols('n', negative=True)
raises(ValueError, lambda: MatrixSymbol('A', n, n))
def test_shape():
assert A.shape == (n, m)
assert (A*B).shape == (n, l)
raises(ShapeError, lambda: B*A)
def test_matexpr():
assert (x*A).shape == A.shape
assert (x*A).__class__ == MatMul
assert 2*A - A - A == ZeroMatrix(*A.shape)
assert (A*B).shape == (n, l)
def test_subs():
A = MatrixSymbol('A', n, m)
B = MatrixSymbol('B', m, l)
C = MatrixSymbol('C', m, l)
assert A.subs(n, m).shape == (m, m)
assert (A*B).subs(B, C) == A*C
assert (A*B).subs(l, n).is_square
A = SparseMatrix([[1, 2], [3, 4]])
B = Matrix([[1, 2], [3, 4]])
C, D = MatrixSymbol('C', 2, 2), MatrixSymbol('D', 2, 2)
assert (C*D).subs({C: A, D: B}) == MatMul(A, B)
def test_addition():
A = MatrixSymbol('A', n, m)
B = MatrixSymbol('B', n, m)
assert isinstance(A + B, MatAdd)
assert (A + B).shape == A.shape
assert isinstance(A - A + 2*B, MatMul)
raises(ShapeError, lambda: A + B.T)
raises(TypeError, lambda: A + 1)
raises(TypeError, lambda: 5 + A)
raises(TypeError, lambda: 5 - A)
assert A + ZeroMatrix(n, m) - A == ZeroMatrix(n, m)
with raises(TypeError):
ZeroMatrix(n,m) + S.Zero
def test_multiplication():
A = MatrixSymbol('A', n, m)
B = MatrixSymbol('B', m, l)
C = MatrixSymbol('C', n, n)
assert (2*A*B).shape == (n, l)
assert (A*0*B) == ZeroMatrix(n, l)
raises(ShapeError, lambda: B*A)
assert (2*A).shape == A.shape
assert A * ZeroMatrix(m, m) * B == ZeroMatrix(n, l)
assert C * Identity(n) * C.I == Identity(n)
assert B/2 == S.Half*B
raises(NotImplementedError, lambda: 2/B)
A = MatrixSymbol('A', n, n)
B = MatrixSymbol('B', n, n)
assert Identity(n) * (A + B) == A + B
assert A**2*A == A**3
assert A**2*(A.I)**3 == A.I
assert A**3*(A.I)**2 == A
def test_MatPow():
A = MatrixSymbol('A', n, n)
AA = MatPow(A, 2)
assert AA.exp == 2
assert AA.base == A
assert (A**n).exp == n
assert A**0 == Identity(n)
assert A**1 == A
assert A**2 == AA
assert A**-1 == Inverse(A)
assert (A**-1)**-1 == A
assert (A**2)**3 == A**6
assert A**S.Half == sqrt(A)
assert A**Rational(1, 3) == cbrt(A)
raises(NonSquareMatrixError, lambda: MatrixSymbol('B', 3, 2)**2)
def test_MatrixSymbol():
n, m, t = symbols('n,m,t')
X = MatrixSymbol('X', n, m)
assert X.shape == (n, m)
raises(TypeError, lambda: MatrixSymbol('X', n, m)(t)) # issue 5855
assert X.doit() == X
def test_dense_conversion():
X = MatrixSymbol('X', 2, 2)
assert ImmutableMatrix(X) == ImmutableMatrix(2, 2, lambda i, j: X[i, j])
assert Matrix(X) == Matrix(2, 2, lambda i, j: X[i, j])
def test_free_symbols():
assert (C*D).free_symbols == {C, D}
def test_zero_matmul():
assert isinstance(S.Zero * MatrixSymbol('X', 2, 2), MatrixExpr)
def test_matadd_simplify():
A = MatrixSymbol('A', 1, 1)
assert simplify(MatAdd(A, ImmutableMatrix([[sin(x)**2 + cos(x)**2]]))) == \
MatAdd(A, Matrix([[1]]))
def test_matmul_simplify():
A = MatrixSymbol('A', 1, 1)
assert simplify(MatMul(A, ImmutableMatrix([[sin(x)**2 + cos(x)**2]]))) == \
MatMul(A, Matrix([[1]]))
def test_invariants():
A = MatrixSymbol('A', n, m)
B = MatrixSymbol('B', m, l)
X = MatrixSymbol('X', n, n)
objs = [Identity(n), ZeroMatrix(m, n), A, MatMul(A, B), MatAdd(A, A),
Transpose(A), Adjoint(A), Inverse(X), MatPow(X, 2), MatPow(X, -1),
MatPow(X, 0)]
for obj in objs:
assert obj == obj.__class__(*obj.args)
def test_indexing():
A = MatrixSymbol('A', n, m)
A[1, 2]
A[l, k]
A[l+1, k+1]
def test_single_indexing():
A = MatrixSymbol('A', 2, 3)
assert A[1] == A[0, 1]
assert A[int(1)] == A[0, 1]
assert A[3] == A[1, 0]
assert list(A[:2, :2]) == [A[0, 0], A[0, 1], A[1, 0], A[1, 1]]
raises(IndexError, lambda: A[6])
raises(IndexError, lambda: A[n])
B = MatrixSymbol('B', n, m)
raises(IndexError, lambda: B[1])
B = MatrixSymbol('B', n, 3)
assert B[3] == B[1, 0]
def test_MatrixElement_commutative():
assert A[0, 1]*A[1, 0] == A[1, 0]*A[0, 1]
def test_MatrixSymbol_determinant():
A = MatrixSymbol('A', 4, 4)
assert A.as_explicit().det() == A[0, 0]*A[1, 1]*A[2, 2]*A[3, 3] - \
A[0, 0]*A[1, 1]*A[2, 3]*A[3, 2] - A[0, 0]*A[1, 2]*A[2, 1]*A[3, 3] + \
A[0, 0]*A[1, 2]*A[2, 3]*A[3, 1] + A[0, 0]*A[1, 3]*A[2, 1]*A[3, 2] - \
A[0, 0]*A[1, 3]*A[2, 2]*A[3, 1] - A[0, 1]*A[1, 0]*A[2, 2]*A[3, 3] + \
A[0, 1]*A[1, 0]*A[2, 3]*A[3, 2] + A[0, 1]*A[1, 2]*A[2, 0]*A[3, 3] - \
A[0, 1]*A[1, 2]*A[2, 3]*A[3, 0] - A[0, 1]*A[1, 3]*A[2, 0]*A[3, 2] + \
A[0, 1]*A[1, 3]*A[2, 2]*A[3, 0] + A[0, 2]*A[1, 0]*A[2, 1]*A[3, 3] - \
A[0, 2]*A[1, 0]*A[2, 3]*A[3, 1] - A[0, 2]*A[1, 1]*A[2, 0]*A[3, 3] + \
A[0, 2]*A[1, 1]*A[2, 3]*A[3, 0] + A[0, 2]*A[1, 3]*A[2, 0]*A[3, 1] - \
A[0, 2]*A[1, 3]*A[2, 1]*A[3, 0] - A[0, 3]*A[1, 0]*A[2, 1]*A[3, 2] + \
A[0, 3]*A[1, 0]*A[2, 2]*A[3, 1] + A[0, 3]*A[1, 1]*A[2, 0]*A[3, 2] - \
A[0, 3]*A[1, 1]*A[2, 2]*A[3, 0] - A[0, 3]*A[1, 2]*A[2, 0]*A[3, 1] + \
A[0, 3]*A[1, 2]*A[2, 1]*A[3, 0]
def test_MatrixElement_diff():
assert (A[3, 0]*A[0, 0]).diff(A[0, 0]) == A[3, 0]
def test_MatrixElement_doit():
u = MatrixSymbol('u', 2, 1)
v = ImmutableMatrix([3, 5])
assert u[0, 0].subs(u, v).doit() == v[0, 0]
def test_identity_powers():
M = Identity(n)
assert MatPow(M, 3).doit() == M**3
assert M**n == M
assert MatPow(M, 0).doit() == M**2
assert M**-2 == M
assert MatPow(M, -2).doit() == M**0
N = Identity(3)
assert MatPow(N, 2).doit() == N**n
assert MatPow(N, 3).doit() == N
assert MatPow(N, -2).doit() == N**4
assert MatPow(N, 2).doit() == N**0
def test_Zero_power():
z1 = ZeroMatrix(n, n)
assert z1**4 == z1
raises(ValueError, lambda:z1**-2)
assert z1**0 == Identity(n)
assert MatPow(z1, 2).doit() == z1**2
raises(ValueError, lambda:MatPow(z1, -2).doit())
z2 = ZeroMatrix(3, 3)
assert MatPow(z2, 4).doit() == z2**4
raises(ValueError, lambda:z2**-3)
assert z2**3 == MatPow(z2, 3).doit()
assert z2**0 == Identity(3)
raises(ValueError, lambda:MatPow(z2, -1).doit())
def test_matrixelement_diff():
dexpr = diff((D*w)[k,0], w[p,0])
assert w[k, p].diff(w[k, p]) == 1
assert w[k, p].diff(w[0, 0]) == KroneckerDelta(0, k, (0, n-1))*KroneckerDelta(0, p, (0, 0))
_i_1 = Dummy("_i_1")
assert dexpr.dummy_eq(Sum(KroneckerDelta(_i_1, p, (0, n-1))*D[k, _i_1], (_i_1, 0, n - 1)))
assert dexpr.doit() == D[k, p]
def test_MatrixElement_with_values():
x, y, z, w = symbols("x y z w")
M = Matrix([[x, y], [z, w]])
i, j = symbols("i, j")
Mij = M[i, j]
assert isinstance(Mij, MatrixElement)
Ms = SparseMatrix([[2, 3], [4, 5]])
msij = Ms[i, j]
assert isinstance(msij, MatrixElement)
for oi, oj in [(0, 0), (0, 1), (1, 0), (1, 1)]:
assert Mij.subs({i: oi, j: oj}) == M[oi, oj]
assert msij.subs({i: oi, j: oj}) == Ms[oi, oj]
A = MatrixSymbol("A", 2, 2)
assert A[0, 0].subs(A, M) == x
assert A[i, j].subs(A, M) == M[i, j]
assert M[i, j].subs(M, A) == A[i, j]
assert isinstance(M[3*i - 2, j], MatrixElement)
assert M[3*i - 2, j].subs({i: 1, j: 0}) == M[1, 0]
assert isinstance(M[i, 0], MatrixElement)
assert M[i, 0].subs(i, 0) == M[0, 0]
assert M[0, i].subs(i, 1) == M[0, 1]
assert M[i, j].diff(x) == Matrix([[1, 0], [0, 0]])[i, j]
raises(ValueError, lambda: M[i, 2])
raises(ValueError, lambda: M[i, -1])
raises(ValueError, lambda: M[2, i])
raises(ValueError, lambda: M[-1, i])
def test_inv():
B = MatrixSymbol('B', 3, 3)
assert B.inv() == B**-1
@XFAIL
def test_factor_expand():
A = MatrixSymbol("A", n, n)
B = MatrixSymbol("B", n, n)
expr1 = (A + B)*(C + D)
expr2 = A*C + B*C + A*D + B*D
assert expr1 != expr2
assert expand(expr1) == expr2
assert factor(expr2) == expr1
expr = B**(-1)*(A**(-1)*B**(-1) - A**(-1)*C*B**(-1))**(-1)*A**(-1)
I = Identity(n)
# Ideally we get the first, but we at least don't want a wrong answer
assert factor(expr) in [I - C, B**-1*(A**-1*(I - C)*B**-1)**-1*A**-1]
def test_issue_2749():
A = MatrixSymbol("A", 5, 2)
assert (A.T * A).I.as_explicit() == Matrix([[(A.T * A).I[0, 0], (A.T * A).I[0, 1]], \
[(A.T * A).I[1, 0], (A.T * A).I[1, 1]]])
def test_issue_2750():
x = MatrixSymbol('x', 1, 1)
assert (x.T*x).as_explicit()**-1 == Matrix([[x[0, 0]**(-2)]])
def test_issue_7842():
A = MatrixSymbol('A', 3, 1)
B = MatrixSymbol('B', 2, 1)
assert Eq(A, B) == False
assert Eq(A[1,0], B[1, 0]).func is Eq
A = ZeroMatrix(2, 3)
B = ZeroMatrix(2, 3)
assert Eq(A, B) == True
def test_issue_21195():
t = symbols('t')
x = Function('x')(t)
dx = x.diff(t)
exp1 = cos(x) + cos(x)*dx
exp2 = sin(x) + tan(x)*(dx.diff(t))
exp3 = sin(x)*sin(t)*(dx.diff(t)).diff(t)
A = Matrix([[exp1], [exp2], [exp3]])
B = Matrix([[exp1.diff(x)], [exp2.diff(x)], [exp3.diff(x)]])
assert A.diff(x) == B
def test_MatMul_postprocessor():
z = zeros(2)
z1 = ZeroMatrix(2, 2)
assert Mul(0, z) == Mul(z, 0) in [z, z1]
M = Matrix([[1, 2], [3, 4]])
Mx = Matrix([[x, 2*x], [3*x, 4*x]])
assert Mul(x, M) == Mul(M, x) == Mx
A = MatrixSymbol("A", 2, 2)
assert Mul(A, M) == MatMul(A, M)
assert Mul(M, A) == MatMul(M, A)
# Scalars should be absorbed into constant matrices
a = Mul(x, M, A)
b = Mul(M, x, A)
c = Mul(M, A, x)
assert a == b == c == MatMul(Mx, A)
a = Mul(x, A, M)
b = Mul(A, x, M)
c = Mul(A, M, x)
assert a == b == c == MatMul(A, Mx)
assert Mul(M, M) == M**2
assert Mul(A, M, M) == MatMul(A, M**2)
assert Mul(M, M, A) == MatMul(M**2, A)
assert Mul(M, A, M) == MatMul(M, A, M)
assert Mul(A, x, M, M, x) == MatMul(A, Mx**2)
@XFAIL
def test_MatAdd_postprocessor_xfail():
# This is difficult to get working because of the way that Add processes
# its args.
z = zeros(2)
assert Add(z, S.NaN) == Add(S.NaN, z)
def test_MatAdd_postprocessor():
# Some of these are nonsensical, but we do not raise errors for Add
# because that breaks algorithms that want to replace matrices with dummy
# symbols.
z = zeros(2)
assert Add(0, z) == Add(z, 0) == z
a = Add(S.Infinity, z)
assert a == Add(z, S.Infinity)
assert isinstance(a, Add)
assert a.args == (S.Infinity, z)
a = Add(S.ComplexInfinity, z)
assert a == Add(z, S.ComplexInfinity)
assert isinstance(a, Add)
assert a.args == (S.ComplexInfinity, z)
a = Add(z, S.NaN)
# assert a == Add(S.NaN, z) # See the XFAIL above
assert isinstance(a, Add)
assert a.args == (S.NaN, z)
M = Matrix([[1, 2], [3, 4]])
a = Add(x, M)
assert a == Add(M, x)
assert isinstance(a, Add)
assert a.args == (x, M)
A = MatrixSymbol("A", 2, 2)
assert Add(A, M) == Add(M, A) == A + M
# Scalars should be absorbed into constant matrices (producing an error)
a = Add(x, M, A)
assert a == Add(M, x, A) == Add(M, A, x) == Add(x, A, M) == Add(A, x, M) == Add(A, M, x)
assert isinstance(a, Add)
assert a.args == (x, A + M)
assert Add(M, M) == 2*M
assert Add(M, A, M) == Add(M, M, A) == Add(A, M, M) == A + 2*M
a = Add(A, x, M, M, x)
assert isinstance(a, Add)
assert a.args == (2*x, A + 2*M)
def test_simplify_matrix_expressions():
# Various simplification functions
assert type(gcd_terms(C*D + D*C)) == MatAdd
a = gcd_terms(2*C*D + 4*D*C)
assert type(a) == MatAdd
assert a.args == (2*C*D, 4*D*C)
def test_exp():
A = MatrixSymbol('A', 2, 2)
B = MatrixSymbol('B', 2, 2)
expr1 = exp(A)*exp(B)
expr2 = exp(B)*exp(A)
assert expr1 != expr2
assert expr1 - expr2 != 0
assert not isinstance(expr1, exp)
assert not isinstance(expr2, exp)
def test_invalid_args():
raises(SympifyError, lambda: MatrixSymbol(1, 2, 'A'))
def test_matrixsymbol_from_symbol():
# The label should be preserved during doit and subs
A_label = Symbol('A', complex=True)
A = MatrixSymbol(A_label, 2, 2)
A_1 = A.doit()
A_2 = A.subs(2, 3)
assert A_1.args == A.args
assert A_2.args[0] == A.args[0]
def test_as_explicit():
Z = MatrixSymbol('Z', 2, 3)
assert Z.as_explicit() == ImmutableMatrix([
[Z[0, 0], Z[0, 1], Z[0, 2]],
[Z[1, 0], Z[1, 1], Z[1, 2]],
])
raises(ValueError, lambda: A.as_explicit())
def test_MatrixSet():
M = MatrixSet(2, 2, set=S.Reals)
assert M.shape == (2, 2)
assert M.set == S.Reals
X = Matrix([[1, 2], [3, 4]])
assert X in M
X = ZeroMatrix(2, 2)
assert X in M
raises(TypeError, lambda: A in M)
raises(TypeError, lambda: 1 in M)
M = MatrixSet(n, m, set=S.Reals)
assert A in M
raises(TypeError, lambda: C in M)
raises(TypeError, lambda: X in M)
M = MatrixSet(2, 2, set={1, 2, 3})
X = Matrix([[1, 2], [3, 4]])
Y = Matrix([[1, 2]])
assert (X in M) == S.false
assert (Y in M) == S.false
raises(ValueError, lambda: MatrixSet(2, -2, S.Reals))
raises(ValueError, lambda: MatrixSet(2.4, -1, S.Reals))
raises(TypeError, lambda: MatrixSet(2, 2, (1, 2, 3)))
def test_matrixsymbol_solving():
A = MatrixSymbol('A', 2, 2)
B = MatrixSymbol('B', 2, 2)
Z = ZeroMatrix(2, 2)
assert -(-A + B) - A + B == Z
assert (-(-A + B) - A + B).simplify() == Z
assert (-(-A + B) - A + B).expand() == Z
assert (-(-A + B) - A + B - Z).simplify() == Z
assert (-(-A + B) - A + B - Z).expand() == Z
|
16d6dda53edb9a8253fc018442417f40386908f486453a41973c5dfd4bf459d8 | 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])],
])
def test_applyfunc_transpose():
af = Xk.applyfunc(sin)
assert af.T.dummy_eq(Xk.T.applyfunc(sin))
|
9112ada911427b236840a458687d4b9c1428fd20cfa245113851506082bde804 | from sympy import Set, symbols, exp, log, S, Wild, Dummy, oo, Float
from sympy.core import Expr, Add
from sympy.core.function import Lambda, _coeff_isneg, FunctionClass
from sympy.logic.boolalg import true
from sympy.multipledispatch import dispatch
from sympy.sets import (imageset, Interval, FiniteSet, Union, ImageSet,
EmptySet, Intersection, Range)
from sympy.sets.fancysets import Integers, Naturals, Reals
from sympy.functions.elementary.exponential import match_real_imag
_x, _y = symbols("x y")
FunctionUnion = (FunctionClass, Lambda)
@dispatch(FunctionClass, Set) # type: ignore # noqa:F811
def _set_function(f, x): # noqa:F811
return None
@dispatch(FunctionUnion, FiniteSet) # type: ignore # noqa:F811
def _set_function(f, x): # noqa:F811
return FiniteSet(*map(f, x))
@dispatch(Lambda, Interval) # type: ignore # noqa:F811
def _set_function(f, x): # noqa:F811
from sympy.functions.elementary.miscellaneous import Min, Max
from sympy.solvers.solveset import solveset
from sympy.core.function import diff, Lambda
from sympy.series import limit
from sympy.calculus.singularities import singularities
from sympy.sets import Complement
# TODO: handle functions with infinitely many solutions (eg, sin, tan)
# TODO: handle multivariate functions
expr = f.expr
if len(expr.free_symbols) > 1 or len(f.variables) != 1:
return
var = f.variables[0]
if not var.is_real:
if expr.subs(var, Dummy(real=True)).is_real is False:
return
if expr.is_Piecewise:
result = S.EmptySet
domain_set = x
for (p_expr, p_cond) in expr.args:
if p_cond is true:
intrvl = domain_set
else:
intrvl = p_cond.as_set()
intrvl = Intersection(domain_set, intrvl)
if p_expr.is_Number:
image = FiniteSet(p_expr)
else:
image = imageset(Lambda(var, p_expr), intrvl)
result = Union(result, image)
# remove the part which has been `imaged`
domain_set = Complement(domain_set, intrvl)
if domain_set is S.EmptySet:
break
return result
if not x.start.is_comparable or not x.end.is_comparable:
return
try:
from sympy.polys.polyutils import _nsort
sing = list(singularities(expr, var, x))
if len(sing) > 1:
sing = _nsort(sing)
except NotImplementedError:
return
if x.left_open:
_start = limit(expr, var, x.start, dir="+")
elif x.start not in sing:
_start = f(x.start)
if x.right_open:
_end = limit(expr, var, x.end, dir="-")
elif x.end not in sing:
_end = f(x.end)
if len(sing) == 0:
soln_expr = solveset(diff(expr, var), var)
if not (isinstance(soln_expr, FiniteSet) or soln_expr is EmptySet):
return
solns = list(soln_expr)
extr = [_start, _end] + [f(i) for i in solns
if i.is_real and i in x]
start, end = Min(*extr), Max(*extr)
left_open, right_open = False, False
if _start <= _end:
# the minimum or maximum value can occur simultaneously
# on both the edge of the interval and in some interior
# point
if start == _start and start not in solns:
left_open = x.left_open
if end == _end and end not in solns:
right_open = x.right_open
else:
if start == _end and start not in solns:
left_open = x.right_open
if end == _start and end not in solns:
right_open = x.left_open
return Interval(start, end, left_open, right_open)
else:
return imageset(f, Interval(x.start, sing[0],
x.left_open, True)) + \
Union(*[imageset(f, Interval(sing[i], sing[i + 1], True, True))
for i in range(0, len(sing) - 1)]) + \
imageset(f, Interval(sing[-1], x.end, True, x.right_open))
@dispatch(FunctionClass, Interval) # type: ignore # noqa:F811
def _set_function(f, x): # noqa:F811
if f == exp:
return Interval(exp(x.start), exp(x.end), x.left_open, x.right_open)
elif f == log:
return Interval(log(x.start), log(x.end), x.left_open, x.right_open)
return ImageSet(Lambda(_x, f(_x)), x)
@dispatch(FunctionUnion, Union) # type: ignore # noqa:F811
def _set_function(f, x): # noqa:F811
return Union(*(imageset(f, arg) for arg in x.args))
@dispatch(FunctionUnion, Intersection) # type: ignore # noqa:F811
def _set_function(f, x): # noqa:F811
from sympy.sets.sets import is_function_invertible_in_set
# If the function is invertible, intersect the maps of the sets.
if is_function_invertible_in_set(f, x):
return Intersection(*(imageset(f, arg) for arg in x.args))
else:
return ImageSet(Lambda(_x, f(_x)), x)
@dispatch(FunctionUnion, type(EmptySet)) # type: ignore # noqa:F811
def _set_function(f, x): # noqa:F811
return x
@dispatch(FunctionUnion, Set) # type: ignore # noqa:F811
def _set_function(f, x): # noqa:F811
return ImageSet(Lambda(_x, f(_x)), x)
@dispatch(FunctionUnion, Range) # type: ignore # noqa:F811
def _set_function(f, self): # noqa:F811
from sympy.core.function import expand_mul
if not self:
return S.EmptySet
if not isinstance(f.expr, Expr):
return
if self.size == 1:
return FiniteSet(f(self[0]))
if f is S.IdentityFunction:
return self
x = f.variables[0]
expr = f.expr
# handle f that is linear in f's variable
if x not in expr.free_symbols or x in expr.diff(x).free_symbols:
return
if self.start.is_finite:
F = f(self.step*x + self.start) # for i in range(len(self))
else:
F = f(-self.step*x + self[-1])
F = expand_mul(F)
if F != expr:
return imageset(x, F, Range(self.size))
@dispatch(FunctionUnion, Integers) # type: ignore # noqa:F811
def _set_function(f, self): # noqa:F811
expr = f.expr
if not isinstance(expr, Expr):
return
n = f.variables[0]
if expr == abs(n):
return S.Naturals0
# f(x) + c and f(-x) + c cover the same integers
# so choose the form that has the fewest negatives
c = f(0)
fx = f(n) - c
f_x = f(-n) - c
neg_count = lambda e: sum(_coeff_isneg(_) for _ in Add.make_args(e))
if neg_count(f_x) < neg_count(fx):
expr = f_x + c
a = Wild('a', exclude=[n])
b = Wild('b', exclude=[n])
match = expr.match(a*n + b)
if match and match[a] and (
not match[a].atoms(Float) and
not match[b].atoms(Float)):
# canonical shift
a, b = match[a], match[b]
if a in [1, -1]:
# drop integer addends in b
nonint = []
for bi in Add.make_args(b):
if not bi.is_integer:
nonint.append(bi)
b = Add(*nonint)
if b.is_number and a.is_real:
# avoid Mod for complex numbers, #11391
br, bi = match_real_imag(b)
if br and br.is_comparable and a.is_comparable:
br %= a
b = br + S.ImaginaryUnit*bi
elif b.is_number and a.is_imaginary:
br, bi = match_real_imag(b)
ai = a/S.ImaginaryUnit
if bi and bi.is_comparable and ai.is_comparable:
bi %= ai
b = br + S.ImaginaryUnit*bi
expr = a*n + b
if expr != f.expr:
return ImageSet(Lambda(n, expr), S.Integers)
@dispatch(FunctionUnion, Naturals) # type: ignore # noqa:F811
def _set_function(f, self): # noqa:F811
expr = f.expr
if not isinstance(expr, Expr):
return
x = f.variables[0]
if not expr.free_symbols - {x}:
if expr == abs(x):
if self is S.Naturals:
return self
return S.Naturals0
step = expr.coeff(x)
c = expr.subs(x, 0)
if c.is_Integer and step.is_Integer and expr == step*x + c:
if self is S.Naturals:
c += step
if step > 0:
if step == 1:
if c == 0:
return S.Naturals0
elif c == 1:
return S.Naturals
return Range(c, oo, step)
return Range(c, -oo, step)
@dispatch(FunctionUnion, Reals) # type: ignore # noqa:F811
def _set_function(f, self): # noqa:F811
expr = f.expr
if not isinstance(expr, Expr):
return
return _set_function(f, Interval(-oo, oo))
|
0e3f1e1ed7fe1cb938a66243015cbecce32d02536a19960ab1d06715a9aaf89c | 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, z
from sympy.core.mod import Mod
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
ip = symbols('ip', positive=True)
assert Range(0, ip, -1) == empty
assert Range(0, -ip, 1) == 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)
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)[0])
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)
assert empty.as_relational(x) is S.false
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(Mod(x, 1), 0)
assert Range(oo, 1, -2).as_relational(x) == (x >= 3) & (x < oo) & Eq(Mod(x + 1, -2), 0)
def test_Range_symbolic():
# symbolic Range
xr = Range(x, x + 4, 5)
sr = Range(x, y, t)
i = Symbol('i', integer=True)
ip = Symbol('i', integer=True, positive=True)
ipr = Range(ip)
inr = Range(0, -ip, -1)
ir = Range(i, i + 19, 2)
ir2 = Range(i, i*8, 3*i)
i = Symbol('i', integer=True)
inf = symbols('inf', infinite=True)
raises(ValueError, lambda: Range(inf))
raises(ValueError, lambda: Range(inf, 0, -1))
raises(ValueError, lambda: Range(inf, inf, 1))
raises(ValueError, lambda: Range(1, 1, inf))
# args
assert xr.args == (x, x + 5, 5)
assert sr.args == (x, y, t)
assert ir.args == (i, i + 20, 2)
assert ir2.args == (i, 10*i, 3*i)
# reversed
raises(ValueError, lambda: xr.reversed)
raises(ValueError, lambda: sr.reversed)
assert ipr.reversed.args == (ip - 1, -1, -1)
assert inr.reversed.args == (-ip + 1, 1, 1)
assert ir.reversed.args == (i + 18, i - 2, -2)
assert ir2.reversed.args == (7*i, -2*i, -3*i)
# contains
assert inf not in sr
assert inf not in ir
assert 0 in ipr
assert 0 in inr
raises(TypeError, lambda: 1 in ipr)
raises(TypeError, lambda: -1 in inr)
assert .1 not in sr
assert .1 not in ir
assert i + 1 not in ir
assert i + 2 in ir
raises(TypeError, lambda: x in xr) # XXX is this what contains is supposed to do?
raises(TypeError, lambda: 1 in sr) # XXX is this what contains is supposed to do?
# iter
raises(ValueError, lambda: next(iter(xr)))
raises(ValueError, lambda: next(iter(sr)))
assert next(iter(ir)) == i
assert next(iter(ir2)) == i
assert sr.intersect(S.Integers) == sr
assert sr.intersect(FiniteSet(x)) == Intersection({x}, sr)
raises(ValueError, lambda: sr[:2])
raises(ValueError, lambda: xr[0])
raises(ValueError, lambda: sr[0])
# len
assert len(ir) == ir.size == 10
assert len(ir2) == ir2.size == 3
raises(ValueError, lambda: len(xr))
raises(ValueError, lambda: xr.size)
raises(ValueError, lambda: len(sr))
raises(ValueError, lambda: sr.size)
# bool
assert bool(Range(0)) == False
assert bool(xr)
assert bool(ir)
assert bool(ipr)
assert bool(inr)
raises(ValueError, lambda: bool(sr))
raises(ValueError, lambda: bool(ir2))
# inf
raises(ValueError, lambda: xr.inf)
raises(ValueError, lambda: sr.inf)
assert ipr.inf == 0
assert inr.inf == -ip + 1
assert ir.inf == i
raises(ValueError, lambda: ir2.inf)
# sup
raises(ValueError, lambda: xr.sup)
raises(ValueError, lambda: sr.sup)
assert ipr.sup == ip - 1
assert inr.sup == 0
assert ir.inf == i
raises(ValueError, lambda: ir2.sup)
# getitem
raises(ValueError, lambda: xr[0])
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
assert ir2[:2] == Range(i, 7*i, 3*i)
assert ir2[0] == i
assert ir2[-2] == 4*i
assert ir2[-1] == 7*i
raises(ValueError, lambda: Range(i)[-1])
assert ipr[0] == ipr.inf == 0
assert ipr[-1] == ipr.sup == ip - 1
assert inr[0] == inr.sup == 0
assert inr[-1] == inr.inf == -ip + 1
raises(ValueError, lambda: ipr[-2])
assert ir.inf == i
assert ir.sup == i + 18
raises(ValueError, lambda: Range(i).inf)
# as_relational
assert ir.as_relational(x) == ((x >= i) & (x <= i + 18) &
Eq(Mod(-i + x, 2), 0))
assert ir2.as_relational(x) == Eq(
Mod(-i + x, 3*i), 0) & (((x >= i) & (x <= 7*i) & (3*i >= 1)) |
((x <= i) & (x >= 7*i) & (3*i <= -1)))
assert Range(i, i + 1).as_relational(x) == Eq(x, i)
assert sr.as_relational(z) == Eq(
Mod(t, 1), 0) & Eq(Mod(x, 1), 0) & Eq(Mod(-x + z, t), 0
) & (((z >= x) & (z <= -t + y) & (t >= 1)) |
((z <= x) & (z >= -t + y) & (t <= -1)))
assert xr.as_relational(z) == Eq(z, x) & Eq(Mod(x, 1), 0)
# symbols can clash if user wants (but it must be integer)
assert xr.as_relational(x) == Eq(Mod(x, 1), 0)
# 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
a, b, c = 0.092, 0.433, 0.341
assert a in imageset(x, a + c*x, S.Integers)
assert b in imageset(x, b + c*x, S.Integers)
_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 S.Rationals.contains(0.5) == Contains(0.5, S.Rationals, evaluate=False)
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])
|
4858b80f5dd95d42651293a7bc0060c68960db6116e793d2e9abc4670d8d81b3 | 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 unchanged(ProductSet, ProductSet(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))
assert Or(x < 0, x > 0).as_set().as_relational(x) == \
And((x > -oo), (x < oo), Ne(x, 0))
assert (Interval.Ropen(1, 3) + Interval.Lopen(3, 5)
).as_relational(x) == And((x > 1), (x < 5), Ne(x, 3))
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) == {(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)) is 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(S(1), S(2) , S(3)) ^ Set(S(2), S(3), S(4)) == Union(Set(S(1), S(2), S(3)) - Set(S(2), S(3), S(4)), \
Set(S(2), S(3), S(4)) - Set(S(1), S(2), S(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)))
def test_issue_20089():
B = FiniteSet(FiniteSet(1, 2), FiniteSet(1))
assert not 1 in B
assert not 1.0 in B
assert not Eq(1, FiniteSet(1, 2))
assert FiniteSet(1) in B
A = FiniteSet(1, 2)
assert A in B
assert B.issubset(B)
assert not A.issubset(B)
assert 1 in A
C = FiniteSet(FiniteSet(1, 2), FiniteSet(1), 1, 2)
assert A.issubset(C)
assert B.issubset(C)
def test_issue_19378():
a = FiniteSet(1, 2)
b = ProductSet(a, a)
c = FiniteSet((1, 1), (1, 2), (2, 1), (2, 2))
assert b.is_subset(c) is True
d = FiniteSet(1)
assert b.is_subset(d) is False
assert Eq(c, b).simplify() is S.true
assert Eq(a, c).simplify() is S.false
assert Eq({1}, {x}).simplify() == Eq({1}, {x})
def test_issue_20379():
#https://github.com/sympy/sympy/issues/20379
x = pi - 3.14159265358979
assert FiniteSet(x).evalf(2) == FiniteSet(Float('3.23108914886517e-15', 2))
def test_finiteset_simplify():
S = FiniteSet(1, cos(1)**2 + sin(1)**2)
assert S.simplify() == {1}
|
e92afef2aa1c2f3e5a46b3aa9c303091b0a3891d812168e81b44c9f49125b7f5 | import os
from tempfile import TemporaryDirectory
from sympy import (
pi, sin, cos, Symbol, Integral, Sum, sqrt, log, exp, Ne, oo, LambertW, I,
meijerg, exp_polar, Piecewise, And, real_root)
from sympy.core.singleton import S
from sympy.core.sympify import sympify
from sympy.external import import_module
from sympy.plotting.plot import (
Plot, plot, plot_parametric, plot3d_parametric_line, plot3d,
plot3d_parametric_surface)
from sympy.plotting.plot import (
unset_show, plot_contour, PlotGrid, DefaultBackend, MatplotlibBackend,
TextBackend, BaseBackend)
from sympy.testing.pytest import skip, raises, warns
from sympy.utilities import lambdify as lambdify_
unset_show()
matplotlib = import_module(
'matplotlib', min_module_version='1.1.0', catch=(RuntimeError,))
class DummyBackendNotOk(BaseBackend):
""" Used to verify if users can create their own backends.
This backend is meant to raise NotImplementedError for methods `show`,
`save`, `close`.
"""
pass
class DummyBackendOk(BaseBackend):
""" Used to verify if users can create their own backends.
This backend is meant to pass all tests.
"""
def show(self):
pass
def save(self):
pass
def close(self):
pass
def test_plot_and_save_1():
if not matplotlib:
skip("Matplotlib not the default backend")
x = Symbol('x')
y = Symbol('y')
with TemporaryDirectory(prefix='sympy_') as tmpdir:
###
# Examples from the 'introduction' notebook
###
p = plot(x, legend=True, label='f1')
p = plot(x*sin(x), x*cos(x), label='f2')
p.extend(p)
p[0].line_color = lambda a: a
p[1].line_color = 'b'
p.title = 'Big title'
p.xlabel = 'the x axis'
p[1].label = 'straight line'
p.legend = True
p.aspect_ratio = (1, 1)
p.xlim = (-15, 20)
filename = 'test_basic_options_and_colors.png'
p.save(os.path.join(tmpdir, filename))
p._backend.close()
p.extend(plot(x + 1))
p.append(plot(x + 3, x**2)[1])
filename = 'test_plot_extend_append.png'
p.save(os.path.join(tmpdir, filename))
p[2] = plot(x**2, (x, -2, 3))
filename = 'test_plot_setitem.png'
p.save(os.path.join(tmpdir, filename))
p._backend.close()
p = plot(sin(x), (x, -2*pi, 4*pi))
filename = 'test_line_explicit.png'
p.save(os.path.join(tmpdir, filename))
p._backend.close()
p = plot(sin(x))
filename = 'test_line_default_range.png'
p.save(os.path.join(tmpdir, filename))
p._backend.close()
p = plot((x**2, (x, -5, 5)), (x**3, (x, -3, 3)))
filename = 'test_line_multiple_range.png'
p.save(os.path.join(tmpdir, filename))
p._backend.close()
raises(ValueError, lambda: plot(x, y))
#Piecewise plots
p = plot(Piecewise((1, x > 0), (0, True)), (x, -1, 1))
filename = 'test_plot_piecewise.png'
p.save(os.path.join(tmpdir, filename))
p._backend.close()
p = plot(Piecewise((x, x < 1), (x**2, True)), (x, -3, 3))
filename = 'test_plot_piecewise_2.png'
p.save(os.path.join(tmpdir, filename))
p._backend.close()
# test issue 7471
p1 = plot(x)
p2 = plot(3)
p1.extend(p2)
filename = 'test_horizontal_line.png'
p.save(os.path.join(tmpdir, filename))
p._backend.close()
# test issue 10925
f = Piecewise((-1, x < -1), (x, And(-1 <= x, x < 0)), \
(x**2, And(0 <= x, x < 1)), (x**3, x >= 1))
p = plot(f, (x, -3, 3))
filename = 'test_plot_piecewise_3.png'
p.save(os.path.join(tmpdir, filename))
p._backend.close()
def test_plot_and_save_2():
if not matplotlib:
skip("Matplotlib not the default backend")
x = Symbol('x')
y = Symbol('y')
z = Symbol('z')
with TemporaryDirectory(prefix='sympy_') as tmpdir:
#parametric 2d plots.
#Single plot with default range.
p = plot_parametric(sin(x), cos(x))
filename = 'test_parametric.png'
p.save(os.path.join(tmpdir, filename))
p._backend.close()
#Single plot with range.
p = plot_parametric(
sin(x), cos(x), (x, -5, 5), legend=True, label='parametric_plot')
filename = 'test_parametric_range.png'
p.save(os.path.join(tmpdir, filename))
p._backend.close()
#Multiple plots with same range.
p = plot_parametric((sin(x), cos(x)), (x, sin(x)))
filename = 'test_parametric_multiple.png'
p.save(os.path.join(tmpdir, filename))
p._backend.close()
#Multiple plots with different ranges.
p = plot_parametric(
(sin(x), cos(x), (x, -3, 3)), (x, sin(x), (x, -5, 5)))
filename = 'test_parametric_multiple_ranges.png'
p.save(os.path.join(tmpdir, filename))
p._backend.close()
#depth of recursion specified.
p = plot_parametric(x, sin(x), depth=13)
filename = 'test_recursion_depth.png'
p.save(os.path.join(tmpdir, filename))
p._backend.close()
#No adaptive sampling.
p = plot_parametric(cos(x), sin(x), adaptive=False, nb_of_points=500)
filename = 'test_adaptive.png'
p.save(os.path.join(tmpdir, filename))
p._backend.close()
#3d parametric plots
p = plot3d_parametric_line(
sin(x), cos(x), x, legend=True, label='3d_parametric_plot')
filename = 'test_3d_line.png'
p.save(os.path.join(tmpdir, filename))
p._backend.close()
p = plot3d_parametric_line(
(sin(x), cos(x), x, (x, -5, 5)), (cos(x), sin(x), x, (x, -3, 3)))
filename = 'test_3d_line_multiple.png'
p.save(os.path.join(tmpdir, filename))
p._backend.close()
p = plot3d_parametric_line(sin(x), cos(x), x, nb_of_points=30)
filename = 'test_3d_line_points.png'
p.save(os.path.join(tmpdir, filename))
p._backend.close()
# 3d surface single plot.
p = plot3d(x * y)
filename = 'test_surface.png'
p.save(os.path.join(tmpdir, filename))
p._backend.close()
# Multiple 3D plots with same range.
p = plot3d(-x * y, x * y, (x, -5, 5))
filename = 'test_surface_multiple.png'
p.save(os.path.join(tmpdir, filename))
p._backend.close()
# Multiple 3D plots with different ranges.
p = plot3d(
(x * y, (x, -3, 3), (y, -3, 3)), (-x * y, (x, -3, 3), (y, -3, 3)))
filename = 'test_surface_multiple_ranges.png'
p.save(os.path.join(tmpdir, filename))
p._backend.close()
# Single Parametric 3D plot
p = plot3d_parametric_surface(sin(x + y), cos(x - y), x - y)
filename = 'test_parametric_surface.png'
p.save(os.path.join(tmpdir, filename))
p._backend.close()
# Multiple Parametric 3D plots.
p = plot3d_parametric_surface(
(x*sin(z), x*cos(z), z, (x, -5, 5), (z, -5, 5)),
(sin(x + y), cos(x - y), x - y, (x, -5, 5), (y, -5, 5)))
filename = 'test_parametric_surface.png'
p.save(os.path.join(tmpdir, filename))
p._backend.close()
# Single Contour plot.
p = plot_contour(sin(x)*sin(y), (x, -5, 5), (y, -5, 5))
filename = 'test_contour_plot.png'
p.save(os.path.join(tmpdir, filename))
p._backend.close()
# Multiple Contour plots with same range.
p = plot_contour(x**2 + y**2, x**3 + y**3, (x, -5, 5), (y, -5, 5))
filename = 'test_contour_plot.png'
p.save(os.path.join(tmpdir, filename))
p._backend.close()
# Multiple Contour plots with different range.
p = plot_contour(
(x**2 + y**2, (x, -5, 5), (y, -5, 5)),
(x**3 + y**3, (x, -3, 3), (y, -3, 3)))
filename = 'test_contour_plot.png'
p.save(os.path.join(tmpdir, filename))
p._backend.close()
def test_plot_and_save_3():
if not matplotlib:
skip("Matplotlib not the default backend")
x = Symbol('x')
y = Symbol('y')
z = Symbol('z')
with TemporaryDirectory(prefix='sympy_') as tmpdir:
###
# Examples from the 'colors' notebook
###
p = plot(sin(x))
p[0].line_color = lambda a: a
filename = 'test_colors_line_arity1.png'
p.save(os.path.join(tmpdir, filename))
p[0].line_color = lambda a, b: b
filename = 'test_colors_line_arity2.png'
p.save(os.path.join(tmpdir, filename))
p._backend.close()
p = plot(x*sin(x), x*cos(x), (x, 0, 10))
p[0].line_color = lambda a: a
filename = 'test_colors_param_line_arity1.png'
p.save(os.path.join(tmpdir, filename))
p[0].line_color = lambda a, b: a
filename = 'test_colors_param_line_arity1.png'
p.save(os.path.join(tmpdir, filename))
p[0].line_color = lambda a, b: b
filename = 'test_colors_param_line_arity2b.png'
p.save(os.path.join(tmpdir, filename))
p._backend.close()
p = plot3d_parametric_line(sin(x) + 0.1*sin(x)*cos(7*x),
cos(x) + 0.1*cos(x)*cos(7*x),
0.1*sin(7*x),
(x, 0, 2*pi))
p[0].line_color = lambdify_(x, sin(4*x))
filename = 'test_colors_3d_line_arity1.png'
p.save(os.path.join(tmpdir, filename))
p[0].line_color = lambda a, b: b
filename = 'test_colors_3d_line_arity2.png'
p.save(os.path.join(tmpdir, filename))
p[0].line_color = lambda a, b, c: c
filename = 'test_colors_3d_line_arity3.png'
p.save(os.path.join(tmpdir, filename))
p._backend.close()
p = plot3d(sin(x)*y, (x, 0, 6*pi), (y, -5, 5))
p[0].surface_color = lambda a: a
filename = 'test_colors_surface_arity1.png'
p.save(os.path.join(tmpdir, filename))
p[0].surface_color = lambda a, b: b
filename = 'test_colors_surface_arity2.png'
p.save(os.path.join(tmpdir, filename))
p[0].surface_color = lambda a, b, c: c
filename = 'test_colors_surface_arity3a.png'
p.save(os.path.join(tmpdir, filename))
p[0].surface_color = lambdify_((x, y, z), sqrt((x - 3*pi)**2 + y**2))
filename = 'test_colors_surface_arity3b.png'
p.save(os.path.join(tmpdir, filename))
p._backend.close()
p = plot3d_parametric_surface(x * cos(4 * y), x * sin(4 * y), y,
(x, -1, 1), (y, -1, 1))
p[0].surface_color = lambda a: a
filename = 'test_colors_param_surf_arity1.png'
p.save(os.path.join(tmpdir, filename))
p[0].surface_color = lambda a, b: a*b
filename = 'test_colors_param_surf_arity2.png'
p.save(os.path.join(tmpdir, filename))
p[0].surface_color = lambdify_((x, y, z), sqrt(x**2 + y**2 + z**2))
filename = 'test_colors_param_surf_arity3.png'
p.save(os.path.join(tmpdir, filename))
p._backend.close()
def test_plot_and_save_4():
if not matplotlib:
skip("Matplotlib not the default backend")
x = Symbol('x')
y = Symbol('y')
###
# Examples from the 'advanced' notebook
###
# XXX: This raises the warning "The evaluation of the expression is
# problematic. We are trying a failback method that may still work. Please
# report this as a bug." It has to use the fallback because using evalf()
# is the only way to evaluate the integral. We should perhaps just remove
# that warning.
with TemporaryDirectory(prefix='sympy_') as tmpdir:
with warns(
UserWarning,
match="The evaluation of the expression is problematic"):
i = Integral(log((sin(x)**2 + 1)*sqrt(x**2 + 1)), (x, 0, y))
p = plot(i, (y, 1, 5))
filename = 'test_advanced_integral.png'
p.save(os.path.join(tmpdir, filename))
p._backend.close()
def test_plot_and_save_5():
if not matplotlib:
skip("Matplotlib not the default backend")
x = Symbol('x')
y = Symbol('y')
with TemporaryDirectory(prefix='sympy_') as tmpdir:
s = Sum(1/x**y, (x, 1, oo))
p = plot(s, (y, 2, 10))
filename = 'test_advanced_inf_sum.png'
p.save(os.path.join(tmpdir, filename))
p._backend.close()
p = plot(Sum(1/x, (x, 1, y)), (y, 2, 10), show=False)
p[0].only_integers = True
p[0].steps = True
filename = 'test_advanced_fin_sum.png'
p.save(os.path.join(tmpdir, filename))
p._backend.close()
def test_plot_and_save_6():
if not matplotlib:
skip("Matplotlib not the default backend")
x = Symbol('x')
with TemporaryDirectory(prefix='sympy_') as tmpdir:
filename = 'test.png'
###
# Test expressions that can not be translated to np and generate complex
# results.
###
p = plot(sin(x) + I*cos(x))
p.save(os.path.join(tmpdir, filename))
p = plot(sqrt(sqrt(-x)))
p.save(os.path.join(tmpdir, filename))
p = plot(LambertW(x))
p.save(os.path.join(tmpdir, filename))
p = plot(sqrt(LambertW(x)))
p.save(os.path.join(tmpdir, filename))
#Characteristic function of a StudentT distribution with nu=10
x1 = 5 * x**2 * exp_polar(-I*pi)/2
m1 = meijerg(((1 / 2,), ()), ((5, 0, 1 / 2), ()), x1)
x2 = 5*x**2 * exp_polar(I*pi)/2
m2 = meijerg(((1/2,), ()), ((5, 0, 1/2), ()), x2)
expr = (m1 + m2) / (48 * pi)
p = plot(expr, (x, 1e-6, 1e-2))
p.save(os.path.join(tmpdir, filename))
def test_plotgrid_and_save():
if not matplotlib:
skip("Matplotlib not the default backend")
x = Symbol('x')
y = Symbol('y')
with TemporaryDirectory(prefix='sympy_') as tmpdir:
p1 = plot(x)
p2 = plot_parametric((sin(x), cos(x)), (x, sin(x)), show=False)
p3 = plot_parametric(
cos(x), sin(x), adaptive=False, nb_of_points=500, show=False)
p4 = plot3d_parametric_line(sin(x), cos(x), x, show=False)
# symmetric grid
p = PlotGrid(2, 2, p1, p2, p3, p4)
filename = 'test_grid1.png'
p.save(os.path.join(tmpdir, filename))
p._backend.close()
# grid size greater than the number of subplots
p = PlotGrid(3, 4, p1, p2, p3, p4)
filename = 'test_grid2.png'
p.save(os.path.join(tmpdir, filename))
p._backend.close()
p5 = plot(cos(x),(x, -pi, pi), show=False)
p5[0].line_color = lambda a: a
p6 = plot(Piecewise((1, x > 0), (0, True)), (x, -1, 1), show=False)
p7 = plot_contour(
(x**2 + y**2, (x, -5, 5), (y, -5, 5)),
(x**3 + y**3, (x, -3, 3), (y, -3, 3)), show=False)
# unsymmetric grid (subplots in one line)
p = PlotGrid(1, 3, p5, p6, p7)
filename = 'test_grid3.png'
p.save(os.path.join(tmpdir, filename))
p._backend.close()
def test_append_issue_7140():
if not matplotlib:
skip("Matplotlib not the default backend")
x = Symbol('x')
p1 = plot(x)
p2 = plot(x**2)
plot(x + 2)
# append a series
p2.append(p1[0])
assert len(p2._series) == 2
with raises(TypeError):
p1.append(p2)
with raises(TypeError):
p1.append(p2._series)
def test_issue_15265():
if not matplotlib:
skip("Matplotlib not the default backend")
x = Symbol('x')
eqn = sin(x)
p = plot(eqn, xlim=(-S.Pi, S.Pi), ylim=(-1, 1))
p._backend.close()
p = plot(eqn, xlim=(-1, 1), ylim=(-S.Pi, S.Pi))
p._backend.close()
p = plot(eqn, xlim=(-1, 1), ylim=(sympify('-3.14'), sympify('3.14')))
p._backend.close()
p = plot(eqn, xlim=(sympify('-3.14'), sympify('3.14')), ylim=(-1, 1))
p._backend.close()
raises(ValueError,
lambda: plot(eqn, xlim=(-S.ImaginaryUnit, 1), ylim=(-1, 1)))
raises(ValueError,
lambda: plot(eqn, xlim=(-1, 1), ylim=(-1, S.ImaginaryUnit)))
raises(ValueError,
lambda: plot(eqn, xlim=(S.NegativeInfinity, 1), ylim=(-1, 1)))
raises(ValueError,
lambda: plot(eqn, xlim=(-1, 1), ylim=(-1, S.Infinity)))
def test_empty_Plot():
if not matplotlib:
skip("Matplotlib not the default backend")
# No exception showing an empty plot
plot()
p = Plot()
p.show()
def test_issue_17405():
if not matplotlib:
skip("Matplotlib not the default backend")
x = Symbol('x')
f = x**0.3 - 10*x**3 + x**2
p = plot(f, (x, -10, 10), show=False)
# Random number of segments, probably more than 100, but we want to see
# that there are segments generated, as opposed to when the bug was present
assert len(p[0].get_data()[0]) >= 30
def test_logplot_PR_16796():
if not matplotlib:
skip("Matplotlib not the default backend")
x = Symbol('x')
p = plot(x, (x, .001, 100), xscale='log', show=False)
# Random number of segments, probably more than 100, but we want to see
# that there are segments generated, as opposed to when the bug was present
assert len(p[0].get_data()[0]) >= 30
assert p[0].end == 100.0
assert p[0].start == .001
def test_issue_16572():
if not matplotlib:
skip("Matplotlib not the default backend")
x = Symbol('x')
p = plot(LambertW(x), show=False)
# Random number of segments, probably more than 50, but we want to see
# that there are segments generated, as opposed to when the bug was present
assert len(p[0].get_data()[0]) >= 30
def test_issue_11865():
if not matplotlib:
skip("Matplotlib not the default backend")
k = Symbol('k', integer=True)
f = Piecewise((-I*exp(I*pi*k)/k + I*exp(-I*pi*k)/k, Ne(k, 0)), (2*pi, True))
p = plot(f, show=False)
# Random number of segments, probably more than 100, but we want to see
# that there are segments generated, as opposed to when the bug was present
# and that there are no exceptions.
assert len(p[0].get_data()[0]) >= 30
def test_issue_11461():
if not matplotlib:
skip("Matplotlib not the default backend")
x = Symbol('x')
p = plot(real_root((log(x/(x-2))), 3), show=False)
# Random number of segments, probably more than 100, but we want to see
# that there are segments generated, as opposed to when the bug was present
# and that there are no exceptions.
assert len(p[0].get_data()[0]) >= 30
def test_issue_11764():
if not matplotlib:
skip("Matplotlib not the default backend")
x = Symbol('x')
p = plot_parametric(cos(x), sin(x), (x, 0, 2 * pi), aspect_ratio=(1,1), show=False)
p.aspect_ratio == (1, 1)
# Random number of segments, probably more than 100, but we want to see
# that there are segments generated, as opposed to when the bug was present
assert len(p[0].get_data()[0]) >= 30
def test_issue_13516():
if not matplotlib:
skip("Matplotlib not the default backend")
x = Symbol('x')
pm = plot(sin(x), backend="matplotlib", show=False)
assert pm.backend == MatplotlibBackend
assert len(pm[0].get_data()[0]) >= 30
pt = plot(sin(x), backend="text", show=False)
assert pt.backend == TextBackend
assert len(pt[0].get_data()[0]) >= 30
pd = plot(sin(x), backend="default", show=False)
assert pd.backend == DefaultBackend
assert len(pd[0].get_data()[0]) >= 30
p = plot(sin(x), show=False)
assert p.backend == DefaultBackend
assert len(p[0].get_data()[0]) >= 30
def test_plot_limits():
if not matplotlib:
skip("Matplotlib not the default backend")
x = Symbol('x')
p = plot(x, x**2, (x, -10, 10))
backend = p._backend
xmin, xmax = backend.ax[0].get_xlim()
assert abs(xmin + 10) < 2
assert abs(xmax - 10) < 2
ymin, ymax = backend.ax[0].get_ylim()
assert abs(ymin + 10) < 10
assert abs(ymax - 100) < 10
def test_plot3d_parametric_line_limits():
if not matplotlib:
skip("Matplotlib not the default backend")
x = Symbol('x')
v1 = (2*cos(x), 2*sin(x), 2*x, (x, -5, 5))
v2 = (sin(x), cos(x), x, (x, -5, 5))
p = plot3d_parametric_line(v1, v2)
backend = p._backend
xmin, xmax = backend.ax[0].get_xlim()
assert abs(xmin + 2) < 1e-2
assert abs(xmax - 2) < 1e-2
ymin, ymax = backend.ax[0].get_ylim()
assert abs(ymin + 2) < 1e-2
assert abs(ymax - 2) < 1e-2
zmin, zmax = backend.ax[0].get_zlim()
assert abs(zmin + 10) < 1e-2
assert abs(zmax - 10) < 1e-2
p = plot3d_parametric_line(v2, v1)
backend = p._backend
xmin, xmax = backend.ax[0].get_xlim()
assert abs(xmin + 2) < 1e-2
assert abs(xmax - 2) < 1e-2
ymin, ymax = backend.ax[0].get_ylim()
assert abs(ymin + 2) < 1e-2
assert abs(ymax - 2) < 1e-2
zmin, zmax = backend.ax[0].get_zlim()
assert abs(zmin + 10) < 1e-2
assert abs(zmax - 10) < 1e-2
def test_plot_size():
if not matplotlib:
skip("Matplotlib not the default backend")
x = Symbol('x')
p1 = plot(sin(x), backend="matplotlib", size=(8, 4))
s1 = p1._backend.fig.get_size_inches()
assert (s1[0] == 8) and (s1[1] == 4)
p2 = plot(sin(x), backend="matplotlib", size=(5, 10))
s2 = p2._backend.fig.get_size_inches()
assert (s2[0] == 5) and (s2[1] == 10)
p3 = PlotGrid(2, 1, p1, p2, size=(6, 2))
s3 = p3._backend.fig.get_size_inches()
assert (s3[0] == 6) and (s3[1] == 2)
with raises(ValueError):
plot(sin(x), backend="matplotlib", size=(-1, 3))
def test_issue_20113():
if not matplotlib:
skip("Matplotlib not the default backend")
x = Symbol('x')
# verify the capability to use custom backends
with raises(TypeError):
plot(sin(x), backend=Plot, show=False)
p2 = plot(sin(x), backend=MatplotlibBackend, show=False)
assert p2.backend == MatplotlibBackend
assert len(p2[0].get_data()[0]) >= 30
p3 = plot(sin(x), backend=DummyBackendOk, show=False)
assert p3.backend == DummyBackendOk
assert len(p3[0].get_data()[0]) >= 30
# test for an improper coded backend
p4 = plot(sin(x), backend=DummyBackendNotOk, show=False)
assert p4.backend == DummyBackendNotOk
assert len(p4[0].get_data()[0]) >= 30
with raises(NotImplementedError):
p4.show()
with raises(NotImplementedError):
p4.save("test/path")
with raises(NotImplementedError):
p4._backend.close()
def test_custom_coloring():
x = Symbol('x')
y = Symbol('y')
plot(cos(x), line_color=lambda a: a)
plot(cos(x), line_color=1)
plot(cos(x), line_color="r")
plot_parametric(cos(x), sin(x), line_color=lambda a: a)
plot_parametric(cos(x), sin(x), line_color=1)
plot_parametric(cos(x), sin(x), line_color="r")
plot3d_parametric_line(cos(x), sin(x), x, line_color=lambda a: a)
plot3d_parametric_line(cos(x), sin(x), x, line_color=1)
plot3d_parametric_line(cos(x), sin(x), x, line_color="r")
plot3d_parametric_surface(cos(x + y), sin(x - y), x - y,
(x, -5, 5), (y, -5, 5),
surface_color=lambda a, b: a**2 + b**2)
plot3d_parametric_surface(cos(x + y), sin(x - y), x - y,
(x, -5, 5), (y, -5, 5),
surface_color=1)
plot3d_parametric_surface(cos(x + y), sin(x - y), x - y,
(x, -5, 5), (y, -5, 5),
surface_color="r")
plot3d(x*y, (x, -5, 5), (y, -5, 5),
surface_color=lambda a, b: a**2 + b**2)
plot3d(x*y, (x, -5, 5), (y, -5, 5), surface_color=1)
plot3d(x*y, (x, -5, 5), (y, -5, 5), surface_color="r")
|
03e08c41d7d90378b41fb2c35c558ed281e9760db26a09c4c6fa40a9a62a2668 | #!/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', 'python-sat']
import sys
import os
def is_stdlib(p):
return ((p.startswith(sys.prefix)
or p.startswith(sys.base_prefix))
and 'site-packages' not in p)
stdlib = {p for p in sys.path if is_stdlib(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()
|
da56bc53a156daed9aa1530f298b5f8b3cda5df377dda470e6cc65163a294916 | #!/usr/bin/env python
""" Update the ``ask_generated.py`` file.
This must be run each time ``known_facts()`` in ``assumptions.facts`` module
is changed.
Should be run from sympy root directory.
$ python bin/ask_update.py
"""
# hook in-tree SymPy into Python path, if possible
import os
import sys
isympy_path = os.path.abspath(__file__)
isympy_dir = os.path.dirname(isympy_path)
sympy_top = os.path.split(isympy_dir)[0]
sympy_dir = os.path.join(sympy_top, 'sympy')
if os.path.isdir(sympy_dir):
sys.path.insert(0, sympy_top)
from sympy.assumptions.cnf import CNF, Literal
from sympy.assumptions.facts import (get_known_facts,
generate_known_facts_dict, get_known_facts_keys)
from sympy.core import Symbol
def generate_code():
from textwrap import dedent, wrap
LINE = ",\n "
HANG = ' '*8
code_string = dedent('''\
"""
Do NOT manually edit this file.
Instead, run ./bin/ask_update.py.
"""
from sympy.assumptions.ask import Q
from sympy.assumptions.cnf import Literal
from sympy.core.cache import cacheit
@cacheit
def get_all_known_facts():
"""
Known facts between unary predicates as CNF clauses.
"""
return {
%s
}
@cacheit
def get_known_facts_dict():
"""
Logical relations between unary predicates as dictionary.
Each key is a predicate, and item is two groups of predicates.
First group contains the predicates which are implied by the key, and
second group contains the predicates which are rejected by the key.
"""
return {
%s
}
''')
x = Symbol('x')
fact = get_known_facts(x)
# Generate CNF of facts between known unary predicates
cnf = CNF.to_CNF(fact)
p = LINE.join(sorted([
'frozenset(('
+ ', '.join(str(Literal(lit.arg.function, lit.is_Not))
for lit in sorted(clause, key=str))
+ '))' for clause in cnf.clauses]))
# Generate dictionary of facts between known unary predicates
keys = [pred(x) for pred in get_known_facts_keys()]
mapping = generate_known_facts_dict(keys, fact)
items = sorted(mapping.items(), key=str)
keys = [str(i[0]) for i in items]
values = ['(set(%s), set(%s))' % (sorted(i[1][0], key=str),
sorted(i[1][1], key=str))
for i in items]
m = LINE.join(['\n'.join(
wrap("{}: {}".format(k, v),
subsequent_indent=HANG,
break_long_words=False))
for k, v in zip(keys, values)]) + ','
return code_string % (p, m)
with open('sympy/assumptions/ask_generated.py', 'w') as f:
code = generate_code()
f.write(code)
|
6ba4ffba65ae8a95846ce3e1836729eb6c1e54ea652aeb4fc52910211f0d367a | """
This module exports all latin and greek letters as Symbols, so you can
conveniently do
>>> from sympy.abc import x, y
instead of the slightly more clunky-looking
>>> from sympy import symbols
>>> x, y = symbols('x y')
Caveats
=======
1. As of the time of writing this, the names ``O``, ``S``, ``I``, ``N``,
``E``, and ``Q`` are colliding with names defined in SymPy. If you import them
from both ``sympy.abc`` and ``sympy``, the second import will "win".
This is an issue only for * imports, which should only be used for short-lived
code such as interactive sessions and throwaway scripts that do not survive
until the next SymPy upgrade, where ``sympy`` may contain a different set of
names.
2. This module does not define symbol names on demand, i.e.
``from sympy.abc import foo`` will be reported as an error because
``sympy.abc`` does not contain the name ``foo``. To get a symbol named ``foo``,
you still need to use ``Symbol('foo')`` or ``symbols('foo')``.
You can freely mix usage of ``sympy.abc`` and ``Symbol``/``symbols``, though
sticking with one and only one way to get the symbols does tend to make the code
more readable.
The module also defines some special names to help detect which names clash
with the default SymPy namespace.
``_clash1`` defines all the single letter variables that clash with
SymPy objects; ``_clash2`` defines the multi-letter clashing symbols;
and ``_clash`` is the union of both. These can be passed for ``locals``
during sympification if one desires Symbols rather than the non-Symbol
objects for those names.
Examples
========
>>> from sympy import S
>>> from sympy.abc import _clash1, _clash2, _clash
>>> S("Q & C", locals=_clash1)
C & Q
>>> S('pi(x)', locals=_clash2)
pi(x)
>>> S('pi(C, Q)', locals=_clash)
pi(C, Q)
"""
from typing import Any, Dict
import string
from .core import Symbol, symbols
from .core.alphabets import greeks
##### Symbol definitions #####
# Implementation note: The easiest way to avoid typos in the symbols()
# parameter is to copy it from the left-hand side of the assignment.
a, b, c, d, e, f, g, h, i, j = symbols('a, b, c, d, e, f, g, h, i, j')
k, l, m, n, o, p, q, r, s, t = symbols('k, l, m, n, o, p, q, r, s, t')
u, v, w, x, y, z = symbols('u, v, w, x, y, z')
A, B, C, D, E, F, G, H, I, J = symbols('A, B, C, D, E, F, G, H, I, J')
K, L, M, N, O, P, Q, R, S, T = symbols('K, L, M, N, O, P, Q, R, S, T')
U, V, W, X, Y, Z = symbols('U, V, W, X, Y, Z')
alpha, beta, gamma, delta = symbols('alpha, beta, gamma, delta')
epsilon, zeta, eta, theta = symbols('epsilon, zeta, eta, theta')
iota, kappa, lamda, mu = symbols('iota, kappa, lamda, mu')
nu, xi, omicron, pi = symbols('nu, xi, omicron, pi')
rho, sigma, tau, upsilon = symbols('rho, sigma, tau, upsilon')
phi, chi, psi, omega = symbols('phi, chi, psi, omega')
##### Clashing-symbols diagnostics #####
# We want to know which names in SymPy collide with those in here.
# This is mostly for diagnosing SymPy's namespace during SymPy development.
_latin = list(string.ascii_letters)
# OSINEQ should not be imported as they clash; gamma, pi and zeta clash, too
_greek = list(greeks) # make a copy, so we can mutate it
# Note: We import lamda since lambda is a reserved keyword in Python
_greek.remove("lambda")
_greek.append("lamda")
ns = {} # type: Dict[str, Any]
exec('from sympy import *', ns)
_clash1 = {}
_clash2 = {}
while ns:
_k, _ = ns.popitem()
if _k in _greek:
_clash2[_k] = None
_greek.remove(_k)
elif _k in _latin:
_clash1[_k] = None
_latin.remove(_k)
_clash = {}
_clash.update(_clash1)
_clash.update(_clash2)
del _latin, _greek, Symbol, _k
|
c9de18437fc320445f244969f2153afb86f086fe2c2066f13314839df0cd11f9 | """
Continuous Random Variables - Prebuilt variables
Contains
========
Arcsin
Benini
Beta
BetaNoncentral
BetaPrime
BoundedPareto
Cauchy
Chi
ChiNoncentral
ChiSquared
Dagum
Erlang
ExGaussian
Exponential
ExponentialPower
FDistribution
FisherZ
Frechet
Gamma
GammaInverse
Gumbel
Gompertz
Kumaraswamy
Laplace
Levy
LogCauchy
Logistic
LogLogistic
LogitNormal
LogNormal
Lomax
Maxwell
Moyal
Nakagami
Normal
Pareto
PowerFunction
QuadraticU
RaisedCosine
Rayleigh
Reciprocal
ShiftedGompertz
StudentT
Trapezoidal
Triangular
Uniform
UniformSum
VonMises
Wald
Weibull
WignerSemicircle
"""
from sympy import beta as beta_fn
from sympy import cos, sin, tan, atan, exp, besseli, besselj, besselk
from sympy import (log, sqrt, pi, S, Dummy, Interval, sympify, gamma, sign,
Piecewise, And, Eq, binomial, factorial, Sum, floor, Abs,
Lambda, Basic, lowergamma, erf, erfc, erfi, erfinv, I, asin,
hyper, uppergamma, sinh, Ne, expint, Rational, integrate)
from sympy.matrices import MatrixBase, MatrixExpr
from sympy.stats.crv import SingleContinuousPSpace, SingleContinuousDistribution
from sympy.stats.rv import _value_check, is_random
oo = S.Infinity
__all__ = ['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',
'LogCauchy',
'Logistic',
'LogLogistic',
'LogitNormal',
'LogNormal',
'Lomax',
'Maxwell',
'Moyal',
'Nakagami',
'Normal',
'GaussianInverse',
'Pareto',
'PowerFunction',
'QuadraticU',
'RaisedCosine',
'Rayleigh',
'Reciprocal',
'StudentT',
'ShiftedGompertz',
'Trapezoidal',
'Triangular',
'Uniform',
'UniformSum',
'VonMises',
'Wald',
'Weibull',
'WignerSemicircle',
]
@is_random.register(MatrixBase)
def _(x):
return any([is_random(i) for i in x])
def rv(symbol, cls, args, **kwargs):
args = list(map(sympify, args))
dist = cls(*args)
if kwargs.pop('check', True):
dist.check(*args)
pspace = SingleContinuousPSpace(symbol, dist)
if any(is_random(arg) for arg in args):
from sympy.stats.compound_rv import CompoundPSpace, CompoundDistribution
pspace = CompoundPSpace(symbol, CompoundDistribution(dist))
return pspace.value
class ContinuousDistributionHandmade(SingleContinuousDistribution):
_argnames = ('pdf',)
def __new__(cls, pdf, set=Interval(-oo, oo)):
return Basic.__new__(cls, pdf, set)
@property
def set(self):
return self.args[1]
@staticmethod
def check(pdf, set):
x = Dummy('x')
val = integrate(pdf(x), (x, set))
_value_check(Eq(val, 1) != S.false, "The pdf on the given set is incorrect.")
def ContinuousRV(symbol, density, set=Interval(-oo, oo), **kwargs):
"""
Create a Continuous Random Variable given the following:
Parameters
==========
symbol : Symbol
Represents name of the random variable.
density : Expression containing symbol
Represents probability density function.
set : set/Interval
Represents the region where the pdf is valid, by default is real line.
check : bool
If True, it will check whether the given density
integrates to 1 over the given set. If False, it
will not perform this check. Default is False.
Returns
=======
RandomSymbol
Many common continuous random variable types are already implemented.
This function should be necessary only very rarely.
Examples
========
>>> from sympy import Symbol, sqrt, exp, pi
>>> from sympy.stats import ContinuousRV, P, E
>>> x = Symbol("x")
>>> pdf = sqrt(2)*exp(-x**2/2)/(2*sqrt(pi)) # Normal distribution
>>> X = ContinuousRV(x, pdf)
>>> E(X)
0
>>> P(X>0)
1/2
"""
pdf = Piecewise((density, set.as_relational(symbol)), (0, True))
pdf = Lambda(symbol, pdf)
# have a default of False while `rv` should have a default of True
kwargs['check'] = kwargs.pop('check', False)
return rv(symbol.name, ContinuousDistributionHandmade, (pdf, set), **kwargs)
########################################
# Continuous Probability Distributions #
########################################
#-------------------------------------------------------------------------------
# Arcsin distribution ----------------------------------------------------------
class ArcsinDistribution(SingleContinuousDistribution):
_argnames = ('a', 'b')
@property
def set(self):
return Interval(self.a, self.b)
def pdf(self, x):
a, b = self.a, self.b
return 1/(pi*sqrt((x - a)*(b - x)))
def _cdf(self, x):
a, b = self.a, self.b
return Piecewise(
(S.Zero, x < a),
(2*asin(sqrt((x - a)/(b - a)))/pi, x <= b),
(S.One, True))
def Arcsin(name, a=0, b=1):
r"""
Create a Continuous Random Variable with an arcsin distribution.
The density of the arcsin distribution is given by
.. math::
f(x) := \frac{1}{\pi\sqrt{(x-a)(b-x)}}
with :math:`x \in (a,b)`. It must hold that :math:`-\infty < a < b < \infty`.
Parameters
==========
a : Real number, the left interval boundary
b : Real number, the right interval boundary
Returns
=======
RandomSymbol
Examples
========
>>> from sympy.stats import Arcsin, density, cdf
>>> from sympy import Symbol
>>> a = Symbol("a", real=True)
>>> b = Symbol("b", real=True)
>>> z = Symbol("z")
>>> X = Arcsin("x", a, b)
>>> density(X)(z)
1/(pi*sqrt((-a + z)*(b - z)))
>>> cdf(X)(z)
Piecewise((0, a > z),
(2*asin(sqrt((-a + z)/(-a + b)))/pi, b >= z),
(1, True))
References
==========
.. [1] https://en.wikipedia.org/wiki/Arcsine_distribution
"""
return rv(name, ArcsinDistribution, (a, b))
#-------------------------------------------------------------------------------
# Benini distribution ----------------------------------------------------------
class BeniniDistribution(SingleContinuousDistribution):
_argnames = ('alpha', 'beta', 'sigma')
@staticmethod
def check(alpha, beta, sigma):
_value_check(alpha > 0, "Shape parameter Alpha must be positive.")
_value_check(beta > 0, "Shape parameter Beta must be positive.")
_value_check(sigma > 0, "Scale parameter Sigma must be positive.")
@property
def set(self):
return Interval(self.sigma, oo)
def pdf(self, x):
alpha, beta, sigma = self.alpha, self.beta, self.sigma
return (exp(-alpha*log(x/sigma) - beta*log(x/sigma)**2)
*(alpha/x + 2*beta*log(x/sigma)/x))
def _moment_generating_function(self, t):
raise NotImplementedError('The moment generating function of the '
'Benini distribution does not exist.')
def Benini(name, alpha, beta, sigma):
r"""
Create a Continuous Random Variable with a Benini distribution.
The density of the Benini distribution is given by
.. math::
f(x) := e^{-\alpha\log{\frac{x}{\sigma}}
-\beta\log^2\left[{\frac{x}{\sigma}}\right]}
\left(\frac{\alpha}{x}+\frac{2\beta\log{\frac{x}{\sigma}}}{x}\right)
This is a heavy-tailed distribution and is also known as the log-Rayleigh
distribution.
Parameters
==========
alpha : Real number, `\alpha > 0`, a shape
beta : Real number, `\beta > 0`, a shape
sigma : Real number, `\sigma > 0`, a scale
Returns
=======
RandomSymbol
Examples
========
>>> from sympy.stats import Benini, density, cdf
>>> from sympy import Symbol, pprint
>>> alpha = Symbol("alpha", positive=True)
>>> beta = Symbol("beta", positive=True)
>>> sigma = Symbol("sigma", positive=True)
>>> z = Symbol("z")
>>> X = Benini("x", alpha, beta, sigma)
>>> D = density(X)(z)
>>> pprint(D, use_unicode=False)
/ / z \\ / z \ 2/ z \
| 2*beta*log|-----|| - alpha*log|-----| - beta*log |-----|
|alpha \sigma/| \sigma/ \sigma/
|----- + -----------------|*e
\ z z /
>>> cdf(X)(z)
Piecewise((1 - exp(-alpha*log(z/sigma) - beta*log(z/sigma)**2), sigma <= z),
(0, True))
References
==========
.. [1] https://en.wikipedia.org/wiki/Benini_distribution
.. [2] http://reference.wolfram.com/legacy/v8/ref/BeniniDistribution.html
"""
return rv(name, BeniniDistribution, (alpha, beta, sigma))
#-------------------------------------------------------------------------------
# Beta distribution ------------------------------------------------------------
class BetaDistribution(SingleContinuousDistribution):
_argnames = ('alpha', 'beta')
set = Interval(0, 1)
@staticmethod
def check(alpha, beta):
_value_check(alpha > 0, "Shape parameter Alpha must be positive.")
_value_check(beta > 0, "Shape parameter Beta must be positive.")
def pdf(self, x):
alpha, beta = self.alpha, self.beta
return x**(alpha - 1) * (1 - x)**(beta - 1) / beta_fn(alpha, beta)
def _characteristic_function(self, t):
return hyper((self.alpha,), (self.alpha + self.beta,), I*t)
def _moment_generating_function(self, t):
return hyper((self.alpha,), (self.alpha + self.beta,), t)
def Beta(name, alpha, beta):
r"""
Create a Continuous Random Variable with a Beta distribution.
The density of the Beta distribution is given by
.. math::
f(x) := \frac{x^{\alpha-1}(1-x)^{\beta-1}} {\mathrm{B}(\alpha,\beta)}
with :math:`x \in [0,1]`.
Parameters
==========
alpha : Real number, `\alpha > 0`, a shape
beta : Real number, `\beta > 0`, a shape
Returns
=======
RandomSymbol
Examples
========
>>> from sympy.stats import Beta, density, E, variance
>>> from sympy import Symbol, simplify, pprint, factor
>>> alpha = Symbol("alpha", positive=True)
>>> beta = Symbol("beta", positive=True)
>>> z = Symbol("z")
>>> X = Beta("x", alpha, beta)
>>> D = density(X)(z)
>>> pprint(D, use_unicode=False)
alpha - 1 beta - 1
z *(1 - z)
--------------------------
B(alpha, beta)
>>> simplify(E(X))
alpha/(alpha + beta)
>>> factor(simplify(variance(X)))
alpha*beta/((alpha + beta)**2*(alpha + beta + 1))
References
==========
.. [1] https://en.wikipedia.org/wiki/Beta_distribution
.. [2] http://mathworld.wolfram.com/BetaDistribution.html
"""
return rv(name, BetaDistribution, (alpha, beta))
#-------------------------------------------------------------------------------
# Noncentral Beta distribution ------------------------------------------------------------
class BetaNoncentralDistribution(SingleContinuousDistribution):
_argnames = ('alpha', 'beta', 'lamda')
set = Interval(0, 1)
@staticmethod
def check(alpha, beta, lamda):
_value_check(alpha > 0, "Shape parameter Alpha must be positive.")
_value_check(beta > 0, "Shape parameter Beta must be positive.")
_value_check(lamda >= 0, "Noncentrality parameter Lambda must be positive")
def pdf(self, x):
alpha, beta, lamda = self.alpha, self.beta, self.lamda
k = Dummy("k")
return Sum(exp(-lamda / 2) * (lamda / 2)**k * x**(alpha + k - 1) *(
1 - x)**(beta - 1) / (factorial(k) * beta_fn(alpha + k, beta)), (k, 0, oo))
def BetaNoncentral(name, alpha, beta, lamda):
r"""
Create a Continuous Random Variable with a Type I Noncentral Beta distribution.
The density of the Noncentral Beta distribution is given by
.. math::
f(x) := \sum_{k=0}^\infty e^{-\lambda/2}\frac{(\lambda/2)^k}{k!}
\frac{x^{\alpha+k-1}(1-x)^{\beta-1}}{\mathrm{B}(\alpha+k,\beta)}
with :math:`x \in [0,1]`.
Parameters
==========
alpha : Real number, `\alpha > 0`, a shape
beta : Real number, `\beta > 0`, a shape
lamda: Real number, `\lambda >= 0`, noncentrality parameter
Returns
=======
RandomSymbol
Examples
========
>>> from sympy.stats import BetaNoncentral, density, cdf
>>> from sympy import Symbol, pprint
>>> alpha = Symbol("alpha", positive=True)
>>> beta = Symbol("beta", positive=True)
>>> lamda = Symbol("lamda", nonnegative=True)
>>> z = Symbol("z")
>>> X = BetaNoncentral("x", alpha, beta, lamda)
>>> D = density(X)(z)
>>> pprint(D, use_unicode=False)
oo
_____
\ `
\ -lamda
\ k -------
\ k + alpha - 1 /lamda\ beta - 1 2
) z *|-----| *(1 - z) *e
/ \ 2 /
/ ------------------------------------------------
/ B(k + alpha, beta)*k!
/____,
k = 0
Compute cdf with specific 'x', 'alpha', 'beta' and 'lamda' values as follows :
>>> cdf(BetaNoncentral("x", 1, 1, 1), evaluate=False)(2).doit()
2*exp(1/2)
The argument evaluate=False prevents an attempt at evaluation
of the sum for general x, before the argument 2 is passed.
References
==========
.. [1] https://en.wikipedia.org/wiki/Noncentral_beta_distribution
.. [2] https://reference.wolfram.com/language/ref/NoncentralBetaDistribution.html
"""
return rv(name, BetaNoncentralDistribution, (alpha, beta, lamda))
#-------------------------------------------------------------------------------
# Beta prime distribution ------------------------------------------------------
class BetaPrimeDistribution(SingleContinuousDistribution):
_argnames = ('alpha', 'beta')
@staticmethod
def check(alpha, beta):
_value_check(alpha > 0, "Shape parameter Alpha must be positive.")
_value_check(beta > 0, "Shape parameter Beta must be positive.")
set = Interval(0, oo)
def pdf(self, x):
alpha, beta = self.alpha, self.beta
return x**(alpha - 1)*(1 + x)**(-alpha - beta)/beta_fn(alpha, beta)
def BetaPrime(name, alpha, beta):
r"""
Create a continuous random variable with a Beta prime distribution.
The density of the Beta prime distribution is given by
.. math::
f(x) := \frac{x^{\alpha-1} (1+x)^{-\alpha -\beta}}{B(\alpha,\beta)}
with :math:`x > 0`.
Parameters
==========
alpha : Real number, `\alpha > 0`, a shape
beta : Real number, `\beta > 0`, a shape
Returns
=======
RandomSymbol
Examples
========
>>> from sympy.stats import BetaPrime, density
>>> from sympy import Symbol, pprint
>>> alpha = Symbol("alpha", positive=True)
>>> beta = Symbol("beta", positive=True)
>>> z = Symbol("z")
>>> X = BetaPrime("x", alpha, beta)
>>> D = density(X)(z)
>>> pprint(D, use_unicode=False)
alpha - 1 -alpha - beta
z *(z + 1)
-------------------------------
B(alpha, beta)
References
==========
.. [1] https://en.wikipedia.org/wiki/Beta_prime_distribution
.. [2] http://mathworld.wolfram.com/BetaPrimeDistribution.html
"""
return rv(name, BetaPrimeDistribution, (alpha, beta))
#-------------------------------------------------------------------------------
# Bounded Pareto Distribution --------------------------------------------------
class BoundedParetoDistribution(SingleContinuousDistribution):
_argnames = ('alpha', 'left', 'right')
@property
def set(self):
return Interval(self.left , self.right)
@staticmethod
def check(alpha, left, right):
_value_check (alpha.is_positive, "Shape must be positive.")
_value_check (left.is_positive, "Left value should be positive.")
_value_check (right > left, "Right should be greater than left.")
def pdf(self, x):
alpha, left, right = self.alpha, self.left, self.right
num = alpha * (left**alpha) * x**(- alpha -1)
den = 1 - (left/right)**alpha
return num/den
def BoundedPareto(name, alpha, left, right):
r"""
Create a continuous random variable with a Bounded Pareto distribution.
The density of the Bounded Pareto distribution is given by
.. math::
f(x) := \frac{\alpha L^{\alpha}x^{-\alpha-1}}{1-(\frac{L}{H})^{\alpha}}
Parameters
==========
alpha : Real Number, `alpha > 0`
Shape parameter
left : Real Number, `left > 0`
Location parameter
right : Real Number, `right > left`
Location parameter
Examples
========
>>> from sympy.stats import BoundedPareto, density, cdf, E
>>> from sympy import symbols
>>> L, H = symbols('L, H', positive=True)
>>> X = BoundedPareto('X', 2, L, H)
>>> x = symbols('x')
>>> density(X)(x)
2*L**2/(x**3*(1 - L**2/H**2))
>>> cdf(X)(x)
Piecewise((-H**2*L**2/(x**2*(H**2 - L**2)) + H**2/(H**2 - L**2), L <= x), (0, True))
>>> E(X).simplify()
2*H*L/(H + L)
Returns
=======
RandomSymbol
References
==========
.. [1] https://en.wikipedia.org/wiki/Pareto_distribution#Bounded_Pareto_distribution
"""
return rv (name, BoundedParetoDistribution, (alpha, left, right))
# ------------------------------------------------------------------------------
# Cauchy distribution ----------------------------------------------------------
class CauchyDistribution(SingleContinuousDistribution):
_argnames = ('x0', 'gamma')
@staticmethod
def check(x0, gamma):
_value_check(gamma > 0, "Scale parameter Gamma must be positive.")
_value_check(x0.is_real, "Location parameter must be real.")
def pdf(self, x):
return 1/(pi*self.gamma*(1 + ((x - self.x0)/self.gamma)**2))
def _cdf(self, x):
x0, gamma = self.x0, self.gamma
return (1/pi)*atan((x - x0)/gamma) + S.Half
def _characteristic_function(self, t):
return exp(self.x0 * I * t - self.gamma * Abs(t))
def _moment_generating_function(self, t):
raise NotImplementedError("The moment generating function for the "
"Cauchy distribution does not exist.")
def _quantile(self, p):
return self.x0 + self.gamma*tan(pi*(p - S.Half))
def Cauchy(name, x0, gamma):
r"""
Create a continuous random variable with a Cauchy distribution.
The density of the Cauchy distribution is given by
.. math::
f(x) := \frac{1}{\pi \gamma [1 + {(\frac{x-x_0}{\gamma})}^2]}
Parameters
==========
x0 : Real number, the location
gamma : Real number, `\gamma > 0`, a scale
Returns
=======
RandomSymbol
Examples
========
>>> from sympy.stats import Cauchy, density
>>> from sympy import Symbol
>>> x0 = Symbol("x0")
>>> gamma = Symbol("gamma", positive=True)
>>> z = Symbol("z")
>>> X = Cauchy("x", x0, gamma)
>>> density(X)(z)
1/(pi*gamma*(1 + (-x0 + z)**2/gamma**2))
References
==========
.. [1] https://en.wikipedia.org/wiki/Cauchy_distribution
.. [2] http://mathworld.wolfram.com/CauchyDistribution.html
"""
return rv(name, CauchyDistribution, (x0, gamma))
#-------------------------------------------------------------------------------
# Chi distribution -------------------------------------------------------------
class ChiDistribution(SingleContinuousDistribution):
_argnames = ('k',)
@staticmethod
def check(k):
_value_check(k > 0, "Number of degrees of freedom (k) must be positive.")
_value_check(k.is_integer, "Number of degrees of freedom (k) must be an integer.")
set = Interval(0, oo)
def pdf(self, x):
return 2**(1 - self.k/2)*x**(self.k - 1)*exp(-x**2/2)/gamma(self.k/2)
def _characteristic_function(self, t):
k = self.k
part_1 = hyper((k/2,), (S.Half,), -t**2/2)
part_2 = I*t*sqrt(2)*gamma((k+1)/2)/gamma(k/2)
part_3 = hyper(((k+1)/2,), (Rational(3, 2),), -t**2/2)
return part_1 + part_2*part_3
def _moment_generating_function(self, t):
k = self.k
part_1 = hyper((k / 2,), (S.Half,), t ** 2 / 2)
part_2 = t * sqrt(2) * gamma((k + 1) / 2) / gamma(k / 2)
part_3 = hyper(((k + 1) / 2,), (S(3) / 2,), t ** 2 / 2)
return part_1 + part_2 * part_3
def Chi(name, k):
r"""
Create a continuous random variable with a Chi distribution.
The density of the Chi distribution is given by
.. math::
f(x) := \frac{2^{1-k/2}x^{k-1}e^{-x^2/2}}{\Gamma(k/2)}
with :math:`x \geq 0`.
Parameters
==========
k : Positive integer, The number of degrees of freedom
Returns
=======
RandomSymbol
Examples
========
>>> from sympy.stats import Chi, density, E
>>> from sympy import Symbol, simplify
>>> k = Symbol("k", integer=True)
>>> z = Symbol("z")
>>> X = Chi("x", k)
>>> density(X)(z)
2**(1 - k/2)*z**(k - 1)*exp(-z**2/2)/gamma(k/2)
>>> simplify(E(X))
sqrt(2)*gamma(k/2 + 1/2)/gamma(k/2)
References
==========
.. [1] https://en.wikipedia.org/wiki/Chi_distribution
.. [2] http://mathworld.wolfram.com/ChiDistribution.html
"""
return rv(name, ChiDistribution, (k,))
#-------------------------------------------------------------------------------
# Non-central Chi distribution -------------------------------------------------
class ChiNoncentralDistribution(SingleContinuousDistribution):
_argnames = ('k', 'l')
@staticmethod
def check(k, l):
_value_check(k > 0, "Number of degrees of freedom (k) must be positive.")
_value_check(k.is_integer, "Number of degrees of freedom (k) must be an integer.")
_value_check(l > 0, "Shift parameter Lambda must be positive.")
set = Interval(0, oo)
def pdf(self, x):
k, l = self.k, self.l
return exp(-(x**2+l**2)/2)*x**k*l / (l*x)**(k/2) * besseli(k/2-1, l*x)
def ChiNoncentral(name, k, l):
r"""
Create a continuous random variable with a non-central Chi distribution.
Explanation
===========
The density of the non-central Chi distribution is given by
.. math::
f(x) := \frac{e^{-(x^2+\lambda^2)/2} x^k\lambda}
{(\lambda x)^{k/2}} I_{k/2-1}(\lambda x)
with `x \geq 0`. Here, `I_\nu (x)` is the
:ref:`modified Bessel function of the first kind <besseli>`.
Parameters
==========
k : A positive Integer, $k > 0$
The number of degrees of freedom.
lambda : Real number, `\lambda > 0`
Shift parameter.
Returns
=======
RandomSymbol
Examples
========
>>> from sympy.stats import ChiNoncentral, density
>>> from sympy import Symbol
>>> k = Symbol("k", integer=True)
>>> l = Symbol("l")
>>> z = Symbol("z")
>>> X = ChiNoncentral("x", k, l)
>>> density(X)(z)
l*z**k*exp(-l**2/2 - z**2/2)*besseli(k/2 - 1, l*z)/(l*z)**(k/2)
References
==========
.. [1] https://en.wikipedia.org/wiki/Noncentral_chi_distribution
"""
return rv(name, ChiNoncentralDistribution, (k, l))
#-------------------------------------------------------------------------------
# Chi squared distribution -----------------------------------------------------
class ChiSquaredDistribution(SingleContinuousDistribution):
_argnames = ('k',)
@staticmethod
def check(k):
_value_check(k > 0, "Number of degrees of freedom (k) must be positive.")
_value_check(k.is_integer, "Number of degrees of freedom (k) must be an integer.")
set = Interval(0, oo)
def pdf(self, x):
k = self.k
return 1/(2**(k/2)*gamma(k/2))*x**(k/2 - 1)*exp(-x/2)
def _cdf(self, x):
k = self.k
return Piecewise(
(S.One/gamma(k/2)*lowergamma(k/2, x/2), x >= 0),
(0, True)
)
def _characteristic_function(self, t):
return (1 - 2*I*t)**(-self.k/2)
def _moment_generating_function(self, t):
return (1 - 2*t)**(-self.k/2)
def ChiSquared(name, k):
r"""
Create a continuous random variable with a Chi-squared distribution.
Explanation
===========
The density of the Chi-squared distribution is given by
.. math::
f(x) := \frac{1}{2^{\frac{k}{2}}\Gamma\left(\frac{k}{2}\right)}
x^{\frac{k}{2}-1} e^{-\frac{x}{2}}
with :math:`x \geq 0`.
Parameters
==========
k : Positive integer
The number of degrees of freedom.
Returns
=======
RandomSymbol
Examples
========
>>> from sympy.stats import ChiSquared, density, E, variance, moment
>>> from sympy import Symbol
>>> k = Symbol("k", integer=True, positive=True)
>>> z = Symbol("z")
>>> X = ChiSquared("x", k)
>>> density(X)(z)
z**(k/2 - 1)*exp(-z/2)/(2**(k/2)*gamma(k/2))
>>> E(X)
k
>>> variance(X)
2*k
>>> moment(X, 3)
k**3 + 6*k**2 + 8*k
References
==========
.. [1] https://en.wikipedia.org/wiki/Chi_squared_distribution
.. [2] http://mathworld.wolfram.com/Chi-SquaredDistribution.html
"""
return rv(name, ChiSquaredDistribution, (k, ))
#-------------------------------------------------------------------------------
# Dagum distribution -----------------------------------------------------------
class DagumDistribution(SingleContinuousDistribution):
_argnames = ('p', 'a', 'b')
set = Interval(0, oo)
@staticmethod
def check(p, a, b):
_value_check(p > 0, "Shape parameter p must be positive.")
_value_check(a > 0, "Shape parameter a must be positive.")
_value_check(b > 0, "Scale parameter b must be positive.")
def pdf(self, x):
p, a, b = self.p, self.a, self.b
return a*p/x*((x/b)**(a*p)/(((x/b)**a + 1)**(p + 1)))
def _cdf(self, x):
p, a, b = self.p, self.a, self.b
return Piecewise(((S.One + (S(x)/b)**-a)**-p, x>=0),
(S.Zero, True))
def Dagum(name, p, a, b):
r"""
Create a continuous random variable with a Dagum distribution.
Explanation
===========
The density of the Dagum distribution is given by
.. math::
f(x) := \frac{a p}{x} \left( \frac{\left(\tfrac{x}{b}\right)^{a p}}
{\left(\left(\tfrac{x}{b}\right)^a + 1 \right)^{p+1}} \right)
with :math:`x > 0`.
Parameters
==========
p : Real number
``p > 0``, a shape.
a : Real number
``a > 0``, a shape.
b : Real number
``b > 0``, a scale.
Returns
=======
RandomSymbol
Examples
========
>>> from sympy.stats import Dagum, density, cdf
>>> from sympy import Symbol
>>> p = Symbol("p", positive=True)
>>> a = Symbol("a", positive=True)
>>> b = Symbol("b", positive=True)
>>> z = Symbol("z")
>>> X = Dagum("x", p, a, b)
>>> density(X)(z)
a*p*(z/b)**(a*p)*((z/b)**a + 1)**(-p - 1)/z
>>> cdf(X)(z)
Piecewise(((1 + (z/b)**(-a))**(-p), z >= 0), (0, True))
References
==========
.. [1] https://en.wikipedia.org/wiki/Dagum_distribution
"""
return rv(name, DagumDistribution, (p, a, b))
#-------------------------------------------------------------------------------
# Erlang distribution ----------------------------------------------------------
def Erlang(name, k, l):
r"""
Create a continuous random variable with an Erlang distribution.
Explanation
===========
The density of the Erlang distribution is given by
.. math::
f(x) := \frac{\lambda^k x^{k-1} e^{-\lambda x}}{(k-1)!}
with :math:`x \in [0,\infty]`.
Parameters
==========
k : Positive integer
l : Real number, `\lambda > 0`, the rate
Returns
=======
RandomSymbol
Examples
========
>>> from sympy.stats import Erlang, density, cdf, E, variance
>>> from sympy import Symbol, simplify, pprint
>>> k = Symbol("k", integer=True, positive=True)
>>> l = Symbol("l", positive=True)
>>> z = Symbol("z")
>>> X = Erlang("x", k, l)
>>> D = density(X)(z)
>>> pprint(D, use_unicode=False)
k k - 1 -l*z
l *z *e
---------------
Gamma(k)
>>> C = cdf(X)(z)
>>> pprint(C, use_unicode=False)
/lowergamma(k, l*z)
|------------------ for z > 0
< Gamma(k)
|
\ 0 otherwise
>>> E(X)
k/l
>>> simplify(variance(X))
k/l**2
References
==========
.. [1] https://en.wikipedia.org/wiki/Erlang_distribution
.. [2] http://mathworld.wolfram.com/ErlangDistribution.html
"""
return rv(name, GammaDistribution, (k, S.One/l))
# -------------------------------------------------------------------------------
# ExGaussian distribution -----------------------------------------------------
class ExGaussianDistribution(SingleContinuousDistribution):
_argnames = ('mean', 'std', 'rate')
set = Interval(-oo, oo)
@staticmethod
def check(mean, std, rate):
_value_check(
std > 0, "Standard deviation of ExGaussian must be positive.")
_value_check(rate > 0, "Rate of ExGaussian must be positive.")
def pdf(self, x):
mean, std, rate = self.mean, self.std, self.rate
term1 = rate/2
term2 = exp(rate * (2 * mean + rate * std**2 - 2*x)/2)
term3 = erfc((mean + rate*std**2 - x)/(sqrt(2)*std))
return term1*term2*term3
def _cdf(self, x):
from sympy.stats import cdf
mean, std, rate = self.mean, self.std, self.rate
u = rate*(x - mean)
v = rate*std
GaussianCDF1 = cdf(Normal('x', 0, v))(u)
GaussianCDF2 = cdf(Normal('x', v**2, v))(u)
return GaussianCDF1 - exp(-u + (v**2/2) + log(GaussianCDF2))
def _characteristic_function(self, t):
mean, std, rate = self.mean, self.std, self.rate
term1 = (1 - I*t/rate)**(-1)
term2 = exp(I*mean*t - std**2*t**2/2)
return term1 * term2
def _moment_generating_function(self, t):
mean, std, rate = self.mean, self.std, self.rate
term1 = (1 - t/rate)**(-1)
term2 = exp(mean*t + std**2*t**2/2)
return term1*term2
def ExGaussian(name, mean, std, rate):
r"""
Create a continuous random variable with an Exponentially modified
Gaussian (EMG) distribution.
Explanation
===========
The density of the exponentially modified Gaussian distribution is given by
.. math::
f(x) := \frac{\lambda}{2}e^{\frac{\lambda}{2}(2\mu+\lambda\sigma^2-2x)}
\text{erfc}(\frac{\mu + \lambda\sigma^2 - x}{\sqrt{2}\sigma})
with $x > 0$. Note that the expected value is `1/\lambda`.
Parameters
==========
mu : A Real number, the mean of Gaussian component
std: A positive Real number,
:math: `\sigma^2 > 0` the variance of Gaussian component
lambda: A positive Real number,
:math: `\lambda > 0` the rate of Exponential component
Returns
=======
RandomSymbol
Examples
========
>>> from sympy.stats import ExGaussian, density, cdf, E
>>> from sympy.stats import variance, skewness
>>> from sympy import Symbol, pprint, simplify
>>> mean = Symbol("mu")
>>> std = Symbol("sigma", positive=True)
>>> rate = Symbol("lamda", positive=True)
>>> z = Symbol("z")
>>> X = ExGaussian("x", mean, std, rate)
>>> pprint(density(X)(z), use_unicode=False)
/ 2 \
lamda*\lamda*sigma + 2*mu - 2*z/
--------------------------------- / ___ / 2 \\
2 |\/ 2 *\lamda*sigma + mu - z/|
lamda*e *erfc|-----------------------------|
\ 2*sigma /
----------------------------------------------------------------------------
2
>>> cdf(X)(z)
-(erf(sqrt(2)*(-lamda**2*sigma**2 + lamda*(-mu + z))/(2*lamda*sigma))/2 + 1/2)*exp(lamda**2*sigma**2/2 - lamda*(-mu + z)) + erf(sqrt(2)*(-mu + z)/(2*sigma))/2 + 1/2
>>> E(X)
(lamda*mu + 1)/lamda
>>> simplify(variance(X))
sigma**2 + lamda**(-2)
>>> simplify(skewness(X))
2/(lamda**2*sigma**2 + 1)**(3/2)
References
==========
.. [1] https://en.wikipedia.org/wiki/Exponentially_modified_Gaussian_distribution
"""
return rv(name, ExGaussianDistribution, (mean, std, rate))
#-------------------------------------------------------------------------------
# Exponential distribution -----------------------------------------------------
class ExponentialDistribution(SingleContinuousDistribution):
_argnames = ('rate',)
set = Interval(0, oo)
@staticmethod
def check(rate):
_value_check(rate > 0, "Rate must be positive.")
def pdf(self, x):
return self.rate * exp(-self.rate*x)
def _cdf(self, x):
return Piecewise(
(S.One - exp(-self.rate*x), x >= 0),
(0, True),
)
def _characteristic_function(self, t):
rate = self.rate
return rate / (rate - I*t)
def _moment_generating_function(self, t):
rate = self.rate
return rate / (rate - t)
def _quantile(self, p):
return -log(1-p)/self.rate
def Exponential(name, rate):
r"""
Create a continuous random variable with an Exponential distribution.
Explanation
===========
The density of the exponential distribution is given by
.. math::
f(x) := \lambda \exp(-\lambda x)
with $x > 0$. Note that the expected value is `1/\lambda`.
Parameters
==========
rate : A positive Real number, `\lambda > 0`, the rate (or inverse scale/inverse mean)
Returns
=======
RandomSymbol
Examples
========
>>> from sympy.stats import Exponential, density, cdf, E
>>> from sympy.stats import variance, std, skewness, quantile
>>> from sympy import Symbol
>>> l = Symbol("lambda", positive=True)
>>> z = Symbol("z")
>>> p = Symbol("p")
>>> X = Exponential("x", l)
>>> density(X)(z)
lambda*exp(-lambda*z)
>>> cdf(X)(z)
Piecewise((1 - exp(-lambda*z), z >= 0), (0, True))
>>> quantile(X)(p)
-log(1 - p)/lambda
>>> E(X)
1/lambda
>>> variance(X)
lambda**(-2)
>>> skewness(X)
2
>>> X = Exponential('x', 10)
>>> density(X)(z)
10*exp(-10*z)
>>> E(X)
1/10
>>> std(X)
1/10
References
==========
.. [1] https://en.wikipedia.org/wiki/Exponential_distribution
.. [2] http://mathworld.wolfram.com/ExponentialDistribution.html
"""
return rv(name, ExponentialDistribution, (rate, ))
# -------------------------------------------------------------------------------
# Exponential Power distribution -----------------------------------------------------
class ExponentialPowerDistribution(SingleContinuousDistribution):
_argnames = ('mu', 'alpha', 'beta')
set = Interval(-oo, oo)
@staticmethod
def check(mu, alpha, beta):
_value_check(alpha > 0, "Scale parameter alpha must be positive.")
_value_check(beta > 0, "Shape parameter beta must be positive.")
def pdf(self, x):
mu, alpha, beta = self.mu, self.alpha, self.beta
num = beta*exp(-(Abs(x - mu)/alpha)**beta)
den = 2*alpha*gamma(1/beta)
return num/den
def _cdf(self, x):
mu, alpha, beta = self.mu, self.alpha, self.beta
num = lowergamma(1/beta, (Abs(x - mu) / alpha)**beta)
den = 2*gamma(1/beta)
return sign(x - mu)*num/den + S.Half
def ExponentialPower(name, mu, alpha, beta):
r"""
Create a Continuous Random Variable with Exponential Power distribution.
This distribution is known also as Generalized Normal
distribution version 1.
Explanation
===========
The density of the Exponential Power distribution is given by
.. math::
f(x) := \frac{\beta}{2\alpha\Gamma(\frac{1}{\beta})}
e^{{-(\frac{|x - \mu|}{\alpha})^{\beta}}}
with :math:`x \in [ - \infty, \infty ]`.
Parameters
==========
mu : Real number
A location.
alpha : Real number,``alpha > 0``
A scale.
beta : Real number, ``beta > 0``
A shape.
Returns
=======
RandomSymbol
Examples
========
>>> from sympy.stats import ExponentialPower, density, cdf
>>> from sympy import Symbol, pprint
>>> z = Symbol("z")
>>> mu = Symbol("mu")
>>> alpha = Symbol("alpha", positive=True)
>>> beta = Symbol("beta", positive=True)
>>> X = ExponentialPower("x", mu, alpha, beta)
>>> pprint(density(X)(z), use_unicode=False)
beta
/|mu - z|\
-|--------|
\ alpha /
beta*e
---------------------
/ 1 \
2*alpha*Gamma|----|
\beta/
>>> cdf(X)(z)
1/2 + lowergamma(1/beta, (Abs(mu - z)/alpha)**beta)*sign(-mu + z)/(2*gamma(1/beta))
References
==========
.. [1] https://reference.wolfram.com/language/ref/ExponentialPowerDistribution.html
.. [2] https://en.wikipedia.org/wiki/Generalized_normal_distribution#Version_1
"""
return rv(name, ExponentialPowerDistribution, (mu, alpha, beta))
#-------------------------------------------------------------------------------
# F distribution ---------------------------------------------------------------
class FDistributionDistribution(SingleContinuousDistribution):
_argnames = ('d1', 'd2')
set = Interval(0, oo)
@staticmethod
def check(d1, d2):
_value_check((d1 > 0, d1.is_integer),
"Degrees of freedom d1 must be positive integer.")
_value_check((d2 > 0, d2.is_integer),
"Degrees of freedom d2 must be positive integer.")
def pdf(self, x):
d1, d2 = self.d1, self.d2
return (sqrt((d1*x)**d1*d2**d2 / (d1*x+d2)**(d1+d2))
/ (x * beta_fn(d1/2, d2/2)))
def _moment_generating_function(self, t):
raise NotImplementedError('The moment generating function for the '
'F-distribution does not exist.')
def FDistribution(name, d1, d2):
r"""
Create a continuous random variable with a F distribution.
Explanation
===========
The density of the F distribution is given by
.. math::
f(x) := \frac{\sqrt{\frac{(d_1 x)^{d_1} d_2^{d_2}}
{(d_1 x + d_2)^{d_1 + d_2}}}}
{x \mathrm{B} \left(\frac{d_1}{2}, \frac{d_2}{2}\right)}
with :math:`x > 0`.
Parameters
==========
d1 : `d_1 > 0`, where d_1 is the degrees of freedom (n_1 - 1)
d2 : `d_2 > 0`, where d_2 is the degrees of freedom (n_2 - 1)
Returns
=======
RandomSymbol
Examples
========
>>> from sympy.stats import FDistribution, density
>>> from sympy import Symbol, pprint
>>> d1 = Symbol("d1", positive=True)
>>> d2 = Symbol("d2", positive=True)
>>> z = Symbol("z")
>>> X = FDistribution("x", d1, d2)
>>> D = density(X)(z)
>>> pprint(D, use_unicode=False)
d2
-- ______________________________
2 / d1 -d1 - d2
d2 *\/ (d1*z) *(d1*z + d2)
--------------------------------------
/d1 d2\
z*B|--, --|
\2 2 /
References
==========
.. [1] https://en.wikipedia.org/wiki/F-distribution
.. [2] http://mathworld.wolfram.com/F-Distribution.html
"""
return rv(name, FDistributionDistribution, (d1, d2))
#-------------------------------------------------------------------------------
# Fisher Z distribution --------------------------------------------------------
class FisherZDistribution(SingleContinuousDistribution):
_argnames = ('d1', 'd2')
set = Interval(-oo, oo)
@staticmethod
def check(d1, d2):
_value_check(d1 > 0, "Degree of freedom d1 must be positive.")
_value_check(d2 > 0, "Degree of freedom d2 must be positive.")
def pdf(self, x):
d1, d2 = self.d1, self.d2
return (2*d1**(d1/2)*d2**(d2/2) / beta_fn(d1/2, d2/2) *
exp(d1*x) / (d1*exp(2*x)+d2)**((d1+d2)/2))
def FisherZ(name, d1, d2):
r"""
Create a Continuous Random Variable with an Fisher's Z distribution.
Explanation
===========
The density of the Fisher's Z distribution is given by
.. math::
f(x) := \frac{2d_1^{d_1/2} d_2^{d_2/2}} {\mathrm{B}(d_1/2, d_2/2)}
\frac{e^{d_1z}}{\left(d_1e^{2z}+d_2\right)^{\left(d_1+d_2\right)/2}}
.. TODO - What is the difference between these degrees of freedom?
Parameters
==========
d1 : ``d_1 > 0``
Degree of freedom.
d2 : ``d_2 > 0``
Degree of freedom.
Returns
=======
RandomSymbol
Examples
========
>>> from sympy.stats import FisherZ, density
>>> from sympy import Symbol, pprint
>>> d1 = Symbol("d1", positive=True)
>>> d2 = Symbol("d2", positive=True)
>>> z = Symbol("z")
>>> X = FisherZ("x", d1, d2)
>>> D = density(X)(z)
>>> pprint(D, use_unicode=False)
d1 d2
d1 d2 - -- - --
-- -- 2 2
2 2 / 2*z \ d1*z
2*d1 *d2 *\d1*e + d2/ *e
-----------------------------------------
/d1 d2\
B|--, --|
\2 2 /
References
==========
.. [1] https://en.wikipedia.org/wiki/Fisher%27s_z-distribution
.. [2] http://mathworld.wolfram.com/Fishersz-Distribution.html
"""
return rv(name, FisherZDistribution, (d1, d2))
#-------------------------------------------------------------------------------
# Frechet distribution ---------------------------------------------------------
class FrechetDistribution(SingleContinuousDistribution):
_argnames = ('a', 's', 'm')
set = Interval(0, oo)
@staticmethod
def check(a, s, m):
_value_check(a > 0, "Shape parameter alpha must be positive.")
_value_check(s > 0, "Scale parameter s must be positive.")
def __new__(cls, a, s=1, m=0):
a, s, m = list(map(sympify, (a, s, m)))
return Basic.__new__(cls, a, s, m)
def pdf(self, x):
a, s, m = self.a, self.s, self.m
return a/s * ((x-m)/s)**(-1-a) * exp(-((x-m)/s)**(-a))
def _cdf(self, x):
a, s, m = self.a, self.s, self.m
return Piecewise((exp(-((x-m)/s)**(-a)), x >= m),
(S.Zero, True))
def Frechet(name, a, s=1, m=0):
r"""
Create a continuous random variable with a Frechet distribution.
Explanation
===========
The density of the Frechet distribution is given by
.. math::
f(x) := \frac{\alpha}{s} \left(\frac{x-m}{s}\right)^{-1-\alpha}
e^{-(\frac{x-m}{s})^{-\alpha}}
with :math:`x \geq m`.
Parameters
==========
a : Real number, :math:`a \in \left(0, \infty\right)` the shape
s : Real number, :math:`s \in \left(0, \infty\right)` the scale
m : Real number, :math:`m \in \left(-\infty, \infty\right)` the minimum
Returns
=======
RandomSymbol
Examples
========
>>> from sympy.stats import Frechet, density, cdf
>>> from sympy import Symbol
>>> a = Symbol("a", positive=True)
>>> s = Symbol("s", positive=True)
>>> m = Symbol("m", real=True)
>>> z = Symbol("z")
>>> X = Frechet("x", a, s, m)
>>> density(X)(z)
a*((-m + z)/s)**(-a - 1)*exp(-1/((-m + z)/s)**a)/s
>>> cdf(X)(z)
Piecewise((exp(-1/((-m + z)/s)**a), m <= z), (0, True))
References
==========
.. [1] https://en.wikipedia.org/wiki/Fr%C3%A9chet_distribution
"""
return rv(name, FrechetDistribution, (a, s, m))
#-------------------------------------------------------------------------------
# Gamma distribution -----------------------------------------------------------
class GammaDistribution(SingleContinuousDistribution):
_argnames = ('k', 'theta')
set = Interval(0, oo)
@staticmethod
def check(k, theta):
_value_check(k > 0, "k must be positive")
_value_check(theta > 0, "Theta must be positive")
def pdf(self, x):
k, theta = self.k, self.theta
return x**(k - 1) * exp(-x/theta) / (gamma(k)*theta**k)
def _cdf(self, x):
k, theta = self.k, self.theta
return Piecewise(
(lowergamma(k, S(x)/theta)/gamma(k), x > 0),
(S.Zero, True))
def _characteristic_function(self, t):
return (1 - self.theta*I*t)**(-self.k)
def _moment_generating_function(self, t):
return (1- self.theta*t)**(-self.k)
def Gamma(name, k, theta):
r"""
Create a continuous random variable with a Gamma distribution.
Explanation
===========
The density of the Gamma distribution is given by
.. math::
f(x) := \frac{1}{\Gamma(k) \theta^k} x^{k - 1} e^{-\frac{x}{\theta}}
with :math:`x \in [0,1]`.
Parameters
==========
k : Real number, ``k > 0``, a shape
theta : Real number, `\theta > 0`, a scale
Returns
=======
RandomSymbol
Examples
========
>>> from sympy.stats import Gamma, density, cdf, E, variance
>>> from sympy import Symbol, pprint, simplify
>>> k = Symbol("k", positive=True)
>>> theta = Symbol("theta", positive=True)
>>> z = Symbol("z")
>>> X = Gamma("x", k, theta)
>>> D = density(X)(z)
>>> pprint(D, use_unicode=False)
-z
-----
-k k - 1 theta
theta *z *e
---------------------
Gamma(k)
>>> C = cdf(X, meijerg=True)(z)
>>> pprint(C, use_unicode=False)
/ / z \
|k*lowergamma|k, -----|
| \ theta/
<---------------------- for z >= 0
| Gamma(k + 1)
|
\ 0 otherwise
>>> E(X)
k*theta
>>> V = simplify(variance(X))
>>> pprint(V, use_unicode=False)
2
k*theta
References
==========
.. [1] https://en.wikipedia.org/wiki/Gamma_distribution
.. [2] http://mathworld.wolfram.com/GammaDistribution.html
"""
return rv(name, GammaDistribution, (k, theta))
#-------------------------------------------------------------------------------
# Inverse Gamma distribution ---------------------------------------------------
class GammaInverseDistribution(SingleContinuousDistribution):
_argnames = ('a', 'b')
set = Interval(0, oo)
@staticmethod
def check(a, b):
_value_check(a > 0, "alpha must be positive")
_value_check(b > 0, "beta must be positive")
def pdf(self, x):
a, b = self.a, self.b
return b**a/gamma(a) * x**(-a-1) * exp(-b/x)
def _cdf(self, x):
a, b = self.a, self.b
return Piecewise((uppergamma(a,b/x)/gamma(a), x > 0),
(S.Zero, True))
def _characteristic_function(self, t):
a, b = self.a, self.b
return 2 * (-I*b*t)**(a/2) * besselk(a, sqrt(-4*I*b*t)) / gamma(a)
def _moment_generating_function(self, t):
raise NotImplementedError('The moment generating function for the '
'gamma inverse distribution does not exist.')
def GammaInverse(name, a, b):
r"""
Create a continuous random variable with an inverse Gamma distribution.
Explanation
===========
The density of the inverse Gamma distribution is given by
.. math::
f(x) := \frac{\beta^\alpha}{\Gamma(\alpha)} x^{-\alpha - 1}
\exp\left(\frac{-\beta}{x}\right)
with :math:`x > 0`.
Parameters
==========
a : Real number, `a > 0` a shape
b : Real number, `b > 0` a scale
Returns
=======
RandomSymbol
Examples
========
>>> from sympy.stats import GammaInverse, density, cdf
>>> from sympy import Symbol, pprint
>>> a = Symbol("a", positive=True)
>>> b = Symbol("b", positive=True)
>>> z = Symbol("z")
>>> X = GammaInverse("x", a, b)
>>> D = density(X)(z)
>>> pprint(D, use_unicode=False)
-b
---
a -a - 1 z
b *z *e
---------------
Gamma(a)
>>> cdf(X)(z)
Piecewise((uppergamma(a, b/z)/gamma(a), z > 0), (0, True))
References
==========
.. [1] https://en.wikipedia.org/wiki/Inverse-gamma_distribution
"""
return rv(name, GammaInverseDistribution, (a, b))
#-------------------------------------------------------------------------------
# Gumbel distribution (Maximum and Minimum) --------------------------------------------------------
class GumbelDistribution(SingleContinuousDistribution):
_argnames = ('beta', 'mu', 'minimum')
set = Interval(-oo, oo)
@staticmethod
def check(beta, mu, minimum):
_value_check(beta > 0, "Scale parameter beta must be positive.")
def pdf(self, x):
beta, mu = self.beta, self.mu
z = (x - mu)/beta
f_max = (1/beta)*exp(-z - exp(-z))
f_min = (1/beta)*exp(z - exp(z))
return Piecewise((f_min, self.minimum), (f_max, not self.minimum))
def _cdf(self, x):
beta, mu = self.beta, self.mu
z = (x - mu)/beta
F_max = exp(-exp(-z))
F_min = 1 - exp(-exp(z))
return Piecewise((F_min, self.minimum), (F_max, not self.minimum))
def _characteristic_function(self, t):
cf_max = gamma(1 - I*self.beta*t) * exp(I*self.mu*t)
cf_min = gamma(1 + I*self.beta*t) * exp(I*self.mu*t)
return Piecewise((cf_min, self.minimum), (cf_max, not self.minimum))
def _moment_generating_function(self, t):
mgf_max = gamma(1 - self.beta*t) * exp(self.mu*t)
mgf_min = gamma(1 + self.beta*t) * exp(self.mu*t)
return Piecewise((mgf_min, self.minimum), (mgf_max, not self.minimum))
def Gumbel(name, beta, mu, minimum=False):
r"""
Create a Continuous Random Variable with Gumbel distribution.
Explanation
===========
The density of the Gumbel distribution is given by
For Maximum
.. math::
f(x) := \dfrac{1}{\beta} \exp \left( -\dfrac{x-\mu}{\beta}
- \exp \left( -\dfrac{x - \mu}{\beta} \right) \right)
with :math:`x \in [ - \infty, \infty ]`.
For Minimum
.. math::
f(x) := \frac{e^{- e^{\frac{- \mu + x}{\beta}} + \frac{- \mu + x}{\beta}}}{\beta}
with :math:`x \in [ - \infty, \infty ]`.
Parameters
==========
mu : Real number, 'mu' is a location
beta : Real number, 'beta > 0' is a scale
minimum : Boolean, by default, False, set to True for enabling minimum distribution
Returns
=======
RandomSymbol
Examples
========
>>> from sympy.stats import Gumbel, density, cdf
>>> from sympy import Symbol
>>> x = Symbol("x")
>>> mu = Symbol("mu")
>>> beta = Symbol("beta", positive=True)
>>> X = Gumbel("x", beta, mu)
>>> density(X)(x)
exp(-exp(-(-mu + x)/beta) - (-mu + x)/beta)/beta
>>> cdf(X)(x)
exp(-exp(-(-mu + x)/beta))
References
==========
.. [1] http://mathworld.wolfram.com/GumbelDistribution.html
.. [2] https://en.wikipedia.org/wiki/Gumbel_distribution
.. [3] http://www.mathwave.com/help/easyfit/html/analyses/distributions/gumbel_max.html
.. [4] http://www.mathwave.com/help/easyfit/html/analyses/distributions/gumbel_min.html
"""
return rv(name, GumbelDistribution, (beta, mu, minimum))
#-------------------------------------------------------------------------------
# Gompertz distribution --------------------------------------------------------
class GompertzDistribution(SingleContinuousDistribution):
_argnames = ('b', 'eta')
set = Interval(0, oo)
@staticmethod
def check(b, eta):
_value_check(b > 0, "b must be positive")
_value_check(eta > 0, "eta must be positive")
def pdf(self, x):
eta, b = self.eta, self.b
return b*eta*exp(b*x)*exp(eta)*exp(-eta*exp(b*x))
def _cdf(self, x):
eta, b = self.eta, self.b
return 1 - exp(eta)*exp(-eta*exp(b*x))
def _moment_generating_function(self, t):
eta, b = self.eta, self.b
return eta * exp(eta) * expint(t/b, eta)
def Gompertz(name, b, eta):
r"""
Create a Continuous Random Variable with Gompertz distribution.
Explanation
===========
The density of the Gompertz distribution is given by
.. math::
f(x) := b \eta e^{b x} e^{\eta} \exp \left(-\eta e^{bx} \right)
with :math: 'x \in [0, \inf)'.
Parameters
==========
b: Real number, 'b > 0' a scale
eta: Real number, 'eta > 0' a shape
Returns
=======
RandomSymbol
Examples
========
>>> from sympy.stats import Gompertz, density
>>> from sympy import Symbol
>>> b = Symbol("b", positive=True)
>>> eta = Symbol("eta", positive=True)
>>> z = Symbol("z")
>>> X = Gompertz("x", b, eta)
>>> density(X)(z)
b*eta*exp(eta)*exp(b*z)*exp(-eta*exp(b*z))
References
==========
.. [1] https://en.wikipedia.org/wiki/Gompertz_distribution
"""
return rv(name, GompertzDistribution, (b, eta))
#-------------------------------------------------------------------------------
# Kumaraswamy distribution -----------------------------------------------------
class KumaraswamyDistribution(SingleContinuousDistribution):
_argnames = ('a', 'b')
set = Interval(0, oo)
@staticmethod
def check(a, b):
_value_check(a > 0, "a must be positive")
_value_check(b > 0, "b must be positive")
def pdf(self, x):
a, b = self.a, self.b
return a * b * x**(a-1) * (1-x**a)**(b-1)
def _cdf(self, x):
a, b = self.a, self.b
return Piecewise(
(S.Zero, x < S.Zero),
(1 - (1 - x**a)**b, x <= S.One),
(S.One, True))
def Kumaraswamy(name, a, b):
r"""
Create a Continuous Random Variable with a Kumaraswamy distribution.
Explanation
===========
The density of the Kumaraswamy distribution is given by
.. math::
f(x) := a b x^{a-1} (1-x^a)^{b-1}
with :math:`x \in [0,1]`.
Parameters
==========
a : Real number, ``a > 0`` a shape
b : Real number, ``b > 0`` a shape
Returns
=======
RandomSymbol
Examples
========
>>> from sympy.stats import Kumaraswamy, density, cdf
>>> from sympy import Symbol, pprint
>>> a = Symbol("a", positive=True)
>>> b = Symbol("b", positive=True)
>>> z = Symbol("z")
>>> X = Kumaraswamy("x", a, b)
>>> D = density(X)(z)
>>> pprint(D, use_unicode=False)
b - 1
a - 1 / a\
a*b*z *\1 - z /
>>> cdf(X)(z)
Piecewise((0, z < 0), (1 - (1 - z**a)**b, z <= 1), (1, True))
References
==========
.. [1] https://en.wikipedia.org/wiki/Kumaraswamy_distribution
"""
return rv(name, KumaraswamyDistribution, (a, b))
#-------------------------------------------------------------------------------
# Laplace distribution ---------------------------------------------------------
class LaplaceDistribution(SingleContinuousDistribution):
_argnames = ('mu', 'b')
set = Interval(-oo, oo)
@staticmethod
def check(mu, b):
_value_check(b > 0, "Scale parameter b must be positive.")
_value_check(mu.is_real, "Location parameter mu should be real")
def pdf(self, x):
mu, b = self.mu, self.b
return 1/(2*b)*exp(-Abs(x - mu)/b)
def _cdf(self, x):
mu, b = self.mu, self.b
return Piecewise(
(S.Half*exp((x - mu)/b), x < mu),
(S.One - S.Half*exp(-(x - mu)/b), x >= mu)
)
def _characteristic_function(self, t):
return exp(self.mu*I*t) / (1 + self.b**2*t**2)
def _moment_generating_function(self, t):
return exp(self.mu*t) / (1 - self.b**2*t**2)
def Laplace(name, mu, b):
r"""
Create a continuous random variable with a Laplace distribution.
Explanation
===========
The density of the Laplace distribution is given by
.. math::
f(x) := \frac{1}{2 b} \exp \left(-\frac{|x-\mu|}b \right)
Parameters
==========
mu : Real number or a list/matrix, the location (mean) or the
location vector
b : Real number or a positive definite matrix, representing a scale
or the covariance matrix.
Returns
=======
RandomSymbol
Examples
========
>>> from sympy.stats import Laplace, density, cdf
>>> from sympy import Symbol, pprint
>>> mu = Symbol("mu")
>>> b = Symbol("b", positive=True)
>>> z = Symbol("z")
>>> X = Laplace("x", mu, b)
>>> density(X)(z)
exp(-Abs(mu - z)/b)/(2*b)
>>> cdf(X)(z)
Piecewise((exp((-mu + z)/b)/2, mu > z), (1 - exp((mu - z)/b)/2, True))
>>> L = Laplace('L', [1, 2], [[1, 0], [0, 1]])
>>> pprint(density(L)(1, 2), use_unicode=False)
5 / ____\
e *besselk\0, \/ 35 /
---------------------
pi
References
==========
.. [1] https://en.wikipedia.org/wiki/Laplace_distribution
.. [2] http://mathworld.wolfram.com/LaplaceDistribution.html
"""
if isinstance(mu, (list, MatrixBase)) and\
isinstance(b, (list, MatrixBase)):
from sympy.stats.joint_rv_types import MultivariateLaplace
return MultivariateLaplace(name, mu, b)
return rv(name, LaplaceDistribution, (mu, b))
#-------------------------------------------------------------------------------
# Levy distribution ---------------------------------------------------------
class LevyDistribution(SingleContinuousDistribution):
_argnames = ('mu', 'c')
@property
def set(self):
return Interval(self.mu, oo)
@staticmethod
def check(mu, c):
_value_check(c > 0, "c (scale parameter) must be positive")
_value_check(mu.is_real, "mu (location paramater) must be real")
def pdf(self, x):
mu, c = self.mu, self.c
return sqrt(c/(2*pi))*exp(-c/(2*(x - mu)))/((x - mu)**(S.One + S.Half))
def _cdf(self, x):
mu, c = self.mu, self.c
return erfc(sqrt(c/(2*(x - mu))))
def _characteristic_function(self, t):
mu, c = self.mu, self.c
return exp(I * mu * t - sqrt(-2 * I * c * t))
def _moment_generating_function(self, t):
raise NotImplementedError('The moment generating function of Levy distribution does not exist.')
def Levy(name, mu, c):
r"""
Create a continuous random variable with a Levy distribution.
The density of the Levy distribution is given by
.. math::
f(x) := \sqrt(\frac{c}{2 \pi}) \frac{\exp -\frac{c}{2 (x - \mu)}}{(x - \mu)^{3/2}}
Parameters
==========
mu : Real number
The location parameter.
c : Real number, ``c > 0``
A scale parameter.
Returns
=======
RandomSymbol
Examples
========
>>> from sympy.stats import Levy, density, cdf
>>> from sympy import Symbol
>>> mu = Symbol("mu", real=True)
>>> c = Symbol("c", positive=True)
>>> z = Symbol("z")
>>> X = Levy("x", mu, c)
>>> density(X)(z)
sqrt(2)*sqrt(c)*exp(-c/(-2*mu + 2*z))/(2*sqrt(pi)*(-mu + z)**(3/2))
>>> cdf(X)(z)
erfc(sqrt(c)*sqrt(1/(-2*mu + 2*z)))
References
==========
.. [1] https://en.wikipedia.org/wiki/L%C3%A9vy_distribution
.. [2] http://mathworld.wolfram.com/LevyDistribution.html
"""
return rv(name, LevyDistribution, (mu, c))
#-------------------------------------------------------------------------------
# Log-Cauchy distribution --------------------------------------------------------
class LogCauchyDistribution(SingleContinuousDistribution):
_argnames = ('mu', 'sigma')
set = Interval.open(0, oo)
@staticmethod
def check(mu, sigma):
_value_check((sigma > 0) != False, "Scale parameter Gamma must be positive.")
_value_check(mu.is_real != False, "Location parameter must be real.")
def pdf(self, x):
mu, sigma = self.mu, self.sigma
return 1/(x*pi)*(sigma/((log(x) - mu)**2 + sigma**2))
def _cdf(self, x):
mu, sigma = self.mu, self.sigma
return (1/pi)*atan((log(x) - mu)/sigma) + S.Half
def _characteristic_function(self, t):
raise NotImplementedError("The characteristic function for the "
"Log-Cauchy distribution does not exist.")
def _moment_generating_function(self, t):
raise NotImplementedError("The moment generating function for the "
"Log-Cauchy distribution does not exist.")
def LogCauchy(name, mu, sigma):
r"""
Create a continuous random variable with a Log-Cauchy distribution.
The density of the Log-Cauchy distribution is given by
.. math::
f(x) := \frac{1}{\pi x} \frac{\sigma}{(log(x)-\mu^2) + \sigma^2}
Parameters
==========
mu : Real number, the location
sigma : Real number, `\sigma > 0`, a scale
Returns
=======
RandomSymbol
Examples
========
>>> from sympy.stats import LogCauchy, density, cdf
>>> from sympy import Symbol, S
>>> mu = 2
>>> sigma = S.One / 5
>>> z = Symbol("z")
>>> X = LogCauchy("x", mu, sigma)
>>> density(X)(z)
1/(5*pi*z*((log(z) - 2)**2 + 1/25))
>>> cdf(X)(z)
atan(5*log(z) - 10)/pi + 1/2
References
==========
.. [1] https://en.wikipedia.org/wiki/Log-Cauchy_distribution
"""
return rv(name, LogCauchyDistribution, (mu, sigma))
#-------------------------------------------------------------------------------
# Logistic distribution --------------------------------------------------------
class LogisticDistribution(SingleContinuousDistribution):
_argnames = ('mu', 's')
set = Interval(-oo, oo)
@staticmethod
def check(mu, s):
_value_check(s > 0, "Scale parameter s must be positive.")
def pdf(self, x):
mu, s = self.mu, self.s
return exp(-(x - mu)/s)/(s*(1 + exp(-(x - mu)/s))**2)
def _cdf(self, x):
mu, s = self.mu, self.s
return S.One/(1 + exp(-(x - mu)/s))
def _characteristic_function(self, t):
return Piecewise((exp(I*t*self.mu) * pi*self.s*t / sinh(pi*self.s*t), Ne(t, 0)), (S.One, True))
def _moment_generating_function(self, t):
return exp(self.mu*t) * beta_fn(1 - self.s*t, 1 + self.s*t)
def _quantile(self, p):
return self.mu - self.s*log(-S.One + S.One/p)
def Logistic(name, mu, s):
r"""
Create a continuous random variable with a logistic distribution.
Explanation
===========
The density of the logistic distribution is given by
.. math::
f(x) := \frac{e^{-(x-\mu)/s}} {s\left(1+e^{-(x-\mu)/s}\right)^2}
Parameters
==========
mu : Real number, the location (mean)
s : Real number, `s > 0` a scale
Returns
=======
RandomSymbol
Examples
========
>>> from sympy.stats import Logistic, density, cdf
>>> from sympy import Symbol
>>> mu = Symbol("mu", real=True)
>>> s = Symbol("s", positive=True)
>>> z = Symbol("z")
>>> X = Logistic("x", mu, s)
>>> density(X)(z)
exp((mu - z)/s)/(s*(exp((mu - z)/s) + 1)**2)
>>> cdf(X)(z)
1/(exp((mu - z)/s) + 1)
References
==========
.. [1] https://en.wikipedia.org/wiki/Logistic_distribution
.. [2] http://mathworld.wolfram.com/LogisticDistribution.html
"""
return rv(name, LogisticDistribution, (mu, s))
#-------------------------------------------------------------------------------
# Log-logistic distribution --------------------------------------------------------
class LogLogisticDistribution(SingleContinuousDistribution):
_argnames = ('alpha', 'beta')
set = Interval(0, oo)
@staticmethod
def check(alpha, beta):
_value_check(alpha > 0, "Scale parameter Alpha must be positive.")
_value_check(beta > 0, "Shape parameter Beta must be positive.")
def pdf(self, x):
a, b = self.alpha, self.beta
return ((b/a)*(x/a)**(b - 1))/(1 + (x/a)**b)**2
def _cdf(self, x):
a, b = self.alpha, self.beta
return 1/(1 + (x/a)**(-b))
def _quantile(self, p):
a, b = self.alpha, self.beta
return a*((p/(1 - p))**(1/b))
def expectation(self, expr, var, **kwargs):
a, b = self.args
return Piecewise((S.NaN, b <= 1), (pi*a/(b*sin(pi/b)), True))
def LogLogistic(name, alpha, beta):
r"""
Create a continuous random variable with a log-logistic distribution.
The distribution is unimodal when ``beta > 1``.
Explanation
===========
The density of the log-logistic distribution is given by
.. math::
f(x) := \frac{(\frac{\beta}{\alpha})(\frac{x}{\alpha})^{\beta - 1}}
{(1 + (\frac{x}{\alpha})^{\beta})^2}
Parameters
==========
alpha : Real number, `\alpha > 0`, scale parameter and median of distribution
beta : Real number, `\beta > 0` a shape parameter
Returns
=======
RandomSymbol
Examples
========
>>> from sympy.stats import LogLogistic, density, cdf, quantile
>>> from sympy import Symbol, pprint
>>> alpha = Symbol("alpha", real=True, positive=True)
>>> beta = Symbol("beta", real=True, positive=True)
>>> p = Symbol("p")
>>> z = Symbol("z", positive=True)
>>> X = LogLogistic("x", alpha, beta)
>>> D = density(X)(z)
>>> pprint(D, use_unicode=False)
beta - 1
/ z \
beta*|-----|
\alpha/
------------------------
2
/ beta \
|/ z \ |
alpha*||-----| + 1|
\\alpha/ /
>>> cdf(X)(z)
1/(1 + (z/alpha)**(-beta))
>>> quantile(X)(p)
alpha*(p/(1 - p))**(1/beta)
References
==========
.. [1] https://en.wikipedia.org/wiki/Log-logistic_distribution
"""
return rv(name, LogLogisticDistribution, (alpha, beta))
#-------------------------------------------------------------------------------
#Logit-Normal distribution------------------------------------------------------
class LogitNormalDistribution(SingleContinuousDistribution):
_argnames = ('mu', 's')
set = Interval.open(0, 1)
@staticmethod
def check(mu, s):
_value_check((s ** 2).is_real is not False and s ** 2 > 0, "Squared scale parameter s must be positive.")
_value_check(mu.is_real is not False, "Location parameter must be real")
def _logit(self, x):
return log(x / (1 - x))
def pdf(self, x):
mu, s = self.mu, self.s
return exp(-(self._logit(x) - mu)**2/(2*s**2))*(S.One/sqrt(2*pi*(s**2)))*(1/(x*(1 - x)))
def _cdf(self, x):
mu, s = self.mu, self.s
return (S.One/2)*(1 + erf((self._logit(x) - mu)/(sqrt(2*s**2))))
def LogitNormal(name, mu, s):
r"""
Create a continuous random variable with a Logit-Normal distribution.
The density of the logistic distribution is given by
.. math::
f(x) := \frac{1}{s \sqrt{2 \pi}} \frac{1}{x(1 - x)} e^{- \frac{(logit(x) - \mu)^2}{s^2}}
where logit(x) = \log(\frac{x}{1 - x})
Parameters
==========
mu : Real number, the location (mean)
s : Real number, `s > 0` a scale
Returns
=======
RandomSymbol
Examples
========
>>> from sympy.stats import LogitNormal, density, cdf
>>> from sympy import Symbol,pprint
>>> mu = Symbol("mu", real=True)
>>> s = Symbol("s", positive=True)
>>> z = Symbol("z")
>>> X = LogitNormal("x",mu,s)
>>> D = density(X)(z)
>>> pprint(D, use_unicode=False)
2
/ / z \\
-|-mu + log|-----||
\ \1 - z//
---------------------
2
___ 2*s
\/ 2 *e
----------------------------
____
2*\/ pi *s*z*(1 - z)
>>> density(X)(z)
sqrt(2)*exp(-(-mu + log(z/(1 - z)))**2/(2*s**2))/(2*sqrt(pi)*s*z*(1 - z))
>>> cdf(X)(z)
erf(sqrt(2)*(-mu + log(z/(1 - z)))/(2*s))/2 + 1/2
References
==========
.. [1] https://en.wikipedia.org/wiki/Logit-normal_distribution
"""
return rv(name, LogitNormalDistribution, (mu, s))
#-------------------------------------------------------------------------------
# Log Normal distribution ------------------------------------------------------
class LogNormalDistribution(SingleContinuousDistribution):
_argnames = ('mean', 'std')
set = Interval(0, oo)
@staticmethod
def check(mean, std):
_value_check(std > 0, "Parameter std must be positive.")
def pdf(self, x):
mean, std = self.mean, self.std
return exp(-(log(x) - mean)**2 / (2*std**2)) / (x*sqrt(2*pi)*std)
def _cdf(self, x):
mean, std = self.mean, self.std
return Piecewise(
(S.Half + S.Half*erf((log(x) - mean)/sqrt(2)/std), x > 0),
(S.Zero, True)
)
def _moment_generating_function(self, t):
raise NotImplementedError('Moment generating function of the log-normal distribution is not defined.')
def LogNormal(name, mean, std):
r"""
Create a continuous random variable with a log-normal distribution.
Explanation
===========
The density of the log-normal distribution is given by
.. math::
f(x) := \frac{1}{x\sqrt{2\pi\sigma^2}}
e^{-\frac{\left(\ln x-\mu\right)^2}{2\sigma^2}}
with :math:`x \geq 0`.
Parameters
==========
mu : Real number
The log-scale.
sigma : Real number
A shape. ($\sigma^2 > 0$)
Returns
=======
RandomSymbol
Examples
========
>>> from sympy.stats import LogNormal, density
>>> from sympy import Symbol, pprint
>>> mu = Symbol("mu", real=True)
>>> sigma = Symbol("sigma", positive=True)
>>> z = Symbol("z")
>>> X = LogNormal("x", mu, sigma)
>>> D = density(X)(z)
>>> pprint(D, use_unicode=False)
2
-(-mu + log(z))
-----------------
2
___ 2*sigma
\/ 2 *e
------------------------
____
2*\/ pi *sigma*z
>>> X = LogNormal('x', 0, 1) # Mean 0, standard deviation 1
>>> density(X)(z)
sqrt(2)*exp(-log(z)**2/2)/(2*sqrt(pi)*z)
References
==========
.. [1] https://en.wikipedia.org/wiki/Lognormal
.. [2] http://mathworld.wolfram.com/LogNormalDistribution.html
"""
return rv(name, LogNormalDistribution, (mean, std))
#-------------------------------------------------------------------------------
# Lomax Distribution -----------------------------------------------------------
class LomaxDistribution(SingleContinuousDistribution):
_argnames = ('alpha', 'lamda',)
set = Interval(0, oo)
@staticmethod
def check(alpha, lamda):
_value_check(alpha.is_real, "Shape parameter should be real.")
_value_check(lamda.is_real, "Scale parameter should be real.")
_value_check(alpha.is_positive, "Shape parameter should be positive.")
_value_check(lamda.is_positive, "Scale parameter should be positive.")
def pdf(self, x):
lamba, alpha = self.lamda, self.alpha
return (alpha/lamba) * (S.One + x/lamba)**(-alpha-1)
def Lomax(name, alpha, lamda):
r"""
Create a continuous random variable with a Lomax distribution.
Explanation
===========
The density of the Lomax distribution is given by
.. math::
f(x) := \frac{\alpha}{\lambda}\left[1+\frac{x}{\lambda}\right]^{-(\alpha+1)}
Parameters
==========
alpha : Real Number, `alpha > 0`
Shape parameter
lamda : Real Number, `lamda > 0`
Scale parameter
Examples
========
>>> from sympy.stats import Lomax, density, cdf, E
>>> from sympy import symbols
>>> a, l = symbols('a, l', positive=True)
>>> X = Lomax('X', a, l)
>>> x = symbols('x')
>>> density(X)(x)
a*(1 + x/l)**(-a - 1)/l
>>> cdf(X)(x)
Piecewise((1 - 1/(1 + x/l)**a, x >= 0), (0, True))
>>> a = 2
>>> X = Lomax('X', a, l)
>>> E(X)
l
Returns
=======
RandomSymbol
References
==========
.. [1] https://en.wikipedia.org/wiki/Lomax_distribution
"""
return rv(name, LomaxDistribution, (alpha, lamda))
#-------------------------------------------------------------------------------
# Maxwell distribution ---------------------------------------------------------
class MaxwellDistribution(SingleContinuousDistribution):
_argnames = ('a',)
set = Interval(0, oo)
@staticmethod
def check(a):
_value_check(a > 0, "Parameter a must be positive.")
def pdf(self, x):
a = self.a
return sqrt(2/pi)*x**2*exp(-x**2/(2*a**2))/a**3
def _cdf(self, x):
a = self.a
return erf(sqrt(2)*x/(2*a)) - sqrt(2)*x*exp(-x**2/(2*a**2))/(sqrt(pi)*a)
def Maxwell(name, a):
r"""
Create a continuous random variable with a Maxwell distribution.
Explanation
===========
The density of the Maxwell distribution is given by
.. math::
f(x) := \sqrt{\frac{2}{\pi}} \frac{x^2 e^{-x^2/(2a^2)}}{a^3}
with :math:`x \geq 0`.
.. TODO - what does the parameter mean?
Parameters
==========
a : Real number, `a > 0`
Returns
=======
RandomSymbol
Examples
========
>>> from sympy.stats import Maxwell, density, E, variance
>>> from sympy import Symbol, simplify
>>> a = Symbol("a", positive=True)
>>> z = Symbol("z")
>>> X = Maxwell("x", a)
>>> density(X)(z)
sqrt(2)*z**2*exp(-z**2/(2*a**2))/(sqrt(pi)*a**3)
>>> E(X)
2*sqrt(2)*a/sqrt(pi)
>>> simplify(variance(X))
a**2*(-8 + 3*pi)/pi
References
==========
.. [1] https://en.wikipedia.org/wiki/Maxwell_distribution
.. [2] http://mathworld.wolfram.com/MaxwellDistribution.html
"""
return rv(name, MaxwellDistribution, (a, ))
#-------------------------------------------------------------------------------
# Moyal Distribution -----------------------------------------------------------
class MoyalDistribution(SingleContinuousDistribution):
_argnames = ('mu', 'sigma')
@staticmethod
def check(mu, sigma):
_value_check(mu.is_real, "Location parameter must be real.")
_value_check(sigma.is_real and sigma > 0, "Scale parameter must be real\
and positive.")
def pdf(self, x):
mu, sigma = self.mu, self.sigma
num = exp(-(exp(-(x - mu)/sigma) + (x - mu)/(sigma))/2)
den = (sqrt(2*pi) * sigma)
return num/den
def _characteristic_function(self, t):
mu, sigma = self.mu, self.sigma
term1 = exp(I*t*mu)
term2 = (2**(-I*sigma*t) * gamma(Rational(1, 2) - I*t*sigma))
return (term1 * term2)/sqrt(pi)
def _moment_generating_function(self, t):
mu, sigma = self.mu, self.sigma
term1 = exp(t*mu)
term2 = (2**(-1*sigma*t) * gamma(Rational(1, 2) - t*sigma))
return (term1 * term2)/sqrt(pi)
def Moyal(name, mu, sigma):
r"""
Create a continuous random variable with a Moyal distribution.
Explanation
===========
The density of the Moyal distribution is given by
.. math::
f(x) := \frac{\exp-\frac{1}{2}\exp-\frac{x-\mu}{\sigma}-\frac{x-\mu}{2\sigma}}{\sqrt{2\pi}\sigma}
with :math:`x \in \mathbb{R}`.
Parameters
==========
mu : Real number
Location parameter
sigma : Real positive number
Scale parameter
Returns
=======
RandomSymbol
Examples
========
>>> from sympy.stats import Moyal, density, cdf
>>> from sympy import Symbol, simplify
>>> mu = Symbol("mu", real=True)
>>> sigma = Symbol("sigma", positive=True, real=True)
>>> z = Symbol("z")
>>> X = Moyal("x", mu, sigma)
>>> density(X)(z)
sqrt(2)*exp(-exp((mu - z)/sigma)/2 - (-mu + z)/(2*sigma))/(2*sqrt(pi)*sigma)
>>> simplify(cdf(X)(z))
1 - erf(sqrt(2)*exp((mu - z)/(2*sigma))/2)
References
==========
.. [1] https://reference.wolfram.com/language/ref/MoyalDistribution.html
.. [2] http://www.stat.rice.edu/~dobelman/textfiles/DistributionsHandbook.pdf
"""
return rv(name, MoyalDistribution, (mu, sigma))
#-------------------------------------------------------------------------------
# Nakagami distribution --------------------------------------------------------
class NakagamiDistribution(SingleContinuousDistribution):
_argnames = ('mu', 'omega')
set = Interval(0, oo)
@staticmethod
def check(mu, omega):
_value_check(mu >= S.Half, "Shape parameter mu must be greater than equal to 1/2.")
_value_check(omega > 0, "Spread parameter omega must be positive.")
def pdf(self, x):
mu, omega = self.mu, self.omega
return 2*mu**mu/(gamma(mu)*omega**mu)*x**(2*mu - 1)*exp(-mu/omega*x**2)
def _cdf(self, x):
mu, omega = self.mu, self.omega
return Piecewise(
(lowergamma(mu, (mu/omega)*x**2)/gamma(mu), x > 0),
(S.Zero, True))
def Nakagami(name, mu, omega):
r"""
Create a continuous random variable with a Nakagami distribution.
Explanation
===========
The density of the Nakagami distribution is given by
.. math::
f(x) := \frac{2\mu^\mu}{\Gamma(\mu)\omega^\mu} x^{2\mu-1}
\exp\left(-\frac{\mu}{\omega}x^2 \right)
with :math:`x > 0`.
Parameters
==========
mu : Real number, `\mu \geq \frac{1}{2}` a shape
omega : Real number, `\omega > 0`, the spread
Returns
=======
RandomSymbol
Examples
========
>>> from sympy.stats import Nakagami, density, E, variance, cdf
>>> from sympy import Symbol, simplify, pprint
>>> mu = Symbol("mu", positive=True)
>>> omega = Symbol("omega", positive=True)
>>> z = Symbol("z")
>>> X = Nakagami("x", mu, omega)
>>> D = density(X)(z)
>>> pprint(D, use_unicode=False)
2
-mu*z
-------
mu -mu 2*mu - 1 omega
2*mu *omega *z *e
----------------------------------
Gamma(mu)
>>> simplify(E(X))
sqrt(mu)*sqrt(omega)*gamma(mu + 1/2)/gamma(mu + 1)
>>> V = simplify(variance(X))
>>> pprint(V, use_unicode=False)
2
omega*Gamma (mu + 1/2)
omega - -----------------------
Gamma(mu)*Gamma(mu + 1)
>>> cdf(X)(z)
Piecewise((lowergamma(mu, mu*z**2/omega)/gamma(mu), z > 0),
(0, True))
References
==========
.. [1] https://en.wikipedia.org/wiki/Nakagami_distribution
"""
return rv(name, NakagamiDistribution, (mu, omega))
#-------------------------------------------------------------------------------
# Normal distribution ----------------------------------------------------------
class NormalDistribution(SingleContinuousDistribution):
_argnames = ('mean', 'std')
@staticmethod
def check(mean, std):
_value_check(std > 0, "Standard deviation must be positive")
def pdf(self, x):
return exp(-(x - self.mean)**2 / (2*self.std**2)) / (sqrt(2*pi)*self.std)
def _cdf(self, x):
mean, std = self.mean, self.std
return erf(sqrt(2)*(-mean + x)/(2*std))/2 + S.Half
def _characteristic_function(self, t):
mean, std = self.mean, self.std
return exp(I*mean*t - std**2*t**2/2)
def _moment_generating_function(self, t):
mean, std = self.mean, self.std
return exp(mean*t + std**2*t**2/2)
def _quantile(self, p):
mean, std = self.mean, self.std
return mean + std*sqrt(2)*erfinv(2*p - 1)
def Normal(name, mean, std):
r"""
Create a continuous random variable with a Normal distribution.
Explanation
===========
The density of the Normal distribution is given by
.. math::
f(x) := \frac{1}{\sigma\sqrt{2\pi}} e^{ -\frac{(x-\mu)^2}{2\sigma^2} }
Parameters
==========
mu : Real number or a list representing the mean or the mean vector
sigma : Real number or a positive definite square matrix,
:math:`\sigma^2 > 0` the variance
Returns
=======
RandomSymbol
Examples
========
>>> from sympy.stats import Normal, density, E, std, cdf, skewness, quantile, marginal_distribution
>>> from sympy import Symbol, simplify, pprint
>>> mu = Symbol("mu")
>>> sigma = Symbol("sigma", positive=True)
>>> z = Symbol("z")
>>> y = Symbol("y")
>>> p = Symbol("p")
>>> X = Normal("x", mu, sigma)
>>> density(X)(z)
sqrt(2)*exp(-(-mu + z)**2/(2*sigma**2))/(2*sqrt(pi)*sigma)
>>> C = simplify(cdf(X))(z) # it needs a little more help...
>>> pprint(C, use_unicode=False)
/ ___ \
|\/ 2 *(-mu + z)|
erf|---------------|
\ 2*sigma / 1
-------------------- + -
2 2
>>> quantile(X)(p)
mu + sqrt(2)*sigma*erfinv(2*p - 1)
>>> simplify(skewness(X))
0
>>> X = Normal("x", 0, 1) # Mean 0, standard deviation 1
>>> density(X)(z)
sqrt(2)*exp(-z**2/2)/(2*sqrt(pi))
>>> E(2*X + 1)
1
>>> simplify(std(2*X + 1))
2
>>> m = Normal('X', [1, 2], [[2, 1], [1, 2]])
>>> pprint(density(m)(y, z), use_unicode=False)
2 2
y y*z z
- -- + --- - -- + z - 1
___ 3 3 3
\/ 3 *e
------------------------------
6*pi
>>> marginal_distribution(m, m[0])(1)
1/(2*sqrt(pi))
References
==========
.. [1] https://en.wikipedia.org/wiki/Normal_distribution
.. [2] http://mathworld.wolfram.com/NormalDistributionFunction.html
"""
if isinstance(mean, (list, MatrixBase, MatrixExpr)) and\
isinstance(std, (list, MatrixBase, MatrixExpr)):
from sympy.stats.joint_rv_types import MultivariateNormal
return MultivariateNormal(name, mean, std)
return rv(name, NormalDistribution, (mean, std))
#-------------------------------------------------------------------------------
# Inverse Gaussian distribution ----------------------------------------------------------
class GaussianInverseDistribution(SingleContinuousDistribution):
_argnames = ('mean', 'shape')
@property
def set(self):
return Interval(0, oo)
@staticmethod
def check(mean, shape):
_value_check(shape > 0, "Shape parameter must be positive")
_value_check(mean > 0, "Mean must be positive")
def pdf(self, x):
mu, s = self.mean, self.shape
return exp(-s*(x - mu)**2 / (2*x*mu**2)) * sqrt(s/(2*pi*x**3))
def _cdf(self, x):
from sympy.stats import cdf
mu, s = self.mean, self.shape
stdNormalcdf = cdf(Normal('x', 0, 1))
first_term = stdNormalcdf(sqrt(s/x) * ((x/mu) - S.One))
second_term = exp(2*s/mu) * stdNormalcdf(-sqrt(s/x)*(x/mu + S.One))
return first_term + second_term
def _characteristic_function(self, t):
mu, s = self.mean, self.shape
return exp((s/mu)*(1 - sqrt(1 - (2*mu**2*I*t)/s)))
def _moment_generating_function(self, t):
mu, s = self.mean, self.shape
return exp((s/mu)*(1 - sqrt(1 - (2*mu**2*t)/s)))
def GaussianInverse(name, mean, shape):
r"""
Create a continuous random variable with an Inverse Gaussian distribution.
Inverse Gaussian distribution is also known as Wald distribution.
Explanation
===========
The density of the Inverse Gaussian distribution is given by
.. math::
f(x) := \sqrt{\frac{\lambda}{2\pi x^3}} e^{-\frac{\lambda(x-\mu)^2}{2x\mu^2}}
Parameters
==========
mu :
Positive number representing the mean.
lambda :
Positive number representing the shape parameter.
Returns
=======
RandomSymbol
Examples
========
>>> from sympy.stats import GaussianInverse, density, E, std, skewness
>>> from sympy import Symbol, pprint
>>> mu = Symbol("mu", positive=True)
>>> lamda = Symbol("lambda", positive=True)
>>> z = Symbol("z", positive=True)
>>> X = GaussianInverse("x", mu, lamda)
>>> D = density(X)(z)
>>> pprint(D, use_unicode=False)
2
-lambda*(-mu + z)
-------------------
2
___ ________ 2*mu *z
\/ 2 *\/ lambda *e
-------------------------------------
____ 3/2
2*\/ pi *z
>>> E(X)
mu
>>> std(X).expand()
mu**(3/2)/sqrt(lambda)
>>> skewness(X).expand()
3*sqrt(mu)/sqrt(lambda)
References
==========
.. [1] https://en.wikipedia.org/wiki/Inverse_Gaussian_distribution
.. [2] http://mathworld.wolfram.com/InverseGaussianDistribution.html
"""
return rv(name, GaussianInverseDistribution, (mean, shape))
Wald = GaussianInverse
#-------------------------------------------------------------------------------
# Pareto distribution ----------------------------------------------------------
class ParetoDistribution(SingleContinuousDistribution):
_argnames = ('xm', 'alpha')
@property
def set(self):
return Interval(self.xm, oo)
@staticmethod
def check(xm, alpha):
_value_check(xm > 0, "Xm must be positive")
_value_check(alpha > 0, "Alpha must be positive")
def pdf(self, x):
xm, alpha = self.xm, self.alpha
return alpha * xm**alpha / x**(alpha + 1)
def _cdf(self, x):
xm, alpha = self.xm, self.alpha
return Piecewise(
(S.One - xm**alpha/x**alpha, x>=xm),
(0, True),
)
def _moment_generating_function(self, t):
xm, alpha = self.xm, self.alpha
return alpha * (-xm*t)**alpha * uppergamma(-alpha, -xm*t)
def _characteristic_function(self, t):
xm, alpha = self.xm, self.alpha
return alpha * (-I * xm * t) ** alpha * uppergamma(-alpha, -I * xm * t)
def Pareto(name, xm, alpha):
r"""
Create a continuous random variable with the Pareto distribution.
Explanation
===========
The density of the Pareto distribution is given by
.. math::
f(x) := \frac{\alpha\,x_m^\alpha}{x^{\alpha+1}}
with :math:`x \in [x_m,\infty]`.
Parameters
==========
xm : Real number, `x_m > 0`, a scale
alpha : Real number, `\alpha > 0`, a shape
Returns
=======
RandomSymbol
Examples
========
>>> from sympy.stats import Pareto, density
>>> from sympy import Symbol
>>> xm = Symbol("xm", positive=True)
>>> beta = Symbol("beta", positive=True)
>>> z = Symbol("z")
>>> X = Pareto("x", xm, beta)
>>> density(X)(z)
beta*xm**beta*z**(-beta - 1)
References
==========
.. [1] https://en.wikipedia.org/wiki/Pareto_distribution
.. [2] http://mathworld.wolfram.com/ParetoDistribution.html
"""
return rv(name, ParetoDistribution, (xm, alpha))
#-------------------------------------------------------------------------------
# PowerFunction distribution ---------------------------------------------------
class PowerFunctionDistribution(SingleContinuousDistribution):
_argnames=('alpha','a','b')
@property
def set(self):
return Interval(self.a, self.b)
@staticmethod
def check(alpha, a, b):
_value_check(a.is_real, "Continuous Boundary parameter should be real.")
_value_check(b.is_real, "Continuous Boundary parameter should be real.")
_value_check(a < b, " 'a' the left Boundary must be smaller than 'b' the right Boundary." )
_value_check(alpha.is_positive, "Continuous Shape parameter should be positive.")
def pdf(self, x):
alpha, a, b = self.alpha, self.a, self.b
num = alpha*(x - a)**(alpha - 1)
den = (b - a)**alpha
return num/den
def PowerFunction(name, alpha, a, b):
r"""
Creates a continuous random variable with a Power Function Distribution.
Explanation
===========
The density of PowerFunction distribution is given by
.. math::
f(x) := \frac{{\alpha}(x - a)^{\alpha - 1}}{(b - a)^{\alpha}}
with :math:`x \in [a,b]`.
Parameters
==========
alpha: Positive number, `0 < alpha` the shape paramater
a : Real number, :math:`-\infty < a` the left boundary
b : Real number, :math:`a < b < \infty` the right boundary
Returns
=======
RandomSymbol
Examples
========
>>> from sympy.stats import PowerFunction, density, cdf, E, variance
>>> from sympy import Symbol
>>> alpha = Symbol("alpha", positive=True)
>>> a = Symbol("a", real=True)
>>> b = Symbol("b", real=True)
>>> z = Symbol("z")
>>> X = PowerFunction("X", 2, a, b)
>>> density(X)(z)
(-2*a + 2*z)/(-a + b)**2
>>> cdf(X)(z)
Piecewise((a**2/(a**2 - 2*a*b + b**2) - 2*a*z/(a**2 - 2*a*b + b**2) +
z**2/(a**2 - 2*a*b + b**2), a <= z), (0, True))
>>> alpha = 2
>>> a = 0
>>> b = 1
>>> Y = PowerFunction("Y", alpha, a, b)
>>> E(Y)
2/3
>>> variance(Y)
1/18
References
==========
.. [1] http://www.mathwave.com/help/easyfit/html/analyses/distributions/power_func.html
"""
return rv(name, PowerFunctionDistribution, (alpha, a, b))
#-------------------------------------------------------------------------------
# QuadraticU distribution ------------------------------------------------------
class QuadraticUDistribution(SingleContinuousDistribution):
_argnames = ('a', 'b')
@property
def set(self):
return Interval(self.a, self.b)
@staticmethod
def check(a, b):
_value_check(b > a, "Parameter b must be in range (%s, oo)."%(a))
def pdf(self, x):
a, b = self.a, self.b
alpha = 12 / (b-a)**3
beta = (a+b) / 2
return Piecewise(
(alpha * (x-beta)**2, And(a<=x, x<=b)),
(S.Zero, True))
def _moment_generating_function(self, t):
a, b = self.a, self.b
return -3 * (exp(a*t) * (4 + (a**2 + 2*a*(-2 + b) + b**2) * t) \
- exp(b*t) * (4 + (-4*b + (a + b)**2) * t)) / ((a-b)**3 * t**2)
def _characteristic_function(self, t):
a, b = self.a, self.b
return -3*I*(exp(I*a*t*exp(I*b*t)) * (4*I - (-4*b + (a+b)**2)*t)) \
/ ((a-b)**3 * t**2)
def QuadraticU(name, a, b):
r"""
Create a Continuous Random Variable with a U-quadratic distribution.
Explanation
===========
The density of the U-quadratic distribution is given by
.. math::
f(x) := \alpha (x-\beta)^2
with :math:`x \in [a,b]`.
Parameters
==========
a : Real number
b : Real number, :math:`a < b`
Returns
=======
RandomSymbol
Examples
========
>>> from sympy.stats import QuadraticU, density
>>> from sympy import Symbol, pprint
>>> a = Symbol("a", real=True)
>>> b = Symbol("b", real=True)
>>> z = Symbol("z")
>>> X = QuadraticU("x", a, b)
>>> D = density(X)(z)
>>> pprint(D, use_unicode=False)
/ 2
| / a b \
|12*|- - - - + z|
| \ 2 2 /
<----------------- for And(b >= z, a <= z)
| 3
| (-a + b)
|
\ 0 otherwise
References
==========
.. [1] https://en.wikipedia.org/wiki/U-quadratic_distribution
"""
return rv(name, QuadraticUDistribution, (a, b))
#-------------------------------------------------------------------------------
# RaisedCosine distribution ----------------------------------------------------
class RaisedCosineDistribution(SingleContinuousDistribution):
_argnames = ('mu', 's')
@property
def set(self):
return Interval(self.mu - self.s, self.mu + self.s)
@staticmethod
def check(mu, s):
_value_check(s > 0, "s must be positive")
def pdf(self, x):
mu, s = self.mu, self.s
return Piecewise(
((1+cos(pi*(x-mu)/s)) / (2*s), And(mu-s<=x, x<=mu+s)),
(S.Zero, True))
def _characteristic_function(self, t):
mu, s = self.mu, self.s
return Piecewise((exp(-I*pi*mu/s)/2, Eq(t, -pi/s)),
(exp(I*pi*mu/s)/2, Eq(t, pi/s)),
(pi**2*sin(s*t)*exp(I*mu*t) / (s*t*(pi**2 - s**2*t**2)), True))
def _moment_generating_function(self, t):
mu, s = self.mu, self.s
return pi**2 * sinh(s*t) * exp(mu*t) / (s*t*(pi**2 + s**2*t**2))
def RaisedCosine(name, mu, s):
r"""
Create a Continuous Random Variable with a raised cosine distribution.
Explanation
===========
The density of the raised cosine distribution is given by
.. math::
f(x) := \frac{1}{2s}\left(1+\cos\left(\frac{x-\mu}{s}\pi\right)\right)
with :math:`x \in [\mu-s,\mu+s]`.
Parameters
==========
mu : Real number
s : Real number, `s > 0`
Returns
=======
RandomSymbol
Examples
========
>>> from sympy.stats import RaisedCosine, density
>>> from sympy import Symbol, pprint
>>> mu = Symbol("mu", real=True)
>>> s = Symbol("s", positive=True)
>>> z = Symbol("z")
>>> X = RaisedCosine("x", mu, s)
>>> D = density(X)(z)
>>> pprint(D, use_unicode=False)
/ /pi*(-mu + z)\
|cos|------------| + 1
| \ s /
<--------------------- for And(z >= mu - s, z <= mu + s)
| 2*s
|
\ 0 otherwise
References
==========
.. [1] https://en.wikipedia.org/wiki/Raised_cosine_distribution
"""
return rv(name, RaisedCosineDistribution, (mu, s))
#-------------------------------------------------------------------------------
# Rayleigh distribution --------------------------------------------------------
class RayleighDistribution(SingleContinuousDistribution):
_argnames = ('sigma',)
set = Interval(0, oo)
@staticmethod
def check(sigma):
_value_check(sigma > 0, "Scale parameter sigma must be positive.")
def pdf(self, x):
sigma = self.sigma
return x/sigma**2*exp(-x**2/(2*sigma**2))
def _cdf(self, x):
sigma = self.sigma
return 1 - exp(-(x**2/(2*sigma**2)))
def _characteristic_function(self, t):
sigma = self.sigma
return 1 - sigma*t*exp(-sigma**2*t**2/2) * sqrt(pi/2) * (erfi(sigma*t/sqrt(2)) - I)
def _moment_generating_function(self, t):
sigma = self.sigma
return 1 + sigma*t*exp(sigma**2*t**2/2) * sqrt(pi/2) * (erf(sigma*t/sqrt(2)) + 1)
def Rayleigh(name, sigma):
r"""
Create a continuous random variable with a Rayleigh distribution.
Explanation
===========
The density of the Rayleigh distribution is given by
.. math ::
f(x) := \frac{x}{\sigma^2} e^{-x^2/2\sigma^2}
with :math:`x > 0`.
Parameters
==========
sigma : Real number, `\sigma > 0`
Returns
=======
RandomSymbol
Examples
========
>>> from sympy.stats import Rayleigh, density, E, variance
>>> from sympy import Symbol
>>> sigma = Symbol("sigma", positive=True)
>>> z = Symbol("z")
>>> X = Rayleigh("x", sigma)
>>> density(X)(z)
z*exp(-z**2/(2*sigma**2))/sigma**2
>>> E(X)
sqrt(2)*sqrt(pi)*sigma/2
>>> variance(X)
-pi*sigma**2/2 + 2*sigma**2
References
==========
.. [1] https://en.wikipedia.org/wiki/Rayleigh_distribution
.. [2] http://mathworld.wolfram.com/RayleighDistribution.html
"""
return rv(name, RayleighDistribution, (sigma, ))
#-------------------------------------------------------------------------------
# Reciprocal distribution --------------------------------------------------------
class ReciprocalDistribution(SingleContinuousDistribution):
_argnames = ('a', 'b')
@property
def set(self):
return Interval(self.a, self.b)
@staticmethod
def check(a, b):
_value_check(a > 0, "Parameter > 0. a = %s"%a)
_value_check((a < b),
"Parameter b must be in range (%s, +oo]. b = %s"%(a, b))
def pdf(self, x):
a, b = self.a, self.b
return 1/(x*(log(b) - log(a)))
def Reciprocal(name, a, b):
r"""Creates a continuous random variable with a reciprocal distribution.
Parameters
==========
a : Real number, :math:`0 < a`
b : Real number, :math:`a < b`
Returns
=======
RandomSymbol
Examples
========
>>> from sympy.stats import Reciprocal, density, cdf
>>> from sympy import symbols
>>> a, b, x = symbols('a, b, x', positive=True)
>>> R = Reciprocal('R', a, b)
>>> density(R)(x)
1/(x*(-log(a) + log(b)))
>>> cdf(R)(x)
Piecewise((log(a)/(log(a) - log(b)) - log(x)/(log(a) - log(b)), a <= x), (0, True))
Reference
=========
.. [1] https://en.wikipedia.org/wiki/Reciprocal_distribution
"""
return rv(name, ReciprocalDistribution, (a, b))
#-------------------------------------------------------------------------------
# Shifted Gompertz distribution ------------------------------------------------
class ShiftedGompertzDistribution(SingleContinuousDistribution):
_argnames = ('b', 'eta')
set = Interval(0, oo)
@staticmethod
def check(b, eta):
_value_check(b > 0, "b must be positive")
_value_check(eta > 0, "eta must be positive")
def pdf(self, x):
b, eta = self.b, self.eta
return b*exp(-b*x)*exp(-eta*exp(-b*x))*(1+eta*(1-exp(-b*x)))
def ShiftedGompertz(name, b, eta):
r"""
Create a continuous random variable with a Shifted Gompertz distribution.
Explanation
===========
The density of the Shifted Gompertz distribution is given by
.. math::
f(x) := b e^{-b x} e^{-\eta \exp(-b x)} \left[1 + \eta(1 - e^(-bx)) \right]
with :math: 'x \in [0, \inf)'.
Parameters
==========
b: Real number, 'b > 0' a scale
eta: Real number, 'eta > 0' a shape
Returns
=======
RandomSymbol
Examples
========
>>> from sympy.stats import ShiftedGompertz, density
>>> from sympy import Symbol
>>> b = Symbol("b", positive=True)
>>> eta = Symbol("eta", positive=True)
>>> x = Symbol("x")
>>> X = ShiftedGompertz("x", b, eta)
>>> density(X)(x)
b*(eta*(1 - exp(-b*x)) + 1)*exp(-b*x)*exp(-eta*exp(-b*x))
References
==========
.. [1] https://en.wikipedia.org/wiki/Shifted_Gompertz_distribution
"""
return rv(name, ShiftedGompertzDistribution, (b, eta))
#-------------------------------------------------------------------------------
# StudentT distribution --------------------------------------------------------
class StudentTDistribution(SingleContinuousDistribution):
_argnames = ('nu',)
set = Interval(-oo, oo)
@staticmethod
def check(nu):
_value_check(nu > 0, "Degrees of freedom nu must be positive.")
def pdf(self, x):
nu = self.nu
return 1/(sqrt(nu)*beta_fn(S.Half, nu/2))*(1 + x**2/nu)**(-(nu + 1)/2)
def _cdf(self, x):
nu = self.nu
return S.Half + x*gamma((nu+1)/2)*hyper((S.Half, (nu+1)/2),
(Rational(3, 2),), -x**2/nu)/(sqrt(pi*nu)*gamma(nu/2))
def _moment_generating_function(self, t):
raise NotImplementedError('The moment generating function for the Student-T distribution is undefined.')
def StudentT(name, nu):
r"""
Create a continuous random variable with a student's t distribution.
Explanation
===========
The density of the student's t distribution is given by
.. math::
f(x) := \frac{\Gamma \left(\frac{\nu+1}{2} \right)}
{\sqrt{\nu\pi}\Gamma \left(\frac{\nu}{2} \right)}
\left(1+\frac{x^2}{\nu} \right)^{-\frac{\nu+1}{2}}
Parameters
==========
nu : Real number, `\nu > 0`, the degrees of freedom
Returns
=======
RandomSymbol
Examples
========
>>> from sympy.stats import StudentT, density, cdf
>>> from sympy import Symbol, pprint
>>> nu = Symbol("nu", positive=True)
>>> z = Symbol("z")
>>> X = StudentT("x", nu)
>>> D = density(X)(z)
>>> pprint(D, use_unicode=False)
nu 1
- -- - -
2 2
/ 2\
| z |
|1 + --|
\ nu/
-----------------
____ / nu\
\/ nu *B|1/2, --|
\ 2 /
>>> cdf(X)(z)
1/2 + z*gamma(nu/2 + 1/2)*hyper((1/2, nu/2 + 1/2), (3/2,),
-z**2/nu)/(sqrt(pi)*sqrt(nu)*gamma(nu/2))
References
==========
.. [1] https://en.wikipedia.org/wiki/Student_t-distribution
.. [2] http://mathworld.wolfram.com/Studentst-Distribution.html
"""
return rv(name, StudentTDistribution, (nu, ))
#-------------------------------------------------------------------------------
# Trapezoidal distribution ------------------------------------------------------
class TrapezoidalDistribution(SingleContinuousDistribution):
_argnames = ('a', 'b', 'c', 'd')
@property
def set(self):
return Interval(self.a, self.d)
@staticmethod
def check(a, b, c, d):
_value_check(a < d, "Lower bound parameter a < %s. a = %s"%(d, a))
_value_check((a <= b, b < c),
"Level start parameter b must be in range [%s, %s). b = %s"%(a, c, b))
_value_check((b < c, c <= d),
"Level end parameter c must be in range (%s, %s]. c = %s"%(b, d, c))
_value_check(d >= c, "Upper bound parameter d > %s. d = %s"%(c, d))
def pdf(self, x):
a, b, c, d = self.a, self.b, self.c, self.d
return Piecewise(
(2*(x-a) / ((b-a)*(d+c-a-b)), And(a <= x, x < b)),
(2 / (d+c-a-b), And(b <= x, x < c)),
(2*(d-x) / ((d-c)*(d+c-a-b)), And(c <= x, x <= d)),
(S.Zero, True))
def Trapezoidal(name, a, b, c, d):
r"""
Create a continuous random variable with a trapezoidal distribution.
Explanation
===========
The density of the trapezoidal distribution is given by
.. math::
f(x) := \begin{cases}
0 & \mathrm{for\ } x < a, \\
\frac{2(x-a)}{(b-a)(d+c-a-b)} & \mathrm{for\ } a \le x < b, \\
\frac{2}{d+c-a-b} & \mathrm{for\ } b \le x < c, \\
\frac{2(d-x)}{(d-c)(d+c-a-b)} & \mathrm{for\ } c \le x < d, \\
0 & \mathrm{for\ } d < x.
\end{cases}
Parameters
==========
a : Real number, :math:`a < d`
b : Real number, :math:`a <= b < c`
c : Real number, :math:`b < c <= d`
d : Real number
Returns
=======
RandomSymbol
Examples
========
>>> from sympy.stats import Trapezoidal, density
>>> from sympy import Symbol, pprint
>>> a = Symbol("a")
>>> b = Symbol("b")
>>> c = Symbol("c")
>>> d = Symbol("d")
>>> z = Symbol("z")
>>> X = Trapezoidal("x", a,b,c,d)
>>> pprint(density(X)(z), use_unicode=False)
/ -2*a + 2*z
|------------------------- for And(a <= z, b > z)
|(-a + b)*(-a - b + c + d)
|
| 2
| -------------- for And(b <= z, c > z)
< -a - b + c + d
|
| 2*d - 2*z
|------------------------- for And(d >= z, c <= z)
|(-c + d)*(-a - b + c + d)
|
\ 0 otherwise
References
==========
.. [1] https://en.wikipedia.org/wiki/Trapezoidal_distribution
"""
return rv(name, TrapezoidalDistribution, (a, b, c, d))
#-------------------------------------------------------------------------------
# Triangular distribution ------------------------------------------------------
class TriangularDistribution(SingleContinuousDistribution):
_argnames = ('a', 'b', 'c')
@property
def set(self):
return Interval(self.a, self.b)
@staticmethod
def check(a, b, c):
_value_check(b > a, "Parameter b > %s. b = %s"%(a, b))
_value_check((a <= c, c <= b),
"Parameter c must be in range [%s, %s]. c = %s"%(a, b, c))
def pdf(self, x):
a, b, c = self.a, self.b, self.c
return Piecewise(
(2*(x - a)/((b - a)*(c - a)), And(a <= x, x < c)),
(2/(b - a), Eq(x, c)),
(2*(b - x)/((b - a)*(b - c)), And(c < x, x <= b)),
(S.Zero, True))
def _characteristic_function(self, t):
a, b, c = self.a, self.b, self.c
return -2 *((b-c) * exp(I*a*t) - (b-a) * exp(I*c*t) + (c-a) * exp(I*b*t)) / ((b-a)*(c-a)*(b-c)*t**2)
def _moment_generating_function(self, t):
a, b, c = self.a, self.b, self.c
return 2 * ((b - c) * exp(a * t) - (b - a) * exp(c * t) + (c - a) * exp(b * t)) / (
(b - a) * (c - a) * (b - c) * t ** 2)
def Triangular(name, a, b, c):
r"""
Create a continuous random variable with a triangular distribution.
Explanation
===========
The density of the triangular distribution is given by
.. math::
f(x) := \begin{cases}
0 & \mathrm{for\ } x < a, \\
\frac{2(x-a)}{(b-a)(c-a)} & \mathrm{for\ } a \le x < c, \\
\frac{2}{b-a} & \mathrm{for\ } x = c, \\
\frac{2(b-x)}{(b-a)(b-c)} & \mathrm{for\ } c < x \le b, \\
0 & \mathrm{for\ } b < x.
\end{cases}
Parameters
==========
a : Real number, :math:`a \in \left(-\infty, \infty\right)`
b : Real number, :math:`a < b`
c : Real number, :math:`a \leq c \leq b`
Returns
=======
RandomSymbol
Examples
========
>>> from sympy.stats import Triangular, density
>>> from sympy import Symbol, pprint
>>> a = Symbol("a")
>>> b = Symbol("b")
>>> c = Symbol("c")
>>> z = Symbol("z")
>>> X = Triangular("x", a,b,c)
>>> pprint(density(X)(z), use_unicode=False)
/ -2*a + 2*z
|----------------- for And(a <= z, c > z)
|(-a + b)*(-a + c)
|
| 2
| ------ for c = z
< -a + b
|
| 2*b - 2*z
|---------------- for And(b >= z, c < z)
|(-a + b)*(b - c)
|
\ 0 otherwise
References
==========
.. [1] https://en.wikipedia.org/wiki/Triangular_distribution
.. [2] http://mathworld.wolfram.com/TriangularDistribution.html
"""
return rv(name, TriangularDistribution, (a, b, c))
#-------------------------------------------------------------------------------
# Uniform distribution ---------------------------------------------------------
class UniformDistribution(SingleContinuousDistribution):
_argnames = ('left', 'right')
@property
def set(self):
return Interval(self.left, self.right)
@staticmethod
def check(left, right):
_value_check(left < right, "Lower limit should be less than Upper limit.")
def pdf(self, x):
left, right = self.left, self.right
return Piecewise(
(S.One/(right - left), And(left <= x, x <= right)),
(S.Zero, True)
)
def _cdf(self, x):
left, right = self.left, self.right
return Piecewise(
(S.Zero, x < left),
((x - left)/(right - left), x <= right),
(S.One, True)
)
def _characteristic_function(self, t):
left, right = self.left, self.right
return Piecewise(((exp(I*t*right) - exp(I*t*left)) / (I*t*(right - left)), Ne(t, 0)),
(S.One, True))
def _moment_generating_function(self, t):
left, right = self.left, self.right
return Piecewise(((exp(t*right) - exp(t*left)) / (t * (right - left)), Ne(t, 0)),
(S.One, True))
def expectation(self, expr, var, **kwargs):
from sympy import Max, Min
kwargs['evaluate'] = True
result = SingleContinuousDistribution.expectation(self, expr, var, **kwargs)
result = result.subs({Max(self.left, self.right): self.right,
Min(self.left, self.right): self.left})
return result
def Uniform(name, left, right):
r"""
Create a continuous random variable with a uniform distribution.
Explanation
===========
The density of the uniform distribution is given by
.. math::
f(x) := \begin{cases}
\frac{1}{b - a} & \text{for } x \in [a,b] \\
0 & \text{otherwise}
\end{cases}
with :math:`x \in [a,b]`.
Parameters
==========
a : Real number, :math:`-\infty < a` the left boundary
b : Real number, :math:`a < b < \infty` the right boundary
Returns
=======
RandomSymbol
Examples
========
>>> from sympy.stats import Uniform, density, cdf, E, variance
>>> from sympy import Symbol, simplify
>>> a = Symbol("a", negative=True)
>>> b = Symbol("b", positive=True)
>>> z = Symbol("z")
>>> X = Uniform("x", a, b)
>>> density(X)(z)
Piecewise((1/(-a + b), (b >= z) & (a <= z)), (0, True))
>>> cdf(X)(z)
Piecewise((0, a > z), ((-a + z)/(-a + b), b >= z), (1, True))
>>> E(X)
a/2 + b/2
>>> simplify(variance(X))
a**2/12 - a*b/6 + b**2/12
References
==========
.. [1] https://en.wikipedia.org/wiki/Uniform_distribution_%28continuous%29
.. [2] http://mathworld.wolfram.com/UniformDistribution.html
"""
return rv(name, UniformDistribution, (left, right))
#-------------------------------------------------------------------------------
# UniformSum distribution ------------------------------------------------------
class UniformSumDistribution(SingleContinuousDistribution):
_argnames = ('n',)
@property
def set(self):
return Interval(0, self.n)
@staticmethod
def check(n):
_value_check((n > 0, n.is_integer),
"Parameter n must be positive integer.")
def pdf(self, x):
n = self.n
k = Dummy("k")
return 1/factorial(
n - 1)*Sum((-1)**k*binomial(n, k)*(x - k)**(n - 1), (k, 0, floor(x)))
def _cdf(self, x):
n = self.n
k = Dummy("k")
return Piecewise((S.Zero, x < 0),
(1/factorial(n)*Sum((-1)**k*binomial(n, k)*(x - k)**(n),
(k, 0, floor(x))), x <= n),
(S.One, True))
def _characteristic_function(self, t):
return ((exp(I*t) - 1) / (I*t))**self.n
def _moment_generating_function(self, t):
return ((exp(t) - 1) / t)**self.n
def UniformSum(name, n):
r"""
Create a continuous random variable with an Irwin-Hall distribution.
Explanation
===========
The probability distribution function depends on a single parameter
$n$ which is an integer.
The density of the Irwin-Hall distribution is given by
.. math ::
f(x) := \frac{1}{(n-1)!}\sum_{k=0}^{\left\lfloor x\right\rfloor}(-1)^k
\binom{n}{k}(x-k)^{n-1}
Parameters
==========
n : A positive Integer, `n > 0`
Returns
=======
RandomSymbol
Examples
========
>>> from sympy.stats import UniformSum, density, cdf
>>> from sympy import Symbol, pprint
>>> n = Symbol("n", integer=True)
>>> z = Symbol("z")
>>> X = UniformSum("x", n)
>>> D = density(X)(z)
>>> pprint(D, use_unicode=False)
floor(z)
___
\ `
\ k n - 1 /n\
) (-1) *(-k + z) *| |
/ \k/
/__,
k = 0
--------------------------------
(n - 1)!
>>> cdf(X)(z)
Piecewise((0, z < 0), (Sum((-1)**_k*(-_k + z)**n*binomial(n, _k),
(_k, 0, floor(z)))/factorial(n), n >= z), (1, True))
Compute cdf with specific 'x' and 'n' values as follows :
>>> cdf(UniformSum("x", 5), evaluate=False)(2).doit()
9/40
The argument evaluate=False prevents an attempt at evaluation
of the sum for general n, before the argument 2 is passed.
References
==========
.. [1] https://en.wikipedia.org/wiki/Uniform_sum_distribution
.. [2] http://mathworld.wolfram.com/UniformSumDistribution.html
"""
return rv(name, UniformSumDistribution, (n, ))
#-------------------------------------------------------------------------------
# VonMises distribution --------------------------------------------------------
class VonMisesDistribution(SingleContinuousDistribution):
_argnames = ('mu', 'k')
set = Interval(0, 2*pi)
@staticmethod
def check(mu, k):
_value_check(k > 0, "k must be positive")
def pdf(self, x):
mu, k = self.mu, self.k
return exp(k*cos(x-mu)) / (2*pi*besseli(0, k))
def VonMises(name, mu, k):
r"""
Create a Continuous Random Variable with a von Mises distribution.
Explanation
===========
The density of the von Mises distribution is given by
.. math::
f(x) := \frac{e^{\kappa\cos(x-\mu)}}{2\pi I_0(\kappa)}
with :math:`x \in [0,2\pi]`.
Parameters
==========
mu : Real number
Measure of location.
k : Real number
Measure of concentration.
Returns
=======
RandomSymbol
Examples
========
>>> from sympy.stats import VonMises, density
>>> from sympy import Symbol, pprint
>>> mu = Symbol("mu")
>>> k = Symbol("k", positive=True)
>>> z = Symbol("z")
>>> X = VonMises("x", mu, k)
>>> D = density(X)(z)
>>> pprint(D, use_unicode=False)
k*cos(mu - z)
e
------------------
2*pi*besseli(0, k)
References
==========
.. [1] https://en.wikipedia.org/wiki/Von_Mises_distribution
.. [2] http://mathworld.wolfram.com/vonMisesDistribution.html
"""
return rv(name, VonMisesDistribution, (mu, k))
#-------------------------------------------------------------------------------
# Weibull distribution ---------------------------------------------------------
class WeibullDistribution(SingleContinuousDistribution):
_argnames = ('alpha', 'beta')
set = Interval(0, oo)
@staticmethod
def check(alpha, beta):
_value_check(alpha > 0, "Alpha must be positive")
_value_check(beta > 0, "Beta must be positive")
def pdf(self, x):
alpha, beta = self.alpha, self.beta
return beta * (x/alpha)**(beta - 1) * exp(-(x/alpha)**beta) / alpha
def Weibull(name, alpha, beta):
r"""
Create a continuous random variable with a Weibull distribution.
Explanation
===========
The density of the Weibull distribution is given by
.. math::
f(x) := \begin{cases}
\frac{k}{\lambda}\left(\frac{x}{\lambda}\right)^{k-1}
e^{-(x/\lambda)^{k}} & x\geq0\\
0 & x<0
\end{cases}
Parameters
==========
lambda : Real number, :math:`\lambda > 0` a scale
k : Real number, ``k > 0`` a shape
Returns
=======
RandomSymbol
Examples
========
>>> from sympy.stats import Weibull, density, E, variance
>>> from sympy import Symbol, simplify
>>> l = Symbol("lambda", positive=True)
>>> k = Symbol("k", positive=True)
>>> z = Symbol("z")
>>> X = Weibull("x", l, k)
>>> density(X)(z)
k*(z/lambda)**(k - 1)*exp(-(z/lambda)**k)/lambda
>>> simplify(E(X))
lambda*gamma(1 + 1/k)
>>> simplify(variance(X))
lambda**2*(-gamma(1 + 1/k)**2 + gamma(1 + 2/k))
References
==========
.. [1] https://en.wikipedia.org/wiki/Weibull_distribution
.. [2] http://mathworld.wolfram.com/WeibullDistribution.html
"""
return rv(name, WeibullDistribution, (alpha, beta))
#-------------------------------------------------------------------------------
# Wigner semicircle distribution -----------------------------------------------
class WignerSemicircleDistribution(SingleContinuousDistribution):
_argnames = ('R',)
@property
def set(self):
return Interval(-self.R, self.R)
@staticmethod
def check(R):
_value_check(R > 0, "Radius R must be positive.")
def pdf(self, x):
R = self.R
return 2/(pi*R**2)*sqrt(R**2 - x**2)
def _characteristic_function(self, t):
return Piecewise((2 * besselj(1, self.R*t) / (self.R*t), Ne(t, 0)),
(S.One, True))
def _moment_generating_function(self, t):
return Piecewise((2 * besseli(1, self.R*t) / (self.R*t), Ne(t, 0)),
(S.One, True))
def WignerSemicircle(name, R):
r"""
Create a continuous random variable with a Wigner semicircle distribution.
Explanation
===========
The density of the Wigner semicircle distribution is given by
.. math::
f(x) := \frac2{\pi R^2}\,\sqrt{R^2-x^2}
with :math:`x \in [-R,R]`.
Parameters
==========
R : Real number, `R > 0`, the radius
Returns
=======
A `RandomSymbol`.
Examples
========
>>> from sympy.stats import WignerSemicircle, density, E
>>> from sympy import Symbol
>>> R = Symbol("R", positive=True)
>>> z = Symbol("z")
>>> X = WignerSemicircle("x", R)
>>> density(X)(z)
2*sqrt(R**2 - z**2)/(pi*R**2)
>>> E(X)
0
References
==========
.. [1] https://en.wikipedia.org/wiki/Wigner_semicircle_distribution
.. [2] http://mathworld.wolfram.com/WignersSemicircleLaw.html
"""
return rv(name, WignerSemicircleDistribution, (R,))
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.