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from sympy.core.compatibility import (default_sort_key, as_int, ordered,
iterable, NotIterable)
from sympy.core.singleton import S
from sympy.utilities.pytest import raises
from sympy.abc import x
def test_default_sort_key():
func = lambda x: x
assert sorted([func, x, func], key=default_sort_key) == [func, func, x]
def test_as_int():
raises(ValueError, lambda : as_int(1.1))
raises(ValueError, lambda : as_int([]))
raises(ValueError, lambda : as_int(S.NaN))
raises(ValueError, lambda : as_int(S.Infinity))
raises(ValueError, lambda : as_int(S.NegativeInfinity))
raises(ValueError, lambda : as_int(S.ComplexInfinity))
# for the following, limited precision makes int(arg) == arg
# but the int value is not necessarily what a user might have
# expected; Q.prime is more nuanced in its response for
# expressions which might be complex representations of an
# integer. This is not -- by design -- as_ints role.
raises(ValueError, lambda : as_int(1e23))
raises(ValueError, lambda : as_int(S('1.'+'0'*20+'1')))
def test_iterable():
assert iterable(0) is False
assert iterable(1) is False
assert iterable(None) is False
class Test1(NotIterable):
pass
assert iterable(Test1()) is False
class Test2(NotIterable):
_iterable = True
assert iterable(Test2()) is True
class Test3(object):
pass
assert iterable(Test3()) is False
class Test4(object):
_iterable = True
assert iterable(Test4()) is True
class Test5(object):
def __iter__(self):
yield 1
assert iterable(Test5()) is True
class Test6(Test5):
_iterable = False
assert iterable(Test6()) is False
def test_ordered():
# Issue 7210 - this had been failing with python2/3 problems
assert (list(ordered([{1:3, 2:4, 9:10}, {1:3}])) == \
[{1: 3}, {1: 3, 2: 4, 9: 10}])
# warnings should not be raised for identical items
l = [1, 1]
assert list(ordered(l, warn=True)) == l
l = [[1], [2], [1]]
assert list(ordered(l, warn=True)) == [[1], [1], [2]]
raises(ValueError, lambda: list(ordered(['a', 'ab'], keys=[lambda x: x[0]],
default=False, warn=True)))
|
7c642b5046f223b124a6b64c3ae52f80d601867773b4d219e30b66a4f5faca8c
|
"""Test whether all elements of cls.args are instances of Basic. """
# NOTE: keep tests sorted by (module, class name) key. If a class can't
# be instantiated, add it here anyway with @SKIP("abstract class) (see
# e.g. Function).
import os
import re
import io
from sympy import (Basic, S, symbols, sqrt, sin, oo, Interval, exp, Lambda, pi,
Eq, log, Function)
from sympy.core.compatibility import range
from sympy.utilities.pytest import XFAIL, SKIP
x, y, z = symbols('x,y,z')
def test_all_classes_are_tested():
this = os.path.split(__file__)[0]
path = os.path.join(this, os.pardir, os.pardir)
sympy_path = os.path.abspath(path)
prefix = os.path.split(sympy_path)[0] + os.sep
re_cls = re.compile(r"^class ([A-Za-z][A-Za-z0-9_]*)\s*\(", re.MULTILINE)
modules = {}
for root, dirs, files in os.walk(sympy_path):
module = root.replace(prefix, "").replace(os.sep, ".")
for file in files:
if file.startswith(("_", "test_", "bench_")):
continue
if not file.endswith(".py"):
continue
with io.open(os.path.join(root, file), "r", encoding='utf-8') as f:
text = f.read()
submodule = module + '.' + file[:-3]
names = re_cls.findall(text)
if not names:
continue
try:
mod = __import__(submodule, fromlist=names)
except ImportError:
continue
def is_Basic(name):
cls = getattr(mod, name)
if hasattr(cls, '_sympy_deprecated_func'):
cls = cls._sympy_deprecated_func
return issubclass(cls, Basic)
names = list(filter(is_Basic, names))
if names:
modules[submodule] = names
ns = globals()
failed = []
for module, names in modules.items():
mod = module.replace('.', '__')
for name in names:
test = 'test_' + mod + '__' + name
if test not in ns:
failed.append(module + '.' + name)
assert not failed, "Missing classes: %s. Please add tests for these to sympy/core/tests/test_args.py." % ", ".join(failed)
def _test_args(obj):
return all(isinstance(arg, Basic) for arg in obj.args)
def test_sympy__assumptions__assume__AppliedPredicate():
from sympy.assumptions.assume import AppliedPredicate, Predicate
from sympy import Q
assert _test_args(AppliedPredicate(Predicate("test"), 2))
assert _test_args(Q.is_true(True))
def test_sympy__assumptions__assume__Predicate():
from sympy.assumptions.assume import Predicate
assert _test_args(Predicate("test"))
def test_sympy__assumptions__sathandlers__UnevaluatedOnFree():
from sympy.assumptions.sathandlers import UnevaluatedOnFree
from sympy import Q
assert _test_args(UnevaluatedOnFree(Q.positive))
assert _test_args(UnevaluatedOnFree(Q.positive(x)))
assert _test_args(UnevaluatedOnFree(Q.positive(x * y)))
def test_sympy__assumptions__sathandlers__AllArgs():
from sympy.assumptions.sathandlers import AllArgs
from sympy import Q
assert _test_args(AllArgs(Q.positive))
assert _test_args(AllArgs(Q.positive(x)))
assert _test_args(AllArgs(Q.positive(x*y)))
def test_sympy__assumptions__sathandlers__AnyArgs():
from sympy.assumptions.sathandlers import AnyArgs
from sympy import Q
assert _test_args(AnyArgs(Q.positive))
assert _test_args(AnyArgs(Q.positive(x)))
assert _test_args(AnyArgs(Q.positive(x*y)))
def test_sympy__assumptions__sathandlers__ExactlyOneArg():
from sympy.assumptions.sathandlers import ExactlyOneArg
from sympy import Q
assert _test_args(ExactlyOneArg(Q.positive))
assert _test_args(ExactlyOneArg(Q.positive(x)))
assert _test_args(ExactlyOneArg(Q.positive(x*y)))
def test_sympy__assumptions__sathandlers__CheckOldAssump():
from sympy.assumptions.sathandlers import CheckOldAssump
from sympy import Q
assert _test_args(CheckOldAssump(Q.positive))
assert _test_args(CheckOldAssump(Q.positive(x)))
assert _test_args(CheckOldAssump(Q.positive(x*y)))
def test_sympy__assumptions__sathandlers__CheckIsPrime():
from sympy.assumptions.sathandlers import CheckIsPrime
from sympy import Q
# Input must be a number
assert _test_args(CheckIsPrime(Q.positive))
assert _test_args(CheckIsPrime(Q.positive(5)))
@SKIP("abstract Class")
def test_sympy__codegen__ast__AssignmentBase():
from sympy.codegen.ast import AssignmentBase
assert _test_args(AssignmentBase(x, 1))
@SKIP("abstract Class")
def test_sympy__codegen__ast__AugmentedAssignment():
from sympy.codegen.ast import AugmentedAssignment
assert _test_args(AugmentedAssignment(x, 1))
def test_sympy__codegen__ast__AddAugmentedAssignment():
from sympy.codegen.ast import AddAugmentedAssignment
assert _test_args(AddAugmentedAssignment(x, 1))
def test_sympy__codegen__ast__SubAugmentedAssignment():
from sympy.codegen.ast import SubAugmentedAssignment
assert _test_args(SubAugmentedAssignment(x, 1))
def test_sympy__codegen__ast__MulAugmentedAssignment():
from sympy.codegen.ast import MulAugmentedAssignment
assert _test_args(MulAugmentedAssignment(x, 1))
def test_sympy__codegen__ast__DivAugmentedAssignment():
from sympy.codegen.ast import DivAugmentedAssignment
assert _test_args(DivAugmentedAssignment(x, 1))
def test_sympy__codegen__ast__ModAugmentedAssignment():
from sympy.codegen.ast import ModAugmentedAssignment
assert _test_args(ModAugmentedAssignment(x, 1))
def test_sympy__codegen__ast__CodeBlock():
from sympy.codegen.ast import CodeBlock, Assignment
assert _test_args(CodeBlock(Assignment(x, 1), Assignment(y, 2)))
def test_sympy__codegen__ast__For():
from sympy.codegen.ast import For, CodeBlock, AddAugmentedAssignment
from sympy import Range
assert _test_args(For(x, Range(10), CodeBlock(AddAugmentedAssignment(y, 1))))
def test_sympy__codegen__ast__Token():
from sympy.codegen.ast import Token
assert _test_args(Token())
def test_sympy__codegen__ast__ContinueToken():
from sympy.codegen.ast import ContinueToken
assert _test_args(ContinueToken())
def test_sympy__codegen__ast__BreakToken():
from sympy.codegen.ast import BreakToken
assert _test_args(BreakToken())
def test_sympy__codegen__ast__NoneToken():
from sympy.codegen.ast import NoneToken
assert _test_args(NoneToken())
def test_sympy__codegen__ast__String():
from sympy.codegen.ast import String
assert _test_args(String('foobar'))
def test_sympy__codegen__ast__QuotedString():
from sympy.codegen.ast import QuotedString
assert _test_args(QuotedString('foobar'))
def test_sympy__codegen__ast__Comment():
from sympy.codegen.ast import Comment
assert _test_args(Comment('this is a comment'))
def test_sympy__codegen__ast__Node():
from sympy.codegen.ast import Node
assert _test_args(Node())
assert _test_args(Node(attrs={1, 2, 3}))
def test_sympy__codegen__ast__Type():
from sympy.codegen.ast import Type
assert _test_args(Type('float128'))
def test_sympy__codegen__ast__IntBaseType():
from sympy.codegen.ast import IntBaseType
assert _test_args(IntBaseType('bigint'))
def test_sympy__codegen__ast___SizedIntType():
from sympy.codegen.ast import _SizedIntType
assert _test_args(_SizedIntType('int128', 128))
def test_sympy__codegen__ast__SignedIntType():
from sympy.codegen.ast import SignedIntType
assert _test_args(SignedIntType('int128_with_sign', 128))
def test_sympy__codegen__ast__UnsignedIntType():
from sympy.codegen.ast import UnsignedIntType
assert _test_args(UnsignedIntType('unt128', 128))
def test_sympy__codegen__ast__FloatBaseType():
from sympy.codegen.ast import FloatBaseType
assert _test_args(FloatBaseType('positive_real'))
def test_sympy__codegen__ast__FloatType():
from sympy.codegen.ast import FloatType
assert _test_args(FloatType('float242', 242, nmant=142, nexp=99))
def test_sympy__codegen__ast__ComplexBaseType():
from sympy.codegen.ast import ComplexBaseType
assert _test_args(ComplexBaseType('positive_cmplx'))
def test_sympy__codegen__ast__ComplexType():
from sympy.codegen.ast import ComplexType
assert _test_args(ComplexType('complex42', 42, nmant=15, nexp=5))
def test_sympy__codegen__ast__Attribute():
from sympy.codegen.ast import Attribute
assert _test_args(Attribute('noexcept'))
def test_sympy__codegen__ast__Variable():
from sympy.codegen.ast import Variable, Type, value_const
assert _test_args(Variable(x))
assert _test_args(Variable(y, Type('float32'), {value_const}))
assert _test_args(Variable(z, type=Type('float64')))
def test_sympy__codegen__ast__Pointer():
from sympy.codegen.ast import Pointer, Type, pointer_const
assert _test_args(Pointer(x))
assert _test_args(Pointer(y, type=Type('float32')))
assert _test_args(Pointer(z, Type('float64'), {pointer_const}))
def test_sympy__codegen__ast__Declaration():
from sympy.codegen.ast import Declaration, Variable, Type
vx = Variable(x, type=Type('float'))
assert _test_args(Declaration(vx))
def test_sympy__codegen__ast__While():
from sympy.codegen.ast import While, AddAugmentedAssignment
assert _test_args(While(abs(x) < 1, [AddAugmentedAssignment(x, -1)]))
def test_sympy__codegen__ast__Scope():
from sympy.codegen.ast import Scope, AddAugmentedAssignment
assert _test_args(Scope([AddAugmentedAssignment(x, -1)]))
def test_sympy__codegen__ast__Stream():
from sympy.codegen.ast import Stream
assert _test_args(Stream('stdin'))
def test_sympy__codegen__ast__Print():
from sympy.codegen.ast import Print
assert _test_args(Print([x, y]))
assert _test_args(Print([x, y], "%d %d"))
def test_sympy__codegen__ast__FunctionPrototype():
from sympy.codegen.ast import FunctionPrototype, real, Declaration, Variable
inp_x = Declaration(Variable(x, type=real))
assert _test_args(FunctionPrototype(real, 'pwer', [inp_x]))
def test_sympy__codegen__ast__FunctionDefinition():
from sympy.codegen.ast import FunctionDefinition, real, Declaration, Variable, Assignment
inp_x = Declaration(Variable(x, type=real))
assert _test_args(FunctionDefinition(real, 'pwer', [inp_x], [Assignment(x, x**2)]))
def test_sympy__codegen__ast__Return():
from sympy.codegen.ast import Return
assert _test_args(Return(x))
def test_sympy__codegen__ast__FunctionCall():
from sympy.codegen.ast import FunctionCall
assert _test_args(FunctionCall('pwer', [x]))
def test_sympy__codegen__ast__Element():
from sympy.codegen.ast import Element
assert _test_args(Element('x', range(3)))
def test_sympy__codegen__cnodes__CommaOperator():
from sympy.codegen.cnodes import CommaOperator
assert _test_args(CommaOperator(1, 2))
def test_sympy__codegen__cnodes__goto():
from sympy.codegen.cnodes import goto
assert _test_args(goto('early_exit'))
def test_sympy__codegen__cnodes__Label():
from sympy.codegen.cnodes import Label
assert _test_args(Label('early_exit'))
def test_sympy__codegen__cnodes__PreDecrement():
from sympy.codegen.cnodes import PreDecrement
assert _test_args(PreDecrement(x))
def test_sympy__codegen__cnodes__PostDecrement():
from sympy.codegen.cnodes import PostDecrement
assert _test_args(PostDecrement(x))
def test_sympy__codegen__cnodes__PreIncrement():
from sympy.codegen.cnodes import PreIncrement
assert _test_args(PreIncrement(x))
def test_sympy__codegen__cnodes__PostIncrement():
from sympy.codegen.cnodes import PostIncrement
assert _test_args(PostIncrement(x))
def test_sympy__codegen__cnodes__struct():
from sympy.codegen.ast import real, Variable
from sympy.codegen.cnodes import struct
assert _test_args(struct(declarations=[
Variable(x, type=real),
Variable(y, type=real)
]))
def test_sympy__codegen__cnodes__union():
from sympy.codegen.ast import float32, int32, Variable
from sympy.codegen.cnodes import union
assert _test_args(union(declarations=[
Variable(x, type=float32),
Variable(y, type=int32)
]))
def test_sympy__codegen__cxxnodes__using():
from sympy.codegen.cxxnodes import using
assert _test_args(using('std::vector'))
assert _test_args(using('std::vector', 'vec'))
def test_sympy__codegen__fnodes__Program():
from sympy.codegen.fnodes import Program
assert _test_args(Program('foobar', []))
def test_sympy__codegen__fnodes__Module():
from sympy.codegen.fnodes import Module
assert _test_args(Module('foobar', [], []))
def test_sympy__codegen__fnodes__Subroutine():
from sympy.codegen.fnodes import Subroutine
x = symbols('x', real=True)
assert _test_args(Subroutine('foo', [x], []))
def test_sympy__codegen__fnodes__GoTo():
from sympy.codegen.fnodes import GoTo
assert _test_args(GoTo([10]))
assert _test_args(GoTo([10, 20], x > 1))
def test_sympy__codegen__fnodes__FortranReturn():
from sympy.codegen.fnodes import FortranReturn
assert _test_args(FortranReturn(10))
def test_sympy__codegen__fnodes__Extent():
from sympy.codegen.fnodes import Extent
assert _test_args(Extent())
assert _test_args(Extent(None))
assert _test_args(Extent(':'))
assert _test_args(Extent(-3, 4))
assert _test_args(Extent(x, y))
def test_sympy__codegen__fnodes__use_rename():
from sympy.codegen.fnodes import use_rename
assert _test_args(use_rename('loc', 'glob'))
def test_sympy__codegen__fnodes__use():
from sympy.codegen.fnodes import use
assert _test_args(use('modfoo', only='bar'))
def test_sympy__codegen__fnodes__SubroutineCall():
from sympy.codegen.fnodes import SubroutineCall
assert _test_args(SubroutineCall('foo', ['bar', 'baz']))
def test_sympy__codegen__fnodes__Do():
from sympy.codegen.fnodes import Do
assert _test_args(Do([], 'i', 1, 42))
def test_sympy__codegen__fnodes__ImpliedDoLoop():
from sympy.codegen.fnodes import ImpliedDoLoop
assert _test_args(ImpliedDoLoop('i', 'i', 1, 42))
def test_sympy__codegen__fnodes__ArrayConstructor():
from sympy.codegen.fnodes import ArrayConstructor
assert _test_args(ArrayConstructor([1, 2, 3]))
from sympy.codegen.fnodes import ImpliedDoLoop
idl = ImpliedDoLoop('i', 'i', 1, 42)
assert _test_args(ArrayConstructor([1, idl, 3]))
def test_sympy__codegen__fnodes__sum_():
from sympy.codegen.fnodes import sum_
assert _test_args(sum_('arr'))
def test_sympy__codegen__fnodes__product_():
from sympy.codegen.fnodes import product_
assert _test_args(product_('arr'))
@XFAIL
def test_sympy__combinatorics__graycode__GrayCode():
from sympy.combinatorics.graycode import GrayCode
# an integer is given and returned from GrayCode as the arg
assert _test_args(GrayCode(3, start='100'))
assert _test_args(GrayCode(3, rank=1))
def test_sympy__combinatorics__subsets__Subset():
from sympy.combinatorics.subsets import Subset
assert _test_args(Subset([0, 1], [0, 1, 2, 3]))
assert _test_args(Subset(['c', 'd'], ['a', 'b', 'c', 'd']))
@XFAIL
def test_sympy__combinatorics__permutations__Permutation():
from sympy.combinatorics.permutations import Permutation
assert _test_args(Permutation([0, 1, 2, 3]))
def test_sympy__combinatorics__perm_groups__PermutationGroup():
from sympy.combinatorics.permutations import Permutation
from sympy.combinatorics.perm_groups import PermutationGroup
assert _test_args(PermutationGroup([Permutation([0, 1])]))
def test_sympy__combinatorics__polyhedron__Polyhedron():
from sympy.combinatorics.permutations import Permutation
from sympy.combinatorics.polyhedron import Polyhedron
from sympy.abc import w, x, y, z
pgroup = [Permutation([[0, 1, 2], [3]]),
Permutation([[0, 1, 3], [2]]),
Permutation([[0, 2, 3], [1]]),
Permutation([[1, 2, 3], [0]]),
Permutation([[0, 1], [2, 3]]),
Permutation([[0, 2], [1, 3]]),
Permutation([[0, 3], [1, 2]]),
Permutation([[0, 1, 2, 3]])]
corners = [w, x, y, z]
faces = [(w, x, y), (w, y, z), (w, z, x), (x, y, z)]
assert _test_args(Polyhedron(corners, faces, pgroup))
@XFAIL
def test_sympy__combinatorics__prufer__Prufer():
from sympy.combinatorics.prufer import Prufer
assert _test_args(Prufer([[0, 1], [0, 2], [0, 3]], 4))
def test_sympy__combinatorics__partitions__Partition():
from sympy.combinatorics.partitions import Partition
assert _test_args(Partition([1]))
@XFAIL
def test_sympy__combinatorics__partitions__IntegerPartition():
from sympy.combinatorics.partitions import IntegerPartition
assert _test_args(IntegerPartition([1]))
def test_sympy__concrete__products__Product():
from sympy.concrete.products import Product
assert _test_args(Product(x, (x, 0, 10)))
assert _test_args(Product(x, (x, 0, y), (y, 0, 10)))
@SKIP("abstract Class")
def test_sympy__concrete__expr_with_limits__ExprWithLimits():
from sympy.concrete.expr_with_limits import ExprWithLimits
assert _test_args(ExprWithLimits(x, (x, 0, 10)))
assert _test_args(ExprWithLimits(x*y, (x, 0, 10.),(y,1.,3)))
@SKIP("abstract Class")
def test_sympy__concrete__expr_with_limits__AddWithLimits():
from sympy.concrete.expr_with_limits import AddWithLimits
assert _test_args(AddWithLimits(x, (x, 0, 10)))
assert _test_args(AddWithLimits(x*y, (x, 0, 10),(y,1,3)))
@SKIP("abstract Class")
def test_sympy__concrete__expr_with_intlimits__ExprWithIntLimits():
from sympy.concrete.expr_with_intlimits import ExprWithIntLimits
assert _test_args(ExprWithIntLimits(x, (x, 0, 10)))
assert _test_args(ExprWithIntLimits(x*y, (x, 0, 10),(y,1,3)))
def test_sympy__concrete__summations__Sum():
from sympy.concrete.summations import Sum
assert _test_args(Sum(x, (x, 0, 10)))
assert _test_args(Sum(x, (x, 0, y), (y, 0, 10)))
def test_sympy__core__add__Add():
from sympy.core.add import Add
assert _test_args(Add(x, y, z, 2))
def test_sympy__core__basic__Atom():
from sympy.core.basic import Atom
assert _test_args(Atom())
def test_sympy__core__basic__Basic():
from sympy.core.basic import Basic
assert _test_args(Basic())
def test_sympy__core__containers__Dict():
from sympy.core.containers import Dict
assert _test_args(Dict({x: y, y: z}))
def test_sympy__core__containers__Tuple():
from sympy.core.containers import Tuple
assert _test_args(Tuple(x, y, z, 2))
def test_sympy__core__expr__AtomicExpr():
from sympy.core.expr import AtomicExpr
assert _test_args(AtomicExpr())
def test_sympy__core__expr__Expr():
from sympy.core.expr import Expr
assert _test_args(Expr())
def test_sympy__core__expr__UnevaluatedExpr():
from sympy.core.expr import UnevaluatedExpr
from sympy.abc import x
assert _test_args(UnevaluatedExpr(x))
def test_sympy__core__function__Application():
from sympy.core.function import Application
assert _test_args(Application(1, 2, 3))
def test_sympy__core__function__AppliedUndef():
from sympy.core.function import AppliedUndef
assert _test_args(AppliedUndef(1, 2, 3))
def test_sympy__core__function__Derivative():
from sympy.core.function import Derivative
assert _test_args(Derivative(2, x, y, 3))
@SKIP("abstract class")
def test_sympy__core__function__Function():
pass
def test_sympy__core__function__Lambda():
assert _test_args(Lambda((x, y), x + y + z))
def test_sympy__core__function__Subs():
from sympy.core.function import Subs
assert _test_args(Subs(x + y, x, 2))
def test_sympy__core__function__WildFunction():
from sympy.core.function import WildFunction
assert _test_args(WildFunction('f'))
def test_sympy__core__mod__Mod():
from sympy.core.mod import Mod
assert _test_args(Mod(x, 2))
def test_sympy__core__mul__Mul():
from sympy.core.mul import Mul
assert _test_args(Mul(2, x, y, z))
def test_sympy__core__numbers__Catalan():
from sympy.core.numbers import Catalan
assert _test_args(Catalan())
def test_sympy__core__numbers__ComplexInfinity():
from sympy.core.numbers import ComplexInfinity
assert _test_args(ComplexInfinity())
def test_sympy__core__numbers__EulerGamma():
from sympy.core.numbers import EulerGamma
assert _test_args(EulerGamma())
def test_sympy__core__numbers__Exp1():
from sympy.core.numbers import Exp1
assert _test_args(Exp1())
def test_sympy__core__numbers__Float():
from sympy.core.numbers import Float
assert _test_args(Float(1.23))
def test_sympy__core__numbers__GoldenRatio():
from sympy.core.numbers import GoldenRatio
assert _test_args(GoldenRatio())
def test_sympy__core__numbers__TribonacciConstant():
from sympy.core.numbers import TribonacciConstant
assert _test_args(TribonacciConstant())
def test_sympy__core__numbers__Half():
from sympy.core.numbers import Half
assert _test_args(Half())
def test_sympy__core__numbers__ImaginaryUnit():
from sympy.core.numbers import ImaginaryUnit
assert _test_args(ImaginaryUnit())
def test_sympy__core__numbers__Infinity():
from sympy.core.numbers import Infinity
assert _test_args(Infinity())
def test_sympy__core__numbers__Integer():
from sympy.core.numbers import Integer
assert _test_args(Integer(7))
@SKIP("abstract class")
def test_sympy__core__numbers__IntegerConstant():
pass
def test_sympy__core__numbers__NaN():
from sympy.core.numbers import NaN
assert _test_args(NaN())
def test_sympy__core__numbers__NegativeInfinity():
from sympy.core.numbers import NegativeInfinity
assert _test_args(NegativeInfinity())
def test_sympy__core__numbers__NegativeOne():
from sympy.core.numbers import NegativeOne
assert _test_args(NegativeOne())
def test_sympy__core__numbers__Number():
from sympy.core.numbers import Number
assert _test_args(Number(1, 7))
def test_sympy__core__numbers__NumberSymbol():
from sympy.core.numbers import NumberSymbol
assert _test_args(NumberSymbol())
def test_sympy__core__numbers__One():
from sympy.core.numbers import One
assert _test_args(One())
def test_sympy__core__numbers__Pi():
from sympy.core.numbers import Pi
assert _test_args(Pi())
def test_sympy__core__numbers__Rational():
from sympy.core.numbers import Rational
assert _test_args(Rational(1, 7))
@SKIP("abstract class")
def test_sympy__core__numbers__RationalConstant():
pass
def test_sympy__core__numbers__Zero():
from sympy.core.numbers import Zero
assert _test_args(Zero())
@SKIP("abstract class")
def test_sympy__core__operations__AssocOp():
pass
@SKIP("abstract class")
def test_sympy__core__operations__LatticeOp():
pass
def test_sympy__core__power__Pow():
from sympy.core.power import Pow
assert _test_args(Pow(x, 2))
def test_sympy__algebras__quaternion__Quaternion():
from sympy.algebras.quaternion import Quaternion
assert _test_args(Quaternion(x, 1, 2, 3))
def test_sympy__core__relational__Equality():
from sympy.core.relational import Equality
assert _test_args(Equality(x, 2))
def test_sympy__core__relational__GreaterThan():
from sympy.core.relational import GreaterThan
assert _test_args(GreaterThan(x, 2))
def test_sympy__core__relational__LessThan():
from sympy.core.relational import LessThan
assert _test_args(LessThan(x, 2))
@SKIP("abstract class")
def test_sympy__core__relational__Relational():
pass
def test_sympy__core__relational__StrictGreaterThan():
from sympy.core.relational import StrictGreaterThan
assert _test_args(StrictGreaterThan(x, 2))
def test_sympy__core__relational__StrictLessThan():
from sympy.core.relational import StrictLessThan
assert _test_args(StrictLessThan(x, 2))
def test_sympy__core__relational__Unequality():
from sympy.core.relational import Unequality
assert _test_args(Unequality(x, 2))
def test_sympy__sandbox__indexed_integrals__IndexedIntegral():
from sympy.tensor import IndexedBase, Idx
from sympy.sandbox.indexed_integrals import IndexedIntegral
A = IndexedBase('A')
i, j = symbols('i j', integer=True)
a1, a2 = symbols('a1:3', cls=Idx)
assert _test_args(IndexedIntegral(A[a1], A[a2]))
assert _test_args(IndexedIntegral(A[i], A[j]))
def test_sympy__calculus__util__AccumulationBounds():
from sympy.calculus.util import AccumulationBounds
assert _test_args(AccumulationBounds(0, 1))
def test_sympy__sets__ordinals__OmegaPower():
from sympy.sets.ordinals import OmegaPower
assert _test_args(OmegaPower(1, 1))
def test_sympy__sets__ordinals__Ordinal():
from sympy.sets.ordinals import Ordinal, OmegaPower
assert _test_args(Ordinal(OmegaPower(2, 1)))
def test_sympy__sets__ordinals__OrdinalOmega():
from sympy.sets.ordinals import OrdinalOmega
assert _test_args(OrdinalOmega())
def test_sympy__sets__ordinals__OrdinalZero():
from sympy.sets.ordinals import OrdinalZero
assert _test_args(OrdinalZero())
def test_sympy__sets__sets__EmptySet():
from sympy.sets.sets import EmptySet
assert _test_args(EmptySet())
def test_sympy__sets__sets__UniversalSet():
from sympy.sets.sets import UniversalSet
assert _test_args(UniversalSet())
def test_sympy__sets__sets__FiniteSet():
from sympy.sets.sets import FiniteSet
assert _test_args(FiniteSet(x, y, z))
def test_sympy__sets__sets__Interval():
from sympy.sets.sets import Interval
assert _test_args(Interval(0, 1))
def test_sympy__sets__sets__ProductSet():
from sympy.sets.sets import ProductSet, Interval
assert _test_args(ProductSet(Interval(0, 1), Interval(0, 1)))
@SKIP("does it make sense to test this?")
def test_sympy__sets__sets__Set():
from sympy.sets.sets import Set
assert _test_args(Set())
def test_sympy__sets__sets__Intersection():
from sympy.sets.sets import Intersection, Interval
assert _test_args(Intersection(Interval(0, 3), Interval(2, 4),
evaluate=False))
def test_sympy__sets__sets__Union():
from sympy.sets.sets import Union, Interval
assert _test_args(Union(Interval(0, 1), Interval(2, 3)))
def test_sympy__sets__sets__Complement():
from sympy.sets.sets import Complement
assert _test_args(Complement(Interval(0, 2), Interval(0, 1)))
def test_sympy__sets__sets__SymmetricDifference():
from sympy.sets.sets import FiniteSet, SymmetricDifference
assert _test_args(SymmetricDifference(FiniteSet(1, 2, 3), \
FiniteSet(2, 3, 4)))
def test_sympy__core__trace__Tr():
from sympy.core.trace import Tr
a, b = symbols('a b')
assert _test_args(Tr(a + b))
def test_sympy__sets__setexpr__SetExpr():
from sympy.sets.setexpr import SetExpr
assert _test_args(SetExpr(Interval(0, 1)))
def test_sympy__sets__fancysets__Naturals():
from sympy.sets.fancysets import Naturals
assert _test_args(Naturals())
def test_sympy__sets__fancysets__Naturals0():
from sympy.sets.fancysets import Naturals0
assert _test_args(Naturals0())
def test_sympy__sets__fancysets__Integers():
from sympy.sets.fancysets import Integers
assert _test_args(Integers())
def test_sympy__sets__fancysets__Reals():
from sympy.sets.fancysets import Reals
assert _test_args(Reals())
def test_sympy__sets__fancysets__Complexes():
from sympy.sets.fancysets import Complexes
assert _test_args(Complexes())
def test_sympy__sets__fancysets__ComplexRegion():
from sympy.sets.fancysets import ComplexRegion
from sympy import S
from sympy.sets import Interval
a = Interval(0, 1)
b = Interval(2, 3)
theta = Interval(0, 2*S.Pi)
assert _test_args(ComplexRegion(a*b))
assert _test_args(ComplexRegion(a*theta, polar=True))
def test_sympy__sets__fancysets__ImageSet():
from sympy.sets.fancysets import ImageSet
from sympy import S, Symbol
x = Symbol('x')
assert _test_args(ImageSet(Lambda(x, x**2), S.Naturals))
def test_sympy__sets__fancysets__Range():
from sympy.sets.fancysets import Range
assert _test_args(Range(1, 5, 1))
def test_sympy__sets__conditionset__ConditionSet():
from sympy.sets.conditionset import ConditionSet
from sympy import S, Symbol
x = Symbol('x')
assert _test_args(ConditionSet(x, Eq(x**2, 1), S.Reals))
def test_sympy__sets__contains__Contains():
from sympy.sets.fancysets import Range
from sympy.sets.contains import Contains
assert _test_args(Contains(x, Range(0, 10, 2)))
# STATS
from sympy.stats.crv_types import NormalDistribution
nd = NormalDistribution(0, 1)
from sympy.stats.frv_types import DieDistribution
die = DieDistribution(6)
def test_sympy__stats__crv__ContinuousDomain():
from sympy.stats.crv import ContinuousDomain
assert _test_args(ContinuousDomain({x}, Interval(-oo, oo)))
def test_sympy__stats__crv__SingleContinuousDomain():
from sympy.stats.crv import SingleContinuousDomain
assert _test_args(SingleContinuousDomain(x, Interval(-oo, oo)))
def test_sympy__stats__crv__ProductContinuousDomain():
from sympy.stats.crv import SingleContinuousDomain, ProductContinuousDomain
D = SingleContinuousDomain(x, Interval(-oo, oo))
E = SingleContinuousDomain(y, Interval(0, oo))
assert _test_args(ProductContinuousDomain(D, E))
def test_sympy__stats__crv__ConditionalContinuousDomain():
from sympy.stats.crv import (SingleContinuousDomain,
ConditionalContinuousDomain)
D = SingleContinuousDomain(x, Interval(-oo, oo))
assert _test_args(ConditionalContinuousDomain(D, x > 0))
def test_sympy__stats__crv__ContinuousPSpace():
from sympy.stats.crv import ContinuousPSpace, SingleContinuousDomain
D = SingleContinuousDomain(x, Interval(-oo, oo))
assert _test_args(ContinuousPSpace(D, nd))
def test_sympy__stats__crv__SingleContinuousPSpace():
from sympy.stats.crv import SingleContinuousPSpace
assert _test_args(SingleContinuousPSpace(x, nd))
@SKIP("abstract class")
def test_sympy__stats__crv__SingleContinuousDistribution():
pass
def test_sympy__stats__drv__SingleDiscreteDomain():
from sympy.stats.drv import SingleDiscreteDomain
assert _test_args(SingleDiscreteDomain(x, S.Naturals))
def test_sympy__stats__drv__ProductDiscreteDomain():
from sympy.stats.drv import SingleDiscreteDomain, ProductDiscreteDomain
X = SingleDiscreteDomain(x, S.Naturals)
Y = SingleDiscreteDomain(y, S.Integers)
assert _test_args(ProductDiscreteDomain(X, Y))
def test_sympy__stats__drv__SingleDiscretePSpace():
from sympy.stats.drv import SingleDiscretePSpace
from sympy.stats.drv_types import PoissonDistribution
assert _test_args(SingleDiscretePSpace(x, PoissonDistribution(1)))
def test_sympy__stats__drv__DiscretePSpace():
from sympy.stats.drv import DiscretePSpace, SingleDiscreteDomain
density = Lambda(x, 2**(-x))
domain = SingleDiscreteDomain(x, S.Naturals)
assert _test_args(DiscretePSpace(domain, density))
def test_sympy__stats__drv__ConditionalDiscreteDomain():
from sympy.stats.drv import ConditionalDiscreteDomain, SingleDiscreteDomain
X = SingleDiscreteDomain(x, S.Naturals0)
assert _test_args(ConditionalDiscreteDomain(X, x > 2))
def test_sympy__stats__joint_rv__JointPSpace():
from sympy.stats.joint_rv import JointPSpace, JointDistribution
assert _test_args(JointPSpace('X', JointDistribution(1)))
def test_sympy__stats__joint_rv__JointRandomSymbol():
from sympy.stats.joint_rv import JointRandomSymbol
assert _test_args(JointRandomSymbol(x))
def test_sympy__stats__joint_rv__JointDistributionHandmade():
from sympy import Indexed
from sympy.stats.joint_rv import JointDistributionHandmade
x1, x2 = (Indexed('x', i) for i in (1, 2))
assert _test_args(JointDistributionHandmade(x1 + x2, S.Reals**2))
def test_sympy__stats__joint_rv__MarginalDistribution():
from sympy.stats.rv import RandomSymbol
from sympy.stats.joint_rv import MarginalDistribution
r = RandomSymbol(S('r'))
assert _test_args(MarginalDistribution(r, (r,)))
def test_sympy__stats__joint_rv__CompoundDistribution():
from sympy.stats.joint_rv import CompoundDistribution
from sympy.stats.drv_types import PoissonDistribution
r = PoissonDistribution(x)
assert _test_args(CompoundDistribution(PoissonDistribution(r)))
@SKIP("abstract class")
def test_sympy__stats__drv__SingleDiscreteDistribution():
pass
@SKIP("abstract class")
def test_sympy__stats__drv__DiscreteDistribution():
pass
@SKIP("abstract class")
def test_sympy__stats__drv__DiscreteDomain():
pass
def test_sympy__stats__rv__RandomDomain():
from sympy.stats.rv import RandomDomain
from sympy.sets.sets import FiniteSet
assert _test_args(RandomDomain(FiniteSet(x), FiniteSet(1, 2, 3)))
def test_sympy__stats__rv__SingleDomain():
from sympy.stats.rv import SingleDomain
from sympy.sets.sets import FiniteSet
assert _test_args(SingleDomain(x, FiniteSet(1, 2, 3)))
def test_sympy__stats__rv__ConditionalDomain():
from sympy.stats.rv import ConditionalDomain, RandomDomain
from sympy.sets.sets import FiniteSet
D = RandomDomain(FiniteSet(x), FiniteSet(1, 2))
assert _test_args(ConditionalDomain(D, x > 1))
def test_sympy__stats__rv__PSpace():
from sympy.stats.rv import PSpace, RandomDomain
from sympy import FiniteSet
D = RandomDomain(FiniteSet(x), FiniteSet(1, 2, 3, 4, 5, 6))
assert _test_args(PSpace(D, die))
@SKIP("abstract Class")
def test_sympy__stats__rv__SinglePSpace():
pass
def test_sympy__stats__rv__RandomSymbol():
from sympy.stats.rv import RandomSymbol
from sympy.stats.crv import SingleContinuousPSpace
A = SingleContinuousPSpace(x, nd)
assert _test_args(RandomSymbol(x, A))
@SKIP("abstract Class")
def test_sympy__stats__rv__ProductPSpace():
pass
def test_sympy__stats__rv__IndependentProductPSpace():
from sympy.stats.rv import IndependentProductPSpace
from sympy.stats.crv import SingleContinuousPSpace
A = SingleContinuousPSpace(x, nd)
B = SingleContinuousPSpace(y, nd)
assert _test_args(IndependentProductPSpace(A, B))
def test_sympy__stats__rv__ProductDomain():
from sympy.stats.rv import ProductDomain, SingleDomain
D = SingleDomain(x, Interval(-oo, oo))
E = SingleDomain(y, Interval(0, oo))
assert _test_args(ProductDomain(D, E))
def test_sympy__stats__symbolic_probability__Probability():
from sympy.stats.symbolic_probability import Probability
from sympy.stats import Normal
X = Normal('X', 0, 1)
assert _test_args(Probability(X > 0))
def test_sympy__stats__symbolic_probability__Expectation():
from sympy.stats.symbolic_probability import Expectation
from sympy.stats import Normal
X = Normal('X', 0, 1)
assert _test_args(Expectation(X > 0))
def test_sympy__stats__symbolic_probability__Covariance():
from sympy.stats.symbolic_probability import Covariance
from sympy.stats import Normal
X = Normal('X', 0, 1)
Y = Normal('Y', 0, 3)
assert _test_args(Covariance(X, Y))
def test_sympy__stats__symbolic_probability__Variance():
from sympy.stats.symbolic_probability import Variance
from sympy.stats import Normal
X = Normal('X', 0, 1)
assert _test_args(Variance(X))
def test_sympy__stats__frv_types__DiscreteUniformDistribution():
from sympy.stats.frv_types import DiscreteUniformDistribution
from sympy.core.containers import Tuple
assert _test_args(DiscreteUniformDistribution(Tuple(*list(range(6)))))
def test_sympy__stats__frv_types__DieDistribution():
assert _test_args(die)
def test_sympy__stats__frv_types__BernoulliDistribution():
from sympy.stats.frv_types import BernoulliDistribution
assert _test_args(BernoulliDistribution(S.Half, 0, 1))
def test_sympy__stats__frv_types__BinomialDistribution():
from sympy.stats.frv_types import BinomialDistribution
assert _test_args(BinomialDistribution(5, S.Half, 1, 0))
def test_sympy__stats__frv_types__HypergeometricDistribution():
from sympy.stats.frv_types import HypergeometricDistribution
assert _test_args(HypergeometricDistribution(10, 5, 3))
def test_sympy__stats__frv_types__RademacherDistribution():
from sympy.stats.frv_types import RademacherDistribution
assert _test_args(RademacherDistribution())
def test_sympy__stats__frv__FiniteDomain():
from sympy.stats.frv import FiniteDomain
assert _test_args(FiniteDomain({(x, 1), (x, 2)})) # x can be 1 or 2
def test_sympy__stats__frv__SingleFiniteDomain():
from sympy.stats.frv import SingleFiniteDomain
assert _test_args(SingleFiniteDomain(x, {1, 2})) # x can be 1 or 2
def test_sympy__stats__frv__ProductFiniteDomain():
from sympy.stats.frv import SingleFiniteDomain, ProductFiniteDomain
xd = SingleFiniteDomain(x, {1, 2})
yd = SingleFiniteDomain(y, {1, 2})
assert _test_args(ProductFiniteDomain(xd, yd))
def test_sympy__stats__frv__ConditionalFiniteDomain():
from sympy.stats.frv import SingleFiniteDomain, ConditionalFiniteDomain
xd = SingleFiniteDomain(x, {1, 2})
assert _test_args(ConditionalFiniteDomain(xd, x > 1))
def test_sympy__stats__frv__FinitePSpace():
from sympy.stats.frv import FinitePSpace, SingleFiniteDomain
xd = SingleFiniteDomain(x, {1, 2, 3, 4, 5, 6})
p = 1.0/6
xd = SingleFiniteDomain(x, {1, 2})
assert _test_args(FinitePSpace(xd, {(x, 1): S.Half, (x, 2): S.Half}))
def test_sympy__stats__frv__SingleFinitePSpace():
from sympy.stats.frv import SingleFinitePSpace
from sympy import Symbol
assert _test_args(SingleFinitePSpace(Symbol('x'), die))
def test_sympy__stats__frv__ProductFinitePSpace():
from sympy.stats.frv import SingleFinitePSpace, ProductFinitePSpace
from sympy import Symbol
xp = SingleFinitePSpace(Symbol('x'), die)
yp = SingleFinitePSpace(Symbol('y'), die)
assert _test_args(ProductFinitePSpace(xp, yp))
@SKIP("abstract class")
def test_sympy__stats__frv__SingleFiniteDistribution():
pass
@SKIP("abstract class")
def test_sympy__stats__crv__ContinuousDistribution():
pass
def test_sympy__stats__frv_types__FiniteDistributionHandmade():
from sympy.stats.frv_types import FiniteDistributionHandmade
assert _test_args(FiniteDistributionHandmade({1: 1}))
def test_sympy__stats__crv__ContinuousDistributionHandmade():
from sympy.stats.crv import ContinuousDistributionHandmade
from sympy import Symbol, Interval
assert _test_args(ContinuousDistributionHandmade(Symbol('x'),
Interval(0, 2)))
def test_sympy__stats__drv__DiscreteDistributionHandmade():
from sympy.stats.drv import DiscreteDistributionHandmade
assert _test_args(DiscreteDistributionHandmade(x, S.Naturals))
def test_sympy__stats__rv__Density():
from sympy.stats.rv import Density
from sympy.stats.crv_types import Normal
assert _test_args(Density(Normal('x', 0, 1)))
def test_sympy__stats__crv_types__ArcsinDistribution():
from sympy.stats.crv_types import ArcsinDistribution
assert _test_args(ArcsinDistribution(0, 1))
def test_sympy__stats__crv_types__BeniniDistribution():
from sympy.stats.crv_types import BeniniDistribution
assert _test_args(BeniniDistribution(1, 1, 1))
def test_sympy__stats__crv_types__BetaDistribution():
from sympy.stats.crv_types import BetaDistribution
assert _test_args(BetaDistribution(1, 1))
def test_sympy__stats__crv_types__BetaPrimeDistribution():
from sympy.stats.crv_types import BetaPrimeDistribution
assert _test_args(BetaPrimeDistribution(1, 1))
def test_sympy__stats__crv_types__CauchyDistribution():
from sympy.stats.crv_types import CauchyDistribution
assert _test_args(CauchyDistribution(0, 1))
def test_sympy__stats__crv_types__ChiDistribution():
from sympy.stats.crv_types import ChiDistribution
assert _test_args(ChiDistribution(1))
def test_sympy__stats__crv_types__ChiNoncentralDistribution():
from sympy.stats.crv_types import ChiNoncentralDistribution
assert _test_args(ChiNoncentralDistribution(1,1))
def test_sympy__stats__crv_types__ChiSquaredDistribution():
from sympy.stats.crv_types import ChiSquaredDistribution
assert _test_args(ChiSquaredDistribution(1))
def test_sympy__stats__crv_types__DagumDistribution():
from sympy.stats.crv_types import DagumDistribution
assert _test_args(DagumDistribution(1, 1, 1))
def test_sympy__stats__crv_types__ExponentialDistribution():
from sympy.stats.crv_types import ExponentialDistribution
assert _test_args(ExponentialDistribution(1))
def test_sympy__stats__crv_types__FDistributionDistribution():
from sympy.stats.crv_types import FDistributionDistribution
assert _test_args(FDistributionDistribution(1, 1))
def test_sympy__stats__crv_types__FisherZDistribution():
from sympy.stats.crv_types import FisherZDistribution
assert _test_args(FisherZDistribution(1, 1))
def test_sympy__stats__crv_types__FrechetDistribution():
from sympy.stats.crv_types import FrechetDistribution
assert _test_args(FrechetDistribution(1, 1, 1))
def test_sympy__stats__crv_types__GammaInverseDistribution():
from sympy.stats.crv_types import GammaInverseDistribution
assert _test_args(GammaInverseDistribution(1, 1))
def test_sympy__stats__crv_types__GammaDistribution():
from sympy.stats.crv_types import GammaDistribution
assert _test_args(GammaDistribution(1, 1))
def test_sympy__stats__crv_types__GumbelDistribution():
from sympy.stats.crv_types import GumbelDistribution
assert _test_args(GumbelDistribution(1, 1))
def test_sympy__stats__crv_types__GompertzDistribution():
from sympy.stats.crv_types import GompertzDistribution
assert _test_args(GompertzDistribution(1, 1))
def test_sympy__stats__crv_types__KumaraswamyDistribution():
from sympy.stats.crv_types import KumaraswamyDistribution
assert _test_args(KumaraswamyDistribution(1, 1))
def test_sympy__stats__crv_types__LaplaceDistribution():
from sympy.stats.crv_types import LaplaceDistribution
assert _test_args(LaplaceDistribution(0, 1))
def test_sympy__stats__crv_types__LogisticDistribution():
from sympy.stats.crv_types import LogisticDistribution
assert _test_args(LogisticDistribution(0, 1))
def test_sympy__stats__crv_types__LogNormalDistribution():
from sympy.stats.crv_types import LogNormalDistribution
assert _test_args(LogNormalDistribution(0, 1))
def test_sympy__stats__crv_types__MaxwellDistribution():
from sympy.stats.crv_types import MaxwellDistribution
assert _test_args(MaxwellDistribution(1))
def test_sympy__stats__crv_types__NakagamiDistribution():
from sympy.stats.crv_types import NakagamiDistribution
assert _test_args(NakagamiDistribution(1, 1))
def test_sympy__stats__crv_types__NormalDistribution():
from sympy.stats.crv_types import NormalDistribution
assert _test_args(NormalDistribution(0, 1))
def test_sympy__stats__crv_types__ParetoDistribution():
from sympy.stats.crv_types import ParetoDistribution
assert _test_args(ParetoDistribution(1, 1))
def test_sympy__stats__crv_types__QuadraticUDistribution():
from sympy.stats.crv_types import QuadraticUDistribution
assert _test_args(QuadraticUDistribution(1, 2))
def test_sympy__stats__crv_types__RaisedCosineDistribution():
from sympy.stats.crv_types import RaisedCosineDistribution
assert _test_args(RaisedCosineDistribution(1, 1))
def test_sympy__stats__crv_types__RayleighDistribution():
from sympy.stats.crv_types import RayleighDistribution
assert _test_args(RayleighDistribution(1))
def test_sympy__stats__crv_types__ShiftedGompertzDistribution():
from sympy.stats.crv_types import ShiftedGompertzDistribution
assert _test_args(ShiftedGompertzDistribution(1, 1))
def test_sympy__stats__crv_types__StudentTDistribution():
from sympy.stats.crv_types import StudentTDistribution
assert _test_args(StudentTDistribution(1))
def test_sympy__stats__crv_types__TrapezoidalDistribution():
from sympy.stats.crv_types import TrapezoidalDistribution
assert _test_args(TrapezoidalDistribution(1, 2, 3, 4))
def test_sympy__stats__crv_types__TriangularDistribution():
from sympy.stats.crv_types import TriangularDistribution
assert _test_args(TriangularDistribution(-1, 0, 1))
def test_sympy__stats__crv_types__UniformDistribution():
from sympy.stats.crv_types import UniformDistribution
assert _test_args(UniformDistribution(0, 1))
def test_sympy__stats__crv_types__UniformSumDistribution():
from sympy.stats.crv_types import UniformSumDistribution
assert _test_args(UniformSumDistribution(1))
def test_sympy__stats__crv_types__VonMisesDistribution():
from sympy.stats.crv_types import VonMisesDistribution
assert _test_args(VonMisesDistribution(1, 1))
def test_sympy__stats__crv_types__WeibullDistribution():
from sympy.stats.crv_types import WeibullDistribution
assert _test_args(WeibullDistribution(1, 1))
def test_sympy__stats__crv_types__WignerSemicircleDistribution():
from sympy.stats.crv_types import WignerSemicircleDistribution
assert _test_args(WignerSemicircleDistribution(1))
def test_sympy__stats__drv_types__GeometricDistribution():
from sympy.stats.drv_types import GeometricDistribution
assert _test_args(GeometricDistribution(.5))
def test_sympy__stats__drv_types__LogarithmicDistribution():
from sympy.stats.drv_types import LogarithmicDistribution
assert _test_args(LogarithmicDistribution(.5))
def test_sympy__stats__drv_types__NegativeBinomialDistribution():
from sympy.stats.drv_types import NegativeBinomialDistribution
assert _test_args(NegativeBinomialDistribution(.5, .5))
def test_sympy__stats__drv_types__PoissonDistribution():
from sympy.stats.drv_types import PoissonDistribution
assert _test_args(PoissonDistribution(1))
def test_sympy__stats__drv_types__YuleSimonDistribution():
from sympy.stats.drv_types import YuleSimonDistribution
assert _test_args(YuleSimonDistribution(.5))
def test_sympy__stats__drv_types__ZetaDistribution():
from sympy.stats.drv_types import ZetaDistribution
assert _test_args(ZetaDistribution(1.5))
def test_sympy__stats__joint_rv__JointDistribution():
from sympy.stats.joint_rv import JointDistribution
assert _test_args(JointDistribution(1, 2, 3, 4))
def test_sympy__stats__joint_rv_types__MultivariateNormalDistribution():
from sympy.stats.joint_rv_types import MultivariateNormalDistribution
assert _test_args(
MultivariateNormalDistribution([0, 1], [[1, 0],[0, 1]]))
def test_sympy__stats__joint_rv_types__MultivariateLaplaceDistribution():
from sympy.stats.joint_rv_types import MultivariateLaplaceDistribution
assert _test_args(MultivariateLaplaceDistribution([0, 1], [[1, 0],[0, 1]]))
def test_sympy__stats__joint_rv_types__MultivariateTDistribution():
from sympy.stats.joint_rv_types import MultivariateTDistribution
assert _test_args(MultivariateTDistribution([0, 1], [[1, 0],[0, 1]], 1))
def test_sympy__stats__joint_rv_types__NormalGammaDistribution():
from sympy.stats.joint_rv_types import NormalGammaDistribution
assert _test_args(NormalGammaDistribution(1, 2, 3, 4))
def test_sympy__core__symbol__Dummy():
from sympy.core.symbol import Dummy
assert _test_args(Dummy('t'))
def test_sympy__core__symbol__Symbol():
from sympy.core.symbol import Symbol
assert _test_args(Symbol('t'))
def test_sympy__core__symbol__Wild():
from sympy.core.symbol import Wild
assert _test_args(Wild('x', exclude=[x]))
@SKIP("abstract class")
def test_sympy__functions__combinatorial__factorials__CombinatorialFunction():
pass
def test_sympy__functions__combinatorial__factorials__FallingFactorial():
from sympy.functions.combinatorial.factorials import FallingFactorial
assert _test_args(FallingFactorial(2, x))
def test_sympy__functions__combinatorial__factorials__MultiFactorial():
from sympy.functions.combinatorial.factorials import MultiFactorial
assert _test_args(MultiFactorial(x))
def test_sympy__functions__combinatorial__factorials__RisingFactorial():
from sympy.functions.combinatorial.factorials import RisingFactorial
assert _test_args(RisingFactorial(2, x))
def test_sympy__functions__combinatorial__factorials__binomial():
from sympy.functions.combinatorial.factorials import binomial
assert _test_args(binomial(2, x))
def test_sympy__functions__combinatorial__factorials__subfactorial():
from sympy.functions.combinatorial.factorials import subfactorial
assert _test_args(subfactorial(1))
def test_sympy__functions__combinatorial__factorials__factorial():
from sympy.functions.combinatorial.factorials import factorial
assert _test_args(factorial(x))
def test_sympy__functions__combinatorial__factorials__factorial2():
from sympy.functions.combinatorial.factorials import factorial2
assert _test_args(factorial2(x))
def test_sympy__functions__combinatorial__numbers__bell():
from sympy.functions.combinatorial.numbers import bell
assert _test_args(bell(x, y))
def test_sympy__functions__combinatorial__numbers__bernoulli():
from sympy.functions.combinatorial.numbers import bernoulli
assert _test_args(bernoulli(x))
def test_sympy__functions__combinatorial__numbers__catalan():
from sympy.functions.combinatorial.numbers import catalan
assert _test_args(catalan(x))
def test_sympy__functions__combinatorial__numbers__genocchi():
from sympy.functions.combinatorial.numbers import genocchi
assert _test_args(genocchi(x))
def test_sympy__functions__combinatorial__numbers__euler():
from sympy.functions.combinatorial.numbers import euler
assert _test_args(euler(x))
def test_sympy__functions__combinatorial__numbers__carmichael():
from sympy.functions.combinatorial.numbers import carmichael
assert _test_args(carmichael(x))
def test_sympy__functions__combinatorial__numbers__fibonacci():
from sympy.functions.combinatorial.numbers import fibonacci
assert _test_args(fibonacci(x))
def test_sympy__functions__combinatorial__numbers__tribonacci():
from sympy.functions.combinatorial.numbers import tribonacci
assert _test_args(tribonacci(x))
def test_sympy__functions__combinatorial__numbers__harmonic():
from sympy.functions.combinatorial.numbers import harmonic
assert _test_args(harmonic(x, 2))
def test_sympy__functions__combinatorial__numbers__lucas():
from sympy.functions.combinatorial.numbers import lucas
assert _test_args(lucas(x))
def test_sympy__functions__combinatorial__numbers__partition():
from sympy.core.symbol import Symbol
from sympy.functions.combinatorial.numbers import partition
assert _test_args(partition(Symbol('a', integer=True)))
def test_sympy__functions__elementary__complexes__Abs():
from sympy.functions.elementary.complexes import Abs
assert _test_args(Abs(x))
def test_sympy__functions__elementary__complexes__adjoint():
from sympy.functions.elementary.complexes import adjoint
assert _test_args(adjoint(x))
def test_sympy__functions__elementary__complexes__arg():
from sympy.functions.elementary.complexes import arg
assert _test_args(arg(x))
def test_sympy__functions__elementary__complexes__conjugate():
from sympy.functions.elementary.complexes import conjugate
assert _test_args(conjugate(x))
def test_sympy__functions__elementary__complexes__im():
from sympy.functions.elementary.complexes import im
assert _test_args(im(x))
def test_sympy__functions__elementary__complexes__re():
from sympy.functions.elementary.complexes import re
assert _test_args(re(x))
def test_sympy__functions__elementary__complexes__sign():
from sympy.functions.elementary.complexes import sign
assert _test_args(sign(x))
def test_sympy__functions__elementary__complexes__polar_lift():
from sympy.functions.elementary.complexes import polar_lift
assert _test_args(polar_lift(x))
def test_sympy__functions__elementary__complexes__periodic_argument():
from sympy.functions.elementary.complexes import periodic_argument
assert _test_args(periodic_argument(x, y))
def test_sympy__functions__elementary__complexes__principal_branch():
from sympy.functions.elementary.complexes import principal_branch
assert _test_args(principal_branch(x, y))
def test_sympy__functions__elementary__complexes__transpose():
from sympy.functions.elementary.complexes import transpose
assert _test_args(transpose(x))
def test_sympy__functions__elementary__exponential__LambertW():
from sympy.functions.elementary.exponential import LambertW
assert _test_args(LambertW(2))
@SKIP("abstract class")
def test_sympy__functions__elementary__exponential__ExpBase():
pass
def test_sympy__functions__elementary__exponential__exp():
from sympy.functions.elementary.exponential import exp
assert _test_args(exp(2))
def test_sympy__functions__elementary__exponential__exp_polar():
from sympy.functions.elementary.exponential import exp_polar
assert _test_args(exp_polar(2))
def test_sympy__functions__elementary__exponential__log():
from sympy.functions.elementary.exponential import log
assert _test_args(log(2))
@SKIP("abstract class")
def test_sympy__functions__elementary__hyperbolic__HyperbolicFunction():
pass
@SKIP("abstract class")
def test_sympy__functions__elementary__hyperbolic__ReciprocalHyperbolicFunction():
pass
@SKIP("abstract class")
def test_sympy__functions__elementary__hyperbolic__InverseHyperbolicFunction():
pass
def test_sympy__functions__elementary__hyperbolic__acosh():
from sympy.functions.elementary.hyperbolic import acosh
assert _test_args(acosh(2))
def test_sympy__functions__elementary__hyperbolic__acoth():
from sympy.functions.elementary.hyperbolic import acoth
assert _test_args(acoth(2))
def test_sympy__functions__elementary__hyperbolic__asinh():
from sympy.functions.elementary.hyperbolic import asinh
assert _test_args(asinh(2))
def test_sympy__functions__elementary__hyperbolic__atanh():
from sympy.functions.elementary.hyperbolic import atanh
assert _test_args(atanh(2))
def test_sympy__functions__elementary__hyperbolic__asech():
from sympy.functions.elementary.hyperbolic import asech
assert _test_args(asech(2))
def test_sympy__functions__elementary__hyperbolic__acsch():
from sympy.functions.elementary.hyperbolic import acsch
assert _test_args(acsch(2))
def test_sympy__functions__elementary__hyperbolic__cosh():
from sympy.functions.elementary.hyperbolic import cosh
assert _test_args(cosh(2))
def test_sympy__functions__elementary__hyperbolic__coth():
from sympy.functions.elementary.hyperbolic import coth
assert _test_args(coth(2))
def test_sympy__functions__elementary__hyperbolic__csch():
from sympy.functions.elementary.hyperbolic import csch
assert _test_args(csch(2))
def test_sympy__functions__elementary__hyperbolic__sech():
from sympy.functions.elementary.hyperbolic import sech
assert _test_args(sech(2))
def test_sympy__functions__elementary__hyperbolic__sinh():
from sympy.functions.elementary.hyperbolic import sinh
assert _test_args(sinh(2))
def test_sympy__functions__elementary__hyperbolic__tanh():
from sympy.functions.elementary.hyperbolic import tanh
assert _test_args(tanh(2))
@SKIP("does this work at all?")
def test_sympy__functions__elementary__integers__RoundFunction():
from sympy.functions.elementary.integers import RoundFunction
assert _test_args(RoundFunction())
def test_sympy__functions__elementary__integers__ceiling():
from sympy.functions.elementary.integers import ceiling
assert _test_args(ceiling(x))
def test_sympy__functions__elementary__integers__floor():
from sympy.functions.elementary.integers import floor
assert _test_args(floor(x))
def test_sympy__functions__elementary__integers__frac():
from sympy.functions.elementary.integers import frac
assert _test_args(frac(x))
def test_sympy__functions__elementary__miscellaneous__IdentityFunction():
from sympy.functions.elementary.miscellaneous import IdentityFunction
assert _test_args(IdentityFunction())
def test_sympy__functions__elementary__miscellaneous__Max():
from sympy.functions.elementary.miscellaneous import Max
assert _test_args(Max(x, 2))
def test_sympy__functions__elementary__miscellaneous__Min():
from sympy.functions.elementary.miscellaneous import Min
assert _test_args(Min(x, 2))
@SKIP("abstract class")
def test_sympy__functions__elementary__miscellaneous__MinMaxBase():
pass
def test_sympy__functions__elementary__piecewise__ExprCondPair():
from sympy.functions.elementary.piecewise import ExprCondPair
assert _test_args(ExprCondPair(1, True))
def test_sympy__functions__elementary__piecewise__Piecewise():
from sympy.functions.elementary.piecewise import Piecewise
assert _test_args(Piecewise((1, x >= 0), (0, True)))
@SKIP("abstract class")
def test_sympy__functions__elementary__trigonometric__TrigonometricFunction():
pass
@SKIP("abstract class")
def test_sympy__functions__elementary__trigonometric__ReciprocalTrigonometricFunction():
pass
@SKIP("abstract class")
def test_sympy__functions__elementary__trigonometric__InverseTrigonometricFunction():
pass
def test_sympy__functions__elementary__trigonometric__acos():
from sympy.functions.elementary.trigonometric import acos
assert _test_args(acos(2))
def test_sympy__functions__elementary__trigonometric__acot():
from sympy.functions.elementary.trigonometric import acot
assert _test_args(acot(2))
def test_sympy__functions__elementary__trigonometric__asin():
from sympy.functions.elementary.trigonometric import asin
assert _test_args(asin(2))
def test_sympy__functions__elementary__trigonometric__asec():
from sympy.functions.elementary.trigonometric import asec
assert _test_args(asec(2))
def test_sympy__functions__elementary__trigonometric__acsc():
from sympy.functions.elementary.trigonometric import acsc
assert _test_args(acsc(2))
def test_sympy__functions__elementary__trigonometric__atan():
from sympy.functions.elementary.trigonometric import atan
assert _test_args(atan(2))
def test_sympy__functions__elementary__trigonometric__atan2():
from sympy.functions.elementary.trigonometric import atan2
assert _test_args(atan2(2, 3))
def test_sympy__functions__elementary__trigonometric__cos():
from sympy.functions.elementary.trigonometric import cos
assert _test_args(cos(2))
def test_sympy__functions__elementary__trigonometric__csc():
from sympy.functions.elementary.trigonometric import csc
assert _test_args(csc(2))
def test_sympy__functions__elementary__trigonometric__cot():
from sympy.functions.elementary.trigonometric import cot
assert _test_args(cot(2))
def test_sympy__functions__elementary__trigonometric__sin():
assert _test_args(sin(2))
def test_sympy__functions__elementary__trigonometric__sinc():
from sympy.functions.elementary.trigonometric import sinc
assert _test_args(sinc(2))
def test_sympy__functions__elementary__trigonometric__sec():
from sympy.functions.elementary.trigonometric import sec
assert _test_args(sec(2))
def test_sympy__functions__elementary__trigonometric__tan():
from sympy.functions.elementary.trigonometric import tan
assert _test_args(tan(2))
@SKIP("abstract class")
def test_sympy__functions__special__bessel__BesselBase():
pass
@SKIP("abstract class")
def test_sympy__functions__special__bessel__SphericalBesselBase():
pass
@SKIP("abstract class")
def test_sympy__functions__special__bessel__SphericalHankelBase():
pass
def test_sympy__functions__special__bessel__besseli():
from sympy.functions.special.bessel import besseli
assert _test_args(besseli(x, 1))
def test_sympy__functions__special__bessel__besselj():
from sympy.functions.special.bessel import besselj
assert _test_args(besselj(x, 1))
def test_sympy__functions__special__bessel__besselk():
from sympy.functions.special.bessel import besselk
assert _test_args(besselk(x, 1))
def test_sympy__functions__special__bessel__bessely():
from sympy.functions.special.bessel import bessely
assert _test_args(bessely(x, 1))
def test_sympy__functions__special__bessel__hankel1():
from sympy.functions.special.bessel import hankel1
assert _test_args(hankel1(x, 1))
def test_sympy__functions__special__bessel__hankel2():
from sympy.functions.special.bessel import hankel2
assert _test_args(hankel2(x, 1))
def test_sympy__functions__special__bessel__jn():
from sympy.functions.special.bessel import jn
assert _test_args(jn(0, x))
def test_sympy__functions__special__bessel__yn():
from sympy.functions.special.bessel import yn
assert _test_args(yn(0, x))
def test_sympy__functions__special__bessel__hn1():
from sympy.functions.special.bessel import hn1
assert _test_args(hn1(0, x))
def test_sympy__functions__special__bessel__hn2():
from sympy.functions.special.bessel import hn2
assert _test_args(hn2(0, x))
def test_sympy__functions__special__bessel__AiryBase():
pass
def test_sympy__functions__special__bessel__airyai():
from sympy.functions.special.bessel import airyai
assert _test_args(airyai(2))
def test_sympy__functions__special__bessel__airybi():
from sympy.functions.special.bessel import airybi
assert _test_args(airybi(2))
def test_sympy__functions__special__bessel__airyaiprime():
from sympy.functions.special.bessel import airyaiprime
assert _test_args(airyaiprime(2))
def test_sympy__functions__special__bessel__airybiprime():
from sympy.functions.special.bessel import airybiprime
assert _test_args(airybiprime(2))
def test_sympy__functions__special__elliptic_integrals__elliptic_k():
from sympy.functions.special.elliptic_integrals import elliptic_k as K
assert _test_args(K(x))
def test_sympy__functions__special__elliptic_integrals__elliptic_f():
from sympy.functions.special.elliptic_integrals import elliptic_f as F
assert _test_args(F(x, y))
def test_sympy__functions__special__elliptic_integrals__elliptic_e():
from sympy.functions.special.elliptic_integrals import elliptic_e as E
assert _test_args(E(x))
assert _test_args(E(x, y))
def test_sympy__functions__special__elliptic_integrals__elliptic_pi():
from sympy.functions.special.elliptic_integrals import elliptic_pi as P
assert _test_args(P(x, y))
assert _test_args(P(x, y, z))
def test_sympy__functions__special__delta_functions__DiracDelta():
from sympy.functions.special.delta_functions import DiracDelta
assert _test_args(DiracDelta(x, 1))
def test_sympy__functions__special__singularity_functions__SingularityFunction():
from sympy.functions.special.singularity_functions import SingularityFunction
assert _test_args(SingularityFunction(x, y, z))
def test_sympy__functions__special__delta_functions__Heaviside():
from sympy.functions.special.delta_functions import Heaviside
assert _test_args(Heaviside(x))
def test_sympy__functions__special__error_functions__erf():
from sympy.functions.special.error_functions import erf
assert _test_args(erf(2))
def test_sympy__functions__special__error_functions__erfc():
from sympy.functions.special.error_functions import erfc
assert _test_args(erfc(2))
def test_sympy__functions__special__error_functions__erfi():
from sympy.functions.special.error_functions import erfi
assert _test_args(erfi(2))
def test_sympy__functions__special__error_functions__erf2():
from sympy.functions.special.error_functions import erf2
assert _test_args(erf2(2, 3))
def test_sympy__functions__special__error_functions__erfinv():
from sympy.functions.special.error_functions import erfinv
assert _test_args(erfinv(2))
def test_sympy__functions__special__error_functions__erfcinv():
from sympy.functions.special.error_functions import erfcinv
assert _test_args(erfcinv(2))
def test_sympy__functions__special__error_functions__erf2inv():
from sympy.functions.special.error_functions import erf2inv
assert _test_args(erf2inv(2, 3))
@SKIP("abstract class")
def test_sympy__functions__special__error_functions__FresnelIntegral():
pass
def test_sympy__functions__special__error_functions__fresnels():
from sympy.functions.special.error_functions import fresnels
assert _test_args(fresnels(2))
def test_sympy__functions__special__error_functions__fresnelc():
from sympy.functions.special.error_functions import fresnelc
assert _test_args(fresnelc(2))
def test_sympy__functions__special__error_functions__erfs():
from sympy.functions.special.error_functions import _erfs
assert _test_args(_erfs(2))
def test_sympy__functions__special__error_functions__Ei():
from sympy.functions.special.error_functions import Ei
assert _test_args(Ei(2))
def test_sympy__functions__special__error_functions__li():
from sympy.functions.special.error_functions import li
assert _test_args(li(2))
def test_sympy__functions__special__error_functions__Li():
from sympy.functions.special.error_functions import Li
assert _test_args(Li(2))
@SKIP("abstract class")
def test_sympy__functions__special__error_functions__TrigonometricIntegral():
pass
def test_sympy__functions__special__error_functions__Si():
from sympy.functions.special.error_functions import Si
assert _test_args(Si(2))
def test_sympy__functions__special__error_functions__Ci():
from sympy.functions.special.error_functions import Ci
assert _test_args(Ci(2))
def test_sympy__functions__special__error_functions__Shi():
from sympy.functions.special.error_functions import Shi
assert _test_args(Shi(2))
def test_sympy__functions__special__error_functions__Chi():
from sympy.functions.special.error_functions import Chi
assert _test_args(Chi(2))
def test_sympy__functions__special__error_functions__expint():
from sympy.functions.special.error_functions import expint
assert _test_args(expint(y, x))
def test_sympy__functions__special__gamma_functions__gamma():
from sympy.functions.special.gamma_functions import gamma
assert _test_args(gamma(x))
def test_sympy__functions__special__gamma_functions__loggamma():
from sympy.functions.special.gamma_functions import loggamma
assert _test_args(loggamma(2))
def test_sympy__functions__special__gamma_functions__lowergamma():
from sympy.functions.special.gamma_functions import lowergamma
assert _test_args(lowergamma(x, 2))
def test_sympy__functions__special__gamma_functions__polygamma():
from sympy.functions.special.gamma_functions import polygamma
assert _test_args(polygamma(x, 2))
def test_sympy__functions__special__gamma_functions__uppergamma():
from sympy.functions.special.gamma_functions import uppergamma
assert _test_args(uppergamma(x, 2))
def test_sympy__functions__special__beta_functions__beta():
from sympy.functions.special.beta_functions import beta
assert _test_args(beta(x, x))
def test_sympy__functions__special__mathieu_functions__MathieuBase():
pass
def test_sympy__functions__special__mathieu_functions__mathieus():
from sympy.functions.special.mathieu_functions import mathieus
assert _test_args(mathieus(1, 1, 1))
def test_sympy__functions__special__mathieu_functions__mathieuc():
from sympy.functions.special.mathieu_functions import mathieuc
assert _test_args(mathieuc(1, 1, 1))
def test_sympy__functions__special__mathieu_functions__mathieusprime():
from sympy.functions.special.mathieu_functions import mathieusprime
assert _test_args(mathieusprime(1, 1, 1))
def test_sympy__functions__special__mathieu_functions__mathieucprime():
from sympy.functions.special.mathieu_functions import mathieucprime
assert _test_args(mathieucprime(1, 1, 1))
@SKIP("abstract class")
def test_sympy__functions__special__hyper__TupleParametersBase():
pass
@SKIP("abstract class")
def test_sympy__functions__special__hyper__TupleArg():
pass
def test_sympy__functions__special__hyper__hyper():
from sympy.functions.special.hyper import hyper
assert _test_args(hyper([1, 2, 3], [4, 5], x))
def test_sympy__functions__special__hyper__meijerg():
from sympy.functions.special.hyper import meijerg
assert _test_args(meijerg([1, 2, 3], [4, 5], [6], [], x))
@SKIP("abstract class")
def test_sympy__functions__special__hyper__HyperRep():
pass
def test_sympy__functions__special__hyper__HyperRep_power1():
from sympy.functions.special.hyper import HyperRep_power1
assert _test_args(HyperRep_power1(x, y))
def test_sympy__functions__special__hyper__HyperRep_power2():
from sympy.functions.special.hyper import HyperRep_power2
assert _test_args(HyperRep_power2(x, y))
def test_sympy__functions__special__hyper__HyperRep_log1():
from sympy.functions.special.hyper import HyperRep_log1
assert _test_args(HyperRep_log1(x))
def test_sympy__functions__special__hyper__HyperRep_atanh():
from sympy.functions.special.hyper import HyperRep_atanh
assert _test_args(HyperRep_atanh(x))
def test_sympy__functions__special__hyper__HyperRep_asin1():
from sympy.functions.special.hyper import HyperRep_asin1
assert _test_args(HyperRep_asin1(x))
def test_sympy__functions__special__hyper__HyperRep_asin2():
from sympy.functions.special.hyper import HyperRep_asin2
assert _test_args(HyperRep_asin2(x))
def test_sympy__functions__special__hyper__HyperRep_sqrts1():
from sympy.functions.special.hyper import HyperRep_sqrts1
assert _test_args(HyperRep_sqrts1(x, y))
def test_sympy__functions__special__hyper__HyperRep_sqrts2():
from sympy.functions.special.hyper import HyperRep_sqrts2
assert _test_args(HyperRep_sqrts2(x, y))
def test_sympy__functions__special__hyper__HyperRep_log2():
from sympy.functions.special.hyper import HyperRep_log2
assert _test_args(HyperRep_log2(x))
def test_sympy__functions__special__hyper__HyperRep_cosasin():
from sympy.functions.special.hyper import HyperRep_cosasin
assert _test_args(HyperRep_cosasin(x, y))
def test_sympy__functions__special__hyper__HyperRep_sinasin():
from sympy.functions.special.hyper import HyperRep_sinasin
assert _test_args(HyperRep_sinasin(x, y))
def test_sympy__functions__special__hyper__appellf1():
from sympy.functions.special.hyper import appellf1
a, b1, b2, c, x, y = symbols('a b1 b2 c x y')
assert _test_args(appellf1(a, b1, b2, c, x, y))
@SKIP("abstract class")
def test_sympy__functions__special__polynomials__OrthogonalPolynomial():
pass
def test_sympy__functions__special__polynomials__jacobi():
from sympy.functions.special.polynomials import jacobi
assert _test_args(jacobi(x, 2, 2, 2))
def test_sympy__functions__special__polynomials__gegenbauer():
from sympy.functions.special.polynomials import gegenbauer
assert _test_args(gegenbauer(x, 2, 2))
def test_sympy__functions__special__polynomials__chebyshevt():
from sympy.functions.special.polynomials import chebyshevt
assert _test_args(chebyshevt(x, 2))
def test_sympy__functions__special__polynomials__chebyshevt_root():
from sympy.functions.special.polynomials import chebyshevt_root
assert _test_args(chebyshevt_root(3, 2))
def test_sympy__functions__special__polynomials__chebyshevu():
from sympy.functions.special.polynomials import chebyshevu
assert _test_args(chebyshevu(x, 2))
def test_sympy__functions__special__polynomials__chebyshevu_root():
from sympy.functions.special.polynomials import chebyshevu_root
assert _test_args(chebyshevu_root(3, 2))
def test_sympy__functions__special__polynomials__hermite():
from sympy.functions.special.polynomials import hermite
assert _test_args(hermite(x, 2))
def test_sympy__functions__special__polynomials__legendre():
from sympy.functions.special.polynomials import legendre
assert _test_args(legendre(x, 2))
def test_sympy__functions__special__polynomials__assoc_legendre():
from sympy.functions.special.polynomials import assoc_legendre
assert _test_args(assoc_legendre(x, 0, y))
def test_sympy__functions__special__polynomials__laguerre():
from sympy.functions.special.polynomials import laguerre
assert _test_args(laguerre(x, 2))
def test_sympy__functions__special__polynomials__assoc_laguerre():
from sympy.functions.special.polynomials import assoc_laguerre
assert _test_args(assoc_laguerre(x, 0, y))
def test_sympy__functions__special__spherical_harmonics__Ynm():
from sympy.functions.special.spherical_harmonics import Ynm
assert _test_args(Ynm(1, 1, x, y))
def test_sympy__functions__special__spherical_harmonics__Znm():
from sympy.functions.special.spherical_harmonics import Znm
assert _test_args(Znm(1, 1, x, y))
def test_sympy__functions__special__tensor_functions__LeviCivita():
from sympy.functions.special.tensor_functions import LeviCivita
assert _test_args(LeviCivita(x, y, 2))
def test_sympy__functions__special__tensor_functions__KroneckerDelta():
from sympy.functions.special.tensor_functions import KroneckerDelta
assert _test_args(KroneckerDelta(x, y))
def test_sympy__functions__special__zeta_functions__dirichlet_eta():
from sympy.functions.special.zeta_functions import dirichlet_eta
assert _test_args(dirichlet_eta(x))
def test_sympy__functions__special__zeta_functions__zeta():
from sympy.functions.special.zeta_functions import zeta
assert _test_args(zeta(101))
def test_sympy__functions__special__zeta_functions__lerchphi():
from sympy.functions.special.zeta_functions import lerchphi
assert _test_args(lerchphi(x, y, z))
def test_sympy__functions__special__zeta_functions__polylog():
from sympy.functions.special.zeta_functions import polylog
assert _test_args(polylog(x, y))
def test_sympy__functions__special__zeta_functions__stieltjes():
from sympy.functions.special.zeta_functions import stieltjes
assert _test_args(stieltjes(x, y))
def test_sympy__integrals__integrals__Integral():
from sympy.integrals.integrals import Integral
assert _test_args(Integral(2, (x, 0, 1)))
def test_sympy__integrals__risch__NonElementaryIntegral():
from sympy.integrals.risch import NonElementaryIntegral
assert _test_args(NonElementaryIntegral(exp(-x**2), x))
@SKIP("abstract class")
def test_sympy__integrals__transforms__IntegralTransform():
pass
def test_sympy__integrals__transforms__MellinTransform():
from sympy.integrals.transforms import MellinTransform
assert _test_args(MellinTransform(2, x, y))
def test_sympy__integrals__transforms__InverseMellinTransform():
from sympy.integrals.transforms import InverseMellinTransform
assert _test_args(InverseMellinTransform(2, x, y, 0, 1))
def test_sympy__integrals__transforms__LaplaceTransform():
from sympy.integrals.transforms import LaplaceTransform
assert _test_args(LaplaceTransform(2, x, y))
def test_sympy__integrals__transforms__InverseLaplaceTransform():
from sympy.integrals.transforms import InverseLaplaceTransform
assert _test_args(InverseLaplaceTransform(2, x, y, 0))
@SKIP("abstract class")
def test_sympy__integrals__transforms__FourierTypeTransform():
pass
def test_sympy__integrals__transforms__InverseFourierTransform():
from sympy.integrals.transforms import InverseFourierTransform
assert _test_args(InverseFourierTransform(2, x, y))
def test_sympy__integrals__transforms__FourierTransform():
from sympy.integrals.transforms import FourierTransform
assert _test_args(FourierTransform(2, x, y))
@SKIP("abstract class")
def test_sympy__integrals__transforms__SineCosineTypeTransform():
pass
def test_sympy__integrals__transforms__InverseSineTransform():
from sympy.integrals.transforms import InverseSineTransform
assert _test_args(InverseSineTransform(2, x, y))
def test_sympy__integrals__transforms__SineTransform():
from sympy.integrals.transforms import SineTransform
assert _test_args(SineTransform(2, x, y))
def test_sympy__integrals__transforms__InverseCosineTransform():
from sympy.integrals.transforms import InverseCosineTransform
assert _test_args(InverseCosineTransform(2, x, y))
def test_sympy__integrals__transforms__CosineTransform():
from sympy.integrals.transforms import CosineTransform
assert _test_args(CosineTransform(2, x, y))
@SKIP("abstract class")
def test_sympy__integrals__transforms__HankelTypeTransform():
pass
def test_sympy__integrals__transforms__InverseHankelTransform():
from sympy.integrals.transforms import InverseHankelTransform
assert _test_args(InverseHankelTransform(2, x, y, 0))
def test_sympy__integrals__transforms__HankelTransform():
from sympy.integrals.transforms import HankelTransform
assert _test_args(HankelTransform(2, x, y, 0))
@XFAIL
def test_sympy__liealgebras__cartan_type__CartanType_generator():
from sympy.liealgebras.cartan_type import CartanType_generator
assert _test_args(CartanType_generator("A2"))
@XFAIL
def test_sympy__liealgebras__cartan_type__Standard_Cartan():
from sympy.liealgebras.cartan_type import Standard_Cartan
assert _test_args(Standard_Cartan("A", 2))
@XFAIL
def test_sympy__liealgebras__weyl_group__WeylGroup():
from sympy.liealgebras.weyl_group import WeylGroup
assert _test_args(WeylGroup("B4"))
@XFAIL
def test_sympy__liealgebras__root_system__RootSystem():
from sympy.liealgebras.root_system import RootSystem
assert _test_args(RootSystem("A2"))
@XFAIL
def test_sympy__liealgebras__type_a__TypeA():
from sympy.liealgebras.type_a import TypeA
assert _test_args(TypeA(2))
@XFAIL
def test_sympy__liealgebras__type_b__TypeB():
from sympy.liealgebras.type_b import TypeB
assert _test_args(TypeB(4))
@XFAIL
def test_sympy__liealgebras__type_c__TypeC():
from sympy.liealgebras.type_c import TypeC
assert _test_args(TypeC(4))
@XFAIL
def test_sympy__liealgebras__type_d__TypeD():
from sympy.liealgebras.type_d import TypeD
assert _test_args(TypeD(4))
@XFAIL
def test_sympy__liealgebras__type_e__TypeE():
from sympy.liealgebras.type_e import TypeE
assert _test_args(TypeE(6))
@XFAIL
def test_sympy__liealgebras__type_f__TypeF():
from sympy.liealgebras.type_f import TypeF
assert _test_args(TypeF(4))
@XFAIL
def test_sympy__liealgebras__type_g__TypeG():
from sympy.liealgebras.type_g import TypeG
assert _test_args(TypeG(2))
def test_sympy__logic__boolalg__And():
from sympy.logic.boolalg import And
assert _test_args(And(x, y, 1))
@SKIP("abstract class")
def test_sympy__logic__boolalg__Boolean():
pass
def test_sympy__logic__boolalg__BooleanFunction():
from sympy.logic.boolalg import BooleanFunction
assert _test_args(BooleanFunction(1, 2, 3))
@SKIP("abstract class")
def test_sympy__logic__boolalg__BooleanAtom():
pass
def test_sympy__logic__boolalg__BooleanTrue():
from sympy.logic.boolalg import true
assert _test_args(true)
def test_sympy__logic__boolalg__BooleanFalse():
from sympy.logic.boolalg import false
assert _test_args(false)
def test_sympy__logic__boolalg__Equivalent():
from sympy.logic.boolalg import Equivalent
assert _test_args(Equivalent(x, 2))
def test_sympy__logic__boolalg__ITE():
from sympy.logic.boolalg import ITE
assert _test_args(ITE(x, y, 1))
def test_sympy__logic__boolalg__Implies():
from sympy.logic.boolalg import Implies
assert _test_args(Implies(x, y))
def test_sympy__logic__boolalg__Nand():
from sympy.logic.boolalg import Nand
assert _test_args(Nand(x, y, 1))
def test_sympy__logic__boolalg__Nor():
from sympy.logic.boolalg import Nor
assert _test_args(Nor(x, y))
def test_sympy__logic__boolalg__Not():
from sympy.logic.boolalg import Not
assert _test_args(Not(x))
def test_sympy__logic__boolalg__Or():
from sympy.logic.boolalg import Or
assert _test_args(Or(x, y))
def test_sympy__logic__boolalg__Xor():
from sympy.logic.boolalg import Xor
assert _test_args(Xor(x, y, 2))
def test_sympy__logic__boolalg__Xnor():
from sympy.logic.boolalg import Xnor
assert _test_args(Xnor(x, y, 2))
def test_sympy__matrices__matrices__DeferredVector():
from sympy.matrices.matrices import DeferredVector
assert _test_args(DeferredVector("X"))
@SKIP("abstract class")
def test_sympy__matrices__expressions__matexpr__MatrixBase():
pass
def test_sympy__matrices__immutable__ImmutableDenseMatrix():
from sympy.matrices.immutable import ImmutableDenseMatrix
m = ImmutableDenseMatrix([[1, 2], [3, 4]])
assert _test_args(m)
assert _test_args(Basic(*list(m)))
m = ImmutableDenseMatrix(1, 1, [1])
assert _test_args(m)
assert _test_args(Basic(*list(m)))
m = ImmutableDenseMatrix(2, 2, lambda i, j: 1)
assert m[0, 0] is S.One
m = ImmutableDenseMatrix(2, 2, lambda i, j: 1/(1 + i) + 1/(1 + j))
assert m[1, 1] is S.One # true div. will give 1.0 if i,j not sympified
assert _test_args(m)
assert _test_args(Basic(*list(m)))
def test_sympy__matrices__immutable__ImmutableSparseMatrix():
from sympy.matrices.immutable import ImmutableSparseMatrix
m = ImmutableSparseMatrix([[1, 2], [3, 4]])
assert _test_args(m)
assert _test_args(Basic(*list(m)))
m = ImmutableSparseMatrix(1, 1, {(0, 0): 1})
assert _test_args(m)
assert _test_args(Basic(*list(m)))
m = ImmutableSparseMatrix(1, 1, [1])
assert _test_args(m)
assert _test_args(Basic(*list(m)))
m = ImmutableSparseMatrix(2, 2, lambda i, j: 1)
assert m[0, 0] is S.One
m = ImmutableSparseMatrix(2, 2, lambda i, j: 1/(1 + i) + 1/(1 + j))
assert m[1, 1] is S.One # true div. will give 1.0 if i,j not sympified
assert _test_args(m)
assert _test_args(Basic(*list(m)))
def test_sympy__matrices__expressions__slice__MatrixSlice():
from sympy.matrices.expressions.slice import MatrixSlice
from sympy.matrices.expressions import MatrixSymbol
X = MatrixSymbol('X', 4, 4)
assert _test_args(MatrixSlice(X, (0, 2), (0, 2)))
def test_sympy__matrices__expressions__applyfunc__ElementwiseApplyFunction():
from sympy.matrices.expressions.applyfunc import ElementwiseApplyFunction
from sympy.matrices.expressions import MatrixSymbol
X = MatrixSymbol("X", x, x)
func = Lambda(x, x**2)
assert _test_args(ElementwiseApplyFunction(func, X))
def test_sympy__matrices__expressions__blockmatrix__BlockDiagMatrix():
from sympy.matrices.expressions.blockmatrix import BlockDiagMatrix
from sympy.matrices.expressions import MatrixSymbol
X = MatrixSymbol('X', x, x)
Y = MatrixSymbol('Y', y, y)
assert _test_args(BlockDiagMatrix(X, Y))
def test_sympy__matrices__expressions__blockmatrix__BlockMatrix():
from sympy.matrices.expressions.blockmatrix import BlockMatrix
from sympy.matrices.expressions import MatrixSymbol, ZeroMatrix
X = MatrixSymbol('X', x, x)
Y = MatrixSymbol('Y', y, y)
Z = MatrixSymbol('Z', x, y)
O = ZeroMatrix(y, x)
assert _test_args(BlockMatrix([[X, Z], [O, Y]]))
def test_sympy__matrices__expressions__inverse__Inverse():
from sympy.matrices.expressions.inverse import Inverse
from sympy.matrices.expressions import MatrixSymbol
assert _test_args(Inverse(MatrixSymbol('A', 3, 3)))
def test_sympy__matrices__expressions__matadd__MatAdd():
from sympy.matrices.expressions.matadd import MatAdd
from sympy.matrices.expressions import MatrixSymbol
X = MatrixSymbol('X', x, y)
Y = MatrixSymbol('Y', x, y)
assert _test_args(MatAdd(X, Y))
def test_sympy__matrices__expressions__matexpr__Identity():
from sympy.matrices.expressions.matexpr import Identity
assert _test_args(Identity(3))
def test_sympy__matrices__expressions__matexpr__GenericIdentity():
from sympy.matrices.expressions.matexpr import GenericIdentity
assert _test_args(GenericIdentity())
@SKIP("abstract class")
def test_sympy__matrices__expressions__matexpr__MatrixExpr():
pass
def test_sympy__matrices__expressions__matexpr__MatrixElement():
from sympy.matrices.expressions.matexpr import MatrixSymbol, MatrixElement
from sympy import S
assert _test_args(MatrixElement(MatrixSymbol('A', 3, 5), S(2), S(3)))
@XFAIL
def test_sympy__matrices__expressions__matexpr__MatrixSymbol():
from sympy.matrices.expressions.matexpr import MatrixSymbol
assert _test_args(MatrixSymbol('A', 3, 5))
def test_sympy__matrices__expressions__matexpr__ZeroMatrix():
from sympy.matrices.expressions.matexpr import ZeroMatrix
assert _test_args(ZeroMatrix(3, 5))
def test_sympy__matrices__expressions__matexpr__GenericZeroMatrix():
from sympy.matrices.expressions.matexpr import GenericZeroMatrix
assert _test_args(GenericZeroMatrix())
def test_sympy__matrices__expressions__matmul__MatMul():
from sympy.matrices.expressions.matmul import MatMul
from sympy.matrices.expressions import MatrixSymbol
X = MatrixSymbol('X', x, y)
Y = MatrixSymbol('Y', y, x)
assert _test_args(MatMul(X, Y))
def test_sympy__matrices__expressions__dotproduct__DotProduct():
from sympy.matrices.expressions.dotproduct import DotProduct
from sympy.matrices.expressions import MatrixSymbol
X = MatrixSymbol('X', x, 1)
Y = MatrixSymbol('Y', x, 1)
assert _test_args(DotProduct(X, Y))
def test_sympy__matrices__expressions__diagonal__DiagonalMatrix():
from sympy.matrices.expressions.diagonal import DiagonalMatrix
from sympy.matrices.expressions import MatrixSymbol
x = MatrixSymbol('x', 10, 1)
assert _test_args(DiagonalMatrix(x))
def test_sympy__matrices__expressions__diagonal__DiagonalOf():
from sympy.matrices.expressions.diagonal import DiagonalOf
from sympy.matrices.expressions import MatrixSymbol
X = MatrixSymbol('x', 10, 10)
assert _test_args(DiagonalOf(X))
def test_sympy__matrices__expressions__hadamard__HadamardProduct():
from sympy.matrices.expressions.hadamard import HadamardProduct
from sympy.matrices.expressions import MatrixSymbol
X = MatrixSymbol('X', x, y)
Y = MatrixSymbol('Y', x, y)
assert _test_args(HadamardProduct(X, Y))
def test_sympy__matrices__expressions__kronecker__KroneckerProduct():
from sympy.matrices.expressions.kronecker import KroneckerProduct
from sympy.matrices.expressions import MatrixSymbol
X = MatrixSymbol('X', x, y)
Y = MatrixSymbol('Y', x, y)
assert _test_args(KroneckerProduct(X, Y))
def test_sympy__matrices__expressions__matpow__MatPow():
from sympy.matrices.expressions.matpow import MatPow
from sympy.matrices.expressions import MatrixSymbol
X = MatrixSymbol('X', x, x)
assert _test_args(MatPow(X, 2))
def test_sympy__matrices__expressions__transpose__Transpose():
from sympy.matrices.expressions.transpose import Transpose
from sympy.matrices.expressions import MatrixSymbol
assert _test_args(Transpose(MatrixSymbol('A', 3, 5)))
def test_sympy__matrices__expressions__adjoint__Adjoint():
from sympy.matrices.expressions.adjoint import Adjoint
from sympy.matrices.expressions import MatrixSymbol
assert _test_args(Adjoint(MatrixSymbol('A', 3, 5)))
def test_sympy__matrices__expressions__trace__Trace():
from sympy.matrices.expressions.trace import Trace
from sympy.matrices.expressions import MatrixSymbol
assert _test_args(Trace(MatrixSymbol('A', 3, 3)))
def test_sympy__matrices__expressions__determinant__Determinant():
from sympy.matrices.expressions.determinant import Determinant
from sympy.matrices.expressions import MatrixSymbol
assert _test_args(Determinant(MatrixSymbol('A', 3, 3)))
def test_sympy__matrices__expressions__funcmatrix__FunctionMatrix():
from sympy.matrices.expressions.funcmatrix import FunctionMatrix
from sympy import symbols
i, j = symbols('i,j')
assert _test_args(FunctionMatrix(3, 3, Lambda((i, j), i - j) ))
def test_sympy__matrices__expressions__fourier__DFT():
from sympy.matrices.expressions.fourier import DFT
from sympy import S
assert _test_args(DFT(S(2)))
def test_sympy__matrices__expressions__fourier__IDFT():
from sympy.matrices.expressions.fourier import IDFT
from sympy import S
assert _test_args(IDFT(S(2)))
from sympy.matrices.expressions import MatrixSymbol
X = MatrixSymbol('X', 10, 10)
def test_sympy__matrices__expressions__factorizations__LofLU():
from sympy.matrices.expressions.factorizations import LofLU
assert _test_args(LofLU(X))
def test_sympy__matrices__expressions__factorizations__UofLU():
from sympy.matrices.expressions.factorizations import UofLU
assert _test_args(UofLU(X))
def test_sympy__matrices__expressions__factorizations__QofQR():
from sympy.matrices.expressions.factorizations import QofQR
assert _test_args(QofQR(X))
def test_sympy__matrices__expressions__factorizations__RofQR():
from sympy.matrices.expressions.factorizations import RofQR
assert _test_args(RofQR(X))
def test_sympy__matrices__expressions__factorizations__LofCholesky():
from sympy.matrices.expressions.factorizations import LofCholesky
assert _test_args(LofCholesky(X))
def test_sympy__matrices__expressions__factorizations__UofCholesky():
from sympy.matrices.expressions.factorizations import UofCholesky
assert _test_args(UofCholesky(X))
def test_sympy__matrices__expressions__factorizations__EigenVectors():
from sympy.matrices.expressions.factorizations import EigenVectors
assert _test_args(EigenVectors(X))
def test_sympy__matrices__expressions__factorizations__EigenValues():
from sympy.matrices.expressions.factorizations import EigenValues
assert _test_args(EigenValues(X))
def test_sympy__matrices__expressions__factorizations__UofSVD():
from sympy.matrices.expressions.factorizations import UofSVD
assert _test_args(UofSVD(X))
def test_sympy__matrices__expressions__factorizations__VofSVD():
from sympy.matrices.expressions.factorizations import VofSVD
assert _test_args(VofSVD(X))
def test_sympy__matrices__expressions__factorizations__SofSVD():
from sympy.matrices.expressions.factorizations import SofSVD
assert _test_args(SofSVD(X))
@SKIP("abstract class")
def test_sympy__matrices__expressions__factorizations__Factorization():
pass
def test_sympy__physics__vector__frame__CoordinateSym():
from sympy.physics.vector import CoordinateSym
from sympy.physics.vector import ReferenceFrame
assert _test_args(CoordinateSym('R_x', ReferenceFrame('R'), 0))
def test_sympy__physics__paulialgebra__Pauli():
from sympy.physics.paulialgebra import Pauli
assert _test_args(Pauli(1))
def test_sympy__physics__quantum__anticommutator__AntiCommutator():
from sympy.physics.quantum.anticommutator import AntiCommutator
assert _test_args(AntiCommutator(x, y))
def test_sympy__physics__quantum__cartesian__PositionBra3D():
from sympy.physics.quantum.cartesian import PositionBra3D
assert _test_args(PositionBra3D(x, y, z))
def test_sympy__physics__quantum__cartesian__PositionKet3D():
from sympy.physics.quantum.cartesian import PositionKet3D
assert _test_args(PositionKet3D(x, y, z))
def test_sympy__physics__quantum__cartesian__PositionState3D():
from sympy.physics.quantum.cartesian import PositionState3D
assert _test_args(PositionState3D(x, y, z))
def test_sympy__physics__quantum__cartesian__PxBra():
from sympy.physics.quantum.cartesian import PxBra
assert _test_args(PxBra(x, y, z))
def test_sympy__physics__quantum__cartesian__PxKet():
from sympy.physics.quantum.cartesian import PxKet
assert _test_args(PxKet(x, y, z))
def test_sympy__physics__quantum__cartesian__PxOp():
from sympy.physics.quantum.cartesian import PxOp
assert _test_args(PxOp(x, y, z))
def test_sympy__physics__quantum__cartesian__XBra():
from sympy.physics.quantum.cartesian import XBra
assert _test_args(XBra(x))
def test_sympy__physics__quantum__cartesian__XKet():
from sympy.physics.quantum.cartesian import XKet
assert _test_args(XKet(x))
def test_sympy__physics__quantum__cartesian__XOp():
from sympy.physics.quantum.cartesian import XOp
assert _test_args(XOp(x))
def test_sympy__physics__quantum__cartesian__YOp():
from sympy.physics.quantum.cartesian import YOp
assert _test_args(YOp(x))
def test_sympy__physics__quantum__cartesian__ZOp():
from sympy.physics.quantum.cartesian import ZOp
assert _test_args(ZOp(x))
def test_sympy__physics__quantum__cg__CG():
from sympy.physics.quantum.cg import CG
from sympy import S
assert _test_args(CG(S(3)/2, S(3)/2, S(1)/2, -S(1)/2, 1, 1))
def test_sympy__physics__quantum__cg__Wigner3j():
from sympy.physics.quantum.cg import Wigner3j
assert _test_args(Wigner3j(6, 0, 4, 0, 2, 0))
def test_sympy__physics__quantum__cg__Wigner6j():
from sympy.physics.quantum.cg import Wigner6j
assert _test_args(Wigner6j(1, 2, 3, 2, 1, 2))
def test_sympy__physics__quantum__cg__Wigner9j():
from sympy.physics.quantum.cg import Wigner9j
assert _test_args(Wigner9j(2, 1, 1, S(3)/2, S(1)/2, 1, S(1)/2, S(1)/2, 0))
def test_sympy__physics__quantum__circuitplot__Mz():
from sympy.physics.quantum.circuitplot import Mz
assert _test_args(Mz(0))
def test_sympy__physics__quantum__circuitplot__Mx():
from sympy.physics.quantum.circuitplot import Mx
assert _test_args(Mx(0))
def test_sympy__physics__quantum__commutator__Commutator():
from sympy.physics.quantum.commutator import Commutator
A, B = symbols('A,B', commutative=False)
assert _test_args(Commutator(A, B))
def test_sympy__physics__quantum__constants__HBar():
from sympy.physics.quantum.constants import HBar
assert _test_args(HBar())
def test_sympy__physics__quantum__dagger__Dagger():
from sympy.physics.quantum.dagger import Dagger
from sympy.physics.quantum.state import Ket
assert _test_args(Dagger(Dagger(Ket('psi'))))
def test_sympy__physics__quantum__gate__CGate():
from sympy.physics.quantum.gate import CGate, Gate
assert _test_args(CGate((0, 1), Gate(2)))
def test_sympy__physics__quantum__gate__CGateS():
from sympy.physics.quantum.gate import CGateS, Gate
assert _test_args(CGateS((0, 1), Gate(2)))
def test_sympy__physics__quantum__gate__CNotGate():
from sympy.physics.quantum.gate import CNotGate
assert _test_args(CNotGate(0, 1))
def test_sympy__physics__quantum__gate__Gate():
from sympy.physics.quantum.gate import Gate
assert _test_args(Gate(0))
def test_sympy__physics__quantum__gate__HadamardGate():
from sympy.physics.quantum.gate import HadamardGate
assert _test_args(HadamardGate(0))
def test_sympy__physics__quantum__gate__IdentityGate():
from sympy.physics.quantum.gate import IdentityGate
assert _test_args(IdentityGate(0))
def test_sympy__physics__quantum__gate__OneQubitGate():
from sympy.physics.quantum.gate import OneQubitGate
assert _test_args(OneQubitGate(0))
def test_sympy__physics__quantum__gate__PhaseGate():
from sympy.physics.quantum.gate import PhaseGate
assert _test_args(PhaseGate(0))
def test_sympy__physics__quantum__gate__SwapGate():
from sympy.physics.quantum.gate import SwapGate
assert _test_args(SwapGate(0, 1))
def test_sympy__physics__quantum__gate__TGate():
from sympy.physics.quantum.gate import TGate
assert _test_args(TGate(0))
def test_sympy__physics__quantum__gate__TwoQubitGate():
from sympy.physics.quantum.gate import TwoQubitGate
assert _test_args(TwoQubitGate(0))
def test_sympy__physics__quantum__gate__UGate():
from sympy.physics.quantum.gate import UGate
from sympy.matrices.immutable import ImmutableDenseMatrix
from sympy import Integer, Tuple
assert _test_args(
UGate(Tuple(Integer(1)), ImmutableDenseMatrix([[1, 0], [0, 2]])))
def test_sympy__physics__quantum__gate__XGate():
from sympy.physics.quantum.gate import XGate
assert _test_args(XGate(0))
def test_sympy__physics__quantum__gate__YGate():
from sympy.physics.quantum.gate import YGate
assert _test_args(YGate(0))
def test_sympy__physics__quantum__gate__ZGate():
from sympy.physics.quantum.gate import ZGate
assert _test_args(ZGate(0))
@SKIP("TODO: sympy.physics")
def test_sympy__physics__quantum__grover__OracleGate():
from sympy.physics.quantum.grover import OracleGate
assert _test_args(OracleGate())
def test_sympy__physics__quantum__grover__WGate():
from sympy.physics.quantum.grover import WGate
assert _test_args(WGate(1))
def test_sympy__physics__quantum__hilbert__ComplexSpace():
from sympy.physics.quantum.hilbert import ComplexSpace
assert _test_args(ComplexSpace(x))
def test_sympy__physics__quantum__hilbert__DirectSumHilbertSpace():
from sympy.physics.quantum.hilbert import DirectSumHilbertSpace, ComplexSpace, FockSpace
c = ComplexSpace(2)
f = FockSpace()
assert _test_args(DirectSumHilbertSpace(c, f))
def test_sympy__physics__quantum__hilbert__FockSpace():
from sympy.physics.quantum.hilbert import FockSpace
assert _test_args(FockSpace())
def test_sympy__physics__quantum__hilbert__HilbertSpace():
from sympy.physics.quantum.hilbert import HilbertSpace
assert _test_args(HilbertSpace())
def test_sympy__physics__quantum__hilbert__L2():
from sympy.physics.quantum.hilbert import L2
from sympy import oo, Interval
assert _test_args(L2(Interval(0, oo)))
def test_sympy__physics__quantum__hilbert__TensorPowerHilbertSpace():
from sympy.physics.quantum.hilbert import TensorPowerHilbertSpace, FockSpace
f = FockSpace()
assert _test_args(TensorPowerHilbertSpace(f, 2))
def test_sympy__physics__quantum__hilbert__TensorProductHilbertSpace():
from sympy.physics.quantum.hilbert import TensorProductHilbertSpace, FockSpace, ComplexSpace
c = ComplexSpace(2)
f = FockSpace()
assert _test_args(TensorProductHilbertSpace(f, c))
def test_sympy__physics__quantum__innerproduct__InnerProduct():
from sympy.physics.quantum import Bra, Ket, InnerProduct
b = Bra('b')
k = Ket('k')
assert _test_args(InnerProduct(b, k))
def test_sympy__physics__quantum__operator__DifferentialOperator():
from sympy.physics.quantum.operator import DifferentialOperator
from sympy import Derivative, Function
f = Function('f')
assert _test_args(DifferentialOperator(1/x*Derivative(f(x), x), f(x)))
def test_sympy__physics__quantum__operator__HermitianOperator():
from sympy.physics.quantum.operator import HermitianOperator
assert _test_args(HermitianOperator('H'))
def test_sympy__physics__quantum__operator__IdentityOperator():
from sympy.physics.quantum.operator import IdentityOperator
assert _test_args(IdentityOperator(5))
def test_sympy__physics__quantum__operator__Operator():
from sympy.physics.quantum.operator import Operator
assert _test_args(Operator('A'))
def test_sympy__physics__quantum__operator__OuterProduct():
from sympy.physics.quantum.operator import OuterProduct
from sympy.physics.quantum import Ket, Bra
b = Bra('b')
k = Ket('k')
assert _test_args(OuterProduct(k, b))
def test_sympy__physics__quantum__operator__UnitaryOperator():
from sympy.physics.quantum.operator import UnitaryOperator
assert _test_args(UnitaryOperator('U'))
def test_sympy__physics__quantum__piab__PIABBra():
from sympy.physics.quantum.piab import PIABBra
assert _test_args(PIABBra('B'))
def test_sympy__physics__quantum__boson__BosonOp():
from sympy.physics.quantum.boson import BosonOp
assert _test_args(BosonOp('a'))
assert _test_args(BosonOp('a', False))
def test_sympy__physics__quantum__boson__BosonFockKet():
from sympy.physics.quantum.boson import BosonFockKet
assert _test_args(BosonFockKet(1))
def test_sympy__physics__quantum__boson__BosonFockBra():
from sympy.physics.quantum.boson import BosonFockBra
assert _test_args(BosonFockBra(1))
def test_sympy__physics__quantum__boson__BosonCoherentKet():
from sympy.physics.quantum.boson import BosonCoherentKet
assert _test_args(BosonCoherentKet(1))
def test_sympy__physics__quantum__boson__BosonCoherentBra():
from sympy.physics.quantum.boson import BosonCoherentBra
assert _test_args(BosonCoherentBra(1))
def test_sympy__physics__quantum__fermion__FermionOp():
from sympy.physics.quantum.fermion import FermionOp
assert _test_args(FermionOp('c'))
assert _test_args(FermionOp('c', False))
def test_sympy__physics__quantum__fermion__FermionFockKet():
from sympy.physics.quantum.fermion import FermionFockKet
assert _test_args(FermionFockKet(1))
def test_sympy__physics__quantum__fermion__FermionFockBra():
from sympy.physics.quantum.fermion import FermionFockBra
assert _test_args(FermionFockBra(1))
def test_sympy__physics__quantum__pauli__SigmaOpBase():
from sympy.physics.quantum.pauli import SigmaOpBase
assert _test_args(SigmaOpBase())
def test_sympy__physics__quantum__pauli__SigmaX():
from sympy.physics.quantum.pauli import SigmaX
assert _test_args(SigmaX())
def test_sympy__physics__quantum__pauli__SigmaY():
from sympy.physics.quantum.pauli import SigmaY
assert _test_args(SigmaY())
def test_sympy__physics__quantum__pauli__SigmaZ():
from sympy.physics.quantum.pauli import SigmaZ
assert _test_args(SigmaZ())
def test_sympy__physics__quantum__pauli__SigmaMinus():
from sympy.physics.quantum.pauli import SigmaMinus
assert _test_args(SigmaMinus())
def test_sympy__physics__quantum__pauli__SigmaPlus():
from sympy.physics.quantum.pauli import SigmaPlus
assert _test_args(SigmaPlus())
def test_sympy__physics__quantum__pauli__SigmaZKet():
from sympy.physics.quantum.pauli import SigmaZKet
assert _test_args(SigmaZKet(0))
def test_sympy__physics__quantum__pauli__SigmaZBra():
from sympy.physics.quantum.pauli import SigmaZBra
assert _test_args(SigmaZBra(0))
def test_sympy__physics__quantum__piab__PIABHamiltonian():
from sympy.physics.quantum.piab import PIABHamiltonian
assert _test_args(PIABHamiltonian('P'))
def test_sympy__physics__quantum__piab__PIABKet():
from sympy.physics.quantum.piab import PIABKet
assert _test_args(PIABKet('K'))
def test_sympy__physics__quantum__qexpr__QExpr():
from sympy.physics.quantum.qexpr import QExpr
assert _test_args(QExpr(0))
def test_sympy__physics__quantum__qft__Fourier():
from sympy.physics.quantum.qft import Fourier
assert _test_args(Fourier(0, 1))
def test_sympy__physics__quantum__qft__IQFT():
from sympy.physics.quantum.qft import IQFT
assert _test_args(IQFT(0, 1))
def test_sympy__physics__quantum__qft__QFT():
from sympy.physics.quantum.qft import QFT
assert _test_args(QFT(0, 1))
def test_sympy__physics__quantum__qft__RkGate():
from sympy.physics.quantum.qft import RkGate
assert _test_args(RkGate(0, 1))
def test_sympy__physics__quantum__qubit__IntQubit():
from sympy.physics.quantum.qubit import IntQubit
assert _test_args(IntQubit(0))
def test_sympy__physics__quantum__qubit__IntQubitBra():
from sympy.physics.quantum.qubit import IntQubitBra
assert _test_args(IntQubitBra(0))
def test_sympy__physics__quantum__qubit__IntQubitState():
from sympy.physics.quantum.qubit import IntQubitState, QubitState
assert _test_args(IntQubitState(QubitState(0, 1)))
def test_sympy__physics__quantum__qubit__Qubit():
from sympy.physics.quantum.qubit import Qubit
assert _test_args(Qubit(0, 0, 0))
def test_sympy__physics__quantum__qubit__QubitBra():
from sympy.physics.quantum.qubit import QubitBra
assert _test_args(QubitBra('1', 0))
def test_sympy__physics__quantum__qubit__QubitState():
from sympy.physics.quantum.qubit import QubitState
assert _test_args(QubitState(0, 1))
def test_sympy__physics__quantum__density__Density():
from sympy.physics.quantum.density import Density
from sympy.physics.quantum.state import Ket
assert _test_args(Density([Ket(0), 0.5], [Ket(1), 0.5]))
@SKIP("TODO: sympy.physics.quantum.shor: Cmod Not Implemented")
def test_sympy__physics__quantum__shor__CMod():
from sympy.physics.quantum.shor import CMod
assert _test_args(CMod())
def test_sympy__physics__quantum__spin__CoupledSpinState():
from sympy.physics.quantum.spin import CoupledSpinState
assert _test_args(CoupledSpinState(1, 0, (1, 1)))
assert _test_args(CoupledSpinState(1, 0, (1, S(1)/2, S(1)/2)))
assert _test_args(CoupledSpinState(
1, 0, (1, S(1)/2, S(1)/2), ((2, 3, S(1)/2), (1, 2, 1)) ))
j, m, j1, j2, j3, j12, x = symbols('j m j1:4 j12 x')
assert CoupledSpinState(
j, m, (j1, j2, j3)).subs(j2, x) == CoupledSpinState(j, m, (j1, x, j3))
assert CoupledSpinState(j, m, (j1, j2, j3), ((1, 3, j12), (1, 2, j)) ).subs(j12, x) == \
CoupledSpinState(j, m, (j1, j2, j3), ((1, 3, x), (1, 2, j)) )
def test_sympy__physics__quantum__spin__J2Op():
from sympy.physics.quantum.spin import J2Op
assert _test_args(J2Op('J'))
def test_sympy__physics__quantum__spin__JminusOp():
from sympy.physics.quantum.spin import JminusOp
assert _test_args(JminusOp('J'))
def test_sympy__physics__quantum__spin__JplusOp():
from sympy.physics.quantum.spin import JplusOp
assert _test_args(JplusOp('J'))
def test_sympy__physics__quantum__spin__JxBra():
from sympy.physics.quantum.spin import JxBra
assert _test_args(JxBra(1, 0))
def test_sympy__physics__quantum__spin__JxBraCoupled():
from sympy.physics.quantum.spin import JxBraCoupled
assert _test_args(JxBraCoupled(1, 0, (1, 1)))
def test_sympy__physics__quantum__spin__JxKet():
from sympy.physics.quantum.spin import JxKet
assert _test_args(JxKet(1, 0))
def test_sympy__physics__quantum__spin__JxKetCoupled():
from sympy.physics.quantum.spin import JxKetCoupled
assert _test_args(JxKetCoupled(1, 0, (1, 1)))
def test_sympy__physics__quantum__spin__JxOp():
from sympy.physics.quantum.spin import JxOp
assert _test_args(JxOp('J'))
def test_sympy__physics__quantum__spin__JyBra():
from sympy.physics.quantum.spin import JyBra
assert _test_args(JyBra(1, 0))
def test_sympy__physics__quantum__spin__JyBraCoupled():
from sympy.physics.quantum.spin import JyBraCoupled
assert _test_args(JyBraCoupled(1, 0, (1, 1)))
def test_sympy__physics__quantum__spin__JyKet():
from sympy.physics.quantum.spin import JyKet
assert _test_args(JyKet(1, 0))
def test_sympy__physics__quantum__spin__JyKetCoupled():
from sympy.physics.quantum.spin import JyKetCoupled
assert _test_args(JyKetCoupled(1, 0, (1, 1)))
def test_sympy__physics__quantum__spin__JyOp():
from sympy.physics.quantum.spin import JyOp
assert _test_args(JyOp('J'))
def test_sympy__physics__quantum__spin__JzBra():
from sympy.physics.quantum.spin import JzBra
assert _test_args(JzBra(1, 0))
def test_sympy__physics__quantum__spin__JzBraCoupled():
from sympy.physics.quantum.spin import JzBraCoupled
assert _test_args(JzBraCoupled(1, 0, (1, 1)))
def test_sympy__physics__quantum__spin__JzKet():
from sympy.physics.quantum.spin import JzKet
assert _test_args(JzKet(1, 0))
def test_sympy__physics__quantum__spin__JzKetCoupled():
from sympy.physics.quantum.spin import JzKetCoupled
assert _test_args(JzKetCoupled(1, 0, (1, 1)))
def test_sympy__physics__quantum__spin__JzOp():
from sympy.physics.quantum.spin import JzOp
assert _test_args(JzOp('J'))
def test_sympy__physics__quantum__spin__Rotation():
from sympy.physics.quantum.spin import Rotation
assert _test_args(Rotation(pi, 0, pi/2))
def test_sympy__physics__quantum__spin__SpinState():
from sympy.physics.quantum.spin import SpinState
assert _test_args(SpinState(1, 0))
def test_sympy__physics__quantum__spin__WignerD():
from sympy.physics.quantum.spin import WignerD
assert _test_args(WignerD(0, 1, 2, 3, 4, 5))
def test_sympy__physics__quantum__state__Bra():
from sympy.physics.quantum.state import Bra
assert _test_args(Bra(0))
def test_sympy__physics__quantum__state__BraBase():
from sympy.physics.quantum.state import BraBase
assert _test_args(BraBase(0))
def test_sympy__physics__quantum__state__Ket():
from sympy.physics.quantum.state import Ket
assert _test_args(Ket(0))
def test_sympy__physics__quantum__state__KetBase():
from sympy.physics.quantum.state import KetBase
assert _test_args(KetBase(0))
def test_sympy__physics__quantum__state__State():
from sympy.physics.quantum.state import State
assert _test_args(State(0))
def test_sympy__physics__quantum__state__StateBase():
from sympy.physics.quantum.state import StateBase
assert _test_args(StateBase(0))
def test_sympy__physics__quantum__state__TimeDepBra():
from sympy.physics.quantum.state import TimeDepBra
assert _test_args(TimeDepBra('psi', 't'))
def test_sympy__physics__quantum__state__TimeDepKet():
from sympy.physics.quantum.state import TimeDepKet
assert _test_args(TimeDepKet('psi', 't'))
def test_sympy__physics__quantum__state__TimeDepState():
from sympy.physics.quantum.state import TimeDepState
assert _test_args(TimeDepState('psi', 't'))
def test_sympy__physics__quantum__state__Wavefunction():
from sympy.physics.quantum.state import Wavefunction
from sympy.functions import sin
from sympy import Piecewise
n = 1
L = 1
g = Piecewise((0, x < 0), (0, x > L), (sqrt(2//L)*sin(n*pi*x/L), True))
assert _test_args(Wavefunction(g, x))
def test_sympy__physics__quantum__tensorproduct__TensorProduct():
from sympy.physics.quantum.tensorproduct import TensorProduct
assert _test_args(TensorProduct(x, y))
def test_sympy__physics__quantum__identitysearch__GateIdentity():
from sympy.physics.quantum.gate import X
from sympy.physics.quantum.identitysearch import GateIdentity
assert _test_args(GateIdentity(X(0), X(0)))
def test_sympy__physics__quantum__sho1d__SHOOp():
from sympy.physics.quantum.sho1d import SHOOp
assert _test_args(SHOOp('a'))
def test_sympy__physics__quantum__sho1d__RaisingOp():
from sympy.physics.quantum.sho1d import RaisingOp
assert _test_args(RaisingOp('a'))
def test_sympy__physics__quantum__sho1d__LoweringOp():
from sympy.physics.quantum.sho1d import LoweringOp
assert _test_args(LoweringOp('a'))
def test_sympy__physics__quantum__sho1d__NumberOp():
from sympy.physics.quantum.sho1d import NumberOp
assert _test_args(NumberOp('N'))
def test_sympy__physics__quantum__sho1d__Hamiltonian():
from sympy.physics.quantum.sho1d import Hamiltonian
assert _test_args(Hamiltonian('H'))
def test_sympy__physics__quantum__sho1d__SHOState():
from sympy.physics.quantum.sho1d import SHOState
assert _test_args(SHOState(0))
def test_sympy__physics__quantum__sho1d__SHOKet():
from sympy.physics.quantum.sho1d import SHOKet
assert _test_args(SHOKet(0))
def test_sympy__physics__quantum__sho1d__SHOBra():
from sympy.physics.quantum.sho1d import SHOBra
assert _test_args(SHOBra(0))
def test_sympy__physics__secondquant__AnnihilateBoson():
from sympy.physics.secondquant import AnnihilateBoson
assert _test_args(AnnihilateBoson(0))
def test_sympy__physics__secondquant__AnnihilateFermion():
from sympy.physics.secondquant import AnnihilateFermion
assert _test_args(AnnihilateFermion(0))
@SKIP("abstract class")
def test_sympy__physics__secondquant__Annihilator():
pass
def test_sympy__physics__secondquant__AntiSymmetricTensor():
from sympy.physics.secondquant import AntiSymmetricTensor
i, j = symbols('i j', below_fermi=True)
a, b = symbols('a b', above_fermi=True)
assert _test_args(AntiSymmetricTensor('v', (a, i), (b, j)))
def test_sympy__physics__secondquant__BosonState():
from sympy.physics.secondquant import BosonState
assert _test_args(BosonState((0, 1)))
@SKIP("abstract class")
def test_sympy__physics__secondquant__BosonicOperator():
pass
def test_sympy__physics__secondquant__Commutator():
from sympy.physics.secondquant import Commutator
assert _test_args(Commutator(x, y))
def test_sympy__physics__secondquant__CreateBoson():
from sympy.physics.secondquant import CreateBoson
assert _test_args(CreateBoson(0))
def test_sympy__physics__secondquant__CreateFermion():
from sympy.physics.secondquant import CreateFermion
assert _test_args(CreateFermion(0))
@SKIP("abstract class")
def test_sympy__physics__secondquant__Creator():
pass
def test_sympy__physics__secondquant__Dagger():
from sympy.physics.secondquant import Dagger
from sympy import I
assert _test_args(Dagger(2*I))
def test_sympy__physics__secondquant__FermionState():
from sympy.physics.secondquant import FermionState
assert _test_args(FermionState((0, 1)))
def test_sympy__physics__secondquant__FermionicOperator():
from sympy.physics.secondquant import FermionicOperator
assert _test_args(FermionicOperator(0))
def test_sympy__physics__secondquant__FockState():
from sympy.physics.secondquant import FockState
assert _test_args(FockState((0, 1)))
def test_sympy__physics__secondquant__FockStateBosonBra():
from sympy.physics.secondquant import FockStateBosonBra
assert _test_args(FockStateBosonBra((0, 1)))
def test_sympy__physics__secondquant__FockStateBosonKet():
from sympy.physics.secondquant import FockStateBosonKet
assert _test_args(FockStateBosonKet((0, 1)))
def test_sympy__physics__secondquant__FockStateBra():
from sympy.physics.secondquant import FockStateBra
assert _test_args(FockStateBra((0, 1)))
def test_sympy__physics__secondquant__FockStateFermionBra():
from sympy.physics.secondquant import FockStateFermionBra
assert _test_args(FockStateFermionBra((0, 1)))
def test_sympy__physics__secondquant__FockStateFermionKet():
from sympy.physics.secondquant import FockStateFermionKet
assert _test_args(FockStateFermionKet((0, 1)))
def test_sympy__physics__secondquant__FockStateKet():
from sympy.physics.secondquant import FockStateKet
assert _test_args(FockStateKet((0, 1)))
def test_sympy__physics__secondquant__InnerProduct():
from sympy.physics.secondquant import InnerProduct
from sympy.physics.secondquant import FockStateKet, FockStateBra
assert _test_args(InnerProduct(FockStateBra((0, 1)), FockStateKet((0, 1))))
def test_sympy__physics__secondquant__NO():
from sympy.physics.secondquant import NO, F, Fd
assert _test_args(NO(Fd(x)*F(y)))
def test_sympy__physics__secondquant__PermutationOperator():
from sympy.physics.secondquant import PermutationOperator
assert _test_args(PermutationOperator(0, 1))
def test_sympy__physics__secondquant__SqOperator():
from sympy.physics.secondquant import SqOperator
assert _test_args(SqOperator(0))
def test_sympy__physics__secondquant__TensorSymbol():
from sympy.physics.secondquant import TensorSymbol
assert _test_args(TensorSymbol(x))
def test_sympy__physics__units__dimensions__Dimension():
from sympy.physics.units.dimensions import Dimension
assert _test_args(Dimension("length", "L"))
def test_sympy__physics__units__dimensions__DimensionSystem():
from sympy.physics.units.dimensions import DimensionSystem
from sympy.physics.units.dimensions import length, time, velocity
assert _test_args(DimensionSystem((length, time), (velocity,)))
def test_sympy__physics__units__quantities__Quantity():
from sympy.physics.units.quantities import Quantity
from sympy.physics.units import length
assert _test_args(Quantity("dam"))
def test_sympy__physics__units__prefixes__Prefix():
from sympy.physics.units.prefixes import Prefix
assert _test_args(Prefix('kilo', 'k', 3))
def test_sympy__core__numbers__AlgebraicNumber():
from sympy.core.numbers import AlgebraicNumber
assert _test_args(AlgebraicNumber(sqrt(2), [1, 2, 3]))
def test_sympy__polys__polytools__GroebnerBasis():
from sympy.polys.polytools import GroebnerBasis
assert _test_args(GroebnerBasis([x, y, z], x, y, z))
def test_sympy__polys__polytools__Poly():
from sympy.polys.polytools import Poly
assert _test_args(Poly(2, x, y))
def test_sympy__polys__polytools__PurePoly():
from sympy.polys.polytools import PurePoly
assert _test_args(PurePoly(2, x, y))
@SKIP('abstract class')
def test_sympy__polys__rootoftools__RootOf():
pass
def test_sympy__polys__rootoftools__ComplexRootOf():
from sympy.polys.rootoftools import ComplexRootOf
assert _test_args(ComplexRootOf(x**3 + x + 1, 0))
def test_sympy__polys__rootoftools__RootSum():
from sympy.polys.rootoftools import RootSum
assert _test_args(RootSum(x**3 + x + 1, sin))
def test_sympy__series__limits__Limit():
from sympy.series.limits import Limit
assert _test_args(Limit(x, x, 0, dir='-'))
def test_sympy__series__order__Order():
from sympy.series.order import Order
assert _test_args(Order(1, x, y))
@SKIP('Abstract Class')
def test_sympy__series__sequences__SeqBase():
pass
def test_sympy__series__sequences__EmptySequence():
from sympy.series.sequences import EmptySequence
assert _test_args(EmptySequence())
@SKIP('Abstract Class')
def test_sympy__series__sequences__SeqExpr():
pass
def test_sympy__series__sequences__SeqPer():
from sympy.series.sequences import SeqPer
assert _test_args(SeqPer((1, 2, 3), (0, 10)))
def test_sympy__series__sequences__SeqFormula():
from sympy.series.sequences import SeqFormula
assert _test_args(SeqFormula(x**2, (0, 10)))
def test_sympy__series__sequences__RecursiveSeq():
from sympy.series.sequences import RecursiveSeq
y = Function("y")
n = symbols("n")
assert _test_args(RecursiveSeq(y(n - 1) + y(n - 2), y, n, (0, 1)))
assert _test_args(RecursiveSeq(y(n - 1) + y(n - 2), y, n))
def test_sympy__series__sequences__SeqExprOp():
from sympy.series.sequences import SeqExprOp, sequence
s1 = sequence((1, 2, 3))
s2 = sequence(x**2)
assert _test_args(SeqExprOp(s1, s2))
def test_sympy__series__sequences__SeqAdd():
from sympy.series.sequences import SeqAdd, sequence
s1 = sequence((1, 2, 3))
s2 = sequence(x**2)
assert _test_args(SeqAdd(s1, s2))
def test_sympy__series__sequences__SeqMul():
from sympy.series.sequences import SeqMul, sequence
s1 = sequence((1, 2, 3))
s2 = sequence(x**2)
assert _test_args(SeqMul(s1, s2))
@SKIP('Abstract Class')
def test_sympy__series__series_class__SeriesBase():
pass
def test_sympy__series__fourier__FourierSeries():
from sympy.series.fourier import fourier_series
assert _test_args(fourier_series(x, (x, -pi, pi)))
def test_sympy__series__fourier__FiniteFourierSeries():
from sympy.series.fourier import fourier_series
assert _test_args(fourier_series(sin(pi*x), (x, -1, 1)))
def test_sympy__series__formal__FormalPowerSeries():
from sympy.series.formal import fps
assert _test_args(fps(log(1 + x), x))
def test_sympy__simplify__hyperexpand__Hyper_Function():
from sympy.simplify.hyperexpand import Hyper_Function
assert _test_args(Hyper_Function([2], [1]))
def test_sympy__simplify__hyperexpand__G_Function():
from sympy.simplify.hyperexpand import G_Function
assert _test_args(G_Function([2], [1], [], []))
@SKIP("abstract class")
def test_sympy__tensor__array__ndim_array__ImmutableNDimArray():
pass
def test_sympy__tensor__array__dense_ndim_array__ImmutableDenseNDimArray():
from sympy.tensor.array.dense_ndim_array import ImmutableDenseNDimArray
densarr = ImmutableDenseNDimArray(range(10, 34), (2, 3, 4))
assert _test_args(densarr)
def test_sympy__tensor__array__sparse_ndim_array__ImmutableSparseNDimArray():
from sympy.tensor.array.sparse_ndim_array import ImmutableSparseNDimArray
sparr = ImmutableSparseNDimArray(range(10, 34), (2, 3, 4))
assert _test_args(sparr)
def test_sympy__tensor__functions__TensorProduct():
from sympy.tensor.functions import TensorProduct
tp = TensorProduct(3, 4, evaluate=False)
assert _test_args(tp)
def test_sympy__tensor__indexed__Idx():
from sympy.tensor.indexed import Idx
assert _test_args(Idx('test'))
assert _test_args(Idx(1, (0, 10)))
def test_sympy__tensor__indexed__Indexed():
from sympy.tensor.indexed import Indexed, Idx
assert _test_args(Indexed('A', Idx('i'), Idx('j')))
def test_sympy__tensor__indexed__IndexedBase():
from sympy.tensor.indexed import IndexedBase
assert _test_args(IndexedBase('A', shape=(x, y)))
assert _test_args(IndexedBase('A', 1))
assert _test_args(IndexedBase('A')[0, 1])
def test_sympy__tensor__tensor__TensorIndexType():
from sympy.tensor.tensor import TensorIndexType
assert _test_args(TensorIndexType('Lorentz', metric=False))
def test_sympy__tensor__tensor__TensorSymmetry():
from sympy.tensor.tensor import TensorSymmetry, get_symmetric_group_sgs
assert _test_args(TensorSymmetry(get_symmetric_group_sgs(2)))
def test_sympy__tensor__tensor__TensorType():
from sympy.tensor.tensor import TensorIndexType, TensorSymmetry, get_symmetric_group_sgs, TensorType
Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
sym = TensorSymmetry(get_symmetric_group_sgs(1))
assert _test_args(TensorType([Lorentz], sym))
def test_sympy__tensor__tensor__TensorHead():
from sympy.tensor.tensor import TensorIndexType, TensorSymmetry, TensorType, get_symmetric_group_sgs, TensorHead
Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
sym = TensorSymmetry(get_symmetric_group_sgs(1))
S1 = TensorType([Lorentz], sym)
assert _test_args(TensorHead('p', S1, 0))
def test_sympy__tensor__tensor__TensorIndex():
from sympy.tensor.tensor import TensorIndexType, TensorIndex
Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
assert _test_args(TensorIndex('i', Lorentz))
@SKIP("abstract class")
def test_sympy__tensor__tensor__TensExpr():
pass
def test_sympy__tensor__tensor__TensAdd():
from sympy.tensor.tensor import TensorIndexType, TensorSymmetry, TensorType, get_symmetric_group_sgs, tensor_indices, TensAdd
Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
a, b = tensor_indices('a,b', Lorentz)
sym = TensorSymmetry(get_symmetric_group_sgs(1))
S1 = TensorType([Lorentz], sym)
p, q = S1('p,q')
t1 = p(a)
t2 = q(a)
assert _test_args(TensAdd(t1, t2))
def test_sympy__tensor__tensor__Tensor():
from sympy.core import S
from sympy.tensor.tensor import TensorIndexType, TensorSymmetry, TensorType, get_symmetric_group_sgs, tensor_indices, TensMul
Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
a, b = tensor_indices('a,b', Lorentz)
sym = TensorSymmetry(get_symmetric_group_sgs(1))
S1 = TensorType([Lorentz], sym)
p = S1('p')
assert _test_args(p(a))
def test_sympy__tensor__tensor__TensMul():
from sympy.core import S
from sympy.tensor.tensor import TensorIndexType, TensorSymmetry, TensorType, get_symmetric_group_sgs, tensor_indices, TensMul
Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
a, b = tensor_indices('a,b', Lorentz)
sym = TensorSymmetry(get_symmetric_group_sgs(1))
S1 = TensorType([Lorentz], sym)
p = S1('p')
q = S1('q')
assert _test_args(3*p(a)*q(b))
def test_sympy__tensor__tensor__TensorElement():
from sympy.tensor.tensor import TensorIndexType, tensorhead, TensorElement
L = TensorIndexType("L")
A = tensorhead("A", [L, L], [[1], [1]])
telem = TensorElement(A(x, y), {x: 1})
assert _test_args(telem)
def test_sympy__tensor__toperators__PartialDerivative():
from sympy.tensor.tensor import TensorIndexType, tensor_indices, tensorhead
from sympy.tensor.toperators import PartialDerivative
Lorentz = TensorIndexType('Lorentz', dummy_fmt='L')
a, b = tensor_indices('a,b', Lorentz)
A = tensorhead("A", [Lorentz], [[1]])
assert _test_args(PartialDerivative(A(a), A(b)))
def test_as_coeff_add():
assert (7, (3*x, 4*x**2)) == (7 + 3*x + 4*x**2).as_coeff_add()
def test_sympy__geometry__curve__Curve():
from sympy.geometry.curve import Curve
assert _test_args(Curve((x, 1), (x, 0, 1)))
def test_sympy__geometry__point__Point():
from sympy.geometry.point import Point
assert _test_args(Point(0, 1))
def test_sympy__geometry__point__Point2D():
from sympy.geometry.point import Point2D
assert _test_args(Point2D(0, 1))
def test_sympy__geometry__point__Point3D():
from sympy.geometry.point import Point3D
assert _test_args(Point3D(0, 1, 2))
def test_sympy__geometry__ellipse__Ellipse():
from sympy.geometry.ellipse import Ellipse
assert _test_args(Ellipse((0, 1), 2, 3))
def test_sympy__geometry__ellipse__Circle():
from sympy.geometry.ellipse import Circle
assert _test_args(Circle((0, 1), 2))
def test_sympy__geometry__parabola__Parabola():
from sympy.geometry.parabola import Parabola
from sympy.geometry.line import Line
assert _test_args(Parabola((0, 0), Line((2, 3), (4, 3))))
@SKIP("abstract class")
def test_sympy__geometry__line__LinearEntity():
pass
def test_sympy__geometry__line__Line():
from sympy.geometry.line import Line
assert _test_args(Line((0, 1), (2, 3)))
def test_sympy__geometry__line__Ray():
from sympy.geometry.line import Ray
assert _test_args(Ray((0, 1), (2, 3)))
def test_sympy__geometry__line__Segment():
from sympy.geometry.line import Segment
assert _test_args(Segment((0, 1), (2, 3)))
@SKIP("abstract class")
def test_sympy__geometry__line__LinearEntity2D():
pass
def test_sympy__geometry__line__Line2D():
from sympy.geometry.line import Line2D
assert _test_args(Line2D((0, 1), (2, 3)))
def test_sympy__geometry__line__Ray2D():
from sympy.geometry.line import Ray2D
assert _test_args(Ray2D((0, 1), (2, 3)))
def test_sympy__geometry__line__Segment2D():
from sympy.geometry.line import Segment2D
assert _test_args(Segment2D((0, 1), (2, 3)))
@SKIP("abstract class")
def test_sympy__geometry__line__LinearEntity3D():
pass
def test_sympy__geometry__line__Line3D():
from sympy.geometry.line import Line3D
assert _test_args(Line3D((0, 1, 1), (2, 3, 4)))
def test_sympy__geometry__line__Segment3D():
from sympy.geometry.line import Segment3D
assert _test_args(Segment3D((0, 1, 1), (2, 3, 4)))
def test_sympy__geometry__line__Ray3D():
from sympy.geometry.line import Ray3D
assert _test_args(Ray3D((0, 1, 1), (2, 3, 4)))
def test_sympy__geometry__plane__Plane():
from sympy.geometry.plane import Plane
assert _test_args(Plane((1, 1, 1), (-3, 4, -2), (1, 2, 3)))
def test_sympy__geometry__polygon__Polygon():
from sympy.geometry.polygon import Polygon
assert _test_args(Polygon((0, 1), (2, 3), (4, 5), (6, 7)))
def test_sympy__geometry__polygon__RegularPolygon():
from sympy.geometry.polygon import RegularPolygon
assert _test_args(RegularPolygon((0, 1), 2, 3, 4))
def test_sympy__geometry__polygon__Triangle():
from sympy.geometry.polygon import Triangle
assert _test_args(Triangle((0, 1), (2, 3), (4, 5)))
def test_sympy__geometry__entity__GeometryEntity():
from sympy.geometry.entity import GeometryEntity
from sympy.geometry.point import Point
assert _test_args(GeometryEntity(Point(1, 0), 1, [1, 2]))
@SKIP("abstract class")
def test_sympy__geometry__entity__GeometrySet():
pass
def test_sympy__diffgeom__diffgeom__Manifold():
from sympy.diffgeom import Manifold
assert _test_args(Manifold('name', 3))
def test_sympy__diffgeom__diffgeom__Patch():
from sympy.diffgeom import Manifold, Patch
assert _test_args(Patch('name', Manifold('name', 3)))
def test_sympy__diffgeom__diffgeom__CoordSystem():
from sympy.diffgeom import Manifold, Patch, CoordSystem
assert _test_args(CoordSystem('name', Patch('name', Manifold('name', 3))))
@XFAIL
def test_sympy__diffgeom__diffgeom__Point():
from sympy.diffgeom import Manifold, Patch, CoordSystem, Point
assert _test_args(Point(
CoordSystem('name', Patch('name', Manifold('name', 3))), [x, y]))
def test_sympy__diffgeom__diffgeom__BaseScalarField():
from sympy.diffgeom import Manifold, Patch, CoordSystem, BaseScalarField
cs = CoordSystem('name', Patch('name', Manifold('name', 3)))
assert _test_args(BaseScalarField(cs, 0))
def test_sympy__diffgeom__diffgeom__BaseVectorField():
from sympy.diffgeom import Manifold, Patch, CoordSystem, BaseVectorField
cs = CoordSystem('name', Patch('name', Manifold('name', 3)))
assert _test_args(BaseVectorField(cs, 0))
def test_sympy__diffgeom__diffgeom__Differential():
from sympy.diffgeom import Manifold, Patch, CoordSystem, BaseScalarField, Differential
cs = CoordSystem('name', Patch('name', Manifold('name', 3)))
assert _test_args(Differential(BaseScalarField(cs, 0)))
def test_sympy__diffgeom__diffgeom__Commutator():
from sympy.diffgeom import Manifold, Patch, CoordSystem, BaseVectorField, Commutator
cs = CoordSystem('name', Patch('name', Manifold('name', 3)))
cs1 = CoordSystem('name1', Patch('name', Manifold('name', 3)))
v = BaseVectorField(cs, 0)
v1 = BaseVectorField(cs1, 0)
assert _test_args(Commutator(v, v1))
def test_sympy__diffgeom__diffgeom__TensorProduct():
from sympy.diffgeom import Manifold, Patch, CoordSystem, BaseScalarField, Differential, TensorProduct
cs = CoordSystem('name', Patch('name', Manifold('name', 3)))
d = Differential(BaseScalarField(cs, 0))
assert _test_args(TensorProduct(d, d))
def test_sympy__diffgeom__diffgeom__WedgeProduct():
from sympy.diffgeom import Manifold, Patch, CoordSystem, BaseScalarField, Differential, WedgeProduct
cs = CoordSystem('name', Patch('name', Manifold('name', 3)))
d = Differential(BaseScalarField(cs, 0))
d1 = Differential(BaseScalarField(cs, 1))
assert _test_args(WedgeProduct(d, d1))
def test_sympy__diffgeom__diffgeom__LieDerivative():
from sympy.diffgeom import Manifold, Patch, CoordSystem, BaseScalarField, Differential, BaseVectorField, LieDerivative
cs = CoordSystem('name', Patch('name', Manifold('name', 3)))
d = Differential(BaseScalarField(cs, 0))
v = BaseVectorField(cs, 0)
assert _test_args(LieDerivative(v, d))
@XFAIL
def test_sympy__diffgeom__diffgeom__BaseCovarDerivativeOp():
from sympy.diffgeom import Manifold, Patch, CoordSystem, BaseCovarDerivativeOp
cs = CoordSystem('name', Patch('name', Manifold('name', 3)))
assert _test_args(BaseCovarDerivativeOp(cs, 0, [[[0, ]*3, ]*3, ]*3))
def test_sympy__diffgeom__diffgeom__CovarDerivativeOp():
from sympy.diffgeom import Manifold, Patch, CoordSystem, BaseVectorField, CovarDerivativeOp
cs = CoordSystem('name', Patch('name', Manifold('name', 3)))
v = BaseVectorField(cs, 0)
_test_args(CovarDerivativeOp(v, [[[0, ]*3, ]*3, ]*3))
def test_sympy__categories__baseclasses__Class():
from sympy.categories.baseclasses import Class
assert _test_args(Class())
def test_sympy__categories__baseclasses__Object():
from sympy.categories import Object
assert _test_args(Object("A"))
@XFAIL
def test_sympy__categories__baseclasses__Morphism():
from sympy.categories import Object, Morphism
assert _test_args(Morphism(Object("A"), Object("B")))
def test_sympy__categories__baseclasses__IdentityMorphism():
from sympy.categories import Object, IdentityMorphism
assert _test_args(IdentityMorphism(Object("A")))
def test_sympy__categories__baseclasses__NamedMorphism():
from sympy.categories import Object, NamedMorphism
assert _test_args(NamedMorphism(Object("A"), Object("B"), "f"))
def test_sympy__categories__baseclasses__CompositeMorphism():
from sympy.categories import Object, NamedMorphism, CompositeMorphism
A = Object("A")
B = Object("B")
C = Object("C")
f = NamedMorphism(A, B, "f")
g = NamedMorphism(B, C, "g")
assert _test_args(CompositeMorphism(f, g))
def test_sympy__categories__baseclasses__Diagram():
from sympy.categories import Object, NamedMorphism, Diagram
A = Object("A")
B = Object("B")
C = Object("C")
f = NamedMorphism(A, B, "f")
d = Diagram([f])
assert _test_args(d)
def test_sympy__categories__baseclasses__Category():
from sympy.categories import Object, NamedMorphism, Diagram, Category
A = Object("A")
B = Object("B")
C = Object("C")
f = NamedMorphism(A, B, "f")
g = NamedMorphism(B, C, "g")
d1 = Diagram([f, g])
d2 = Diagram([f])
K = Category("K", commutative_diagrams=[d1, d2])
assert _test_args(K)
def test_sympy__ntheory__factor___totient():
from sympy.ntheory.factor_ import totient
k = symbols('k', integer=True)
t = totient(k)
assert _test_args(t)
def test_sympy__ntheory__factor___reduced_totient():
from sympy.ntheory.factor_ import reduced_totient
k = symbols('k', integer=True)
t = reduced_totient(k)
assert _test_args(t)
def test_sympy__ntheory__factor___divisor_sigma():
from sympy.ntheory.factor_ import divisor_sigma
k = symbols('k', integer=True)
n = symbols('n', integer=True)
t = divisor_sigma(n, k)
assert _test_args(t)
def test_sympy__ntheory__factor___udivisor_sigma():
from sympy.ntheory.factor_ import udivisor_sigma
k = symbols('k', integer=True)
n = symbols('n', integer=True)
t = udivisor_sigma(n, k)
assert _test_args(t)
def test_sympy__ntheory__factor___primenu():
from sympy.ntheory.factor_ import primenu
n = symbols('n', integer=True)
t = primenu(n)
assert _test_args(t)
def test_sympy__ntheory__factor___primeomega():
from sympy.ntheory.factor_ import primeomega
n = symbols('n', integer=True)
t = primeomega(n)
assert _test_args(t)
def test_sympy__ntheory__residue_ntheory__mobius():
from sympy.ntheory import mobius
assert _test_args(mobius(2))
def test_sympy__ntheory__generate__primepi():
from sympy.ntheory import primepi
n = symbols('n')
t = primepi(n)
assert _test_args(t)
def test_sympy__physics__optics__waves__TWave():
from sympy.physics.optics import TWave
A, f, phi = symbols('A, f, phi')
assert _test_args(TWave(A, f, phi))
def test_sympy__physics__optics__gaussopt__BeamParameter():
from sympy.physics.optics import BeamParameter
assert _test_args(BeamParameter(530e-9, 1, w=1e-3))
def test_sympy__physics__optics__medium__Medium():
from sympy.physics.optics import Medium
assert _test_args(Medium('m'))
def test_sympy__codegen__array_utils__CodegenArrayContraction():
from sympy.codegen.array_utils import CodegenArrayContraction
from sympy import IndexedBase
A = symbols("A", cls=IndexedBase)
assert _test_args(CodegenArrayContraction(A, (0, 1)))
def test_sympy__codegen__array_utils__CodegenArrayDiagonal():
from sympy.codegen.array_utils import CodegenArrayDiagonal
from sympy import IndexedBase
A = symbols("A", cls=IndexedBase)
assert _test_args(CodegenArrayDiagonal(A, (0, 1)))
def test_sympy__codegen__array_utils__CodegenArrayTensorProduct():
from sympy.codegen.array_utils import CodegenArrayTensorProduct
from sympy import IndexedBase
A, B = symbols("A B", cls=IndexedBase)
assert _test_args(CodegenArrayTensorProduct(A, B))
def test_sympy__codegen__array_utils__CodegenArrayElementwiseAdd():
from sympy.codegen.array_utils import CodegenArrayElementwiseAdd
from sympy import IndexedBase
A, B = symbols("A B", cls=IndexedBase)
assert _test_args(CodegenArrayElementwiseAdd(A, B))
def test_sympy__codegen__array_utils__CodegenArrayPermuteDims():
from sympy.codegen.array_utils import CodegenArrayPermuteDims
from sympy import IndexedBase
A = symbols("A", cls=IndexedBase)
assert _test_args(CodegenArrayPermuteDims(A, (1, 0)))
def test_sympy__codegen__ast__Assignment():
from sympy.codegen.ast import Assignment
assert _test_args(Assignment(x, y))
def test_sympy__codegen__cfunctions__expm1():
from sympy.codegen.cfunctions import expm1
assert _test_args(expm1(x))
def test_sympy__codegen__cfunctions__log1p():
from sympy.codegen.cfunctions import log1p
assert _test_args(log1p(x))
def test_sympy__codegen__cfunctions__exp2():
from sympy.codegen.cfunctions import exp2
assert _test_args(exp2(x))
def test_sympy__codegen__cfunctions__log2():
from sympy.codegen.cfunctions import log2
assert _test_args(log2(x))
def test_sympy__codegen__cfunctions__fma():
from sympy.codegen.cfunctions import fma
assert _test_args(fma(x, y, z))
def test_sympy__codegen__cfunctions__log10():
from sympy.codegen.cfunctions import log10
assert _test_args(log10(x))
def test_sympy__codegen__cfunctions__Sqrt():
from sympy.codegen.cfunctions import Sqrt
assert _test_args(Sqrt(x))
def test_sympy__codegen__cfunctions__Cbrt():
from sympy.codegen.cfunctions import Cbrt
assert _test_args(Cbrt(x))
def test_sympy__codegen__cfunctions__hypot():
from sympy.codegen.cfunctions import hypot
assert _test_args(hypot(x, y))
def test_sympy__codegen__fnodes__FFunction():
from sympy.codegen.fnodes import FFunction
assert _test_args(FFunction('f'))
def test_sympy__codegen__fnodes__F95Function():
from sympy.codegen.fnodes import F95Function
assert _test_args(F95Function('f'))
def test_sympy__codegen__fnodes__isign():
from sympy.codegen.fnodes import isign
assert _test_args(isign(1, x))
def test_sympy__codegen__fnodes__dsign():
from sympy.codegen.fnodes import dsign
assert _test_args(dsign(1, x))
def test_sympy__codegen__fnodes__cmplx():
from sympy.codegen.fnodes import cmplx
assert _test_args(cmplx(x, y))
def test_sympy__codegen__fnodes__kind():
from sympy.codegen.fnodes import kind
assert _test_args(kind(x))
def test_sympy__codegen__fnodes__merge():
from sympy.codegen.fnodes import merge
assert _test_args(merge(1, 2, Eq(x, 0)))
def test_sympy__codegen__fnodes___literal():
from sympy.codegen.fnodes import _literal
assert _test_args(_literal(1))
def test_sympy__codegen__fnodes__literal_sp():
from sympy.codegen.fnodes import literal_sp
assert _test_args(literal_sp(1))
def test_sympy__codegen__fnodes__literal_dp():
from sympy.codegen.fnodes import literal_dp
assert _test_args(literal_dp(1))
def test_sympy__vector__coordsysrect__CoordSys3D():
from sympy.vector.coordsysrect import CoordSys3D
assert _test_args(CoordSys3D('C'))
def test_sympy__vector__point__Point():
from sympy.vector.point import Point
assert _test_args(Point('P'))
def test_sympy__vector__basisdependent__BasisDependent():
from sympy.vector.basisdependent import BasisDependent
#These classes have been created to maintain an OOP hierarchy
#for Vectors and Dyadics. Are NOT meant to be initialized
def test_sympy__vector__basisdependent__BasisDependentMul():
from sympy.vector.basisdependent import BasisDependentMul
#These classes have been created to maintain an OOP hierarchy
#for Vectors and Dyadics. Are NOT meant to be initialized
def test_sympy__vector__basisdependent__BasisDependentAdd():
from sympy.vector.basisdependent import BasisDependentAdd
#These classes have been created to maintain an OOP hierarchy
#for Vectors and Dyadics. Are NOT meant to be initialized
def test_sympy__vector__basisdependent__BasisDependentZero():
from sympy.vector.basisdependent import BasisDependentZero
#These classes have been created to maintain an OOP hierarchy
#for Vectors and Dyadics. Are NOT meant to be initialized
def test_sympy__vector__vector__BaseVector():
from sympy.vector.vector import BaseVector
from sympy.vector.coordsysrect import CoordSys3D
C = CoordSys3D('C')
assert _test_args(BaseVector(0, C, ' ', ' '))
def test_sympy__vector__vector__VectorAdd():
from sympy.vector.vector import VectorAdd, VectorMul
from sympy.vector.coordsysrect import CoordSys3D
C = CoordSys3D('C')
from sympy.abc import a, b, c, x, y, z
v1 = a*C.i + b*C.j + c*C.k
v2 = x*C.i + y*C.j + z*C.k
assert _test_args(VectorAdd(v1, v2))
assert _test_args(VectorMul(x, v1))
def test_sympy__vector__vector__VectorMul():
from sympy.vector.vector import VectorMul
from sympy.vector.coordsysrect import CoordSys3D
C = CoordSys3D('C')
from sympy.abc import a
assert _test_args(VectorMul(a, C.i))
def test_sympy__vector__vector__VectorZero():
from sympy.vector.vector import VectorZero
assert _test_args(VectorZero())
def test_sympy__vector__vector__Vector():
from sympy.vector.vector import Vector
#Vector is never to be initialized using args
pass
def test_sympy__vector__vector__Cross():
from sympy.vector.vector import Cross
from sympy.vector.coordsysrect import CoordSys3D
C = CoordSys3D('C')
_test_args(Cross(C.i, C.j))
def test_sympy__vector__vector__Dot():
from sympy.vector.vector import Dot
from sympy.vector.coordsysrect import CoordSys3D
C = CoordSys3D('C')
_test_args(Dot(C.i, C.j))
def test_sympy__vector__dyadic__Dyadic():
from sympy.vector.dyadic import Dyadic
#Dyadic is never to be initialized using args
pass
def test_sympy__vector__dyadic__BaseDyadic():
from sympy.vector.dyadic import BaseDyadic
from sympy.vector.coordsysrect import CoordSys3D
C = CoordSys3D('C')
assert _test_args(BaseDyadic(C.i, C.j))
def test_sympy__vector__dyadic__DyadicMul():
from sympy.vector.dyadic import BaseDyadic, DyadicMul
from sympy.vector.coordsysrect import CoordSys3D
C = CoordSys3D('C')
assert _test_args(DyadicMul(3, BaseDyadic(C.i, C.j)))
def test_sympy__vector__dyadic__DyadicAdd():
from sympy.vector.dyadic import BaseDyadic, DyadicAdd
from sympy.vector.coordsysrect import CoordSys3D
C = CoordSys3D('C')
assert _test_args(2 * DyadicAdd(BaseDyadic(C.i, C.i),
BaseDyadic(C.i, C.j)))
def test_sympy__vector__dyadic__DyadicZero():
from sympy.vector.dyadic import DyadicZero
assert _test_args(DyadicZero())
def test_sympy__vector__deloperator__Del():
from sympy.vector.deloperator import Del
assert _test_args(Del())
def test_sympy__vector__operators__Curl():
from sympy.vector.operators import Curl
from sympy.vector.coordsysrect import CoordSys3D
C = CoordSys3D('C')
assert _test_args(Curl(C.i))
def test_sympy__vector__operators__Laplacian():
from sympy.vector.operators import Laplacian
from sympy.vector.coordsysrect import CoordSys3D
C = CoordSys3D('C')
assert _test_args(Laplacian(C.i))
def test_sympy__vector__operators__Divergence():
from sympy.vector.operators import Divergence
from sympy.vector.coordsysrect import CoordSys3D
C = CoordSys3D('C')
assert _test_args(Divergence(C.i))
def test_sympy__vector__operators__Gradient():
from sympy.vector.operators import Gradient
from sympy.vector.coordsysrect import CoordSys3D
C = CoordSys3D('C')
assert _test_args(Gradient(C.x))
def test_sympy__vector__orienters__Orienter():
from sympy.vector.orienters import Orienter
#Not to be initialized
def test_sympy__vector__orienters__ThreeAngleOrienter():
from sympy.vector.orienters import ThreeAngleOrienter
#Not to be initialized
def test_sympy__vector__orienters__AxisOrienter():
from sympy.vector.orienters import AxisOrienter
from sympy.vector.coordsysrect import CoordSys3D
C = CoordSys3D('C')
assert _test_args(AxisOrienter(x, C.i))
def test_sympy__vector__orienters__BodyOrienter():
from sympy.vector.orienters import BodyOrienter
assert _test_args(BodyOrienter(x, y, z, '123'))
def test_sympy__vector__orienters__SpaceOrienter():
from sympy.vector.orienters import SpaceOrienter
assert _test_args(SpaceOrienter(x, y, z, '123'))
def test_sympy__vector__orienters__QuaternionOrienter():
from sympy.vector.orienters import QuaternionOrienter
a, b, c, d = symbols('a b c d')
assert _test_args(QuaternionOrienter(a, b, c, d))
def test_sympy__vector__scalar__BaseScalar():
from sympy.vector.scalar import BaseScalar
from sympy.vector.coordsysrect import CoordSys3D
C = CoordSys3D('C')
assert _test_args(BaseScalar(0, C, ' ', ' '))
def test_sympy__physics__wigner__Wigner3j():
from sympy.physics.wigner import Wigner3j
assert _test_args(Wigner3j(0, 0, 0, 0, 0, 0))
def test_sympy__integrals__rubi__symbol__matchpyWC():
from sympy.integrals.rubi.symbol import matchpyWC
assert _test_args(matchpyWC(1, True, 'a'))
def test_sympy__integrals__rubi__utility_function__rubi_unevaluated_expr():
from sympy.integrals.rubi.utility_function import rubi_unevaluated_expr
a = symbols('a')
assert _test_args(rubi_unevaluated_expr(a))
def test_sympy__integrals__rubi__utility_function__exp():
from sympy.integrals.rubi.utility_function import exp
assert _test_args(exp(5))
def test_sympy__integrals__rubi__utility_function__log():
from sympy.integrals.rubi.utility_function import log
assert _test_args(log(5))
def test_sympy__integrals__rubi__utility_function__Int():
from sympy.integrals.rubi.utility_function import Int
assert _test_args(Int(5, x))
def test_sympy__integrals__rubi__utility_function__Util_Coefficient():
from sympy.integrals.rubi.utility_function import Util_Coefficient
a, x = symbols('a x')
assert _test_args(Util_Coefficient(a, x))
def test_sympy__integrals__rubi__utility_function__Gamma():
from sympy.integrals.rubi.utility_function import Gamma
assert _test_args(Gamma(5))
def test_sympy__integrals__rubi__utility_function__Util_Part():
from sympy.integrals.rubi.utility_function import Util_Part
a, b = symbols('a b')
assert _test_args(Util_Part(a + b, 0))
def test_sympy__integrals__rubi__utility_function__PolyGamma():
from sympy.integrals.rubi.utility_function import PolyGamma
assert _test_args(PolyGamma(1, 1))
def test_sympy__integrals__rubi__utility_function__ProductLog():
from sympy.integrals.rubi.utility_function import ProductLog
assert _test_args(ProductLog(1))
|
136894290c2615746314cf62325c52904a68320aba9f9819e96825fab59f575a
|
import decimal
from sympy import (Rational, Symbol, Float, I, sqrt, cbrt, oo, nan, pi, E,
Integer, S, factorial, Catalan, EulerGamma, GoldenRatio,
TribonacciConstant, cos, exp,
Number, zoo, log, Mul, Pow, Tuple, latex, Gt, Lt, Ge, Le,
AlgebraicNumber, simplify, sin, fibonacci, RealField,
sympify, srepr)
from sympy.core.compatibility import long
from sympy.core.power import integer_nthroot, isqrt, integer_log
from sympy.core.logic import fuzzy_not
from sympy.core.numbers import (igcd, ilcm, igcdex, seterr,
igcd2, igcd_lehmer, mpf_norm, comp, mod_inverse)
from sympy.core.mod import Mod
from sympy.polys.domains.groundtypes import PythonRational
from sympy.utilities.decorator import conserve_mpmath_dps
from sympy.utilities.iterables import permutations
from sympy.utilities.pytest import XFAIL, raises
from mpmath import mpf
from mpmath.rational import mpq
import mpmath
t = Symbol('t', real=False)
def same_and_same_prec(a, b):
# stricter matching for Floats
return a == b and a._prec == b._prec
def test_seterr():
seterr(divide=True)
raises(ValueError, lambda: S.Zero/S.Zero)
seterr(divide=False)
assert S.Zero / S.Zero == S.NaN
def test_mod():
x = Rational(1, 2)
y = Rational(3, 4)
z = Rational(5, 18043)
assert x % x == 0
assert x % y == 1/S(2)
assert x % z == 3/S(36086)
assert y % x == 1/S(4)
assert y % y == 0
assert y % z == 9/S(72172)
assert z % x == 5/S(18043)
assert z % y == 5/S(18043)
assert z % z == 0
a = Float(2.6)
assert (a % .2) == 0
assert (a % 2).round(15) == 0.6
assert (a % 0.5).round(15) == 0.1
p = Symbol('p', infinite=True)
assert oo % oo == nan
assert zoo % oo == nan
assert 5 % oo == nan
assert p % 5 == nan
# In these two tests, if the precision of m does
# not match the precision of the ans, then it is
# likely that the change made now gives an answer
# with degraded accuracy.
r = Rational(500, 41)
f = Float('.36', 3)
m = r % f
ans = Float(r % Rational(f), 3)
assert m == ans and m._prec == ans._prec
f = Float('8.36', 3)
m = f % r
ans = Float(Rational(f) % r, 3)
assert m == ans and m._prec == ans._prec
s = S.Zero
assert s % float(1) == S.Zero
# No rounding required since these numbers can be represented
# exactly.
assert Rational(3, 4) % Float(1.1) == 0.75
assert Float(1.5) % Rational(5, 4) == 0.25
assert Rational(5, 4).__rmod__(Float('1.5')) == 0.25
assert Float('1.5').__rmod__(Float('2.75')) == Float('1.25')
assert 2.75 % Float('1.5') == Float('1.25')
a = Integer(7)
b = Integer(4)
assert type(a % b) == Integer
assert a % b == Integer(3)
assert Integer(1) % Rational(2, 3) == Rational(1, 3)
assert Rational(7, 5) % Integer(1) == Rational(2, 5)
assert Integer(2) % 1.5 == 0.5
assert Integer(3).__rmod__(Integer(10)) == Integer(1)
assert Integer(10) % 4 == Integer(2)
assert 15 % Integer(4) == Integer(3)
def test_divmod():
assert divmod(S(12), S(8)) == Tuple(1, 4)
assert divmod(-S(12), S(8)) == Tuple(-2, 4)
assert divmod(S(0), S(1)) == Tuple(0, 0)
raises(ZeroDivisionError, lambda: divmod(S(0), S(0)))
raises(ZeroDivisionError, lambda: divmod(S(1), S(0)))
assert divmod(S(12), 8) == Tuple(1, 4)
assert divmod(12, S(8)) == Tuple(1, 4)
assert divmod(S("2"), S("3/2")) == Tuple(S("1"), S("1/2"))
assert divmod(S("3/2"), S("2")) == Tuple(S("0"), S("3/2"))
assert divmod(S("2"), S("3.5")) == Tuple(S("0"), S("2"))
assert divmod(S("3.5"), S("2")) == Tuple(S("1"), S("1.5"))
assert divmod(S("2"), S("1/3")) == Tuple(S("6"), S("0"))
assert divmod(S("1/3"), S("2")) == Tuple(S("0"), S("1/3"))
assert divmod(S("2"), S("0.1")) == Tuple(S("20"), S("0"))
assert divmod(S("0.1"), S("2")) == Tuple(S("0"), S("0.1"))
assert divmod(S("2"), 2) == Tuple(S("1"), S("0"))
assert divmod(2, S("2")) == Tuple(S("1"), S("0"))
assert divmod(S("2"), 1.5) == Tuple(S("1"), S("0.5"))
assert divmod(1.5, S("2")) == Tuple(S("0"), S("1.5"))
assert divmod(0.3, S("2")) == Tuple(S("0"), S("0.3"))
assert divmod(S("3/2"), S("3.5")) == Tuple(S("0"), S("3/2"))
assert divmod(S("3.5"), S("3/2")) == Tuple(S("2"), S("0.5"))
assert divmod(S("3/2"), S("1/3")) == Tuple(S("4"), Float("1/6"))
assert divmod(S("1/3"), S("3/2")) == Tuple(S("0"), S("1/3"))
assert divmod(S("3/2"), S("0.1")) == Tuple(S("15"), S("0"))
assert divmod(S("0.1"), S("3/2")) == Tuple(S("0"), S("0.1"))
assert divmod(S("3/2"), 2) == Tuple(S("0"), S("3/2"))
assert divmod(2, S("3/2")) == Tuple(S("1"), S("0.5"))
assert divmod(S("3/2"), 1.5) == Tuple(S("1"), S("0"))
assert divmod(1.5, S("3/2")) == Tuple(S("1"), S("0"))
assert divmod(S("3/2"), 0.3) == Tuple(S("5"), S("0"))
assert divmod(0.3, S("3/2")) == Tuple(S("0"), S("0.3"))
assert divmod(S("1/3"), S("3.5")) == Tuple(S("0"), S("1/3"))
assert divmod(S("3.5"), S("0.1")) == Tuple(S("35"), S("0"))
assert divmod(S("0.1"), S("3.5")) == Tuple(S("0"), S("0.1"))
assert divmod(S("3.5"), 2) == Tuple(S("1"), S("1.5"))
assert divmod(2, S("3.5")) == Tuple(S("0"), S("2"))
assert divmod(S("3.5"), 1.5) == Tuple(S("2"), S("0.5"))
assert divmod(1.5, S("3.5")) == Tuple(S("0"), S("1.5"))
assert divmod(0.3, S("3.5")) == Tuple(S("0"), S("0.3"))
assert divmod(S("0.1"), S("1/3")) == Tuple(S("0"), S("0.1"))
assert divmod(S("1/3"), 2) == Tuple(S("0"), S("1/3"))
assert divmod(2, S("1/3")) == Tuple(S("6"), S("0"))
assert divmod(S("1/3"), 1.5) == Tuple(S("0"), S("1/3"))
assert divmod(0.3, S("1/3")) == Tuple(S("0"), S("0.3"))
assert divmod(S("0.1"), 2) == Tuple(S("0"), S("0.1"))
assert divmod(2, S("0.1")) == Tuple(S("20"), S("0"))
assert divmod(S("0.1"), 1.5) == Tuple(S("0"), S("0.1"))
assert divmod(1.5, S("0.1")) == Tuple(S("15"), S("0"))
assert divmod(S("0.1"), 0.3) == Tuple(S("0"), S("0.1"))
assert str(divmod(S("2"), 0.3)) == '(6, 0.2)'
assert str(divmod(S("3.5"), S("1/3"))) == '(10, 0.166666666666667)'
assert str(divmod(S("3.5"), 0.3)) == '(11, 0.2)'
assert str(divmod(S("1/3"), S("0.1"))) == '(3, 0.0333333333333333)'
assert str(divmod(1.5, S("1/3"))) == '(4, 0.166666666666667)'
assert str(divmod(S("1/3"), 0.3)) == '(1, 0.0333333333333333)'
assert str(divmod(0.3, S("0.1"))) == '(2, 0.1)'
assert divmod(-3, S(2)) == (-2, 1)
assert divmod(S(-3), S(2)) == (-2, 1)
assert divmod(S(-3), 2) == (-2, 1)
assert divmod(S(4), S(-3.1)) == Tuple(-2, -2.2)
assert divmod(S(4), S(-2.1)) == divmod(4, -2.1)
assert divmod(S(-8), S(-2.5) ) == Tuple(3 , -0.5)
def test_igcd():
assert igcd(0, 0) == 0
assert igcd(0, 1) == 1
assert igcd(1, 0) == 1
assert igcd(0, 7) == 7
assert igcd(7, 0) == 7
assert igcd(7, 1) == 1
assert igcd(1, 7) == 1
assert igcd(-1, 0) == 1
assert igcd(0, -1) == 1
assert igcd(-1, -1) == 1
assert igcd(-1, 7) == 1
assert igcd(7, -1) == 1
assert igcd(8, 2) == 2
assert igcd(4, 8) == 4
assert igcd(8, 16) == 8
assert igcd(7, -3) == 1
assert igcd(-7, 3) == 1
assert igcd(-7, -3) == 1
assert igcd(*[10, 20, 30]) == 10
raises(TypeError, lambda: igcd())
raises(TypeError, lambda: igcd(2))
raises(ValueError, lambda: igcd(0, None))
raises(ValueError, lambda: igcd(1, 2.2))
for args in permutations((45.1, 1, 30)):
raises(ValueError, lambda: igcd(*args))
for args in permutations((1, 2, None)):
raises(ValueError, lambda: igcd(*args))
def test_igcd_lehmer():
a, b = fibonacci(10001), fibonacci(10000)
# len(str(a)) == 2090
# small divisors, long Euclidean sequence
assert igcd_lehmer(a, b) == 1
c = fibonacci(100)
assert igcd_lehmer(a*c, b*c) == c
# big divisor
assert igcd_lehmer(a, 10**1000) == 1
# swapping argmument
assert igcd_lehmer(1, 2) == igcd_lehmer(2, 1)
def test_igcd2():
# short loop
assert igcd2(2**100 - 1, 2**99 - 1) == 1
# Lehmer's algorithm
a, b = int(fibonacci(10001)), int(fibonacci(10000))
assert igcd2(a, b) == 1
def test_ilcm():
assert ilcm(0, 0) == 0
assert ilcm(1, 0) == 0
assert ilcm(0, 1) == 0
assert ilcm(1, 1) == 1
assert ilcm(2, 1) == 2
assert ilcm(8, 2) == 8
assert ilcm(8, 6) == 24
assert ilcm(8, 7) == 56
assert ilcm(*[10, 20, 30]) == 60
raises(ValueError, lambda: ilcm(8.1, 7))
raises(ValueError, lambda: ilcm(8, 7.1))
raises(TypeError, lambda: ilcm(8))
def test_igcdex():
assert igcdex(2, 3) == (-1, 1, 1)
assert igcdex(10, 12) == (-1, 1, 2)
assert igcdex(100, 2004) == (-20, 1, 4)
assert igcdex(0, 0) == (0, 1, 0)
assert igcdex(1, 0) == (1, 0, 1)
def _strictly_equal(a, b):
return (a.p, a.q, type(a.p), type(a.q)) == \
(b.p, b.q, type(b.p), type(b.q))
def _test_rational_new(cls):
"""
Tests that are common between Integer and Rational.
"""
assert cls(0) is S.Zero
assert cls(1) is S.One
assert cls(-1) is S.NegativeOne
# These look odd, but are similar to int():
assert cls('1') is S.One
assert cls(u'-1') is S.NegativeOne
i = Integer(10)
assert _strictly_equal(i, cls('10'))
assert _strictly_equal(i, cls(u'10'))
assert _strictly_equal(i, cls(long(10)))
assert _strictly_equal(i, cls(i))
raises(TypeError, lambda: cls(Symbol('x')))
def test_Integer_new():
"""
Test for Integer constructor
"""
_test_rational_new(Integer)
assert _strictly_equal(Integer(0.9), S.Zero)
assert _strictly_equal(Integer(10.5), Integer(10))
raises(ValueError, lambda: Integer("10.5"))
assert Integer(Rational('1.' + '9'*20)) == 1
def test_Rational_new():
""""
Test for Rational constructor
"""
_test_rational_new(Rational)
n1 = Rational(1, 2)
assert n1 == Rational(Integer(1), 2)
assert n1 == Rational(Integer(1), Integer(2))
assert n1 == Rational(1, Integer(2))
assert n1 == Rational(Rational(1, 2))
assert 1 == Rational(n1, n1)
assert Rational(3, 2) == Rational(Rational(1, 2), Rational(1, 3))
assert Rational(3, 1) == Rational(1, Rational(1, 3))
n3_4 = Rational(3, 4)
assert Rational('3/4') == n3_4
assert -Rational('-3/4') == n3_4
assert Rational('.76').limit_denominator(4) == n3_4
assert Rational(19, 25).limit_denominator(4) == n3_4
assert Rational('19/25').limit_denominator(4) == n3_4
assert Rational(1.0, 3) == Rational(1, 3)
assert Rational(1, 3.0) == Rational(1, 3)
assert Rational(Float(0.5)) == Rational(1, 2)
assert Rational('1e2/1e-2') == Rational(10000)
assert Rational('1 234') == Rational(1234)
assert Rational('1/1 234') == Rational(1, 1234)
assert Rational(-1, 0) == S.ComplexInfinity
assert Rational(1, 0) == S.ComplexInfinity
# Make sure Rational doesn't lose precision on Floats
assert Rational(pi.evalf(100)).evalf(100) == pi.evalf(100)
raises(TypeError, lambda: Rational('3**3'))
raises(TypeError, lambda: Rational('1/2 + 2/3'))
# handle fractions.Fraction instances
try:
import fractions
assert Rational(fractions.Fraction(1, 2)) == Rational(1, 2)
except ImportError:
pass
assert Rational(mpq(2, 6)) == Rational(1, 3)
assert Rational(PythonRational(2, 6)) == Rational(1, 3)
def test_Number_new():
""""
Test for Number constructor
"""
# Expected behavior on numbers and strings
assert Number(1) is S.One
assert Number(2).__class__ is Integer
assert Number(-622).__class__ is Integer
assert Number(5, 3).__class__ is Rational
assert Number(5.3).__class__ is Float
assert Number('1') is S.One
assert Number('2').__class__ is Integer
assert Number('-622').__class__ is Integer
assert Number('5/3').__class__ is Rational
assert Number('5.3').__class__ is Float
raises(ValueError, lambda: Number('cos'))
raises(TypeError, lambda: Number(cos))
a = Rational(3, 5)
assert Number(a) is a # Check idempotence on Numbers
def test_Number_cmp():
n1 = Number(1)
n2 = Number(2)
n3 = Number(-3)
assert n1 < n2
assert n1 <= n2
assert n3 < n1
assert n2 > n3
assert n2 >= n3
raises(TypeError, lambda: n1 < S.NaN)
raises(TypeError, lambda: n1 <= S.NaN)
raises(TypeError, lambda: n1 > S.NaN)
raises(TypeError, lambda: n1 >= S.NaN)
def test_Rational_cmp():
n1 = Rational(1, 4)
n2 = Rational(1, 3)
n3 = Rational(2, 4)
n4 = Rational(2, -4)
n5 = Rational(0)
n6 = Rational(1)
n7 = Rational(3)
n8 = Rational(-3)
assert n8 < n5
assert n5 < n6
assert n6 < n7
assert n8 < n7
assert n7 > n8
assert (n1 + 1)**n2 < 2
assert ((n1 + n6)/n7) < 1
assert n4 < n3
assert n2 < n3
assert n1 < n2
assert n3 > n1
assert not n3 < n1
assert not (Rational(-1) > 0)
assert Rational(-1) < 0
raises(TypeError, lambda: n1 < S.NaN)
raises(TypeError, lambda: n1 <= S.NaN)
raises(TypeError, lambda: n1 > S.NaN)
raises(TypeError, lambda: n1 >= S.NaN)
def test_Float():
def eq(a, b):
t = Float("1.0E-15")
return (-t < a - b < t)
a = Float(2) ** Float(3)
assert eq(a.evalf(), Float(8))
assert eq((pi ** -1).evalf(), Float("0.31830988618379067"))
a = Float(2) ** Float(4)
assert eq(a.evalf(), Float(16))
assert (S(.3) == S(.5)) is False
x_str = Float((0, '13333333333333', -52, 53))
x2_str = Float((0, '26666666666666', -53, 53))
x_hex = Float((0, long(0x13333333333333), -52, 53))
x_dec = Float((0, 5404319552844595, -52, 53))
assert x_str == x_hex == x_dec == Float(1.2)
# This looses a binary digit of precision, so it isn't equal to the above,
# but check that it normalizes correctly
x2_hex = Float((0, long(0x13333333333333)*2, -53, 53))
assert x2_hex._mpf_ == (0, 5404319552844595, -52, 52)
# XXX: Should this test also hold?
# assert x2_hex._prec == 52
# x2_str and 1.2 are superficially the same
assert str(x2_str) == str(Float(1.2))
# but are different at the mpf level
assert Float(1.2)._mpf_ == (0, long(5404319552844595), -52, 53)
assert x2_str._mpf_ == (0, long(10808639105689190), -53, 53)
assert Float((0, long(0), -123, -1)) == Float('nan')
assert Float((0, long(0), -456, -2)) == Float('inf') == Float('+inf')
assert Float((1, long(0), -789, -3)) == Float('-inf')
raises(ValueError, lambda: Float((0, 7, 1, 3), ''))
assert Float('+inf').is_finite is False
assert Float('+inf').is_negative is False
assert Float('+inf').is_positive is True
assert Float('+inf').is_infinite is True
assert Float('+inf').is_zero is False
assert Float('-inf').is_finite is False
assert Float('-inf').is_negative is True
assert Float('-inf').is_positive is False
assert Float('-inf').is_infinite is True
assert Float('-inf').is_zero is False
assert Float('0.0').is_finite is True
assert Float('0.0').is_negative is False
assert Float('0.0').is_positive is False
assert Float('0.0').is_infinite is False
assert Float('0.0').is_zero is True
# rationality properties
assert Float(1).is_rational is None
assert Float(1).is_irrational is None
assert sqrt(2).n(15).is_rational is None
assert sqrt(2).n(15).is_irrational is None
# do not automatically evalf
def teq(a):
assert (a.evalf() == a) is False
assert (a.evalf() != a) is True
assert (a == a.evalf()) is False
assert (a != a.evalf()) is True
teq(pi)
teq(2*pi)
teq(cos(0.1, evaluate=False))
# long integer
i = 12345678901234567890
assert same_and_same_prec(Float(12, ''), Float('12', ''))
assert same_and_same_prec(Float(Integer(i), ''), Float(i, ''))
assert same_and_same_prec(Float(i, ''), Float(str(i), 20))
assert same_and_same_prec(Float(str(i)), Float(i, ''))
assert same_and_same_prec(Float(i), Float(i, ''))
# inexact floats (repeating binary = denom not multiple of 2)
# cannot have precision greater than 15
assert Float(.125, 22) == .125
assert Float(2.0, 22) == 2
assert float(Float('.12500000000000001', '')) == .125
raises(ValueError, lambda: Float(.12500000000000001, ''))
# allow spaces
Float('123 456.123 456') == Float('123456.123456')
Integer('123 456') == Integer('123456')
Rational('123 456.123 456') == Rational('123456.123456')
assert Float(' .3e2') == Float('0.3e2')
# allow auto precision detection
assert Float('.1', '') == Float(.1, 1)
assert Float('.125', '') == Float(.125, 3)
assert Float('.100', '') == Float(.1, 3)
assert Float('2.0', '') == Float('2', 2)
raises(ValueError, lambda: Float("12.3d-4", ""))
raises(ValueError, lambda: Float(12.3, ""))
raises(ValueError, lambda: Float('.'))
raises(ValueError, lambda: Float('-.'))
zero = Float('0.0')
assert Float('-0') == zero
assert Float('.0') == zero
assert Float('-.0') == zero
assert Float('-0.0') == zero
assert Float(0.0) == zero
assert Float(0) == zero
assert Float(0, '') == Float('0', '')
assert Float(1) == Float(1.0)
assert Float(S.Zero) == zero
assert Float(S.One) == Float(1.0)
assert Float(decimal.Decimal('0.1'), 3) == Float('.1', 3)
assert Float(decimal.Decimal('nan')) == S.NaN
assert Float(decimal.Decimal('Infinity')) == S.Infinity
assert Float(decimal.Decimal('-Infinity')) == S.NegativeInfinity
assert '{0:.3f}'.format(Float(4.236622)) == '4.237'
assert '{0:.35f}'.format(Float(pi.n(40), 40)) == \
'3.14159265358979323846264338327950288'
assert Float(oo) == Float('+inf')
assert Float(-oo) == Float('-inf')
# unicode
assert Float(u'0.73908513321516064100000000') == \
Float('0.73908513321516064100000000')
assert Float(u'0.73908513321516064100000000', 28) == \
Float('0.73908513321516064100000000', 28)
# binary precision
# Decimal value 0.1 cannot be expressed precisely as a base 2 fraction
a = Float(S(1)/10, dps=15)
b = Float(S(1)/10, dps=16)
p = Float(S(1)/10, precision=53)
q = Float(S(1)/10, precision=54)
assert a._mpf_ == p._mpf_
assert not a._mpf_ == q._mpf_
assert not b._mpf_ == q._mpf_
# Precision specifying errors
raises(ValueError, lambda: Float("1.23", dps=3, precision=10))
raises(ValueError, lambda: Float("1.23", dps="", precision=10))
raises(ValueError, lambda: Float("1.23", dps=3, precision=""))
raises(ValueError, lambda: Float("1.23", dps="", precision=""))
# from NumberSymbol
assert same_and_same_prec(Float(pi, 32), pi.evalf(32))
assert same_and_same_prec(Float(Catalan), Catalan.evalf())
@conserve_mpmath_dps
def test_float_mpf():
import mpmath
mpmath.mp.dps = 100
mp_pi = mpmath.pi()
assert Float(mp_pi, 100) == Float(mp_pi._mpf_, 100) == pi.evalf(100)
mpmath.mp.dps = 15
assert Float(mp_pi, 100) == Float(mp_pi._mpf_, 100) == pi.evalf(100)
def test_Float_RealElement():
repi = RealField(dps=100)(pi.evalf(100))
# We still have to pass the precision because Float doesn't know what
# RealElement is, but make sure it keeps full precision from the result.
assert Float(repi, 100) == pi.evalf(100)
def test_Float_default_to_highprec_from_str():
s = str(pi.evalf(128))
assert same_and_same_prec(Float(s), Float(s, ''))
def test_Float_eval():
a = Float(3.2)
assert (a**2).is_Float
def test_Float_issue_2107():
a = Float(0.1, 10)
b = Float("0.1", 10)
assert a - a == 0
assert a + (-a) == 0
assert S.Zero + a - a == 0
assert S.Zero + a + (-a) == 0
assert b - b == 0
assert b + (-b) == 0
assert S.Zero + b - b == 0
assert S.Zero + b + (-b) == 0
def test_issue_14289():
from sympy.polys.numberfields import to_number_field
a = 1 - sqrt(2)
b = to_number_field(a)
assert b.as_expr() == a
assert b.minpoly(a).expand() == 0
def test_Float_from_tuple():
a = Float((0, '1L', 0, 1))
b = Float((0, '1', 0, 1))
assert a == b
def test_Infinity():
assert oo != 1
assert 1*oo == oo
assert 1 != oo
assert oo != -oo
assert oo != Symbol("x")**3
assert oo + 1 == oo
assert 2 + oo == oo
assert 3*oo + 2 == oo
assert S.Half**oo == 0
assert S.Half**(-oo) == oo
assert -oo*3 == -oo
assert oo + oo == oo
assert -oo + oo*(-5) == -oo
assert 1/oo == 0
assert 1/(-oo) == 0
assert 8/oo == 0
assert oo % 2 == nan
assert 2 % oo == nan
assert oo/oo == nan
assert oo/-oo == nan
assert -oo/oo == nan
assert -oo/-oo == nan
assert oo - oo == nan
assert oo - -oo == oo
assert -oo - oo == -oo
assert -oo - -oo == nan
assert oo + -oo == nan
assert -oo + oo == nan
assert oo + oo == oo
assert -oo + oo == nan
assert oo + -oo == nan
assert -oo + -oo == -oo
assert oo*oo == oo
assert -oo*oo == -oo
assert oo*-oo == -oo
assert -oo*-oo == oo
assert oo/0 == oo
assert -oo/0 == -oo
assert 0/oo == 0
assert 0/-oo == 0
assert oo*0 == nan
assert -oo*0 == nan
assert 0*oo == nan
assert 0*-oo == nan
assert oo + 0 == oo
assert -oo + 0 == -oo
assert 0 + oo == oo
assert 0 + -oo == -oo
assert oo - 0 == oo
assert -oo - 0 == -oo
assert 0 - oo == -oo
assert 0 - -oo == oo
assert oo/2 == oo
assert -oo/2 == -oo
assert oo/-2 == -oo
assert -oo/-2 == oo
assert oo*2 == oo
assert -oo*2 == -oo
assert oo*-2 == -oo
assert 2/oo == 0
assert 2/-oo == 0
assert -2/oo == 0
assert -2/-oo == 0
assert 2*oo == oo
assert 2*-oo == -oo
assert -2*oo == -oo
assert -2*-oo == oo
assert 2 + oo == oo
assert 2 - oo == -oo
assert -2 + oo == oo
assert -2 - oo == -oo
assert 2 + -oo == -oo
assert 2 - -oo == oo
assert -2 + -oo == -oo
assert -2 - -oo == oo
assert S(2) + oo == oo
assert S(2) - oo == -oo
assert oo/I == -oo*I
assert -oo/I == oo*I
assert oo*float(1) == Float('inf') and (oo*float(1)).is_Float
assert -oo*float(1) == Float('-inf') and (-oo*float(1)).is_Float
assert oo/float(1) == Float('inf') and (oo/float(1)).is_Float
assert -oo/float(1) == Float('-inf') and (-oo/float(1)).is_Float
assert oo*float(-1) == Float('-inf') and (oo*float(-1)).is_Float
assert -oo*float(-1) == Float('inf') and (-oo*float(-1)).is_Float
assert oo/float(-1) == Float('-inf') and (oo/float(-1)).is_Float
assert -oo/float(-1) == Float('inf') and (-oo/float(-1)).is_Float
assert oo + float(1) == Float('inf') and (oo + float(1)).is_Float
assert -oo + float(1) == Float('-inf') and (-oo + float(1)).is_Float
assert oo - float(1) == Float('inf') and (oo - float(1)).is_Float
assert -oo - float(1) == Float('-inf') and (-oo - float(1)).is_Float
assert float(1)*oo == Float('inf') and (float(1)*oo).is_Float
assert float(1)*-oo == Float('-inf') and (float(1)*-oo).is_Float
assert float(1)/oo == 0
assert float(1)/-oo == 0
assert float(-1)*oo == Float('-inf') and (float(-1)*oo).is_Float
assert float(-1)*-oo == Float('inf') and (float(-1)*-oo).is_Float
assert float(-1)/oo == 0
assert float(-1)/-oo == 0
assert float(1) + oo == Float('inf')
assert float(1) + -oo == Float('-inf')
assert float(1) - oo == Float('-inf')
assert float(1) - -oo == Float('inf')
assert Float('nan') == nan
assert nan*1.0 == nan
assert -1.0*nan == nan
assert nan*oo == nan
assert nan*-oo == nan
assert nan/oo == nan
assert nan/-oo == nan
assert nan + oo == nan
assert nan + -oo == nan
assert nan - oo == nan
assert nan - -oo == nan
assert -oo * S.Zero == nan
assert oo*nan == nan
assert -oo*nan == nan
assert oo/nan == nan
assert -oo/nan == nan
assert oo + nan == nan
assert -oo + nan == nan
assert oo - nan == nan
assert -oo - nan == nan
assert S.Zero * oo == nan
assert oo.is_Rational is False
assert isinstance(oo, Rational) is False
assert S.One/oo == 0
assert -S.One/oo == 0
assert S.One/-oo == 0
assert -S.One/-oo == 0
assert S.One*oo == oo
assert -S.One*oo == -oo
assert S.One*-oo == -oo
assert -S.One*-oo == oo
assert S.One/nan == nan
assert S.One - -oo == oo
assert S.One + nan == nan
assert S.One - nan == nan
assert nan - S.One == nan
assert nan/S.One == nan
assert -oo - S.One == -oo
def test_Infinity_2():
x = Symbol('x')
assert oo*x != oo
assert oo*(pi - 1) == oo
assert oo*(1 - pi) == -oo
assert (-oo)*x != -oo
assert (-oo)*(pi - 1) == -oo
assert (-oo)*(1 - pi) == oo
assert (-1)**S.NaN is S.NaN
assert oo - Float('inf') is S.NaN
assert oo + Float('-inf') is S.NaN
assert oo*0 is S.NaN
assert oo/Float('inf') is S.NaN
assert oo/Float('-inf') is S.NaN
assert oo**S.NaN is S.NaN
assert -oo + Float('inf') is S.NaN
assert -oo - Float('-inf') is S.NaN
assert -oo*S.NaN is S.NaN
assert -oo*0 is S.NaN
assert -oo/Float('inf') is S.NaN
assert -oo/Float('-inf') is S.NaN
assert -oo/S.NaN is S.NaN
assert abs(-oo) == oo
assert all((-oo)**i is S.NaN for i in (oo, -oo, S.NaN))
assert (-oo)**3 == -oo
assert (-oo)**2 == oo
assert abs(S.ComplexInfinity) == oo
def test_Mul_Infinity_Zero():
assert 0*Float('inf') == nan
assert 0*Float('-inf') == nan
assert 0*Float('inf') == nan
assert 0*Float('-inf') == nan
assert Float('inf')*0 == nan
assert Float('-inf')*0 == nan
assert Float('inf')*0 == nan
assert Float('-inf')*0 == nan
assert Float(0)*Float('inf') == nan
assert Float(0)*Float('-inf') == nan
assert Float(0)*Float('inf') == nan
assert Float(0)*Float('-inf') == nan
assert Float('inf')*Float(0) == nan
assert Float('-inf')*Float(0) == nan
assert Float('inf')*Float(0) == nan
assert Float('-inf')*Float(0) == nan
def test_Div_By_Zero():
assert 1/S(0) == zoo
assert 1/Float(0) == Float('inf')
assert 0/S(0) == nan
assert 0/Float(0) == nan
assert S(0)/0 == nan
assert Float(0)/0 == nan
assert -1/S(0) == zoo
assert -1/Float(0) == Float('-inf')
def test_Infinity_inequations():
assert oo > pi
assert not (oo < pi)
assert exp(-3) < oo
assert Float('+inf') > pi
assert not (Float('+inf') < pi)
assert exp(-3) < Float('+inf')
raises(TypeError, lambda: oo < I)
raises(TypeError, lambda: oo <= I)
raises(TypeError, lambda: oo > I)
raises(TypeError, lambda: oo >= I)
raises(TypeError, lambda: -oo < I)
raises(TypeError, lambda: -oo <= I)
raises(TypeError, lambda: -oo > I)
raises(TypeError, lambda: -oo >= I)
raises(TypeError, lambda: I < oo)
raises(TypeError, lambda: I <= oo)
raises(TypeError, lambda: I > oo)
raises(TypeError, lambda: I >= oo)
raises(TypeError, lambda: I < -oo)
raises(TypeError, lambda: I <= -oo)
raises(TypeError, lambda: I > -oo)
raises(TypeError, lambda: I >= -oo)
assert oo > -oo and oo >= -oo
assert (oo < -oo) == False and (oo <= -oo) == False
assert -oo < oo and -oo <= oo
assert (-oo > oo) == False and (-oo >= oo) == False
assert (oo < oo) == False # issue 7775
assert (oo > oo) == False
assert (-oo > -oo) == False and (-oo < -oo) == False
assert oo >= oo and oo <= oo and -oo >= -oo and -oo <= -oo
assert (-oo < -Float('inf')) == False
assert (oo > Float('inf')) == False
assert -oo >= -Float('inf')
assert oo <= Float('inf')
x = Symbol('x')
b = Symbol('b', finite=True, real=True)
assert (x < oo) == Lt(x, oo) # issue 7775
assert b < oo and b > -oo and b <= oo and b >= -oo
assert oo > b and oo >= b and (oo < b) == False and (oo <= b) == False
assert (-oo > b) == False and (-oo >= b) == False and -oo < b and -oo <= b
assert (oo < x) == Lt(oo, x) and (oo > x) == Gt(oo, x)
assert (oo <= x) == Le(oo, x) and (oo >= x) == Ge(oo, x)
assert (-oo < x) == Lt(-oo, x) and (-oo > x) == Gt(-oo, x)
assert (-oo <= x) == Le(-oo, x) and (-oo >= x) == Ge(-oo, x)
def test_NaN():
assert nan == nan
assert nan != 1
assert 1*nan == nan
assert 1 != nan
assert nan == -nan
assert oo != Symbol("x")**3
assert nan + 1 == nan
assert 2 + nan == nan
assert 3*nan + 2 == nan
assert -nan*3 == nan
assert nan + nan == nan
assert -nan + nan*(-5) == nan
assert 1/nan == nan
assert 1/(-nan) == nan
assert 8/nan == nan
raises(TypeError, lambda: nan > 0)
raises(TypeError, lambda: nan < 0)
raises(TypeError, lambda: nan >= 0)
raises(TypeError, lambda: nan <= 0)
raises(TypeError, lambda: 0 < nan)
raises(TypeError, lambda: 0 > nan)
raises(TypeError, lambda: 0 <= nan)
raises(TypeError, lambda: 0 >= nan)
assert S.One + nan == nan
assert S.One - nan == nan
assert S.One*nan == nan
assert S.One/nan == nan
assert nan - S.One == nan
assert nan*S.One == nan
assert nan + S.One == nan
assert nan/S.One == nan
assert nan**0 == 1 # as per IEEE 754
assert 1**nan == nan # IEEE 754 is not the best choice for symbolic work
# test Pow._eval_power's handling of NaN
assert Pow(nan, 0, evaluate=False)**2 == 1
def test_special_numbers():
assert isinstance(S.NaN, Number) is True
assert isinstance(S.Infinity, Number) is True
assert isinstance(S.NegativeInfinity, Number) is True
assert S.NaN.is_number is True
assert S.Infinity.is_number is True
assert S.NegativeInfinity.is_number is True
assert S.ComplexInfinity.is_number is True
assert isinstance(S.NaN, Rational) is False
assert isinstance(S.Infinity, Rational) is False
assert isinstance(S.NegativeInfinity, Rational) is False
assert S.NaN.is_rational is not True
assert S.Infinity.is_rational is not True
assert S.NegativeInfinity.is_rational is not True
def test_powers():
assert integer_nthroot(1, 2) == (1, True)
assert integer_nthroot(1, 5) == (1, True)
assert integer_nthroot(2, 1) == (2, True)
assert integer_nthroot(2, 2) == (1, False)
assert integer_nthroot(2, 5) == (1, False)
assert integer_nthroot(4, 2) == (2, True)
assert integer_nthroot(123**25, 25) == (123, True)
assert integer_nthroot(123**25 + 1, 25) == (123, False)
assert integer_nthroot(123**25 - 1, 25) == (122, False)
assert integer_nthroot(1, 1) == (1, True)
assert integer_nthroot(0, 1) == (0, True)
assert integer_nthroot(0, 3) == (0, True)
assert integer_nthroot(10000, 1) == (10000, True)
assert integer_nthroot(4, 2) == (2, True)
assert integer_nthroot(16, 2) == (4, True)
assert integer_nthroot(26, 2) == (5, False)
assert integer_nthroot(1234567**7, 7) == (1234567, True)
assert integer_nthroot(1234567**7 + 1, 7) == (1234567, False)
assert integer_nthroot(1234567**7 - 1, 7) == (1234566, False)
b = 25**1000
assert integer_nthroot(b, 1000) == (25, True)
assert integer_nthroot(b + 1, 1000) == (25, False)
assert integer_nthroot(b - 1, 1000) == (24, False)
c = 10**400
c2 = c**2
assert integer_nthroot(c2, 2) == (c, True)
assert integer_nthroot(c2 + 1, 2) == (c, False)
assert integer_nthroot(c2 - 1, 2) == (c - 1, False)
assert integer_nthroot(2, 10**10) == (1, False)
p, r = integer_nthroot(int(factorial(10000)), 100)
assert p % (10**10) == 5322420655
assert not r
# Test that this is fast
assert integer_nthroot(2, 10**10) == (1, False)
# output should be int if possible
assert type(integer_nthroot(2**61, 2)[0]) is int
def test_integer_nthroot_overflow():
assert integer_nthroot(10**(50*50), 50) == (10**50, True)
assert integer_nthroot(10**100000, 10000) == (10**10, True)
def test_integer_log():
raises(ValueError, lambda: integer_log(2, 1))
raises(ValueError, lambda: integer_log(0, 2))
raises(ValueError, lambda: integer_log(1.1, 2))
raises(ValueError, lambda: integer_log(1, 2.2))
assert integer_log(1, 2) == (0, True)
assert integer_log(1, 3) == (0, True)
assert integer_log(2, 3) == (0, False)
assert integer_log(3, 3) == (1, True)
assert integer_log(3*2, 3) == (1, False)
assert integer_log(3**2, 3) == (2, True)
assert integer_log(3*4, 3) == (2, False)
assert integer_log(3**3, 3) == (3, True)
assert integer_log(27, 5) == (2, False)
assert integer_log(2, 3) == (0, False)
assert integer_log(-4, -2) == (2, False)
assert integer_log(27, -3) == (3, False)
assert integer_log(-49, 7) == (0, False)
assert integer_log(-49, -7) == (2, False)
def test_isqrt():
from math import sqrt as _sqrt
limit = 17984395633462800708566937239551
assert int(_sqrt(limit)) == integer_nthroot(limit, 2)[0]
assert int(_sqrt(limit + 1)) != integer_nthroot(limit + 1, 2)[0]
assert isqrt(limit + 1) == integer_nthroot(limit + 1, 2)[0]
assert isqrt(limit + 1 + S.Half) == integer_nthroot(limit + 1, 2)[0]
def test_powers_Integer():
"""Test Integer._eval_power"""
# check infinity
assert S(1) ** S.Infinity == S.NaN
assert S(-1)** S.Infinity == S.NaN
assert S(2) ** S.Infinity == S.Infinity
assert S(-2)** S.Infinity == S.Infinity + S.Infinity * S.ImaginaryUnit
assert S(0) ** S.Infinity == 0
# check Nan
assert S(1) ** S.NaN == S.NaN
assert S(-1) ** S.NaN == S.NaN
# check for exact roots
assert S(-1) ** Rational(6, 5) == - (-1)**(S(1)/5)
assert sqrt(S(4)) == 2
assert sqrt(S(-4)) == I * 2
assert S(16) ** Rational(1, 4) == 2
assert S(-16) ** Rational(1, 4) == 2 * (-1)**Rational(1, 4)
assert S(9) ** Rational(3, 2) == 27
assert S(-9) ** Rational(3, 2) == -27*I
assert S(27) ** Rational(2, 3) == 9
assert S(-27) ** Rational(2, 3) == 9 * (S(-1) ** Rational(2, 3))
assert (-2) ** Rational(-2, 1) == Rational(1, 4)
# not exact roots
assert sqrt(-3) == I*sqrt(3)
assert (3) ** (S(3)/2) == 3 * sqrt(3)
assert (-3) ** (S(3)/2) == - 3 * sqrt(-3)
assert (-3) ** (S(5)/2) == 9 * I * sqrt(3)
assert (-3) ** (S(7)/2) == - I * 27 * sqrt(3)
assert (2) ** (S(3)/2) == 2 * sqrt(2)
assert (2) ** (S(-3)/2) == sqrt(2) / 4
assert (81) ** (S(2)/3) == 9 * (S(3) ** (S(2)/3))
assert (-81) ** (S(2)/3) == 9 * (S(-3) ** (S(2)/3))
assert (-3) ** Rational(-7, 3) == \
-(-1)**Rational(2, 3)*3**Rational(2, 3)/27
assert (-3) ** Rational(-2, 3) == \
-(-1)**Rational(1, 3)*3**Rational(1, 3)/3
# join roots
assert sqrt(6) + sqrt(24) == 3*sqrt(6)
assert sqrt(2) * sqrt(3) == sqrt(6)
# separate symbols & constansts
x = Symbol("x")
assert sqrt(49 * x) == 7 * sqrt(x)
assert sqrt((3 - sqrt(pi)) ** 2) == 3 - sqrt(pi)
# check that it is fast for big numbers
assert (2**64 + 1) ** Rational(4, 3)
assert (2**64 + 1) ** Rational(17, 25)
# negative rational power and negative base
assert (-3) ** Rational(-7, 3) == \
-(-1)**Rational(2, 3)*3**Rational(2, 3)/27
assert (-3) ** Rational(-2, 3) == \
-(-1)**Rational(1, 3)*3**Rational(1, 3)/3
assert (-2) ** Rational(-10, 3) == \
(-1)**Rational(2, 3)*2**Rational(2, 3)/16
assert abs(Pow(-2, Rational(-10, 3)).n() -
Pow(-2, Rational(-10, 3), evaluate=False).n()) < 1e-16
# negative base and rational power with some simplification
assert (-8) ** Rational(2, 5) == \
2*(-1)**Rational(2, 5)*2**Rational(1, 5)
assert (-4) ** Rational(9, 5) == \
-8*(-1)**Rational(4, 5)*2**Rational(3, 5)
assert S(1234).factors() == {617: 1, 2: 1}
assert Rational(2*3, 3*5*7).factors() == {2: 1, 5: -1, 7: -1}
# test that eval_power factors numbers bigger than
# the current limit in factor_trial_division (2**15)
from sympy import nextprime
n = nextprime(2**15)
assert sqrt(n**2) == n
assert sqrt(n**3) == n*sqrt(n)
assert sqrt(4*n) == 2*sqrt(n)
# check that factors of base with powers sharing gcd with power are removed
assert (2**4*3)**Rational(1, 6) == 2**Rational(2, 3)*3**Rational(1, 6)
assert (2**4*3)**Rational(5, 6) == 8*2**Rational(1, 3)*3**Rational(5, 6)
# check that bases sharing a gcd are exptracted
assert 2**Rational(1, 3)*3**Rational(1, 4)*6**Rational(1, 5) == \
2**Rational(8, 15)*3**Rational(9, 20)
assert sqrt(8)*24**Rational(1, 3)*6**Rational(1, 5) == \
4*2**Rational(7, 10)*3**Rational(8, 15)
assert sqrt(8)*(-24)**Rational(1, 3)*(-6)**Rational(1, 5) == \
4*(-3)**Rational(8, 15)*2**Rational(7, 10)
assert 2**Rational(1, 3)*2**Rational(8, 9) == 2*2**Rational(2, 9)
assert 2**Rational(2, 3)*6**Rational(1, 3) == 2*3**Rational(1, 3)
assert 2**Rational(2, 3)*6**Rational(8, 9) == \
2*2**Rational(5, 9)*3**Rational(8, 9)
assert (-2)**Rational(2, S(3))*(-4)**Rational(1, S(3)) == -2*2**Rational(1, 3)
assert 3*Pow(3, 2, evaluate=False) == 3**3
assert 3*Pow(3, -1/S(3), evaluate=False) == 3**(2/S(3))
assert (-2)**(1/S(3))*(-3)**(1/S(4))*(-5)**(5/S(6)) == \
-(-1)**Rational(5, 12)*2**Rational(1, 3)*3**Rational(1, 4) * \
5**Rational(5, 6)
assert Integer(-2)**Symbol('', even=True) == \
Integer(2)**Symbol('', even=True)
assert (-1)**Float(.5) == 1.0*I
def test_powers_Rational():
"""Test Rational._eval_power"""
# check infinity
assert Rational(1, 2) ** S.Infinity == 0
assert Rational(3, 2) ** S.Infinity == S.Infinity
assert Rational(-1, 2) ** S.Infinity == 0
assert Rational(-3, 2) ** S.Infinity == \
S.Infinity + S.Infinity * S.ImaginaryUnit
# check Nan
assert Rational(3, 4) ** S.NaN == S.NaN
assert Rational(-2, 3) ** S.NaN == S.NaN
# exact roots on numerator
assert sqrt(Rational(4, 3)) == 2 * sqrt(3) / 3
assert Rational(4, 3) ** Rational(3, 2) == 8 * sqrt(3) / 9
assert sqrt(Rational(-4, 3)) == I * 2 * sqrt(3) / 3
assert Rational(-4, 3) ** Rational(3, 2) == - I * 8 * sqrt(3) / 9
assert Rational(27, 2) ** Rational(1, 3) == 3 * (2 ** Rational(2, 3)) / 2
assert Rational(5**3, 8**3) ** Rational(4, 3) == Rational(5**4, 8**4)
# exact root on denominator
assert sqrt(Rational(1, 4)) == Rational(1, 2)
assert sqrt(Rational(1, -4)) == I * Rational(1, 2)
assert sqrt(Rational(3, 4)) == sqrt(3) / 2
assert sqrt(Rational(3, -4)) == I * sqrt(3) / 2
assert Rational(5, 27) ** Rational(1, 3) == (5 ** Rational(1, 3)) / 3
# not exact roots
assert sqrt(Rational(1, 2)) == sqrt(2) / 2
assert sqrt(Rational(-4, 7)) == I * sqrt(Rational(4, 7))
assert Rational(-3, 2)**Rational(-7, 3) == \
-4*(-1)**Rational(2, 3)*2**Rational(1, 3)*3**Rational(2, 3)/27
assert Rational(-3, 2)**Rational(-2, 3) == \
-(-1)**Rational(1, 3)*2**Rational(2, 3)*3**Rational(1, 3)/3
assert Rational(-3, 2)**Rational(-10, 3) == \
8*(-1)**Rational(2, 3)*2**Rational(1, 3)*3**Rational(2, 3)/81
assert abs(Pow(Rational(-2, 3), Rational(-7, 4)).n() -
Pow(Rational(-2, 3), Rational(-7, 4), evaluate=False).n()) < 1e-16
# negative integer power and negative rational base
assert Rational(-2, 3) ** Rational(-2, 1) == Rational(9, 4)
a = Rational(1, 10)
assert a**Float(a, 2) == Float(a, 2)**Float(a, 2)
assert Rational(-2, 3)**Symbol('', even=True) == \
Rational(2, 3)**Symbol('', even=True)
def test_powers_Float():
assert str((S('-1/10')**S('3/10')).n()) == str(Float(-.1)**(.3))
def test_abs1():
assert Rational(1, 6) != Rational(-1, 6)
assert abs(Rational(1, 6)) == abs(Rational(-1, 6))
def test_accept_int():
assert Float(4) == 4
def test_dont_accept_str():
assert Float("0.2") != "0.2"
assert not (Float("0.2") == "0.2")
def test_int():
a = Rational(5)
assert int(a) == 5
a = Rational(9, 10)
assert int(a) == int(-a) == 0
assert 1/(-1)**Rational(2, 3) == -(-1)**Rational(1, 3)
assert int(pi) == 3
assert int(E) == 2
assert int(GoldenRatio) == 1
assert int(TribonacciConstant) == 2
# issue 10368
a = S(32442016954)/78058255275
assert type(int(a)) is type(int(-a)) is int
def test_long():
a = Rational(5)
assert long(a) == 5
a = Rational(9, 10)
assert long(a) == long(-a) == 0
a = Integer(2**100)
assert long(a) == a
assert long(pi) == 3
assert long(E) == 2
assert long(GoldenRatio) == 1
assert long(TribonacciConstant) == 2
def test_real_bug():
x = Symbol("x")
assert str(2.0*x*x) in ["(2.0*x)*x", "2.0*x**2", "2.00000000000000*x**2"]
assert str(2.1*x*x) != "(2.0*x)*x"
def test_bug_sqrt():
assert ((sqrt(Rational(2)) + 1)*(sqrt(Rational(2)) - 1)).expand() == 1
def test_pi_Pi():
"Test that pi (instance) is imported, but Pi (class) is not"
from sympy import pi
with raises(ImportError):
from sympy import Pi
def test_no_len():
# there should be no len for numbers
raises(TypeError, lambda: len(Rational(2)))
raises(TypeError, lambda: len(Rational(2, 3)))
raises(TypeError, lambda: len(Integer(2)))
def test_issue_3321():
assert sqrt(Rational(1, 5)) == sqrt(Rational(1, 5))
assert 5 * sqrt(Rational(1, 5)) == sqrt(5)
def test_issue_3692():
assert ((-1)**Rational(1, 6)).expand(complex=True) == I/2 + sqrt(3)/2
assert ((-5)**Rational(1, 6)).expand(complex=True) == \
5**Rational(1, 6)*I/2 + 5**Rational(1, 6)*sqrt(3)/2
assert ((-64)**Rational(1, 6)).expand(complex=True) == I + sqrt(3)
def test_issue_3423():
x = Symbol("x")
assert sqrt(x - 1).as_base_exp() == (x - 1, S.Half)
assert sqrt(x - 1) != I*sqrt(1 - x)
def test_issue_3449():
x = Symbol("x")
assert sqrt(x - 1).subs(x, 5) == 2
def test_issue_13890():
x = Symbol("x")
e = (-x/4 - S(1)/12)**x - 1
f = simplify(e)
a = S(9)/5
assert abs(e.subs(x,a).evalf() - f.subs(x,a).evalf()) < 1e-15
def test_Integer_factors():
def F(i):
return Integer(i).factors()
assert F(1) == {}
assert F(2) == {2: 1}
assert F(3) == {3: 1}
assert F(4) == {2: 2}
assert F(5) == {5: 1}
assert F(6) == {2: 1, 3: 1}
assert F(7) == {7: 1}
assert F(8) == {2: 3}
assert F(9) == {3: 2}
assert F(10) == {2: 1, 5: 1}
assert F(11) == {11: 1}
assert F(12) == {2: 2, 3: 1}
assert F(13) == {13: 1}
assert F(14) == {2: 1, 7: 1}
assert F(15) == {3: 1, 5: 1}
assert F(16) == {2: 4}
assert F(17) == {17: 1}
assert F(18) == {2: 1, 3: 2}
assert F(19) == {19: 1}
assert F(20) == {2: 2, 5: 1}
assert F(21) == {3: 1, 7: 1}
assert F(22) == {2: 1, 11: 1}
assert F(23) == {23: 1}
assert F(24) == {2: 3, 3: 1}
assert F(25) == {5: 2}
assert F(26) == {2: 1, 13: 1}
assert F(27) == {3: 3}
assert F(28) == {2: 2, 7: 1}
assert F(29) == {29: 1}
assert F(30) == {2: 1, 3: 1, 5: 1}
assert F(31) == {31: 1}
assert F(32) == {2: 5}
assert F(33) == {3: 1, 11: 1}
assert F(34) == {2: 1, 17: 1}
assert F(35) == {5: 1, 7: 1}
assert F(36) == {2: 2, 3: 2}
assert F(37) == {37: 1}
assert F(38) == {2: 1, 19: 1}
assert F(39) == {3: 1, 13: 1}
assert F(40) == {2: 3, 5: 1}
assert F(41) == {41: 1}
assert F(42) == {2: 1, 3: 1, 7: 1}
assert F(43) == {43: 1}
assert F(44) == {2: 2, 11: 1}
assert F(45) == {3: 2, 5: 1}
assert F(46) == {2: 1, 23: 1}
assert F(47) == {47: 1}
assert F(48) == {2: 4, 3: 1}
assert F(49) == {7: 2}
assert F(50) == {2: 1, 5: 2}
assert F(51) == {3: 1, 17: 1}
def test_Rational_factors():
def F(p, q, visual=None):
return Rational(p, q).factors(visual=visual)
assert F(2, 3) == {2: 1, 3: -1}
assert F(2, 9) == {2: 1, 3: -2}
assert F(2, 15) == {2: 1, 3: -1, 5: -1}
assert F(6, 10) == {3: 1, 5: -1}
def test_issue_4107():
assert pi*(E + 10) + pi*(-E - 10) != 0
assert pi*(E + 10**10) + pi*(-E - 10**10) != 0
assert pi*(E + 10**20) + pi*(-E - 10**20) != 0
assert pi*(E + 10**80) + pi*(-E - 10**80) != 0
assert (pi*(E + 10) + pi*(-E - 10)).expand() == 0
assert (pi*(E + 10**10) + pi*(-E - 10**10)).expand() == 0
assert (pi*(E + 10**20) + pi*(-E - 10**20)).expand() == 0
assert (pi*(E + 10**80) + pi*(-E - 10**80)).expand() == 0
def test_IntegerInteger():
a = Integer(4)
b = Integer(a)
assert a == b
def test_Rational_gcd_lcm_cofactors():
assert Integer(4).gcd(2) == Integer(2)
assert Integer(4).lcm(2) == Integer(4)
assert Integer(4).gcd(Integer(2)) == Integer(2)
assert Integer(4).lcm(Integer(2)) == Integer(4)
a, b = 720**99911, 480**12342
assert Integer(a).lcm(b) == a*b/Integer(a).gcd(b)
assert Integer(4).gcd(3) == Integer(1)
assert Integer(4).lcm(3) == Integer(12)
assert Integer(4).gcd(Integer(3)) == Integer(1)
assert Integer(4).lcm(Integer(3)) == Integer(12)
assert Rational(4, 3).gcd(2) == Rational(2, 3)
assert Rational(4, 3).lcm(2) == Integer(4)
assert Rational(4, 3).gcd(Integer(2)) == Rational(2, 3)
assert Rational(4, 3).lcm(Integer(2)) == Integer(4)
assert Integer(4).gcd(Rational(2, 9)) == Rational(2, 9)
assert Integer(4).lcm(Rational(2, 9)) == Integer(4)
assert Rational(4, 3).gcd(Rational(2, 9)) == Rational(2, 9)
assert Rational(4, 3).lcm(Rational(2, 9)) == Rational(4, 3)
assert Rational(4, 5).gcd(Rational(2, 9)) == Rational(2, 45)
assert Rational(4, 5).lcm(Rational(2, 9)) == Integer(4)
assert Rational(5, 9).lcm(Rational(3, 7)) == Rational(Integer(5).lcm(3),Integer(9).gcd(7))
assert Integer(4).cofactors(2) == (Integer(2), Integer(2), Integer(1))
assert Integer(4).cofactors(Integer(2)) == \
(Integer(2), Integer(2), Integer(1))
assert Integer(4).gcd(Float(2.0)) == S.One
assert Integer(4).lcm(Float(2.0)) == Float(8.0)
assert Integer(4).cofactors(Float(2.0)) == (S.One, Integer(4), Float(2.0))
assert Rational(1, 2).gcd(Float(2.0)) == S.One
assert Rational(1, 2).lcm(Float(2.0)) == Float(1.0)
assert Rational(1, 2).cofactors(Float(2.0)) == \
(S.One, Rational(1, 2), Float(2.0))
def test_Float_gcd_lcm_cofactors():
assert Float(2.0).gcd(Integer(4)) == S.One
assert Float(2.0).lcm(Integer(4)) == Float(8.0)
assert Float(2.0).cofactors(Integer(4)) == (S.One, Float(2.0), Integer(4))
assert Float(2.0).gcd(Rational(1, 2)) == S.One
assert Float(2.0).lcm(Rational(1, 2)) == Float(1.0)
assert Float(2.0).cofactors(Rational(1, 2)) == \
(S.One, Float(2.0), Rational(1, 2))
def test_issue_4611():
assert abs(pi._evalf(50) - 3.14159265358979) < 1e-10
assert abs(E._evalf(50) - 2.71828182845905) < 1e-10
assert abs(Catalan._evalf(50) - 0.915965594177219) < 1e-10
assert abs(EulerGamma._evalf(50) - 0.577215664901533) < 1e-10
assert abs(GoldenRatio._evalf(50) - 1.61803398874989) < 1e-10
assert abs(TribonacciConstant._evalf(50) - 1.83928675521416) < 1e-10
x = Symbol("x")
assert (pi + x).evalf() == pi.evalf() + x
assert (E + x).evalf() == E.evalf() + x
assert (Catalan + x).evalf() == Catalan.evalf() + x
assert (EulerGamma + x).evalf() == EulerGamma.evalf() + x
assert (GoldenRatio + x).evalf() == GoldenRatio.evalf() + x
assert (TribonacciConstant + x).evalf() == TribonacciConstant.evalf() + x
@conserve_mpmath_dps
def test_conversion_to_mpmath():
assert mpmath.mpmathify(Integer(1)) == mpmath.mpf(1)
assert mpmath.mpmathify(Rational(1, 2)) == mpmath.mpf(0.5)
assert mpmath.mpmathify(Float('1.23', 15)) == mpmath.mpf('1.23')
assert mpmath.mpmathify(I) == mpmath.mpc(1j)
assert mpmath.mpmathify(1 + 2*I) == mpmath.mpc(1 + 2j)
assert mpmath.mpmathify(1.0 + 2*I) == mpmath.mpc(1 + 2j)
assert mpmath.mpmathify(1 + 2.0*I) == mpmath.mpc(1 + 2j)
assert mpmath.mpmathify(1.0 + 2.0*I) == mpmath.mpc(1 + 2j)
assert mpmath.mpmathify(Rational(1, 2) + Rational(1, 2)*I) == mpmath.mpc(0.5 + 0.5j)
assert mpmath.mpmathify(2*I) == mpmath.mpc(2j)
assert mpmath.mpmathify(2.0*I) == mpmath.mpc(2j)
assert mpmath.mpmathify(Rational(1, 2)*I) == mpmath.mpc(0.5j)
mpmath.mp.dps = 100
assert mpmath.mpmathify(pi.evalf(100) + pi.evalf(100)*I) == mpmath.pi + mpmath.pi*mpmath.j
assert mpmath.mpmathify(pi.evalf(100)*I) == mpmath.pi*mpmath.j
def test_relational():
# real
x = S(.1)
assert (x != cos) is True
assert (x == cos) is False
# rational
x = Rational(1, 3)
assert (x != cos) is True
assert (x == cos) is False
# integer defers to rational so these tests are omitted
# number symbol
x = pi
assert (x != cos) is True
assert (x == cos) is False
def test_Integer_as_index():
assert 'hello'[Integer(2):] == 'llo'
def test_Rational_int():
assert int( Rational(7, 5)) == 1
assert int( Rational(1, 2)) == 0
assert int(-Rational(1, 2)) == 0
assert int(-Rational(7, 5)) == -1
def test_zoo():
b = Symbol('b', finite=True)
nz = Symbol('nz', nonzero=True)
p = Symbol('p', positive=True)
n = Symbol('n', negative=True)
im = Symbol('i', imaginary=True)
c = Symbol('c', complex=True)
pb = Symbol('pb', positive=True, finite=True)
nb = Symbol('nb', negative=True, finite=True)
imb = Symbol('ib', imaginary=True, finite=True)
for i in [I, S.Infinity, S.NegativeInfinity, S.Zero, S.One, S.Pi, S.Half, S(3), log(3),
b, nz, p, n, im, pb, nb, imb, c]:
if i.is_finite and (i.is_real or i.is_imaginary):
assert i + zoo is zoo
assert i - zoo is zoo
assert zoo + i is zoo
assert zoo - i is zoo
elif i.is_finite is not False:
assert (i + zoo).is_Add
assert (i - zoo).is_Add
assert (zoo + i).is_Add
assert (zoo - i).is_Add
else:
assert (i + zoo) is S.NaN
assert (i - zoo) is S.NaN
assert (zoo + i) is S.NaN
assert (zoo - i) is S.NaN
if fuzzy_not(i.is_zero) and (i.is_real or i.is_imaginary):
assert i*zoo is zoo
assert zoo*i is zoo
elif i.is_zero:
assert i*zoo is S.NaN
assert zoo*i is S.NaN
else:
assert (i*zoo).is_Mul
assert (zoo*i).is_Mul
if fuzzy_not((1/i).is_zero) and (i.is_real or i.is_imaginary):
assert zoo/i is zoo
elif (1/i).is_zero:
assert zoo/i is S.NaN
elif i.is_zero:
assert zoo/i is zoo
else:
assert (zoo/i).is_Mul
assert (I*oo).is_Mul # allow directed infinity
assert zoo + zoo is S.NaN
assert zoo * zoo is zoo
assert zoo - zoo is S.NaN
assert zoo/zoo is S.NaN
assert zoo**zoo is S.NaN
assert zoo**0 is S.One
assert zoo**2 is zoo
assert 1/zoo is S.Zero
assert Mul.flatten([S(-1), oo, S(0)]) == ([S.NaN], [], None)
def test_issue_4122():
x = Symbol('x', nonpositive=True)
assert (oo + x).is_Add
x = Symbol('x', finite=True)
assert (oo + x).is_Add # x could be imaginary
x = Symbol('x', nonnegative=True)
assert oo + x == oo
x = Symbol('x', finite=True, real=True)
assert oo + x == oo
# similarly for negative infinity
x = Symbol('x', nonnegative=True)
assert (-oo + x).is_Add
x = Symbol('x', finite=True)
assert (-oo + x).is_Add
x = Symbol('x', nonpositive=True)
assert -oo + x == -oo
x = Symbol('x', finite=True, real=True)
assert -oo + x == -oo
def test_GoldenRatio_expand():
assert GoldenRatio.expand(func=True) == S.Half + sqrt(5)/2
def test_TribonacciConstant_expand():
assert TribonacciConstant.expand(func=True) == \
(1 + cbrt(19 - 3*sqrt(33)) + cbrt(19 + 3*sqrt(33))) / 3
def test_as_content_primitive():
assert S.Zero.as_content_primitive() == (1, 0)
assert S.Half.as_content_primitive() == (S.Half, 1)
assert (-S.Half).as_content_primitive() == (S.Half, -1)
assert S(3).as_content_primitive() == (3, 1)
assert S(3.1).as_content_primitive() == (1, 3.1)
def test_hashing_sympy_integers():
# Test for issue 5072
assert set([Integer(3)]) == set([int(3)])
assert hash(Integer(4)) == hash(int(4))
def test_issue_4172():
assert int((E**100).round()) == \
26881171418161354484126255515800135873611119
assert int((pi**100).round()) == \
51878483143196131920862615246303013562686760680406
assert int((Rational(1)/EulerGamma**100).round()) == \
734833795660954410469466
@XFAIL
def test_mpmath_issues():
from mpmath.libmp.libmpf import _normalize
import mpmath.libmp as mlib
rnd = mlib.round_nearest
mpf = (0, long(0), -123, -1, 53, rnd) # nan
assert _normalize(mpf, 53) != (0, long(0), 0, 0)
mpf = (0, long(0), -456, -2, 53, rnd) # +inf
assert _normalize(mpf, 53) != (0, long(0), 0, 0)
mpf = (1, long(0), -789, -3, 53, rnd) # -inf
assert _normalize(mpf, 53) != (0, long(0), 0, 0)
from mpmath.libmp.libmpf import fnan
assert mlib.mpf_eq(fnan, fnan)
def test_Catalan_EulerGamma_prec():
n = GoldenRatio
f = Float(n.n(), 5)
assert f._mpf_ == (0, long(212079), -17, 18)
assert f._prec == 20
assert n._as_mpf_val(20) == f._mpf_
n = EulerGamma
f = Float(n.n(), 5)
assert f._mpf_ == (0, long(302627), -19, 19)
assert f._prec == 20
assert n._as_mpf_val(20) == f._mpf_
def test_Float_eq():
assert Float(.12, 3) != Float(.12, 4)
assert Float(.12, 3) == .12
assert 0.12 == Float(.12, 3)
assert Float('.12', 22) != .12
def test_int_NumberSymbols():
assert [int(i) for i in [pi, EulerGamma, E, GoldenRatio, Catalan]] == \
[3, 0, 2, 1, 0]
def test_issue_6640():
from mpmath.libmp.libmpf import finf, fninf
# fnan is not included because Float no longer returns fnan,
# but otherwise, the same sort of test could apply
assert Float(finf).is_zero is False
assert Float(fninf).is_zero is False
assert bool(Float(0)) is False
def test_issue_6349():
assert Float('23.e3', '')._prec == 10
assert Float('23e3', '')._prec == 20
assert Float('23000', '')._prec == 20
assert Float('-23000', '')._prec == 20
def test_mpf_norm():
assert mpf_norm((1, 0, 1, 0), 10) == mpf('0')._mpf_
assert Float._new((1, 0, 1, 0), 10)._mpf_ == mpf('0')._mpf_
def test_latex():
assert latex(pi) == r"\pi"
assert latex(E) == r"e"
assert latex(GoldenRatio) == r"\phi"
assert latex(TribonacciConstant) == r"\text{TribonacciConstant}"
assert latex(EulerGamma) == r"\gamma"
assert latex(oo) == r"\infty"
assert latex(-oo) == r"-\infty"
assert latex(zoo) == r"\tilde{\infty}"
assert latex(nan) == r"\text{NaN}"
assert latex(I) == r"i"
def test_issue_7742():
assert -oo % 1 == nan
def test_simplify_AlgebraicNumber():
A = AlgebraicNumber
e = 3**(S(1)/6)*(3 + (135 + 78*sqrt(3))**(S(2)/3))/(45 + 26*sqrt(3))**(S(1)/3)
assert simplify(A(e)) == A(12) # wester test_C20
e = (41 + 29*sqrt(2))**(S(1)/5)
assert simplify(A(e)) == A(1 + sqrt(2)) # wester test_C21
e = (3 + 4*I)**(Rational(3, 2))
assert simplify(A(e)) == A(2 + 11*I) # issue 4401
def test_Float_idempotence():
x = Float('1.23', '')
y = Float(x)
z = Float(x, 15)
assert same_and_same_prec(y, x)
assert not same_and_same_prec(z, x)
x = Float(10**20)
y = Float(x)
z = Float(x, 15)
assert same_and_same_prec(y, x)
assert not same_and_same_prec(z, x)
def test_comp():
# sqrt(2) = 1.414213 5623730950...
a = sqrt(2).n(7)
assert comp(a, 1.41421346) is False
assert comp(a, 1.41421347)
assert comp(a, 1.41421366)
assert comp(a, 1.41421367) is False
assert comp(sqrt(2).n(2), '1.4')
assert comp(sqrt(2).n(2), Float(1.4, 2), '')
raises(ValueError, lambda: comp(sqrt(2).n(2), 1.4, ''))
assert comp(sqrt(2).n(2), Float(1.4, 3), '') is False
def test_issue_9491():
assert oo**zoo == nan
def test_issue_10063():
assert 2**Float(3) == Float(8)
def test_issue_10020():
assert oo**I is S.NaN
assert oo**(1 + I) is S.ComplexInfinity
assert oo**(-1 + I) is S.Zero
assert (-oo)**I is S.NaN
assert (-oo)**(-1 + I) is S.Zero
assert oo**t == Pow(oo, t, evaluate=False)
assert (-oo)**t == Pow(-oo, t, evaluate=False)
def test_invert_numbers():
assert S(2).invert(5) == 3
assert S(2).invert(S(5)/2) == S.Half
assert S(2).invert(5.) == 0.5
assert S(2).invert(S(5)) == 3
assert S(2.).invert(5) == 0.5
assert S(sqrt(2)).invert(5) == 1/sqrt(2)
assert S(sqrt(2)).invert(sqrt(3)) == 1/sqrt(2)
def test_mod_inverse():
assert mod_inverse(3, 11) == 4
assert mod_inverse(5, 11) == 9
assert mod_inverse(21124921, 521512) == 7713
assert mod_inverse(124215421, 5125) == 2981
assert mod_inverse(214, 12515) == 1579
assert mod_inverse(5823991, 3299) == 1442
assert mod_inverse(123, 44) == 39
assert mod_inverse(2, 5) == 3
assert mod_inverse(-2, 5) == 2
assert mod_inverse(2, -5) == -2
assert mod_inverse(-2, -5) == -3
assert mod_inverse(-3, -7) == -5
x = Symbol('x')
assert S(2).invert(x) == S.Half
raises(TypeError, lambda: mod_inverse(2, x))
raises(ValueError, lambda: mod_inverse(2, S.Half))
raises(ValueError, lambda: mod_inverse(2, cos(1)**2 + sin(1)**2))
def test_golden_ratio_rewrite_as_sqrt():
assert GoldenRatio.rewrite(sqrt) == S.Half + sqrt(5)*S.Half
def test_tribonacci_constant_rewrite_as_sqrt():
assert TribonacciConstant.rewrite(sqrt) == \
(1 + cbrt(19 - 3*sqrt(33)) + cbrt(19 + 3*sqrt(33))) / 3
def test_comparisons_with_unknown_type():
class Foo(object):
"""
Class that is unaware of Basic, and relies on both classes returning
the NotImplemented singleton for equivalence to evaluate to False.
"""
ni, nf, nr = Integer(3), Float(1.0), Rational(1, 3)
foo = Foo()
for n in ni, nf, nr, oo, -oo, zoo, nan:
assert n != foo
assert foo != n
assert not n == foo
assert not foo == n
raises(TypeError, lambda: n < foo)
raises(TypeError, lambda: foo > n)
raises(TypeError, lambda: n > foo)
raises(TypeError, lambda: foo < n)
raises(TypeError, lambda: n <= foo)
raises(TypeError, lambda: foo >= n)
raises(TypeError, lambda: n >= foo)
raises(TypeError, lambda: foo <= n)
class Bar(object):
"""
Class that considers itself equal to any instance of Number except
infinities and nans, and relies on sympy types returning the
NotImplemented singleton for symmetric equality relations.
"""
def __eq__(self, other):
if other in (oo, -oo, zoo, nan):
return False
if isinstance(other, Number):
return True
return NotImplemented
def __ne__(self, other):
return not self == other
bar = Bar()
for n in ni, nf, nr:
assert n == bar
assert bar == n
assert not n != bar
assert not bar != n
for n in oo, -oo, zoo, nan:
assert n != bar
assert bar != n
assert not n == bar
assert not bar == n
for n in ni, nf, nr, oo, -oo, zoo, nan:
raises(TypeError, lambda: n < bar)
raises(TypeError, lambda: bar > n)
raises(TypeError, lambda: n > bar)
raises(TypeError, lambda: bar < n)
raises(TypeError, lambda: n <= bar)
raises(TypeError, lambda: bar >= n)
raises(TypeError, lambda: n >= bar)
raises(TypeError, lambda: bar <= n)
def test_NumberSymbol_comparison():
rpi = Rational('905502432259640373/288230376151711744')
fpi = Float(float(pi))
assert (rpi == pi) == (pi == rpi)
assert (rpi != pi) == (pi != rpi)
assert (rpi < pi) == (pi > rpi)
assert (rpi <= pi) == (pi >= rpi)
assert (rpi > pi) == (pi < rpi)
assert (rpi >= pi) == (pi <= rpi)
assert (fpi == pi) == (pi == fpi)
assert (fpi != pi) == (pi != fpi)
assert (fpi < pi) == (pi > fpi)
assert (fpi <= pi) == (pi >= fpi)
assert (fpi > pi) == (pi < fpi)
assert (fpi >= pi) == (pi <= fpi)
def test_Integer_precision():
# Make sure Integer inputs for keyword args work
assert Float('1.0', dps=Integer(15))._prec == 53
assert Float('1.0', precision=Integer(15))._prec == 15
assert type(Float('1.0', precision=Integer(15))._prec) == int
assert sympify(srepr(Float('1.0', precision=15))) == Float('1.0', precision=15)
def test_numpy_to_float():
from sympy.utilities.pytest import skip
from sympy.external import import_module
np = import_module('numpy')
if not np:
skip('numpy not installed. Abort numpy tests.')
def check_prec_and_relerr(npval, ratval):
prec = np.finfo(npval).nmant + 1
x = Float(npval)
assert x._prec == prec
y = Float(ratval, precision=prec)
assert abs((x - y)/y) < 2**(-(prec + 1))
check_prec_and_relerr(np.float16(2.0/3), S(2)/3)
check_prec_and_relerr(np.float32(2.0/3), S(2)/3)
check_prec_and_relerr(np.float64(2.0/3), S(2)/3)
# extended precision, on some arch/compilers:
x = np.longdouble(2)/3
check_prec_and_relerr(x, S(2)/3)
y = Float(x, precision=10)
assert same_and_same_prec(y, Float(S(2)/3, precision=10))
raises(TypeError, lambda: Float(np.complex64(1+2j)))
raises(TypeError, lambda: Float(np.complex128(1+2j)))
def test_Integer_ceiling_floor():
a = Integer(4)
assert(a.floor() == a)
assert(a.ceiling() == a)
def test_ComplexInfinity():
assert((zoo).floor() == zoo)
assert((zoo).ceiling() == zoo)
assert(zoo**zoo == S.NaN)
def test_Infinity_floor_ceiling_power():
assert((oo).floor() == oo)
assert((oo).ceiling() == oo)
assert((oo)**S.NaN == S.NaN)
assert((oo)**zoo == S.NaN)
def test_One_power():
assert((S.One)**12 == S.One)
assert((S.NegativeOne)**S.NaN == S.NaN)
def test_NegativeInfinity():
assert((-oo).floor() == -oo)
assert((-oo).ceiling() == -oo)
assert((-oo)**11 == -oo)
assert((-oo)**12 == oo)
def test_issue_6133():
raises(TypeError, lambda: (-oo < None))
raises(TypeError, lambda: (S(-2) < None))
raises(TypeError, lambda: (oo < None))
raises(TypeError, lambda: (oo > None))
raises(TypeError, lambda: (S(2) < None))
|
32dd06ba4b27c2b1a0233237f36580474de2b974efaaed2afc68e37ef64a80a8
|
from sympy import (Lambda, Symbol, Function, Derivative, Subs, sqrt,
log, exp, Rational, Float, sin, cos, acos, diff, I, re, im,
E, expand, pi, O, Sum, S, polygamma, loggamma, expint,
Tuple, Dummy, Eq, Expr, symbols, nfloat, Piecewise, Indexed,
Matrix, Basic)
from sympy.utilities.pytest import XFAIL, raises
from sympy.core.basic import _aresame
from sympy.core.function import PoleError, _mexpand, arity
from sympy.core.sympify import sympify
from sympy.sets.sets import FiniteSet
from sympy.solvers.solveset import solveset
from sympy.utilities.iterables import subsets, variations
from sympy.core.cache import clear_cache
from sympy.core.compatibility import range
from sympy.tensor.array import NDimArray
from sympy.abc import t, w, x, y, z
f, g, h = symbols('f g h', cls=Function)
_xi_1, _xi_2, _xi_3 = [Dummy() for i in range(3)]
def test_f_expand_complex():
x = Symbol('x', real=True)
assert f(x).expand(complex=True) == I*im(f(x)) + re(f(x))
assert exp(x).expand(complex=True) == exp(x)
assert exp(I*x).expand(complex=True) == cos(x) + I*sin(x)
assert exp(z).expand(complex=True) == cos(im(z))*exp(re(z)) + \
I*sin(im(z))*exp(re(z))
def test_bug1():
e = sqrt(-log(w))
assert e.subs(log(w), -x) == sqrt(x)
e = sqrt(-5*log(w))
assert e.subs(log(w), -x) == sqrt(5*x)
def test_general_function():
nu = Function('nu')
e = nu(x)
edx = e.diff(x)
edy = e.diff(y)
edxdx = e.diff(x).diff(x)
edxdy = e.diff(x).diff(y)
assert e == nu(x)
assert edx != nu(x)
assert edx == diff(nu(x), x)
assert edy == 0
assert edxdx == diff(diff(nu(x), x), x)
assert edxdy == 0
def test_general_function_nullary():
nu = Function('nu')
e = nu()
edx = e.diff(x)
edxdx = e.diff(x).diff(x)
assert e == nu()
assert edx != nu()
assert edx == 0
assert edxdx == 0
def test_derivative_subs_bug():
e = diff(g(x), x)
assert e.subs(g(x), f(x)) != e
assert e.subs(g(x), f(x)) == Derivative(f(x), x)
assert e.subs(g(x), -f(x)) == Derivative(-f(x), x)
assert e.subs(x, y) == Derivative(g(y), y)
def test_derivative_subs_self_bug():
d = diff(f(x), x)
assert d.subs(d, y) == y
def test_derivative_linearity():
assert diff(-f(x), x) == -diff(f(x), x)
assert diff(8*f(x), x) == 8*diff(f(x), x)
assert diff(8*f(x), x) != 7*diff(f(x), x)
assert diff(8*f(x)*x, x) == 8*f(x) + 8*x*diff(f(x), x)
assert diff(8*f(x)*y*x, x).expand() == 8*y*f(x) + 8*y*x*diff(f(x), x)
def test_derivative_evaluate():
assert Derivative(sin(x), x) != diff(sin(x), x)
assert Derivative(sin(x), x).doit() == diff(sin(x), x)
assert Derivative(Derivative(f(x), x), x) == diff(f(x), x, x)
assert Derivative(sin(x), x, 0) == sin(x)
assert Derivative(sin(x), (x, y), (x, -y)) == sin(x)
def test_diff_symbols():
assert diff(f(x, y, z), x, y, z) == Derivative(f(x, y, z), x, y, z)
assert diff(f(x, y, z), x, x, x) == Derivative(f(x, y, z), x, x, x) == Derivative(f(x, y, z), (x, 3))
assert diff(f(x, y, z), x, 3) == Derivative(f(x, y, z), x, 3)
# issue 5028
assert [diff(-z + x/y, sym) for sym in (z, x, y)] == [-1, 1/y, -x/y**2]
assert diff(f(x, y, z), x, y, z, 2) == Derivative(f(x, y, z), x, y, z, z)
assert diff(f(x, y, z), x, y, z, 2, evaluate=False) == \
Derivative(f(x, y, z), x, y, z, z)
assert Derivative(f(x, y, z), x, y, z)._eval_derivative(z) == \
Derivative(f(x, y, z), x, y, z, z)
assert Derivative(Derivative(f(x, y, z), x), y)._eval_derivative(z) == \
Derivative(f(x, y, z), x, y, z)
raises(TypeError, lambda: cos(x).diff((x, y)).variables)
assert cos(x).diff((x, y))._wrt_variables == [x]
def test_Function():
class myfunc(Function):
@classmethod
def eval(cls): # zero args
return
assert myfunc.nargs == FiniteSet(0)
assert myfunc().nargs == FiniteSet(0)
raises(TypeError, lambda: myfunc(x).nargs)
class myfunc(Function):
@classmethod
def eval(cls, x): # one arg
return
assert myfunc.nargs == FiniteSet(1)
assert myfunc(x).nargs == FiniteSet(1)
raises(TypeError, lambda: myfunc(x, y).nargs)
class myfunc(Function):
@classmethod
def eval(cls, *x): # star args
return
assert myfunc.nargs == S.Naturals0
assert myfunc(x).nargs == S.Naturals0
def test_nargs():
f = Function('f')
assert f.nargs == S.Naturals0
assert f(1).nargs == S.Naturals0
assert Function('f', nargs=2)(1, 2).nargs == FiniteSet(2)
assert sin.nargs == FiniteSet(1)
assert sin(2).nargs == FiniteSet(1)
assert log.nargs == FiniteSet(1, 2)
assert log(2).nargs == FiniteSet(1, 2)
assert Function('f', nargs=2).nargs == FiniteSet(2)
assert Function('f', nargs=0).nargs == FiniteSet(0)
assert Function('f', nargs=(0, 1)).nargs == FiniteSet(0, 1)
assert Function('f', nargs=None).nargs == S.Naturals0
raises(ValueError, lambda: Function('f', nargs=()))
def test_arity():
f = lambda x, y: 1
assert arity(f) == 2
def f(x, y, z=None):
pass
assert arity(f) == (2, 3)
assert arity(lambda *x: x) is None
assert arity(log) == (1, 2)
def test_Lambda():
e = Lambda(x, x**2)
assert e(4) == 16
assert e(x) == x**2
assert e(y) == y**2
assert Lambda((), 42)() == 42
assert Lambda((), 42) == Lambda((), 42)
assert Lambda((), 42) != Lambda((), 43)
assert Lambda((), f(x))() == f(x)
assert Lambda((), 42).nargs == FiniteSet(0)
assert Lambda(x, x**2) == Lambda(x, x**2)
assert Lambda(x, x**2) == Lambda(y, y**2)
assert Lambda(x, x**2) != Lambda(y, y**2 + 1)
assert Lambda((x, y), x**y) == Lambda((y, x), y**x)
assert Lambda((x, y), x**y) != Lambda((x, y), y**x)
assert Lambda((x, y), x**y)(x, y) == x**y
assert Lambda((x, y), x**y)(3, 3) == 3**3
assert Lambda((x, y), x**y)(x, 3) == x**3
assert Lambda((x, y), x**y)(3, y) == 3**y
assert Lambda(x, f(x))(x) == f(x)
assert Lambda(x, x**2)(e(x)) == x**4
assert e(e(x)) == x**4
x1, x2 = (Indexed('x', i) for i in (1, 2))
assert Lambda((x1, x2), x1 + x2)(x, y) == x + y
assert Lambda((x, y), x + y).nargs == FiniteSet(2)
p = x, y, z, t
assert Lambda(p, t*(x + y + z))(*p) == t * (x + y + z)
assert Lambda(x, 2*x) + Lambda(y, 2*y) == 2*Lambda(x, 2*x)
assert Lambda(x, 2*x) not in [ Lambda(x, x) ]
raises(TypeError, lambda: Lambda(1, x))
assert Lambda(x, 1)(1) is S.One
def test_IdentityFunction():
assert Lambda(x, x) is Lambda(y, y) is S.IdentityFunction
assert Lambda(x, 2*x) is not S.IdentityFunction
assert Lambda((x, y), x) is not S.IdentityFunction
def test_Lambda_symbols():
assert Lambda(x, 2*x).free_symbols == set()
assert Lambda(x, x*y).free_symbols == {y}
assert Lambda((), 42).free_symbols == set()
assert Lambda((), x*y).free_symbols == {x,y}
def test_functionclas_symbols():
assert f.free_symbols == set()
def test_Lambda_arguments():
raises(TypeError, lambda: Lambda(x, 2*x)(x, y))
raises(TypeError, lambda: Lambda((x, y), x + y)(x))
raises(TypeError, lambda: Lambda((), 42)(x))
def test_Lambda_equality():
assert Lambda(x, 2*x) == Lambda(y, 2*y)
# although variables are casts as Dummies, the expressions
# should still compare equal
assert Lambda((x, y), 2*x) == Lambda((x, y), 2*x)
assert Lambda(x, 2*x) != Lambda((x, y), 2*x)
assert Lambda(x, 2*x) != 2*x
def test_Subs():
assert Subs(1, (), ()) is S.One
# check null subs influence on hashing
assert Subs(x, y, z) != Subs(x, y, 1)
# neutral subs works
assert Subs(x, x, 1).subs(x, y).has(y)
# self mapping var/point
assert Subs(Derivative(f(x), (x, 2)), x, x).doit() == f(x).diff(x, x)
assert Subs(x, x, 0).has(x) # it's a structural answer
assert not Subs(x, x, 0).free_symbols
assert Subs(Subs(x + y, x, 2), y, 1) == Subs(x + y, (x, y), (2, 1))
assert Subs(x, (x,), (0,)) == Subs(x, x, 0)
assert Subs(x, x, 0) == Subs(y, y, 0)
assert Subs(x, x, 0).subs(x, 1) == Subs(x, x, 0)
assert Subs(y, x, 0).subs(y, 1) == Subs(1, x, 0)
assert Subs(f(x), x, 0).doit() == f(0)
assert Subs(f(x**2), x**2, 0).doit() == f(0)
assert Subs(f(x, y, z), (x, y, z), (0, 1, 1)) != \
Subs(f(x, y, z), (x, y, z), (0, 0, 1))
assert Subs(x, y, 2).subs(x, y).doit() == 2
assert Subs(f(x, y), (x, y, z), (0, 1, 1)) != \
Subs(f(x, y) + z, (x, y, z), (0, 1, 0))
assert Subs(f(x, y), (x, y), (0, 1)).doit() == f(0, 1)
assert Subs(Subs(f(x, y), x, 0), y, 1).doit() == f(0, 1)
raises(ValueError, lambda: Subs(f(x, y), (x, y), (0, 0, 1)))
raises(ValueError, lambda: Subs(f(x, y), (x, x, y), (0, 0, 1)))
assert len(Subs(f(x, y), (x, y), (0, 1)).variables) == 2
assert Subs(f(x, y), (x, y), (0, 1)).point == Tuple(0, 1)
assert Subs(f(x), x, 0) == Subs(f(y), y, 0)
assert Subs(f(x, y), (x, y), (0, 1)) == Subs(f(x, y), (y, x), (1, 0))
assert Subs(f(x)*y, (x, y), (0, 1)) == Subs(f(y)*x, (y, x), (0, 1))
assert Subs(f(x)*y, (x, y), (1, 1)) == Subs(f(y)*x, (x, y), (1, 1))
assert Subs(f(x), x, 0).subs(x, 1).doit() == f(0)
assert Subs(f(x), x, y).subs(y, 0) == Subs(f(x), x, 0)
assert Subs(y*f(x), x, y).subs(y, 2) == Subs(2*f(x), x, 2)
assert (2 * Subs(f(x), x, 0)).subs(Subs(f(x), x, 0), y) == 2*y
assert Subs(f(x), x, 0).free_symbols == set([])
assert Subs(f(x, y), x, z).free_symbols == {y, z}
assert Subs(f(x).diff(x), x, 0).doit(), Subs(f(x).diff(x), x, 0)
assert Subs(1 + f(x).diff(x), x, 0).doit(), 1 + Subs(f(x).diff(x), x, 0)
assert Subs(y*f(x, y).diff(x), (x, y), (0, 2)).doit() == \
2*Subs(Derivative(f(x, 2), x), x, 0)
assert Subs(y**2*f(x), x, 0).diff(y) == 2*y*f(0)
e = Subs(y**2*f(x), x, y)
assert e.diff(y) == e.doit().diff(y) == y**2*Derivative(f(y), y) + 2*y*f(y)
assert Subs(f(x), x, 0) + Subs(f(x), x, 0) == 2*Subs(f(x), x, 0)
e1 = Subs(z*f(x), x, 1)
e2 = Subs(z*f(y), y, 1)
assert e1 + e2 == 2*e1
assert e1.__hash__() == e2.__hash__()
assert Subs(z*f(x + 1), x, 1) not in [ e1, e2 ]
assert Derivative(f(x), x).subs(x, g(x)) == Derivative(f(g(x)), g(x))
assert Derivative(f(x), x).subs(x, x + y) == Subs(Derivative(f(x), x),
x, x + y)
assert Subs(f(x)*cos(y) + z, (x, y), (0, pi/3)).n(2) == \
Subs(f(x)*cos(y) + z, (x, y), (0, pi/3)).evalf(2) == \
z + Rational('1/2').n(2)*f(0)
assert f(x).diff(x).subs(x, 0).subs(x, y) == f(x).diff(x).subs(x, 0)
assert (x*f(x).diff(x).subs(x, 0)).subs(x, y) == y*f(x).diff(x).subs(x, 0)
assert Subs(Derivative(g(x)**2, g(x), x), g(x), exp(x)
).doit() == 2*exp(x)
assert Subs(Derivative(g(x)**2, g(x), x), g(x), exp(x)
).doit(deep=False) == 2*Derivative(exp(x), x)
assert Derivative(f(x, g(x)), x).doit() == Derivative(
f(x, g(x)), g(x))*Derivative(g(x), x) + Subs(Derivative(
f(y, g(x)), y), y, x)
def test_doitdoit():
done = Derivative(f(x, g(x)), x, g(x)).doit()
assert done == done.doit()
@XFAIL
def test_Subs2():
# this reflects a limitation of subs(), probably won't fix
assert Subs(f(x), x**2, x).doit() == f(sqrt(x))
def test_expand_function():
assert expand(x + y) == x + y
assert expand(x + y, complex=True) == I*im(x) + I*im(y) + re(x) + re(y)
assert expand((x + y)**11, modulus=11) == x**11 + y**11
def test_function_comparable():
assert sin(x).is_comparable is False
assert cos(x).is_comparable is False
assert sin(Float('0.1')).is_comparable is True
assert cos(Float('0.1')).is_comparable is True
assert sin(E).is_comparable is True
assert cos(E).is_comparable is True
assert sin(Rational(1, 3)).is_comparable is True
assert cos(Rational(1, 3)).is_comparable is True
@XFAIL
def test_function_comparable_infinities():
assert sin(oo).is_comparable is False
assert sin(-oo).is_comparable is False
assert sin(zoo).is_comparable is False
assert sin(nan).is_comparable is False
def test_deriv1():
# These all requre derivatives evaluated at a point (issue 4719) to work.
# See issue 4624
assert f(2*x).diff(x) == 2*Subs(Derivative(f(x), x), x, 2*x)
assert (f(x)**3).diff(x) == 3*f(x)**2*f(x).diff(x)
assert (f(2*x)**3).diff(x) == 6*f(2*x)**2*Subs(
Derivative(f(x), x), x, 2*x)
assert f(2 + x).diff(x) == Subs(Derivative(f(x), x), x, x + 2)
assert f(2 + 3*x).diff(x) == 3*Subs(
Derivative(f(x), x), x, 3*x + 2)
assert f(3*sin(x)).diff(x) == 3*cos(x)*Subs(
Derivative(f(x), x), x, 3*sin(x))
# See issue 8510
assert f(x, x + z).diff(x) == (
Subs(Derivative(f(y, x + z), y), y, x) +
Subs(Derivative(f(x, y), y), y, x + z))
assert f(x, x**2).diff(x) == (
2*x*Subs(Derivative(f(x, y), y), y, x**2) +
Subs(Derivative(f(y, x**2), y), y, x))
# but Subs is not always necessary
assert f(x, g(y)).diff(g(y)) == Derivative(f(x, g(y)), g(y))
def test_deriv2():
assert (x**3).diff(x) == 3*x**2
assert (x**3).diff(x, evaluate=False) != 3*x**2
assert (x**3).diff(x, evaluate=False) == Derivative(x**3, x)
assert diff(x**3, x) == 3*x**2
assert diff(x**3, x, evaluate=False) != 3*x**2
assert diff(x**3, x, evaluate=False) == Derivative(x**3, x)
def test_func_deriv():
assert f(x).diff(x) == Derivative(f(x), x)
# issue 4534
assert f(x, y).diff(x, y) - f(x, y).diff(y, x) == 0
assert Derivative(f(x, y), x, y).args[1:] == ((x, 1), (y, 1))
assert Derivative(f(x, y), y, x).args[1:] == ((y, 1), (x, 1))
assert (Derivative(f(x, y), x, y) - Derivative(f(x, y), y, x)).doit() == 0
def test_suppressed_evaluation():
a = sin(0, evaluate=False)
assert a != 0
assert a.func is sin
assert a.args == (0,)
def test_function_evalf():
def eq(a, b, eps):
return abs(a - b) < eps
assert eq(sin(1).evalf(15), Float("0.841470984807897"), 1e-13)
assert eq(
sin(2).evalf(25), Float("0.9092974268256816953960199", 25), 1e-23)
assert eq(sin(1 + I).evalf(
15), Float("1.29845758141598") + Float("0.634963914784736")*I, 1e-13)
assert eq(exp(1 + I).evalf(15), Float(
"1.46869393991588") + Float("2.28735528717884239")*I, 1e-13)
assert eq(exp(-0.5 + 1.5*I).evalf(15), Float(
"0.0429042815937374") + Float("0.605011292285002")*I, 1e-13)
assert eq(log(pi + sqrt(2)*I).evalf(
15), Float("1.23699044022052") + Float("0.422985442737893")*I, 1e-13)
assert eq(cos(100).evalf(15), Float("0.86231887228768"), 1e-13)
def test_extensibility_eval():
class MyFunc(Function):
@classmethod
def eval(cls, *args):
return (0, 0, 0)
assert MyFunc(0) == (0, 0, 0)
def test_function_non_commutative():
x = Symbol('x', commutative=False)
assert f(x).is_commutative is False
assert sin(x).is_commutative is False
assert exp(x).is_commutative is False
assert log(x).is_commutative is False
assert f(x).is_complex is False
assert sin(x).is_complex is False
assert exp(x).is_complex is False
assert log(x).is_complex is False
def test_function_complex():
x = Symbol('x', complex=True)
assert f(x).is_commutative is True
assert sin(x).is_commutative is True
assert exp(x).is_commutative is True
assert log(x).is_commutative is True
assert f(x).is_complex is True
assert sin(x).is_complex is True
assert exp(x).is_complex is True
assert log(x).is_complex is True
def test_function__eval_nseries():
n = Symbol('n')
assert sin(x)._eval_nseries(x, 2, None) == x + O(x**2)
assert sin(x + 1)._eval_nseries(x, 2, None) == x*cos(1) + sin(1) + O(x**2)
assert sin(pi*(1 - x))._eval_nseries(x, 2, None) == pi*x + O(x**2)
assert acos(1 - x**2)._eval_nseries(x, 2, None) == sqrt(2)*sqrt(x**2) + O(x**2)
assert polygamma(n, x + 1)._eval_nseries(x, 2, None) == \
polygamma(n, 1) + polygamma(n + 1, 1)*x + O(x**2)
raises(PoleError, lambda: sin(1/x)._eval_nseries(x, 2, None))
assert acos(1 - x)._eval_nseries(x, 2, None) == sqrt(2)*sqrt(x) + O(x)
assert acos(1 + x)._eval_nseries(x, 2, None) == sqrt(2)*sqrt(-x) + O(x) # XXX: wrong, branch cuts
assert loggamma(1/x)._eval_nseries(x, 0, None) == \
log(x)/2 - log(x)/x - 1/x + O(1, x)
assert loggamma(log(1/x)).nseries(x, n=1, logx=y) == loggamma(-y)
# issue 6725:
assert expint(S(3)/2, -x)._eval_nseries(x, 5, None) == \
2 - 2*sqrt(pi)*sqrt(-x) - 2*x - x**2/3 - x**3/15 - x**4/84 + O(x**5)
assert sin(sqrt(x))._eval_nseries(x, 3, None) == \
sqrt(x) - x**(S(3)/2)/6 + x**(S(5)/2)/120 + O(x**3)
def test_doit():
n = Symbol('n', integer=True)
f = Sum(2 * n * x, (n, 1, 3))
d = Derivative(f, x)
assert d.doit() == 12
assert d.doit(deep=False) == Sum(2*n, (n, 1, 3))
def test_evalf_default():
from sympy.functions.special.gamma_functions import polygamma
assert type(sin(4.0)) == Float
assert type(re(sin(I + 1.0))) == Float
assert type(im(sin(I + 1.0))) == Float
assert type(sin(4)) == sin
assert type(polygamma(2.0, 4.0)) == Float
assert type(sin(Rational(1, 4))) == sin
def test_issue_5399():
args = [x, y, S(2), S.Half]
def ok(a):
"""Return True if the input args for diff are ok"""
if not a:
return False
if a[0].is_Symbol is False:
return False
s_at = [i for i in range(len(a)) if a[i].is_Symbol]
n_at = [i for i in range(len(a)) if not a[i].is_Symbol]
# every symbol is followed by symbol or int
# every number is followed by a symbol
return (all(a[i + 1].is_Symbol or a[i + 1].is_Integer
for i in s_at if i + 1 < len(a)) and
all(a[i + 1].is_Symbol
for i in n_at if i + 1 < len(a)))
eq = x**10*y**8
for a in subsets(args):
for v in variations(a, len(a)):
if ok(v):
noraise = eq.diff(*v)
else:
raises(ValueError, lambda: eq.diff(*v))
def test_derivative_numerically():
from random import random
z0 = random() + I*random()
assert abs(Derivative(sin(x), x).doit_numerically(z0) - cos(z0)) < 1e-15
def test_fdiff_argument_index_error():
from sympy.core.function import ArgumentIndexError
class myfunc(Function):
nargs = 1 # define since there is no eval routine
def fdiff(self, idx):
raise ArgumentIndexError
mf = myfunc(x)
assert mf.diff(x) == Derivative(mf, x)
raises(TypeError, lambda: myfunc(x, x))
def test_deriv_wrt_function():
x = f(t)
xd = diff(x, t)
xdd = diff(xd, t)
y = g(t)
yd = diff(y, t)
assert diff(x, t) == xd
assert diff(2 * x + 4, t) == 2 * xd
assert diff(2 * x + 4 + y, t) == 2 * xd + yd
assert diff(2 * x + 4 + y * x, t) == 2 * xd + x * yd + xd * y
assert diff(2 * x + 4 + y * x, x) == 2 + y
assert (diff(4 * x**2 + 3 * x + x * y, t) == 3 * xd + x * yd + xd * y +
8 * x * xd)
assert (diff(4 * x**2 + 3 * xd + x * y, t) == 3 * xdd + x * yd + xd * y +
8 * x * xd)
assert diff(4 * x**2 + 3 * xd + x * y, xd) == 3
assert diff(4 * x**2 + 3 * xd + x * y, xdd) == 0
assert diff(sin(x), t) == xd * cos(x)
assert diff(exp(x), t) == xd * exp(x)
assert diff(sqrt(x), t) == xd / (2 * sqrt(x))
def test_diff_wrt_value():
assert Expr()._diff_wrt is False
assert x._diff_wrt is True
assert f(x)._diff_wrt is True
assert Derivative(f(x), x)._diff_wrt is True
assert Derivative(x**2, x)._diff_wrt is False
def test_diff_wrt():
fx = f(x)
dfx = diff(f(x), x)
ddfx = diff(f(x), x, x)
assert diff(sin(fx) + fx**2, fx) == cos(fx) + 2*fx
assert diff(sin(dfx) + dfx**2, dfx) == cos(dfx) + 2*dfx
assert diff(sin(ddfx) + ddfx**2, ddfx) == cos(ddfx) + 2*ddfx
assert diff(fx**2, dfx) == 0
assert diff(fx**2, ddfx) == 0
assert diff(dfx**2, fx) == 0
assert diff(dfx**2, ddfx) == 0
assert diff(ddfx**2, dfx) == 0
assert diff(fx*dfx*ddfx, fx) == dfx*ddfx
assert diff(fx*dfx*ddfx, dfx) == fx*ddfx
assert diff(fx*dfx*ddfx, ddfx) == fx*dfx
assert diff(f(x), x).diff(f(x)) == 0
assert (sin(f(x)) - cos(diff(f(x), x))).diff(f(x)) == cos(f(x))
assert diff(sin(fx), fx, x) == diff(sin(fx), x, fx)
# Chain rule cases
assert f(g(x)).diff(x) == \
Derivative(g(x), x)*Derivative(f(g(x)), g(x))
assert diff(f(g(x), h(y)), x) == \
Derivative(g(x), x)*Derivative(f(g(x), h(y)), g(x))
assert diff(f(g(x), h(x)), x) == (
Subs(Derivative(f(y, h(x)), y), y, g(x))*Derivative(g(x), x) +
Subs(Derivative(f(g(x), y), y), y, h(x))*Derivative(h(x), x))
assert f(
sin(x)).diff(x) == cos(x)*Subs(Derivative(f(x), x), x, sin(x))
assert diff(f(g(x)), g(x)) == Derivative(f(g(x)), g(x))
def test_diff_wrt_func_subs():
assert f(g(x)).diff(x).subs(g, Lambda(x, 2*x)).doit() == f(2*x).diff(x)
def test_subs_in_derivative():
expr = sin(x*exp(y))
u = Function('u')
v = Function('v')
assert Derivative(expr, y).subs(expr, y) == Derivative(y, y)
assert Derivative(expr, y).subs(y, x).doit() == \
Derivative(expr, y).doit().subs(y, x)
assert Derivative(f(x, y), y).subs(y, x) == Subs(Derivative(f(x, y), y), y, x)
assert Derivative(f(x, y), y).subs(x, y) == Subs(Derivative(f(x, y), y), x, y)
assert Derivative(f(x, y), y).subs(y, g(x, y)) == Subs(Derivative(f(x, y), y), y, g(x, y)).doit()
assert Derivative(f(x, y), y).subs(x, g(x, y)) == Subs(Derivative(f(x, y), y), x, g(x, y))
assert Derivative(f(x, y), g(y)).subs(x, g(x, y)) == Derivative(f(g(x, y), y), g(y))
assert Derivative(f(u(x), h(y)), h(y)).subs(h(y), g(x, y)) == \
Subs(Derivative(f(u(x), h(y)), h(y)), h(y), g(x, y)).doit()
assert Derivative(f(x, y), y).subs(y, z) == Derivative(f(x, z), z)
assert Derivative(f(x, y), y).subs(y, g(y)) == Derivative(f(x, g(y)), g(y))
assert Derivative(f(g(x), h(y)), h(y)).subs(h(y), u(y)) == \
Derivative(f(g(x), u(y)), u(y))
assert Derivative(f(x, f(x, x)), f(x, x)).subs(
f, Lambda((x, y), x + y)) == Subs(
Derivative(z + x, z), z, 2*x)
assert Subs(Derivative(f(f(x)), x), f, cos).doit() == sin(x)*sin(cos(x))
assert Subs(Derivative(f(f(x)), f(x)), f, cos).doit() == -sin(cos(x))
# Issue 13791. No comparison (it's a long formula) but this used to raise an exception.
assert isinstance(v(x, y, u(x, y)).diff(y).diff(x).diff(y), Expr)
# This is also related to issues 13791 and 13795; issue 15190
F = Lambda((x, y), exp(2*x + 3*y))
abstract = f(x, f(x, x)).diff(x, 2)
concrete = F(x, F(x, x)).diff(x, 2)
assert (abstract.subs(f, F).doit() - concrete).simplify() == 0
# don't introduce a new symbol if not necessary
assert x in f(x).diff(x).subs(x, 0).atoms()
# case (4)
assert Derivative(f(x,f(x,y)), x, y).subs(x, g(y)
) == Subs(Derivative(f(x, f(x, y)), x, y), x, g(y))
assert Derivative(f(x, x), x).subs(x, 0
) == Subs(Derivative(f(x, x), x), x, 0)
# issue 15194
assert Derivative(f(y, g(x)), (x, z)).subs(z, x
) == Derivative(f(y, g(x)), (x, x))
df = f(x).diff(x)
assert df.subs(df, 1) is S.One
assert df.diff(df) is S.One
dxy = Derivative(f(x, y), x, y)
dyx = Derivative(f(x, y), y, x)
assert dxy.subs(Derivative(f(x, y), y, x), 1) is S.One
assert dxy.diff(dyx) is S.One
assert Derivative(f(x, y), x, 2, y, 3).subs(
dyx, g(x, y)) == Derivative(g(x, y), x, 1, y, 2)
assert Derivative(f(x, x - y), y).subs(x, x + y) == Subs(
Derivative(f(x, x - y), y), x, x + y)
def test_diff_wrt_not_allowed():
# issue 7027 included
for wrt in (
cos(x), re(x), x**2, x*y, 1 + x,
Derivative(cos(x), x), Derivative(f(f(x)), x)):
raises(ValueError, lambda: diff(f(x), wrt))
# if we don't differentiate wrt then don't raise error
assert diff(exp(x*y), x*y, 0) == exp(x*y)
def test_klein_gordon_lagrangian():
m = Symbol('m')
phi = f(x, t)
L = -(diff(phi, t)**2 - diff(phi, x)**2 - m**2*phi**2)/2
eqna = Eq(
diff(L, phi) - diff(L, diff(phi, x), x) - diff(L, diff(phi, t), t), 0)
eqnb = Eq(diff(phi, t, t) - diff(phi, x, x) + m**2*phi, 0)
assert eqna == eqnb
def test_sho_lagrangian():
m = Symbol('m')
k = Symbol('k')
x = f(t)
L = m*diff(x, t)**2/2 - k*x**2/2
eqna = Eq(diff(L, x), diff(L, diff(x, t), t))
eqnb = Eq(-k*x, m*diff(x, t, t))
assert eqna == eqnb
assert diff(L, x, t) == diff(L, t, x)
assert diff(L, diff(x, t), t) == m*diff(x, t, 2)
assert diff(L, t, diff(x, t)) == -k*x + m*diff(x, t, 2)
def test_straight_line():
F = f(x)
Fd = F.diff(x)
L = sqrt(1 + Fd**2)
assert diff(L, F) == 0
assert diff(L, Fd) == Fd/sqrt(1 + Fd**2)
def test_sort_variable():
vsort = Derivative._sort_variable_count
def vsort0(*v, **kw):
reverse = kw.get('reverse', False)
return [i[0] for i in vsort([(i, 0) for i in (
reversed(v) if reverse else v)])]
for R in range(2):
assert vsort0(y, x, reverse=R) == [x, y]
assert vsort0(f(x), x, reverse=R) == [x, f(x)]
assert vsort0(f(y), f(x), reverse=R) == [f(x), f(y)]
assert vsort0(g(x), f(y), reverse=R) == [f(y), g(x)]
assert vsort0(f(x, y), f(x), reverse=R) == [f(x), f(x, y)]
fx = f(x).diff(x)
assert vsort0(fx, y, reverse=R) == [y, fx]
fy = f(y).diff(y)
assert vsort0(fy, fx, reverse=R) == [fx, fy]
fxx = fx.diff(x)
assert vsort0(fxx, fx, reverse=R) == [fx, fxx]
assert vsort0(Basic(x), f(x), reverse=R) == [f(x), Basic(x)]
assert vsort0(Basic(y), Basic(x), reverse=R) == [Basic(x), Basic(y)]
assert vsort0(Basic(y, z), Basic(x), reverse=R) == [
Basic(x), Basic(y, z)]
assert vsort0(fx, x, reverse=R) == [
x, fx] if R else [fx, x]
assert vsort0(Basic(x), x, reverse=R) == [
x, Basic(x)] if R else [Basic(x), x]
assert vsort0(Basic(f(x)), f(x), reverse=R) == [
f(x), Basic(f(x))] if R else [Basic(f(x)), f(x)]
assert vsort0(Basic(x, z), Basic(x), reverse=R) == [
Basic(x), Basic(x, z)] if R else [Basic(x, z), Basic(x)]
assert vsort([]) == []
assert _aresame(vsort([(x, 1)]), [Tuple(x, 1)])
assert vsort([(x, y), (x, z)]) == [(x, y + z)]
assert vsort([(y, 1), (x, 1 + y)]) == [(x, 1 + y), (y, 1)]
# coverage complete; legacy tests below
assert vsort([(x, 3), (y, 2), (z, 1)]) == [(x, 3), (y, 2), (z, 1)]
assert vsort([(h(x), 1), (g(x), 1), (f(x), 1)]) == [
(f(x), 1), (g(x), 1), (h(x), 1)]
assert vsort([(z, 1), (y, 2), (x, 3), (h(x), 1), (g(x), 1),
(f(x), 1)]) == [(x, 3), (y, 2), (z, 1), (f(x), 1), (g(x), 1),
(h(x), 1)]
assert vsort([(x, 1), (f(x), 1), (y, 1), (f(y), 1)]) == [(x, 1),
(y, 1), (f(x), 1), (f(y), 1)]
assert vsort([(y, 1), (x, 2), (g(x), 1), (f(x), 1), (z, 1),
(h(x), 1), (y, 2), (x, 1)]) == [(x, 3), (y, 3), (z, 1),
(f(x), 1), (g(x), 1), (h(x), 1)]
assert vsort([(z, 1), (y, 1), (f(x), 1), (x, 1), (f(x), 1),
(g(x), 1)]) == [(x, 1), (y, 1), (z, 1), (f(x), 2), (g(x), 1)]
assert vsort([(z, 1), (y, 2), (f(x), 1), (x, 2), (f(x), 2),
(g(x), 1), (z, 2), (z, 1), (y, 1), (x, 1)]) == [(x, 3), (y, 3),
(z, 4), (f(x), 3), (g(x), 1)]
assert vsort(((y, 2), (x, 1), (y, 1), (x, 1))) == [(x, 2), (y, 3)]
assert isinstance(vsort([(x, 3), (y, 2), (z, 1)])[0], Tuple)
assert vsort([(x, 1), (f(x), 1), (x, 1)]) == [(x, 2), (f(x), 1)]
assert vsort([(y, 2), (x, 3), (z, 1)]) == [(x, 3), (y, 2), (z, 1)]
assert vsort([(h(y), 1), (g(x), 1), (f(x), 1)]) == [
(f(x), 1), (g(x), 1), (h(y), 1)]
assert vsort([(x, 1), (y, 1), (x, 1)]) == [(x, 2), (y, 1)]
assert vsort([(f(x), 1), (f(y), 1), (f(x), 1)]) == [
(f(x), 2), (f(y), 1)]
dfx = f(x).diff(x)
self = [(dfx, 1), (x, 1)]
assert vsort(self) == self
assert vsort([
(dfx, 1), (y, 1), (f(x), 1), (x, 1), (f(y), 1), (x, 1)]) == [
(y, 1), (f(x), 1), (f(y), 1), (dfx, 1), (x, 2)]
dfy = f(y).diff(y)
assert vsort([(dfy, 1), (dfx, 1)]) == [(dfx, 1), (dfy, 1)]
d2fx = dfx.diff(x)
assert vsort([(d2fx, 1), (dfx, 1)]) == [(dfx, 1), (d2fx, 1)]
def test_multiple_derivative():
# Issue #15007
assert f(x, y).diff(y, y, x, y, x
) == Derivative(f(x, y), (x, 2), (y, 3))
def test_unhandled():
class MyExpr(Expr):
def _eval_derivative(self, s):
if not s.name.startswith('xi'):
return self
else:
return None
d = Dummy()
eq = MyExpr(f(x), y, z)
assert diff(eq, x, y, f(x), z) == Derivative(eq, f(x))
assert diff(eq, f(x), x) == Derivative(eq, f(x))
assert f(x, y).diff(x,(y, z)) == Derivative(f(x, y), x, (y, z))
assert f(x, y).diff(x,(y, 0)) == Derivative(f(x, y), x)
def test_nfloat():
from sympy.core.basic import _aresame
from sympy.polys.rootoftools import rootof
x = Symbol("x")
eq = x**(S(4)/3) + 4*x**(S(1)/3)/3
assert _aresame(nfloat(eq), x**(S(4)/3) + (4.0/3)*x**(S(1)/3))
assert _aresame(nfloat(eq, exponent=True), x**(4.0/3) + (4.0/3)*x**(1.0/3))
eq = x**(S(4)/3) + 4*x**(x/3)/3
assert _aresame(nfloat(eq), x**(S(4)/3) + (4.0/3)*x**(x/3))
big = 12345678901234567890
# specify precision to match value used in nfloat
Float_big = Float(big, 15)
assert _aresame(nfloat(big), Float_big)
assert _aresame(nfloat(big*x), Float_big*x)
assert _aresame(nfloat(x**big, exponent=True), x**Float_big)
assert nfloat({x: sqrt(2)}) == {x: nfloat(sqrt(2))}
assert nfloat({sqrt(2): x}) == {sqrt(2): x}
assert nfloat(cos(x + sqrt(2))) == cos(x + nfloat(sqrt(2)))
# issue 6342
f = S('x*lamda + lamda**3*(x/2 + 1/2) + lamda**2 + 1/4')
assert not any(a.free_symbols for a in solveset(f.subs(x, -0.139)))
# issue 6632
assert nfloat(-100000*sqrt(2500000001) + 5000000001) == \
9.99999999800000e-11
# issue 7122
eq = cos(3*x**4 + y)*rootof(x**5 + 3*x**3 + 1, 0)
assert str(nfloat(eq, exponent=False, n=1)) == '-0.7*cos(3.0*x**4 + y)'
def test_issue_7068():
from sympy.abc import a, b
f = Function('f')
y1 = Dummy('y')
y2 = Dummy('y')
func1 = f(a + y1 * b)
func2 = f(a + y2 * b)
func1_y = func1.diff(y1)
func2_y = func2.diff(y2)
assert func1_y != func2_y
z1 = Subs(f(a), a, y1)
z2 = Subs(f(a), a, y2)
assert z1 != z2
def test_issue_7231():
from sympy.abc import a
ans1 = f(x).series(x, a)
res = (f(a) + (-a + x)*Subs(Derivative(f(y), y), y, a) +
(-a + x)**2*Subs(Derivative(f(y), y, y), y, a)/2 +
(-a + x)**3*Subs(Derivative(f(y), y, y, y),
y, a)/6 +
(-a + x)**4*Subs(Derivative(f(y), y, y, y, y),
y, a)/24 +
(-a + x)**5*Subs(Derivative(f(y), y, y, y, y, y),
y, a)/120 + O((-a + x)**6, (x, a)))
assert res == ans1
ans2 = f(x).series(x, a)
assert res == ans2
def test_issue_7687():
from sympy.core.function import Function
from sympy.abc import x
f = Function('f')(x)
ff = Function('f')(x)
match_with_cache = ff.matches(f)
assert isinstance(f, type(ff))
clear_cache()
ff = Function('f')(x)
assert isinstance(f, type(ff))
assert match_with_cache == ff.matches(f)
def test_issue_7688():
from sympy.core.function import Function, UndefinedFunction
f = Function('f') # actually an UndefinedFunction
clear_cache()
class A(UndefinedFunction):
pass
a = A('f')
assert isinstance(a, type(f))
def test_mexpand():
from sympy.abc import x
assert _mexpand(None) is None
assert _mexpand(1) is S.One
assert _mexpand(x*(x + 1)**2) == (x*(x + 1)**2).expand()
def test_issue_8469():
# This should not take forever to run
N = 40
def g(w, theta):
return 1/(1+exp(w-theta))
ws = symbols(['w%i'%i for i in range(N)])
import functools
expr = functools.reduce(g,ws)
def test_issue_12996():
# foo=True imitates the sort of arguments that Derivative can get
# from Integral when it passes doit to the expression
assert Derivative(im(x), x).doit(foo=True) == Derivative(im(x), x)
def test_should_evalf():
# This should not take forever to run (see #8506)
assert isinstance(sin((1.0 + 1.0*I)**10000 + 1), sin)
def test_Derivative_as_finite_difference():
# Central 1st derivative at gridpoint
x, h = symbols('x h', real=True)
dfdx = f(x).diff(x)
assert (dfdx.as_finite_difference([x-2, x-1, x, x+1, x+2]) -
(S(1)/12*(f(x-2)-f(x+2)) + S(2)/3*(f(x+1)-f(x-1)))).simplify() == 0
# Central 1st derivative "half-way"
assert (dfdx.as_finite_difference() -
(f(x + S(1)/2)-f(x - S(1)/2))).simplify() == 0
assert (dfdx.as_finite_difference(h) -
(f(x + h/S(2))-f(x - h/S(2)))/h).simplify() == 0
assert (dfdx.as_finite_difference([x - 3*h, x-h, x+h, x + 3*h]) -
(S(9)/(8*2*h)*(f(x+h) - f(x-h)) +
S(1)/(24*2*h)*(f(x - 3*h) - f(x + 3*h)))).simplify() == 0
# One sided 1st derivative at gridpoint
assert (dfdx.as_finite_difference([0, 1, 2], 0) -
(-S(3)/2*f(0) + 2*f(1) - f(2)/2)).simplify() == 0
assert (dfdx.as_finite_difference([x, x+h], x) -
(f(x+h) - f(x))/h).simplify() == 0
assert (dfdx.as_finite_difference([x-h, x, x+h], x-h) -
(-S(3)/(2*h)*f(x-h) + 2/h*f(x) -
S(1)/(2*h)*f(x+h))).simplify() == 0
# One sided 1st derivative "half-way"
assert (dfdx.as_finite_difference([x-h, x+h, x + 3*h, x + 5*h, x + 7*h])
- 1/(2*h)*(-S(11)/(12)*f(x-h) + S(17)/(24)*f(x+h)
+ S(3)/8*f(x + 3*h) - S(5)/24*f(x + 5*h)
+ S(1)/24*f(x + 7*h))).simplify() == 0
d2fdx2 = f(x).diff(x, 2)
# Central 2nd derivative at gridpoint
assert (d2fdx2.as_finite_difference([x-h, x, x+h]) -
h**-2 * (f(x-h) + f(x+h) - 2*f(x))).simplify() == 0
assert (d2fdx2.as_finite_difference([x - 2*h, x-h, x, x+h, x + 2*h]) -
h**-2 * (-S(1)/12*(f(x - 2*h) + f(x + 2*h)) +
S(4)/3*(f(x+h) + f(x-h)) - S(5)/2*f(x))).simplify() == 0
# Central 2nd derivative "half-way"
assert (d2fdx2.as_finite_difference([x - 3*h, x-h, x+h, x + 3*h]) -
(2*h)**-2 * (S(1)/2*(f(x - 3*h) + f(x + 3*h)) -
S(1)/2*(f(x+h) + f(x-h)))).simplify() == 0
# One sided 2nd derivative at gridpoint
assert (d2fdx2.as_finite_difference([x, x+h, x + 2*h, x + 3*h]) -
h**-2 * (2*f(x) - 5*f(x+h) +
4*f(x+2*h) - f(x+3*h))).simplify() == 0
# One sided 2nd derivative at "half-way"
assert (d2fdx2.as_finite_difference([x-h, x+h, x + 3*h, x + 5*h]) -
(2*h)**-2 * (S(3)/2*f(x-h) - S(7)/2*f(x+h) + S(5)/2*f(x + 3*h) -
S(1)/2*f(x + 5*h))).simplify() == 0
d3fdx3 = f(x).diff(x, 3)
# Central 3rd derivative at gridpoint
assert (d3fdx3.as_finite_difference() -
(-f(x - 3/S(2)) + 3*f(x - 1/S(2)) -
3*f(x + 1/S(2)) + f(x + 3/S(2)))).simplify() == 0
assert (d3fdx3.as_finite_difference(
[x - 3*h, x - 2*h, x-h, x, x+h, x + 2*h, x + 3*h]) -
h**-3 * (S(1)/8*(f(x - 3*h) - f(x + 3*h)) - f(x - 2*h) +
f(x + 2*h) + S(13)/8*(f(x-h) - f(x+h)))).simplify() == 0
# Central 3rd derivative at "half-way"
assert (d3fdx3.as_finite_difference([x - 3*h, x-h, x+h, x + 3*h]) -
(2*h)**-3 * (f(x + 3*h)-f(x - 3*h) +
3*(f(x-h)-f(x+h)))).simplify() == 0
# One sided 3rd derivative at gridpoint
assert (d3fdx3.as_finite_difference([x, x+h, x + 2*h, x + 3*h]) -
h**-3 * (f(x + 3*h)-f(x) + 3*(f(x+h)-f(x + 2*h)))).simplify() == 0
# One sided 3rd derivative at "half-way"
assert (d3fdx3.as_finite_difference([x-h, x+h, x + 3*h, x + 5*h]) -
(2*h)**-3 * (f(x + 5*h)-f(x-h) +
3*(f(x+h)-f(x + 3*h)))).simplify() == 0
# issue 11007
y = Symbol('y', real=True)
d2fdxdy = f(x, y).diff(x, y)
ref0 = Derivative(f(x + S(1)/2, y), y) - Derivative(f(x - S(1)/2, y), y)
assert (d2fdxdy.as_finite_difference(wrt=x) - ref0).simplify() == 0
half = S(1)/2
xm, xp, ym, yp = x-half, x+half, y-half, y+half
ref2 = f(xm, ym) + f(xp, yp) - f(xp, ym) - f(xm, yp)
assert (d2fdxdy.as_finite_difference() - ref2).simplify() == 0
def test_issue_11159():
# Tests Application._eval_subs
expr1 = E
expr0 = expr1 * expr1
expr1 = expr0.subs(expr1,expr0)
assert expr0 == expr1
def test_issue_12005():
e1 = Subs(Derivative(f(x), x), x, x)
assert e1.diff(x) == Derivative(f(x), x, x)
e2 = Subs(Derivative(f(x), x), x, x**2 + 1)
assert e2.diff(x) == 2*x*Subs(Derivative(f(x), x, x), x, x**2 + 1)
e3 = Subs(Derivative(f(x) + y**2 - y, y), y, y**2)
assert e3.diff(y) == 4*y
e4 = Subs(Derivative(f(x + y), y), y, (x**2))
assert e4.diff(y) == S.Zero
e5 = Subs(Derivative(f(x), x), (y, z), (y, z))
assert e5.diff(x) == Derivative(f(x), x, x)
assert f(g(x)).diff(g(x), g(x)) == Derivative(f(g(x)), g(x), g(x))
def test_issue_13843():
x = symbols('x')
f = Function('f')
m, n = symbols('m n', integer=True)
assert Derivative(Derivative(f(x), (x, m)), (x, n)) == Derivative(f(x), (x, m + n))
assert Derivative(Derivative(f(x), (x, m+5)), (x, n+3)) == Derivative(f(x), (x, m + n + 8))
assert Derivative(f(x), (x, n)).doit() == Derivative(f(x), (x, n))
def test_order_could_be_zero():
x, y = symbols('x, y')
n = symbols('n', integer=True, nonnegative=True)
m = symbols('m', integer=True, positive=True)
assert diff(y, (x, n)) == Piecewise((y, Eq(n, 0)), (0, True))
assert diff(y, (x, n + 1)) == S.Zero
assert diff(y, (x, m)) == S.Zero
def test_undefined_function_eq():
f = Function('f')
f2 = Function('f')
g = Function('g')
f_real = Function('f', is_real=True)
# This test may only be meaningful if the cache is turned off
assert f == f2
assert hash(f) == hash(f2)
assert f == f
assert f != g
assert f != f_real
def test_function_assumptions():
x = Symbol('x')
f = Function('f')
f_real = Function('f', real=True)
assert f != f_real
assert f(x) != f_real(x)
assert f(x).is_real is None
assert f_real(x).is_real is True
# Can also do it this way, but it won't be equal to f_real because of the
# way UndefinedFunction.__new__ works.
f_real2 = Function('f', is_real=True)
assert f_real2(x).is_real is True
def test_undef_fcn_float_issue_6938():
f = Function('ceil')
assert not f(0.3).is_number
f = Function('sin')
assert not f(0.3).is_number
assert not f(pi).evalf().is_number
x = Symbol('x')
assert not f(x).evalf(subs={x:1.2}).is_number
def test_undefined_function_eval():
# Issue 15170. Make sure UndefinedFunction with eval defined works
# properly. The issue there was that the hash was determined before _nargs
# was set, which is included in the hash, hence changing the hash. The
# class is added to sympy.core.core.all_classes before the hash is
# changed, meaning "temp in all_classes" would fail, causing sympify(temp(t))
# to give a new class. We will eventually remove all_classes, but make
# sure this continues to work.
fdiff = lambda self, argindex=1: cos(self.args[argindex - 1])
eval = classmethod(lambda cls, t: None)
_imp_ = classmethod(lambda cls, t: sin(t))
temp = Function('temp', fdiff=fdiff, eval=eval, _imp_=_imp_)
expr = temp(t)
assert sympify(expr) == expr
assert type(sympify(expr)).fdiff.__name__ == "<lambda>"
assert expr.diff(t) == cos(t)
def test_issue_15241():
F = f(x)
Fx = F.diff(x)
assert (F + x*Fx).diff(x, Fx) == 2
assert (F + x*Fx).diff(Fx, x) == 1
assert (x*F + x*Fx*F).diff(F, x) == x*Fx.diff(x) + Fx + 1
assert (x*F + x*Fx*F).diff(x, F) == x*Fx.diff(x) + Fx + 1
y = f(x)
G = f(y)
Gy = G.diff(y)
assert (G + y*Gy).diff(y, Gy) == 2
assert (G + y*Gy).diff(Gy, y) == 1
assert (y*G + y*Gy*G).diff(G, y) == y*Gy.diff(y) + Gy + 1
assert (y*G + y*Gy*G).diff(y, G) == y*Gy.diff(y) + Gy + 1
def test_issue_15226():
assert Subs(Derivative(f(y), x, y), y, g(x)).doit() != 0
def test_issue_7027():
for wrt in (cos(x), re(x), Derivative(cos(x), x)):
raises(ValueError, lambda: diff(f(x), wrt))
def test_derivative_quick_exit():
assert f(x).diff(y) == 0
assert f(x).diff(y, f(x)) == 0
assert f(x).diff(x, f(y)) == 0
assert f(f(x)).diff(x, f(x), f(y)) == 0
assert f(f(x)).diff(x, f(x), y) == 0
assert f(x).diff(g(x)) == 0
assert f(x).diff(x, f(x).diff(x)) == 1
df = f(x).diff(x)
assert f(x).diff(df) == 0
dg = g(x).diff(x)
assert dg.diff(df).doit() == 0
def test_issue_15084_13166():
eq = f(x, g(x))
assert eq.diff((g(x), y)) == Derivative(f(x, g(x)), (g(x), y))
# issue 13166
assert eq.diff(x, 2).doit() == (
(Derivative(f(x, g(x)), (g(x), 2))*Derivative(g(x), x) +
Subs(Derivative(f(x, _xi_2), _xi_2, x), _xi_2, g(x)))*Derivative(g(x),
x) + Derivative(f(x, g(x)), g(x))*Derivative(g(x), (x, 2)) +
Derivative(g(x), x)*Subs(Derivative(f(_xi_1, g(x)), _xi_1, g(x)),
_xi_1, x) + Subs(Derivative(f(_xi_1, g(x)), (_xi_1, 2)), _xi_1, x))
# issue 6681
assert diff(f(x, t, g(x, t)), x).doit() == (
Derivative(f(x, t, g(x, t)), g(x, t))*Derivative(g(x, t), x) +
Subs(Derivative(f(_xi_1, t, g(x, t)), _xi_1), _xi_1, x))
# make sure the order doesn't matter when using diff
assert eq.diff(x, g(x)) == eq.diff(g(x), x)
def test_negative_counts():
# issue 13873
raises(ValueError, lambda: sin(x).diff(x, -1))
def test_Derivative__new__():
raises(TypeError, lambda: f(x).diff((x, 2), 0))
assert f(x, y).diff([(x, y), 0]) == f(x, y)
assert f(x, y).diff([(x, y), 1]) == NDimArray([
Derivative(f(x, y), x), Derivative(f(x, y), y)])
assert f(x,y).diff(y, (x, z), y, x) == Derivative(
f(x, y), (x, z + 1), (y, 2))
assert Matrix([x]).diff(x, 2) == Matrix([0]) # is_zero exit
def test_issue_14719_10150():
class V(Expr):
_diff_wrt = True
is_scalar = False
assert V().diff(V()) == Derivative(V(), V())
assert (2*V()).diff(V()) == 2*Derivative(V(), V())
class X(Expr):
_diff_wrt = True
assert X().diff(X()) == 1
assert (2*X()).diff(X()) == 2
def test_noncommutative_issue_15131():
x = Symbol('x', commutative=False)
t = Symbol('t', commutative=False)
fx = Function('Fx', commutative=False)(x)
ft = Function('Ft', commutative=False)(t)
A = Symbol('A', commutative=False)
eq = fx * A * ft
eqdt = eq.diff(t)
assert eqdt.args[-1] == ft.diff(t)
def test_Subs_Derivative():
a = Derivative(f(g(x), h(x)), g(x), h(x),x)
b = Derivative(Derivative(f(g(x), h(x)), g(x), h(x)),x)
c = f(g(x), h(x)).diff(g(x), h(x), x)
d = f(g(x), h(x)).diff(g(x), h(x)).diff(x)
e = Derivative(f(g(x), h(x)), x)
eqs = (a, b, c, d, e)
subs = lambda arg: arg.subs(f, Lambda((x, y), exp(x + y))
).subs(g(x), 1/x).subs(h(x), x**3)
ans = 3*x**2*exp(1/x)*exp(x**3) - exp(1/x)*exp(x**3)/x**2
assert all(subs(i).doit().expand() == ans for i in eqs)
assert all(subs(i.doit()).doit().expand() == ans for i in eqs)
def test_issue_15360():
f = Function('f')
assert f.name == 'f'
|
a276e52310021055bb8a3e59a5252ea33035fb108e772a20d6c078f7e7819304
|
"""This tests sympy/core/basic.py with (ideally) no reference to subclasses
of Basic or Atom."""
import collections
import sys
from sympy.core.basic import (Basic, Atom, preorder_traversal, as_Basic,
_atomic)
from sympy.core.singleton import S
from sympy.core.symbol import symbols, Symbol
from sympy.core.function import Function, Lambda
from sympy.core.compatibility import default_sort_key
from sympy import sin, Q, cos, gamma, Tuple, Integral, Sum
from sympy.functions.elementary.exponential import exp
from sympy.utilities.pytest import raises
from sympy.core import I, pi
b1 = Basic()
b2 = Basic(b1)
b3 = Basic(b2)
b21 = Basic(b2, b1)
def test_structure():
assert b21.args == (b2, b1)
assert b21.func(*b21.args) == b21
assert bool(b1)
def test_equality():
instances = [b1, b2, b3, b21, Basic(b1, b1, b1), Basic]
for i, b_i in enumerate(instances):
for j, b_j in enumerate(instances):
assert (b_i == b_j) == (i == j)
assert (b_i != b_j) == (i != j)
assert Basic() != []
assert not(Basic() == [])
assert Basic() != 0
assert not(Basic() == 0)
class Foo(object):
"""
Class that is unaware of Basic, and relies on both classes returning
the NotImplemented singleton for equivalence to evaluate to False.
"""
b = Basic()
foo = Foo()
assert b != foo
assert foo != b
assert not b == foo
assert not foo == b
class Bar(object):
"""
Class that considers itself equal to any instance of Basic, and relies
on Basic returning the NotImplemented singleton in order to achieve
a symmetric equivalence relation.
"""
def __eq__(self, other):
if isinstance(other, Basic):
return True
return NotImplemented
def __ne__(self, other):
return not self == other
bar = Bar()
assert b == bar
assert bar == b
assert not b != bar
assert not bar != b
def test_matches_basic():
instances = [Basic(b1, b1, b2), Basic(b1, b2, b1), Basic(b2, b1, b1),
Basic(b1, b2), Basic(b2, b1), b2, b1]
for i, b_i in enumerate(instances):
for j, b_j in enumerate(instances):
if i == j:
assert b_i.matches(b_j) == {}
else:
assert b_i.matches(b_j) is None
assert b1.match(b1) == {}
def test_has():
assert b21.has(b1)
assert b21.has(b3, b1)
assert b21.has(Basic)
assert not b1.has(b21, b3)
assert not b21.has()
def test_subs():
assert b21.subs(b2, b1) == Basic(b1, b1)
assert b21.subs(b2, b21) == Basic(b21, b1)
assert b3.subs(b2, b1) == b2
assert b21.subs([(b2, b1), (b1, b2)]) == Basic(b2, b2)
assert b21.subs({b1: b2, b2: b1}) == Basic(b2, b2)
if sys.version_info >= (3, 4):
assert b21.subs(collections.ChainMap({b1: b2}, {b2: b1})) == Basic(b2, b2)
assert b21.subs(collections.OrderedDict([(b2, b1), (b1, b2)])) == Basic(b2, b2)
raises(ValueError, lambda: b21.subs('bad arg'))
raises(ValueError, lambda: b21.subs(b1, b2, b3))
# dict(b1=foo) creates a string 'b1' but leaves foo unchanged; subs
# will convert the first to a symbol but will raise an error if foo
# cannot be sympified; sympification is strict if foo is not string
raises(ValueError, lambda: b21.subs(b1='bad arg'))
assert Symbol(u"text").subs({u"text": b1}) == b1
assert Symbol(u"s").subs({u"s": 1}) == 1
def test_subs_with_unicode_symbols():
expr = Symbol('var1')
replaced = expr.subs('var1', u'x')
assert replaced.name == 'x'
replaced = expr.subs('var1', 'x')
assert replaced.name == 'x'
def test_atoms():
assert b21.atoms() == set()
def test_free_symbols_empty():
assert b21.free_symbols == set()
def test_doit():
assert b21.doit() == b21
assert b21.doit(deep=False) == b21
def test_S():
assert repr(S) == 'S'
def test_xreplace():
assert b21.xreplace({b2: b1}) == Basic(b1, b1)
assert b21.xreplace({b2: b21}) == Basic(b21, b1)
assert b3.xreplace({b2: b1}) == b2
assert Basic(b1, b2).xreplace({b1: b2, b2: b1}) == Basic(b2, b1)
assert Atom(b1).xreplace({b1: b2}) == Atom(b1)
assert Atom(b1).xreplace({Atom(b1): b2}) == b2
raises(TypeError, lambda: b1.xreplace())
raises(TypeError, lambda: b1.xreplace([b1, b2]))
for f in (exp, Function('f')):
assert f.xreplace({}) == f
assert f.xreplace({}, hack2=True) == f
assert f.xreplace({f: b1}) == b1
assert f.xreplace({f: b1}, hack2=True) == b1
def test_preorder_traversal():
expr = Basic(b21, b3)
assert list(
preorder_traversal(expr)) == [expr, b21, b2, b1, b1, b3, b2, b1]
assert list(preorder_traversal(('abc', ('d', 'ef')))) == [
('abc', ('d', 'ef')), 'abc', ('d', 'ef'), 'd', 'ef']
result = []
pt = preorder_traversal(expr)
for i in pt:
result.append(i)
if i == b2:
pt.skip()
assert result == [expr, b21, b2, b1, b3, b2]
w, x, y, z = symbols('w:z')
expr = z + w*(x + y)
assert list(preorder_traversal([expr], keys=default_sort_key)) == \
[[w*(x + y) + z], w*(x + y) + z, z, w*(x + y), w, x + y, x, y]
assert list(preorder_traversal((x + y)*z, keys=True)) == \
[z*(x + y), z, x + y, x, y]
def test_sorted_args():
x = symbols('x')
assert b21._sorted_args == b21.args
raises(AttributeError, lambda: x._sorted_args)
def test_call():
x, y = symbols('x y')
# See the long history of this in issues 5026 and 5105.
raises(TypeError, lambda: sin(x)({ x : 1, sin(x) : 2}))
raises(TypeError, lambda: sin(x)(1))
# No effect as there are no callables
assert sin(x).rcall(1) == sin(x)
assert (1 + sin(x)).rcall(1) == 1 + sin(x)
# Effect in the pressence of callables
l = Lambda(x, 2*x)
assert (l + x).rcall(y) == 2*y + x
assert (x**l).rcall(2) == x**4
# TODO UndefinedFunction does not subclass Expr
#f = Function('f')
#assert (2*f)(x) == 2*f(x)
assert (Q.real & Q.positive).rcall(x) == Q.real(x) & Q.positive(x)
def test_rewrite():
x, y, z = symbols('x y z')
a, b = symbols('a b')
f1 = sin(x) + cos(x)
assert f1.rewrite(cos,exp) == exp(I*x)/2 + sin(x) + exp(-I*x)/2
assert f1.rewrite([cos],sin) == sin(x) + sin(x + pi/2, evaluate=False)
f2 = sin(x) + cos(y)/gamma(z)
assert f2.rewrite(sin,exp) == -I*(exp(I*x) - exp(-I*x))/2 + cos(y)/gamma(z)
assert f1.rewrite() == f1
def test_literal_evalf_is_number_is_zero_is_comparable():
from sympy.integrals.integrals import Integral
from sympy.core.symbol import symbols
from sympy.core.function import Function
from sympy.functions.elementary.trigonometric import cos, sin
x = symbols('x')
f = Function('f')
# issue 5033
assert f.is_number is False
# issue 6646
assert f(1).is_number is False
i = Integral(0, (x, x, x))
# expressions that are symbolically 0 can be difficult to prove
# so in case there is some easy way to know if something is 0
# it should appear in the is_zero property for that object;
# if is_zero is true evalf should always be able to compute that
# zero
assert i.n() == 0
assert i.is_zero
assert i.is_number is False
assert i.evalf(2, strict=False) == 0
# issue 10268
n = sin(1)**2 + cos(1)**2 - 1
assert n.is_comparable is False
assert n.n(2).is_comparable is False
assert n.n(2).n(2).is_comparable
def test_as_Basic():
assert as_Basic(1) is S.One
assert as_Basic(()) == Tuple()
raises(TypeError, lambda: as_Basic([]))
def test_atomic():
g, h = map(Function, 'gh')
x = symbols('x')
assert _atomic(g(x + h(x))) == {g(x + h(x))}
assert _atomic(g(x + h(x)), recursive=True) == {h(x), x, g(x + h(x))}
assert _atomic(1) == set()
assert _atomic(Basic(1,2)) == {Basic(1, 2)}
def test_as_dummy():
u, v, x, y, z, _0, _1 = symbols('u v x y z _0 _1')
assert Lambda(x, x + 1).as_dummy() == Lambda(_0, _0 + 1)
assert Lambda(x, x + _0).as_dummy() == Lambda(_1, _0 + _1)
assert (1 + Sum(x, (x, 1, x))).as_dummy() == 1 + Sum(_0, (_0, 1, x))
def test_canonical_variables():
x, i0, i1 = symbols('x _:2')
assert Integral(x, (x, x + 1)).canonical_variables == {x: i0}
assert Integral(x, (x, x + i0)).canonical_variables == {x: i1}
def test_replace_exceptions():
from sympy import Wild
x, y = symbols('x y')
e = (x**2 + x*y)
raises(TypeError, lambda: e.replace(sin, 2))
b = Wild('b')
c = Wild('c')
raises(TypeError, lambda: e.replace(b*c, c.is_real))
raises(TypeError, lambda: e.replace(b.is_real, 1))
raises(TypeError, lambda: e.replace(lambda d: d.is_Number, 1))
|
fcd3e2fe70c30c99a79d00af58ab1507142a10624087cfdae84edb5e2f6af9bf
|
from sympy.concrete.summations import Sum
from sympy.core.expr import Expr
from sympy.core.function import (Derivative, Function, diff, Subs)
from sympy.core.numbers import (I, Rational, pi)
from sympy.core.relational import Eq
from sympy.core.singleton import S
from sympy.core.symbol import Symbol
from sympy.functions.combinatorial.factorials import (FallingFactorial,
factorial)
from sympy.functions.elementary.complexes import (im, re)
from sympy.functions.elementary.exponential import (exp, log)
from sympy.functions.elementary.miscellaneous import Max
from sympy.functions.elementary.piecewise import Piecewise
from sympy.functions.elementary.trigonometric import (cos, cot, sin, tan)
from sympy.logic.boolalg import And
from sympy.tensor.array.ndim_array import NDimArray
from sympy.utilities.pytest import raises
from sympy.abc import a, b, c, x, y, z
def test_diff():
assert Rational(1, 3).diff(x) is S.Zero
assert I.diff(x) is S.Zero
assert pi.diff(x) is S.Zero
assert x.diff(x, 0) == x
assert (x**2).diff(x, 2, x) == 0
assert (x**2).diff((x, 2), x) == 0
assert (x**2).diff((x, 1), x) == 2
assert (x**2).diff((x, 1), (x, 1)) == 2
assert (x**2).diff((x, 2)) == 2
assert (x**2).diff(x, y, 0) == 2*x
assert (x**2).diff(x, (y, 0)) == 2*x
assert (x**2).diff(x, y) == 0
raises(ValueError, lambda: x.diff(1, x))
p = Rational(5)
e = a*b + b**p
assert e.diff(a) == b
assert e.diff(b) == a + 5*b**4
assert e.diff(b).diff(a) == Rational(1)
e = a*(b + c)
assert e.diff(a) == b + c
assert e.diff(b) == a
assert e.diff(b).diff(a) == Rational(1)
e = c**p
assert e.diff(c, 6) == Rational(0)
assert e.diff(c, 5) == Rational(120)
e = c**Rational(2)
assert e.diff(c) == 2*c
e = a*b*c
assert e.diff(c) == a*b
def test_diff2():
n3 = Rational(3)
n2 = Rational(2)
n6 = Rational(6)
e = n3*(-n2 + x**n2)*cos(x) + x*(-n6 + x**n2)*sin(x)
assert e == 3*(-2 + x**2)*cos(x) + x*(-6 + x**2)*sin(x)
assert e.diff(x).expand() == x**3*cos(x)
e = (x + 1)**3
assert e.diff(x) == 3*(x + 1)**2
e = x*(x + 1)**3
assert e.diff(x) == (x + 1)**3 + 3*x*(x + 1)**2
e = 2*exp(x*x)*x
assert e.diff(x) == 2*exp(x**2) + 4*x**2*exp(x**2)
def test_diff3():
p = Rational(5)
e = a*b + sin(b**p)
assert e == a*b + sin(b**5)
assert e.diff(a) == b
assert e.diff(b) == a + 5*b**4*cos(b**5)
e = tan(c)
assert e == tan(c)
assert e.diff(c) in [cos(c)**(-2), 1 + sin(c)**2/cos(c)**2, 1 + tan(c)**2]
e = c*log(c) - c
assert e == -c + c*log(c)
assert e.diff(c) == log(c)
e = log(sin(c))
assert e == log(sin(c))
assert e.diff(c) in [sin(c)**(-1)*cos(c), cot(c)]
e = (Rational(2)**a/log(Rational(2)))
assert e == 2**a*log(Rational(2))**(-1)
assert e.diff(a) == 2**a
def test_diff_no_eval_derivative():
class My(Expr):
def __new__(cls, x):
return Expr.__new__(cls, x)
# My doesn't have its own _eval_derivative method
assert My(x).diff(x).func is Derivative
assert My(x).diff(x, 3).func is Derivative
assert re(x).diff(x, 2) == Derivative(re(x), (x, 2)) # issue 15518
assert diff(NDimArray([re(x), im(x)]), (x, 2)) == NDimArray(
[Derivative(re(x), (x, 2)), Derivative(im(x), (x, 2))])
# it doesn't have y so it shouldn't need a method for this case
assert My(x).diff(y) == 0
def test_speed():
# this should return in 0.0s. If it takes forever, it's wrong.
assert x.diff(x, 10**8) == 0
def test_deriv_noncommutative():
A = Symbol("A", commutative=False)
f = Function("f")
assert A*f(x)*A == f(x)*A**2
assert A*f(x).diff(x)*A == f(x).diff(x) * A**2
def test_diff_nth_derivative():
f = Function("f")
n = Symbol("n", integer=True)
expr = diff(sin(x), (x, n))
expr2 = diff(f(x), (x, 2))
expr3 = diff(f(x), (x, n))
assert expr.subs(sin(x), cos(-x)) == Derivative(cos(-x), (x, n))
assert expr.subs(n, 1).doit() == cos(x)
assert expr.subs(n, 2).doit() == -sin(x)
assert expr2.subs(Derivative(f(x), x), y) == Derivative(y, x)
# Currently not supported (cannot determine if `n > 1`):
#assert expr3.subs(Derivative(f(x), x), y) == Derivative(y, (x, n-1))
assert expr3 == Derivative(f(x), (x, n))
assert diff(x, (x, n)) == Piecewise((x, Eq(n, 0)), (1, Eq(n, 1)), (0, True))
assert diff(2*x, (x, n)).dummy_eq(
Sum(Piecewise((2*x*factorial(n)/(factorial(y)*factorial(-y + n)),
Eq(y, 0) & Eq(Max(0, -y + n), 0)),
(2*factorial(n)/(factorial(y)*factorial(-y + n)), Eq(y, 0) & Eq(Max(0,
-y + n), 1)), (0, True)), (y, 0, n)))
# TODO: assert diff(x**2, (x, n)) == x**(2-n)*ff(2, n)
exprm = x*sin(x)
mul_diff = diff(exprm, (x, n))
assert isinstance(mul_diff, Sum)
for i in range(5):
assert mul_diff.subs(n, i).doit() == exprm.diff((x, i)).expand()
exprm2 = 2*y*x*sin(x)*cos(x)*log(x)*exp(x)
dex = exprm2.diff((x, n))
assert isinstance(dex, Sum)
for i in range(7):
assert dex.subs(n, i).doit().expand() == \
exprm2.diff((x, i)).expand()
assert (cos(x)*sin(y)).diff([[x, y, z]]) == NDimArray([
-sin(x)*sin(y), cos(x)*cos(y), 0])
def test_issue_16160():
assert Derivative(x**3, (x, x)).subs(x, 2) == Subs(
Derivative(x**3, (x, 2)), x, 2)
assert Derivative(1 + x**3, (x, x)).subs(x, 0
) == Derivative(1 + y**3, (y, 0)).subs(y, 0)
|
b1d641378f7489f4dcd23a2cdfbaa28caca88727363a78fd5193e2a95d69f262
|
from sympy import Symbol, Mul, symbols, Basic
from sympy.utilities.pytest import XFAIL
class SymbolInMulOnce(Symbol):
# Test class for a symbol that can only appear once in a `Mul` expression.
pass
Basic._constructor_postprocessor_mapping[SymbolInMulOnce] = {
"Mul": [lambda x: x],
"Pow": [lambda x: x.base if isinstance(x.base, SymbolInMulOnce) else x],
"Add": [lambda x: x],
}
def _postprocess_SymbolRemovesOtherSymbols(expr):
args = tuple(i for i in expr.args if not isinstance(i, Symbol) or isinstance(i, SymbolRemovesOtherSymbols))
if args == expr.args:
return expr
return Mul.fromiter(args)
class SymbolRemovesOtherSymbols(Symbol):
# Test class for a symbol that removes other symbols in `Mul`.
pass
Basic._constructor_postprocessor_mapping[SymbolRemovesOtherSymbols] = {
"Mul": [_postprocess_SymbolRemovesOtherSymbols],
}
class SubclassSymbolInMulOnce(SymbolInMulOnce):
pass
class SubclassSymbolRemovesOtherSymbols(SymbolRemovesOtherSymbols):
pass
def test_constructor_postprocessors1():
x = SymbolInMulOnce("x")
y = SymbolInMulOnce("y")
assert isinstance(3*x, Mul)
assert (3*x).args == (3, x)
assert x*x == x
assert 3*x*x == 3*x
assert 2*x*x + x == 3*x
assert x**3*y*y == x*y
assert x**5 + y*x**3 == x + x*y
w = SymbolRemovesOtherSymbols("w")
assert x*w == w
assert (3*w).args == (3, w)
assert set((w + x).args) == set((x, w))
def test_constructor_postprocessors2():
x = SubclassSymbolInMulOnce("x")
y = SubclassSymbolInMulOnce("y")
assert isinstance(3*x, Mul)
assert (3*x).args == (3, x)
assert x*x == x
assert 3*x*x == 3*x
assert 2*x*x + x == 3*x
assert x**3*y*y == x*y
assert x**5 + y*x**3 == x + x*y
w = SubclassSymbolRemovesOtherSymbols("w")
assert x*w == w
assert (3*w).args == (3, w)
assert set((w + x).args) == set((x, w))
@XFAIL
def test_subexpression_postprocessors():
# The postprocessors used to work with subexpressions, but the
# functionality was removed. See #15948.
a = symbols("a")
x = SymbolInMulOnce("x")
w = SymbolRemovesOtherSymbols("w")
assert 3*a*w**2 == 3*w**2
assert 3*a*x**3*w**2 == 3*w**2
x = SubclassSymbolInMulOnce("x")
w = SubclassSymbolRemovesOtherSymbols("w")
assert 3*a*w**2 == 3*w**2
assert 3*a*x**3*w**2 == 3*w**2
|
273f9bb2509cdd953f8ab4c47e3caf84944534e5c841752209b5dc4514464e33
|
from sympy import I, sqrt, log, exp, sin, asin, factorial, Mod, pi
from sympy.core import Symbol, S, Rational, Integer, Dummy, Wild, Pow
from sympy.core.facts import InconsistentAssumptions
from sympy import simplify
from sympy.core.compatibility import range
from sympy.utilities.pytest import raises, XFAIL
def test_symbol_unset():
x = Symbol('x', real=True, integer=True)
assert x.is_real is True
assert x.is_integer is True
assert x.is_imaginary is False
assert x.is_noninteger is False
assert x.is_number is False
def test_zero():
z = Integer(0)
assert z.is_commutative is True
assert z.is_integer is True
assert z.is_rational is True
assert z.is_algebraic is True
assert z.is_transcendental is False
assert z.is_real is True
assert z.is_complex is True
assert z.is_noninteger is False
assert z.is_irrational is False
assert z.is_imaginary is False
assert z.is_positive is False
assert z.is_negative is False
assert z.is_nonpositive is True
assert z.is_nonnegative is True
assert z.is_even is True
assert z.is_odd is False
assert z.is_finite is True
assert z.is_infinite is False
assert z.is_comparable is True
assert z.is_prime is False
assert z.is_composite is False
assert z.is_number is True
def test_one():
z = Integer(1)
assert z.is_commutative is True
assert z.is_integer is True
assert z.is_rational is True
assert z.is_algebraic is True
assert z.is_transcendental is False
assert z.is_real is True
assert z.is_complex is True
assert z.is_noninteger is False
assert z.is_irrational is False
assert z.is_imaginary is False
assert z.is_positive is True
assert z.is_negative is False
assert z.is_nonpositive is False
assert z.is_nonnegative is True
assert z.is_even is False
assert z.is_odd is True
assert z.is_finite is True
assert z.is_infinite is False
assert z.is_comparable is True
assert z.is_prime is False
assert z.is_number is True
assert z.is_composite is False # issue 8807
def test_negativeone():
z = Integer(-1)
assert z.is_commutative is True
assert z.is_integer is True
assert z.is_rational is True
assert z.is_algebraic is True
assert z.is_transcendental is False
assert z.is_real is True
assert z.is_complex is True
assert z.is_noninteger is False
assert z.is_irrational is False
assert z.is_imaginary is False
assert z.is_positive is False
assert z.is_negative is True
assert z.is_nonpositive is True
assert z.is_nonnegative is False
assert z.is_even is False
assert z.is_odd is True
assert z.is_finite is True
assert z.is_infinite is False
assert z.is_comparable is True
assert z.is_prime is False
assert z.is_composite is False
assert z.is_number is True
def test_infinity():
oo = S.Infinity
assert oo.is_commutative is True
assert oo.is_integer is None
assert oo.is_rational is None
assert oo.is_algebraic is None
assert oo.is_transcendental is None
assert oo.is_real is True
assert oo.is_complex is True
assert oo.is_noninteger is None
assert oo.is_irrational is None
assert oo.is_imaginary is False
assert oo.is_positive is True
assert oo.is_negative is False
assert oo.is_nonpositive is False
assert oo.is_nonnegative is True
assert oo.is_even is None
assert oo.is_odd is None
assert oo.is_finite is False
assert oo.is_infinite is True
assert oo.is_comparable is True
assert oo.is_prime is False
assert oo.is_composite is None
assert oo.is_number is True
def test_neg_infinity():
mm = S.NegativeInfinity
assert mm.is_commutative is True
assert mm.is_integer is None
assert mm.is_rational is None
assert mm.is_algebraic is None
assert mm.is_transcendental is None
assert mm.is_real is True
assert mm.is_complex is True
assert mm.is_noninteger is None
assert mm.is_irrational is None
assert mm.is_imaginary is False
assert mm.is_positive is False
assert mm.is_negative is True
assert mm.is_nonpositive is True
assert mm.is_nonnegative is False
assert mm.is_even is None
assert mm.is_odd is None
assert mm.is_finite is False
assert mm.is_infinite is True
assert mm.is_comparable is True
assert mm.is_prime is False
assert mm.is_composite is False
assert mm.is_number is True
def test_zoo():
zoo = S.ComplexInfinity
assert zoo.is_complex
assert zoo.is_real is False
assert zoo.is_prime is False
def test_nan():
nan = S.NaN
assert nan.is_commutative is True
assert nan.is_integer is None
assert nan.is_rational is None
assert nan.is_algebraic is None
assert nan.is_transcendental is None
assert nan.is_real is None
assert nan.is_complex is None
assert nan.is_noninteger is None
assert nan.is_irrational is None
assert nan.is_imaginary is None
assert nan.is_positive is None
assert nan.is_negative is None
assert nan.is_nonpositive is None
assert nan.is_nonnegative is None
assert nan.is_even is None
assert nan.is_odd is None
assert nan.is_finite is None
assert nan.is_infinite is None
assert nan.is_comparable is False
assert nan.is_prime is None
assert nan.is_composite is None
assert nan.is_number is True
def test_pos_rational():
r = Rational(3, 4)
assert r.is_commutative is True
assert r.is_integer is False
assert r.is_rational is True
assert r.is_algebraic is True
assert r.is_transcendental is False
assert r.is_real is True
assert r.is_complex is True
assert r.is_noninteger is True
assert r.is_irrational is False
assert r.is_imaginary is False
assert r.is_positive is True
assert r.is_negative is False
assert r.is_nonpositive is False
assert r.is_nonnegative is True
assert r.is_even is False
assert r.is_odd is False
assert r.is_finite is True
assert r.is_infinite is False
assert r.is_comparable is True
assert r.is_prime is False
assert r.is_composite is False
r = Rational(1, 4)
assert r.is_nonpositive is False
assert r.is_positive is True
assert r.is_negative is False
assert r.is_nonnegative is True
r = Rational(5, 4)
assert r.is_negative is False
assert r.is_positive is True
assert r.is_nonpositive is False
assert r.is_nonnegative is True
r = Rational(5, 3)
assert r.is_nonnegative is True
assert r.is_positive is True
assert r.is_negative is False
assert r.is_nonpositive is False
def test_neg_rational():
r = Rational(-3, 4)
assert r.is_positive is False
assert r.is_nonpositive is True
assert r.is_negative is True
assert r.is_nonnegative is False
r = Rational(-1, 4)
assert r.is_nonpositive is True
assert r.is_positive is False
assert r.is_negative is True
assert r.is_nonnegative is False
r = Rational(-5, 4)
assert r.is_negative is True
assert r.is_positive is False
assert r.is_nonpositive is True
assert r.is_nonnegative is False
r = Rational(-5, 3)
assert r.is_nonnegative is False
assert r.is_positive is False
assert r.is_negative is True
assert r.is_nonpositive is True
def test_pi():
z = S.Pi
assert z.is_commutative is True
assert z.is_integer is False
assert z.is_rational is False
assert z.is_algebraic is False
assert z.is_transcendental is True
assert z.is_real is True
assert z.is_complex is True
assert z.is_noninteger is True
assert z.is_irrational is True
assert z.is_imaginary is False
assert z.is_positive is True
assert z.is_negative is False
assert z.is_nonpositive is False
assert z.is_nonnegative is True
assert z.is_even is False
assert z.is_odd is False
assert z.is_finite is True
assert z.is_infinite is False
assert z.is_comparable is True
assert z.is_prime is False
assert z.is_composite is False
def test_E():
z = S.Exp1
assert z.is_commutative is True
assert z.is_integer is False
assert z.is_rational is False
assert z.is_algebraic is False
assert z.is_transcendental is True
assert z.is_real is True
assert z.is_complex is True
assert z.is_noninteger is True
assert z.is_irrational is True
assert z.is_imaginary is False
assert z.is_positive is True
assert z.is_negative is False
assert z.is_nonpositive is False
assert z.is_nonnegative is True
assert z.is_even is False
assert z.is_odd is False
assert z.is_finite is True
assert z.is_infinite is False
assert z.is_comparable is True
assert z.is_prime is False
assert z.is_composite is False
def test_I():
z = S.ImaginaryUnit
assert z.is_commutative is True
assert z.is_integer is False
assert z.is_rational is False
assert z.is_algebraic is True
assert z.is_transcendental is False
assert z.is_real is False
assert z.is_complex is True
assert z.is_noninteger is False
assert z.is_irrational is False
assert z.is_imaginary is True
assert z.is_positive is False
assert z.is_negative is False
assert z.is_nonpositive is False
assert z.is_nonnegative is False
assert z.is_even is False
assert z.is_odd is False
assert z.is_finite is True
assert z.is_infinite is False
assert z.is_comparable is False
assert z.is_prime is False
assert z.is_composite is False
def test_symbol_real():
# issue 3848
a = Symbol('a', real=False)
assert a.is_real is False
assert a.is_integer is False
assert a.is_negative is False
assert a.is_positive is False
assert a.is_nonnegative is False
assert a.is_nonpositive is False
assert a.is_zero is False
def test_symbol_imaginary():
a = Symbol('a', imaginary=True)
assert a.is_real is False
assert a.is_integer is False
assert a.is_negative is False
assert a.is_positive is False
assert a.is_nonnegative is False
assert a.is_nonpositive is False
assert a.is_zero is False
assert a.is_nonzero is False # since nonzero -> real
def test_symbol_zero():
x = Symbol('x', zero=True)
assert x.is_positive is False
assert x.is_nonpositive
assert x.is_negative is False
assert x.is_nonnegative
assert x.is_zero is True
# TODO Change to x.is_nonzero is None
# See https://github.com/sympy/sympy/pull/9583
assert x.is_nonzero is False
assert x.is_finite is True
def test_symbol_positive():
x = Symbol('x', positive=True)
assert x.is_positive is True
assert x.is_nonpositive is False
assert x.is_negative is False
assert x.is_nonnegative is True
assert x.is_zero is False
assert x.is_nonzero is True
def test_neg_symbol_positive():
x = -Symbol('x', positive=True)
assert x.is_positive is False
assert x.is_nonpositive is True
assert x.is_negative is True
assert x.is_nonnegative is False
assert x.is_zero is False
assert x.is_nonzero is True
def test_symbol_nonpositive():
x = Symbol('x', nonpositive=True)
assert x.is_positive is False
assert x.is_nonpositive is True
assert x.is_negative is None
assert x.is_nonnegative is None
assert x.is_zero is None
assert x.is_nonzero is None
def test_neg_symbol_nonpositive():
x = -Symbol('x', nonpositive=True)
assert x.is_positive is None
assert x.is_nonpositive is None
assert x.is_negative is False
assert x.is_nonnegative is True
assert x.is_zero is None
assert x.is_nonzero is None
def test_symbol_falsepositive():
x = Symbol('x', positive=False)
assert x.is_positive is False
assert x.is_nonpositive is None
assert x.is_negative is None
assert x.is_nonnegative is None
assert x.is_zero is None
assert x.is_nonzero is None
def test_symbol_falsepositive_mul():
# To test pull request 9379
# Explicit handling of arg.is_positive=False was added to Mul._eval_is_positive
x = 2*Symbol('x', positive=False)
assert x.is_positive is False # This was None before
assert x.is_nonpositive is None
assert x.is_negative is None
assert x.is_nonnegative is None
assert x.is_zero is None
assert x.is_nonzero is None
def test_neg_symbol_falsepositive():
x = -Symbol('x', positive=False)
assert x.is_positive is None
assert x.is_nonpositive is None
assert x.is_negative is False
assert x.is_nonnegative is None
assert x.is_zero is None
assert x.is_nonzero is None
def test_neg_symbol_falsenegative():
# To test pull request 9379
# Explicit handling of arg.is_negative=False was added to Mul._eval_is_positive
x = -Symbol('x', negative=False)
assert x.is_positive is False # This was None before
assert x.is_nonpositive is None
assert x.is_negative is None
assert x.is_nonnegative is None
assert x.is_zero is None
assert x.is_nonzero is None
def test_symbol_falsepositive_real():
x = Symbol('x', positive=False, real=True)
assert x.is_positive is False
assert x.is_nonpositive is True
assert x.is_negative is None
assert x.is_nonnegative is None
assert x.is_zero is None
assert x.is_nonzero is None
def test_neg_symbol_falsepositive_real():
x = -Symbol('x', positive=False, real=True)
assert x.is_positive is None
assert x.is_nonpositive is None
assert x.is_negative is False
assert x.is_nonnegative is True
assert x.is_zero is None
assert x.is_nonzero is None
def test_symbol_falsenonnegative():
x = Symbol('x', nonnegative=False)
assert x.is_positive is False
assert x.is_nonpositive is None
assert x.is_negative is None
assert x.is_nonnegative is False
assert x.is_zero is False
assert x.is_nonzero is None
@XFAIL
def test_neg_symbol_falsenonnegative():
x = -Symbol('x', nonnegative=False)
assert x.is_positive is None
assert x.is_nonpositive is False # this currently returns None
assert x.is_negative is False # this currently returns None
assert x.is_nonnegative is None
assert x.is_zero is False # this currently returns None
assert x.is_nonzero is True # this currently returns None
def test_symbol_falsenonnegative_real():
x = Symbol('x', nonnegative=False, real=True)
assert x.is_positive is False
assert x.is_nonpositive is True
assert x.is_negative is True
assert x.is_nonnegative is False
assert x.is_zero is False
assert x.is_nonzero is True
def test_neg_symbol_falsenonnegative_real():
x = -Symbol('x', nonnegative=False, real=True)
assert x.is_positive is True
assert x.is_nonpositive is False
assert x.is_negative is False
assert x.is_nonnegative is True
assert x.is_zero is False
assert x.is_nonzero is True
def test_prime():
assert S(-1).is_prime is False
assert S(-2).is_prime is False
assert S(-4).is_prime is False
assert S(0).is_prime is False
assert S(1).is_prime is False
assert S(2).is_prime is True
assert S(17).is_prime is True
assert S(4).is_prime is False
def test_composite():
assert S(-1).is_composite is False
assert S(-2).is_composite is False
assert S(-4).is_composite is False
assert S(0).is_composite is False
assert S(2).is_composite is False
assert S(17).is_composite is False
assert S(4).is_composite is True
x = Dummy(integer=True, positive=True, prime=False)
assert x.is_composite is None # x could be 1
assert (x + 1).is_composite is None
x = Dummy(positive=True, even=True, prime=False)
assert x.is_integer is True
assert x.is_composite is True
def test_prime_symbol():
x = Symbol('x', prime=True)
assert x.is_prime is True
assert x.is_integer is True
assert x.is_positive is True
assert x.is_negative is False
assert x.is_nonpositive is False
assert x.is_nonnegative is True
x = Symbol('x', prime=False)
assert x.is_prime is False
assert x.is_integer is None
assert x.is_positive is None
assert x.is_negative is None
assert x.is_nonpositive is None
assert x.is_nonnegative is None
def test_symbol_noncommutative():
x = Symbol('x', commutative=True)
assert x.is_complex is None
x = Symbol('x', commutative=False)
assert x.is_integer is False
assert x.is_rational is False
assert x.is_algebraic is False
assert x.is_irrational is False
assert x.is_real is False
assert x.is_complex is False
def test_other_symbol():
x = Symbol('x', integer=True)
assert x.is_integer is True
assert x.is_real is True
x = Symbol('x', integer=True, nonnegative=True)
assert x.is_integer is True
assert x.is_nonnegative is True
assert x.is_negative is False
assert x.is_positive is None
x = Symbol('x', integer=True, nonpositive=True)
assert x.is_integer is True
assert x.is_nonpositive is True
assert x.is_positive is False
assert x.is_negative is None
x = Symbol('x', odd=True)
assert x.is_odd is True
assert x.is_even is False
assert x.is_integer is True
x = Symbol('x', odd=False)
assert x.is_odd is False
assert x.is_even is None
assert x.is_integer is None
x = Symbol('x', even=True)
assert x.is_even is True
assert x.is_odd is False
assert x.is_integer is True
x = Symbol('x', even=False)
assert x.is_even is False
assert x.is_odd is None
assert x.is_integer is None
x = Symbol('x', integer=True, nonnegative=True)
assert x.is_integer is True
assert x.is_nonnegative is True
x = Symbol('x', integer=True, nonpositive=True)
assert x.is_integer is True
assert x.is_nonpositive is True
with raises(AttributeError):
x.is_real = False
x = Symbol('x', algebraic=True)
assert x.is_transcendental is False
x = Symbol('x', transcendental=True)
assert x.is_algebraic is False
assert x.is_rational is False
assert x.is_integer is False
def test_issue_3825():
"""catch: hash instability"""
x = Symbol("x")
y = Symbol("y")
a1 = x + y
a2 = y + x
a2.is_comparable
h1 = hash(a1)
h2 = hash(a2)
assert h1 == h2
def test_issue_4822():
z = (-1)**Rational(1, 3)*(1 - I*sqrt(3))
assert z.is_real in [True, None]
def test_hash_vs_typeinfo():
"""seemingly different typeinfo, but in fact equal"""
# the following two are semantically equal
x1 = Symbol('x', even=True)
x2 = Symbol('x', integer=True, odd=False)
assert hash(x1) == hash(x2)
assert x1 == x2
def test_hash_vs_typeinfo_2():
"""different typeinfo should mean !eq"""
# the following two are semantically different
x = Symbol('x')
x1 = Symbol('x', even=True)
assert x != x1
assert hash(x) != hash(x1) # This might fail with very low probability
def test_hash_vs_eq():
"""catch: different hash for equal objects"""
a = 1 + S.Pi # important: do not fold it into a Number instance
ha = hash(a) # it should be Add/Mul/... to trigger the bug
a.is_positive # this uses .evalf() and deduces it is positive
assert a.is_positive is True
# be sure that hash stayed the same
assert ha == hash(a)
# now b should be the same expression
b = a.expand(trig=True)
hb = hash(b)
assert a == b
assert ha == hb
def test_Add_is_pos_neg():
# these cover lines not covered by the rest of tests in core
n = Symbol('n', negative=True, infinite=True)
nn = Symbol('n', nonnegative=True, infinite=True)
np = Symbol('n', nonpositive=True, infinite=True)
p = Symbol('p', positive=True, infinite=True)
r = Dummy(real=True, finite=False)
x = Symbol('x')
xf = Symbol('xb', finite=True)
assert (n + p).is_positive is None
assert (n + x).is_positive is None
assert (p + x).is_positive is None
assert (n + p).is_negative is None
assert (n + x).is_negative is None
assert (p + x).is_negative is None
assert (n + xf).is_positive is False
assert (p + xf).is_positive is True
assert (n + xf).is_negative is True
assert (p + xf).is_negative is False
assert (x - S.Infinity).is_negative is None # issue 7798
# issue 8046, 16.2
assert (p + nn).is_positive
assert (n + np).is_negative
assert (p + r).is_positive is None
def test_Add_is_imaginary():
nn = Dummy(nonnegative=True)
assert (I*nn + I).is_imaginary # issue 8046, 17
def test_Add_is_algebraic():
a = Symbol('a', algebraic=True)
b = Symbol('a', algebraic=True)
na = Symbol('na', algebraic=False)
nb = Symbol('nb', algebraic=False)
x = Symbol('x')
assert (a + b).is_algebraic
assert (na + nb).is_algebraic is None
assert (a + na).is_algebraic is False
assert (a + x).is_algebraic is None
assert (na + x).is_algebraic is None
def test_Mul_is_algebraic():
a = Symbol('a', algebraic=True)
b = Symbol('a', algebraic=True)
na = Symbol('na', algebraic=False)
an = Symbol('an', algebraic=True, nonzero=True)
nb = Symbol('nb', algebraic=False)
x = Symbol('x')
assert (a*b).is_algebraic
assert (na*nb).is_algebraic is None
assert (a*na).is_algebraic is None
assert (an*na).is_algebraic is False
assert (a*x).is_algebraic is None
assert (na*x).is_algebraic is None
def test_Pow_is_algebraic():
e = Symbol('e', algebraic=True)
assert Pow(1, e, evaluate=False).is_algebraic
assert Pow(0, e, evaluate=False).is_algebraic
a = Symbol('a', algebraic=True)
na = Symbol('na', algebraic=False)
ia = Symbol('ia', algebraic=True, irrational=True)
ib = Symbol('ib', algebraic=True, irrational=True)
r = Symbol('r', rational=True)
x = Symbol('x')
assert (a**r).is_algebraic
assert (a**x).is_algebraic is None
assert (na**r).is_algebraic is None
assert (ia**r).is_algebraic
assert (ia**ib).is_algebraic is False
assert (a**e).is_algebraic is None
# Gelfond-Schneider constant:
assert Pow(2, sqrt(2), evaluate=False).is_algebraic is False
assert Pow(S.GoldenRatio, sqrt(3), evaluate=False).is_algebraic is False
# issue 8649
t = Symbol('t', real=True, transcendental=True)
n = Symbol('n', integer=True)
assert (t**n).is_algebraic is None
assert (t**n).is_integer is None
assert (pi**3).is_algebraic is False
r = Symbol('r', zero=True)
assert (pi**r).is_algebraic is True
def test_Mul_is_prime_composite():
from sympy import Mul
x = Symbol('x', positive=True, integer=True)
y = Symbol('y', positive=True, integer=True)
assert (x*y).is_prime is None
assert ( (x+1)*(y+1) ).is_prime is False
assert ( (x+1)*(y+1) ).is_composite is True
x = Symbol('x', positive=True)
assert ( (x+1)*(y+1) ).is_prime is None
assert ( (x+1)*(y+1) ).is_composite is None
def test_Pow_is_pos_neg():
z = Symbol('z', real=True)
w = Symbol('w', nonpositive=True)
assert (S(-1)**S(2)).is_positive is True
assert (S(1)**z).is_positive is True
assert (S(-1)**S(3)).is_positive is False
assert (S(0)**S(0)).is_positive is True # 0**0 is 1
assert (w**S(3)).is_positive is False
assert (w**S(2)).is_positive is None
assert (I**2).is_positive is False
assert (I**4).is_positive is True
# tests emerging from #16332 issue
p = Symbol('p', zero=True)
q = Symbol('q', zero=False, real=True)
j = Symbol('j', zero=False, even=True)
x = Symbol('x', zero=True)
y = Symbol('y', zero=True)
assert (p**q).is_positive is False
assert (p**q).is_negative is False
assert (p**j).is_positive is False
assert (x**y).is_positive is True # 0**0
assert (x**y).is_negative is False
def test_Pow_is_prime_composite():
from sympy import Pow
x = Symbol('x', positive=True, integer=True)
y = Symbol('y', positive=True, integer=True)
assert (x**y).is_prime is None
assert ( x**(y+1) ).is_prime is False
assert ( x**(y+1) ).is_composite is None
assert ( (x+1)**(y+1) ).is_composite is True
assert ( (-x-1)**(2*y) ).is_composite is True
x = Symbol('x', positive=True)
assert (x**y).is_prime is None
def test_Mul_is_infinite():
x = Symbol('x')
f = Symbol('f', finite=True)
i = Symbol('i', infinite=True)
z = Dummy(zero=True)
nzf = Dummy(finite=True, zero=False)
from sympy import Mul
assert (x*f).is_finite is None
assert (x*i).is_finite is None
assert (f*i).is_finite is False
assert (x*f*i).is_finite is None
assert (z*i).is_finite is False
assert (nzf*i).is_finite is False
assert (z*f).is_finite is True
assert Mul(0, f, evaluate=False).is_finite is True
assert Mul(0, i, evaluate=False).is_finite is False
assert (x*f).is_infinite is None
assert (x*i).is_infinite is None
assert (f*i).is_infinite is None
assert (x*f*i).is_infinite is None
assert (z*i).is_infinite is S.NaN.is_infinite
assert (nzf*i).is_infinite is True
assert (z*f).is_infinite is False
assert Mul(0, f, evaluate=False).is_infinite is False
assert Mul(0, i, evaluate=False).is_infinite is S.NaN.is_infinite
def test_special_is_rational():
i = Symbol('i', integer=True)
i2 = Symbol('i2', integer=True)
ni = Symbol('ni', integer=True, nonzero=True)
r = Symbol('r', rational=True)
rn = Symbol('r', rational=True, nonzero=True)
nr = Symbol('nr', irrational=True)
x = Symbol('x')
assert sqrt(3).is_rational is False
assert (3 + sqrt(3)).is_rational is False
assert (3*sqrt(3)).is_rational is False
assert exp(3).is_rational is False
assert exp(ni).is_rational is False
assert exp(rn).is_rational is False
assert exp(x).is_rational is None
assert exp(log(3), evaluate=False).is_rational is True
assert log(exp(3), evaluate=False).is_rational is True
assert log(3).is_rational is False
assert log(ni + 1).is_rational is False
assert log(rn + 1).is_rational is False
assert log(x).is_rational is None
assert (sqrt(3) + sqrt(5)).is_rational is None
assert (sqrt(3) + S.Pi).is_rational is False
assert (x**i).is_rational is None
assert (i**i).is_rational is True
assert (i**i2).is_rational is None
assert (r**i).is_rational is None
assert (r**r).is_rational is None
assert (r**x).is_rational is None
assert (nr**i).is_rational is None # issue 8598
assert (nr**Symbol('z', zero=True)).is_rational
assert sin(1).is_rational is False
assert sin(ni).is_rational is False
assert sin(rn).is_rational is False
assert sin(x).is_rational is None
assert asin(r).is_rational is False
assert sin(asin(3), evaluate=False).is_rational is True
@XFAIL
def test_issue_6275():
x = Symbol('x')
# both zero or both Muls...but neither "change would be very appreciated.
# This is similar to x/x => 1 even though if x = 0, it is really nan.
assert isinstance(x*0, type(0*S.Infinity))
if 0*S.Infinity is S.NaN:
b = Symbol('b', finite=None)
assert (b*0).is_zero is None
def test_sanitize_assumptions():
# issue 6666
for cls in (Symbol, Dummy, Wild):
x = cls('x', real=1, positive=0)
assert x.is_real is True
assert x.is_positive is False
assert cls('', real=True, positive=None).is_positive is None
raises(ValueError, lambda: cls('', commutative=None))
raises(ValueError, lambda: Symbol._sanitize(dict(commutative=None)))
def test_special_assumptions():
e = -3 - sqrt(5) + (-sqrt(10)/2 - sqrt(2)/2)**2
assert simplify(e < 0) is S.false
assert simplify(e > 0) is S.false
assert (e == 0) is False # it's not a literal 0
assert e.equals(0) is True
def test_inconsistent():
# cf. issues 5795 and 5545
raises(InconsistentAssumptions, lambda: Symbol('x', real=True,
commutative=False))
def test_issue_6631():
assert ((-1)**(I)).is_real is True
assert ((-1)**(I*2)).is_real is True
assert ((-1)**(I/2)).is_real is True
assert ((-1)**(I*S.Pi)).is_real is True
assert (I**(I + 2)).is_real is True
def test_issue_2730():
assert (1/(1 + I)).is_real is False
def test_issue_4149():
assert (3 + I).is_complex
assert (3 + I).is_imaginary is False
assert (3*I + S.Pi*I).is_imaginary
# as Zero.is_imaginary is False, see issue 7649
y = Symbol('y', real=True)
assert (3*I + S.Pi*I + y*I).is_imaginary is None
p = Symbol('p', positive=True)
assert (3*I + S.Pi*I + p*I).is_imaginary
n = Symbol('n', negative=True)
assert (-3*I - S.Pi*I + n*I).is_imaginary
i = Symbol('i', imaginary=True)
assert ([(i**a).is_imaginary for a in range(4)] ==
[False, True, False, True])
# tests from the PR #7887:
e = S("-sqrt(3)*I/2 + 0.866025403784439*I")
assert e.is_real is False
assert e.is_imaginary
def test_issue_2920():
n = Symbol('n', negative=True)
assert sqrt(n).is_imaginary
def test_issue_7899():
x = Symbol('x', real=True)
assert (I*x).is_real is None
assert ((x - I)*(x - 1)).is_zero is None
assert ((x - I)*(x - 1)).is_real is None
@XFAIL
def test_issue_7993():
x = Dummy(integer=True)
y = Dummy(noninteger=True)
assert (x - y).is_zero is False
def test_issue_8075():
raises(InconsistentAssumptions, lambda: Dummy(zero=True, finite=False))
raises(InconsistentAssumptions, lambda: Dummy(zero=True, infinite=True))
def test_issue_8642():
x = Symbol('x', real=True, integer=False)
assert (x*2).is_integer is None
def test_issues_8632_8633_8638_8675_8992():
p = Dummy(integer=True, positive=True)
nn = Dummy(integer=True, nonnegative=True)
assert (p - S.Half).is_positive
assert (p - 1).is_nonnegative
assert (nn + 1).is_positive
assert (-p + 1).is_nonpositive
assert (-nn - 1).is_negative
prime = Dummy(prime=True)
assert (prime - 2).is_nonnegative
assert (prime - 3).is_nonnegative is None
even = Dummy(positive=True, even=True)
assert (even - 2).is_nonnegative
p = Dummy(positive=True)
assert (p/(p + 1) - 1).is_negative
assert ((p + 2)**3 - S.Half).is_positive
n = Dummy(negative=True)
assert (n - 3).is_nonpositive
def test_issue_9115_9150():
n = Dummy('n', integer=True, nonnegative=True)
assert (factorial(n) >= 1) == True
assert (factorial(n) < 1) == False
assert factorial(n + 1).is_even is None
assert factorial(n + 2).is_even is True
assert factorial(n + 2) >= 2
def test_issue_9165():
z = Symbol('z', zero=True)
f = Symbol('f', finite=False)
assert 0/z == S.NaN
assert 0*(1/z) == S.NaN
assert 0*f == S.NaN
def test_issue_10024():
x = Dummy('x')
assert Mod(x, 2*pi).is_zero is None
def test_issue_10302():
x = Symbol('x')
r = Symbol('r', real=True)
u = -(3*2**pi)**(1/pi) + 2*3**(1/pi)
i = u + u*I
assert i.is_real is None # w/o simplification this should fail
assert (u + i).is_zero is None
assert (1 + i).is_zero is False
a = Dummy('a', zero=True)
assert (a + I).is_zero is False
assert (a + r*I).is_zero is None
assert (a + I).is_imaginary
assert (a + x + I).is_imaginary is None
assert (a + r*I + I).is_imaginary is None
def test_complex_reciprocal_imaginary():
assert (1 / (4 + 3*I)).is_imaginary is False
def test_issue_16313():
x = Symbol('x', real=False)
k = Symbol('k', real=True)
l = Symbol('l', real=True, zero=False)
assert (-x).is_real is False
assert (k*x).is_real is None # k can be zero also
assert (l*x).is_real is False
assert (l*x*x).is_real is None # since x*x can be a real number
assert (-x).is_positive is False
|
c1a8ae78fa84b8c9a0c807265d2e0254d261adcb9066f5670f806e9f344f2fdd
|
from sympy.utilities.pytest import XFAIL, raises
from sympy import (S, Symbol, symbols, nan, oo, I, pi, Float, And, Or,
Not, Implies, Xor, zoo, sqrt, Rational, simplify, Function,
log, cos, sin, Add, floor, ceiling)
from sympy.core.compatibility import range
from sympy.core.relational import (Relational, Equality, Unequality,
GreaterThan, LessThan, StrictGreaterThan,
StrictLessThan, Rel, Eq, Lt, Le,
Gt, Ge, Ne)
from sympy.sets.sets import Interval, FiniteSet
from itertools import combinations
x, y, z, t = symbols('x,y,z,t')
def test_rel_ne():
assert Relational(x, y, '!=') == Ne(x, y)
# issue 6116
p = Symbol('p', positive=True)
assert Ne(p, 0) is S.true
def test_rel_subs():
e = Relational(x, y, '==')
e = e.subs(x, z)
assert isinstance(e, Equality)
assert e.lhs == z
assert e.rhs == y
e = Relational(x, y, '>=')
e = e.subs(x, z)
assert isinstance(e, GreaterThan)
assert e.lhs == z
assert e.rhs == y
e = Relational(x, y, '<=')
e = e.subs(x, z)
assert isinstance(e, LessThan)
assert e.lhs == z
assert e.rhs == y
e = Relational(x, y, '>')
e = e.subs(x, z)
assert isinstance(e, StrictGreaterThan)
assert e.lhs == z
assert e.rhs == y
e = Relational(x, y, '<')
e = e.subs(x, z)
assert isinstance(e, StrictLessThan)
assert e.lhs == z
assert e.rhs == y
e = Eq(x, 0)
assert e.subs(x, 0) is S.true
assert e.subs(x, 1) is S.false
def test_wrappers():
e = x + x**2
res = Relational(y, e, '==')
assert Rel(y, x + x**2, '==') == res
assert Eq(y, x + x**2) == res
res = Relational(y, e, '<')
assert Lt(y, x + x**2) == res
res = Relational(y, e, '<=')
assert Le(y, x + x**2) == res
res = Relational(y, e, '>')
assert Gt(y, x + x**2) == res
res = Relational(y, e, '>=')
assert Ge(y, x + x**2) == res
res = Relational(y, e, '!=')
assert Ne(y, x + x**2) == res
def test_Eq():
assert Eq(x**2) == Eq(x**2, 0)
assert Eq(x**2) != Eq(x**2, 1)
assert Eq(x, x) # issue 5719
# issue 6116
p = Symbol('p', positive=True)
assert Eq(p, 0) is S.false
# issue 13348
assert Eq(True, 1) is S.false
def test_rel_Infinity():
# NOTE: All of these are actually handled by sympy.core.Number, and do
# not create Relational objects.
assert (oo > oo) is S.false
assert (oo > -oo) is S.true
assert (oo > 1) is S.true
assert (oo < oo) is S.false
assert (oo < -oo) is S.false
assert (oo < 1) is S.false
assert (oo >= oo) is S.true
assert (oo >= -oo) is S.true
assert (oo >= 1) is S.true
assert (oo <= oo) is S.true
assert (oo <= -oo) is S.false
assert (oo <= 1) is S.false
assert (-oo > oo) is S.false
assert (-oo > -oo) is S.false
assert (-oo > 1) is S.false
assert (-oo < oo) is S.true
assert (-oo < -oo) is S.false
assert (-oo < 1) is S.true
assert (-oo >= oo) is S.false
assert (-oo >= -oo) is S.true
assert (-oo >= 1) is S.false
assert (-oo <= oo) is S.true
assert (-oo <= -oo) is S.true
assert (-oo <= 1) is S.true
def test_bool():
assert Eq(0, 0) is S.true
assert Eq(1, 0) is S.false
assert Ne(0, 0) is S.false
assert Ne(1, 0) is S.true
assert Lt(0, 1) is S.true
assert Lt(1, 0) is S.false
assert Le(0, 1) is S.true
assert Le(1, 0) is S.false
assert Le(0, 0) is S.true
assert Gt(1, 0) is S.true
assert Gt(0, 1) is S.false
assert Ge(1, 0) is S.true
assert Ge(0, 1) is S.false
assert Ge(1, 1) is S.true
assert Eq(I, 2) is S.false
assert Ne(I, 2) is S.true
raises(TypeError, lambda: Gt(I, 2))
raises(TypeError, lambda: Ge(I, 2))
raises(TypeError, lambda: Lt(I, 2))
raises(TypeError, lambda: Le(I, 2))
a = Float('.000000000000000000001', '')
b = Float('.0000000000000000000001', '')
assert Eq(pi + a, pi + b) is S.false
def test_rich_cmp():
assert (x < y) == Lt(x, y)
assert (x <= y) == Le(x, y)
assert (x > y) == Gt(x, y)
assert (x >= y) == Ge(x, y)
def test_doit():
from sympy import Symbol
p = Symbol('p', positive=True)
n = Symbol('n', negative=True)
np = Symbol('np', nonpositive=True)
nn = Symbol('nn', nonnegative=True)
assert Gt(p, 0).doit() is S.true
assert Gt(p, 1).doit() == Gt(p, 1)
assert Ge(p, 0).doit() is S.true
assert Le(p, 0).doit() is S.false
assert Lt(n, 0).doit() is S.true
assert Le(np, 0).doit() is S.true
assert Gt(nn, 0).doit() == Gt(nn, 0)
assert Lt(nn, 0).doit() is S.false
assert Eq(x, 0).doit() == Eq(x, 0)
def test_new_relational():
x = Symbol('x')
assert Eq(x) == Relational(x, 0) # None ==> Equality
assert Eq(x) == Relational(x, 0, '==')
assert Eq(x) == Relational(x, 0, 'eq')
assert Eq(x) == Equality(x, 0)
assert Eq(x, -1) == Relational(x, -1) # None ==> Equality
assert Eq(x, -1) == Relational(x, -1, '==')
assert Eq(x, -1) == Relational(x, -1, 'eq')
assert Eq(x, -1) == Equality(x, -1)
assert Eq(x) != Relational(x, 1) # None ==> Equality
assert Eq(x) != Relational(x, 1, '==')
assert Eq(x) != Relational(x, 1, 'eq')
assert Eq(x) != Equality(x, 1)
assert Eq(x, -1) != Relational(x, 1) # None ==> Equality
assert Eq(x, -1) != Relational(x, 1, '==')
assert Eq(x, -1) != Relational(x, 1, 'eq')
assert Eq(x, -1) != Equality(x, 1)
assert Ne(x, 0) == Relational(x, 0, '!=')
assert Ne(x, 0) == Relational(x, 0, '<>')
assert Ne(x, 0) == Relational(x, 0, 'ne')
assert Ne(x, 0) == Unequality(x, 0)
assert Ne(x, 0) != Relational(x, 1, '!=')
assert Ne(x, 0) != Relational(x, 1, '<>')
assert Ne(x, 0) != Relational(x, 1, 'ne')
assert Ne(x, 0) != Unequality(x, 1)
assert Ge(x, 0) == Relational(x, 0, '>=')
assert Ge(x, 0) == Relational(x, 0, 'ge')
assert Ge(x, 0) == GreaterThan(x, 0)
assert Ge(x, 1) != Relational(x, 0, '>=')
assert Ge(x, 1) != Relational(x, 0, 'ge')
assert Ge(x, 1) != GreaterThan(x, 0)
assert (x >= 1) == Relational(x, 1, '>=')
assert (x >= 1) == Relational(x, 1, 'ge')
assert (x >= 1) == GreaterThan(x, 1)
assert (x >= 0) != Relational(x, 1, '>=')
assert (x >= 0) != Relational(x, 1, 'ge')
assert (x >= 0) != GreaterThan(x, 1)
assert Le(x, 0) == Relational(x, 0, '<=')
assert Le(x, 0) == Relational(x, 0, 'le')
assert Le(x, 0) == LessThan(x, 0)
assert Le(x, 1) != Relational(x, 0, '<=')
assert Le(x, 1) != Relational(x, 0, 'le')
assert Le(x, 1) != LessThan(x, 0)
assert (x <= 1) == Relational(x, 1, '<=')
assert (x <= 1) == Relational(x, 1, 'le')
assert (x <= 1) == LessThan(x, 1)
assert (x <= 0) != Relational(x, 1, '<=')
assert (x <= 0) != Relational(x, 1, 'le')
assert (x <= 0) != LessThan(x, 1)
assert Gt(x, 0) == Relational(x, 0, '>')
assert Gt(x, 0) == Relational(x, 0, 'gt')
assert Gt(x, 0) == StrictGreaterThan(x, 0)
assert Gt(x, 1) != Relational(x, 0, '>')
assert Gt(x, 1) != Relational(x, 0, 'gt')
assert Gt(x, 1) != StrictGreaterThan(x, 0)
assert (x > 1) == Relational(x, 1, '>')
assert (x > 1) == Relational(x, 1, 'gt')
assert (x > 1) == StrictGreaterThan(x, 1)
assert (x > 0) != Relational(x, 1, '>')
assert (x > 0) != Relational(x, 1, 'gt')
assert (x > 0) != StrictGreaterThan(x, 1)
assert Lt(x, 0) == Relational(x, 0, '<')
assert Lt(x, 0) == Relational(x, 0, 'lt')
assert Lt(x, 0) == StrictLessThan(x, 0)
assert Lt(x, 1) != Relational(x, 0, '<')
assert Lt(x, 1) != Relational(x, 0, 'lt')
assert Lt(x, 1) != StrictLessThan(x, 0)
assert (x < 1) == Relational(x, 1, '<')
assert (x < 1) == Relational(x, 1, 'lt')
assert (x < 1) == StrictLessThan(x, 1)
assert (x < 0) != Relational(x, 1, '<')
assert (x < 0) != Relational(x, 1, 'lt')
assert (x < 0) != StrictLessThan(x, 1)
# finally, some fuzz testing
from random import randint
from sympy.core.compatibility import unichr
for i in range(100):
while 1:
strtype, length = (unichr, 65535) if randint(0, 1) else (chr, 255)
relation_type = strtype(randint(0, length))
if randint(0, 1):
relation_type += strtype(randint(0, length))
if relation_type not in ('==', 'eq', '!=', '<>', 'ne', '>=', 'ge',
'<=', 'le', '>', 'gt', '<', 'lt', ':=',
'+=', '-=', '*=', '/=', '%='):
break
raises(ValueError, lambda: Relational(x, 1, relation_type))
assert all(Relational(x, 0, op).rel_op == '==' for op in ('eq', '=='))
assert all(Relational(x, 0, op).rel_op == '!='
for op in ('ne', '<>', '!='))
assert all(Relational(x, 0, op).rel_op == '>' for op in ('gt', '>'))
assert all(Relational(x, 0, op).rel_op == '<' for op in ('lt', '<'))
assert all(Relational(x, 0, op).rel_op == '>=' for op in ('ge', '>='))
assert all(Relational(x, 0, op).rel_op == '<=' for op in ('le', '<='))
def test_relational_bool_output():
# https://github.com/sympy/sympy/issues/5931
raises(TypeError, lambda: bool(x > 3))
raises(TypeError, lambda: bool(x >= 3))
raises(TypeError, lambda: bool(x < 3))
raises(TypeError, lambda: bool(x <= 3))
raises(TypeError, lambda: bool(Eq(x, 3)))
raises(TypeError, lambda: bool(Ne(x, 3)))
def test_relational_logic_symbols():
# See issue 6204
assert (x < y) & (z < t) == And(x < y, z < t)
assert (x < y) | (z < t) == Or(x < y, z < t)
assert ~(x < y) == Not(x < y)
assert (x < y) >> (z < t) == Implies(x < y, z < t)
assert (x < y) << (z < t) == Implies(z < t, x < y)
assert (x < y) ^ (z < t) == Xor(x < y, z < t)
assert isinstance((x < y) & (z < t), And)
assert isinstance((x < y) | (z < t), Or)
assert isinstance(~(x < y), GreaterThan)
assert isinstance((x < y) >> (z < t), Implies)
assert isinstance((x < y) << (z < t), Implies)
assert isinstance((x < y) ^ (z < t), (Or, Xor))
def test_univariate_relational_as_set():
assert (x > 0).as_set() == Interval(0, oo, True, True)
assert (x >= 0).as_set() == Interval(0, oo)
assert (x < 0).as_set() == Interval(-oo, 0, True, True)
assert (x <= 0).as_set() == Interval(-oo, 0)
assert Eq(x, 0).as_set() == FiniteSet(0)
assert Ne(x, 0).as_set() == Interval(-oo, 0, True, True) + \
Interval(0, oo, True, True)
assert (x**2 >= 4).as_set() == Interval(-oo, -2) + Interval(2, oo)
@XFAIL
def test_multivariate_relational_as_set():
assert (x*y >= 0).as_set() == Interval(0, oo)*Interval(0, oo) + \
Interval(-oo, 0)*Interval(-oo, 0)
def test_Not():
assert Not(Equality(x, y)) == Unequality(x, y)
assert Not(Unequality(x, y)) == Equality(x, y)
assert Not(StrictGreaterThan(x, y)) == LessThan(x, y)
assert Not(StrictLessThan(x, y)) == GreaterThan(x, y)
assert Not(GreaterThan(x, y)) == StrictLessThan(x, y)
assert Not(LessThan(x, y)) == StrictGreaterThan(x, y)
def test_evaluate():
assert str(Eq(x, x, evaluate=False)) == 'Eq(x, x)'
assert Eq(x, x, evaluate=False).doit() == S.true
assert str(Ne(x, x, evaluate=False)) == 'Ne(x, x)'
assert Ne(x, x, evaluate=False).doit() == S.false
assert str(Ge(x, x, evaluate=False)) == 'x >= x'
assert str(Le(x, x, evaluate=False)) == 'x <= x'
assert str(Gt(x, x, evaluate=False)) == 'x > x'
assert str(Lt(x, x, evaluate=False)) == 'x < x'
def assert_all_ineq_raise_TypeError(a, b):
raises(TypeError, lambda: a > b)
raises(TypeError, lambda: a >= b)
raises(TypeError, lambda: a < b)
raises(TypeError, lambda: a <= b)
raises(TypeError, lambda: b > a)
raises(TypeError, lambda: b >= a)
raises(TypeError, lambda: b < a)
raises(TypeError, lambda: b <= a)
def assert_all_ineq_give_class_Inequality(a, b):
"""All inequality operations on `a` and `b` result in class Inequality."""
from sympy.core.relational import _Inequality as Inequality
assert isinstance(a > b, Inequality)
assert isinstance(a >= b, Inequality)
assert isinstance(a < b, Inequality)
assert isinstance(a <= b, Inequality)
assert isinstance(b > a, Inequality)
assert isinstance(b >= a, Inequality)
assert isinstance(b < a, Inequality)
assert isinstance(b <= a, Inequality)
def test_imaginary_compare_raises_TypeError():
# See issue #5724
assert_all_ineq_raise_TypeError(I, x)
def test_complex_compare_not_real():
# two cases which are not real
y = Symbol('y', imaginary=True)
z = Symbol('z', complex=True, real=False)
for w in (y, z):
assert_all_ineq_raise_TypeError(2, w)
# some cases which should remain un-evaluated
t = Symbol('t')
x = Symbol('x', real=True)
z = Symbol('z', complex=True)
for w in (x, z, t):
assert_all_ineq_give_class_Inequality(2, w)
def test_imaginary_and_inf_compare_raises_TypeError():
# See pull request #7835
y = Symbol('y', imaginary=True)
assert_all_ineq_raise_TypeError(oo, y)
assert_all_ineq_raise_TypeError(-oo, y)
def test_complex_pure_imag_not_ordered():
raises(TypeError, lambda: 2*I < 3*I)
# more generally
x = Symbol('x', real=True, nonzero=True)
y = Symbol('y', imaginary=True)
z = Symbol('z', complex=True)
assert_all_ineq_raise_TypeError(I, y)
t = I*x # an imaginary number, should raise errors
assert_all_ineq_raise_TypeError(2, t)
t = -I*y # a real number, so no errors
assert_all_ineq_give_class_Inequality(2, t)
t = I*z # unknown, should be unevaluated
assert_all_ineq_give_class_Inequality(2, t)
def test_x_minus_y_not_same_as_x_lt_y():
"""
A consequence of pull request #7792 is that `x - y < 0` and `x < y`
are not synonymous.
"""
x = I + 2
y = I + 3
raises(TypeError, lambda: x < y)
assert x - y < 0
ineq = Lt(x, y, evaluate=False)
raises(TypeError, lambda: ineq.doit())
assert ineq.lhs - ineq.rhs < 0
t = Symbol('t', imaginary=True)
x = 2 + t
y = 3 + t
ineq = Lt(x, y, evaluate=False)
raises(TypeError, lambda: ineq.doit())
assert ineq.lhs - ineq.rhs < 0
# this one should give error either way
x = I + 2
y = 2*I + 3
raises(TypeError, lambda: x < y)
raises(TypeError, lambda: x - y < 0)
def test_nan_equality_exceptions():
# See issue #7774
import random
assert Equality(nan, nan) is S.false
assert Unequality(nan, nan) is S.true
# See issue #7773
A = (x, S(0), S(1)/3, pi, oo, -oo)
assert Equality(nan, random.choice(A)) is S.false
assert Equality(random.choice(A), nan) is S.false
assert Unequality(nan, random.choice(A)) is S.true
assert Unequality(random.choice(A), nan) is S.true
def test_nan_inequality_raise_errors():
# See discussion in pull request #7776. We test inequalities with
# a set including examples of various classes.
for q in (x, S(0), S(10), S(1)/3, pi, S(1.3), oo, -oo, nan):
assert_all_ineq_raise_TypeError(q, nan)
def test_nan_complex_inequalities():
# Comparisons of NaN with non-real raise errors, we're not too
# fussy whether its the NaN error or complex error.
for r in (I, zoo, Symbol('z', imaginary=True)):
assert_all_ineq_raise_TypeError(r, nan)
def test_complex_infinity_inequalities():
raises(TypeError, lambda: zoo > 0)
raises(TypeError, lambda: zoo >= 0)
raises(TypeError, lambda: zoo < 0)
raises(TypeError, lambda: zoo <= 0)
def test_inequalities_symbol_name_same():
"""Using the operator and functional forms should give same results."""
# We test all combinations from a set
# FIXME: could replace with random selection after test passes
A = (x, y, S(0), S(1)/3, pi, oo, -oo)
for a in A:
for b in A:
assert Gt(a, b) == (a > b)
assert Lt(a, b) == (a < b)
assert Ge(a, b) == (a >= b)
assert Le(a, b) == (a <= b)
for b in (y, S(0), S(1)/3, pi, oo, -oo):
assert Gt(x, b, evaluate=False) == (x > b)
assert Lt(x, b, evaluate=False) == (x < b)
assert Ge(x, b, evaluate=False) == (x >= b)
assert Le(x, b, evaluate=False) == (x <= b)
for b in (y, S(0), S(1)/3, pi, oo, -oo):
assert Gt(b, x, evaluate=False) == (b > x)
assert Lt(b, x, evaluate=False) == (b < x)
assert Ge(b, x, evaluate=False) == (b >= x)
assert Le(b, x, evaluate=False) == (b <= x)
def test_inequalities_symbol_name_same_complex():
"""Using the operator and functional forms should give same results.
With complex non-real numbers, both should raise errors.
"""
# FIXME: could replace with random selection after test passes
for a in (x, S(0), S(1)/3, pi, oo):
raises(TypeError, lambda: Gt(a, I))
raises(TypeError, lambda: a > I)
raises(TypeError, lambda: Lt(a, I))
raises(TypeError, lambda: a < I)
raises(TypeError, lambda: Ge(a, I))
raises(TypeError, lambda: a >= I)
raises(TypeError, lambda: Le(a, I))
raises(TypeError, lambda: a <= I)
def test_inequalities_cant_sympify_other():
# see issue 7833
from operator import gt, lt, ge, le
bar = "foo"
for a in (x, S(0), S(1)/3, pi, I, zoo, oo, -oo, nan):
for op in (lt, gt, le, ge):
raises(TypeError, lambda: op(a, bar))
def test_ineq_avoid_wild_symbol_flip():
# see issue #7951, we try to avoid this internally, e.g., by using
# __lt__ instead of "<".
from sympy.core.symbol import Wild
p = symbols('p', cls=Wild)
# x > p might flip, but Gt should not:
assert Gt(x, p) == Gt(x, p, evaluate=False)
# Previously failed as 'p > x':
e = Lt(x, y).subs({y: p})
assert e == Lt(x, p, evaluate=False)
# Previously failed as 'p <= x':
e = Ge(x, p).doit()
assert e == Ge(x, p, evaluate=False)
def test_issue_8245():
a = S("6506833320952669167898688709329/5070602400912917605986812821504")
q = a.n(10)
assert (a == q) is True
assert (a != q) is False
assert (a > q) == False
assert (a < q) == False
assert (a >= q) == True
assert (a <= q) == True
a = sqrt(2)
r = Rational(str(a.n(30)))
assert (r == a) is False
assert (r != a) is True
assert (r > a) == True
assert (r < a) == False
assert (r >= a) == True
assert (r <= a) == False
a = sqrt(2)
r = Rational(str(a.n(29)))
assert (r == a) is False
assert (r != a) is True
assert (r > a) == False
assert (r < a) == True
assert (r >= a) == False
assert (r <= a) == True
assert Eq(log(cos(2)**2 + sin(2)**2), 0) == True
def test_issue_8449():
p = Symbol('p', nonnegative=True)
assert Lt(-oo, p)
assert Ge(-oo, p) is S.false
assert Gt(oo, -p)
assert Le(oo, -p) is S.false
def test_simplify_relational():
assert simplify(x*(y + 1) - x*y - x + 1 < x) == (x > 1)
r = S(1) < x
# canonical operations are not the same as simplification,
# so if there is no simplification, canonicalization will
# be done unless the measure forbids it
assert simplify(r) == r.canonical
assert simplify(r, ratio=0) != r.canonical
# this is not a random test; in _eval_simplify
# this will simplify to S.false and that is the
# reason for the 'if r.is_Relational' in Relational's
# _eval_simplify routine
assert simplify(-(2**(3*pi/2) + 6**pi)**(1/pi) +
2*(2**(pi/2) + 3**pi)**(1/pi) < 0) is S.false
# canonical at least
for f in (Eq, Ne):
f(y, x).simplify() == f(x, y)
f(x - 1, 0).simplify() == f(x, 1)
f(x - 1, x).simplify() == S.false
f(2*x - 1, x).simplify() == f(x, 1)
f(2*x, 4).simplify() == f(x, 2)
z = cos(1)**2 + sin(1)**2 - 1 # z.is_zero is None
f(z*x, 0).simplify() == f(z*x, 0)
def test_equals():
w, x, y, z = symbols('w:z')
f = Function('f')
assert Eq(x, 1).equals(Eq(x*(y + 1) - x*y - x + 1, x))
assert Eq(x, y).equals(x < y, True) == False
assert Eq(x, f(1)).equals(Eq(x, f(2)), True) == f(1) - f(2)
assert Eq(f(1), y).equals(Eq(f(2), y), True) == f(1) - f(2)
assert Eq(x, f(1)).equals(Eq(f(2), x), True) == f(1) - f(2)
assert Eq(f(1), x).equals(Eq(x, f(2)), True) == f(1) - f(2)
assert Eq(w, x).equals(Eq(y, z), True) == False
assert Eq(f(1), f(2)).equals(Eq(f(3), f(4)), True) == f(1) - f(3)
assert (x < y).equals(y > x, True) == True
assert (x < y).equals(y >= x, True) == False
assert (x < y).equals(z < y, True) == False
assert (x < y).equals(x < z, True) == False
assert (x < f(1)).equals(x < f(2), True) == f(1) - f(2)
assert (f(1) < x).equals(f(2) < x, True) == f(1) - f(2)
def test_reversed():
assert (x < y).reversed == (y > x)
assert (x <= y).reversed == (y >= x)
assert Eq(x, y, evaluate=False).reversed == Eq(y, x, evaluate=False)
assert Ne(x, y, evaluate=False).reversed == Ne(y, x, evaluate=False)
assert (x >= y).reversed == (y <= x)
assert (x > y).reversed == (y < x)
def test_canonical():
c = [i.canonical for i in (
x + y < z,
x + 2 > 3,
x < 2,
S(2) > x,
x**2 > -x/y,
Gt(3, 2, evaluate=False)
)]
assert [i.canonical for i in c] == c
assert [i.reversed.canonical for i in c] == c
assert not any(i.lhs.is_Number and not i.rhs.is_Number for i in c)
c = [i.reversed.func(i.rhs, i.lhs, evaluate=False).canonical for i in c]
assert [i.canonical for i in c] == c
assert [i.reversed.canonical for i in c] == c
assert not any(i.lhs.is_Number and not i.rhs.is_Number for i in c)
@XFAIL
def test_issue_8444_nonworkingtests():
x = symbols('x', real=True)
assert (x <= oo) == (x >= -oo) == True
x = symbols('x')
assert x >= floor(x)
assert (x < floor(x)) == False
assert x <= ceiling(x)
assert (x > ceiling(x)) == False
def test_issue_8444_workingtests():
x = symbols('x')
assert Gt(x, floor(x)) == Gt(x, floor(x), evaluate=False)
assert Ge(x, floor(x)) == Ge(x, floor(x), evaluate=False)
assert Lt(x, ceiling(x)) == Lt(x, ceiling(x), evaluate=False)
assert Le(x, ceiling(x)) == Le(x, ceiling(x), evaluate=False)
i = symbols('i', integer=True)
assert (i > floor(i)) == False
assert (i < ceiling(i)) == False
def test_issue_10304():
d = cos(1)**2 + sin(1)**2 - 1
assert d.is_comparable is False # if this fails, find a new d
e = 1 + d*I
assert simplify(Eq(e, 0)) is S.false
def test_issue_10401():
x = symbols('x')
fin = symbols('inf', finite=True)
inf = symbols('inf', infinite=True)
inf2 = symbols('inf2', infinite=True)
zero = symbols('z', zero=True)
nonzero = symbols('nz', zero=False, finite=True)
assert Eq(1/(1/x + 1), 1).func is Eq
assert Eq(1/(1/x + 1), 1).subs(x, S.ComplexInfinity) is S.true
assert Eq(1/(1/fin + 1), 1) is S.false
T, F = S.true, S.false
assert Eq(fin, inf) is F
assert Eq(inf, inf2) is T and inf != inf2
assert Eq(inf/inf2, 0) is F
assert Eq(inf/fin, 0) is F
assert Eq(fin/inf, 0) is T
assert Eq(zero/nonzero, 0) is T and ((zero/nonzero) != 0)
assert Eq(inf, -inf) is F
assert Eq(fin/(fin + 1), 1) is S.false
o = symbols('o', odd=True)
assert Eq(o, 2*o) is S.false
p = symbols('p', positive=True)
assert Eq(p/(p - 1), 1) is F
def test_issue_10633():
assert Eq(True, False) == False
assert Eq(False, True) == False
assert Eq(True, True) == True
assert Eq(False, False) == True
def test_issue_10927():
x = symbols('x')
assert str(Eq(x, oo)) == 'Eq(x, oo)'
assert str(Eq(x, -oo)) == 'Eq(x, -oo)'
def test_issues_13081_12583_12534():
# 13081
r = Rational('905502432259640373/288230376151711744')
assert (r < pi) is S.false
assert (r > pi) is S.true
# 12583
v = sqrt(2)
u = sqrt(v) + 2/sqrt(10 - 8/sqrt(2 - v) + 4*v*(1/sqrt(2 - v) - 1))
assert (u >= 0) is S.true
# 12534; Rational vs NumberSymbol
# here are some precisions for which Rational forms
# at a lower and higher precision bracket the value of pi
# e.g. for p = 20:
# Rational(pi.n(p + 1)).n(25) = 3.14159265358979323846 2834
# pi.n(25) = 3.14159265358979323846 2643
# Rational(pi.n(p )).n(25) = 3.14159265358979323846 1987
assert [p for p in range(20, 50) if
(Rational(pi.n(p)) < pi) and
(pi < Rational(pi.n(p + 1)))] == [20, 24, 27, 33, 37, 43, 48]
# pick one such precision and affirm that the reversed operation
# gives the opposite result, i.e. if x < y is true then x > y
# must be false
p = 20
# Rational vs NumberSymbol
G = [Rational(pi.n(i)) > pi for i in (p, p + 1)]
L = [Rational(pi.n(i)) < pi for i in (p, p + 1)]
assert G == [False, True]
assert all(i is not j for i, j in zip(L, G))
# Float vs NumberSymbol
G = [pi.n(i) > pi for i in (p, p + 1)]
L = [pi.n(i) < pi for i in (p, p + 1)]
assert G == [False, True]
assert all(i is not j for i, j in zip(L, G))
# Float vs Float
G = [pi.n(p) > pi.n(p + 1)]
L = [pi.n(p) < pi.n(p + 1)]
assert G == [True]
assert all(i is not j for i, j in zip(L, G))
# Float vs Rational
# the rational form is less than the floating representation
# at the same precision
assert [i for i in range(15, 50) if Rational(pi.n(i)) > pi.n(i)] == []
# this should be the same if we reverse the relational
assert [i for i in range(15, 50) if pi.n(i) < Rational(pi.n(i))] == []
def test_binary_symbols():
ans = set([x])
for f in Eq, Ne:
for t in S.true, S.false:
eq = f(x, S.true)
assert eq.binary_symbols == ans
assert eq.reversed.binary_symbols == ans
assert f(x, 1).binary_symbols == set()
def test_rel_args():
# can't have Boolean args; this is automatic with Python 3
# so this test and the __lt__, etc..., definitions in
# relational.py and boolalg.py which are marked with ///
# can be removed.
for op in ['<', '<=', '>', '>=']:
for b in (S.true, x < 1, And(x, y)):
for v in (0.1, 1, 2**32, t, S(1)):
raises(TypeError, lambda: Relational(b, v, op))
def test_Equality_rewrite_as_Add():
eq = Eq(x + y, y - x)
assert eq.rewrite(Add) == 2*x
assert eq.rewrite(Add, evaluate=None).args == (x, x, y, -y)
assert eq.rewrite(Add, evaluate=False).args == (x, y, x, -y)
def test_issue_15847():
a = Ne(x*(x+y), x**2 + x*y)
assert simplify(a) == False
def test_negated_property():
eq = Eq(x, y)
assert eq.negated == Ne(x, y)
eq = Ne(x, y)
assert eq.negated == Eq(x, y)
eq = Ge(x + y, y - x)
assert eq.negated == Lt(x + y, y - x)
for f in (Eq, Ne, Ge, Gt, Le, Lt):
assert f(x, y).negated.negated == f(x, y)
def test_reversedsign_property():
eq = Eq(x, y)
assert eq.reversedsign == Eq(-x, -y)
eq = Ne(x, y)
assert eq.reversedsign == Ne(-x, -y)
eq = Ge(x + y, y - x)
assert eq.reversedsign == Le(-x - y, x - y)
for f in (Eq, Ne, Ge, Gt, Le, Lt):
assert f(x, y).reversedsign.reversedsign == f(x, y)
for f in (Eq, Ne, Ge, Gt, Le, Lt):
assert f(-x, y).reversedsign.reversedsign == f(-x, y)
for f in (Eq, Ne, Ge, Gt, Le, Lt):
assert f(x, -y).reversedsign.reversedsign == f(x, -y)
for f in (Eq, Ne, Ge, Gt, Le, Lt):
assert f(-x, -y).reversedsign.reversedsign == f(-x, -y)
def test_reversed_reversedsign_property():
for f in (Eq, Ne, Ge, Gt, Le, Lt):
assert f(x, y).reversed.reversedsign == f(x, y).reversedsign.reversed
for f in (Eq, Ne, Ge, Gt, Le, Lt):
assert f(-x, y).reversed.reversedsign == f(-x, y).reversedsign.reversed
for f in (Eq, Ne, Ge, Gt, Le, Lt):
assert f(x, -y).reversed.reversedsign == f(x, -y).reversedsign.reversed
for f in (Eq, Ne, Ge, Gt, Le, Lt):
assert f(-x, -y).reversed.reversedsign == \
f(-x, -y).reversedsign.reversed
def test_improved_canonical():
def test_different_forms(listofforms):
for form1, form2 in combinations(listofforms, 2):
assert form1.canonical == form2.canonical
def generate_forms(expr):
return [expr, expr.reversed, expr.reversedsign,
expr.reversed.reversedsign]
test_different_forms(generate_forms(x > -y))
test_different_forms(generate_forms(x >= -y))
test_different_forms(generate_forms(Eq(x, -y)))
test_different_forms(generate_forms(Ne(x, -y)))
test_different_forms(generate_forms(pi < x))
test_different_forms(generate_forms(pi - 5*y < -x + 2*y**2 - 7))
assert (pi >= x).canonical == (x <= pi)
|
16fc36fd78f4a1320b30d132f3acbb41e193325dc1158e4726be0dc30510c2cd
|
from sympy import (Symbol, exp, Integer, Float, sin, cos, log, Poly, Lambda,
Function, I, S, N, sqrt, srepr, Rational, Tuple, Matrix, Interval, Add, Mul,
Pow, Or, true, false, Abs, pi, Range, Xor)
from sympy.abc import x, y
from sympy.core.sympify import sympify, _sympify, SympifyError, kernS
from sympy.core.decorators import _sympifyit
from sympy.external import import_module
from sympy.utilities.pytest import raises, XFAIL, skip
from sympy.utilities.decorator import conserve_mpmath_dps
from sympy.geometry import Point, Line
from sympy.functions.combinatorial.factorials import factorial, factorial2
from sympy.abc import _clash, _clash1, _clash2
from sympy.core.compatibility import exec_, HAS_GMPY, PY3
from sympy.sets import FiniteSet, EmptySet
from sympy.tensor.array.dense_ndim_array import ImmutableDenseNDimArray
from sympy.external import import_module
import mpmath
from mpmath.rational import mpq
numpy = import_module('numpy')
def test_issue_3538():
v = sympify("exp(x)")
assert v == exp(x)
assert type(v) == type(exp(x))
assert str(type(v)) == str(type(exp(x)))
def test_sympify1():
assert sympify("x") == Symbol("x")
assert sympify(" x") == Symbol("x")
assert sympify(" x ") == Symbol("x")
# issue 4877
n1 = Rational(1, 2)
assert sympify('--.5') == n1
assert sympify('-1/2') == -n1
assert sympify('-+--.5') == -n1
assert sympify('-.[3]') == Rational(-1, 3)
assert sympify('.[3]') == Rational(1, 3)
assert sympify('+.[3]') == Rational(1, 3)
assert sympify('+0.[3]*10**-2') == Rational(1, 300)
assert sympify('.[052631578947368421]') == Rational(1, 19)
assert sympify('.0[526315789473684210]') == Rational(1, 19)
assert sympify('.034[56]') == Rational(1711, 49500)
# options to make reals into rationals
assert sympify('1.22[345]', rational=True) == \
1 + Rational(22, 100) + Rational(345, 99900)
assert sympify('2/2.6', rational=True) == Rational(10, 13)
assert sympify('2.6/2', rational=True) == Rational(13, 10)
assert sympify('2.6e2/17', rational=True) == Rational(260, 17)
assert sympify('2.6e+2/17', rational=True) == Rational(260, 17)
assert sympify('2.6e-2/17', rational=True) == Rational(26, 17000)
assert sympify('2.1+3/4', rational=True) == \
Rational(21, 10) + Rational(3, 4)
assert sympify('2.234456', rational=True) == Rational(279307, 125000)
assert sympify('2.234456e23', rational=True) == 223445600000000000000000
assert sympify('2.234456e-23', rational=True) == \
Rational(279307, 12500000000000000000000000000)
assert sympify('-2.234456e-23', rational=True) == \
Rational(-279307, 12500000000000000000000000000)
assert sympify('12345678901/17', rational=True) == \
Rational(12345678901, 17)
assert sympify('1/.3 + x', rational=True) == Rational(10, 3) + x
# make sure longs in fractions work
assert sympify('222222222222/11111111111') == \
Rational(222222222222, 11111111111)
# ... even if they come from repetend notation
assert sympify('1/.2[123456789012]') == Rational(333333333333, 70781892967)
# ... or from high precision reals
assert sympify('.1234567890123456', rational=True) == \
Rational(19290123283179, 156250000000000)
def test_sympify_Fraction():
try:
import fractions
except ImportError:
pass
else:
value = sympify(fractions.Fraction(101, 127))
assert value == Rational(101, 127) and type(value) is Rational
def test_sympify_gmpy():
if HAS_GMPY:
if HAS_GMPY == 2:
import gmpy2 as gmpy
elif HAS_GMPY == 1:
import gmpy
value = sympify(gmpy.mpz(1000001))
assert value == Integer(1000001) and type(value) is Integer
value = sympify(gmpy.mpq(101, 127))
assert value == Rational(101, 127) and type(value) is Rational
@conserve_mpmath_dps
def test_sympify_mpmath():
value = sympify(mpmath.mpf(1.0))
assert value == Float(1.0) and type(value) is Float
mpmath.mp.dps = 12
assert sympify(
mpmath.pi).epsilon_eq(Float("3.14159265359"), Float("1e-12")) == True
assert sympify(
mpmath.pi).epsilon_eq(Float("3.14159265359"), Float("1e-13")) == False
mpmath.mp.dps = 6
assert sympify(
mpmath.pi).epsilon_eq(Float("3.14159"), Float("1e-5")) == True
assert sympify(
mpmath.pi).epsilon_eq(Float("3.14159"), Float("1e-6")) == False
assert sympify(mpmath.mpc(1.0 + 2.0j)) == Float(1.0) + Float(2.0)*I
assert sympify(mpq(1, 2)) == S.Half
def test_sympify2():
class A:
def _sympy_(self):
return Symbol("x")**3
a = A()
assert _sympify(a) == x**3
assert sympify(a) == x**3
assert a == x**3
def test_sympify3():
assert sympify("x**3") == x**3
assert sympify("x^3") == x**3
assert sympify("1/2") == Integer(1)/2
raises(SympifyError, lambda: _sympify('x**3'))
raises(SympifyError, lambda: _sympify('1/2'))
def test_sympify_keywords():
raises(SympifyError, lambda: sympify('if'))
raises(SympifyError, lambda: sympify('for'))
raises(SympifyError, lambda: sympify('while'))
raises(SympifyError, lambda: sympify('lambda'))
def test_sympify_float():
assert sympify("1e-64") != 0
assert sympify("1e-20000") != 0
def test_sympify_bool():
assert sympify(True) is true
assert sympify(False) is false
def test_sympyify_iterables():
ans = [Rational(3, 10), Rational(1, 5)]
assert sympify(['.3', '.2'], rational=True) == ans
assert sympify(tuple(['.3', '.2']), rational=True) == Tuple(*ans)
assert sympify(dict(x=0, y=1)) == {x: 0, y: 1}
assert sympify(['1', '2', ['3', '4']]) == [S(1), S(2), [S(3), S(4)]]
def test_sympify4():
class A:
def _sympy_(self):
return Symbol("x")
a = A()
assert _sympify(a)**3 == x**3
assert sympify(a)**3 == x**3
assert a == x
def test_sympify_text():
assert sympify('some') == Symbol('some')
assert sympify('core') == Symbol('core')
assert sympify('True') is True
assert sympify('False') is False
assert sympify('Poly') == Poly
assert sympify('sin') == sin
def test_sympify_function():
assert sympify('factor(x**2-1, x)') == -(1 - x)*(x + 1)
assert sympify('sin(pi/2)*cos(pi)') == -Integer(1)
def test_sympify_poly():
p = Poly(x**2 + x + 1, x)
assert _sympify(p) is p
assert sympify(p) is p
def test_sympify_factorial():
assert sympify('x!') == factorial(x)
assert sympify('(x+1)!') == factorial(x + 1)
assert sympify('(1 + y*(x + 1))!') == factorial(1 + y*(x + 1))
assert sympify('(1 + y*(x + 1)!)^2') == (1 + y*factorial(x + 1))**2
assert sympify('y*x!') == y*factorial(x)
assert sympify('x!!') == factorial2(x)
assert sympify('(x+1)!!') == factorial2(x + 1)
assert sympify('(1 + y*(x + 1))!!') == factorial2(1 + y*(x + 1))
assert sympify('(1 + y*(x + 1)!!)^2') == (1 + y*factorial2(x + 1))**2
assert sympify('y*x!!') == y*factorial2(x)
assert sympify('factorial2(x)!') == factorial(factorial2(x))
raises(SympifyError, lambda: sympify("+!!"))
raises(SympifyError, lambda: sympify(")!!"))
raises(SympifyError, lambda: sympify("!"))
raises(SympifyError, lambda: sympify("(!)"))
raises(SympifyError, lambda: sympify("x!!!"))
def test_sage():
# how to effectivelly test for the _sage_() method without having SAGE
# installed?
assert hasattr(x, "_sage_")
assert hasattr(Integer(3), "_sage_")
assert hasattr(sin(x), "_sage_")
assert hasattr(cos(x), "_sage_")
assert hasattr(x**2, "_sage_")
assert hasattr(x + y, "_sage_")
assert hasattr(exp(x), "_sage_")
assert hasattr(log(x), "_sage_")
def test_issue_3595():
assert sympify("a_") == Symbol("a_")
assert sympify("_a") == Symbol("_a")
def test_lambda():
x = Symbol('x')
assert sympify('lambda: 1') == Lambda((), 1)
assert sympify('lambda x: x') == Lambda(x, x)
assert sympify('lambda x: 2*x') == Lambda(x, 2*x)
assert sympify('lambda x, y: 2*x+y') == Lambda([x, y], 2*x + y)
def test_lambda_raises():
raises(SympifyError, lambda: sympify("lambda *args: args")) # args argument error
raises(SympifyError, lambda: sympify("lambda **kwargs: kwargs[0]")) # kwargs argument error
raises(SympifyError, lambda: sympify("lambda x = 1: x")) # Keyword argument error
with raises(SympifyError):
_sympify('lambda: 1')
def test_sympify_raises():
raises(SympifyError, lambda: sympify("fx)"))
def test__sympify():
x = Symbol('x')
f = Function('f')
# positive _sympify
assert _sympify(x) is x
assert _sympify(f) is f
assert _sympify(1) == Integer(1)
assert _sympify(0.5) == Float("0.5")
assert _sympify(1 + 1j) == 1.0 + I*1.0
class A:
def _sympy_(self):
return Integer(5)
a = A()
assert _sympify(a) == Integer(5)
# negative _sympify
raises(SympifyError, lambda: _sympify('1'))
raises(SympifyError, lambda: _sympify([1, 2, 3]))
def test_sympifyit():
x = Symbol('x')
y = Symbol('y')
@_sympifyit('b', NotImplemented)
def add(a, b):
return a + b
assert add(x, 1) == x + 1
assert add(x, 0.5) == x + Float('0.5')
assert add(x, y) == x + y
assert add(x, '1') == NotImplemented
@_sympifyit('b')
def add_raises(a, b):
return a + b
assert add_raises(x, 1) == x + 1
assert add_raises(x, 0.5) == x + Float('0.5')
assert add_raises(x, y) == x + y
raises(SympifyError, lambda: add_raises(x, '1'))
def test_int_float():
class F1_1(object):
def __float__(self):
return 1.1
class F1_1b(object):
"""
This class is still a float, even though it also implements __int__().
"""
def __float__(self):
return 1.1
def __int__(self):
return 1
class F1_1c(object):
"""
This class is still a float, because it implements _sympy_()
"""
def __float__(self):
return 1.1
def __int__(self):
return 1
def _sympy_(self):
return Float(1.1)
class I5(object):
def __int__(self):
return 5
class I5b(object):
"""
This class implements both __int__() and __float__(), so it will be
treated as Float in SymPy. One could change this behavior, by using
float(a) == int(a), but deciding that integer-valued floats represent
exact numbers is arbitrary and often not correct, so we do not do it.
If, in the future, we decide to do it anyway, the tests for I5b need to
be changed.
"""
def __float__(self):
return 5.0
def __int__(self):
return 5
class I5c(object):
"""
This class implements both __int__() and __float__(), but also
a _sympy_() method, so it will be Integer.
"""
def __float__(self):
return 5.0
def __int__(self):
return 5
def _sympy_(self):
return Integer(5)
i5 = I5()
i5b = I5b()
i5c = I5c()
f1_1 = F1_1()
f1_1b = F1_1b()
f1_1c = F1_1c()
assert sympify(i5) == 5
assert isinstance(sympify(i5), Integer)
assert sympify(i5b) == 5
assert isinstance(sympify(i5b), Float)
assert sympify(i5c) == 5
assert isinstance(sympify(i5c), Integer)
assert abs(sympify(f1_1) - 1.1) < 1e-5
assert abs(sympify(f1_1b) - 1.1) < 1e-5
assert abs(sympify(f1_1c) - 1.1) < 1e-5
assert _sympify(i5) == 5
assert isinstance(_sympify(i5), Integer)
assert _sympify(i5b) == 5
assert isinstance(_sympify(i5b), Float)
assert _sympify(i5c) == 5
assert isinstance(_sympify(i5c), Integer)
assert abs(_sympify(f1_1) - 1.1) < 1e-5
assert abs(_sympify(f1_1b) - 1.1) < 1e-5
assert abs(_sympify(f1_1c) - 1.1) < 1e-5
def test_evaluate_false():
cases = {
'2 + 3': Add(2, 3, evaluate=False),
'2**2 / 3': Mul(Pow(2, 2, evaluate=False), Pow(3, -1, evaluate=False), evaluate=False),
'2 + 3 * 5': Add(2, Mul(3, 5, evaluate=False), evaluate=False),
'2 - 3 * 5': Add(2, Mul(-1, Mul(3, 5,evaluate=False), evaluate=False), evaluate=False),
'1 / 3': Mul(1, Pow(3, -1, evaluate=False), evaluate=False),
'True | False': Or(True, False, evaluate=False),
'1 + 2 + 3 + 5*3 + integrate(x)': Add(1, 2, 3, Mul(5, 3, evaluate=False), x**2/2, evaluate=False),
'2 * 4 * 6 + 8': Add(Mul(2, 4, 6, evaluate=False), 8, evaluate=False),
'2 - 8 / 4': Add(2, Mul(-1, Mul(8, Pow(4, -1, evaluate=False), evaluate=False), evaluate=False), evaluate=False),
'2 - 2**2': Add(2, Mul(-1, Pow(2, 2, evaluate=False), evaluate=False), evaluate=False),
}
for case, result in cases.items():
assert sympify(case, evaluate=False) == result
def test_issue_4133():
a = sympify('Integer(4)')
assert a == Integer(4)
assert a.is_Integer
def test_issue_3982():
a = [3, 2.0]
assert sympify(a) == [Integer(3), Float(2.0)]
assert sympify(tuple(a)) == Tuple(Integer(3), Float(2.0))
assert sympify(set(a)) == FiniteSet(Integer(3), Float(2.0))
def test_S_sympify():
assert S(1)/2 == sympify(1)/2
assert (-2)**(S(1)/2) == sqrt(2)*I
def test_issue_4788():
assert srepr(S(1.0 + 0J)) == srepr(S(1.0)) == srepr(Float(1.0))
def test_issue_4798_None():
assert S(None) is None
def test_issue_3218():
assert sympify("x+\ny") == x + y
def test_issue_4988_builtins():
C = Symbol('C')
vars = {'C': C}
exp1 = sympify('C')
assert exp1 == C # Make sure it did not get mixed up with sympy.C
exp2 = sympify('C', vars)
assert exp2 == C # Make sure it did not get mixed up with sympy.C
def test_geometry():
p = sympify(Point(0, 1))
assert p == Point(0, 1) and isinstance(p, Point)
L = sympify(Line(p, (1, 0)))
assert L == Line((0, 1), (1, 0)) and isinstance(L, Line)
def test_kernS():
s = '-1 - 2*(-(-x + 1/x)/(x*(x - 1/x)**2) - 1/(x*(x - 1/x)))'
# when 1497 is fixed, this no longer should pass: the expression
# should be unchanged
assert -1 - 2*(-(-x + 1/x)/(x*(x - 1/x)**2) - 1/(x*(x - 1/x))) == -1
# sympification should not allow the constant to enter a Mul
# or else the structure can change dramatically
ss = kernS(s)
assert ss != -1 and ss.simplify() == -1
s = '-1 - 2*(-(-x + 1/x)/(x*(x - 1/x)**2) - 1/(x*(x - 1/x)))'.replace(
'x', '_kern')
ss = kernS(s)
assert ss != -1 and ss.simplify() == -1
# issue 6687
assert kernS('Interval(-1,-2 - 4*(-3))') == Interval(-1, 10)
assert kernS('_kern') == Symbol('_kern')
assert kernS('E**-(x)') == exp(-x)
e = 2*(x + y)*y
assert kernS(['2*(x + y)*y', ('2*(x + y)*y',)]) == [e, (e,)]
assert kernS('-(2*sin(x)**2 + 2*sin(x)*cos(x))*y/2') == \
-y*(2*sin(x)**2 + 2*sin(x)*cos(x))/2
# issue 15132
assert kernS('(1 - x)/(1 - x*(1-y))') == kernS('(1-x)/(1-(1-y)*x)')
assert kernS('(1-2**-(4+1)*(1-y)*x)') == (1 - x*(1 - y)/32)
assert kernS('(1-2**(4+1)*(1-y)*x)') == (1 - 32*x*(1 - y))
assert kernS('(1-2.*(1-y)*x)') == 1 - 2.*x*(1 - y)
one = kernS('x - (x - 1)')
assert one != 1 and one.expand() == 1
def test_issue_6540_6552():
assert S('[[1/3,2], (2/5,)]') == [[Rational(1, 3), 2], (Rational(2, 5),)]
assert S('[[2/6,2], (2/4,)]') == [[Rational(1, 3), 2], (Rational(1, 2),)]
assert S('[[[2*(1)]]]') == [[[2]]]
assert S('Matrix([2*(1)])') == Matrix([2])
def test_issue_6046():
assert str(S("Q & C", locals=_clash1)) == 'C & Q'
assert str(S('pi(x)', locals=_clash2)) == 'pi(x)'
assert str(S('pi(C, Q)', locals=_clash)) == 'pi(C, Q)'
locals = {}
exec_("from sympy.abc import Q, C", locals)
assert str(S('C&Q', locals)) == 'C & Q'
def test_issue_8821_highprec_from_str():
s = str(pi.evalf(128))
p = sympify(s)
assert Abs(sin(p)) < 1e-127
def test_issue_10295():
if not numpy:
skip("numpy not installed.")
A = numpy.array([[1, 3, -1],
[0, 1, 7]])
sA = S(A)
assert sA.shape == (2, 3)
for (ri, ci), val in numpy.ndenumerate(A):
assert sA[ri, ci] == val
B = numpy.array([-7, x, 3*y**2])
sB = S(B)
assert B[0] == -7
assert B[1] == x
assert B[2] == 3*y**2
C = numpy.arange(0, 24)
C.resize(2,3,4)
sC = S(C)
assert sC[0, 0, 0].is_integer
assert sC[0, 0, 0] == 0
a1 = numpy.array([1, 2, 3])
a2 = numpy.array([i for i in range(24)])
a2.resize(2, 4, 3)
assert sympify(a1) == ImmutableDenseNDimArray([1, 2, 3])
assert sympify(a2) == ImmutableDenseNDimArray([i for i in range(24)], (2, 4, 3))
def test_Range():
# Only works in Python 3 where range returns a range type
if PY3:
builtin_range = range
else:
builtin_range = xrange
assert sympify(builtin_range(10)) == Range(10)
assert _sympify(builtin_range(10)) == Range(10)
def test_sympify_set():
n = Symbol('n')
assert sympify({n}) == FiniteSet(n)
assert sympify(set()) == EmptySet()
def test_sympify_numpy():
if not numpy:
skip('numpy not installed. Abort numpy tests.')
np = numpy
def equal(x, y):
return x == y and type(x) == type(y)
assert sympify(np.bool_(1)) is S(True)
try:
assert equal(
sympify(np.int_(1234567891234567891)), S(1234567891234567891))
assert equal(
sympify(np.intp(1234567891234567891)), S(1234567891234567891))
except OverflowError:
# May fail on 32-bit systems: Python int too large to convert to C long
pass
assert equal(sympify(np.intc(1234567891)), S(1234567891))
assert equal(sympify(np.int8(-123)), S(-123))
assert equal(sympify(np.int16(-12345)), S(-12345))
assert equal(sympify(np.int32(-1234567891)), S(-1234567891))
assert equal(
sympify(np.int64(-1234567891234567891)), S(-1234567891234567891))
assert equal(sympify(np.uint8(123)), S(123))
assert equal(sympify(np.uint16(12345)), S(12345))
assert equal(sympify(np.uint32(1234567891)), S(1234567891))
assert equal(
sympify(np.uint64(1234567891234567891)), S(1234567891234567891))
assert equal(sympify(np.float32(1.123456)), Float(1.123456, precision=24))
assert equal(sympify(np.float64(1.1234567891234)),
Float(1.1234567891234, precision=53))
assert equal(sympify(np.longdouble(1.123456789)),
Float(1.123456789, precision=80))
assert equal(sympify(np.complex64(1 + 2j)), S(1.0 + 2.0*I))
assert equal(sympify(np.complex128(1 + 2j)), S(1.0 + 2.0*I))
assert equal(sympify(np.longcomplex(1 + 2j)), S(1.0 + 2.0*I))
#float96 does not exist on all platforms
if hasattr(np, 'float96'):
assert equal(sympify(np.float96(1.123456789)),
Float(1.123456789, precision=80))
#float128 does not exist on all platforms
if hasattr(np, 'float128'):
assert equal(sympify(np.float128(1.123456789123)),
Float(1.123456789123, precision=80))
@XFAIL
def test_sympify_rational_numbers_set():
ans = [Rational(3, 10), Rational(1, 5)]
assert sympify({'.3', '.2'}, rational=True) == FiniteSet(*ans)
def test_issue_13924():
if not numpy:
skip("numpy not installed.")
a = sympify(numpy.array([1]))
assert isinstance(a, ImmutableDenseNDimArray)
assert a[0] == 1
def test_numpy_sympify_args():
# Issue 15098. Make sure sympify args work with numpy types (like numpy.str_)
if not numpy:
skip("numpy not installed.")
a = sympify(numpy.str_('a'))
assert type(a) is Symbol
assert a == Symbol('a')
class CustomSymbol(Symbol):
pass
a = sympify(numpy.str_('a'), {"Symbol": CustomSymbol})
assert isinstance(a, CustomSymbol)
a = sympify(numpy.str_('x^y'))
assert a == x**y
a = sympify(numpy.str_('x^y'), convert_xor=False)
assert a == Xor(x, y)
raises(SympifyError, lambda: sympify(numpy.str_('x'), strict=True))
a = sympify(numpy.str_('1.1'))
assert isinstance(a, Float)
assert a == 1.1
a = sympify(numpy.str_('1.1'), rational=True)
assert isinstance(a, Rational)
assert a == Rational(11, 10)
a = sympify(numpy.str_('x + x'))
assert isinstance(a, Mul)
assert a == 2*x
a = sympify(numpy.str_('x + x'), evaluate=False)
assert isinstance(a, Add)
assert a == Add(x, x, evaluate=False)
def test_issue_5939():
a = Symbol('a')
b = Symbol('b')
assert sympify('''a+\nb''') == a + b
|
a6b8fbfdcb6e561f8b8c9a8475d55e179bad78b346abf0240b72808dd278d797
|
from sympy import (Abs, Add, atan, ceiling, cos, E, Eq, exp, factor,
factorial, fibonacci, floor, Function, GoldenRatio, I, Integral,
integrate, log, Mul, N, oo, pi, Pow, product, Product,
Rational, S, Sum, simplify, sin, sqrt, sstr, sympify, Symbol, Max, nfloat)
from sympy.core.evalf import (complex_accuracy, PrecisionExhausted,
scaled_zero, get_integer_part, as_mpmath, evalf)
from mpmath import inf, ninf
from mpmath.libmp.libmpf import from_float
from sympy.core.compatibility import long, range
from sympy.utilities.pytest import raises, XFAIL
from sympy.abc import n, x, y
def NS(e, n=15, **options):
return sstr(sympify(e).evalf(n, **options), full_prec=True)
def test_evalf_helpers():
assert complex_accuracy((from_float(2.0), None, 35, None)) == 35
assert complex_accuracy((from_float(2.0), from_float(10.0), 35, 100)) == 37
assert complex_accuracy(
(from_float(2.0), from_float(1000.0), 35, 100)) == 43
assert complex_accuracy((from_float(2.0), from_float(10.0), 100, 35)) == 35
assert complex_accuracy(
(from_float(2.0), from_float(1000.0), 100, 35)) == 35
def test_evalf_basic():
assert NS('pi', 15) == '3.14159265358979'
assert NS('2/3', 10) == '0.6666666667'
assert NS('355/113-pi', 6) == '2.66764e-7'
assert NS('16*atan(1/5)-4*atan(1/239)', 15) == '3.14159265358979'
def test_cancellation():
assert NS(Add(pi, Rational(1, 10**1000), -pi, evaluate=False), 15,
maxn=1200) == '1.00000000000000e-1000'
def test_evalf_powers():
assert NS('pi**(10**20)', 10) == '1.339148777e+49714987269413385435'
assert NS(pi**(10**100), 10) == ('4.946362032e+4971498726941338543512682882'
'9089887365167832438044244613405349992494711208'
'95526746555473864642912223')
assert NS('2**(1/10**50)', 15) == '1.00000000000000'
assert NS('2**(1/10**50)-1', 15) == '6.93147180559945e-51'
# Evaluation of Rump's ill-conditioned polynomial
def test_evalf_rump():
a = 1335*y**6/4 + x**2*(11*x**2*y**2 - y**6 - 121*y**4 - 2) + 11*y**8/2 + x/(2*y)
assert NS(a, 15, subs={x: 77617, y: 33096}) == '-0.827396059946821'
def test_evalf_complex():
assert NS('2*sqrt(pi)*I', 10) == '3.544907702*I'
assert NS('3+3*I', 15) == '3.00000000000000 + 3.00000000000000*I'
assert NS('E+pi*I', 15) == '2.71828182845905 + 3.14159265358979*I'
assert NS('pi * (3+4*I)', 15) == '9.42477796076938 + 12.5663706143592*I'
assert NS('I*(2+I)', 15) == '-1.00000000000000 + 2.00000000000000*I'
@XFAIL
def test_evalf_complex_bug():
assert NS('(pi+E*I)*(E+pi*I)', 15) in ('0.e-15 + 17.25866050002*I',
'0.e-17 + 17.25866050002*I', '-0.e-17 + 17.25866050002*I')
def test_evalf_complex_powers():
assert NS('(E+pi*I)**100000000000000000') == \
'-3.58896782867793e+61850354284995199 + 4.58581754997159e+61850354284995199*I'
# XXX: rewrite if a+a*I simplification introduced in sympy
#assert NS('(pi + pi*I)**2') in ('0.e-15 + 19.7392088021787*I', '0.e-16 + 19.7392088021787*I')
assert NS('(pi + pi*I)**2', chop=True) == '19.7392088021787*I'
assert NS(
'(pi + 1/10**8 + pi*I)**2') == '6.2831853e-8 + 19.7392088650106*I'
assert NS('(pi + 1/10**12 + pi*I)**2') == '6.283e-12 + 19.7392088021850*I'
assert NS('(pi + pi*I)**4', chop=True) == '-389.636364136010'
assert NS(
'(pi + 1/10**8 + pi*I)**4') == '-389.636366616512 + 2.4805021e-6*I'
assert NS('(pi + 1/10**12 + pi*I)**4') == '-389.636364136258 + 2.481e-10*I'
assert NS(
'(10000*pi + 10000*pi*I)**4', chop=True) == '-3.89636364136010e+18'
@XFAIL
def test_evalf_complex_powers_bug():
assert NS('(pi + pi*I)**4') == '-389.63636413601 + 0.e-14*I'
def test_evalf_exponentiation():
assert NS(sqrt(-pi)) == '1.77245385090552*I'
assert NS(Pow(pi*I, Rational(
1, 2), evaluate=False)) == '1.25331413731550 + 1.25331413731550*I'
assert NS(pi**I) == '0.413292116101594 + 0.910598499212615*I'
assert NS(pi**(E + I/3)) == '20.8438653991931 + 8.36343473930031*I'
assert NS((pi + I/3)**(E + I/3)) == '17.2442906093590 + 13.6839376767037*I'
assert NS(exp(pi)) == '23.1406926327793'
assert NS(exp(pi + E*I)) == '-21.0981542849657 + 9.50576358282422*I'
assert NS(pi**pi) == '36.4621596072079'
assert NS((-pi)**pi) == '-32.9138577418939 - 15.6897116534332*I'
assert NS((-pi)**(-pi)) == '-0.0247567717232697 + 0.0118013091280262*I'
# An example from Smith, "Multiple Precision Complex Arithmetic and Functions"
def test_evalf_complex_cancellation():
A = Rational('63287/100000')
B = Rational('52498/100000')
C = Rational('69301/100000')
D = Rational('83542/100000')
F = Rational('2231321613/2500000000')
# XXX: the number of returned mantissa digits in the real part could
# change with the implementation. What matters is that the returned digits are
# correct; those that are showing now are correct.
# >>> ((A+B*I)*(C+D*I)).expand()
# 64471/10000000000 + 2231321613*I/2500000000
# >>> 2231321613*4
# 8925286452L
assert NS((A + B*I)*(C + D*I), 6) == '6.44710e-6 + 0.892529*I'
assert NS((A + B*I)*(C + D*I), 10) == '6.447100000e-6 + 0.8925286452*I'
assert NS((A + B*I)*(
C + D*I) - F*I, 5) in ('6.4471e-6 + 0.e-14*I', '6.4471e-6 - 0.e-14*I')
def test_evalf_logs():
assert NS("log(3+pi*I)", 15) == '1.46877619736226 + 0.808448792630022*I'
assert NS("log(pi*I)", 15) == '1.14472988584940 + 1.57079632679490*I'
assert NS('log(-1 + 0.00001)', 2) == '-1.0e-5 + 3.1*I'
assert NS('log(100, 10, evaluate=False)', 15) == '2.00000000000000'
assert NS('-2*I*log(-(-1)**(S(1)/9))', 15) == '-5.58505360638185'
def test_evalf_trig():
assert NS('sin(1)', 15) == '0.841470984807897'
assert NS('cos(1)', 15) == '0.540302305868140'
assert NS('sin(10**-6)', 15) == '9.99999999999833e-7'
assert NS('cos(10**-6)', 15) == '0.999999999999500'
assert NS('sin(E*10**100)', 15) == '0.409160531722613'
# Some input near roots
assert NS(sin(exp(pi*sqrt(163))*pi), 15) == '-2.35596641936785e-12'
assert NS(sin(pi*10**100 + Rational(7, 10**5), evaluate=False), 15, maxn=120) == \
'6.99999999428333e-5'
assert NS(sin(Rational(7, 10**5), evaluate=False), 15) == \
'6.99999999428333e-5'
# Check detection of various false identities
def test_evalf_near_integers():
# Binet's formula
f = lambda n: ((1 + sqrt(5))**n)/(2**n * sqrt(5))
assert NS(f(5000) - fibonacci(5000), 10, maxn=1500) == '5.156009964e-1046'
# Some near-integer identities from
# http://mathworld.wolfram.com/AlmostInteger.html
assert NS('sin(2017*2**(1/5))', 15) == '-1.00000000000000'
assert NS('sin(2017*2**(1/5))', 20) == '-0.99999999999999997857'
assert NS('1+sin(2017*2**(1/5))', 15) == '2.14322287389390e-17'
assert NS('45 - 613*E/37 + 35/991', 15) == '6.03764498766326e-11'
def test_evalf_ramanujan():
assert NS(exp(pi*sqrt(163)) - 640320**3 - 744, 10) == '-7.499274028e-13'
# A related identity
A = 262537412640768744*exp(-pi*sqrt(163))
B = 196884*exp(-2*pi*sqrt(163))
C = 103378831900730205293632*exp(-3*pi*sqrt(163))
assert NS(1 - A - B + C, 10) == '1.613679005e-59'
# Input that for various reasons have failed at some point
def test_evalf_bugs():
assert NS(sin(1) + exp(-10**10), 10) == NS(sin(1), 10)
assert NS(exp(10**10) + sin(1), 10) == NS(exp(10**10), 10)
assert NS('expand_log(log(1+1/10**50))', 20) == '1.0000000000000000000e-50'
assert NS('log(10**100,10)', 10) == '100.0000000'
assert NS('log(2)', 10) == '0.6931471806'
assert NS(
'(sin(x)-x)/x**3', 15, subs={x: '1/10**50'}) == '-0.166666666666667'
assert NS(sin(1) + Rational(
1, 10**100)*I, 15) == '0.841470984807897 + 1.00000000000000e-100*I'
assert x.evalf() == x
assert NS((1 + I)**2*I, 6) == '-2.00000'
d = {n: (
-1)**Rational(6, 7), y: (-1)**Rational(4, 7), x: (-1)**Rational(2, 7)}
assert NS((x*(1 + y*(1 + n))).subs(d).evalf(), 6) == '0.346011 + 0.433884*I'
assert NS(((-I - sqrt(2)*I)**2).evalf()) == '-5.82842712474619'
assert NS((1 + I)**2*I, 15) == '-2.00000000000000'
# issue 4758 (1/2):
assert NS(pi.evalf(69) - pi) == '-4.43863937855894e-71'
# issue 4758 (2/2): With the bug present, this still only fails if the
# terms are in the order given here. This is not generally the case,
# because the order depends on the hashes of the terms.
assert NS(20 - 5008329267844*n**25 - 477638700*n**37 - 19*n,
subs={n: .01}) == '19.8100000000000'
assert NS(((x - 1)*((1 - x))**1000).n()
) == '(1.00000000000000 - x)**1000*(x - 1.00000000000000)'
assert NS((-x).n()) == '-x'
assert NS((-2*x).n()) == '-2.00000000000000*x'
assert NS((-2*x*y).n()) == '-2.00000000000000*x*y'
assert cos(x).n(subs={x: 1+I}) == cos(x).subs(x, 1+I).n()
# issue 6660. Also NaN != mpmath.nan
# In this order:
# 0*nan, 0/nan, 0*inf, 0/inf
# 0+nan, 0-nan, 0+inf, 0-inf
# >>> n = Some Number
# n*nan, n/nan, n*inf, n/inf
# n+nan, n-nan, n+inf, n-inf
assert (0*E**(oo)).n() == S.NaN
assert (0/E**(oo)).n() == S.Zero
assert (0+E**(oo)).n() == S.Infinity
assert (0-E**(oo)).n() == S.NegativeInfinity
assert (5*E**(oo)).n() == S.Infinity
assert (5/E**(oo)).n() == S.Zero
assert (5+E**(oo)).n() == S.Infinity
assert (5-E**(oo)).n() == S.NegativeInfinity
#issue 7416
assert as_mpmath(0.0, 10, {'chop': True}) == 0
#issue 5412
assert ((oo*I).n() == S.Infinity*I)
assert ((oo+oo*I).n() == S.Infinity + S.Infinity*I)
#issue 11518
assert NS(2*x**2.5, 5) == '2.0000*x**2.5000'
#issue 13076
assert NS(Mul(Max(0, y), x, evaluate=False).evalf()) == 'x*Max(0, y)'
def test_evalf_integer_parts():
a = floor(log(8)/log(2) - exp(-1000), evaluate=False)
b = floor(log(8)/log(2), evaluate=False)
assert a.evalf() == 3
assert b.evalf() == 3
# equals, as a fallback, can still fail but it might succeed as here
assert ceiling(10*(sin(1)**2 + cos(1)**2)) == 10
assert int(floor(factorial(50)/E, evaluate=False).evalf(70)) == \
long(11188719610782480504630258070757734324011354208865721592720336800)
assert int(ceiling(factorial(50)/E, evaluate=False).evalf(70)) == \
long(11188719610782480504630258070757734324011354208865721592720336801)
assert int(floor((GoldenRatio**999 / sqrt(5) + Rational(1, 2)))
.evalf(1000)) == fibonacci(999)
assert int(floor((GoldenRatio**1000 / sqrt(5) + Rational(1, 2)))
.evalf(1000)) == fibonacci(1000)
assert ceiling(x).evalf(subs={x: 3}) == 3
assert ceiling(x).evalf(subs={x: 3*I}) == 3*I
assert ceiling(x).evalf(subs={x: 2 + 3*I}) == 2 + 3*I
assert ceiling(x).evalf(subs={x: 3.}) == 3
assert ceiling(x).evalf(subs={x: 3.*I}) == 3*I
assert ceiling(x).evalf(subs={x: 2. + 3*I}) == 2 + 3*I
assert float((floor(1.5, evaluate=False)+1/9).evalf()) == 1 + 1/9
assert float((floor(0.5, evaluate=False)+20).evalf()) == 20
def test_evalf_trig_zero_detection():
a = sin(160*pi, evaluate=False)
t = a.evalf(maxn=100)
assert abs(t) < 1e-100
assert t._prec < 2
assert a.evalf(chop=True) == 0
raises(PrecisionExhausted, lambda: a.evalf(strict=True))
def test_evalf_sum():
assert Sum(n,(n,1,2)).evalf() == 3.
assert Sum(n,(n,1,2)).doit().evalf() == 3.
# the next test should return instantly
assert Sum(1/n,(n,1,2)).evalf() == 1.5
# issue 8219
assert Sum(E/factorial(n), (n, 0, oo)).evalf() == (E*E).evalf()
# issue 8254
assert Sum(2**n*n/factorial(n), (n, 0, oo)).evalf() == (2*E*E).evalf()
# issue 8411
s = Sum(1/x**2, (x, 100, oo))
assert s.n() == s.doit().n()
def test_evalf_divergent_series():
raises(ValueError, lambda: Sum(1/n, (n, 1, oo)).evalf())
raises(ValueError, lambda: Sum(n/(n**2 + 1), (n, 1, oo)).evalf())
raises(ValueError, lambda: Sum((-1)**n, (n, 1, oo)).evalf())
raises(ValueError, lambda: Sum((-1)**n, (n, 1, oo)).evalf())
raises(ValueError, lambda: Sum(n**2, (n, 1, oo)).evalf())
raises(ValueError, lambda: Sum(2**n, (n, 1, oo)).evalf())
raises(ValueError, lambda: Sum((-2)**n, (n, 1, oo)).evalf())
raises(ValueError, lambda: Sum((2*n + 3)/(3*n**2 + 4), (n, 0, oo)).evalf())
raises(ValueError, lambda: Sum((0.5*n**3)/(n**4 + 1), (n, 0, oo)).evalf())
def test_evalf_product():
assert Product(n, (n, 1, 10)).evalf() == 3628800.
assert Product(1 - S.Half**2/n**2, (n, 1, oo)).evalf(5)==0.63662
assert Product(n, (n, -1, 3)).evalf() == 0
def test_evalf_py_methods():
assert abs(float(pi + 1) - 4.1415926535897932) < 1e-10
assert abs(complex(pi + 1) - 4.1415926535897932) < 1e-10
assert abs(
complex(pi + E*I) - (3.1415926535897931 + 2.7182818284590451j)) < 1e-10
raises(TypeError, lambda: float(pi + x))
def test_evalf_power_subs_bugs():
assert (x**2).evalf(subs={x: 0}) == 0
assert sqrt(x).evalf(subs={x: 0}) == 0
assert (x**Rational(2, 3)).evalf(subs={x: 0}) == 0
assert (x**x).evalf(subs={x: 0}) == 1
assert (3**x).evalf(subs={x: 0}) == 1
assert exp(x).evalf(subs={x: 0}) == 1
assert ((2 + I)**x).evalf(subs={x: 0}) == 1
assert (0**x).evalf(subs={x: 0}) == 1
def test_evalf_arguments():
raises(TypeError, lambda: pi.evalf(method="garbage"))
def test_implemented_function_evalf():
from sympy.utilities.lambdify import implemented_function
f = Function('f')
f = implemented_function(f, lambda x: x + 1)
assert str(f(x)) == "f(x)"
assert str(f(2)) == "f(2)"
assert f(2).evalf() == 3
assert f(x).evalf() == f(x)
f = implemented_function(Function('sin'), lambda x: x + 1)
assert f(2).evalf() != sin(2)
del f._imp_ # XXX: due to caching _imp_ would influence all other tests
def test_evaluate_false():
for no in [0, False]:
assert Add(3, 2, evaluate=no).is_Add
assert Mul(3, 2, evaluate=no).is_Mul
assert Pow(3, 2, evaluate=no).is_Pow
assert Pow(y, 2, evaluate=True) - Pow(y, 2, evaluate=True) == 0
def test_evalf_relational():
assert Eq(x/5, y/10).evalf() == Eq(0.2*x, 0.1*y)
def test_issue_5486():
assert not cos(sqrt(0.5 + I)).n().is_Function
def test_issue_5486_bug():
from sympy import I, Expr
assert abs(Expr._from_mpmath(I._to_mpmath(15), 15) - I) < 1.0e-15
def test_bugs():
from sympy import polar_lift, re
assert abs(re((1 + I)**2)) < 1e-15
# anything that evalf's to 0 will do in place of polar_lift
assert abs(polar_lift(0)).n() == 0
def test_subs():
assert NS('besseli(-x, y) - besseli(x, y)', subs={x: 3.5, y: 20.0}) == \
'-4.92535585957223e-10'
assert NS('Piecewise((x, x>0)) + Piecewise((1-x, x>0))', subs={x: 0.1}) == \
'1.00000000000000'
raises(TypeError, lambda: x.evalf(subs=(x, 1)))
def test_issue_4956_5204():
# issue 4956
v = S('''(-27*12**(1/3)*sqrt(31)*I +
27*2**(2/3)*3**(1/3)*sqrt(31)*I)/(-2511*2**(2/3)*3**(1/3) +
(29*18**(1/3) + 9*2**(1/3)*3**(2/3)*sqrt(31)*I +
87*2**(1/3)*3**(1/6)*I)**2)''')
assert NS(v, 1) == '0.e-118 - 0.e-118*I'
# issue 5204
v = S('''-(357587765856 + 18873261792*249**(1/2) + 56619785376*I*83**(1/2) +
108755765856*I*3**(1/2) + 41281887168*6**(1/3)*(1422 +
54*249**(1/2))**(1/3) - 1239810624*6**(1/3)*249**(1/2)*(1422 +
54*249**(1/2))**(1/3) - 3110400000*I*6**(1/3)*83**(1/2)*(1422 +
54*249**(1/2))**(1/3) + 13478400000*I*3**(1/2)*6**(1/3)*(1422 +
54*249**(1/2))**(1/3) + 1274950152*6**(2/3)*(1422 +
54*249**(1/2))**(2/3) + 32347944*6**(2/3)*249**(1/2)*(1422 +
54*249**(1/2))**(2/3) - 1758790152*I*3**(1/2)*6**(2/3)*(1422 +
54*249**(1/2))**(2/3) - 304403832*I*6**(2/3)*83**(1/2)*(1422 +
4*249**(1/2))**(2/3))/(175732658352 + (1106028 + 25596*249**(1/2) +
76788*I*83**(1/2))**2)''')
assert NS(v, 5) == '0.077284 + 1.1104*I'
assert NS(v, 1) == '0.08 + 1.*I'
def test_old_docstring():
a = (E + pi*I)*(E - pi*I)
assert NS(a) == '17.2586605000200'
assert a.n() == 17.25866050002001
def test_issue_4806():
assert integrate(atan(x)**2, (x, -1, 1)).evalf().round(1) == 0.5
assert atan(0, evaluate=False).n() == 0
def test_evalf_mul():
# sympy should not try to expand this; it should be handled term-wise
# in evalf through mpmath
assert NS(product(1 + sqrt(n)*I, (n, 1, 500)), 1) == '5.e+567 + 2.e+568*I'
def test_scaled_zero():
a, b = (([0], 1, 100, 1), -1)
assert scaled_zero(100) == (a, b)
assert scaled_zero(a) == (0, 1, 100, 1)
a, b = (([1], 1, 100, 1), -1)
assert scaled_zero(100, -1) == (a, b)
assert scaled_zero(a) == (1, 1, 100, 1)
raises(ValueError, lambda: scaled_zero(scaled_zero(100)))
raises(ValueError, lambda: scaled_zero(100, 2))
raises(ValueError, lambda: scaled_zero(100, 0))
raises(ValueError, lambda: scaled_zero((1, 5, 1, 3)))
def test_chop_value():
for i in range(-27, 28):
assert (Pow(10, i)*2).n(chop=10**i) and not (Pow(10, i)).n(chop=10**i)
def test_infinities():
assert oo.evalf(chop=True) == inf
assert (-oo).evalf(chop=True) == ninf
def test_to_mpmath():
assert sqrt(3)._to_mpmath(20)._mpf_ == (0, long(908093), -19, 20)
assert S(3.2)._to_mpmath(20)._mpf_ == (0, long(838861), -18, 20)
def test_issue_6632_evalf():
add = (-100000*sqrt(2500000001) + 5000000001)
assert add.n() == 9.999999998e-11
assert (add*add).n() == 9.999999996e-21
def test_issue_4945():
from sympy.abc import H
from sympy import zoo
assert (H/0).evalf(subs={H:1}) == zoo*H
def test_evalf_integral():
# test that workprec has to increase in order to get a result other than 0
eps = Rational(1, 1000000)
assert Integral(sin(x), (x, -pi, pi + eps)).n(2)._prec == 10
def test_issue_8821_highprec_from_str():
s = str(pi.evalf(128))
p = N(s)
assert Abs(sin(p)) < 1e-15
p = N(s, 64)
assert Abs(sin(p)) < 1e-64
def test_issue_8853():
p = Symbol('x', even=True, positive=True)
assert floor(-p - S.Half).is_even == False
assert floor(-p + S.Half).is_even == True
assert ceiling(p - S.Half).is_even == True
assert ceiling(p + S.Half).is_even == False
assert get_integer_part(S.Half, -1, {}, True) == (0, 0)
assert get_integer_part(S.Half, 1, {}, True) == (1, 0)
assert get_integer_part(-S.Half, -1, {}, True) == (-1, 0)
assert get_integer_part(-S.Half, 1, {}, True) == (0, 0)
def test_issue_9326():
from sympy import Dummy
d1 = Dummy('d')
d2 = Dummy('d')
e = d1 + d2
assert e.evalf(subs = {d1: 1, d2: 2}) == 3
def test_issue_10323():
assert ceiling(sqrt(2**30 + 1)) == 2**15 + 1
def test_AssocOp_Function():
# the first arg of Min is not comparable in the imaginary part
raises(ValueError, lambda: S('''
Min(-sqrt(3)*cos(pi/18)/6 + re(1/((-1/2 - sqrt(3)*I/2)*(1/6 +
sqrt(3)*I/18)**(1/3)))/3 + sin(pi/18)/2 + 2 + I*(-cos(pi/18)/2 -
sqrt(3)*sin(pi/18)/6 + im(1/((-1/2 - sqrt(3)*I/2)*(1/6 +
sqrt(3)*I/18)**(1/3)))/3), re(1/((-1/2 + sqrt(3)*I/2)*(1/6 +
sqrt(3)*I/18)**(1/3)))/3 - sqrt(3)*cos(pi/18)/6 - sin(pi/18)/2 + 2 +
I*(im(1/((-1/2 + sqrt(3)*I/2)*(1/6 + sqrt(3)*I/18)**(1/3)))/3 -
sqrt(3)*sin(pi/18)/6 + cos(pi/18)/2))'''))
# if that is changed so a non-comparable number remains as
# an arg, then the Min/Max instantiation needs to be changed
# to watch out for non-comparable args when making simplifications
# and the following test should be added instead (with e being
# the sympified expression above):
# raises(ValueError, lambda: e._eval_evalf(2))
def test_issue_10395():
eq = x*Max(0, y)
assert nfloat(eq) == eq
eq = x*Max(y, -1.1)
assert nfloat(eq) == eq
assert Max(y, 4).n() == Max(4.0, y)
def test_issue_13098():
assert floor(log(S('9.'+'9'*20), 10)) == 0
assert ceiling(log(S('9.'+'9'*20), 10)) == 1
assert floor(log(20 - S('9.'+'9'*20), 10)) == 1
assert ceiling(log(20 - S('9.'+'9'*20), 10)) == 2
def test_issue_14601():
e = 5*x*y/2 - y*(35*(x**3)/2 - 15*x/2)
subst = {x:0.0, y:0.0}
e2 = e.evalf(subs=subst)
assert float(e2) == 0.0
assert float((x + x*(x**2 + x)).evalf(subs={x: 0.0})) == 0.0
def test_issue_11151():
z = S.Zero
e = Sum(z, (x, 1, 2))
assert e != z # it shouldn't evaluate
# when it does evaluate, this is what it should give
assert evalf(e, 15, {}) == \
evalf(z, 15, {}) == (None, None, 15, None)
# so this shouldn't fail
assert (e/2).n() == 0
# this was where the issue appeared
expr0 = Sum(x**2 + x, (x, 1, 2))
expr1 = Sum(0, (x, 1, 2))
expr2 = expr1/expr0
assert simplify(factor(expr2) - expr2) == 0
|
1f67b9494cbef051e00fbb71b48d64d55de8740975aa2304e06033623bf71ef7
|
from sympy import (Basic, Symbol, sin, cos, exp, sqrt, Rational, Float, re, pi,
sympify, Add, Mul, Pow, Mod, I, log, S, Max, symbols, oo, zoo, Integer,
sign, im, nan, Dummy, factorial, comp, refine
)
from sympy.core.compatibility import long, range
from sympy.core.expr import unchanged
from sympy.utilities.iterables import cartes
from sympy.utilities.pytest import XFAIL, raises
from sympy.utilities.randtest import verify_numerically
a, c, x, y, z = symbols('a,c,x,y,z')
b = Symbol("b", positive=True)
def same_and_same_prec(a, b):
# stricter matching for Floats
return a == b and a._prec == b._prec
def test_bug1():
assert re(x) != x
x.series(x, 0, 1)
assert re(x) != x
def test_Symbol():
e = a*b
assert e == a*b
assert a*b*b == a*b**2
assert a*b*b + c == c + a*b**2
assert a*b*b - c == -c + a*b**2
x = Symbol('x', complex=True, real=False)
assert x.is_imaginary is None # could be I or 1 + I
x = Symbol('x', complex=True, imaginary=False)
assert x.is_real is None # could be 1 or 1 + I
x = Symbol('x', real=True)
assert x.is_complex
x = Symbol('x', imaginary=True)
assert x.is_complex
x = Symbol('x', real=False, imaginary=False)
assert x.is_complex is None # might be a non-number
def test_arit0():
p = Rational(5)
e = a*b
assert e == a*b
e = a*b + b*a
assert e == 2*a*b
e = a*b + b*a + a*b + p*b*a
assert e == 8*a*b
e = a*b + b*a + a*b + p*b*a + a
assert e == a + 8*a*b
e = a + a
assert e == 2*a
e = a + b + a
assert e == b + 2*a
e = a + b*b + a + b*b
assert e == 2*a + 2*b**2
e = a + Rational(2) + b*b + a + b*b + p
assert e == 7 + 2*a + 2*b**2
e = (a + b*b + a + b*b)*p
assert e == 5*(2*a + 2*b**2)
e = (a*b*c + c*b*a + b*a*c)*p
assert e == 15*a*b*c
e = (a*b*c + c*b*a + b*a*c)*p - Rational(15)*a*b*c
assert e == Rational(0)
e = Rational(50)*(a - a)
assert e == Rational(0)
e = b*a - b - a*b + b
assert e == Rational(0)
e = a*b + c**p
assert e == a*b + c**5
e = a/b
assert e == a*b**(-1)
e = a*2*2
assert e == 4*a
e = 2 + a*2/2
assert e == 2 + a
e = 2 - a - 2
assert e == -a
e = 2*a*2
assert e == 4*a
e = 2/a/2
assert e == a**(-1)
e = 2**a**2
assert e == 2**(a**2)
e = -(1 + a)
assert e == -1 - a
e = Rational(1, 2)*(1 + a)
assert e == Rational(1, 2) + a/2
def test_div():
e = a/b
assert e == a*b**(-1)
e = a/b + c/2
assert e == a*b**(-1) + Rational(1)/2*c
e = (1 - b)/(b - 1)
assert e == (1 + -b)*((-1) + b)**(-1)
def test_pow():
n1 = Rational(1)
n2 = Rational(2)
n5 = Rational(5)
e = a*a
assert e == a**2
e = a*a*a
assert e == a**3
e = a*a*a*a**Rational(6)
assert e == a**9
e = a*a*a*a**Rational(6) - a**Rational(9)
assert e == Rational(0)
e = a**(b - b)
assert e == Rational(1)
e = (a + Rational(1) - a)**b
assert e == Rational(1)
e = (a + b + c)**n2
assert e == (a + b + c)**2
assert e.expand() == 2*b*c + 2*a*c + 2*a*b + a**2 + c**2 + b**2
e = (a + b)**n2
assert e == (a + b)**2
assert e.expand() == 2*a*b + a**2 + b**2
e = (a + b)**(n1/n2)
assert e == sqrt(a + b)
assert e.expand() == sqrt(a + b)
n = n5**(n1/n2)
assert n == sqrt(5)
e = n*a*b - n*b*a
assert e == Rational(0)
e = n*a*b + n*b*a
assert e == 2*a*b*sqrt(5)
assert e.diff(a) == 2*b*sqrt(5)
assert e.diff(a) == 2*b*sqrt(5)
e = a/b**2
assert e == a*b**(-2)
assert sqrt(2*(1 + sqrt(2))) == (2*(1 + 2**Rational(1, 2)))**Rational(1, 2)
x = Symbol('x')
y = Symbol('y')
assert ((x*y)**3).expand() == y**3 * x**3
assert ((x*y)**-3).expand() == y**-3 * x**-3
assert (x**5*(3*x)**(3)).expand() == 27 * x**8
assert (x**5*(-3*x)**(3)).expand() == -27 * x**8
assert (x**5*(3*x)**(-3)).expand() == Rational(1, 27) * x**2
assert (x**5*(-3*x)**(-3)).expand() == -Rational(1, 27) * x**2
# expand_power_exp
assert (x**(y**(x + exp(x + y)) + z)).expand(deep=False) == \
x**z*x**(y**(x + exp(x + y)))
assert (x**(y**(x + exp(x + y)) + z)).expand() == \
x**z*x**(y**x*y**(exp(x)*exp(y)))
n = Symbol('n', even=False)
k = Symbol('k', even=True)
o = Symbol('o', odd=True)
assert (-1)**x == (-1)**x
assert (-1)**n == (-1)**n
assert (-2)**k == 2**k
assert (-1)**k == 1
def test_pow2():
# x**(2*y) is always (x**y)**2 but is only (x**2)**y if
# x.is_positive or y.is_integer
# let x = 1 to see why the following are not true.
assert (-x)**Rational(2, 3) != x**Rational(2, 3)
assert (-x)**Rational(5, 7) != -x**Rational(5, 7)
assert ((-x)**2)**Rational(1, 3) != ((-x)**Rational(1, 3))**2
assert sqrt(x**2) != x
def test_pow3():
assert sqrt(2)**3 == 2 * sqrt(2)
assert sqrt(2)**3 == sqrt(8)
def test_mod_pow():
for s, t, u, v in [(4, 13, 497, 445), (4, -3, 497, 365),
(3.2, 2.1, 1.9, 0.1031015682350942), (S(3)/2, 5, S(5)/6, S(3)/32)]:
assert pow(S(s), t, u) == v
assert pow(S(s), S(t), u) == v
assert pow(S(s), t, S(u)) == v
assert pow(S(s), S(t), S(u)) == v
assert pow(S(2), S(10000000000), S(3)) == 1
assert pow(x, y, z) == x**y%z
raises(TypeError, lambda: pow(S(4), "13", 497))
raises(TypeError, lambda: pow(S(4), 13, "497"))
def test_pow_E():
assert 2**(y/log(2)) == S.Exp1**y
assert 2**(y/log(2)/3) == S.Exp1**(y/3)
assert 3**(1/log(-3)) != S.Exp1
assert (3 + 2*I)**(1/(log(-3 - 2*I) + I*pi)) == S.Exp1
assert (4 + 2*I)**(1/(log(-4 - 2*I) + I*pi)) == S.Exp1
assert (3 + 2*I)**(1/(log(-3 - 2*I, 3)/2 + I*pi/log(3)/2)) == 9
assert (3 + 2*I)**(1/(log(3 + 2*I, 3)/2)) == 9
# every time tests are run they will affirm with a different random
# value that this identity holds
while 1:
b = x._random()
r, i = b.as_real_imag()
if i:
break
assert verify_numerically(b**(1/(log(-b) + sign(i)*I*pi).n()), S.Exp1)
def test_pow_issue_3516():
assert 4**Rational(1, 4) == sqrt(2)
def test_pow_im():
for m in (-2, -1, 2):
for d in (3, 4, 5):
b = m*I
for i in range(1, 4*d + 1):
e = Rational(i, d)
assert (b**e - b.n()**e.n()).n(2, chop=1e-10) == 0
e = Rational(7, 3)
assert (2*x*I)**e == 4*2**Rational(1, 3)*(I*x)**e # same as Wolfram Alpha
im = symbols('im', imaginary=True)
assert (2*im*I)**e == 4*2**Rational(1, 3)*(I*im)**e
args = [I, I, I, I, 2]
e = Rational(1, 3)
ans = 2**e
assert Mul(*args, evaluate=False)**e == ans
assert Mul(*args)**e == ans
args = [I, I, I, 2]
e = Rational(1, 3)
ans = 2**e*(-I)**e
assert Mul(*args, evaluate=False)**e == ans
assert Mul(*args)**e == ans
args.append(-3)
ans = (6*I)**e
assert Mul(*args, evaluate=False)**e == ans
assert Mul(*args)**e == ans
args.append(-1)
ans = (-6*I)**e
assert Mul(*args, evaluate=False)**e == ans
assert Mul(*args)**e == ans
args = [I, I, 2]
e = Rational(1, 3)
ans = (-2)**e
assert Mul(*args, evaluate=False)**e == ans
assert Mul(*args)**e == ans
args.append(-3)
ans = (6)**e
assert Mul(*args, evaluate=False)**e == ans
assert Mul(*args)**e == ans
args.append(-1)
ans = (-6)**e
assert Mul(*args, evaluate=False)**e == ans
assert Mul(*args)**e == ans
assert Mul(Pow(-1, Rational(3, 2), evaluate=False), I, I) == I
assert Mul(I*Pow(I, S.Half, evaluate=False)) == sqrt(I)*I
def test_real_mul():
assert Float(0) * pi * x == Float(0)
assert set((Float(1) * pi * x).args) == {Float(1), pi, x}
def test_ncmul():
A = Symbol("A", commutative=False)
B = Symbol("B", commutative=False)
C = Symbol("C", commutative=False)
assert A*B != B*A
assert A*B*C != C*B*A
assert A*b*B*3*C == 3*b*A*B*C
assert A*b*B*3*C != 3*b*B*A*C
assert A*b*B*3*C == 3*A*B*C*b
assert A + B == B + A
assert (A + B)*C != C*(A + B)
assert C*(A + B)*C != C*C*(A + B)
assert A*A == A**2
assert (A + B)*(A + B) == (A + B)**2
assert A**-1 * A == 1
assert A/A == 1
assert A/(A**2) == 1/A
assert A/(1 + A) == A/(1 + A)
assert set((A + B + 2*(A + B)).args) == \
{A, B, 2*(A + B)}
def test_ncpow():
x = Symbol('x', commutative=False)
y = Symbol('y', commutative=False)
z = Symbol('z', commutative=False)
a = Symbol('a')
b = Symbol('b')
c = Symbol('c')
assert (x**2)*(y**2) != (y**2)*(x**2)
assert (x**-2)*y != y*(x**2)
assert 2**x*2**y != 2**(x + y)
assert 2**x*2**y*2**z != 2**(x + y + z)
assert 2**x*2**(2*x) == 2**(3*x)
assert 2**x*2**(2*x)*2**x == 2**(4*x)
assert exp(x)*exp(y) != exp(y)*exp(x)
assert exp(x)*exp(y)*exp(z) != exp(y)*exp(x)*exp(z)
assert exp(x)*exp(y)*exp(z) != exp(x + y + z)
assert x**a*x**b != x**(a + b)
assert x**a*x**b*x**c != x**(a + b + c)
assert x**3*x**4 == x**7
assert x**3*x**4*x**2 == x**9
assert x**a*x**(4*a) == x**(5*a)
assert x**a*x**(4*a)*x**a == x**(6*a)
def test_powerbug():
x = Symbol("x")
assert x**1 != (-x)**1
assert x**2 == (-x)**2
assert x**3 != (-x)**3
assert x**4 == (-x)**4
assert x**5 != (-x)**5
assert x**6 == (-x)**6
assert x**128 == (-x)**128
assert x**129 != (-x)**129
assert (2*x)**2 == (-2*x)**2
def test_Mul_doesnt_expand_exp():
x = Symbol('x')
y = Symbol('y')
assert unchanged(Mul, exp(x), exp(y))
assert unchanged(Mul, 2**x, 2**y)
assert x**2*x**3 == x**5
assert 2**x*3**x == 6**x
assert x**(y)*x**(2*y) == x**(3*y)
assert sqrt(2)*sqrt(2) == 2
assert 2**x*2**(2*x) == 2**(3*x)
assert sqrt(2)*2**Rational(1, 4)*5**Rational(3, 4) == 10**Rational(3, 4)
assert (x**(-log(5)/log(3))*x)/(x*x**( - log(5)/log(3))) == sympify(1)
def test_Add_Mul_is_integer():
x = Symbol('x')
k = Symbol('k', integer=True)
n = Symbol('n', integer=True)
assert (2*k).is_integer is True
assert (-k).is_integer is True
assert (k/3).is_integer is None
assert (x*k*n).is_integer is None
assert (k + n).is_integer is True
assert (k + x).is_integer is None
assert (k + n*x).is_integer is None
assert (k + n/3).is_integer is None
assert ((1 + sqrt(3))*(-sqrt(3) + 1)).is_integer is not False
assert (1 + (1 + sqrt(3))*(-sqrt(3) + 1)).is_integer is not False
def test_Add_Mul_is_finite():
x = Symbol('x', real=True, finite=False)
assert sin(x).is_finite is True
assert (x*sin(x)).is_finite is False
assert (1024*sin(x)).is_finite is True
assert (sin(x)*exp(x)).is_finite is not True
assert (sin(x)*cos(x)).is_finite is True
assert (x*sin(x)*exp(x)).is_finite is not True
assert (sin(x) - 67).is_finite is True
assert (sin(x) + exp(x)).is_finite is not True
assert (1 + x).is_finite is False
assert (1 + x**2 + (1 + x)*(1 - x)).is_finite is None
assert (sqrt(2)*(1 + x)).is_finite is False
assert (sqrt(2)*(1 + x)*(1 - x)).is_finite is False
def test_Mul_is_even_odd():
x = Symbol('x', integer=True)
y = Symbol('y', integer=True)
k = Symbol('k', odd=True)
n = Symbol('n', odd=True)
m = Symbol('m', even=True)
assert (2*x).is_even is True
assert (2*x).is_odd is False
assert (3*x).is_even is None
assert (3*x).is_odd is None
assert (k/3).is_integer is None
assert (k/3).is_even is None
assert (k/3).is_odd is None
assert (2*n).is_even is True
assert (2*n).is_odd is False
assert (2*m).is_even is True
assert (2*m).is_odd is False
assert (-n).is_even is False
assert (-n).is_odd is True
assert (k*n).is_even is False
assert (k*n).is_odd is True
assert (k*m).is_even is True
assert (k*m).is_odd is False
assert (k*n*m).is_even is True
assert (k*n*m).is_odd is False
assert (k*m*x).is_even is True
assert (k*m*x).is_odd is False
# issue 6791:
assert (x/2).is_integer is None
assert (k/2).is_integer is False
assert (m/2).is_integer is True
assert (x*y).is_even is None
assert (x*x).is_even is None
assert (x*(x + k)).is_even is True
assert (x*(x + m)).is_even is None
assert (x*y).is_odd is None
assert (x*x).is_odd is None
assert (x*(x + k)).is_odd is False
assert (x*(x + m)).is_odd is None
@XFAIL
def test_evenness_in_ternary_integer_product_with_odd():
# Tests that oddness inference is independent of term ordering.
# Term ordering at the point of testing depends on SymPy's symbol order, so
# we try to force a different order by modifying symbol names.
x = Symbol('x', integer=True)
y = Symbol('y', integer=True)
k = Symbol('k', odd=True)
assert (x*y*(y + k)).is_even is True
assert (y*x*(x + k)).is_even is True
def test_evenness_in_ternary_integer_product_with_even():
x = Symbol('x', integer=True)
y = Symbol('y', integer=True)
m = Symbol('m', even=True)
assert (x*y*(y + m)).is_even is None
@XFAIL
def test_oddness_in_ternary_integer_product_with_odd():
# Tests that oddness inference is independent of term ordering.
# Term ordering at the point of testing depends on SymPy's symbol order, so
# we try to force a different order by modifying symbol names.
x = Symbol('x', integer=True)
y = Symbol('y', integer=True)
k = Symbol('k', odd=True)
assert (x*y*(y + k)).is_odd is False
assert (y*x*(x + k)).is_odd is False
def test_oddness_in_ternary_integer_product_with_even():
x = Symbol('x', integer=True)
y = Symbol('y', integer=True)
m = Symbol('m', even=True)
assert (x*y*(y + m)).is_odd is None
def test_Mul_is_rational():
x = Symbol('x')
n = Symbol('n', integer=True)
m = Symbol('m', integer=True, nonzero=True)
assert (n/m).is_rational is True
assert (x/pi).is_rational is None
assert (x/n).is_rational is None
assert (m/pi).is_rational is False
r = Symbol('r', rational=True)
assert (pi*r).is_rational is None
# issue 8008
z = Symbol('z', zero=True)
i = Symbol('i', imaginary=True)
assert (z*i).is_rational is None
bi = Symbol('i', imaginary=True, finite=True)
assert (z*bi).is_zero is True
def test_Add_is_rational():
x = Symbol('x')
n = Symbol('n', rational=True)
m = Symbol('m', rational=True)
assert (n + m).is_rational is True
assert (x + pi).is_rational is None
assert (x + n).is_rational is None
assert (n + pi).is_rational is False
def test_Add_is_even_odd():
x = Symbol('x', integer=True)
k = Symbol('k', odd=True)
n = Symbol('n', odd=True)
m = Symbol('m', even=True)
assert (k + 7).is_even is True
assert (k + 7).is_odd is False
assert (-k + 7).is_even is True
assert (-k + 7).is_odd is False
assert (k - 12).is_even is False
assert (k - 12).is_odd is True
assert (-k - 12).is_even is False
assert (-k - 12).is_odd is True
assert (k + n).is_even is True
assert (k + n).is_odd is False
assert (k + m).is_even is False
assert (k + m).is_odd is True
assert (k + n + m).is_even is True
assert (k + n + m).is_odd is False
assert (k + n + x + m).is_even is None
assert (k + n + x + m).is_odd is None
def test_Mul_is_negative_positive():
x = Symbol('x', real=True)
y = Symbol('y', real=False, complex=True)
z = Symbol('z', zero=True)
e = 2*z
assert e.is_Mul and e.is_positive is False and e.is_negative is False
neg = Symbol('neg', negative=True)
pos = Symbol('pos', positive=True)
nneg = Symbol('nneg', nonnegative=True)
npos = Symbol('npos', nonpositive=True)
assert neg.is_negative is True
assert (-neg).is_negative is False
assert (2*neg).is_negative is True
assert (2*pos)._eval_is_negative() is False
assert (2*pos).is_negative is False
assert pos.is_negative is False
assert (-pos).is_negative is True
assert (2*pos).is_negative is False
assert (pos*neg).is_negative is True
assert (2*pos*neg).is_negative is True
assert (-pos*neg).is_negative is False
assert (pos*neg*y).is_negative is False # y.is_real=F; !real -> !neg
assert nneg.is_negative is False
assert (-nneg).is_negative is None
assert (2*nneg).is_negative is False
assert npos.is_negative is None
assert (-npos).is_negative is False
assert (2*npos).is_negative is None
assert (nneg*npos).is_negative is None
assert (neg*nneg).is_negative is None
assert (neg*npos).is_negative is False
assert (pos*nneg).is_negative is False
assert (pos*npos).is_negative is None
assert (npos*neg*nneg).is_negative is False
assert (npos*pos*nneg).is_negative is None
assert (-npos*neg*nneg).is_negative is None
assert (-npos*pos*nneg).is_negative is False
assert (17*npos*neg*nneg).is_negative is False
assert (17*npos*pos*nneg).is_negative is None
assert (neg*npos*pos*nneg).is_negative is False
assert (x*neg).is_negative is None
assert (nneg*npos*pos*x*neg).is_negative is None
assert neg.is_positive is False
assert (-neg).is_positive is True
assert (2*neg).is_positive is False
assert pos.is_positive is True
assert (-pos).is_positive is False
assert (2*pos).is_positive is True
assert (pos*neg).is_positive is False
assert (2*pos*neg).is_positive is False
assert (-pos*neg).is_positive is True
assert (-pos*neg*y).is_positive is False # y.is_real=F; !real -> !neg
assert nneg.is_positive is None
assert (-nneg).is_positive is False
assert (2*nneg).is_positive is None
assert npos.is_positive is False
assert (-npos).is_positive is None
assert (2*npos).is_positive is False
assert (nneg*npos).is_positive is False
assert (neg*nneg).is_positive is False
assert (neg*npos).is_positive is None
assert (pos*nneg).is_positive is None
assert (pos*npos).is_positive is False
assert (npos*neg*nneg).is_positive is None
assert (npos*pos*nneg).is_positive is False
assert (-npos*neg*nneg).is_positive is False
assert (-npos*pos*nneg).is_positive is None
assert (17*npos*neg*nneg).is_positive is None
assert (17*npos*pos*nneg).is_positive is False
assert (neg*npos*pos*nneg).is_positive is None
assert (x*neg).is_positive is None
assert (nneg*npos*pos*x*neg).is_positive is None
def test_Mul_is_negative_positive_2():
a = Symbol('a', nonnegative=True)
b = Symbol('b', nonnegative=True)
c = Symbol('c', nonpositive=True)
d = Symbol('d', nonpositive=True)
assert (a*b).is_nonnegative is True
assert (a*b).is_negative is False
assert (a*b).is_zero is None
assert (a*b).is_positive is None
assert (c*d).is_nonnegative is True
assert (c*d).is_negative is False
assert (c*d).is_zero is None
assert (c*d).is_positive is None
assert (a*c).is_nonpositive is True
assert (a*c).is_positive is False
assert (a*c).is_zero is None
assert (a*c).is_negative is None
def test_Mul_is_nonpositive_nonnegative():
x = Symbol('x', real=True)
k = Symbol('k', negative=True)
n = Symbol('n', positive=True)
u = Symbol('u', nonnegative=True)
v = Symbol('v', nonpositive=True)
assert k.is_nonpositive is True
assert (-k).is_nonpositive is False
assert (2*k).is_nonpositive is True
assert n.is_nonpositive is False
assert (-n).is_nonpositive is True
assert (2*n).is_nonpositive is False
assert (n*k).is_nonpositive is True
assert (2*n*k).is_nonpositive is True
assert (-n*k).is_nonpositive is False
assert u.is_nonpositive is None
assert (-u).is_nonpositive is True
assert (2*u).is_nonpositive is None
assert v.is_nonpositive is True
assert (-v).is_nonpositive is None
assert (2*v).is_nonpositive is True
assert (u*v).is_nonpositive is True
assert (k*u).is_nonpositive is True
assert (k*v).is_nonpositive is None
assert (n*u).is_nonpositive is None
assert (n*v).is_nonpositive is True
assert (v*k*u).is_nonpositive is None
assert (v*n*u).is_nonpositive is True
assert (-v*k*u).is_nonpositive is True
assert (-v*n*u).is_nonpositive is None
assert (17*v*k*u).is_nonpositive is None
assert (17*v*n*u).is_nonpositive is True
assert (k*v*n*u).is_nonpositive is None
assert (x*k).is_nonpositive is None
assert (u*v*n*x*k).is_nonpositive is None
assert k.is_nonnegative is False
assert (-k).is_nonnegative is True
assert (2*k).is_nonnegative is False
assert n.is_nonnegative is True
assert (-n).is_nonnegative is False
assert (2*n).is_nonnegative is True
assert (n*k).is_nonnegative is False
assert (2*n*k).is_nonnegative is False
assert (-n*k).is_nonnegative is True
assert u.is_nonnegative is True
assert (-u).is_nonnegative is None
assert (2*u).is_nonnegative is True
assert v.is_nonnegative is None
assert (-v).is_nonnegative is True
assert (2*v).is_nonnegative is None
assert (u*v).is_nonnegative is None
assert (k*u).is_nonnegative is None
assert (k*v).is_nonnegative is True
assert (n*u).is_nonnegative is True
assert (n*v).is_nonnegative is None
assert (v*k*u).is_nonnegative is True
assert (v*n*u).is_nonnegative is None
assert (-v*k*u).is_nonnegative is None
assert (-v*n*u).is_nonnegative is True
assert (17*v*k*u).is_nonnegative is True
assert (17*v*n*u).is_nonnegative is None
assert (k*v*n*u).is_nonnegative is True
assert (x*k).is_nonnegative is None
assert (u*v*n*x*k).is_nonnegative is None
def test_Add_is_negative_positive():
x = Symbol('x', real=True)
k = Symbol('k', negative=True)
n = Symbol('n', positive=True)
u = Symbol('u', nonnegative=True)
v = Symbol('v', nonpositive=True)
assert (k - 2).is_negative is True
assert (k + 17).is_negative is None
assert (-k - 5).is_negative is None
assert (-k + 123).is_negative is False
assert (k - n).is_negative is True
assert (k + n).is_negative is None
assert (-k - n).is_negative is None
assert (-k + n).is_negative is False
assert (k - n - 2).is_negative is True
assert (k + n + 17).is_negative is None
assert (-k - n - 5).is_negative is None
assert (-k + n + 123).is_negative is False
assert (-2*k + 123*n + 17).is_negative is False
assert (k + u).is_negative is None
assert (k + v).is_negative is True
assert (n + u).is_negative is False
assert (n + v).is_negative is None
assert (u - v).is_negative is False
assert (u + v).is_negative is None
assert (-u - v).is_negative is None
assert (-u + v).is_negative is None
assert (u - v + n + 2).is_negative is False
assert (u + v + n + 2).is_negative is None
assert (-u - v + n + 2).is_negative is None
assert (-u + v + n + 2).is_negative is None
assert (k + x).is_negative is None
assert (k + x - n).is_negative is None
assert (k - 2).is_positive is False
assert (k + 17).is_positive is None
assert (-k - 5).is_positive is None
assert (-k + 123).is_positive is True
assert (k - n).is_positive is False
assert (k + n).is_positive is None
assert (-k - n).is_positive is None
assert (-k + n).is_positive is True
assert (k - n - 2).is_positive is False
assert (k + n + 17).is_positive is None
assert (-k - n - 5).is_positive is None
assert (-k + n + 123).is_positive is True
assert (-2*k + 123*n + 17).is_positive is True
assert (k + u).is_positive is None
assert (k + v).is_positive is False
assert (n + u).is_positive is True
assert (n + v).is_positive is None
assert (u - v).is_positive is None
assert (u + v).is_positive is None
assert (-u - v).is_positive is None
assert (-u + v).is_positive is False
assert (u - v - n - 2).is_positive is None
assert (u + v - n - 2).is_positive is None
assert (-u - v - n - 2).is_positive is None
assert (-u + v - n - 2).is_positive is False
assert (n + x).is_positive is None
assert (n + x - k).is_positive is None
z = (-3 - sqrt(5) + (-sqrt(10)/2 - sqrt(2)/2)**2)
assert z.is_zero
z = sqrt(1 + sqrt(3)) + sqrt(3 + 3*sqrt(3)) - sqrt(10 + 6*sqrt(3))
assert z.is_zero
def test_Add_is_nonpositive_nonnegative():
x = Symbol('x', real=True)
k = Symbol('k', negative=True)
n = Symbol('n', positive=True)
u = Symbol('u', nonnegative=True)
v = Symbol('v', nonpositive=True)
assert (u - 2).is_nonpositive is None
assert (u + 17).is_nonpositive is False
assert (-u - 5).is_nonpositive is True
assert (-u + 123).is_nonpositive is None
assert (u - v).is_nonpositive is None
assert (u + v).is_nonpositive is None
assert (-u - v).is_nonpositive is None
assert (-u + v).is_nonpositive is True
assert (u - v - 2).is_nonpositive is None
assert (u + v + 17).is_nonpositive is None
assert (-u - v - 5).is_nonpositive is None
assert (-u + v - 123).is_nonpositive is True
assert (-2*u + 123*v - 17).is_nonpositive is True
assert (k + u).is_nonpositive is None
assert (k + v).is_nonpositive is True
assert (n + u).is_nonpositive is False
assert (n + v).is_nonpositive is None
assert (k - n).is_nonpositive is True
assert (k + n).is_nonpositive is None
assert (-k - n).is_nonpositive is None
assert (-k + n).is_nonpositive is False
assert (k - n + u + 2).is_nonpositive is None
assert (k + n + u + 2).is_nonpositive is None
assert (-k - n + u + 2).is_nonpositive is None
assert (-k + n + u + 2).is_nonpositive is False
assert (u + x).is_nonpositive is None
assert (v - x - n).is_nonpositive is None
assert (u - 2).is_nonnegative is None
assert (u + 17).is_nonnegative is True
assert (-u - 5).is_nonnegative is False
assert (-u + 123).is_nonnegative is None
assert (u - v).is_nonnegative is True
assert (u + v).is_nonnegative is None
assert (-u - v).is_nonnegative is None
assert (-u + v).is_nonnegative is None
assert (u - v + 2).is_nonnegative is True
assert (u + v + 17).is_nonnegative is None
assert (-u - v - 5).is_nonnegative is None
assert (-u + v - 123).is_nonnegative is False
assert (2*u - 123*v + 17).is_nonnegative is True
assert (k + u).is_nonnegative is None
assert (k + v).is_nonnegative is False
assert (n + u).is_nonnegative is True
assert (n + v).is_nonnegative is None
assert (k - n).is_nonnegative is False
assert (k + n).is_nonnegative is None
assert (-k - n).is_nonnegative is None
assert (-k + n).is_nonnegative is True
assert (k - n - u - 2).is_nonnegative is False
assert (k + n - u - 2).is_nonnegative is None
assert (-k - n - u - 2).is_nonnegative is None
assert (-k + n - u - 2).is_nonnegative is None
assert (u - x).is_nonnegative is None
assert (v + x + n).is_nonnegative is None
def test_Pow_is_integer():
x = Symbol('x')
k = Symbol('k', integer=True)
n = Symbol('n', integer=True, nonnegative=True)
m = Symbol('m', integer=True, positive=True)
assert (k**2).is_integer is True
assert (k**(-2)).is_integer is None
assert ((m + 1)**(-2)).is_integer is False
assert (m**(-1)).is_integer is None # issue 8580
assert (2**k).is_integer is None
assert (2**(-k)).is_integer is None
assert (2**n).is_integer is True
assert (2**(-n)).is_integer is None
assert (2**m).is_integer is True
assert (2**(-m)).is_integer is False
assert (x**2).is_integer is None
assert (2**x).is_integer is None
assert (k**n).is_integer is True
assert (k**(-n)).is_integer is None
assert (k**x).is_integer is None
assert (x**k).is_integer is None
assert (k**(n*m)).is_integer is True
assert (k**(-n*m)).is_integer is None
assert sqrt(3).is_integer is False
assert sqrt(.3).is_integer is False
assert Pow(3, 2, evaluate=False).is_integer is True
assert Pow(3, 0, evaluate=False).is_integer is True
assert Pow(3, -2, evaluate=False).is_integer is False
assert Pow(S.Half, 3, evaluate=False).is_integer is False
# decided by re-evaluating
assert Pow(3, S.Half, evaluate=False).is_integer is False
assert Pow(3, S.Half, evaluate=False).is_integer is False
assert Pow(4, S.Half, evaluate=False).is_integer is True
assert Pow(S.Half, -2, evaluate=False).is_integer is True
assert ((-1)**k).is_integer
x = Symbol('x', real=True, integer=False)
assert (x**2).is_integer is None # issue 8641
def test_Pow_is_real():
x = Symbol('x', real=True)
y = Symbol('y', real=True, positive=True)
assert (x**2).is_real is True
assert (x**3).is_real is True
assert (x**x).is_real is None
assert (y**x).is_real is True
assert (x**Rational(1, 3)).is_real is None
assert (y**Rational(1, 3)).is_real is True
assert sqrt(-1 - sqrt(2)).is_real is False
i = Symbol('i', imaginary=True)
assert (i**i).is_real is None
assert (I**i).is_real is True
assert ((-I)**i).is_real is True
assert (2**i).is_real is None # (2**(pi/log(2) * I)) is real, 2**I is not
assert (2**I).is_real is False
assert (2**-I).is_real is False
assert (i**2).is_real is True
assert (i**3).is_real is False
assert (i**x).is_real is None # could be (-I)**(2/3)
e = Symbol('e', even=True)
o = Symbol('o', odd=True)
k = Symbol('k', integer=True)
assert (i**e).is_real is True
assert (i**o).is_real is False
assert (i**k).is_real is None
assert (i**(4*k)).is_real is True
x = Symbol("x", nonnegative=True)
y = Symbol("y", nonnegative=True)
assert im(x**y).expand(complex=True) is S.Zero
assert (x**y).is_real is True
i = Symbol('i', imaginary=True)
assert (exp(i)**I).is_real is True
assert log(exp(i)).is_imaginary is None # i could be 2*pi*I
c = Symbol('c', complex=True)
assert log(c).is_real is None # c could be 0 or 2, too
assert log(exp(c)).is_real is None # log(0), log(E), ...
n = Symbol('n', negative=False)
assert log(n).is_real is None
n = Symbol('n', nonnegative=True)
assert log(n).is_real is None
assert sqrt(-I).is_real is False # issue 7843
def test_real_Pow():
k = Symbol('k', integer=True, nonzero=True)
assert (k**(I*pi/log(k))).is_real
def test_Pow_is_finite():
x = Symbol('x', real=True)
p = Symbol('p', positive=True)
n = Symbol('n', negative=True)
assert (x**2).is_finite is None # x could be oo
assert (x**x).is_finite is None # ditto
assert (p**x).is_finite is None # ditto
assert (n**x).is_finite is None # ditto
assert (1/S.Pi).is_finite
assert (sin(x)**2).is_finite is True
assert (sin(x)**x).is_finite is None
assert (sin(x)**exp(x)).is_finite is None
assert (1/sin(x)).is_finite is None # if zero, no, otherwise yes
assert (1/exp(x)).is_finite is None # x could be -oo
def test_Pow_is_even_odd():
x = Symbol('x')
k = Symbol('k', even=True)
n = Symbol('n', odd=True)
m = Symbol('m', integer=True, nonnegative=True)
p = Symbol('p', integer=True, positive=True)
assert ((-1)**n).is_odd
assert ((-1)**k).is_odd
assert ((-1)**(m - p)).is_odd
assert (k**2).is_even is True
assert (n**2).is_even is False
assert (2**k).is_even is None
assert (x**2).is_even is None
assert (k**m).is_even is None
assert (n**m).is_even is False
assert (k**p).is_even is True
assert (n**p).is_even is False
assert (m**k).is_even is None
assert (p**k).is_even is None
assert (m**n).is_even is None
assert (p**n).is_even is None
assert (k**x).is_even is None
assert (n**x).is_even is None
assert (k**2).is_odd is False
assert (n**2).is_odd is True
assert (3**k).is_odd is None
assert (k**m).is_odd is None
assert (n**m).is_odd is True
assert (k**p).is_odd is False
assert (n**p).is_odd is True
assert (m**k).is_odd is None
assert (p**k).is_odd is None
assert (m**n).is_odd is None
assert (p**n).is_odd is None
assert (k**x).is_odd is None
assert (n**x).is_odd is None
def test_Pow_is_negative_positive():
r = Symbol('r', real=True)
k = Symbol('k', integer=True, positive=True)
n = Symbol('n', even=True)
m = Symbol('m', odd=True)
x = Symbol('x')
assert (2**r).is_positive is True
assert ((-2)**r).is_positive is None
assert ((-2)**n).is_positive is True
assert ((-2)**m).is_positive is False
assert (k**2).is_positive is True
assert (k**(-2)).is_positive is True
assert (k**r).is_positive is True
assert ((-k)**r).is_positive is None
assert ((-k)**n).is_positive is True
assert ((-k)**m).is_positive is False
assert (2**r).is_negative is False
assert ((-2)**r).is_negative is None
assert ((-2)**n).is_negative is False
assert ((-2)**m).is_negative is True
assert (k**2).is_negative is False
assert (k**(-2)).is_negative is False
assert (k**r).is_negative is False
assert ((-k)**r).is_negative is None
assert ((-k)**n).is_negative is False
assert ((-k)**m).is_negative is True
assert (2**x).is_positive is None
assert (2**x).is_negative is None
def test_Pow_is_zero():
z = Symbol('z', zero=True)
e = z**2
assert e.is_zero
assert e.is_positive is False
assert e.is_negative is False
assert Pow(0, 0, evaluate=False).is_zero is False
assert Pow(0, 3, evaluate=False).is_zero
assert Pow(0, oo, evaluate=False).is_zero
assert Pow(0, -3, evaluate=False).is_zero is False
assert Pow(0, -oo, evaluate=False).is_zero is False
assert Pow(2, 2, evaluate=False).is_zero is False
a = Symbol('a', zero=False)
assert Pow(a, 3).is_zero is False # issue 7965
assert Pow(2, oo, evaluate=False).is_zero is False
assert Pow(2, -oo, evaluate=False).is_zero
assert Pow(S.Half, oo, evaluate=False).is_zero
assert Pow(S.Half, -oo, evaluate=False).is_zero is False
def test_Pow_is_nonpositive_nonnegative():
x = Symbol('x', real=True)
k = Symbol('k', integer=True, nonnegative=True)
l = Symbol('l', integer=True, positive=True)
n = Symbol('n', even=True)
m = Symbol('m', odd=True)
assert (x**(4*k)).is_nonnegative is True
assert (2**x).is_nonnegative is True
assert ((-2)**x).is_nonnegative is None
assert ((-2)**n).is_nonnegative is True
assert ((-2)**m).is_nonnegative is False
assert (k**2).is_nonnegative is True
assert (k**(-2)).is_nonnegative is None
assert (k**k).is_nonnegative is True
assert (k**x).is_nonnegative is None # NOTE (0**x).is_real = U
assert (l**x).is_nonnegative is True
assert (l**x).is_positive is True
assert ((-k)**x).is_nonnegative is None
assert ((-k)**m).is_nonnegative is None
assert (2**x).is_nonpositive is False
assert ((-2)**x).is_nonpositive is None
assert ((-2)**n).is_nonpositive is False
assert ((-2)**m).is_nonpositive is True
assert (k**2).is_nonpositive is None
assert (k**(-2)).is_nonpositive is None
assert (k**x).is_nonpositive is None
assert ((-k)**x).is_nonpositive is None
assert ((-k)**n).is_nonpositive is None
assert (x**2).is_nonnegative is True
i = symbols('i', imaginary=True)
assert (i**2).is_nonpositive is True
assert (i**4).is_nonpositive is False
assert (i**3).is_nonpositive is False
assert (I**i).is_nonnegative is True
assert (exp(I)**i).is_nonnegative is True
assert ((-k)**n).is_nonnegative is True
assert ((-k)**m).is_nonpositive is True
def test_Mul_is_imaginary_real():
r = Symbol('r', real=True)
p = Symbol('p', positive=True)
i = Symbol('i', imaginary=True)
ii = Symbol('ii', imaginary=True)
x = Symbol('x')
assert I.is_imaginary is True
assert I.is_real is False
assert (-I).is_imaginary is True
assert (-I).is_real is False
assert (3*I).is_imaginary is True
assert (3*I).is_real is False
assert (I*I).is_imaginary is False
assert (I*I).is_real is True
e = (p + p*I)
j = Symbol('j', integer=True, zero=False)
assert (e**j).is_real is None
assert (e**(2*j)).is_real is None
assert (e**j).is_imaginary is None
assert (e**(2*j)).is_imaginary is None
assert (e**-1).is_imaginary is False
assert (e**2).is_imaginary
assert (e**3).is_imaginary is False
assert (e**4).is_imaginary is False
assert (e**5).is_imaginary is False
assert (e**-1).is_real is False
assert (e**2).is_real is False
assert (e**3).is_real is False
assert (e**4).is_real
assert (e**5).is_real is False
assert (e**3).is_complex
assert (r*i).is_imaginary is None
assert (r*i).is_real is None
assert (x*i).is_imaginary is None
assert (x*i).is_real is None
assert (i*ii).is_imaginary is False
assert (i*ii).is_real is True
assert (r*i*ii).is_imaginary is False
assert (r*i*ii).is_real is True
# Github's issue 5874:
nr = Symbol('nr', real=False, complex=True) # e.g. I or 1 + I
a = Symbol('a', real=True, nonzero=True)
b = Symbol('b', real=True)
assert (i*nr).is_real is None
assert (a*nr).is_real is False
assert (b*nr).is_real is None
ni = Symbol('ni', imaginary=False, complex=True) # e.g. 2 or 1 + I
a = Symbol('a', real=True, nonzero=True)
b = Symbol('b', real=True)
assert (i*ni).is_real is False
assert (a*ni).is_real is None
assert (b*ni).is_real is None
def test_Mul_hermitian_antihermitian():
a = Symbol('a', hermitian=True, zero=False)
b = Symbol('b', hermitian=True)
c = Symbol('c', hermitian=False)
d = Symbol('d', antihermitian=True)
e1 = Mul(a, b, c, evaluate=False)
e2 = Mul(b, a, c, evaluate=False)
e3 = Mul(a, b, c, d, evaluate=False)
e4 = Mul(b, a, c, d, evaluate=False)
e5 = Mul(a, c, evaluate=False)
e6 = Mul(a, c, d, evaluate=False)
assert e1.is_hermitian is None
assert e2.is_hermitian is None
assert e1.is_antihermitian is None
assert e2.is_antihermitian is None
assert e3.is_antihermitian is None
assert e4.is_antihermitian is None
assert e5.is_antihermitian is None
assert e6.is_antihermitian is None
def test_Add_is_comparable():
assert (x + y).is_comparable is False
assert (x + 1).is_comparable is False
assert (Rational(1, 3) - sqrt(8)).is_comparable is True
def test_Mul_is_comparable():
assert (x*y).is_comparable is False
assert (x*2).is_comparable is False
assert (sqrt(2)*Rational(1, 3)).is_comparable is True
def test_Pow_is_comparable():
assert (x**y).is_comparable is False
assert (x**2).is_comparable is False
assert (sqrt(Rational(1, 3))).is_comparable is True
def test_Add_is_positive_2():
e = Rational(1, 3) - sqrt(8)
assert e.is_positive is False
assert e.is_negative is True
e = pi - 1
assert e.is_positive is True
assert e.is_negative is False
def test_Add_is_irrational():
i = Symbol('i', irrational=True)
assert i.is_irrational is True
assert i.is_rational is False
assert (i + 1).is_irrational is True
assert (i + 1).is_rational is False
@XFAIL
def test_issue_3531():
class MightyNumeric(tuple):
def __rdiv__(self, other):
return "something"
def __rtruediv__(self, other):
return "something"
assert sympify(1)/MightyNumeric((1, 2)) == "something"
def test_issue_3531b():
class Foo:
def __init__(self):
self.field = 1.0
def __mul__(self, other):
self.field = self.field * other
def __rmul__(self, other):
self.field = other * self.field
f = Foo()
x = Symbol("x")
assert f*x == x*f
def test_bug3():
a = Symbol("a")
b = Symbol("b", positive=True)
e = 2*a + b
f = b + 2*a
assert e == f
def test_suppressed_evaluation():
a = Add(0, 3, 2, evaluate=False)
b = Mul(1, 3, 2, evaluate=False)
c = Pow(3, 2, evaluate=False)
assert a != 6
assert a.func is Add
assert a.args == (3, 2)
assert b != 6
assert b.func is Mul
assert b.args == (3, 2)
assert c != 9
assert c.func is Pow
assert c.args == (3, 2)
def test_Add_as_coeff_mul():
# issue 5524. These should all be (1, self)
assert (x + 1).as_coeff_mul() == (1, (x + 1,))
assert (x + 2).as_coeff_mul() == (1, (x + 2,))
assert (x + 3).as_coeff_mul() == (1, (x + 3,))
assert (x - 1).as_coeff_mul() == (1, (x - 1,))
assert (x - 2).as_coeff_mul() == (1, (x - 2,))
assert (x - 3).as_coeff_mul() == (1, (x - 3,))
n = Symbol('n', integer=True)
assert (n + 1).as_coeff_mul() == (1, (n + 1,))
assert (n + 2).as_coeff_mul() == (1, (n + 2,))
assert (n + 3).as_coeff_mul() == (1, (n + 3,))
assert (n - 1).as_coeff_mul() == (1, (n - 1,))
assert (n - 2).as_coeff_mul() == (1, (n - 2,))
assert (n - 3).as_coeff_mul() == (1, (n - 3,))
def test_Pow_as_coeff_mul_doesnt_expand():
assert exp(x + y).as_coeff_mul() == (1, (exp(x + y),))
assert exp(x + exp(x + y)) != exp(x + exp(x)*exp(y))
def test_issue_3514():
assert sqrt(S.Half) * sqrt(6) == 2 * sqrt(3)/2
assert S(1)/2*sqrt(6)*sqrt(2) == sqrt(3)
assert sqrt(6)/2*sqrt(2) == sqrt(3)
assert sqrt(6)*sqrt(2)/2 == sqrt(3)
def test_make_args():
assert Add.make_args(x) == (x,)
assert Mul.make_args(x) == (x,)
assert Add.make_args(x*y*z) == (x*y*z,)
assert Mul.make_args(x*y*z) == (x*y*z).args
assert Add.make_args(x + y + z) == (x + y + z).args
assert Mul.make_args(x + y + z) == (x + y + z,)
assert Add.make_args((x + y)**z) == ((x + y)**z,)
assert Mul.make_args((x + y)**z) == ((x + y)**z,)
def test_issue_5126():
assert (-2)**x*(-3)**x != 6**x
i = Symbol('i', integer=1)
assert (-2)**i*(-3)**i == 6**i
def test_Rational_as_content_primitive():
c, p = S(1), S(0)
assert (c*p).as_content_primitive() == (c, p)
c, p = S(1)/2, S(1)
assert (c*p).as_content_primitive() == (c, p)
def test_Add_as_content_primitive():
assert (x + 2).as_content_primitive() == (1, x + 2)
assert (3*x + 2).as_content_primitive() == (1, 3*x + 2)
assert (3*x + 3).as_content_primitive() == (3, x + 1)
assert (3*x + 6).as_content_primitive() == (3, x + 2)
assert (3*x + 2*y).as_content_primitive() == (1, 3*x + 2*y)
assert (3*x + 3*y).as_content_primitive() == (3, x + y)
assert (3*x + 6*y).as_content_primitive() == (3, x + 2*y)
assert (3/x + 2*x*y*z**2).as_content_primitive() == (1, 3/x + 2*x*y*z**2)
assert (3/x + 3*x*y*z**2).as_content_primitive() == (3, 1/x + x*y*z**2)
assert (3/x + 6*x*y*z**2).as_content_primitive() == (3, 1/x + 2*x*y*z**2)
assert (2*x/3 + 4*y/9).as_content_primitive() == \
(Rational(2, 9), 3*x + 2*y)
assert (2*x/3 + 2.5*y).as_content_primitive() == \
(Rational(1, 3), 2*x + 7.5*y)
# the coefficient may sort to a position other than 0
p = 3 + x + y
assert (2*p).expand().as_content_primitive() == (2, p)
assert (2.0*p).expand().as_content_primitive() == (1, 2.*p)
p *= -1
assert (2*p).expand().as_content_primitive() == (2, p)
def test_Mul_as_content_primitive():
assert (2*x).as_content_primitive() == (2, x)
assert (x*(2 + 2*x)).as_content_primitive() == (2, x*(1 + x))
assert (x*(2 + 2*y)*(3*x + 3)**2).as_content_primitive() == \
(18, x*(1 + y)*(x + 1)**2)
assert ((2 + 2*x)**2*(3 + 6*x) + S.Half).as_content_primitive() == \
(S.Half, 24*(x + 1)**2*(2*x + 1) + 1)
def test_Pow_as_content_primitive():
assert (x**y).as_content_primitive() == (1, x**y)
assert ((2*x + 2)**y).as_content_primitive() == \
(1, (Mul(2, (x + 1), evaluate=False))**y)
assert ((2*x + 2)**3).as_content_primitive() == (8, (x + 1)**3)
def test_issue_5460():
u = Mul(2, (1 + x), evaluate=False)
assert (2 + u).args == (2, u)
def test_product_irrational():
from sympy import I, pi
assert (I*pi).is_irrational is False
# The following used to be deduced from the above bug:
assert (I*pi).is_positive is False
def test_issue_5919():
assert (x/(y*(1 + y))).expand() == x/(y**2 + y)
def test_Mod():
assert Mod(x, 1).func is Mod
assert pi % pi == S.Zero
assert Mod(5, 3) == 2
assert Mod(-5, 3) == 1
assert Mod(5, -3) == -1
assert Mod(-5, -3) == -2
assert type(Mod(3.2, 2, evaluate=False)) == Mod
assert 5 % x == Mod(5, x)
assert x % 5 == Mod(x, 5)
assert x % y == Mod(x, y)
assert (x % y).subs({x: 5, y: 3}) == 2
assert Mod(nan, 1) == nan
assert Mod(1, nan) == nan
assert Mod(nan, nan) == nan
Mod(0, x) == 0
with raises(ZeroDivisionError):
Mod(x, 0)
k = Symbol('k', integer=True)
m = Symbol('m', integer=True, positive=True)
assert (x**m % x).func is Mod
assert (k**(-m) % k).func is Mod
assert k**m % k == 0
assert (-2*k)**m % k == 0
# Float handling
point3 = Float(3.3) % 1
assert (x - 3.3) % 1 == Mod(1.*x + 1 - point3, 1)
assert Mod(-3.3, 1) == 1 - point3
assert Mod(0.7, 1) == Float(0.7)
e = Mod(1.3, 1)
assert comp(e, .3) and e.is_Float
e = Mod(1.3, .7)
assert comp(e, .6) and e.is_Float
e = Mod(1.3, Rational(7, 10))
assert comp(e, .6) and e.is_Float
e = Mod(Rational(13, 10), 0.7)
assert comp(e, .6) and e.is_Float
e = Mod(Rational(13, 10), Rational(7, 10))
assert comp(e, .6) and e.is_Rational
# check that sign is right
r2 = sqrt(2)
r3 = sqrt(3)
for i in [-r3, -r2, r2, r3]:
for j in [-r3, -r2, r2, r3]:
assert verify_numerically(i % j, i.n() % j.n())
for _x in range(4):
for _y in range(9):
reps = [(x, _x), (y, _y)]
assert Mod(3*x + y, 9).subs(reps) == (3*_x + _y) % 9
# denesting
t = Symbol('t', real=True)
assert Mod(Mod(x, t), t) == Mod(x, t)
assert Mod(-Mod(x, t), t) == Mod(-x, t)
assert Mod(Mod(x, 2*t), t) == Mod(x, t)
assert Mod(-Mod(x, 2*t), t) == Mod(-x, t)
assert Mod(Mod(x, t), 2*t) == Mod(x, t)
assert Mod(-Mod(x, t), -2*t) == -Mod(x, t)
for i in [-4, -2, 2, 4]:
for j in [-4, -2, 2, 4]:
for k in range(4):
assert Mod(Mod(x, i), j).subs({x: k}) == (k % i) % j
assert Mod(-Mod(x, i), j).subs({x: k}) == -(k % i) % j
# known difference
assert Mod(5*sqrt(2), sqrt(5)) == 5*sqrt(2) - 3*sqrt(5)
p = symbols('p', positive=True)
assert Mod(2, p + 3) == 2
assert Mod(-2, p + 3) == p + 1
assert Mod(2, -p - 3) == -p - 1
assert Mod(-2, -p - 3) == -2
assert Mod(p + 5, p + 3) == 2
assert Mod(-p - 5, p + 3) == p + 1
assert Mod(p + 5, -p - 3) == -p - 1
assert Mod(-p - 5, -p - 3) == -2
assert Mod(p + 1, p - 1).func is Mod
# handling sums
assert (x + 3) % 1 == Mod(x, 1)
assert (x + 3.0) % 1 == Mod(1.*x, 1)
assert (x - S(33)/10) % 1 == Mod(x + S(7)/10, 1)
a = Mod(.6*x + y, .3*y)
b = Mod(0.1*y + 0.6*x, 0.3*y)
# Test that a, b are equal, with 1e-14 accuracy in coefficients
eps = 1e-14
assert abs((a.args[0] - b.args[0]).subs({x: 1, y: 1})) < eps
assert abs((a.args[1] - b.args[1]).subs({x: 1, y: 1})) < eps
assert (x + 1) % x == 1 % x
assert (x + y) % x == y % x
assert (x + y + 2) % x == (y + 2) % x
assert (a + 3*x + 1) % (2*x) == Mod(a + x + 1, 2*x)
assert (12*x + 18*y) % (3*x) == 3*Mod(6*y, x)
# gcd extraction
assert (-3*x) % (-2*y) == -Mod(3*x, 2*y)
assert (.6*pi) % (.3*x*pi) == 0.3*pi*Mod(2, x)
assert (.6*pi) % (.31*x*pi) == pi*Mod(0.6, 0.31*x)
assert (6*pi) % (.3*x*pi) == 0.3*pi*Mod(20, x)
assert (6*pi) % (.31*x*pi) == pi*Mod(6, 0.31*x)
assert (6*pi) % (.42*x*pi) == pi*Mod(6, 0.42*x)
assert (12*x) % (2*y) == 2*Mod(6*x, y)
assert (12*x) % (3*5*y) == 3*Mod(4*x, 5*y)
assert (12*x) % (15*x*y) == 3*x*Mod(4, 5*y)
assert (-2*pi) % (3*pi) == pi
assert (2*x + 2) % (x + 1) == 0
assert (x*(x + 1)) % (x + 1) == (x + 1)*Mod(x, 1)
assert Mod(5.0*x, 0.1*y) == 0.1*Mod(50*x, y)
i = Symbol('i', integer=True)
assert (3*i*x) % (2*i*y) == i*Mod(3*x, 2*y)
assert Mod(4*i, 4) == 0
# issue 8677
n = Symbol('n', integer=True, positive=True)
assert factorial(n) % n == 0
assert factorial(n + 2) % n == 0
assert (factorial(n + 4) % (n + 5)).func is Mod
# modular exponentiation
assert Mod(Pow(4, 13, evaluate=False), 497) == Mod(Pow(4, 13), 497)
assert Mod(Pow(2, 10000000000, evaluate=False), 3) == 1
assert Mod(Pow(32131231232, 9**10**6, evaluate=False),10**12) == pow(32131231232,9**10**6,10**12)
assert Mod(Pow(33284959323, 123**999, evaluate=False),11**13) == pow(33284959323,123**999,11**13)
assert Mod(Pow(78789849597, 333**555, evaluate=False),12**9) == pow(78789849597,333**555,12**9)
# Wilson's theorem
factorial(18042, evaluate=False) % 18043 == 18042
p = Symbol('n', prime=True)
factorial(p - 1) % p == p - 1
factorial(p - 1) % -p == -1
(factorial(3, evaluate=False) % 4).doit() == 2
n = Symbol('n', composite=True, odd=True)
factorial(n - 1) % n == 0
# symbolic with known parity
n = Symbol('n', even=True)
assert Mod(n, 2) == 0
n = Symbol('n', odd=True)
assert Mod(n, 2) == 1
# issue 10963
assert (x**6000%400).args[1] == 400
#issue 13543
assert Mod(Mod(x + 1, 2) + 1 , 2) == Mod(x,2)
assert Mod(Mod(x + 2, 4)*(x + 4), 4) == Mod(x*(x + 2), 4)
assert Mod(Mod(x + 2, 4)*4, 4) == 0
# issue 15493
i, j = symbols('i j', integer=True, positive=True)
assert Mod(3*i, 2) == Mod(i, 2)
assert Mod(8*i/j, 4) == 4*Mod(2*i/j, 1)
assert Mod(8*i, 4) == 0
def test_Mod_is_integer():
p = Symbol('p', integer=True)
q1 = Symbol('q1', integer=True)
q2 = Symbol('q2', integer=True, nonzero=True)
assert Mod(x, y).is_integer is None
assert Mod(p, q1).is_integer is None
assert Mod(x, q2).is_integer is None
assert Mod(p, q2).is_integer
def test_Mod_is_nonposneg():
n = Symbol('n', integer=True)
k = Symbol('k', integer=True, positive=True)
assert (n%3).is_nonnegative
assert Mod(n, -3).is_nonpositive
assert Mod(n, k).is_nonnegative
assert Mod(n, -k).is_nonpositive
assert Mod(k, n).is_nonnegative is None
def test_issue_6001():
A = Symbol("A", commutative=False)
eq = A + A**2
# it doesn't matter whether it's True or False; they should
# just all be the same
assert (
eq.is_commutative ==
(eq + 1).is_commutative ==
(A + 1).is_commutative)
B = Symbol("B", commutative=False)
# Although commutative terms could cancel we return True
# meaning "there are non-commutative symbols; aftersubstitution
# that definition can change, e.g. (A*B).subs(B,A**-1) -> 1
assert (sqrt(2)*A).is_commutative is False
assert (sqrt(2)*A*B).is_commutative is False
def test_polar():
from sympy import polar_lift
p = Symbol('p', polar=True)
x = Symbol('x')
assert p.is_polar
assert x.is_polar is None
assert S(1).is_polar is None
assert (p**x).is_polar is True
assert (x**p).is_polar is None
assert ((2*p)**x).is_polar is True
assert (2*p).is_polar is True
assert (-2*p).is_polar is not True
assert (polar_lift(-2)*p).is_polar is True
q = Symbol('q', polar=True)
assert (p*q)**2 == p**2 * q**2
assert (2*q)**2 == 4 * q**2
assert ((p*q)**x).expand() == p**x * q**x
def test_issue_6040():
a, b = Pow(1, 2, evaluate=False), S.One
assert a != b
assert b != a
assert not (a == b)
assert not (b == a)
def test_issue_6082():
# Comparison is symmetric
assert Basic.compare(Max(x, 1), Max(x, 2)) == \
- Basic.compare(Max(x, 2), Max(x, 1))
# Equal expressions compare equal
assert Basic.compare(Max(x, 1), Max(x, 1)) == 0
# Basic subtypes (such as Max) compare different than standard types
assert Basic.compare(Max(1, x), frozenset((1, x))) != 0
def test_issue_6077():
assert x**2.0/x == x**1.0
assert x/x**2.0 == x**-1.0
assert x*x**2.0 == x**3.0
assert x**1.5*x**2.5 == x**4.0
assert 2**(2.0*x)/2**x == 2**(1.0*x)
assert 2**x/2**(2.0*x) == 2**(-1.0*x)
assert 2**x*2**(2.0*x) == 2**(3.0*x)
assert 2**(1.5*x)*2**(2.5*x) == 2**(4.0*x)
def test_mul_flatten_oo():
p = symbols('p', positive=True)
n, m = symbols('n,m', negative=True)
x_im = symbols('x_im', imaginary=True)
assert n*oo == -oo
assert n*m*oo == oo
assert p*oo == oo
assert x_im*oo != I*oo # i could be +/- 3*I -> +/-oo
def test_add_flatten():
# see https://github.com/sympy/sympy/issues/2633#issuecomment-29545524
a = oo + I*oo
b = oo - I*oo
assert a + b == nan
assert a - b == nan
assert (1/a).simplify() == (1/b).simplify() == 0
a = Pow(2, 3, evaluate=False)
assert a + a == 16
def test_issue_5160_6087_6089_6090():
# issue 6087
assert ((-2*x*y**y)**3.2).n(2) == (2**3.2*(-x*y**y)**3.2).n(2)
# issue 6089
A, B, C = symbols('A,B,C', commutative=False)
assert (2.*B*C)**3 == 8.0*(B*C)**3
assert (-2.*B*C)**3 == -8.0*(B*C)**3
assert (-2*B*C)**2 == 4*(B*C)**2
# issue 5160
assert sqrt(-1.0*x) == 1.0*sqrt(-x)
assert sqrt(1.0*x) == 1.0*sqrt(x)
# issue 6090
assert (-2*x*y*A*B)**2 == 4*x**2*y**2*(A*B)**2
def test_float_int():
assert int(float(sqrt(10))) == int(sqrt(10))
assert int(pi**1000) % 10 == 2
assert int(Float('1.123456789012345678901234567890e20', '')) == \
long(112345678901234567890)
assert int(Float('1.123456789012345678901234567890e25', '')) == \
long(11234567890123456789012345)
# decimal forces float so it's not an exact integer ending in 000000
assert int(Float('1.123456789012345678901234567890e35', '')) == \
112345678901234567890123456789000192
assert int(Float('123456789012345678901234567890e5', '')) == \
12345678901234567890123456789000000
assert Integer(Float('1.123456789012345678901234567890e20', '')) == \
112345678901234567890
assert Integer(Float('1.123456789012345678901234567890e25', '')) == \
11234567890123456789012345
# decimal forces float so it's not an exact integer ending in 000000
assert Integer(Float('1.123456789012345678901234567890e35', '')) == \
112345678901234567890123456789000192
assert Integer(Float('123456789012345678901234567890e5', '')) == \
12345678901234567890123456789000000
assert same_and_same_prec(Float('123000e-2',''), Float('1230.00', ''))
assert same_and_same_prec(Float('123000e2',''), Float('12300000', ''))
assert int(1 + Rational('.9999999999999999999999999')) == 1
assert int(pi/1e20) == 0
assert int(1 + pi/1e20) == 1
assert int(Add(1.2, -2, evaluate=False)) == int(1.2 - 2)
assert int(Add(1.2, +2, evaluate=False)) == int(1.2 + 2)
assert int(Add(1 + Float('.99999999999999999', ''), evaluate=False)) == 1
raises(TypeError, lambda: float(x))
raises(TypeError, lambda: float(sqrt(-1)))
assert int(12345678901234567890 + cos(1)**2 + sin(1)**2) == \
12345678901234567891
def test_issue_6611a():
assert Mul.flatten([3**Rational(1, 3),
Pow(-Rational(1, 9), Rational(2, 3), evaluate=False)]) == \
([Rational(1, 3), (-1)**Rational(2, 3)], [], None)
def test_denest_add_mul():
# when working with evaluated expressions make sure they denest
eq = x + 1
eq = Add(eq, 2, evaluate=False)
eq = Add(eq, 2, evaluate=False)
assert Add(*eq.args) == x + 5
eq = x*2
eq = Mul(eq, 2, evaluate=False)
eq = Mul(eq, 2, evaluate=False)
assert Mul(*eq.args) == 8*x
# but don't let them denest unecessarily
eq = Mul(-2, x - 2, evaluate=False)
assert 2*eq == Mul(-4, x - 2, evaluate=False)
assert -eq == Mul(2, x - 2, evaluate=False)
def test_mul_coeff():
# It is important that all Numbers be removed from the seq;
# This can be tricky when powers combine to produce those numbers
p = exp(I*pi/3)
assert p**2*x*p*y*p*x*p**2 == x**2*y
def test_mul_zero_detection():
nz = Dummy(real=True, zero=False, finite=True)
r = Dummy(real=True)
c = Dummy(real=False, complex=True, finite=True)
c2 = Dummy(real=False, complex=True, finite=True)
i = Dummy(imaginary=True, finite=True)
e = nz*r*c
assert e.is_imaginary is None
assert e.is_real is None
e = nz*c
assert e.is_imaginary is None
assert e.is_real is False
e = nz*i*c
assert e.is_imaginary is False
assert e.is_real is None
# check for more than one complex; it is important to use
# uniquely named Symbols to ensure that two factors appear
# e.g. if the symbols have the same name they just become
# a single factor, a power.
e = nz*i*c*c2
assert e.is_imaginary is None
assert e.is_real is None
# _eval_is_real and _eval_is_zero both employ trapping of the
# zero value so args should be tested in both directions and
# TO AVOID GETTING THE CACHED RESULT, Dummy MUST BE USED
# real is unknonwn
def test(z, b, e):
if z.is_zero and b.is_finite:
assert e.is_real and e.is_zero
else:
assert e.is_real is None
if b.is_finite:
if z.is_zero:
assert e.is_zero
else:
assert e.is_zero is None
elif b.is_finite is False:
if z.is_zero is None:
assert e.is_zero is None
else:
assert e.is_zero is False
for iz, ib in cartes(*[[True, False, None]]*2):
z = Dummy('z', nonzero=iz)
b = Dummy('f', finite=ib)
e = Mul(z, b, evaluate=False)
test(z, b, e)
z = Dummy('nz', nonzero=iz)
b = Dummy('f', finite=ib)
e = Mul(b, z, evaluate=False)
test(z, b, e)
# real is True
def test(z, b, e):
if z.is_zero and not b.is_finite:
assert e.is_real is None
else:
assert e.is_real
for iz, ib in cartes(*[[True, False, None]]*2):
z = Dummy('z', nonzero=iz, real=True)
b = Dummy('b', finite=ib, real=True)
e = Mul(z, b, evaluate=False)
test(z, b, e)
z = Dummy('z', nonzero=iz, real=True)
b = Dummy('b', finite=ib, real=True)
e = Mul(b, z, evaluate=False)
test(z, b, e)
def test_Mul_with_zero_infinite():
zer = Dummy(zero=True)
inf = Dummy(finite=False)
e = Mul(zer, inf, evaluate=False)
assert e.is_positive is None
assert e.is_hermitian is None
e = Mul(inf, zer, evaluate=False)
assert e.is_positive is None
assert e.is_hermitian is None
def test_Mul_does_not_cancel_infinities():
a, b = symbols('a b')
assert ((zoo + 3*a)/(3*a + zoo)) is nan
assert ((b - oo)/(b - oo)) is nan
# issue 13904
expr = (1/(a+b) + 1/(a-b))/(1/(a+b) - 1/(a-b))
assert expr.subs(b, a) is nan
def test_Mul_does_not_distribute_infinity():
a, b = symbols('a b')
assert ((1 + I)*oo).is_Mul
assert ((a + b)*(-oo)).is_Mul
assert ((a + 1)*zoo).is_Mul
assert ((1 + I)*oo).is_finite is False
z = (1 + I)*oo
assert ((1 - I)*z).expand() is oo
def test_issue_8247_8354():
from sympy import tan
z = sqrt(1 + sqrt(3)) + sqrt(3 + 3*sqrt(3)) - sqrt(10 + 6*sqrt(3))
assert z.is_positive is False # it's 0
z = S('''-2**(1/3)*(3*sqrt(93) + 29)**2 - 4*(3*sqrt(93) + 29)**(4/3) +
12*sqrt(93)*(3*sqrt(93) + 29)**(1/3) + 116*(3*sqrt(93) + 29)**(1/3) +
174*2**(1/3)*sqrt(93) + 1678*2**(1/3)''')
assert z.is_positive is False # it's 0
z = 2*(-3*tan(19*pi/90) + sqrt(3))*cos(11*pi/90)*cos(19*pi/90) - \
sqrt(3)*(-3 + 4*cos(19*pi/90)**2)
assert z.is_positive is not True # it's zero and it shouldn't hang
z = S('''9*(3*sqrt(93) + 29)**(2/3)*((3*sqrt(93) +
29)**(1/3)*(-2**(2/3)*(3*sqrt(93) + 29)**(1/3) - 2) - 2*2**(1/3))**3 +
72*(3*sqrt(93) + 29)**(2/3)*(81*sqrt(93) + 783) + (162*sqrt(93) +
1566)*((3*sqrt(93) + 29)**(1/3)*(-2**(2/3)*(3*sqrt(93) + 29)**(1/3) -
2) - 2*2**(1/3))**2''')
assert z.is_positive is False # it's 0 (and a single _mexpand isn't enough)
def test_Add_is_zero():
x, y = symbols('x y', zero=True)
assert (x + y).is_zero
# Issue 15873
e = -2*I + (1 + I)**2
assert e.is_zero is None
def test_issue_14392():
assert (sin(zoo)**2).as_real_imag() == (nan, nan)
def test_divmod():
assert divmod(x, y) == (x//y, x % y)
assert divmod(x, 3) == (x//3, x % 3)
assert divmod(3, x) == (3//x, 3 % x)
|
20f33c7e500de28ad2dd72529d491ab59de83a4e18f6bdd61957ef06b3768bcb
|
from sympy.core.compatibility import PY3
from sympy.core.logic import (fuzzy_not, Logic, And, Or, Not, fuzzy_and,
fuzzy_or, _fuzzy_group, _torf)
from sympy.utilities.pytest import raises
T = True
F = False
U = None
def test_torf():
from sympy.utilities.iterables import cartes
v = [T, F, U]
for i in cartes(*[v]*3):
assert _torf(i) is (True if all(j for j in i) else
(False if all(j is False for j in i) else None))
def test_fuzzy_group():
from sympy.utilities.iterables import cartes
v = [T, F, U]
for i in cartes(*[v]*3):
assert _fuzzy_group(i) is (None if None in i else
(True if all(j for j in i) else False))
assert _fuzzy_group(i, quick_exit=True) is \
(None if (i.count(False) > 1) else
(None if None in i else (True if all(j for j in i) else False)))
it = (True if (i == 0) else None for i in range(2))
assert _torf(it) is None
it = (True if (i == 1) else None for i in range(2))
assert _torf(it) is None
def test_fuzzy_not():
assert fuzzy_not(T) == F
assert fuzzy_not(F) == T
assert fuzzy_not(U) == U
def test_fuzzy_and():
assert fuzzy_and([T, T]) == T
assert fuzzy_and([T, F]) == F
assert fuzzy_and([T, U]) == U
assert fuzzy_and([F, F]) == F
assert fuzzy_and([F, U]) == F
assert fuzzy_and([U, U]) == U
assert [fuzzy_and([w]) for w in [U, T, F]] == [U, T, F]
assert fuzzy_and([T, F, U]) == F
assert fuzzy_and([]) == T
raises(TypeError, lambda: fuzzy_and())
def test_fuzzy_or():
assert fuzzy_or([T, T]) == T
assert fuzzy_or([T, F]) == T
assert fuzzy_or([T, U]) == T
assert fuzzy_or([F, F]) == F
assert fuzzy_or([F, U]) == U
assert fuzzy_or([U, U]) == U
assert [fuzzy_or([w]) for w in [U, T, F]] == [U, T, F]
assert fuzzy_or([T, F, U]) == T
assert fuzzy_or([]) == F
raises(TypeError, lambda: fuzzy_or())
def test_logic_cmp():
l1 = And('a', Not('b'))
l2 = And('a', Not('b'))
assert hash(l1) == hash(l2)
assert (l1 == l2) == T
assert (l1 != l2) == F
assert And('a', 'b', 'c') == And('b', 'a', 'c')
assert And('a', 'b', 'c') == And('c', 'b', 'a')
assert And('a', 'b', 'c') == And('c', 'a', 'b')
assert Not('a') < Not('b')
assert (Not('b') < Not('a')) is False
if PY3:
assert (Not('a') < 2) is False
def test_logic_onearg():
assert And() is True
assert Or() is False
assert And(T) == T
assert And(F) == F
assert Or(T) == T
assert Or(F) == F
assert And('a') == 'a'
assert Or('a') == 'a'
def test_logic_xnotx():
assert And('a', Not('a')) == F
assert Or('a', Not('a')) == T
def test_logic_eval_TF():
assert And(F, F) == F
assert And(F, T) == F
assert And(T, F) == F
assert And(T, T) == T
assert Or(F, F) == F
assert Or(F, T) == T
assert Or(T, F) == T
assert Or(T, T) == T
assert And('a', T) == 'a'
assert And('a', F) == F
assert Or('a', T) == T
assert Or('a', F) == 'a'
def test_logic_combine_args():
assert And('a', 'b', 'a') == And('a', 'b')
assert Or('a', 'b', 'a') == Or('a', 'b')
assert And(And('a', 'b'), And('c', 'd')) == And('a', 'b', 'c', 'd')
assert Or(Or('a', 'b'), Or('c', 'd')) == Or('a', 'b', 'c', 'd')
assert Or('t', And('n', 'p', 'r'), And('n', 'r'), And('n', 'p', 'r'), 't',
And('n', 'r')) == Or('t', And('n', 'p', 'r'), And('n', 'r'))
def test_logic_expand():
t = And(Or('a', 'b'), 'c')
assert t.expand() == Or(And('a', 'c'), And('b', 'c'))
t = And(Or('a', Not('b')), 'b')
assert t.expand() == And('a', 'b')
t = And(Or('a', 'b'), Or('c', 'd'))
assert t.expand() == \
Or(And('a', 'c'), And('a', 'd'), And('b', 'c'), And('b', 'd'))
def test_logic_fromstring():
S = Logic.fromstring
assert S('a') == 'a'
assert S('!a') == Not('a')
assert S('a & b') == And('a', 'b')
assert S('a | b') == Or('a', 'b')
assert S('a | b & c') == And(Or('a', 'b'), 'c')
assert S('a & b | c') == Or(And('a', 'b'), 'c')
assert S('a & b & c') == And('a', 'b', 'c')
assert S('a | b | c') == Or('a', 'b', 'c')
raises(ValueError, lambda: S('| a'))
raises(ValueError, lambda: S('& a'))
raises(ValueError, lambda: S('a | | b'))
raises(ValueError, lambda: S('a | & b'))
raises(ValueError, lambda: S('a & & b'))
raises(ValueError, lambda: S('a |'))
raises(ValueError, lambda: S('a|b'))
raises(ValueError, lambda: S('!'))
raises(ValueError, lambda: S('! a'))
raises(ValueError, lambda: S('!(a + 1)'))
raises(ValueError, lambda: S(''))
def test_logic_not():
assert Not('a') != '!a'
assert Not('!a') != 'a'
assert Not(True) == False
assert Not(False) == True
# NOTE: we may want to change default Not behaviour and put this
# functionality into some method.
assert Not(And('a', 'b')) == Or(Not('a'), Not('b'))
assert Not(Or('a', 'b')) == And(Not('a'), Not('b'))
S = Logic.fromstring
raises(ValueError, lambda: Not(1))
def test_formatting():
S = Logic.fromstring
raises(ValueError, lambda: S('a&b'))
raises(ValueError, lambda: S('a|b'))
raises(ValueError, lambda: S('! a'))
|
57834f69feeb4bb67d808ac15912d9a372fbc8d9c477d64ba785856bf001442c
|
"""Tests for Dixon's and Macaulay's classes. """
from sympy import Matrix
from sympy.core import symbols
from sympy.tensor.indexed import IndexedBase
from sympy.polys.multivariate_resultants import (DixonResultant,
MacaulayResultant)
c, d = symbols("a, b")
x, y = symbols("x, y")
p = c * x + y
q = x + d * y
dixon = DixonResultant(polynomials=[p, q], variables=[x, y])
macaulay = MacaulayResultant(polynomials=[p, q], variables=[x, y])
def test_dixon_resultant_init():
"""Test init method of DixonResultant."""
a = IndexedBase("alpha")
assert dixon.polynomials == [p, q]
assert dixon.variables == [x, y]
assert dixon.n == 2
assert dixon.m == 2
assert dixon.dummy_variables == [a[0], a[1]]
assert dixon.max_degrees == [1, 1]
def test_get_dixon_polynomial_numerical():
"""Test Dixon's polynomial for a numerical example."""
a = IndexedBase("alpha")
p = x + y
q = x ** 2 + y **3
h = x ** 2 + y
dixon = DixonResultant([p, q, h], [x, y])
polynomial = -x * y ** 2 * a[0] - x * y ** 2 * a[1] - x * y * a[0] \
* a[1] - x * y * a[1] ** 2 - x * a[0] * a[1] ** 2 + x * a[0] - \
y ** 2 * a[0] * a[1] + y ** 2 * a[1] - y * a[0] * a[1] ** 2 + y * \
a[1] ** 2
assert dixon.get_dixon_polynomial().factor() == polynomial
def test_get_upper_degree():
"""Tests upper degree function."""
h = c * x ** 2 + y
dixon = DixonResultant(polynomials=[h, q], variables=[x, y])
assert dixon.get_upper_degree() == 3
def test_get_dixon_matrix_example_two():
"""Test Dixon's matrix for example from [Palancz08]_."""
x, y, z = symbols('x, y, z')
f = x ** 2 + y ** 2 - 1 + z * 0
g = x ** 2 + z ** 2 - 1 + y * 0
h = y ** 2 + z ** 2 - 1
example_two = DixonResultant([f, g, h], [y, z])
poly = example_two.get_dixon_polynomial()
matrix = example_two.get_dixon_matrix(poly)
expr = 1 - 8 * x ** 2 + 24 * x ** 4 - 32 * x ** 6 + 16 * x ** 8
assert (matrix.det() - expr).expand() == 0
def test_get_dixon_matrix():
"""Test Dixon's resultant for a numerical example."""
x, y = symbols('x, y')
p = x + y
q = x ** 2 + y ** 3
h = x ** 2 + y
dixon = DixonResultant([p, q, h], [x, y])
polynomial = dixon.get_dixon_polynomial()
assert dixon.get_dixon_matrix(polynomial).det() == 0
def test_macaulay_resultant_init():
"""Test init method of MacaulayResultant."""
a = IndexedBase("alpha")
assert macaulay.polynomials == [p, q]
assert macaulay.variables == [x, y]
assert macaulay.n == 2
assert macaulay.degrees == [1, 1]
assert macaulay.degree_m == 1
assert macaulay.monomials_size == 2
def test_get_degree_m():
assert macaulay._get_degree_m() == 1
def test_get_size():
assert macaulay.get_size() == 2
def test_macaulay_example_one():
"""Tests the Macaulay for example from [Bruce97]_"""
x, y, z = symbols('x, y, z')
a_1_1, a_1_2, a_1_3 = symbols('a_1_1, a_1_2, a_1_3')
a_2_2, a_2_3, a_3_3 = symbols('a_2_2, a_2_3, a_3_3')
b_1_1, b_1_2, b_1_3 = symbols('b_1_1, b_1_2, b_1_3')
b_2_2, b_2_3, b_3_3 = symbols('b_2_2, b_2_3, b_3_3')
c_1, c_2, c_3 = symbols('c_1, c_2, c_3')
f_1 = a_1_1 * x ** 2 + a_1_2 * x * y + a_1_3 * x * z + \
a_2_2 * y ** 2 + a_2_3 * y * z + a_3_3 * z ** 2
f_2 = b_1_1 * x ** 2 + b_1_2 * x * y + b_1_3 * x * z + \
b_2_2 * y ** 2 + b_2_3 * y * z + b_3_3 * z ** 2
f_3 = c_1 * x + c_2 * y + c_3 * z
mac = MacaulayResultant([f_1, f_2, f_3], [x, y, z])
assert mac.degrees == [2, 2, 1]
assert mac.degree_m == 3
assert mac.monomial_set == [x ** 3, x ** 2 * y, x ** 2 * z,
x * y ** 2,
x * y * z, x * z ** 2, y ** 3,
y ** 2 *z, y * z ** 2, z ** 3]
assert mac.monomials_size == 10
assert mac.get_row_coefficients() == [[x, y, z], [x, y, z],
[x * y, x * z, y * z, z ** 2]]
matrix = mac.get_matrix()
assert matrix.shape == (mac.monomials_size, mac.monomials_size)
assert mac.get_submatrix(matrix) == Matrix([[a_1_1, a_2_2],
[b_1_1, b_2_2]])
def test_macaulay_example_two():
"""Tests the Macaulay formulation for example from [Stiller96]_."""
x, y, z = symbols('x, y, z')
a_0, a_1, a_2 = symbols('a_0, a_1, a_2')
b_0, b_1, b_2 = symbols('b_0, b_1, b_2')
c_0, c_1, c_2, c_3, c_4 = symbols('c_0, c_1, c_2, c_3, c_4')
f = a_0 * y - a_1 * x + a_2 * z
g = b_1 * x ** 2 + b_0 * y ** 2 - b_2 * z ** 2
h = c_0 * y - c_1 * x ** 3 + c_2 * x ** 2 * z - c_3 * x * z ** 2 + \
c_4 * z ** 3
mac = MacaulayResultant([f, g, h], [x, y, z])
assert mac.degrees == [1, 2, 3]
assert mac.degree_m == 4
assert mac.monomials_size == 15
assert len(mac.get_row_coefficients()) == mac.n
matrix = mac.get_matrix()
assert matrix.shape == (mac.monomials_size, mac.monomials_size)
assert mac.get_submatrix(matrix) == Matrix([[-a_1, a_0, a_2, 0],
[0, -a_1, 0, 0],
[0, 0, -a_1, 0],
[0, 0, 0, -a_1]])
|
0a84f3da131950a5f53a86a7ea893c8b8fdb3385d8819744d450071a7c31e5dc
|
"""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.utilities.pytest import raises, slow
from sympy.core.expr import unchanged
from sympy.core.compatibility import range
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()
n = 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)]
def test_CRootOf_all_roots():
assert Poly(x**5 + x + 1).all_roots() == [
rootof(x**3 - x**2 + 1, 0),
-S(1)/2 - sqrt(3)*I/2,
-S(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 r in roots:
assert isinstance(r, Rational)
roots = [str(r.n(17)) for r 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))) == S(11)/3
assert RootSum(f, Lambda(x, y/(x + x**2))) == S(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 - 125*z/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] == [
-S(21)/220, S(15)/256 - 805*I/256, S(15)/256 + 805*I/256]
t[0]._reset()
assert [i.eval_rational(1e-1, 1e-4) for i in t] == [
-S(21)/220, S(3275)/65536 - 414645*I/131072,
S(3275)/65536 + 414645*I/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] == [
-S(2001)/20020, S(6545)/131072 - 414645*I/131072,
S(6545)/131072 + 414645*I/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] == [
-S(202201)/2024022, S(104755)/2097152 - 6634255*I/2097152,
S(104755)/2097152 + 6634255*I/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] == [
-S(202201)/2024022, S(1676045)/33554432 - 106148135*I/33554432,
S(1676045)/33554432 + 106148135*I/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)
|
422553eca1ed763c0a4cfee238053ff63ca7e78d0eadfec4e4721a07a61fec13
|
"""Tests for user-friendly public interface to polynomial functions. """
from sympy.polys.polytools import (
Poly, PurePoly, poly,
parallel_poly_from_expr,
degree, degree_list,
total_degree,
LC, LM, LT,
pdiv, prem, pquo, pexquo,
div, rem, quo, exquo,
half_gcdex, gcdex, invert,
subresultants,
resultant, discriminant,
terms_gcd, cofactors,
gcd, gcd_list,
lcm, lcm_list,
trunc,
monic, content, primitive,
compose, decompose,
sturm,
gff_list, gff,
sqf_norm, sqf_part, sqf_list, sqf,
factor_list, factor,
intervals, refine_root, count_roots,
real_roots, nroots, ground_roots,
nth_power_roots_poly,
cancel, reduced, groebner,
GroebnerBasis, is_zero_dimensional,
_torational_factor_list,
to_rational_coeffs)
from sympy.polys.polyerrors import (
MultivariatePolynomialError,
ExactQuotientFailed,
PolificationFailed,
ComputationFailed,
UnificationFailed,
RefinementFailed,
GeneratorsNeeded,
GeneratorsError,
PolynomialError,
CoercionFailed,
DomainError,
OptionError,
FlagError)
from sympy.polys.polyclasses import DMP
from sympy.polys.fields import field
from sympy.polys.domains import FF, ZZ, QQ, RR, EX
from sympy.polys.domains.realfield import RealField
from sympy.polys.orderings import lex, grlex, grevlex
from sympy import (
S, Integer, Rational, Float, Mul, Symbol, sqrt, Piecewise, Derivative,
exp, sin, tanh, expand, oo, I, pi, re, im, rootof, Eq, Tuple, Expr, diff)
from sympy.core.basic import _aresame
from sympy.core.compatibility import iterable, PY3
from sympy.core.mul import _keep_coeff
from sympy.utilities.pytest import raises, XFAIL
from sympy.simplify import simplify
from sympy.abc import a, b, c, d, p, q, t, w, x, y, z
from sympy import MatrixSymbol
def _epsilon_eq(a, b):
for x, y in zip(a, b):
if abs(x - y) > 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_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(0)))
raises(GeneratorsNeeded, lambda: Poly.from_expr(S(7)))
F3 = FF(3)
assert Poly.from_expr(x + 5, domain=F3).rep == DMP([F3(1), F3(2)], F3)
assert Poly.from_expr(y + 5, domain=F3).rep == DMP([F3(1), F3(2)], F3)
assert Poly.from_expr(x + 5, x, domain=F3).rep == DMP([F3(1), F3(2)], F3)
assert Poly.from_expr(y + 5, y, domain=F3).rep == DMP([F3(1), F3(2)], F3)
assert Poly.from_expr(x + y, domain=F3).rep == DMP([[F3(1)], [F3(1), F3(0)]], F3)
assert Poly.from_expr(x + y, x, y, domain=F3).rep == DMP([[F3(1)], [F3(1), F3(0)]], F3)
assert Poly.from_expr(x + 5).rep == DMP([1, 5], ZZ)
assert Poly.from_expr(y + 5).rep == DMP([1, 5], ZZ)
assert Poly.from_expr(x + 5, x).rep == DMP([1, 5], ZZ)
assert Poly.from_expr(y + 5, y).rep == DMP([1, 5], ZZ)
assert Poly.from_expr(x + 5, domain=ZZ).rep == DMP([1, 5], ZZ)
assert Poly.from_expr(y + 5, domain=ZZ).rep == DMP([1, 5], ZZ)
assert Poly.from_expr(x + 5, x, domain=ZZ).rep == DMP([1, 5], ZZ)
assert Poly.from_expr(y + 5, y, domain=ZZ).rep == DMP([1, 5], ZZ)
assert Poly.from_expr(x + 5, x, y, domain=ZZ).rep == DMP([[1], [5]], ZZ)
assert Poly.from_expr(y + 5, x, y, domain=ZZ).rep == DMP([[1, 5]], ZZ)
def test_Poly__new__():
raises(GeneratorsError, lambda: Poly(x + 1, x, x))
raises(GeneratorsError, lambda: Poly(x + y, x, y, domain=ZZ[x]))
raises(GeneratorsError, lambda: Poly(x + y, x, y, domain=ZZ[y]))
raises(OptionError, lambda: Poly(x, x, symmetric=True))
raises(OptionError, lambda: Poly(x + 2, x, modulus=3, domain=QQ))
raises(OptionError, lambda: Poly(x + 2, x, domain=ZZ, gaussian=True))
raises(OptionError, lambda: Poly(x + 2, x, modulus=3, gaussian=True))
raises(OptionError, lambda: Poly(x + 2, x, domain=ZZ, extension=[sqrt(3)]))
raises(OptionError, lambda: Poly(x + 2, x, modulus=3, extension=[sqrt(3)]))
raises(OptionError, lambda: Poly(x + 2, x, domain=ZZ, extension=True))
raises(OptionError, lambda: Poly(x + 2, x, modulus=3, extension=True))
raises(OptionError, lambda: Poly(x + 2, x, domain=ZZ, greedy=True))
raises(OptionError, lambda: Poly(x + 2, x, domain=QQ, field=True))
raises(OptionError, lambda: Poly(x + 2, x, domain=ZZ, greedy=False))
raises(OptionError, lambda: Poly(x + 2, x, domain=QQ, field=False))
raises(NotImplementedError, lambda: Poly(x + 1, x, modulus=3, order='grlex'))
raises(NotImplementedError, lambda: Poly(x + 1, x, order='grlex'))
raises(GeneratorsNeeded, lambda: Poly({1: 2, 0: 1}))
raises(GeneratorsNeeded, lambda: Poly([2, 1]))
raises(GeneratorsNeeded, lambda: Poly((2, 1)))
raises(GeneratorsNeeded, lambda: Poly(1))
f = a*x**2 + b*x + c
assert Poly({2: a, 1: b, 0: c}, x) == f
assert Poly(iter([a, b, c]), x) == f
assert Poly([a, b, c], x) == f
assert Poly((a, b, c), x) == f
f = Poly({}, x, y, z)
assert f.gens == (x, y, z) and f.as_expr() == 0
assert Poly(Poly(a*x + b*y, x, y), x) == Poly(a*x + b*y, x)
assert Poly(3*x**2 + 2*x + 1, domain='ZZ').all_coeffs() == [3, 2, 1]
assert Poly(3*x**2 + 2*x + 1, domain='QQ').all_coeffs() == [3, 2, 1]
assert Poly(3*x**2 + 2*x + 1, domain='RR').all_coeffs() == [3.0, 2.0, 1.0]
raises(CoercionFailed, lambda: Poly(3*x**2/5 + 2*x/5 + 1, domain='ZZ'))
assert Poly(
3*x**2/5 + 2*x/5 + 1, domain='QQ').all_coeffs() == [S(3)/5, S(2)/5, 1]
assert _epsilon_eq(
Poly(3*x**2/5 + 2*x/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() == [S(31)/10, S(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,)
def test_Poly__gens():
assert Poly((x - p)*(x - q), x).gens == (x,)
assert Poly((x - p)*(x - q), p).gens == (p,)
assert Poly((x - p)*(x - q), q).gens == (q,)
assert Poly((x - p)*(x - q), x, p).gens == (x, p)
assert Poly((x - p)*(x - q), x, q).gens == (x, q)
assert Poly((x - p)*(x - q), x, p, q).gens == (x, p, q)
assert Poly((x - p)*(x - q), p, x, q).gens == (p, x, q)
assert Poly((x - p)*(x - q), p, q, x).gens == (p, q, x)
assert Poly((x - p)*(x - q)).gens == (x, p, q)
assert Poly((x - p)*(x - q), sort='x > p > q').gens == (x, p, q)
assert Poly((x - p)*(x - q), sort='p > x > q').gens == (p, x, q)
assert Poly((x - p)*(x - q), sort='p > q > x').gens == (p, q, x)
assert Poly((x - p)*(x - q), x, p, q, sort='p > q > x').gens == (x, p, q)
assert Poly((x - p)*(x - q), wrt='x').gens == (x, p, q)
assert Poly((x - p)*(x - q), wrt='p').gens == (p, x, q)
assert Poly((x - p)*(x - q), wrt='q').gens == (q, x, p)
assert Poly((x - p)*(x - q), wrt=x).gens == (x, p, q)
assert Poly((x - p)*(x - q), wrt=p).gens == (p, x, q)
assert Poly((x - p)*(x - q), wrt=q).gens == (q, x, p)
assert Poly((x - p)*(x - q), x, p, q, wrt='p').gens == (x, p, q)
assert Poly((x - p)*(x - q), wrt='p', sort='q > x').gens == (p, q, x)
assert Poly((x - p)*(x - q), wrt='q', sort='p > x').gens == (q, p, x)
def test_Poly_zero():
assert Poly(x).zero == Poly(0, x, domain=ZZ)
assert Poly(x/2).zero == Poly(0, x, domain=QQ)
def test_Poly_one():
assert Poly(x).one == Poly(1, x, domain=ZZ)
assert Poly(x/2).one == Poly(1, x, domain=QQ)
def test_Poly__unify():
raises(UnificationFailed, lambda: Poly(x)._unify(y))
F3 = FF(3)
F5 = FF(5)
assert Poly(x, x, modulus=3)._unify(Poly(y, y, modulus=3))[2:] == (
DMP([[F3(1)], []], F3), DMP([[F3(1), F3(0)]], F3))
assert Poly(x, x, modulus=3)._unify(Poly(y, y, modulus=5))[2:] == (
DMP([[F5(1)], []], F5), DMP([[F5(1), F5(0)]], F5))
assert Poly(y, x, y)._unify(Poly(x, x, modulus=3))[2:] == (DMP([[F3(1), F3(0)]], F3), DMP([[F3(1)], []], F3))
assert Poly(x, x, modulus=3)._unify(Poly(y, x, y))[2:] == (DMP([[F3(1)], []], F3), DMP([[F3(1), F3(0)]], F3))
assert Poly(x + 1, x)._unify(Poly(x + 2, x))[2:] == (DMP([1, 1], ZZ), DMP([1, 2], ZZ))
assert Poly(x + 1, x, domain='QQ')._unify(Poly(x + 2, x))[2:] == (DMP([1, 1], QQ), DMP([1, 2], QQ))
assert Poly(x + 1, x)._unify(Poly(x + 2, x, domain='QQ'))[2:] == (DMP([1, 1], QQ), DMP([1, 2], QQ))
assert Poly(x + 1, x)._unify(Poly(x + 2, x, y))[2:] == (DMP([[1], [1]], ZZ), DMP([[1], [2]], ZZ))
assert Poly(x + 1, x, domain='QQ')._unify(Poly(x + 2, x, y))[2:] == (DMP([[1], [1]], QQ), DMP([[1], [2]], QQ))
assert Poly(x + 1, x)._unify(Poly(x + 2, x, y, domain='QQ'))[2:] == (DMP([[1], [1]], QQ), DMP([[1], [2]], QQ))
assert Poly(x + 1, x, y)._unify(Poly(x + 2, x))[2:] == (DMP([[1], [1]], ZZ), DMP([[1], [2]], ZZ))
assert Poly(x + 1, x, y, domain='QQ')._unify(Poly(x + 2, x))[2:] == (DMP([[1], [1]], QQ), DMP([[1], [2]], QQ))
assert Poly(x + 1, x, y)._unify(Poly(x + 2, x, domain='QQ'))[2:] == (DMP([[1], [1]], QQ), DMP([[1], [2]], QQ))
assert Poly(x + 1, x, y)._unify(Poly(x + 2, x, y))[2:] == (DMP([[1], [1]], ZZ), DMP([[1], [2]], ZZ))
assert Poly(x + 1, x, y, domain='QQ')._unify(Poly(x + 2, x, y))[2:] == (DMP([[1], [1]], QQ), DMP([[1], [2]], QQ))
assert Poly(x + 1, x, y)._unify(Poly(x + 2, x, y, domain='QQ'))[2:] == (DMP([[1], [1]], QQ), DMP([[1], [2]], QQ))
assert Poly(x + 1, x)._unify(Poly(x + 2, y, x))[2:] == (DMP([[1, 1]], ZZ), DMP([[1, 2]], ZZ))
assert Poly(x + 1, x, domain='QQ')._unify(Poly(x + 2, y, x))[2:] == (DMP([[1, 1]], QQ), DMP([[1, 2]], QQ))
assert Poly(x + 1, x)._unify(Poly(x + 2, y, x, domain='QQ'))[2:] == (DMP([[1, 1]], QQ), DMP([[1, 2]], QQ))
assert Poly(x + 1, y, x)._unify(Poly(x + 2, x))[2:] == (DMP([[1, 1]], ZZ), DMP([[1, 2]], ZZ))
assert Poly(x + 1, y, x, domain='QQ')._unify(Poly(x + 2, x))[2:] == (DMP([[1, 1]], QQ), DMP([[1, 2]], QQ))
assert Poly(x + 1, y, x)._unify(Poly(x + 2, x, domain='QQ'))[2:] == (DMP([[1, 1]], QQ), DMP([[1, 2]], QQ))
assert Poly(x + 1, x, y)._unify(Poly(x + 2, y, x))[2:] == (DMP([[1], [1]], ZZ), DMP([[1], [2]], ZZ))
assert Poly(x + 1, x, y, domain='QQ')._unify(Poly(x + 2, y, x))[2:] == (DMP([[1], [1]], QQ), DMP([[1], [2]], QQ))
assert Poly(x + 1, x, y)._unify(Poly(x + 2, y, x, domain='QQ'))[2:] == (DMP([[1], [1]], QQ), DMP([[1], [2]], QQ))
assert Poly(x + 1, y, x)._unify(Poly(x + 2, x, y))[2:] == (DMP([[1, 1]], ZZ), DMP([[1, 2]], ZZ))
assert Poly(x + 1, y, x, domain='QQ')._unify(Poly(x + 2, x, y))[2:] == (DMP([[1, 1]], QQ), DMP([[1, 2]], QQ))
assert Poly(x + 1, y, x)._unify(Poly(x + 2, x, y, domain='QQ'))[2:] == (DMP([[1, 1]], QQ), DMP([[1, 2]], QQ))
F, A, B = field("a,b", ZZ)
assert Poly(a*x, x, domain='ZZ[a]')._unify(Poly(a*b*x, x, domain='ZZ(a,b)'))[2:] == \
(DMP([A, F(0)], F.to_domain()), DMP([A*B, F(0)], F.to_domain()))
assert Poly(a*x, x, domain='ZZ(a)')._unify(Poly(a*b*x, x, domain='ZZ(a,b)'))[2:] == \
(DMP([A, F(0)], F.to_domain()), DMP([A*B, F(0)], F.to_domain()))
raises(CoercionFailed, lambda: Poly(Poly(x**2 + x**2*z, y, field=True), domain='ZZ(x)'))
f = Poly(t**2 + t/3 + x, t, domain='QQ(x)')
g = Poly(t**2 + t/3 + x, t, domain='QQ[x]')
assert f._unify(g)[2:] == (f.rep, f.rep)
def test_Poly_free_symbols():
assert Poly(x**2 + 1).free_symbols == {x}
assert Poly(x**2 + y*z).free_symbols == {x, y, z}
assert Poly(x**2 + y*z, x).free_symbols == {x, y, z}
assert Poly(x**2 + sin(y*z)).free_symbols == {x, y, z}
assert Poly(x**2 + sin(y*z), x).free_symbols == {x, y, z}
assert Poly(x**2 + sin(y*z), x, domain=EX).free_symbols == {x, y, z}
assert Poly(1 + x + x**2, x, y, z).free_symbols == {x}
assert Poly(x + sin(y), z).free_symbols == {x, y}
def test_PurePoly_free_symbols():
assert PurePoly(x**2 + 1).free_symbols == set([])
assert PurePoly(x**2 + y*z).free_symbols == set([])
assert PurePoly(x**2 + y*z, x).free_symbols == {y, z}
assert PurePoly(x**2 + sin(y*z)).free_symbols == set([])
assert PurePoly(x**2 + sin(y*z), x).free_symbols == {y, z}
assert PurePoly(x**2 + sin(y*z), x, domain=EX).free_symbols == {y, z}
def test_Poly__eq__():
assert (Poly(x, x) == Poly(x, x)) is True
assert (Poly(x, x, domain=QQ) == Poly(x, x)) is True
assert (Poly(x, x) == Poly(x, x, domain=QQ)) is True
assert (Poly(x, x, domain=ZZ[a]) == Poly(x, x)) is True
assert (Poly(x, x) == Poly(x, x, domain=ZZ[a])) is True
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 True
assert f.ne(g) is False
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 True
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(S(2)/10*x + S(1)/10).set_domain('RR') == Poly(0.2*x + 0.1)
assert Poly(0.2*x + 0.1).set_domain('QQ') == Poly(S(2)/10*x + S(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)
assert Poly(1, x) + sin(x) == 1 + 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)
assert Poly(1, x) - sin(x) == 1 - 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)
assert Poly(1, x) * sin(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)
assert Poly(x*y + 1, x, y)**(-1) == (x*y + 1)**(-1)
assert Poly(x*y + 1, x, y)**x == (x*y + 1)**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[w, t]')
assert g.eject(z, t, w) == Poly(ex, x, y, domain='ZZ[w, t, z]')
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() == -oo
assert Poly(1, x).degree() == 0
assert Poly(x, x).degree() == 1
assert Poly(0, x).degree(gen=0) == -oo
assert Poly(1, x).degree(gen=0) == 0
assert Poly(x, x).degree(gen=0) == 1
assert Poly(0, x).degree(gen=x) == -oo
assert Poly(1, x).degree(gen=x) == 0
assert Poly(x, x).degree(gen=x) == 1
assert Poly(0, x).degree(gen='x') == -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) == -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) == -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() == -oo
assert Poly(1, x, y).homogeneous_order() == 0
assert Poly(x, x, y).homogeneous_order() == 1
assert Poly(x*y, x, y).homogeneous_order() == 2
assert Poly(x + 1, x, y).homogeneous_order() is None
assert Poly(x*y + x, x, y).homogeneous_order() is None
assert Poly(x**5 + 2*x**3*y**2 + 9*x*y**4).homogeneous_order() == 5
assert Poly(x**5 + 2*x**3*y**3 + 9*x*y**4).homogeneous_order() is None
def test_Poly_LC():
assert Poly(0, x).LC() == 0
assert Poly(1, x).LC() == 1
assert Poly(2*x**2 + x, x).LC() == 2
assert Poly(x*y**7 + 2*x**2*y**3).LC('lex') == 2
assert Poly(x*y**7 + 2*x**2*y**3).LC('grlex') == 1
assert LC(x*y**7 + 2*x**2*y**3, order='lex') == 2
assert LC(x*y**7 + 2*x**2*y**3, order='grlex') == 1
def test_Poly_TC():
assert Poly(0, x).TC() == 0
assert Poly(1, x).TC() == 1
assert Poly(2*x**2 + x, x).TC() == 0
def test_Poly_EC():
assert Poly(0, x).EC() == 0
assert Poly(1, x).EC() == 1
assert Poly(2*x**2 + x, x).EC() == 1
assert Poly(x*y**7 + 2*x**2*y**3).EC('lex') == 1
assert Poly(x*y**7 + 2*x**2*y**3).EC('grlex') == 2
def test_Poly_coeff():
assert Poly(0, x).coeff_monomial(1) == 0
assert Poly(0, x).coeff_monomial(x) == 0
assert Poly(1, x).coeff_monomial(1) == 1
assert Poly(1, x).coeff_monomial(x) == 0
assert Poly(x**8, x).coeff_monomial(1) == 0
assert Poly(x**8, x).coeff_monomial(x**7) == 0
assert Poly(x**8, x).coeff_monomial(x**8) == 1
assert Poly(x**8, x).coeff_monomial(x**9) == 0
assert Poly(3*x*y**2 + 1, x, y).coeff_monomial(1) == 1
assert Poly(3*x*y**2 + 1, x, y).coeff_monomial(x*y**2) == 3
p = Poly(24*x*y*exp(8) + 23*x, x, y)
assert p.coeff_monomial(x) == 23
assert p.coeff_monomial(y) == 0
assert p.coeff_monomial(x*y) == 24*exp(8)
assert p.as_expr().coeff(x) == 24*y*exp(8) + 23
raises(NotImplementedError, lambda: p.coeff(x))
raises(ValueError, lambda: Poly(x + 1).coeff_monomial(0))
raises(ValueError, lambda: Poly(x + 1).coeff_monomial(3*x))
raises(ValueError, lambda: Poly(x + 1).coeff_monomial(3*x*y))
def test_Poly_nth():
assert Poly(0, x).nth(0) == 0
assert Poly(0, x).nth(1) == 0
assert Poly(1, x).nth(0) == 1
assert Poly(1, x).nth(1) == 0
assert Poly(x**8, x).nth(0) == 0
assert Poly(x**8, x).nth(7) == 0
assert Poly(x**8, x).nth(8) == 1
assert Poly(x**8, x).nth(9) == 0
assert Poly(3*x*y**2 + 1, x, y).nth(0, 0) == 1
assert Poly(3*x*y**2 + 1, x, y).nth(1, 2) == 3
raises(ValueError, lambda: Poly(x*y + 1, x, y).nth(1))
def test_Poly_LM():
assert Poly(0, x).LM() == (0,)
assert Poly(1, x).LM() == (0,)
assert Poly(2*x**2 + x, x).LM() == (2,)
assert Poly(x*y**7 + 2*x**2*y**3).LM('lex') == (2, 3)
assert Poly(x*y**7 + 2*x**2*y**3).LM('grlex') == (1, 7)
assert LM(x*y**7 + 2*x**2*y**3, order='lex') == x**2*y**3
assert LM(x*y**7 + 2*x**2*y**3, order='grlex') == x*y**7
def test_Poly_LM_custom_order():
f = Poly(x**2*y**3*z + x**2*y*z**3 + x*y*z + 1)
rev_lex = lambda monom: tuple(reversed(monom))
assert f.LM(order='lex') == (2, 3, 1)
assert f.LM(order=rev_lex) == (2, 1, 3)
def test_Poly_EM():
assert Poly(0, x).EM() == (0,)
assert Poly(1, x).EM() == (0,)
assert Poly(2*x**2 + x, x).EM() == (1,)
assert Poly(x*y**7 + 2*x**2*y**3).EM('lex') == (1, 7)
assert Poly(x*y**7 + 2*x**2*y**3).EM('grlex') == (2, 3)
def test_Poly_LT():
assert Poly(0, x).LT() == ((0,), 0)
assert Poly(1, x).LT() == ((0,), 1)
assert Poly(2*x**2 + x, x).LT() == ((2,), 2)
assert Poly(x*y**7 + 2*x**2*y**3).LT('lex') == ((2, 3), 2)
assert Poly(x*y**7 + 2*x**2*y**3).LT('grlex') == ((1, 7), 1)
assert LT(x*y**7 + 2*x**2*y**3, order='lex') == 2*x**2*y**3
assert LT(x*y**7 + 2*x**2*y**3, order='grlex') == x*y**7
def test_Poly_ET():
assert Poly(0, x).ET() == ((0,), 0)
assert Poly(1, x).ET() == ((0,), 1)
assert Poly(2*x**2 + x, x).ET() == ((1,), 1)
assert Poly(x*y**7 + 2*x**2*y**3).ET('lex') == ((1, 7), 1)
assert Poly(x*y**7 + 2*x**2*y**3).ET('grlex') == ((2, 3), 2)
def test_Poly_max_norm():
assert Poly(-1, x).max_norm() == 1
assert Poly( 0, x).max_norm() == 0
assert Poly( 1, x).max_norm() == 1
def test_Poly_l1_norm():
assert Poly(-1, x).l1_norm() == 1
assert Poly( 0, x).l1_norm() == 0
assert Poly( 1, x).l1_norm() == 1
def test_Poly_clear_denoms():
coeff, poly = Poly(x + 2, x).clear_denoms()
assert coeff == 1 and poly == Poly(
x + 2, x, domain='ZZ') and poly.get_domain() == ZZ
coeff, poly = Poly(x/2 + 1, x).clear_denoms()
assert coeff == 2 and poly == Poly(
x + 2, x, domain='QQ') and poly.get_domain() == QQ
coeff, poly = Poly(x/2 + 1, x).clear_denoms(convert=True)
assert coeff == 2 and poly == Poly(
x + 2, x, domain='ZZ') and poly.get_domain() == ZZ
coeff, poly = Poly(x/y + 1, x).clear_denoms(convert=True)
assert coeff == y and poly == Poly(
x + y, x, domain='ZZ[y]') and poly.get_domain() == ZZ[y]
coeff, poly = Poly(x/3 + sqrt(2), x, domain='EX').clear_denoms()
assert coeff == 3 and poly == Poly(
x + 3*sqrt(2), x, domain='EX') and poly.get_domain() == EX
coeff, poly = Poly(
x/3 + sqrt(2), x, domain='EX').clear_denoms(convert=True)
assert coeff == 3 and poly == Poly(
x + 3*sqrt(2), x, domain='EX') and poly.get_domain() == EX
def test_Poly_rat_clear_denoms():
f = Poly(x**2/y + 1, x)
g = Poly(x**3 + y, x)
assert f.rat_clear_denoms(g) == \
(Poly(x**2 + y, x), Poly(y*x**3 + y**2, x))
f = f.set_domain(EX)
g = g.set_domain(EX)
assert f.rat_clear_denoms(g) == (f, g)
def test_Poly_integrate():
assert Poly(x + 1).integrate() == Poly(x**2/2 + x)
assert Poly(x + 1).integrate(x) == Poly(x**2/2 + x)
assert Poly(x + 1).integrate((x, 1)) == Poly(x**2/2 + x)
assert Poly(x*y + 1).integrate(x) == Poly(x**2*y/2 + x)
assert Poly(x*y + 1).integrate(y) == Poly(x*y**2/2 + y)
assert Poly(x*y + 1).integrate(x, x) == Poly(x**3*y/6 + x**2/2)
assert Poly(x*y + 1).integrate(y, y) == Poly(x*y**3/6 + y**2/2)
assert Poly(x*y + 1).integrate((x, 2)) == Poly(x**3*y/6 + x**2/2)
assert Poly(x*y + 1).integrate((y, 2)) == Poly(x*y**3/6 + y**2/2)
assert Poly(x*y + 1).integrate(x, y) == Poly(x**2*y**2/4 + x*y)
assert Poly(x*y + 1).integrate(y, x) == Poly(x**2*y**2/4 + x*y)
def test_Poly_diff():
assert Poly(x**2 + x).diff() == Poly(2*x + 1)
assert Poly(x**2 + x).diff(x) == Poly(2*x + 1)
assert Poly(x**2 + x).diff((x, 1)) == Poly(2*x + 1)
assert Poly(x**2*y**2 + x*y).diff(x) == Poly(2*x*y**2 + y)
assert Poly(x**2*y**2 + x*y).diff(y) == Poly(2*x**2*y + x)
assert Poly(x**2*y**2 + x*y).diff(x, x) == Poly(2*y**2, x, y)
assert Poly(x**2*y**2 + x*y).diff(y, y) == Poly(2*x**2, x, y)
assert Poly(x**2*y**2 + x*y).diff((x, 2)) == Poly(2*y**2, x, y)
assert Poly(x**2*y**2 + x*y).diff((y, 2)) == Poly(2*x**2, x, y)
assert Poly(x**2*y**2 + x*y).diff(x, y) == Poly(4*x*y + 1)
assert Poly(x**2*y**2 + x*y).diff(y, x) == Poly(4*x*y + 1)
def test_issue_9585():
assert diff(Poly(x**2 + x)) == Poly(2*x + 1)
assert diff(Poly(x**2 + x), x, evaluate=False) == \
Derivative(Poly(x**2 + x), x)
assert Derivative(Poly(x**2 + x), x).doit() == Poly(2*x + 1)
def test_Poly_eval():
assert Poly(0, x).eval(7) == 0
assert Poly(1, x).eval(7) == 1
assert Poly(x, x).eval(7) == 7
assert Poly(0, x).eval(0, 7) == 0
assert Poly(1, x).eval(0, 7) == 1
assert Poly(x, x).eval(0, 7) == 7
assert Poly(0, x).eval(x, 7) == 0
assert Poly(1, x).eval(x, 7) == 1
assert Poly(x, x).eval(x, 7) == 7
assert Poly(0, x).eval('x', 7) == 0
assert Poly(1, x).eval('x', 7) == 1
assert Poly(x, x).eval('x', 7) == 7
raises(PolynomialError, lambda: Poly(1, x).eval(1, 7))
raises(PolynomialError, lambda: Poly(1, x).eval(y, 7))
raises(PolynomialError, lambda: Poly(1, x).eval('y', 7))
assert Poly(123, x, y).eval(7) == Poly(123, y)
assert Poly(2*y, x, y).eval(7) == Poly(2*y, y)
assert Poly(x*y, x, y).eval(7) == Poly(7*y, y)
assert Poly(123, x, y).eval(x, 7) == Poly(123, y)
assert Poly(2*y, x, y).eval(x, 7) == Poly(2*y, y)
assert Poly(x*y, x, y).eval(x, 7) == Poly(7*y, y)
assert Poly(123, x, y).eval(y, 7) == Poly(123, x)
assert Poly(2*y, x, y).eval(y, 7) == Poly(14, x)
assert Poly(x*y, x, y).eval(y, 7) == Poly(7*x, x)
assert Poly(x*y + y, x, y).eval({x: 7}) == Poly(8*y, y)
assert Poly(x*y + y, x, y).eval({y: 7}) == Poly(7*x + 7, x)
assert Poly(x*y + y, x, y).eval({x: 6, y: 7}) == 49
assert Poly(x*y + y, x, y).eval({x: 7, y: 6}) == 48
assert Poly(x*y + y, x, y).eval((6, 7)) == 49
assert Poly(x*y + y, x, y).eval([6, 7]) == 49
assert Poly(x + 1, domain='ZZ').eval(S(1)/2) == S(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(1)/2, auto=False))
# issue 6344
alpha = Symbol('alpha')
result = (2*alpha*z - 2*alpha + z**2 + 3)/(z**2 - 2*z + 1)
f = Poly(x**2 + (alpha - 1)*x - alpha + 1, x, domain='ZZ[alpha]')
assert f.eval((z + 1)/(z - 1)) == result
g = Poly(x**2 + (alpha - 1)*x - alpha + 1, x, y, domain='ZZ[alpha]')
assert g.eval((z + 1)/(z - 1)) == Poly(result, y, domain='ZZ(alpha,z)')
def test_Poly___call__():
f = Poly(2*x*y + 3*x + y + 2*z)
assert f(2) == Poly(5*y + 2*z + 6)
assert f(2, 5) == Poly(2*z + 31)
assert f(2, 5, 7) == 45
def test_parallel_poly_from_expr():
assert parallel_poly_from_expr(
[x - 1, x**2 - 1], x)[0] == [Poly(x - 1, x), Poly(x**2 - 1, x)]
assert parallel_poly_from_expr(
[Poly(x - 1, x), x**2 - 1], x)[0] == [Poly(x - 1, x), Poly(x**2 - 1, x)]
assert parallel_poly_from_expr(
[x - 1, Poly(x**2 - 1, x)], x)[0] == [Poly(x - 1, x), Poly(x**2 - 1, x)]
assert parallel_poly_from_expr([Poly(
x - 1, x), Poly(x**2 - 1, x)], x)[0] == [Poly(x - 1, x), Poly(x**2 - 1, x)]
assert parallel_poly_from_expr(
[x - 1, x**2 - 1], x, y)[0] == [Poly(x - 1, x, y), Poly(x**2 - 1, x, y)]
assert parallel_poly_from_expr([Poly(
x - 1, x), x**2 - 1], x, y)[0] == [Poly(x - 1, x, y), Poly(x**2 - 1, x, y)]
assert parallel_poly_from_expr([x - 1, Poly(
x**2 - 1, x)], x, y)[0] == [Poly(x - 1, x, y), Poly(x**2 - 1, x, y)]
assert parallel_poly_from_expr([Poly(x - 1, x), Poly(
x**2 - 1, x)], x, y)[0] == [Poly(x - 1, x, y), Poly(x**2 - 1, x, y)]
assert parallel_poly_from_expr(
[x - 1, x**2 - 1])[0] == [Poly(x - 1, x), Poly(x**2 - 1, x)]
assert parallel_poly_from_expr(
[Poly(x - 1, x), x**2 - 1])[0] == [Poly(x - 1, x), Poly(x**2 - 1, x)]
assert parallel_poly_from_expr(
[x - 1, Poly(x**2 - 1, x)])[0] == [Poly(x - 1, x), Poly(x**2 - 1, x)]
assert parallel_poly_from_expr(
[Poly(x - 1, x), Poly(x**2 - 1, x)])[0] == [Poly(x - 1, x), Poly(x**2 - 1, x)]
assert parallel_poly_from_expr(
[1, x**2 - 1])[0] == [Poly(1, x), Poly(x**2 - 1, x)]
assert parallel_poly_from_expr(
[1, x**2 - 1])[0] == [Poly(1, x), Poly(x**2 - 1, x)]
assert parallel_poly_from_expr(
[1, Poly(x**2 - 1, x)])[0] == [Poly(1, x), Poly(x**2 - 1, x)]
assert parallel_poly_from_expr(
[1, Poly(x**2 - 1, x)])[0] == [Poly(1, x), Poly(x**2 - 1, x)]
assert parallel_poly_from_expr(
[x**2 - 1, 1])[0] == [Poly(x**2 - 1, x), Poly(1, x)]
assert parallel_poly_from_expr(
[x**2 - 1, 1])[0] == [Poly(x**2 - 1, x), Poly(1, x)]
assert parallel_poly_from_expr(
[Poly(x**2 - 1, x), 1])[0] == [Poly(x**2 - 1, x), Poly(1, x)]
assert parallel_poly_from_expr(
[Poly(x**2 - 1, x), 1])[0] == [Poly(x**2 - 1, x), Poly(1, x)]
assert parallel_poly_from_expr([Poly(x, x, y), Poly(y, x, y)], x, y, order='lex')[0] == \
[Poly(x, x, y, domain='ZZ'), Poly(y, x, y, domain='ZZ')]
raises(PolificationFailed, lambda: parallel_poly_from_expr([0, 1]))
def test_pdiv():
f, g = x**2 - y**2, x - y
q, r = x + y, 0
F, G, Q, R = [ Poly(h, x, y) for h in (f, g, q, r) ]
assert F.pdiv(G) == (Q, R)
assert F.prem(G) == R
assert F.pquo(G) == Q
assert F.pexquo(G) == Q
assert pdiv(f, g) == (q, r)
assert prem(f, g) == r
assert pquo(f, g) == q
assert pexquo(f, g) == q
assert pdiv(f, g, x, y) == (q, r)
assert prem(f, g, x, y) == r
assert pquo(f, g, x, y) == q
assert pexquo(f, g, x, y) == q
assert pdiv(f, g, (x, y)) == (q, r)
assert prem(f, g, (x, y)) == r
assert pquo(f, g, (x, y)) == q
assert pexquo(f, g, (x, y)) == q
assert pdiv(F, G) == (Q, R)
assert prem(F, G) == R
assert pquo(F, G) == Q
assert pexquo(F, G) == Q
assert pdiv(f, g, polys=True) == (Q, R)
assert prem(f, g, polys=True) == R
assert pquo(f, g, polys=True) == Q
assert pexquo(f, g, polys=True) == Q
assert pdiv(F, G, polys=False) == (q, r)
assert prem(F, G, polys=False) == r
assert pquo(F, G, polys=False) == q
assert pexquo(F, G, polys=False) == q
raises(ComputationFailed, lambda: pdiv(4, 2))
raises(ComputationFailed, lambda: prem(4, 2))
raises(ComputationFailed, lambda: pquo(4, 2))
raises(ComputationFailed, lambda: pexquo(4, 2))
def test_div():
f, g = x**2 - y**2, x - y
q, r = x + y, 0
F, G, Q, R = [ Poly(h, x, y) for h in (f, g, q, r) ]
assert F.div(G) == (Q, R)
assert F.rem(G) == R
assert F.quo(G) == Q
assert F.exquo(G) == Q
assert div(f, g) == (q, r)
assert rem(f, g) == r
assert quo(f, g) == q
assert exquo(f, g) == q
assert div(f, g, x, y) == (q, r)
assert rem(f, g, x, y) == r
assert quo(f, g, x, y) == q
assert exquo(f, g, x, y) == q
assert div(f, g, (x, y)) == (q, r)
assert rem(f, g, (x, y)) == r
assert quo(f, g, (x, y)) == q
assert exquo(f, g, (x, y)) == q
assert div(F, G) == (Q, R)
assert rem(F, G) == R
assert quo(F, G) == Q
assert exquo(F, G) == Q
assert div(f, g, polys=True) == (Q, R)
assert rem(f, g, polys=True) == R
assert quo(f, g, polys=True) == Q
assert exquo(f, g, polys=True) == Q
assert div(F, G, polys=False) == (q, r)
assert rem(F, G, polys=False) == r
assert quo(F, G, polys=False) == q
assert exquo(F, G, polys=False) == q
raises(ComputationFailed, lambda: div(4, 2))
raises(ComputationFailed, lambda: rem(4, 2))
raises(ComputationFailed, lambda: quo(4, 2))
raises(ComputationFailed, lambda: exquo(4, 2))
f, g = x**2 + 1, 2*x - 4
qz, rz = 0, x**2 + 1
qq, rq = x/2 + 1, 5
assert div(f, g) == (qq, rq)
assert div(f, g, auto=True) == (qq, rq)
assert div(f, g, auto=False) == (qz, rz)
assert div(f, g, domain=ZZ) == (qz, rz)
assert div(f, g, domain=QQ) == (qq, rq)
assert div(f, g, domain=ZZ, auto=True) == (qq, rq)
assert div(f, g, domain=ZZ, auto=False) == (qz, rz)
assert div(f, g, domain=QQ, auto=True) == (qq, rq)
assert div(f, g, domain=QQ, auto=False) == (qq, rq)
assert rem(f, g) == rq
assert rem(f, g, auto=True) == rq
assert rem(f, g, auto=False) == rz
assert rem(f, g, domain=ZZ) == rz
assert rem(f, g, domain=QQ) == rq
assert rem(f, g, domain=ZZ, auto=True) == rq
assert rem(f, g, domain=ZZ, auto=False) == rz
assert rem(f, g, domain=QQ, auto=True) == rq
assert rem(f, g, domain=QQ, auto=False) == rq
assert quo(f, g) == qq
assert quo(f, g, auto=True) == qq
assert quo(f, g, auto=False) == qz
assert quo(f, g, domain=ZZ) == qz
assert quo(f, g, domain=QQ) == qq
assert quo(f, g, domain=ZZ, auto=True) == qq
assert quo(f, g, domain=ZZ, auto=False) == qz
assert quo(f, g, domain=QQ, auto=True) == qq
assert quo(f, g, domain=QQ, auto=False) == qq
f, g, q = x**2, 2*x, x/2
assert exquo(f, g) == q
assert exquo(f, g, auto=True) == q
raises(ExactQuotientFailed, lambda: exquo(f, g, auto=False))
raises(ExactQuotientFailed, lambda: exquo(f, g, domain=ZZ))
assert exquo(f, g, domain=QQ) == q
assert exquo(f, g, domain=ZZ, auto=True) == q
raises(ExactQuotientFailed, lambda: exquo(f, g, domain=ZZ, auto=False))
assert exquo(f, g, domain=QQ, auto=True) == q
assert exquo(f, g, domain=QQ, auto=False) == q
f, g = Poly(x**2), Poly(x)
q, r = f.div(g)
assert q.get_domain().is_ZZ and r.get_domain().is_ZZ
r = f.rem(g)
assert r.get_domain().is_ZZ
q = f.quo(g)
assert q.get_domain().is_ZZ
q = f.exquo(g)
assert q.get_domain().is_ZZ
f, g = Poly(x+y, x), Poly(2*x+y, x)
q, r = f.div(g)
assert q.get_domain().is_Frac and r.get_domain().is_Frac
def test_gcdex():
f, g = 2*x, x**2 - 16
s, t, h = x/32, -Rational(1, 16), 1
F, G, S, T, H = [ Poly(u, x, domain='QQ') for u in (f, g, s, t, h) ]
assert F.half_gcdex(G) == (S, H)
assert F.gcdex(G) == (S, T, H)
assert F.invert(G) == S
assert half_gcdex(f, g) == (s, h)
assert gcdex(f, g) == (s, t, h)
assert invert(f, g) == s
assert half_gcdex(f, g, x) == (s, h)
assert gcdex(f, g, x) == (s, t, h)
assert invert(f, g, x) == s
assert half_gcdex(f, g, (x,)) == (s, h)
assert gcdex(f, g, (x,)) == (s, t, h)
assert invert(f, g, (x,)) == s
assert half_gcdex(F, G) == (S, H)
assert gcdex(F, G) == (S, T, H)
assert invert(F, G) == S
assert half_gcdex(f, g, polys=True) == (S, H)
assert gcdex(f, g, polys=True) == (S, T, H)
assert invert(f, g, polys=True) == S
assert half_gcdex(F, G, polys=False) == (s, h)
assert gcdex(F, G, polys=False) == (s, t, h)
assert invert(F, G, polys=False) == s
assert half_gcdex(100, 2004) == (-20, 4)
assert gcdex(100, 2004) == (-20, 1, 4)
assert invert(3, 7) == 5
raises(DomainError, lambda: half_gcdex(x + 1, 2*x + 1, auto=False))
raises(DomainError, lambda: gcdex(x + 1, 2*x + 1, auto=False))
raises(DomainError, lambda: invert(x + 1, 2*x + 1, auto=False))
def test_revert():
f = Poly(1 - x**2/2 + x**4/24 - x**6/720)
g = Poly(61*x**6/720 + 5*x**4/24 + x**2/2 + 1)
assert f.revert(8) == g
def test_subresultants():
f, g, h = x**2 - 2*x + 1, x**2 - 1, 2*x - 2
F, G, H = Poly(f), Poly(g), Poly(h)
assert F.subresultants(G) == [F, G, H]
assert subresultants(f, g) == [f, g, h]
assert subresultants(f, g, x) == [f, g, h]
assert subresultants(f, g, (x,)) == [f, g, h]
assert subresultants(F, G) == [F, G, H]
assert subresultants(f, g, polys=True) == [F, G, H]
assert subresultants(F, G, polys=False) == [f, g, h]
raises(ComputationFailed, lambda: subresultants(4, 2))
def test_resultant():
f, g, h = x**2 - 2*x + 1, x**2 - 1, 0
F, G = Poly(f), Poly(g)
assert F.resultant(G) == h
assert resultant(f, g) == h
assert resultant(f, g, x) == h
assert resultant(f, g, (x,)) == h
assert resultant(F, G) == h
assert resultant(f, g, polys=True) == h
assert resultant(F, G, polys=False) == h
assert resultant(f, g, includePRS=True) == (h, [f, g, 2*x - 2])
f, g, h = x - a, x - b, a - b
F, G, H = Poly(f), Poly(g), Poly(h)
assert F.resultant(G) == H
assert resultant(f, g) == h
assert resultant(f, g, x) == h
assert resultant(f, g, (x,)) == h
assert resultant(F, G) == H
assert resultant(f, g, polys=True) == H
assert resultant(F, G, polys=False) == h
raises(ComputationFailed, lambda: resultant(4, 2))
def test_discriminant():
f, g = x**3 + 3*x**2 + 9*x - 13, -11664
F = Poly(f)
assert F.discriminant() == g
assert discriminant(f) == g
assert discriminant(f, x) == g
assert discriminant(f, (x,)) == g
assert discriminant(F) == g
assert discriminant(f, polys=True) == g
assert discriminant(F, polys=False) == g
f, g = a*x**2 + b*x + c, b**2 - 4*a*c
F, G = Poly(f), Poly(g)
assert F.discriminant() == G
assert discriminant(f) == g
assert discriminant(f, x, a, b, c) == g
assert discriminant(f, (x, a, b, c)) == g
assert discriminant(F) == G
assert discriminant(f, polys=True) == G
assert discriminant(F, polys=False) == g
raises(ComputationFailed, lambda: discriminant(4))
def test_dispersion():
# We test only the API here. For more mathematical
# tests see the dedicated test file.
fp = poly((x + 1)*(x + 2), x)
assert sorted(fp.dispersionset()) == [0, 1]
assert fp.dispersion() == 1
fp = poly(x**4 - 3*x**2 + 1, x)
gp = fp.shift(-3)
assert sorted(fp.dispersionset(gp)) == [2, 3, 4]
assert fp.dispersion(gp) == 4
def test_gcd_list():
F = [x**3 - 1, x**2 - 1, x**2 - 3*x + 2]
assert gcd_list(F) == x - 1
assert gcd_list(F, polys=True) == Poly(x - 1)
assert gcd_list([]) == 0
assert gcd_list([1, 2]) == 1
assert gcd_list([4, 6, 8]) == 2
assert gcd_list([x*(y + 42) - x*y - x*42]) == 0
gcd = gcd_list([], x)
assert gcd.is_Number and gcd is S.Zero
gcd = gcd_list([], x, polys=True)
assert gcd.is_Poly and gcd.is_zero
raises(ComputationFailed, lambda: gcd_list([], polys=True))
def test_lcm_list():
F = [x**3 - 1, x**2 - 1, x**2 - 3*x + 2]
assert lcm_list(F) == x**5 - x**4 - 2*x**3 - x**2 + x + 2
assert lcm_list(F, polys=True) == Poly(x**5 - x**4 - 2*x**3 - x**2 + x + 2)
assert lcm_list([]) == 1
assert lcm_list([1, 2]) == 2
assert lcm_list([4, 6, 8]) == 24
assert lcm_list([x*(y + 42) - x*y - x*42]) == 0
lcm = lcm_list([], x)
assert lcm.is_Number and lcm is S.One
lcm = lcm_list([], x, polys=True)
assert lcm.is_Poly and lcm.is_one
raises(ComputationFailed, lambda: lcm_list([], polys=True))
def test_gcd():
f, g = x**3 - 1, x**2 - 1
s, t = x**2 + x + 1, x + 1
h, r = x - 1, x**4 + x**3 - x - 1
F, G, S, T, H, R = [ Poly(u) for u in (f, g, s, t, h, r) ]
assert F.cofactors(G) == (H, S, T)
assert F.gcd(G) == H
assert F.lcm(G) == R
assert cofactors(f, g) == (h, s, t)
assert gcd(f, g) == h
assert lcm(f, g) == r
assert cofactors(f, g, x) == (h, s, t)
assert gcd(f, g, x) == h
assert lcm(f, g, x) == r
assert cofactors(f, g, (x,)) == (h, s, t)
assert gcd(f, g, (x,)) == h
assert lcm(f, g, (x,)) == r
assert cofactors(F, G) == (H, S, T)
assert gcd(F, G) == H
assert lcm(F, G) == R
assert cofactors(f, g, polys=True) == (H, S, T)
assert gcd(f, g, polys=True) == H
assert lcm(f, g, polys=True) == R
assert cofactors(F, G, polys=False) == (h, s, t)
assert gcd(F, G, polys=False) == h
assert lcm(F, G, polys=False) == r
f, g = 1.0*x**2 - 1.0, 1.0*x - 1.0
h, s, t = g, 1.0*x + 1.0, 1.0
assert cofactors(f, g) == (h, s, t)
assert gcd(f, g) == h
assert lcm(f, g) == f
f, g = 1.0*x**2 - 1.0, 1.0*x - 1.0
h, s, t = g, 1.0*x + 1.0, 1.0
assert cofactors(f, g) == (h, s, t)
assert gcd(f, g) == h
assert lcm(f, g) == f
assert cofactors(8, 6) == (2, 4, 3)
assert gcd(8, 6) == 2
assert lcm(8, 6) == 24
f, g = x**2 - 3*x - 4, x**3 - 4*x**2 + x - 4
l = x**4 - 3*x**3 - 3*x**2 - 3*x - 4
h, s, t = x - 4, x + 1, x**2 + 1
assert cofactors(f, g, modulus=11) == (h, s, t)
assert gcd(f, g, modulus=11) == h
assert lcm(f, g, modulus=11) == l
f, g = x**2 + 8*x + 7, x**3 + 7*x**2 + x + 7
l = x**4 + 8*x**3 + 8*x**2 + 8*x + 7
h, s, t = x + 7, x + 1, x**2 + 1
assert cofactors(f, g, modulus=11, symmetric=False) == (h, s, t)
assert gcd(f, g, modulus=11, symmetric=False) == h
assert lcm(f, g, modulus=11, symmetric=False) == l
raises(TypeError, lambda: gcd(x))
raises(TypeError, lambda: lcm(x))
def test_gcd_numbers_vs_polys():
assert isinstance(gcd(3, 9), Integer)
assert isinstance(gcd(3*x, 9), Integer)
assert gcd(3, 9) == 3
assert gcd(3*x, 9) == 3
assert isinstance(gcd(S(3)/2, S(9)/4), Rational)
assert isinstance(gcd(S(3)/2*x, S(9)/4), Rational)
assert gcd(S(3)/2, S(9)/4) == S(3)/4
assert gcd(S(3)/2*x, S(9)/4) == 1
assert isinstance(gcd(3.0, 9.0), Float)
assert isinstance(gcd(3.0*x, 9.0), Float)
assert gcd(3.0, 9.0) == 1.0
assert gcd(3.0*x, 9.0) == 1.0
def test_terms_gcd():
assert terms_gcd(1) == 1
assert terms_gcd(1, x) == 1
assert terms_gcd(x - 1) == x - 1
assert terms_gcd(-x - 1) == -x - 1
assert terms_gcd(2*x + 3) == 2*x + 3
assert terms_gcd(6*x + 4) == Mul(2, 3*x + 2, evaluate=False)
assert terms_gcd(x**3*y + x*y**3) == x*y*(x**2 + y**2)
assert terms_gcd(2*x**3*y + 2*x*y**3) == 2*x*y*(x**2 + y**2)
assert terms_gcd(x**3*y/2 + x*y**3/2) == x*y/2*(x**2 + y**2)
assert terms_gcd(x**3*y + 2*x*y**3) == x*y*(x**2 + 2*y**2)
assert terms_gcd(2*x**3*y + 4*x*y**3) == 2*x*y*(x**2 + 2*y**2)
assert terms_gcd(2*x**3*y/3 + 4*x*y**3/5) == 2*x*y/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
assert terms_gcd(eq, deep=True) == Eq(2*x, 2*y*(z + 1))
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(1)/2
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 + S(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(1)/2), Poly(x - 1)) == \
Poly(S(9)/4, x) == \
cancel((x - 1)**2*(x**2 - 2*x + 1).subs(x, (x + S(1)/2)/(x - 1)))
assert Poly(x**2 - 2*x + 1, x).transform(Poly(x + 1), Poly(x - S(1)/2)) == \
Poly(S(9)/4, x) == \
cancel((x - S(1)/2)**2*(x**2 - 2*x + 1).subs(x, (x + 1)/(x - S(1)/2)))
# Unify ZZ, QQ, and RR
assert Poly(x**2 - 2*x + 1, x).transform(Poly(x + 1.0), Poly(x - S(1)/2)) == \
Poly(S(9)/4, x) == \
cancel((x - S(1)/2)**2*(x**2 - 2*x + 1).subs(x, (x + 1.0)/(x - S(1)/2)))
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
+ S(1)/62500*x
- S(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((S(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)) == S(2)/3*((3*x - 5)/(x - 2))
assert sqf(Poly(x**2 - 2*x + 1)) == (x - 1)**2
f = 3 + x - x*(1 + x) + x**2
assert sqf(f) == 3
f = (x**2 + 2*x + 1)**20000000000
assert sqf(f) == (x + 1)**40000000000
assert sqf_list(f) == (1, [(x + 1, 40000000000)])
def test_factor():
f = x**5 - x**3 - x**2 + 1
u = x + 1
v = x - 1
w = x**2 + x + 1
F, U, V, W = map(Poly, (f, u, v, w))
assert F.factor_list() == (1, [(U, 1), (V, 2), (W, 1)])
assert factor_list(f) == (1, [(u, 1), (v, 2), (w, 1)])
assert factor_list(f, x) == (1, [(u, 1), (v, 2), (w, 1)])
assert factor_list(f, (x,)) == (1, [(u, 1), (v, 2), (w, 1)])
assert factor_list(F) == (1, [(U, 1), (V, 2), (W, 1)])
assert factor_list(f, polys=True) == (1, [(U, 1), (V, 2), (W, 1)])
assert factor_list(F, polys=False) == (1, [(u, 1), (v, 2), (w, 1)])
assert F.factor_list_include() == [(U, 1), (V, 2), (W, 1)]
assert factor_list(1) == (1, [])
assert factor_list(6) == (6, [])
assert factor_list(sqrt(3), x) == (sqrt(3), [])
assert factor_list((-1)**x, x) == (1, [(-1, x)])
assert factor_list((2*x)**y, x) == (1, [(2, y), (x, y)])
assert factor_list(sqrt(x*y), x) == (1, [(x*y, S.Half)])
assert factor(6) == 6 and factor(6).is_Integer
assert factor_list(3*x) == (3, [(x, 1)])
assert factor_list(3*x**2) == (3, [(x, 2)])
assert factor(3*x) == 3*x
assert factor(3*x**2) == 3*x**2
assert factor((2*x**2 + 2)**7) == 128*(x**2 + 1)**7
assert factor(f) == u*v**2*w
assert factor(f, x) == u*v**2*w
assert factor(f, (x,)) == u*v**2*w
g, p, q, r = x**2 - y**2, x - y, x + y, x**2 + 1
assert factor(f/g) == (u*v**2*w)/(p*q)
assert factor(f/g, x) == (u*v**2*w)/(p*q)
assert factor(f/g, (x,)) == (u*v**2*w)/(p*q)
p = Symbol('p', positive=True)
i = Symbol('i', integer=True)
r = Symbol('r', real=True)
assert factor(sqrt(x*y)).is_Pow is True
assert factor(sqrt(3*x**2 - 3)) == sqrt(3)*sqrt((x - 1)*(x + 1))
assert factor(sqrt(3*x**2 + 3)) == sqrt(3)*sqrt(x**2 + 1)
assert factor((y*x**2 - y)**i) == y**i*(x - 1)**i*(x + 1)**i
assert factor((y*x**2 + y)**i) == y**i*(x**2 + 1)**i
assert factor((y*x**2 - y)**t) == (y*(x - 1)*(x + 1))**t
assert factor((y*x**2 + y)**t) == (y*(x**2 + 1))**t
f = sqrt(expand((r**2 + 1)*(p + 1)*(p - 1)*(p - 2)**3))
g = sqrt((p - 2)**3*(p - 1))*sqrt(p + 1)*sqrt(r**2 + 1)
assert factor(f) == g
assert factor(g) == g
g = (x - 1)**5*(r**2 + 1)
f = sqrt(expand(g))
assert factor(f) == sqrt(g)
f = Poly(sin(1)*x + 1, x, domain=EX)
assert f.factor_list() == (1, [(f, 1)])
f = x**4 + 1
assert factor(f) == f
assert factor(f, extension=I) == (x**2 - I)*(x**2 + I)
assert factor(f, gaussian=True) == (x**2 - I)*(x**2 + I)
assert factor(
f, extension=sqrt(2)) == (x**2 + sqrt(2)*x + 1)*(x**2 - sqrt(2)*x + 1)
f = x**2 + 2*sqrt(2)*x + 2
assert factor(f, extension=sqrt(2)) == (x + sqrt(2))**2
assert factor(f**3, extension=sqrt(2)) == (x + sqrt(2))**6
assert factor(x**2 - 2*y**2, extension=sqrt(2)) == \
(x + sqrt(2)*y)*(x - sqrt(2)*y)
assert factor(2*x**2 - 4*y**2, extension=sqrt(2)) == \
2*((x + sqrt(2)*y)*(x - sqrt(2)*y))
assert factor(x - 1) == x - 1
assert factor(-x - 1) == -x - 1
assert factor(x - 1) == x - 1
assert factor(6*x - 10) == Mul(2, 3*x - 5, evaluate=False)
assert factor(x**11 + x + 1, modulus=65537, symmetric=True) == \
(x**2 + x + 1)*(x**9 - x**8 + x**6 - x**5 + x**3 - x** 2 + 1)
assert factor(x**11 + x + 1, modulus=65537, symmetric=False) == \
(x**2 + x + 1)*(x**9 + 65536*x**8 + x**6 + 65536*x**5 +
x**3 + 65536*x** 2 + 1)
f = x/pi + x*sin(x)/pi
g = y/(pi**2 + 2*pi + 1) + y*sin(x)/(pi**2 + 2*pi + 1)
assert factor(f) == x*(sin(x) + 1)/pi
assert factor(g) == y*(sin(x) + 1)/(pi + 1)**2
assert factor(Eq(
x**2 + 2*x + 1, x**3 + 1)) == Eq((x + 1)**2, (x + 1)*(x**2 - x + 1))
f = (x**2 - 1)/(x**2 + 4*x + 4)
assert factor(f) == (x + 1)*(x - 1)/(x + 2)**2
assert factor(f, x) == (x + 1)*(x - 1)/(x + 2)**2
f = 3 + x - x*(1 + x) + x**2
assert factor(f) == 3
assert factor(f, x) == 3
assert factor(1/(x**2 + 2*x + 1/x) - 1) == -((1 - x + 2*x**2 +
x**3)/(1 + 2*x**2 + x**3))
assert factor(f, expand=False) == f
raises(PolynomialError, lambda: factor(f, x, expand=False))
raises(FlagError, lambda: factor(x**2 - 1, polys=True))
assert factor([x, Eq(x**2 - y**2, Tuple(x**2 - z**2, 1/x + 1/y))]) == \
[x, Eq((x - y)*(x + y), Tuple((x - z)*(x + z), (x + y)/x/y))]
assert not isinstance(
Poly(x**3 + x + 1).factor_list()[1][0][0], PurePoly) is True
assert isinstance(
PurePoly(x**3 + x + 1).factor_list()[1][0][0], PurePoly) is True
assert factor(sqrt(-x)) == sqrt(-x)
# issue 5917
e = (-2*x*(-x + 1)*(x - 1)*(-x*(-x + 1)*(x - 1) - x*(x - 1)**2)*(x**2*(x -
1) - x*(x - 1) - x) - (-2*x**2*(x - 1)**2 - x*(-x + 1)*(-x*(-x + 1) +
x*(x - 1)))*(x**2*(x - 1)**4 - x*(-x*(-x + 1)*(x - 1) - x*(x - 1)**2)))
assert factor(e) == 0
# deep option
assert factor(sin(x**2 + x) + x, deep=True) == sin(x*(x + 1)) + x
assert factor(sin(x**2 + x)*x, deep=True) == sin(x*(x + 1))*x
assert factor(sqrt(x**2)) == sqrt(x**2)
# issue 13149
assert factor(expand((0.5*x+1)*(0.5*y+1))) == Mul(1.0, 0.5*x + 1.0,
0.5*y + 1.0, evaluate = False)
assert factor(expand((0.5*x+0.5)**2)) == 0.25*(1.0*x + 1.0)**2
eq = x**2*y**2 + 11*x**2*y + 30*x**2 + 7*x*y**2 + 77*x*y + 210*x + 12*y**2 + 132*y + 360
assert factor(eq, x) == (x + 3)*(x + 4)*(y**2 + 11*y + 30)
assert factor(eq, x, deep=True) == (x + 3)*(x + 4)*(y**2 + 11*y + 30)
assert factor(eq, y, deep=True) == (y + 5)*(y + 6)*(x**2 + 7*x + 12)
# fraction option
f = 5*x + 3*exp(2 - 7*x)
assert factor(f, deep=True) == factor(f, deep=True, fraction=True)
assert factor(f, deep=True, fraction=False) == 5*x + 3*exp(2)*exp(-7*x)
def test_factor_large():
f = (x**2 + 4*x + 4)**10000000*(x**2 + 1)*(x**2 + 2*x + 1)**1234567
g = ((x**2 + 2*x + 1)**3000*y**2 + (x**2 + 2*x + 1)**3000*2*y + (
x**2 + 2*x + 1)**3000)
assert factor(f) == (x + 2)**20000000*(x**2 + 1)*(x + 1)**2469134
assert factor(g) == (x + 1)**6000*(y + 1)**2
assert factor_list(
f) == (1, [(x + 1, 2469134), (x + 2, 20000000), (x**2 + 1, 1)])
assert factor_list(g) == (1, [(y + 1, 2), (x + 1, 6000)])
f = (x**2 - y**2)**200000*(x**7 + 1)
g = (x**2 + y**2)**200000*(x**7 + 1)
assert factor(f) == \
(x + 1)*(x - y)**200000*(x + y)**200000*(x**6 - x**5 +
x**4 - x**3 + x**2 - x + 1)
assert factor(g, gaussian=True) == \
(x + 1)*(x - I*y)**200000*(x + I*y)**200000*(x**6 - x**5 +
x**4 - x**3 + x**2 - x + 1)
assert factor_list(f) == \
(1, [(x + 1, 1), (x - y, 200000), (x + y, 200000), (x**6 -
x**5 + x**4 - x**3 + x**2 - x + 1, 1)])
assert factor_list(g, gaussian=True) == \
(1, [(x + 1, 1), (x - I*y, 200000), (x + I*y, 200000), (
x**6 - x**5 + x**4 - x**3 + x**2 - x + 1, 1)])
def test_factor_noeval():
assert factor(6*x - 10) == Mul(2, 3*x - 5, evaluate=False)
assert factor((6*x - 10)/(3*x - 6)) == Mul(S(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((2*x/5 - S(17)/3)*(4*x + S(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=S(1)/10) == f.intervals(eps=0.1) == \
[((-S(1)/258, 0), 1), ((S(85)/6, S(85)/6), 1)]
assert f.intervals(eps=S(1)/100) == f.intervals(eps=0.01) == \
[((-S(1)/258, 0), 1), ((S(85)/6, S(85)/6), 1)]
assert f.intervals(eps=S(1)/1000) == f.intervals(eps=0.001) == \
[((-S(1)/1002, 0), 1), ((S(85)/6, S(85)/6), 1)]
assert f.intervals(eps=S(1)/10000) == f.intervals(eps=0.0001) == \
[((-S(1)/1028, -S(1)/1028), 1), ((S(85)/6, S(85)/6), 1)]
f = (2*x/5 - S(17)/3)*(4*x + S(1)/257)
assert intervals(f, sqf=True) == [(-1, 0), (14, 15)]
assert intervals(f) == [((-1, 0), 1), ((14, 15), 1)]
assert intervals(f, eps=S(1)/10) == intervals(f, eps=0.1) == \
[((-S(1)/258, 0), 1), ((S(85)/6, S(85)/6), 1)]
assert intervals(f, eps=S(1)/100) == intervals(f, eps=0.01) == \
[((-S(1)/258, 0), 1), ((S(85)/6, S(85)/6), 1)]
assert intervals(f, eps=S(1)/1000) == intervals(f, eps=0.001) == \
[((-S(1)/1002, 0), 1), ((S(85)/6, S(85)/6), 1)]
assert intervals(f, eps=S(1)/10000) == intervals(f, eps=0.0001) == \
[((-S(1)/1028, -S(1)/1028), 1), ((S(85)/6, S(85)/6), 1)]
f = Poly((x**2 - 2)*(x**2 - 3)**7*(x + 1)*(7*x + 3)**3)
assert f.intervals() == \
[((-2, -S(3)/2), 7), ((-S(3)/2, -1), 1),
((-1, -1), 1), ((-1, 0), 3),
((1, S(3)/2), 1), ((S(3)/2, 2), 7)]
assert intervals([x**5 - 200, x**5 - 201]) == \
[((S(75)/26, S(101)/35), {0: 1}), ((S(309)/107, S(26)/9), {1: 1})]
assert intervals([x**5 - 200, x**5 - 201], fast=True) == \
[((S(75)/26, S(101)/35), {0: 1}), ((S(309)/107, S(26)/9), {1: 1})]
assert intervals([x**2 - 200, x**2 - 201]) == \
[((-S(71)/5, -S(85)/6), {1: 1}), ((-S(85)/6, -14), {0: 1}),
((14, S(85)/6), {0: 1}), ((S(85)/6, S(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=S(7)/4, sqf=True) == []
assert intervals(f, inf=S(7)/5, sqf=True) == [(S(7)/5, S(3)/2)]
assert intervals(f, sup=S(7)/4, sqf=True) == [(-2, -1), (1, S(3)/2)]
assert intervals(f, sup=S(7)/5, sqf=True) == [(-2, -1)]
assert intervals(g, inf=S(7)/4) == []
assert intervals(g, inf=S(7)/5) == [((S(7)/5, S(3)/2), 2)]
assert intervals(g, sup=S(7)/4) == [((-2, -1), 2), ((1, S(3)/2), 2)]
assert intervals(g, sup=S(7)/5) == [((-2, -1), 2)]
assert intervals([g, h], inf=S(7)/4) == []
assert intervals([g, h], inf=S(7)/5) == [((S(7)/5, S(3)/2), {0: 2})]
assert intervals([g, h], sup=S(
7)/4) == [((-2, -1), {0: 2}), ((1, 1), {1: 1}), ((1, S(3)/2), {0: 2})]
assert intervals(
[g, h], sup=S(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}), ((-S(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 == [(-S(40)/7 - 40*I/7, 0), (-S(40)/7, 40*I/7),
(-40*I/7, S(40)/7), (0, S(40)/7 + 40*I/7)]
real_part, complex_part = intervals(f, all=True, sqf=True, eps=S(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, S(3)/2)
assert f.refine_root(-2, -1, steps=None) == (-S(3)/2, -1)
assert f.refine_root(1, 2, steps=1) == (1, S(3)/2)
assert f.refine_root(-2, -1, steps=1) == (-S(3)/2, -1)
assert f.refine_root(1, 2, steps=1, fast=True) == (1, S(3)/2)
assert f.refine_root(-2, -1, steps=1, fast=True) == (-S(3)/2, -1)
assert f.refine_root(1, 2, eps=S(1)/100) == (S(24)/17, S(17)/12)
assert f.refine_root(1, 2, eps=1e-2) == (S(24)/17, S(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, S(3)/2)
assert refine_root(f, -2, -1, steps=1) == (-S(3)/2, -1)
assert refine_root(f, 1, 2, steps=1, fast=True) == (1, S(3)/2)
assert refine_root(f, -2, -1, steps=1, fast=True) == (-S(3)/2, -1)
assert refine_root(f, 1, 2, eps=S(1)/100) == (S(24)/17, S(17)/12)
assert refine_root(f, 1, 2, eps=1e-2) == (S(24)/17, S(17)/12)
raises(PolynomialError, lambda: refine_root(1, 7, 8, eps=S(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) == -S(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() == [-S(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() == [-S(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 - S(1)/3, x).nroots()
assert roots == [-1.0, 1.0]
roots = Poly(x**2/3 + S(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(1): 2, S(0): 2}
assert ground_roots(f) == {S(1): 2, S(0): 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) == oo
assert cancel((2, 3)) == (1, 2, 3)
assert cancel((1, 0), x) == (1, 1, 0)
assert cancel((0, 1), x) == (1, 0, 1)
f, g, p, q = 4*x**2 - 4, 2*x - 2, 2*x + 2, 1
F, G, P, Q = [ Poly(u, x) for u in (f, g, p, q) ]
assert F.cancel(G) == (1, P, Q)
assert cancel((f, g)) == (1, p, q)
assert cancel((f, g), x) == (1, p, q)
assert cancel((f, g), (x,)) == (1, p, q)
assert cancel((F, G)) == (1, P, Q)
assert cancel((f, g), polys=True) == (1, P, Q)
assert cancel((F, G), polys=False) == (1, p, q)
f = (x**2 - 2)/(x + sqrt(2))
assert cancel(f) == f
assert cancel(f, greedy=False) == x - sqrt(2)
f = (x**2 - 2)/(x - sqrt(2))
assert cancel(f) == f
assert cancel(f, greedy=False) == x + sqrt(2)
assert cancel((x**2/4 - 1, x/2 - 1)) == (S(1)/2, 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(1)/2)/(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)
G = Poly(1, x)
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)**(S(3)/2))
assert cancel(f).is_Mul == True
# issue 7022
A = Symbol('A', commutative=False)
p1 = Piecewise((A*(x**2 - 1)/(x + 1), x > 1), ((x + 2)/(x**2 + 2*x), True))
p2 = Piecewise((A*(x - 1), x > 1), (1/x, True))
assert cancel(p1) == p2
assert cancel(2*p1) == 2*p2
assert cancel(1 + p1) == 1 + p2
assert cancel((x**2 - 1)/(x + 1)*p1) == (x - 1)*p2
assert cancel((x**2 - 1)/(x + 1) + p1) == (x - 1) + p2
p3 = Piecewise(((x**2 - 1)/(x + 1), x > 1), ((x + 2)/(x**2 + 2*x), True))
p4 = Piecewise(((x - 1), x > 1), (1/x, True))
assert cancel(p3) == p4
assert cancel(2*p3) == 2*p4
assert cancel(1 + p3) == 1 + p4
assert cancel((x**2 - 1)/(x + 1)*p3) == (x - 1)*p4
assert cancel((x**2 - 1)/(x + 1) + p3) == (x - 1) + p4
# issue 9363
M = MatrixSymbol('M', 5, 5)
assert cancel(M[0,0] + 7) == M[0,0] + 7
expr = sin(M[1, 4] + M[2, 1] * 5 * M[4, 0]) - 5 * M[1, 2] / z
assert cancel(expr) == (z*sin(M[1, 4] + M[2, 1] * 5 * M[4, 0]) - 5 * M[1, 2]) / z
def test_reduced():
f = 2*x**4 + y**2 - x**2 + y**3
G = [x**3 - x, y**3 - y]
Q = [2*x, 1]
r = x**2 + y**2 + y
assert reduced(f, G) == (Q, r)
assert reduced(f, G, x, y) == (Q, r)
H = groebner(G)
assert H.reduce(f) == (Q, r)
Q = [Poly(2*x, x, y), Poly(1, x, y)]
r = Poly(x**2 + y**2 + y, x, y)
assert _strict_eq(reduced(f, G, polys=True), (Q, r))
assert _strict_eq(reduced(f, G, x, y, polys=True), (Q, r))
H = groebner(G, polys=True)
assert _strict_eq(H.reduce(f), (Q, r))
f = 2*x**3 + y**3 + 3*y
G = groebner([x**2 + y**2 - 1, x*y - 2])
Q = [x**2 - x*y**3/2 + x*y/2 + y**6/4 - y**4/2 + y**2/4, -y**5/4 + y**3/2 + 3*y/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(1), x) == x
assert _keep_coeff(S(-1), 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(1), 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
# @XFAIL
# Seems to pass on Python 3.X, but not on Python 2.7
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)
if not PY3:
test_poly_matching_consistency = XFAIL(test_poly_matching_consistency)
@XFAIL
def test_issue_5786():
assert expand(factor(expand(
(x - I*y)*(z - I*t)), extension=[I])) == -I*t*x - t*y + x*z - I*y*z
def test_noncommutative():
class foo(Expr):
is_commutative=False
e = x/(x + x*y)
c = 1/( 1 + y)
assert cancel(foo(e)) == foo(c)
assert cancel(e + foo(e)) == c + foo(c)
assert cancel(e*foo(c)) == c*foo(c)
def test_to_rational_coeffs():
assert to_rational_coeffs(
Poly(x**3 + y*x**2 + sqrt(y), x, domain='EX')) is None
def test_factor_terms():
# issue 7067
assert factor_list(x*(x + y)) == (1, [(x, 1), (x + y, 1)])
assert sqf_list(x*(x + y)) == (1, [(x, 1), (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) == S(3)/10 * (1 + sqrt(3))
assert gcd(sqrt(5)*S(4)/7, sqrt(5)*S(2)/3) == sqrt(5)*S(2)/21
assert lcm(S(2)/3*sqrt(3), S(5)/6*sqrt(3)) == S(10)*sqrt(3)/3
assert lcm(3*sqrt(3), S(4)/sqrt(3)) == 12*sqrt(3)
assert lcm(S(5)*(1 + 2**(S(1)/3))/6, S(3)*(1 + 2**(S(1)/3))/8) == S(15)/2 * (1 + 2**(S(1)/3))
assert gcd(S(2)/3*sqrt(3), S(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]) == S(2)*sqrt(47)/35
assert gcd([S(6)*(1 + sqrt(7))/5, S(2)*(1 + sqrt(7))/7, S(4)*(1 + sqrt(7))/13]) == S(2)/455 * (1 + sqrt(7))
assert lcm((S(7)/sqrt(15)/2, S(5)/sqrt(15)/6, S(5)/sqrt(15)/8)) == S(35)/(2*sqrt(15))
assert lcm([S(5)*(2 + 2**(S(5)/7))/6, S(7)*(2 + 2**(S(5)/7))/2, S(13)*(2 + 2**(S(5)/7))/4]) == S(455)/2 * (2 + 2**(S(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**(S(4)/5)*x*(-x**2 + sqrt(8*x**2 + (x**2 - 2)**2) + 2)**(S(3)/5) + 10*x)
assert factor(expr, deep=True) == x*(x**2 + 2)
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993d6eb3ac2aab807a6fdc1ddd1eafe54266dccdc98337d2cf008e59a8a909c7
|
from sympy.core import S
from sympy.simplify import simplify, trigsimp
from sympy import pi, sqrt, symbols, ImmutableMatrix as Matrix, \
sin, cos, Function, Integral, Derivative, diff
from sympy.vector.vector import Vector, BaseVector, VectorAdd, \
VectorMul, VectorZero
from sympy.vector.coordsysrect import CoordSys3D
from sympy.vector.vector import Cross, Dot, dot, cross
from sympy.utilities.pytest import raises
C = CoordSys3D('C')
i, j, k = C.base_vectors()
a, b, c = symbols('a b c')
def test_cross():
v1 = C.x * i + C.z * C.z * j
v2 = C.x * i + C.y * j + C.z * k
assert Cross(v1, v2) == Cross(C.x*C.i + C.z**2*C.j, C.x*C.i + C.y*C.j + C.z*C.k)
assert Cross(v1, v2).doit() == C.z**3*C.i + (-C.x*C.z)*C.j + (C.x*C.y - C.x*C.z**2)*C.k
assert cross(v1, v2) == C.z**3*C.i + (-C.x*C.z)*C.j + (C.x*C.y - C.x*C.z**2)*C.k
assert Cross(v1, v2) == -Cross(v2, v1)
assert Cross(v1, v2) + Cross(v2, v1) == Vector.zero
def test_dot():
v1 = C.x * i + C.z * C.z * j
v2 = C.x * i + C.y * j + C.z * k
assert Dot(v1, v2) == Dot(C.x*C.i + C.z**2*C.j, C.x*C.i + C.y*C.j + C.z*C.k)
assert Dot(v1, v2).doit() == C.x**2 + C.y*C.z**2
assert Dot(v1, v2).doit() == C.x**2 + C.y*C.z**2
assert Dot(v1, v2) == Dot(v2, v1)
def test_vector_sympy():
"""
Test whether the Vector framework confirms to the hashing
and equality testing properties of SymPy.
"""
v1 = 3*j
assert v1 == j*3
assert v1.components == {j: 3}
v2 = 3*i + 4*j + 5*k
v3 = 2*i + 4*j + i + 4*k + k
assert v3 == v2
assert v3.__hash__() == v2.__hash__()
def test_vector():
assert isinstance(i, BaseVector)
assert i != j
assert j != k
assert k != i
assert i - i == Vector.zero
assert i + Vector.zero == i
assert i - Vector.zero == i
assert Vector.zero != 0
assert -Vector.zero == Vector.zero
v1 = a*i + b*j + c*k
v2 = a**2*i + b**2*j + c**2*k
v3 = v1 + v2
v4 = 2 * v1
v5 = a * i
assert isinstance(v1, VectorAdd)
assert v1 - v1 == Vector.zero
assert v1 + Vector.zero == v1
assert v1.dot(i) == a
assert v1.dot(j) == b
assert v1.dot(k) == c
assert i.dot(v2) == a**2
assert j.dot(v2) == b**2
assert k.dot(v2) == c**2
assert v3.dot(i) == a**2 + a
assert v3.dot(j) == b**2 + b
assert v3.dot(k) == c**2 + c
assert v1 + v2 == v2 + v1
assert v1 - v2 == -1 * (v2 - v1)
assert a * v1 == v1 * a
assert isinstance(v5, VectorMul)
assert v5.base_vector == i
assert v5.measure_number == a
assert isinstance(v4, Vector)
assert isinstance(v4, VectorAdd)
assert isinstance(v4, Vector)
assert isinstance(Vector.zero, VectorZero)
assert isinstance(Vector.zero, Vector)
assert isinstance(v1 * 0, VectorZero)
assert v1.to_matrix(C) == Matrix([[a], [b], [c]])
assert i.components == {i: 1}
assert v5.components == {i: a}
assert v1.components == {i: a, j: b, k: c}
assert VectorAdd(v1, Vector.zero) == v1
assert VectorMul(a, v1) == v1*a
assert VectorMul(1, i) == i
assert VectorAdd(v1, Vector.zero) == v1
assert VectorMul(0, Vector.zero) == Vector.zero
raises(TypeError, lambda: v1.outer(1))
raises(TypeError, lambda: v1.dot(1))
def test_vector_magnitude_normalize():
assert Vector.zero.magnitude() == 0
assert Vector.zero.normalize() == Vector.zero
assert i.magnitude() == 1
assert j.magnitude() == 1
assert k.magnitude() == 1
assert i.normalize() == i
assert j.normalize() == j
assert k.normalize() == k
v1 = a * i
assert v1.normalize() == (a/sqrt(a**2))*i
assert v1.magnitude() == sqrt(a**2)
v2 = a*i + b*j + c*k
assert v2.magnitude() == sqrt(a**2 + b**2 + c**2)
assert v2.normalize() == v2 / v2.magnitude()
v3 = i + j
assert v3.normalize() == (sqrt(2)/2)*C.i + (sqrt(2)/2)*C.j
def test_vector_simplify():
A, s, k, m = symbols('A, s, k, m')
test1 = (1 / a + 1 / b) * i
assert (test1 & i) != (a + b) / (a * b)
test1 = simplify(test1)
assert (test1 & i) == (a + b) / (a * b)
assert test1.simplify() == simplify(test1)
test2 = (A**2 * s**4 / (4 * pi * k * m**3)) * i
test2 = simplify(test2)
assert (test2 & i) == (A**2 * s**4 / (4 * pi * k * m**3))
test3 = ((4 + 4 * a - 2 * (2 + 2 * a)) / (2 + 2 * a)) * i
test3 = simplify(test3)
assert (test3 & i) == 0
test4 = ((-4 * a * b**2 - 2 * b**3 - 2 * a**2 * b) / (a + b)**2) * i
test4 = simplify(test4)
assert (test4 & i) == -2 * b
v = (sin(a)+cos(a))**2*i - j
assert trigsimp(v) == (2*sin(a + pi/4)**2)*i + (-1)*j
assert trigsimp(v) == v.trigsimp()
assert simplify(Vector.zero) == Vector.zero
def test_vector_dot():
assert i.dot(Vector.zero) == 0
assert Vector.zero.dot(i) == 0
assert i & Vector.zero == 0
assert i.dot(i) == 1
assert i.dot(j) == 0
assert i.dot(k) == 0
assert i & i == 1
assert i & j == 0
assert i & k == 0
assert j.dot(i) == 0
assert j.dot(j) == 1
assert j.dot(k) == 0
assert j & i == 0
assert j & j == 1
assert j & k == 0
assert k.dot(i) == 0
assert k.dot(j) == 0
assert k.dot(k) == 1
assert k & i == 0
assert k & j == 0
assert k & k == 1
raises(TypeError, lambda: k.dot(1))
def test_vector_cross():
assert i.cross(Vector.zero) == Vector.zero
assert Vector.zero.cross(i) == Vector.zero
assert i.cross(i) == Vector.zero
assert i.cross(j) == k
assert i.cross(k) == -j
assert i ^ i == Vector.zero
assert i ^ j == k
assert i ^ k == -j
assert j.cross(i) == -k
assert j.cross(j) == Vector.zero
assert j.cross(k) == i
assert j ^ i == -k
assert j ^ j == Vector.zero
assert j ^ k == i
assert k.cross(i) == j
assert k.cross(j) == -i
assert k.cross(k) == Vector.zero
assert k ^ i == j
assert k ^ j == -i
assert k ^ k == Vector.zero
assert k.cross(1) == Cross(k, 1)
def test_projection():
v1 = i + j + k
v2 = 3*i + 4*j
v3 = 0*i + 0*j
assert v1.projection(v1) == i + j + k
assert v1.projection(v2) == S(7)/3*C.i + S(7)/3*C.j + S(7)/3*C.k
assert v1.projection(v1, scalar=True) == 1
assert v1.projection(v2, scalar=True) == S(7)/3
assert v3.projection(v1) == Vector.zero
def test_vector_diff_integrate():
f = Function('f')
v = f(a)*C.i + a**2*C.j - C.k
assert Derivative(v, a) == Derivative((f(a))*C.i +
a**2*C.j + (-1)*C.k, a)
assert (diff(v, a) == v.diff(a) == Derivative(v, a).doit() ==
(Derivative(f(a), a))*C.i + 2*a*C.j)
assert (Integral(v, a) == (Integral(f(a), a))*C.i +
(Integral(a**2, a))*C.j + (Integral(-1, a))*C.k)
def test_vector_args():
raises(ValueError, lambda: BaseVector(3, C))
raises(TypeError, lambda: BaseVector(0, Vector.zero))
|
35c92aac95a79f460b1e4135aac5e26e9b9b8811109bfd147fa4e08951e8854b
|
from sympy.vector import CoordSys3D, Gradient, Divergence, Curl, VectorZero, Laplacian
from sympy.printing.repr import srepr
R = CoordSys3D('R')
s1 = R.x*R.y*R.z
s2 = R.x + 3*R.y**2
s3 = R.x**2 + R.y**2 + R.z**2
v1 = R.x*R.i + R.z*R.z*R.j
v2 = R.x*R.i + R.y*R.j + R.z*R.k
v3 = R.x**2*R.i + R.y**2*R.j + R.z**2*R.k
def test_Gradient():
assert Gradient(s1) == Gradient(R.x*R.y*R.z)
assert Gradient(s2) == Gradient(R.x + 3*R.y**2)
assert Gradient(s1).doit() == R.y*R.z*R.i + R.x*R.z*R.j + R.x*R.y*R.k
assert Gradient(s2).doit() == R.i + 6*R.y*R.j
def test_Divergence():
assert Divergence(v1) == Divergence(R.x*R.i + R.z*R.z*R.j)
assert Divergence(v2) == Divergence(R.x*R.i + R.y*R.j + R.z*R.k)
assert Divergence(v1).doit() == 1
assert Divergence(v2).doit() == 3
def test_Curl():
assert Curl(v1) == Curl(R.x*R.i + R.z*R.z*R.j)
assert Curl(v2) == Curl(R.x*R.i + R.y*R.j + R.z*R.k)
assert Curl(v1).doit() == (-2*R.z)*R.i
assert Curl(v2).doit() == VectorZero()
def test_Laplacian():
assert Laplacian(s3) == Laplacian(R.x**2 + R.y**2 + R.z**2)
assert Laplacian(v3) == Laplacian(R.x**2*R.i + R.y**2*R.j + R.z**2*R.k)
assert Laplacian(s3).doit() == 6
assert Laplacian(v3).doit() == 2*R.i + 2*R.j + 2*R.k
assert srepr(Laplacian(s3)) == \
'Laplacian(Add(Pow(R.x, Integer(2)), Pow(R.y, Integer(2)), Pow(R.z, Integer(2))))'
|
d1da03a6807c3ec86cf0692f6ea82ce918c6f9b67d8396a77e3f21befc9ec50d
|
# -*- coding: utf-8 -*-
from sympy import Integral, latex, Function
from sympy import pretty as xpretty
from sympy.vector import CoordSys3D, Vector, express
from sympy.abc import a, b, c
from sympy.core.compatibility import u_decode as u
from sympy.utilities.pytest import XFAIL
def pretty(expr):
"""ASCII pretty-printing"""
return xpretty(expr, use_unicode=False, wrap_line=False)
def upretty(expr):
"""Unicode pretty-printing"""
return xpretty(expr, use_unicode=True, wrap_line=False)
# Initialize the basic and tedious vector/dyadic expressions
# needed for testing.
# Some of the pretty forms shown denote how the expressions just
# above them should look with pretty printing.
N = CoordSys3D('N')
C = N.orient_new_axis('C', a, N.k)
v = []
d = []
v.append(Vector.zero)
v.append(N.i)
v.append(-N.i)
v.append(N.i + N.j)
v.append(a*N.i)
v.append(a*N.i - b*N.j)
v.append((a**2 + N.x)*N.i + N.k)
v.append((a**2 + b)*N.i + 3*(C.y - c)*N.k)
f = Function('f')
v.append(N.j - (Integral(f(b)) - C.x**2)*N.k)
upretty_v_8 = u(
"""\
⎛ 2 ⌠ ⎞ \n\
j_N + ⎜x_C - ⎮ f(b) db⎟ k_N\n\
⎝ ⌡ ⎠ \
""")
pretty_v_8 = u(
"""\
j_N + / / \\\n\
| 2 | |\n\
|x_C - | f(b) db|\n\
| | |\n\
\\ / / \
""")
v.append(N.i + C.k)
v.append(express(N.i, C))
v.append((a**2 + b)*N.i + (Integral(f(b)))*N.k)
upretty_v_11 = u(
"""\
⎛ 2 ⎞ ⎛⌠ ⎞ \n\
⎝a + b⎠ i_N + ⎜⎮ f(b) db⎟ k_N\n\
⎝⌡ ⎠ \
""")
pretty_v_11 = u(
"""\
/ 2 \\ + / / \\\n\
\\a + b/ i_N| | |\n\
| | f(b) db|\n\
| | |\n\
\\/ / \
""")
for x in v:
d.append(x | N.k)
s = 3*N.x**2*C.y
upretty_s = u(
"""\
2\n\
3⋅y_C⋅x_N \
""")
pretty_s = u(
"""\
2\n\
3*y_C*x_N \
""")
# This is the pretty form for ((a**2 + b)*N.i + 3*(C.y - c)*N.k) | N.k
upretty_d_7 = u(
"""\
⎛ 2 ⎞ \n\
⎝a + b⎠ (i_N|k_N) + (3⋅y_C - 3⋅c) (k_N|k_N)\
""")
pretty_d_7 = u(
"""\
/ 2 \\ (i_N|k_N) + (3*y_C - 3*c) (k_N|k_N)\n\
\\a + b/ \
""")
def test_str_printing():
assert str(v[0]) == '0'
assert str(v[1]) == 'N.i'
assert str(v[2]) == '(-1)*N.i'
assert str(v[3]) == 'N.i + N.j'
assert str(v[8]) == 'N.j + (C.x**2 - Integral(f(b), b))*N.k'
assert str(v[9]) == 'C.k + N.i'
assert str(s) == '3*C.y*N.x**2'
assert str(d[0]) == '0'
assert str(d[1]) == '(N.i|N.k)'
assert str(d[4]) == 'a*(N.i|N.k)'
assert str(d[5]) == 'a*(N.i|N.k) + (-b)*(N.j|N.k)'
assert str(d[8]) == ('(N.j|N.k) + (C.x**2 - ' +
'Integral(f(b), b))*(N.k|N.k)')
@XFAIL
def test_pretty_printing_ascii():
assert pretty(v[0]) == u'0'
assert pretty(v[1]) == u'i_N'
assert pretty(v[5]) == u'(a) i_N + (-b) j_N'
assert pretty(v[8]) == pretty_v_8
assert pretty(v[2]) == u'(-1) i_N'
assert pretty(v[11]) == pretty_v_11
assert pretty(s) == pretty_s
assert pretty(d[0]) == u'(0|0)'
assert pretty(d[5]) == u'(a) (i_N|k_N) + (-b) (j_N|k_N)'
assert pretty(d[7]) == pretty_d_7
assert pretty(d[10]) == u'(cos(a)) (i_C|k_N) + (-sin(a)) (j_C|k_N)'
def test_pretty_print_unicode_v():
assert upretty(v[0]) == u'0'
assert upretty(v[1]) == u'i_N'
assert upretty(v[5]) == u'(a) i_N + (-b) j_N'
# Make sure the printing works in other objects
assert upretty(v[5].args) == u'((a) i_N, (-b) j_N)'
assert upretty(v[8]) == upretty_v_8
assert upretty(v[2]) == u'(-1) i_N'
assert upretty(v[11]) == upretty_v_11
assert upretty(s) == upretty_s
assert upretty(d[0]) == u'(0|0)'
assert upretty(d[5]) == u'(a) (i_N|k_N) + (-b) (j_N|k_N)'
assert upretty(d[7]) == upretty_d_7
assert upretty(d[10]) == u'(cos(a)) (i_C|k_N) + (-sin(a)) (j_C|k_N)'
def test_latex_printing():
assert latex(v[0]) == '\\mathbf{\\hat{0}}'
assert latex(v[1]) == '\\mathbf{\\hat{i}_{N}}'
assert latex(v[2]) == '- \\mathbf{\\hat{i}_{N}}'
assert latex(v[5]) == ('(a)\\mathbf{\\hat{i}_{N}} + ' +
'(- b)\\mathbf{\\hat{j}_{N}}')
assert latex(v[6]) == ('(\\mathbf{{x}_{N}} + a^{2})\\mathbf{\\hat{i}_' +
'{N}} + \\mathbf{\\hat{k}_{N}}')
assert latex(v[8]) == ('\\mathbf{\\hat{j}_{N}} + (\\mathbf{{x}_' +
'{C}}^{2} - \\int f{\\left(b \\right)}\\,' +
' db)\\mathbf{\\hat{k}_{N}}')
assert latex(s) == '3 \\mathbf{{y}_{C}} \\mathbf{{x}_{N}}^{2}'
assert latex(d[0]) == '(\\mathbf{\\hat{0}}|\\mathbf{\\hat{0}})'
assert latex(d[4]) == ('(a)(\\mathbf{\\hat{i}_{N}}{|}\\mathbf' +
'{\\hat{k}_{N}})')
assert latex(d[9]) == ('(\\mathbf{\\hat{k}_{C}}{|}\\mathbf{\\' +
'hat{k}_{N}}) + (\\mathbf{\\hat{i}_{N}}{|' +
'}\\mathbf{\\hat{k}_{N}})')
assert latex(d[11]) == ('(a^{2} + b)(\\mathbf{\\hat{i}_{N}}{|}\\' +
'mathbf{\\hat{k}_{N}}) + (\\int f{\\left(' +
'b \\right)}\\, db)(\\mathbf{\\hat{k}_{N}' +
'}{|}\\mathbf{\\hat{k}_{N}})')
def test_custom_names():
A = CoordSys3D('A', vector_names=['x', 'y', 'z'],
variable_names=['i', 'j', 'k'])
assert A.i.__str__() == 'A.i'
assert A.x.__str__() == 'A.x'
assert A.i._pretty_form == 'i_A'
assert A.x._pretty_form == 'x_A'
assert A.i._latex_form == r'\mathbf{{i}_{A}}'
assert A.x._latex_form == r"\mathbf{\hat{x}_{A}}"
|
a3aa19e6eca251d65c7b4620f137d545e40cc1038272ada171d699fa8c4ff9ca
|
from sympy import Dummy, S, symbols, pi, sqrt, asin, sin, cos
from sympy.geometry import Line, Point, Ray, Segment, Point3D, Line3D, Ray3D, Segment3D, Plane
from sympy.geometry.util import are_coplanar
from sympy.utilities.pytest import raises
def test_plane():
x, y, z, u, v = symbols('x y z u v', real=True)
p1 = Point3D(0, 0, 0)
p2 = Point3D(1, 1, 1)
p3 = Point3D(1, 2, 3)
pl3 = Plane(p1, p2, p3)
pl4 = Plane(p1, normal_vector=(1, 1, 1))
pl4b = Plane(p1, p2)
pl5 = Plane(p3, normal_vector=(1, 2, 3))
pl6 = Plane(Point3D(2, 3, 7), normal_vector=(2, 2, 2))
pl7 = Plane(Point3D(1, -5, -6), normal_vector=(1, -2, 1))
pl8 = Plane(p1, normal_vector=(0, 0, 1))
pl9 = Plane(p1, normal_vector=(0, 12, 0))
pl10 = Plane(p1, normal_vector=(-2, 0, 0))
pl11 = Plane(p2, normal_vector=(0, 0, 1))
l1 = Line3D(Point3D(5, 0, 0), Point3D(1, -1, 1))
l2 = Line3D(Point3D(0, -2, 0), Point3D(3, 1, 1))
l3 = Line3D(Point3D(0, -1, 0), Point3D(5, -1, 9))
assert Plane(p1, p2, p3) != Plane(p1, p3, p2)
assert Plane(p1, p2, p3).is_coplanar(Plane(p1, p3, p2))
assert pl3 == Plane(Point3D(0, 0, 0), normal_vector=(1, -2, 1))
assert pl3 != pl4
assert pl4 == pl4b
assert pl5 == Plane(Point3D(1, 2, 3), normal_vector=(1, 2, 3))
assert pl5.equation(x, y, z) == x + 2*y + 3*z - 14
assert pl3.equation(x, y, z) == x - 2*y + z
assert pl3.p1 == p1
assert pl4.p1 == p1
assert pl5.p1 == p3
assert pl4.normal_vector == (1, 1, 1)
assert pl5.normal_vector == (1, 2, 3)
assert p1 in pl3
assert p1 in pl4
assert p3 in pl5
assert pl3.projection(Point(0, 0)) == p1
p = pl3.projection(Point3D(1, 1, 0))
assert p == Point3D(S(7)/6, S(2)/3, S(1)/6)
assert p in pl3
l = pl3.projection_line(Line(Point(0, 0), Point(1, 1)))
assert l == Line3D(Point3D(0, 0, 0), Point3D(S(7)/6, S(2)/3, S(1)/6))
assert l in pl3
# get a segment that does not intersect the plane which is also
# parallel to pl3's normal veector
t = Dummy()
r = pl3.random_point()
a = pl3.perpendicular_line(r).arbitrary_point(t)
s = Segment3D(a.subs(t, 1), a.subs(t, 2))
assert s.p1 not in pl3 and s.p2 not in pl3
assert pl3.projection_line(s).equals(r)
assert pl3.projection_line(Segment(Point(1, 0), Point(1, 1))) == \
Segment3D(Point3D(S(5)/6, S(1)/3, -S(1)/6), Point3D(S(7)/6, S(2)/3, S(1)/6))
assert pl6.projection_line(Ray(Point(1, 0), Point(1, 1))) == \
Ray3D(Point3D(S(14)/3, S(11)/3, S(11)/3), Point3D(S(13)/3, S(13)/3, S(10)/3))
assert pl3.perpendicular_line(r.args) == pl3.perpendicular_line(r)
assert pl3.is_parallel(pl6) is False
assert pl4.is_parallel(pl6)
assert pl6.is_parallel(l1) is False
assert pl3.is_perpendicular(pl6)
assert pl4.is_perpendicular(pl7)
assert pl6.is_perpendicular(pl7)
assert pl6.is_perpendicular(l1) is False
assert pl6.distance(pl6.arbitrary_point(u, v)) == 0
assert pl7.distance(pl7.arbitrary_point(u, v)) == 0
assert pl6.distance(pl6.arbitrary_point(t)) == 0
assert pl7.distance(pl7.arbitrary_point(t)) == 0
assert pl6.p1.distance(pl6.arbitrary_point(t)).simplify() == 1
assert pl7.p1.distance(pl7.arbitrary_point(t)).simplify() == 1
assert pl3.arbitrary_point(t) == Point3D(-sqrt(30)*sin(t)/30 + \
2*sqrt(5)*cos(t)/5, sqrt(30)*sin(t)/15 + sqrt(5)*cos(t)/5, sqrt(30)*sin(t)/6)
assert pl3.arbitrary_point(u, v) == Point3D(2*u - v, u + 2*v, 5*v)
assert pl7.distance(Point3D(1, 3, 5)) == 5*sqrt(6)/6
assert pl6.distance(Point3D(0, 0, 0)) == 4*sqrt(3)
assert pl6.distance(pl6.p1) == 0
assert pl7.distance(pl6) == 0
assert pl7.distance(l1) == 0
assert pl6.distance(Segment3D(Point3D(2, 3, 1), Point3D(1, 3, 4))) == 0
pl6.distance(Plane(Point3D(5, 5, 5), normal_vector=(8, 8, 8))) == sqrt(3)
assert pl6.angle_between(pl3) == pi/2
assert pl6.angle_between(pl6) == 0
assert pl6.angle_between(pl4) == 0
assert pl7.angle_between(Line3D(Point3D(2, 3, 5), Point3D(2, 4, 6))) == \
-asin(sqrt(3)/6)
assert pl6.angle_between(Ray3D(Point3D(2, 4, 1), Point3D(6, 5, 3))) == \
asin(sqrt(7)/3)
assert pl7.angle_between(Segment3D(Point3D(5, 6, 1), Point3D(1, 2, 4))) == \
asin(7*sqrt(246)/246)
assert are_coplanar(l1, l2, l3) is False
assert are_coplanar(l1) is False
assert are_coplanar(Point3D(2, 7, 2), Point3D(0, 0, 2),
Point3D(1, 1, 2), Point3D(1, 2, 2))
assert are_coplanar(Plane(p1, p2, p3), Plane(p1, p3, p2))
assert Plane.are_concurrent(pl3, pl4, pl5) is False
assert Plane.are_concurrent(pl6) is False
raises(ValueError, lambda: Plane.are_concurrent(Point3D(0, 0, 0)))
raises(ValueError, lambda: Plane((1, 2, 3), normal_vector=(0, 0, 0)))
assert pl3.parallel_plane(Point3D(1, 2, 5)) == Plane(Point3D(1, 2, 5), \
normal_vector=(1, -2, 1))
# perpendicular_plane
p = Plane((0, 0, 0), (1, 0, 0))
# default
assert p.perpendicular_plane() == Plane(Point3D(0, 0, 0), (0, 1, 0))
# 1 pt
assert p.perpendicular_plane(Point3D(1, 0, 1)) == \
Plane(Point3D(1, 0, 1), (0, 1, 0))
# pts as tuples
assert p.perpendicular_plane((1, 0, 1), (1, 1, 1)) == \
Plane(Point3D(1, 0, 1), (0, 0, -1))
a, b = Point3D(0, 0, 0), Point3D(0, 1, 0)
Z = (0, 0, 1)
p = Plane(a, normal_vector=Z)
# case 4
assert p.perpendicular_plane(a, b) == Plane(a, (1, 0, 0))
n = Point3D(*Z)
# case 1
assert p.perpendicular_plane(a, n) == Plane(a, (-1, 0, 0))
# case 2
assert Plane(a, normal_vector=b.args).perpendicular_plane(a, a + b) == \
Plane(Point3D(0, 0, 0), (1, 0, 0))
# case 1&3
assert Plane(b, normal_vector=Z).perpendicular_plane(b, b + n) == \
Plane(Point3D(0, 1, 0), (-1, 0, 0))
# case 2&3
assert Plane(b, normal_vector=b.args).perpendicular_plane(n, n + b) == \
Plane(Point3D(0, 0, 1), (1, 0, 0))
assert pl6.intersection(pl6) == [pl6]
assert pl4.intersection(pl4.p1) == [pl4.p1]
assert pl3.intersection(pl6) == [
Line3D(Point3D(8, 4, 0), Point3D(2, 4, 6))]
assert pl3.intersection(Line3D(Point3D(1,2,4), Point3D(4,4,2))) == [
Point3D(2, S(8)/3, S(10)/3)]
assert pl3.intersection(Plane(Point3D(6, 0, 0), normal_vector=(2, -5, 3))
) == [Line3D(Point3D(-24, -12, 0), Point3D(-25, -13, -1))]
assert pl6.intersection(Ray3D(Point3D(2, 3, 1), Point3D(1, 3, 4))) == [
Point3D(-1, 3, 10)]
assert pl6.intersection(Segment3D(Point3D(2, 3, 1), Point3D(1, 3, 4))) == [
Point3D(-1, 3, 10)]
assert pl7.intersection(Line(Point(2, 3), Point(4, 2))) == [
Point3D(S(13)/2, S(3)/4, 0)]
r = Ray(Point(2, 3), Point(4, 2))
assert Plane((1,2,0), normal_vector=(0,0,1)).intersection(r) == [
Ray3D(Point(2, 3), Point(4, 2))]
assert pl9.intersection(pl8) == [Line3D(Point3D(0, 0, 0), Point3D(12, 0, 0))]
assert pl10.intersection(pl11) == [Line3D(Point3D(0, 0, 1), Point3D(0, 2, 1))]
assert pl4.intersection(pl8) == [Line3D(Point3D(0, 0, 0), Point3D(1, -1, 0))]
assert pl11.intersection(pl8) == []
assert pl9.intersection(pl11) == [Line3D(Point3D(0, 0, 1), Point3D(12, 0, 1))]
assert pl9.intersection(pl4) == [Line3D(Point3D(0, 0, 0), Point3D(12, 0, -12))]
assert pl3.random_point() in pl3
# test geometrical entity using equals
assert pl4.intersection(pl4.p1)[0].equals(pl4.p1)
assert pl3.intersection(pl6)[0].equals(Line3D(Point3D(8, 4, 0), Point3D(2, 4, 6)))
pl8 = Plane((1, 2, 0), normal_vector=(0, 0, 1))
assert pl8.intersection(Line3D(p1, (1, 12, 0)))[0].equals(Line((0, 0, 0), (0.1, 1.2, 0)))
assert pl8.intersection(Ray3D(p1, (1, 12, 0)))[0].equals(Ray((0, 0, 0), (1, 12, 0)))
assert pl8.intersection(Segment3D(p1, (21, 1, 0)))[0].equals(Segment3D(p1, (21, 1, 0)))
assert pl8.intersection(Plane(p1, normal_vector=(0, 0, 112)))[0].equals(pl8)
assert pl8.intersection(Plane(p1, normal_vector=(0, 12, 0)))[0].equals(
Line3D(p1, direction_ratio=(112 * pi, 0, 0)))
assert pl8.intersection(Plane(p1, normal_vector=(11, 0, 1)))[0].equals(
Line3D(p1, direction_ratio=(0, -11, 0)))
assert pl8.intersection(Plane(p1, normal_vector=(1, 0, 11)))[0].equals(
Line3D(p1, direction_ratio=(0, 11, 0)))
assert pl8.intersection(Plane(p1, normal_vector=(-1, -1, -11)))[0].equals(
Line3D(p1, direction_ratio=(1, -1, 0)))
assert pl3.random_point() in pl3
assert len(pl8.intersection(Ray3D(Point3D(0, 2, 3), Point3D(1, 0, 3)))) is 0
# check if two plane are equals
assert pl6.intersection(pl6)[0].equals(pl6)
assert pl8.equals(Plane(p1, normal_vector=(0, 12, 0))) is False
assert pl8.equals(pl8)
assert pl8.equals(Plane(p1, normal_vector=(0, 0, -12)))
assert pl8.equals(Plane(p1, normal_vector=(0, 0, -12*sqrt(3))))
# issue 8570
l2 = Line3D(Point3D(S(50000004459633)/5000000000000,
-S(891926590718643)/1000000000000000,
S(231800966893633)/100000000000000),
Point3D(S(50000004459633)/50000000000000,
-S(222981647679771)/250000000000000,
S(231800966893633)/100000000000000))
p2 = Plane(Point3D(S(402775636372767)/100000000000000,
-S(97224357654973)/100000000000000,
S(216793600814789)/100000000000000),
(-S('9.00000087501922'), -S('4.81170658872543e-13'),
S('0.0')))
assert str([i.n(2) for i in p2.intersection(l2)]) == \
'[Point3D(4.0, -0.89, 2.3)]'
def test_dimension_normalization():
A = Plane(Point3D(1, 1, 2), normal_vector=(1, 1, 1))
b = Point(1, 1)
assert A.projection(b) == Point(S(5)/3, S(5)/3, S(2)/3)
a, b = Point(0, 0), Point3D(0, 1)
Z = (0, 0, 1)
p = Plane(a, normal_vector=Z)
assert p.perpendicular_plane(a, b) == Plane(Point3D(0, 0, 0), (1, 0, 0))
assert Plane((1, 2, 1), (2, 1, 0), (3, 1, 2)
).intersection((2, 1)) == [Point(2, 1, 0)]
def test_parameter_value():
t, u, v = symbols("t, u v")
p = Plane((0, 0, 0), (0, 0, 1), (0, 1, 0))
assert p.parameter_value((0, -3, 2), t) == {t: asin(2*sqrt(13)/13)}
assert p.parameter_value((0, -3, 2), u, v) == {u: 3, v: 2}
raises(ValueError, lambda: p.parameter_value((1, 0, 0), t))
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ca6aa8ff45a2b4decedb5d4264296f9bf1cb760e0c1d1ac732b1cadf56e66344
|
from sympy import Rational, S, Symbol, symbols, pi, sqrt, oo, Point2D, Segment2D, I
from sympy.core.compatibility import range
from sympy.geometry import (Circle, Ellipse, GeometryError, Line, Point, Polygon, Ray, RegularPolygon, Segment,
Triangle, intersection)
from sympy.utilities.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(-S(1)/2, -S(2)/3), 5*sqrt(37)/6)
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 = Rational(1, 2)
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(GeometryError, lambda: Circle(Point(0, 0)))
# 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 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
# 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(S(3)/2, 1), Point(S(3)/2, S(1)/2))]
assert e2.tangent_lines(p1_3) == [Line(Point(1, 2), Point(S(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(S(77)/25, S(132)/25)),
Line(Point(0, 0), Point(S(33)/5, S(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))), ]
# 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(-S(51)/26, -S(1)/5), Point(-S(25)/26, S(17)/83)),
Line(Point(S(28)/29, -S(7)/8), Point(S(57)/29, -S(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(-S(341)/171, -S(1)/13), Point(-S(170)/171, S(5)/64)),
Line(Point(S(26)/15, -S(1)/2), Point(S(41)/15, -S(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(-S(64)/33, -S(20)/71), Point(-S(31)/33, S(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(1)/2).intersection(
Triangle((-1, 0), (1, 0), (0, 1))) == [
Point(-S(1)/2, 0), Point(S(1)/2, 0)]
raises(TypeError, lambda: intersection(e2, Line((0, 0, 0), (0, 0, 1))))
raises(TypeError, lambda: intersection(e2, Rational(12)))
# 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 = S(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, 2*c/17 + a/2), Point(c/68 + a, -2*c/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(S(14)/5, S(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(1)/2, 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(1)/2 + sqrt(3)/2, S(1)/2 + 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
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)
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(S(1)/3, S(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)))
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((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]
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)
# degenerate ellipse
assert Ellipse(None, 1, None, 1).length == 2
# 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():
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)
|
6e512d23835b2dd4e261e4dab07a35b3a7af027a2ea69d8c148668fd6344d772
|
from sympy import Abs, Rational, Float, S, Symbol, symbols, cos, pi, sqrt, oo
from sympy.functions.elementary.trigonometric import tan
from sympy.geometry import (Circle, Ellipse, GeometryError, Point, Point2D, \
Polygon, Ray, RegularPolygon, Segment, Triangle, \
are_similar,convex_hull, intersection, Line)
from sympy.utilities.pytest import raises, slow, warns
from sympy.utilities.randtest import verify_numerically
from sympy.geometry.polygon import rad, deg
from sympy import integrate
def feq(a, b):
"""Test if two floating point values are 'equal'."""
t_float = Float("1.0E-10")
return -t_float < a - b < t_float
@slow
def test_polygon():
x = Symbol('x', real=True)
y = Symbol('y', real=True)
q = Symbol('q', real=True)
u = Symbol('u', real=True)
v = Symbol('v', real=True)
w = Symbol('w', real=True)
x1 = Symbol('x1', real=True)
half = Rational(1, 2)
a, b, c = Point(0, 0), Point(2, 0), Point(3, 3)
t = Triangle(a, b, c)
assert Polygon(a, Point(1, 0), b, c) == t
assert Polygon(Point(1, 0), b, c, a) == t
assert Polygon(b, c, a, Point(1, 0)) == t
# 2 "remove folded" tests
assert Polygon(a, Point(3, 0), b, c) == t
assert Polygon(a, b, Point(3, -1), b, c) == t
# remove multiple collinear points
assert Polygon(Point(-4, 15), Point(-11, 15), Point(-15, 15),
Point(-15, 33/5), Point(-15, -87/10), Point(-15, -15),
Point(-42/5, -15), Point(-2, -15), Point(7, -15), Point(15, -15),
Point(15, -3), Point(15, 10), Point(15, 15)) == \
Polygon(Point(-15,-15), Point(15,-15), Point(15,15), Point(-15,15))
p1 = Polygon(
Point(0, 0), Point(3, -1),
Point(6, 0), Point(4, 5),
Point(2, 3), Point(0, 3))
p2 = Polygon(
Point(6, 0), Point(3, -1),
Point(0, 0), Point(0, 3),
Point(2, 3), Point(4, 5))
p3 = Polygon(
Point(0, 0), Point(3, 0),
Point(5, 2), Point(4, 4))
p4 = Polygon(
Point(0, 0), Point(4, 4),
Point(5, 2), Point(3, 0))
p5 = Polygon(
Point(0, 0), Point(4, 4),
Point(0, 4))
p6 = Polygon(
Point(-11, 1), Point(-9, 6.6),
Point(-4, -3), Point(-8.4, -8.7))
p7 = Polygon(
Point(x, y), Point(q, u),
Point(v, w))
p8 = Polygon(
Point(x, y), Point(v, w),
Point(q, u))
p9 = Polygon(
Point(0, 0), Point(4, 4),
Point(3, 0), Point(5, 2))
p10 = Polygon(
Point(0, 2), Point(2, 2),
Point(0, 0), Point(2, 0))
p11 = Polygon(Point(0, 0), 1, n=3)
r = Ray(Point(-9,6.6), Point(-9,5.5))
#
# General polygon
#
assert p1 == p2
assert len(p1.args) == 6
assert len(p1.sides) == 6
assert p1.perimeter == 5 + 2*sqrt(10) + sqrt(29) + sqrt(8)
assert p1.area == 22
assert not p1.is_convex()
assert Polygon((-1, 1), (2, -1), (2, 1), (-1, -1), (3, 0)
).is_convex() is False
# ensure convex for both CW and CCW point specification
assert p3.is_convex()
assert p4.is_convex()
dict5 = p5.angles
assert dict5[Point(0, 0)] == pi / 4
assert dict5[Point(0, 4)] == pi / 2
assert p5.encloses_point(Point(x, y)) is None
assert p5.encloses_point(Point(1, 3))
assert p5.encloses_point(Point(0, 0)) is False
assert p5.encloses_point(Point(4, 0)) is False
assert p1.encloses(Circle(Point(2.5,2.5),5)) is False
assert p1.encloses(Ellipse(Point(2.5,2),5,6)) is False
p5.plot_interval('x') == [x, 0, 1]
assert p5.distance(
Polygon(Point(10, 10), Point(14, 14), Point(10, 14))) == 6 * sqrt(2)
assert p5.distance(
Polygon(Point(1, 8), Point(5, 8), Point(8, 12), Point(1, 12))) == 4
with warns(UserWarning, \
match="Polygons may intersect producing erroneous output"):
Polygon(Point(0, 0), Point(1, 0), Point(1, 1)).distance(
Polygon(Point(0, 0), Point(0, 1), Point(1, 1)))
assert hash(p5) == hash(Polygon(Point(0, 0), Point(4, 4), Point(0, 4)))
assert hash(p1) == hash(p2)
assert hash(p7) == hash(p8)
assert hash(p3) != hash(p9)
assert p5 == Polygon(Point(4, 4), Point(0, 4), Point(0, 0))
assert Polygon(Point(4, 4), Point(0, 4), Point(0, 0)) in p5
assert p5 != Point(0, 4)
assert Point(0, 1) in p5
assert p5.arbitrary_point('t').subs(Symbol('t', real=True), 0) == \
Point(0, 0)
raises(ValueError, lambda: Polygon(
Point(x, 0), Point(0, y), Point(x, y)).arbitrary_point('x'))
assert p6.intersection(r) == [Point(-9, -S(84)/13), Point(-9, S(33)/5)]
assert p10.area == 0
assert p11 == RegularPolygon(Point(0, 0), 1, 3, 0)
assert p11.vertices[0] == Point(1, 0)
assert p11.args[0] == Point(0, 0)
p11.spin(pi/2)
assert p11.vertices[0] == Point(0, 1)
#
# Regular polygon
#
p1 = RegularPolygon(Point(0, 0), 10, 5)
p2 = RegularPolygon(Point(0, 0), 5, 5)
raises(GeometryError, lambda: RegularPolygon(Point(0, 0), Point(0,
1), Point(1, 1)))
raises(GeometryError, lambda: RegularPolygon(Point(0, 0), 1, 2))
raises(ValueError, lambda: RegularPolygon(Point(0, 0), 1, 2.5))
assert p1 != p2
assert p1.interior_angle == 3*pi/5
assert p1.exterior_angle == 2*pi/5
assert p2.apothem == 5*cos(pi/5)
assert p2.circumcenter == p1.circumcenter == Point(0, 0)
assert p1.circumradius == p1.radius == 10
assert p2.circumcircle == Circle(Point(0, 0), 5)
assert p2.incircle == Circle(Point(0, 0), p2.apothem)
assert p2.inradius == p2.apothem == (5 * (1 + sqrt(5)) / 4)
p2.spin(pi / 10)
dict1 = p2.angles
assert dict1[Point(0, 5)] == 3 * pi / 5
assert p1.is_convex()
assert p1.rotation == 0
assert p1.encloses_point(Point(0, 0))
assert p1.encloses_point(Point(11, 0)) is False
assert p2.encloses_point(Point(0, 4.9))
p1.spin(pi/3)
assert p1.rotation == pi/3
assert p1.vertices[0] == Point(5, 5*sqrt(3))
for var in p1.args:
if isinstance(var, Point):
assert var == Point(0, 0)
else:
assert var == 5 or var == 10 or var == pi / 3
assert p1 != Point(0, 0)
assert p1 != p5
# while spin works in place (notice that rotation is 2pi/3 below)
# rotate returns a new object
p1_old = p1
assert p1.rotate(pi/3) == RegularPolygon(Point(0, 0), 10, 5, 2*pi/3)
assert p1 == p1_old
assert p1.area == (-250*sqrt(5) + 1250)/(4*tan(pi/5))
assert p1.length == 20*sqrt(-sqrt(5)/8 + S(5)/8)
assert p1.scale(2, 2) == \
RegularPolygon(p1.center, p1.radius*2, p1._n, p1.rotation)
assert RegularPolygon((0, 0), 1, 4).scale(2, 3) == \
Polygon(Point(2, 0), Point(0, 3), Point(-2, 0), Point(0, -3))
assert repr(p1) == str(p1)
#
# Angles
#
angles = p4.angles
assert feq(angles[Point(0, 0)].evalf(), Float("0.7853981633974483"))
assert feq(angles[Point(4, 4)].evalf(), Float("1.2490457723982544"))
assert feq(angles[Point(5, 2)].evalf(), Float("1.8925468811915388"))
assert feq(angles[Point(3, 0)].evalf(), Float("2.3561944901923449"))
angles = p3.angles
assert feq(angles[Point(0, 0)].evalf(), Float("0.7853981633974483"))
assert feq(angles[Point(4, 4)].evalf(), Float("1.2490457723982544"))
assert feq(angles[Point(5, 2)].evalf(), Float("1.8925468811915388"))
assert feq(angles[Point(3, 0)].evalf(), Float("2.3561944901923449"))
#
# Triangle
#
p1 = Point(0, 0)
p2 = Point(5, 0)
p3 = Point(0, 5)
t1 = Triangle(p1, p2, p3)
t2 = Triangle(p1, p2, Point(Rational(5, 2), sqrt(Rational(75, 4))))
t3 = Triangle(p1, Point(x1, 0), Point(0, x1))
s1 = t1.sides
assert Triangle(p1, p2, p1) == Polygon(p1, p2, p1) == Segment(p1, p2)
raises(GeometryError, lambda: Triangle(Point(0, 0)))
# Basic stuff
assert Triangle(p1, p1, p1) == p1
assert Triangle(p2, p2*2, p2*3) == Segment(p2, p2*3)
assert t1.area == Rational(25, 2)
assert t1.is_right()
assert t2.is_right() is False
assert t3.is_right()
assert p1 in t1
assert t1.sides[0] in t1
assert Segment((0, 0), (1, 0)) in t1
assert Point(5, 5) not in t2
assert t1.is_convex()
assert feq(t1.angles[p1].evalf(), pi.evalf()/2)
assert t1.is_equilateral() is False
assert t2.is_equilateral()
assert t3.is_equilateral() is False
assert are_similar(t1, t2) is False
assert are_similar(t1, t3)
assert are_similar(t2, t3) is False
assert t1.is_similar(Point(0, 0)) is False
assert t1.is_similar(t2) is False
# Bisectors
bisectors = t1.bisectors()
assert bisectors[p1] == Segment(p1, Point(Rational(5, 2), Rational(5, 2)))
ic = (250 - 125*sqrt(2)) / 50
assert t1.incenter == Point(ic, ic)
# Inradius
assert t1.inradius == t1.incircle.radius == 5 - 5*sqrt(2)/2
assert t2.inradius == t2.incircle.radius == 5*sqrt(3)/6
assert t3.inradius == t3.incircle.radius == x1**2/((2 + sqrt(2))*Abs(x1))
# Exradius
assert t1.exradii[t1.sides[2]] == 5*sqrt(2)/2
# Circumcircle
assert t1.circumcircle.center == Point(2.5, 2.5)
# Medians + Centroid
m = t1.medians
assert t1.centroid == Point(Rational(5, 3), Rational(5, 3))
assert m[p1] == Segment(p1, Point(Rational(5, 2), Rational(5, 2)))
assert t3.medians[p1] == Segment(p1, Point(x1/2, x1/2))
assert intersection(m[p1], m[p2], m[p3]) == [t1.centroid]
assert t1.medial == Triangle(Point(2.5, 0), Point(0, 2.5), Point(2.5, 2.5))
# Nine-point circle
assert t1.nine_point_circle == Circle(Point(2.5, 0),
Point(0, 2.5), Point(2.5, 2.5))
assert t1.nine_point_circle == Circle(Point(0, 0),
Point(0, 2.5), Point(2.5, 2.5))
# Perpendicular
altitudes = t1.altitudes
assert altitudes[p1] == Segment(p1, Point(Rational(5, 2), Rational(5, 2)))
assert altitudes[p2].equals(s1[0])
assert altitudes[p3] == s1[2]
assert t1.orthocenter == p1
t = S('''Triangle(
Point(100080156402737/5000000000000, 79782624633431/500000000000),
Point(39223884078253/2000000000000, 156345163124289/1000000000000),
Point(31241359188437/1250000000000, 338338270939941/1000000000000000))''')
assert t.orthocenter == S('''Point(-780660869050599840216997'''
'''79471538701955848721853/80368430960602242240789074233100000000000000,'''
'''20151573611150265741278060334545897615974257/16073686192120448448157'''
'''8148466200000000000)''')
# Ensure
assert len(intersection(*bisectors.values())) == 1
assert len(intersection(*altitudes.values())) == 1
assert len(intersection(*m.values())) == 1
# Distance
p1 = Polygon(
Point(0, 0), Point(1, 0),
Point(1, 1), Point(0, 1))
p2 = Polygon(
Point(0, Rational(5)/4), Point(1, Rational(5)/4),
Point(1, Rational(9)/4), Point(0, Rational(9)/4))
p3 = Polygon(
Point(1, 2), Point(2, 2),
Point(2, 1))
p4 = Polygon(
Point(1, 1), Point(Rational(6)/5, 1),
Point(1, Rational(6)/5))
pt1 = Point(half, half)
pt2 = Point(1, 1)
'''Polygon to Point'''
assert p1.distance(pt1) == half
assert p1.distance(pt2) == 0
assert p2.distance(pt1) == Rational(3)/4
assert p3.distance(pt2) == sqrt(2)/2
'''Polygon to Polygon'''
# p1.distance(p2) emits a warning
with warns(UserWarning, \
match="Polygons may intersect producing erroneous output"):
assert p1.distance(p2) == half/2
assert p1.distance(p3) == sqrt(2)/2
# p3.distance(p4) emits a warning
with warns(UserWarning, \
match="Polygons may intersect producing erroneous output"):
assert p3.distance(p4) == (sqrt(2)/2 - sqrt(Rational(2)/25)/2)
def test_convex_hull():
p = [Point(-5, -1), Point(-2, 1), Point(-2, -1), Point(-1, -3), \
Point(0, 0), Point(1, 1), Point(2, 2), Point(2, -1), Point(3, 1), \
Point(4, -1), Point(6, 2)]
ch = Polygon(p[0], p[3], p[9], p[10], p[6], p[1])
#test handling of duplicate points
p.append(p[3])
#more than 3 collinear points
another_p = [Point(-45, -85), Point(-45, 85), Point(-45, 26), \
Point(-45, -24)]
ch2 = Segment(another_p[0], another_p[1])
assert convex_hull(*another_p) == ch2
assert convex_hull(*p) == ch
assert convex_hull(p[0]) == p[0]
assert convex_hull(p[0], p[1]) == Segment(p[0], p[1])
# no unique points
assert convex_hull(*[p[-1]]*3) == p[-1]
# collection of items
assert convex_hull(*[Point(0, 0), \
Segment(Point(1, 0), Point(1, 1)), \
RegularPolygon(Point(2, 0), 2, 4)]) == \
Polygon(Point(0, 0), Point(2, -2), Point(4, 0), Point(2, 2))
def test_encloses():
# square with a dimpled left side
s = Polygon(Point(0, 0), Point(1, 0), Point(1, 1), Point(0, 1), \
Point(S.Half, S.Half))
# the following is True if the polygon isn't treated as closing on itself
assert s.encloses(Point(0, S.Half)) is False
assert s.encloses(Point(S.Half, S.Half)) is False # it's a vertex
assert s.encloses(Point(Rational(3, 4), S.Half)) is True
def test_triangle_kwargs():
assert Triangle(sss=(3, 4, 5)) == \
Triangle(Point(0, 0), Point(3, 0), Point(3, 4))
assert Triangle(asa=(30, 2, 30)) == \
Triangle(Point(0, 0), Point(2, 0), Point(1, sqrt(3)/3))
assert Triangle(sas=(1, 45, 2)) == \
Triangle(Point(0, 0), Point(2, 0), Point(sqrt(2)/2, sqrt(2)/2))
assert Triangle(sss=(1, 2, 5)) is None
assert deg(rad(180)) == 180
def test_transform():
pts = [Point(0, 0), Point(S(1)/2, S(1)/4), Point(1, 1)]
pts_out = [Point(-4, -10), Point(-3, -S(37)/4), Point(-2, -7)]
assert Triangle(*pts).scale(2, 3, (4, 5)) == Triangle(*pts_out)
assert RegularPolygon((0, 0), 1, 4).scale(2, 3, (4, 5)) == \
Polygon(Point(-2, -10), Point(-4, -7), Point(-6, -10), Point(-4, -13))
def test_reflect():
x = Symbol('x', real=True)
y = Symbol('y', real=True)
b = Symbol('b')
m = Symbol('m')
l = Line((0, b), slope=m)
p = Point(x, y)
r = p.reflect(l)
dp = l.perpendicular_segment(p).length
dr = l.perpendicular_segment(r).length
assert verify_numerically(dp, dr)
t = Triangle((0, 0), (1, 0), (2, 3))
assert Polygon((1, 0), (2, 0), (2, 2)).reflect(Line((3, 0), slope=oo)) \
== Triangle(Point(5, 0), Point(4, 0), Point(4, 2))
assert Polygon((1, 0), (2, 0), (2, 2)).reflect(Line((0, 3), slope=oo)) \
== Triangle(Point(-1, 0), Point(-2, 0), Point(-2, 2))
assert Polygon((1, 0), (2, 0), (2, 2)).reflect(Line((0, 3), slope=0)) \
== Triangle(Point(1, 6), Point(2, 6), Point(2, 4))
assert Polygon((1, 0), (2, 0), (2, 2)).reflect(Line((3, 0), slope=0)) \
== Triangle(Point(1, 0), Point(2, 0), Point(2, -2))
def test_bisectors():
p1, p2, p3 = Point(0, 0), Point(1, 0), Point(0, 1)
t = Triangle(p1, p2, p3)
assert t.bisectors()[p2] == Segment(Point(1, 0), Point(0, sqrt(2) - 1))
def test_incenter():
assert Triangle(Point(0, 0), Point(1, 0), Point(0, 1)).incenter \
== Point(1 - sqrt(2)/2, 1 - sqrt(2)/2)
def test_inradius():
assert Triangle(Point(0, 0), Point(4, 0), Point(0, 3)).inradius == 1
def test_incircle():
assert Triangle(Point(0, 0), Point(2, 0), Point(0, 2)).incircle \
== Circle(Point(2 - sqrt(2), 2 - sqrt(2)), 2 - sqrt(2))
def test_exradii():
t = Triangle(Point(0, 0), Point(6, 0), Point(0, 2))
assert t.exradii[t.sides[2]] == (-2 + sqrt(10))
def test_medians():
t = Triangle(Point(0, 0), Point(1, 0), Point(0, 1))
assert t.medians[Point(0, 0)] == Segment(Point(0, 0), Point(S(1)/2, S(1)/2))
def test_medial():
assert Triangle(Point(0, 0), Point(1, 0), Point(0, 1)).medial \
== Triangle(Point(S(1)/2, 0), Point(S(1)/2, S(1)/2), Point(0, S(1)/2))
def test_nine_point_circle():
assert Triangle(Point(0, 0), Point(1, 0), Point(0, 1)).nine_point_circle \
== Circle(Point2D(S(1)/4, S(1)/4), sqrt(2)/4)
def test_eulerline():
assert Triangle(Point(0, 0), Point(1, 0), Point(0, 1)).eulerline \
== Line(Point2D(0, 0), Point2D(S(1)/2, S(1)/2))
assert Triangle(Point(0, 0), Point(10, 0), Point(5, 5*sqrt(3))).eulerline \
== Point2D(5, 5*sqrt(3)/3)
assert Triangle(Point(4, -6), Point(4, -1), Point(-3, 3)).eulerline \
== Line(Point2D(S(64)/7, 3), Point2D(-S(29)/14, -S(7)/2))
def test_intersection():
poly1 = Triangle(Point(0, 0), Point(1, 0), Point(0, 1))
poly2 = Polygon(Point(0, 1), Point(-5, 0),
Point(0, -4), Point(0, S(1)/5),
Point(S(1)/2, -0.1), Point(1,0), Point(0, 1))
assert poly1.intersection(poly2) == [Point2D(S(1)/3, 0),
Segment(Point(0, S(1)/5), Point(0, 0)),
Segment(Point(1, 0), Point(0, 1))]
assert poly2.intersection(poly1) == [Point(S(1)/3, 0),
Segment(Point(0, 0), Point(0, S(1)/5)),
Segment(Point(1, 0), Point(0, 1))]
assert poly1.intersection(Point(0, 0)) == [Point(0, 0)]
assert poly1.intersection(Point(-12, -43)) == []
assert poly2.intersection(Line((-12, 0), (12, 0))) == [Point(-5, 0),
Point(0, 0),Point(S(1)/3, 0), Point(1, 0)]
assert poly2.intersection(Line((-12, 12), (12, 12))) == []
assert poly2.intersection(Ray((-3,4), (1,0))) == [Segment(Point(1, 0),
Point(0, 1))]
assert poly2.intersection(Circle((0, -1), 1)) == [Point(0, -2),
Point(0, 0)]
assert poly1.intersection(poly1) == [Segment(Point(0, 0), Point(1, 0)),
Segment(Point(0, 1), Point(0, 0)), Segment(Point(1, 0), Point(0, 1))]
assert poly2.intersection(poly2) == [Segment(Point(-5, 0), Point(0, -4)),
Segment(Point(0, -4), Point(0, S(1)/5)),
Segment(Point(0, S(1)/5), Point(S(1)/2, -S(1)/10)),
Segment(Point(0, 1), Point(-5, 0)),
Segment(Point(S(1)/2, -S(1)/10), Point(1, 0)),
Segment(Point(1, 0), Point(0, 1))]
assert poly2.intersection(Triangle(Point(0, 1), Point(1, 0), Point(-1, 1))) \
== [Point(-S(5)/7, S(6)/7), Segment(Point2D(0, 1), Point(1, 0))]
assert poly1.intersection(RegularPolygon((-12, -15), 3, 3)) == []
def test_parameter_value():
t = Symbol('t')
sq = Polygon((0, 0), (0, 1), (1, 1), (1, 0))
assert sq.parameter_value((0.5, 1), t) == {t: S(3)/8}
q = Polygon((0, 0), (2, 1), (2, 4), (4, 0))
assert q.parameter_value((4, 0), t) == {t: -6 + 3*sqrt(5)} # ~= 0.708
raises(ValueError, lambda: sq.parameter_value((5, 6), t))
def test_issue_12966():
poly = Polygon(Point(0, 0), Point(0, 10), Point(5, 10), Point(5, 5),
Point(10, 5), Point(10, 0))
t = Symbol('t')
pt = poly.arbitrary_point(t)
DELTA = 5/poly.perimeter
assert [pt.subs(t, DELTA*i) for i in range(int(1/DELTA))] == [
Point(0, 0), Point(0, 5), Point(0, 10), Point(5, 10),
Point(5, 5), Point(10, 5), Point(10, 0), Point(5, 0)]
def test_second_moment_of_area():
x, y = symbols('x, y')
# triangle
p1, p2, p3 = [(0, 0), (4, 0), (0, 2)]
p = (0, 0)
# equation of hypotenuse
eq_y = (1-x/4)*2
I_yy = integrate((x**2) * (integrate(1, (y, 0, eq_y))), (x, 0, 4))
I_xx = integrate(1 * (integrate(y**2, (y, 0, eq_y))), (x, 0, 4))
I_xy = integrate(x * (integrate(y, (y, 0, eq_y))), (x, 0, 4))
triangle = Polygon(p1, p2, p3)
assert (I_xx - triangle.second_moment_of_area(p)[0]) == 0
assert (I_yy - triangle.second_moment_of_area(p)[1]) == 0
assert (I_xy - triangle.second_moment_of_area(p)[2]) == 0
# rectangle
p1, p2, p3, p4=[(0, 0), (4, 0), (4, 2), (0, 2)]
I_yy = integrate((x**2) * integrate(1, (y, 0, 2)), (x, 0, 4))
I_xx = integrate(1 * integrate(y**2, (y, 0, 2)), (x, 0, 4))
I_xy = integrate(x * integrate(y, (y, 0, 2)), (x, 0, 4))
rectangle = Polygon(p1, p2, p3, p4)
assert (I_xx - rectangle.second_moment_of_area(p)[0]) == 0
assert (I_yy - rectangle.second_moment_of_area(p)[1]) == 0
assert (I_xy - rectangle.second_moment_of_area(p)[2]) == 0
|
5d0c8dc5dcae82cdddcfd6dfd8b76097965cc510327c8f5540d107001d0b49ac
|
import os
from sympy import sin, cos
from sympy.external import import_module
from sympy.utilities.pytest import skip
from sympy.parsing.autolev import parse_autolev
antlr4 = import_module("antlr4")
if not antlr4:
disabled = True
FILE_DIR = os.path.dirname(
os.path.dirname(os.path.abspath(os.path.realpath(__file__))))
def _test_examples(in_filename, out_filename, test_name=""):
in_file_path = os.path.join(FILE_DIR, 'autolev', 'test-examples',
in_filename)
correct_file_path = os.path.join(FILE_DIR, 'autolev', 'test-examples',
out_filename)
with open(in_file_path) as f:
generated_code = parse_autolev(f, include_numeric=True)
with open(correct_file_path) as f:
for idx, line1 in enumerate(f):
if line1.startswith("#"):
break
try:
line2 = generated_code.split('\n')[idx]
assert line1.rstrip() == line2.rstrip()
except Exception:
msg = 'mismatch in ' + test_name + ' in line no: {0}'
raise AssertionError(msg.format(idx+1))
def test_rule_tests():
l = ["ruletest1", "ruletest2", "ruletest3", "ruletest4", "ruletest5",
"ruletest6", "ruletest7", "ruletest8", "ruletest9", "ruletest10",
"ruletest11", "ruletest12"]
for i in l:
in_filepath = i + ".al"
out_filepath = i + ".py"
_test_examples(in_filepath, out_filepath, i)
def test_pydy_examples():
l = ["mass_spring_damper", "chaos_pendulum", "double_pendulum",
"non_min_pendulum"]
for i in l:
in_filepath = os.path.join("pydy-example-repo", i + ".al")
out_filepath = os.path.join("pydy-example-repo", i + ".py")
_test_examples(in_filepath, out_filepath, i)
def test_autolev_tutorial():
dir_path = os.path.join(FILE_DIR, 'autolev', 'test-examples',
'autolev-tutorial')
if os.path.isdir(dir_path):
l = ["tutor1", "tutor2", "tutor3", "tutor4", "tutor5", "tutor6",
"tutor7"]
for i in l:
in_filepath = os.path.join("autolev-tutorial", i + ".al")
out_filepath = os.path.join("autolev-tutorial", i + ".py")
_test_examples(in_filepath, out_filepath, i)
def test_dynamics_online():
dir_path = os.path.join(FILE_DIR, 'autolev', 'test-examples',
'dynamics-online')
if os.path.isdir(dir_path):
ch1 = ["1-4", "1-5", "1-6", "1-7", "1-8", "1-9_1", "1-9_2", "1-9_3"]
ch2 = ["2-1", "2-2", "2-3", "2-4", "2-5", "2-6", "2-7", "2-8", "2-9",
"circular"]
ch3 = ["3-1_1", "3-1_2", "3-2_1", "3-2_2", "3-2_3", "3-2_4", "3-2_5",
"3-3"]
ch4 = ["4-1_1", "4-2_1", "4-4_1", "4-4_2", "4-5_1", "4-5_2"]
chapters = [(ch1, "ch1"), (ch2, "ch2"), (ch3, "ch3"), (ch4, "ch4")]
for ch, name in chapters:
for i in ch:
in_filepath = os.path.join("dynamics-online", name, i + ".al")
out_filepath = os.path.join("dynamics-online", name, i + ".py")
_test_examples(in_filepath, out_filepath, i)
def test_output_01():
"""Autolev example calculates the position, velocity, and accleration of a
point and expresses in a single reference frame::
(1) FRAMES C,D,F
(2) VARIABLES FD'',DC''
(3) CONSTANTS R,L
(4) POINTS O,E
(5) SIMPROT(F,D,1,FD)
-> (6) F_D = [1, 0, 0; 0, COS(FD), -SIN(FD); 0, SIN(FD), COS(FD)]
(7) SIMPROT(D,C,2,DC)
-> (8) D_C = [COS(DC), 0, SIN(DC); 0, 1, 0; -SIN(DC), 0, COS(DC)]
(9) W_C_F> = EXPRESS(W_C_F>, F)
-> (10) W_C_F> = FD'*F1> + COS(FD)*DC'*F2> + SIN(FD)*DC'*F3>
(11) P_O_E>=R*D2>-L*C1>
(12) P_O_E>=EXPRESS(P_O_E>, D)
-> (13) P_O_E> = -L*COS(DC)*D1> + R*D2> + L*SIN(DC)*D3>
(14) V_E_F>=EXPRESS(DT(P_O_E>,F),D)
-> (15) V_E_F> = L*SIN(DC)*DC'*D1> - L*SIN(DC)*FD'*D2> + (R*FD'+L*COS(DC)*DC')*D3>
(16) A_E_F>=EXPRESS(DT(V_E_F>,F),D)
-> (17) A_E_F> = L*(COS(DC)*DC'^2+SIN(DC)*DC'')*D1> + (-R*FD'^2-2*L*COS(DC)*DC'*FD'-L*SIN(DC)*FD'')*D2> + (R*FD''+L*COS(DC)*DC''-L*SIN(DC)*DC'^2-L*SIN(DC)*FD'^2)*D3>
"""
if not antlr4:
skip('Test skipped: antlr4 is not installed.')
autolev_input = """\
FRAMES C,D,F
VARIABLES FD'',DC''
CONSTANTS R,L
POINTS O,E
SIMPROT(F,D,1,FD)
SIMPROT(D,C,2,DC)
W_C_F>=EXPRESS(W_C_F>,F)
P_O_E>=R*D2>-L*C1>
P_O_E>=EXPRESS(P_O_E>,D)
V_E_F>=EXPRESS(DT(P_O_E>,F),D)
A_E_F>=EXPRESS(DT(V_E_F>,F),D)\
"""
sympy_input = parse_autolev(autolev_input)
g = {}
l = {}
exec(sympy_input, g, l)
w_c_f = l['frame_c'].ang_vel_in(l['frame_f'])
# P_O_E> means "the position of point E wrt to point O"
p_o_e = l['point_e'].pos_from(l['point_o'])
v_e_f = l['point_e'].vel(l['frame_f'])
a_e_f = l['point_e'].acc(l['frame_f'])
# NOTE : The Autolev outputs above were manually transformed into
# equivalent SymPy physics vector expressions. Would be nice to automate
# this transformation.
expected_w_c_f = (l['fd'].diff()*l['frame_f'].x +
cos(l['fd'])*l['dc'].diff()*l['frame_f'].y +
sin(l['fd'])*l['dc'].diff()*l['frame_f'].z)
assert (w_c_f - expected_w_c_f).simplify() == 0
expected_p_o_e = (-l['l']*cos(l['dc'])*l['frame_d'].x +
l['r']*l['frame_d'].y +
l['l']*sin(l['dc'])*l['frame_d'].z)
assert (p_o_e - expected_p_o_e).simplify() == 0
expected_v_e_f = (l['l']*sin(l['dc'])*l['dc'].diff()*l['frame_d'].x -
l['l']*sin(l['dc'])*l['fd'].diff()*l['frame_d'].y +
(l['r']*l['fd'].diff() +
l['l']*cos(l['dc'])*l['dc'].diff())*l['frame_d'].z)
assert (v_e_f - expected_v_e_f).simplify() == 0
expected_a_e_f = (l['l']*(cos(l['dc'])*l['dc'].diff()**2 +
sin(l['dc'])*l['dc'].diff().diff())*l['frame_d'].x +
(-l['r']*l['fd'].diff()**2 -
2*l['l']*cos(l['dc'])*l['dc'].diff()*l['fd'].diff() -
l['l']*sin(l['dc'])*l['fd'].diff().diff())*l['frame_d'].y +
(l['r']*l['fd'].diff().diff() +
l['l']*cos(l['dc'])*l['dc'].diff().diff() -
l['l']*sin(l['dc'])*l['dc'].diff()**2 -
l['l']*sin(l['dc'])*l['fd'].diff()**2)*l['frame_d'].z)
assert (a_e_f - expected_a_e_f).simplify() == 0
|
8812d1f2bd57cfe68c19d5f7d9aa1ca0cb8da6d8ae2e743bd225229bd0e81996
|
import collections
import sys
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" + str(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):
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:
a = 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:
a = 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")
|
68d2ef8271be2ae3a8e40ff6e32e93cceb57da07c096ff80456c7a61820a6c49
|
import sys
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 parse_autolev(autolev_code, include_numeric):
antlr4 = import_module('antlr4', warn_not_installed=True)
if not antlr4:
raise ImportError("Autolev parsing requires the antlr4 python package,"
" provided by pip (antlr4-python2-runtime or"
" antlr4-python3-runtime) or"
" conda (antlr-python-runtime)")
try:
l = autolev_code.readlines()
input_stream = antlr4.InputStream("".join(l))
except Exception:
input_stream = antlr4.InputStream(autolev_code)
if AutolevListener:
from ._listener_autolev_antlr import MyListener
lexer = AutolevLexer(input_stream)
token_stream = antlr4.CommonTokenStream(lexer)
parser = AutolevParser(token_stream)
tree = parser.prog()
my_listener = MyListener(include_numeric)
walker = antlr4.ParseTreeWalker()
walker.walk(my_listener, tree)
return "".join(my_listener.output_code)
|
f9c9a5ecf38edf9afb08d0840da5856784cdf04b52fa41706f0984ce59b4da5a
|
"""Dirac notation for states."""
from __future__ import print_function, division
from sympy import (cacheit, conjugate, Expr, Function, integrate, oo, sqrt,
Tuple)
from sympy.core.compatibility import range
from sympy.printing.pretty.stringpict import stringPict
from sympy.physics.quantum.qexpr import QExpr, dispatch_method
__all__ = [
'KetBase',
'BraBase',
'StateBase',
'State',
'Ket',
'Bra',
'TimeDepState',
'TimeDepBra',
'TimeDepKet',
'Wavefunction'
]
#-----------------------------------------------------------------------------
# States, bras and kets.
#-----------------------------------------------------------------------------
# ASCII brackets
_lbracket = "<"
_rbracket = ">"
_straight_bracket = "|"
# Unicode brackets
# MATHEMATICAL ANGLE BRACKETS
_lbracket_ucode = u"\N{MATHEMATICAL LEFT ANGLE BRACKET}"
_rbracket_ucode = u"\N{MATHEMATICAL RIGHT ANGLE BRACKET}"
# LIGHT VERTICAL BAR
_straight_bracket_ucode = u"\N{LIGHT VERTICAL BAR}"
# Other options for unicode printing of <, > and | for Dirac notation.
# LEFT-POINTING ANGLE BRACKET
# _lbracket = u"\u2329"
# _rbracket = u"\u232A"
# LEFT ANGLE BRACKET
# _lbracket = u"\u3008"
# _rbracket = u"\u3009"
# VERTICAL LINE
# _straight_bracket = u"\u007C"
class StateBase(QExpr):
"""Abstract base class for general abstract states in quantum mechanics.
All other state classes defined will need to inherit from this class. It
carries the basic structure for all other states such as dual, _eval_adjoint
and label.
This is an abstract base class and you should not instantiate it directly,
instead use State.
"""
@classmethod
def _operators_to_state(self, ops, **options):
""" Returns the eigenstate instance for the passed operators.
This method should be overridden in subclasses. It will handle being
passed either an Operator instance or set of Operator instances. It
should return the corresponding state INSTANCE or simply raise a
NotImplementedError. See cartesian.py for an example.
"""
raise NotImplementedError("Cannot map operators to states in this class. Method not implemented!")
def _state_to_operators(self, op_classes, **options):
""" Returns the operators which this state instance is an eigenstate
of.
This method should be overridden in subclasses. It will be called on
state instances and be passed the operator classes that we wish to make
into instances. The state instance will then transform the classes
appropriately, or raise a NotImplementedError if it cannot return
operator instances. See cartesian.py for examples,
"""
raise NotImplementedError(
"Cannot map this state to operators. Method not implemented!")
@property
def operators(self):
"""Return the operator(s) that this state is an eigenstate of"""
from .operatorset import state_to_operators # import internally to avoid circular import errors
return state_to_operators(self)
def _enumerate_state(self, num_states, **options):
raise NotImplementedError("Cannot enumerate this state!")
def _represent_default_basis(self, **options):
return self._represent(basis=self.operators)
#-------------------------------------------------------------------------
# Dagger/dual
#-------------------------------------------------------------------------
@property
def dual(self):
"""Return the dual state of this one."""
return self.dual_class()._new_rawargs(self.hilbert_space, *self.args)
@classmethod
def dual_class(self):
"""Return the class used to construct the dual."""
raise NotImplementedError(
'dual_class must be implemented in a subclass'
)
def _eval_adjoint(self):
"""Compute the dagger of this state using the dual."""
return self.dual
#-------------------------------------------------------------------------
# Printing
#-------------------------------------------------------------------------
def _pretty_brackets(self, height, use_unicode=True):
# Return pretty printed brackets for the state
# Ideally, this could be done by pform.parens but it does not support the angled < and >
# Setup for unicode vs ascii
if use_unicode:
lbracket, rbracket = self.lbracket_ucode, self.rbracket_ucode
slash, bslash, vert = u'\N{BOX DRAWINGS LIGHT DIAGONAL UPPER RIGHT TO LOWER LEFT}', \
u'\N{BOX DRAWINGS LIGHT DIAGONAL UPPER LEFT TO LOWER RIGHT}', \
u'\N{BOX DRAWINGS LIGHT VERTICAL}'
else:
lbracket, rbracket = self.lbracket, self.rbracket
slash, bslash, vert = '/', '\\', '|'
# If height is 1, just return brackets
if height == 1:
return stringPict(lbracket), stringPict(rbracket)
# Make height even
height += (height % 2)
brackets = []
for bracket in lbracket, rbracket:
# Create left bracket
if bracket in {_lbracket, _lbracket_ucode}:
bracket_args = [ ' ' * (height//2 - i - 1) +
slash for i in range(height // 2)]
bracket_args.extend(
[ ' ' * i + bslash for i in range(height // 2)])
# Create right bracket
elif bracket in {_rbracket, _rbracket_ucode}:
bracket_args = [ ' ' * i + bslash for i in range(height // 2)]
bracket_args.extend([ ' ' * (
height//2 - i - 1) + slash for i in range(height // 2)])
# Create straight bracket
elif bracket in {_straight_bracket, _straight_bracket_ucode}:
bracket_args = [vert for i in range(height)]
else:
raise ValueError(bracket)
brackets.append(
stringPict('\n'.join(bracket_args), baseline=height//2))
return brackets
def _sympystr(self, printer, *args):
contents = self._print_contents(printer, *args)
return '%s%s%s' % (self.lbracket, contents, self.rbracket)
def _pretty(self, printer, *args):
from sympy.printing.pretty.stringpict import prettyForm
# Get brackets
pform = self._print_contents_pretty(printer, *args)
lbracket, rbracket = self._pretty_brackets(
pform.height(), printer._use_unicode)
# Put together state
pform = prettyForm(*pform.left(lbracket))
pform = prettyForm(*pform.right(rbracket))
return pform
def _latex(self, printer, *args):
contents = self._print_contents_latex(printer, *args)
# The extra {} brackets are needed to get matplotlib's latex
# rendered to render this properly.
return '{%s%s%s}' % (self.lbracket_latex, contents, self.rbracket_latex)
class KetBase(StateBase):
"""Base class for Kets.
This class defines the dual property and the brackets for printing. This is
an abstract base class and you should not instantiate it directly, instead
use Ket.
"""
lbracket = _straight_bracket
rbracket = _rbracket
lbracket_ucode = _straight_bracket_ucode
rbracket_ucode = _rbracket_ucode
lbracket_latex = r'\left|'
rbracket_latex = r'\right\rangle '
@classmethod
def default_args(self):
return ("psi",)
@classmethod
def dual_class(self):
return BraBase
def __mul__(self, other):
"""KetBase*other"""
from sympy.physics.quantum.operator import OuterProduct
if isinstance(other, BraBase):
return OuterProduct(self, other)
else:
return Expr.__mul__(self, other)
def __rmul__(self, other):
"""other*KetBase"""
from sympy.physics.quantum.innerproduct import InnerProduct
if isinstance(other, BraBase):
return InnerProduct(other, self)
else:
return Expr.__rmul__(self, other)
#-------------------------------------------------------------------------
# _eval_* methods
#-------------------------------------------------------------------------
def _eval_innerproduct(self, bra, **hints):
"""Evaluate the inner product between this ket and a bra.
This is called to compute <bra|ket>, where the ket is ``self``.
This method will dispatch to sub-methods having the format::
``def _eval_innerproduct_BraClass(self, **hints):``
Subclasses should define these methods (one for each BraClass) to
teach the ket how to take inner products with bras.
"""
return dispatch_method(self, '_eval_innerproduct', bra, **hints)
def _apply_operator(self, op, **options):
"""Apply an Operator to this Ket.
This method will dispatch to methods having the format::
``def _apply_operator_OperatorName(op, **options):``
Subclasses should define these methods (one for each OperatorName) to
teach the Ket how operators act on it.
Parameters
==========
op : Operator
The Operator that is acting on the Ket.
options : dict
A dict of key/value pairs that control how the operator is applied
to the Ket.
"""
return dispatch_method(self, '_apply_operator', op, **options)
class BraBase(StateBase):
"""Base class for Bras.
This class defines the dual property and the brackets for printing. This
is an abstract base class and you should not instantiate it directly,
instead use Bra.
"""
lbracket = _lbracket
rbracket = _straight_bracket
lbracket_ucode = _lbracket_ucode
rbracket_ucode = _straight_bracket_ucode
lbracket_latex = r'\left\langle '
rbracket_latex = r'\right|'
@classmethod
def _operators_to_state(self, ops, **options):
state = self.dual_class().operators_to_state(ops, **options)
return state.dual
def _state_to_operators(self, op_classes, **options):
return self.dual._state_to_operators(op_classes, **options)
def _enumerate_state(self, num_states, **options):
dual_states = self.dual._enumerate_state(num_states, **options)
return [x.dual for x in dual_states]
@classmethod
def default_args(self):
return self.dual_class().default_args()
@classmethod
def dual_class(self):
return KetBase
def __mul__(self, other):
"""BraBase*other"""
from sympy.physics.quantum.innerproduct import InnerProduct
if isinstance(other, KetBase):
return InnerProduct(self, other)
else:
return Expr.__mul__(self, other)
def __rmul__(self, other):
"""other*BraBase"""
from sympy.physics.quantum.operator import OuterProduct
if isinstance(other, KetBase):
return OuterProduct(other, self)
else:
return Expr.__rmul__(self, other)
def _represent(self, **options):
"""A default represent that uses the Ket's version."""
from sympy.physics.quantum.dagger import Dagger
return Dagger(self.dual._represent(**options))
class State(StateBase):
"""General abstract quantum state used as a base class for Ket and Bra."""
pass
class Ket(State, KetBase):
"""A general time-independent Ket in quantum mechanics.
Inherits from State and KetBase. This class should be used as the base
class for all physical, time-independent Kets in a system. This class
and its subclasses will be the main classes that users will use for
expressing Kets in Dirac notation [1]_.
Parameters
==========
args : tuple
The list of numbers or parameters that uniquely specify the
ket. This will usually be its symbol or its quantum numbers. For
time-dependent state, this will include the time.
Examples
========
Create a simple Ket and looking at its properties::
>>> from sympy.physics.quantum import Ket, Bra
>>> from sympy import symbols, I
>>> k = Ket('psi')
>>> k
|psi>
>>> k.hilbert_space
H
>>> k.is_commutative
False
>>> k.label
(psi,)
Ket's know about their associated bra::
>>> k.dual
<psi|
>>> k.dual_class()
<class 'sympy.physics.quantum.state.Bra'>
Take a linear combination of two kets::
>>> k0 = Ket(0)
>>> k1 = Ket(1)
>>> 2*I*k0 - 4*k1
2*I*|0> - 4*|1>
Compound labels are passed as tuples::
>>> n, m = symbols('n,m')
>>> k = Ket(n,m)
>>> k
|nm>
References
==========
.. [1] https://en.wikipedia.org/wiki/Bra-ket_notation
"""
@classmethod
def dual_class(self):
return Bra
class Bra(State, BraBase):
"""A general time-independent Bra in quantum mechanics.
Inherits from State and BraBase. A Bra is the dual of a Ket [1]_. This
class and its subclasses will be the main classes that users will use for
expressing Bras in Dirac notation.
Parameters
==========
args : tuple
The list of numbers or parameters that uniquely specify the
ket. This will usually be its symbol or its quantum numbers. For
time-dependent state, this will include the time.
Examples
========
Create a simple Bra and look at its properties::
>>> from sympy.physics.quantum import Ket, Bra
>>> from sympy import symbols, I
>>> b = Bra('psi')
>>> b
<psi|
>>> b.hilbert_space
H
>>> b.is_commutative
False
Bra's know about their dual Ket's::
>>> b.dual
|psi>
>>> b.dual_class()
<class 'sympy.physics.quantum.state.Ket'>
Like Kets, Bras can have compound labels and be manipulated in a similar
manner::
>>> n, m = symbols('n,m')
>>> b = Bra(n,m) - I*Bra(m,n)
>>> b
-I*<mn| + <nm|
Symbols in a Bra can be substituted using ``.subs``::
>>> b.subs(n,m)
<mm| - I*<mm|
References
==========
.. [1] https://en.wikipedia.org/wiki/Bra-ket_notation
"""
@classmethod
def dual_class(self):
return Ket
#-----------------------------------------------------------------------------
# Time dependent states, bras and kets.
#-----------------------------------------------------------------------------
class TimeDepState(StateBase):
"""Base class for a general time-dependent quantum state.
This class is used as a base class for any time-dependent state. The main
difference between this class and the time-independent state is that this
class takes a second argument that is the time in addition to the usual
label argument.
Parameters
==========
args : tuple
The list of numbers or parameters that uniquely specify the ket. This
will usually be its symbol or its quantum numbers. For time-dependent
state, this will include the time as the final argument.
"""
#-------------------------------------------------------------------------
# Initialization
#-------------------------------------------------------------------------
@classmethod
def default_args(self):
return ("psi", "t")
#-------------------------------------------------------------------------
# Properties
#-------------------------------------------------------------------------
@property
def label(self):
"""The label of the state."""
return self.args[:-1]
@property
def time(self):
"""The time of the state."""
return self.args[-1]
#-------------------------------------------------------------------------
# Printing
#-------------------------------------------------------------------------
def _print_time(self, printer, *args):
return printer._print(self.time, *args)
_print_time_repr = _print_time
_print_time_latex = _print_time
def _print_time_pretty(self, printer, *args):
pform = printer._print(self.time, *args)
return pform
def _print_contents(self, printer, *args):
label = self._print_label(printer, *args)
time = self._print_time(printer, *args)
return '%s;%s' % (label, time)
def _print_label_repr(self, printer, *args):
label = self._print_sequence(self.label, ',', printer, *args)
time = self._print_time_repr(printer, *args)
return '%s,%s' % (label, time)
def _print_contents_pretty(self, printer, *args):
label = self._print_label_pretty(printer, *args)
time = self._print_time_pretty(printer, *args)
return printer._print_seq((label, time), delimiter=';')
def _print_contents_latex(self, printer, *args):
label = self._print_sequence(
self.label, self._label_separator, printer, *args)
time = self._print_time_latex(printer, *args)
return '%s;%s' % (label, time)
class TimeDepKet(TimeDepState, KetBase):
"""General time-dependent Ket in quantum mechanics.
This inherits from ``TimeDepState`` and ``KetBase`` and is the main class
that should be used for Kets that vary with time. Its dual is a
``TimeDepBra``.
Parameters
==========
args : tuple
The list of numbers or parameters that uniquely specify the ket. This
will usually be its symbol or its quantum numbers. For time-dependent
state, this will include the time as the final argument.
Examples
========
Create a TimeDepKet and look at its attributes::
>>> from sympy.physics.quantum import TimeDepKet
>>> k = TimeDepKet('psi', 't')
>>> k
|psi;t>
>>> k.time
t
>>> k.label
(psi,)
>>> k.hilbert_space
H
TimeDepKets know about their dual bra::
>>> k.dual
<psi;t|
>>> k.dual_class()
<class 'sympy.physics.quantum.state.TimeDepBra'>
"""
@classmethod
def dual_class(self):
return TimeDepBra
class TimeDepBra(TimeDepState, BraBase):
"""General time-dependent Bra in quantum mechanics.
This inherits from TimeDepState and BraBase and is the main class that
should be used for Bras that vary with time. Its dual is a TimeDepBra.
Parameters
==========
args : tuple
The list of numbers or parameters that uniquely specify the ket. This
will usually be its symbol or its quantum numbers. For time-dependent
state, this will include the time as the final argument.
Examples
========
>>> from sympy.physics.quantum import TimeDepBra
>>> from sympy import symbols, I
>>> b = TimeDepBra('psi', 't')
>>> b
<psi;t|
>>> b.time
t
>>> b.label
(psi,)
>>> b.hilbert_space
H
>>> b.dual
|psi;t>
"""
@classmethod
def dual_class(self):
return TimeDepKet
class Wavefunction(Function):
"""Class for representations in continuous bases
This class takes an expression and coordinates in its constructor. It can
be used to easily calculate normalizations and probabilities.
Parameters
==========
expr : Expr
The expression representing the functional form of the w.f.
coords : Symbol or tuple
The coordinates to be integrated over, and their bounds
Examples
========
Particle in a box, specifying bounds in the more primitive way of using
Piecewise:
>>> from sympy import Symbol, Piecewise, pi, N
>>> from sympy.functions import sqrt, sin
>>> from sympy.physics.quantum.state import Wavefunction
>>> x = Symbol('x', real=True)
>>> n = 1
>>> L = 1
>>> g = Piecewise((0, x < 0), (0, x > L), (sqrt(2//L)*sin(n*pi*x/L), True))
>>> f = Wavefunction(g, x)
>>> f.norm
1
>>> f.is_normalized
True
>>> p = f.prob()
>>> p(0)
0
>>> p(L)
0
>>> p(0.5)
2
>>> p(0.85*L)
2*sin(0.85*pi)**2
>>> N(p(0.85*L))
0.412214747707527
Additionally, you can specify the bounds of the function and the indices in
a more compact way:
>>> from sympy import symbols, pi, diff
>>> from sympy.functions import sqrt, sin
>>> from sympy.physics.quantum.state import Wavefunction
>>> x, L = symbols('x,L', positive=True)
>>> n = symbols('n', integer=True, positive=True)
>>> g = sqrt(2/L)*sin(n*pi*x/L)
>>> f = Wavefunction(g, (x, 0, L))
>>> f.norm
1
>>> f(L+1)
0
>>> f(L-1)
sqrt(2)*sin(pi*n*(L - 1)/L)/sqrt(L)
>>> f(-1)
0
>>> f(0.85)
sqrt(2)*sin(0.85*pi*n/L)/sqrt(L)
>>> f(0.85, n=1, L=1)
sqrt(2)*sin(0.85*pi)
>>> f.is_commutative
False
All arguments are automatically sympified, so you can define the variables
as strings rather than symbols:
>>> expr = x**2
>>> f = Wavefunction(expr, 'x')
>>> type(f.variables[0])
<class 'sympy.core.symbol.Symbol'>
Derivatives of Wavefunctions will return Wavefunctions:
>>> diff(f, x)
Wavefunction(2*x, x)
"""
#Any passed tuples for coordinates and their bounds need to be
#converted to Tuples before Function's constructor is called, to
#avoid errors from calling is_Float in the constructor
def __new__(cls, *args, **options):
new_args = [None for i in args]
ct = 0
for arg in args:
if isinstance(arg, tuple):
new_args[ct] = Tuple(*arg)
else:
new_args[ct] = arg
ct += 1
return super(Wavefunction, cls).__new__(cls, *new_args, **options)
def __call__(self, *args, **options):
var = self.variables
if len(args) != len(var):
raise NotImplementedError(
"Incorrect number of arguments to function!")
ct = 0
#If the passed value is outside the specified bounds, return 0
for v in var:
lower, upper = self.limits[v]
#Do the comparison to limits only if the passed symbol is actually
#a symbol present in the limits;
#Had problems with a comparison of x > L
if isinstance(args[ct], Expr) and \
not (lower in args[ct].free_symbols
or upper in args[ct].free_symbols):
continue
if (args[ct] < lower) == True or (args[ct] > upper) == True:
return 0
ct += 1
expr = self.expr
#Allows user to make a call like f(2, 4, m=1, n=1)
for symbol in list(expr.free_symbols):
if str(symbol) in options.keys():
val = options[str(symbol)]
expr = expr.subs(symbol, val)
return expr.subs(zip(var, args))
def _eval_derivative(self, symbol):
expr = self.expr
deriv = expr._eval_derivative(symbol)
return Wavefunction(deriv, *self.args[1:])
def _eval_conjugate(self):
return Wavefunction(conjugate(self.expr), *self.args[1:])
def _eval_transpose(self):
return self
@property
def free_symbols(self):
return self.expr.free_symbols
@property
def is_commutative(self):
"""
Override Function's is_commutative so that order is preserved in
represented expressions
"""
return False
@classmethod
def eval(self, *args):
return None
@property
def variables(self):
"""
Return the coordinates which the wavefunction depends on
Examples
========
>>> from sympy.physics.quantum.state import Wavefunction
>>> from sympy import symbols
>>> x,y = symbols('x,y')
>>> f = Wavefunction(x*y, x, y)
>>> f.variables
(x, y)
>>> g = Wavefunction(x*y, x)
>>> g.variables
(x,)
"""
var = [g[0] if isinstance(g, Tuple) else g for g in self._args[1:]]
return tuple(var)
@property
def limits(self):
"""
Return the limits of the coordinates which the w.f. depends on If no
limits are specified, defaults to ``(-oo, oo)``.
Examples
========
>>> from sympy.physics.quantum.state import Wavefunction
>>> from sympy import symbols
>>> x, y = symbols('x, y')
>>> f = Wavefunction(x**2, (x, 0, 1))
>>> f.limits
{x: (0, 1)}
>>> f = Wavefunction(x**2, x)
>>> f.limits
{x: (-oo, oo)}
>>> f = Wavefunction(x**2 + y**2, x, (y, -1, 2))
>>> f.limits
{x: (-oo, oo), y: (-1, 2)}
"""
limits = [(g[1], g[2]) if isinstance(g, Tuple) else (-oo, oo)
for g in self._args[1:]]
return dict(zip(self.variables, tuple(limits)))
@property
def expr(self):
"""
Return the expression which is the functional form of the Wavefunction
Examples
========
>>> from sympy.physics.quantum.state import Wavefunction
>>> from sympy import symbols
>>> x, y = symbols('x, y')
>>> f = Wavefunction(x**2, x)
>>> f.expr
x**2
"""
return self._args[0]
@property
def is_normalized(self):
"""
Returns true if the Wavefunction is properly normalized
Examples
========
>>> from sympy import symbols, pi
>>> from sympy.functions import sqrt, sin
>>> from sympy.physics.quantum.state import Wavefunction
>>> x, L = symbols('x,L', positive=True)
>>> n = symbols('n', integer=True, positive=True)
>>> g = sqrt(2/L)*sin(n*pi*x/L)
>>> f = Wavefunction(g, (x, 0, L))
>>> f.is_normalized
True
"""
return (self.norm == 1.0)
@property
@cacheit
def norm(self):
"""
Return the normalization of the specified functional form.
This function integrates over the coordinates of the Wavefunction, with
the bounds specified.
Examples
========
>>> from sympy import symbols, pi
>>> from sympy.functions import sqrt, sin
>>> from sympy.physics.quantum.state import Wavefunction
>>> x, L = symbols('x,L', positive=True)
>>> n = symbols('n', integer=True, positive=True)
>>> g = sqrt(2/L)*sin(n*pi*x/L)
>>> f = Wavefunction(g, (x, 0, L))
>>> f.norm
1
>>> g = sin(n*pi*x/L)
>>> f = Wavefunction(g, (x, 0, L))
>>> f.norm
sqrt(2)*sqrt(L)/2
"""
exp = self.expr*conjugate(self.expr)
var = self.variables
limits = self.limits
for v in var:
curr_limits = limits[v]
exp = integrate(exp, (v, curr_limits[0], curr_limits[1]))
return sqrt(exp)
def normalize(self):
"""
Return a normalized version of the Wavefunction
Examples
========
>>> from sympy import symbols, pi
>>> from sympy.functions import sqrt, sin
>>> from sympy.physics.quantum.state import Wavefunction
>>> x = symbols('x', real=True)
>>> L = symbols('L', positive=True)
>>> n = symbols('n', integer=True, positive=True)
>>> g = sin(n*pi*x/L)
>>> f = Wavefunction(g, (x, 0, L))
>>> f.normalize()
Wavefunction(sqrt(2)*sin(pi*n*x/L)/sqrt(L), (x, 0, L))
"""
const = self.norm
if const == oo:
raise NotImplementedError("The function is not normalizable!")
else:
return Wavefunction((const)**(-1)*self.expr, *self.args[1:])
def prob(self):
r"""
Return the absolute magnitude of the w.f., `|\psi(x)|^2`
Examples
========
>>> from sympy import symbols, pi
>>> from sympy.functions import sqrt, sin
>>> from sympy.physics.quantum.state import Wavefunction
>>> x, L = symbols('x,L', real=True)
>>> n = symbols('n', integer=True)
>>> g = sin(n*pi*x/L)
>>> f = Wavefunction(g, (x, 0, L))
>>> f.prob()
Wavefunction(sin(pi*n*x/L)**2, x)
"""
return Wavefunction(self.expr*conjugate(self.expr), *self.variables)
|
74b60718fd816894364d4ea7d8899c8d002ec03e1b164f84dbf083155c353097
|
"""An implementation of gates that act on qubits.
Gates are unitary operators that act on the space of qubits.
Medium Term Todo:
* Optimize Gate._apply_operators_Qubit to remove the creation of many
intermediate Qubit objects.
* Add commutation relationships to all operators and use this in gate_sort.
* Fix gate_sort and gate_simp.
* Get multi-target UGates plotting properly.
* Get UGate to work with either sympy/numpy matrices and output either
format. This should also use the matrix slots.
"""
from __future__ import print_function, division
from itertools import chain
import random
from sympy import Add, I, Integer, Mul, Pow, sqrt, Tuple
from sympy.core.numbers import Number
from sympy.core.compatibility import is_sequence, unicode, range
from sympy.printing.pretty.stringpict import prettyForm, stringPict
from sympy.physics.quantum.anticommutator import AntiCommutator
from sympy.physics.quantum.commutator import Commutator
from sympy.physics.quantum.qexpr import QuantumError
from sympy.physics.quantum.hilbert import ComplexSpace
from sympy.physics.quantum.operator import (UnitaryOperator, Operator,
HermitianOperator)
from sympy.physics.quantum.matrixutils import matrix_tensor_product, matrix_eye
from sympy.physics.quantum.matrixcache import matrix_cache
from sympy.matrices.matrices import MatrixBase
from sympy.utilities import default_sort_key
__all__ = [
'Gate',
'CGate',
'UGate',
'OneQubitGate',
'TwoQubitGate',
'IdentityGate',
'HadamardGate',
'XGate',
'YGate',
'ZGate',
'TGate',
'PhaseGate',
'SwapGate',
'CNotGate',
# Aliased gate names
'CNOT',
'SWAP',
'H',
'X',
'Y',
'Z',
'T',
'S',
'Phase',
'normalized',
'gate_sort',
'gate_simp',
'random_circuit',
'CPHASE',
'CGateS',
]
#-----------------------------------------------------------------------------
# Gate Super-Classes
#-----------------------------------------------------------------------------
_normalized = True
def _max(*args, **kwargs):
if "key" not in kwargs:
kwargs["key"] = default_sort_key
return max(*args, **kwargs)
def _min(*args, **kwargs):
if "key" not in kwargs:
kwargs["key"] = default_sort_key
return min(*args, **kwargs)
def normalized(normalize):
"""Set flag controlling normalization of Hadamard gates by 1/sqrt(2).
This is a global setting that can be used to simplify the look of various
expressions, by leaving off the leading 1/sqrt(2) of the Hadamard gate.
Parameters
----------
normalize : bool
Should the Hadamard gate include the 1/sqrt(2) normalization factor?
When True, the Hadamard gate will have the 1/sqrt(2). When False, the
Hadamard gate will not have this factor.
"""
global _normalized
_normalized = normalize
def _validate_targets_controls(tandc):
tandc = list(tandc)
# Check for integers
for bit in tandc:
if not bit.is_Integer and not bit.is_Symbol:
raise TypeError('Integer expected, got: %r' % tandc[bit])
# Detect duplicates
if len(list(set(tandc))) != len(tandc):
raise QuantumError(
'Target/control qubits in a gate cannot be duplicated'
)
class Gate(UnitaryOperator):
"""Non-controlled unitary gate operator that acts on qubits.
This is a general abstract gate that needs to be subclassed to do anything
useful.
Parameters
----------
label : tuple, int
A list of the target qubits (as ints) that the gate will apply to.
Examples
========
"""
_label_separator = ','
gate_name = u'G'
gate_name_latex = u'G'
#-------------------------------------------------------------------------
# Initialization/creation
#-------------------------------------------------------------------------
@classmethod
def _eval_args(cls, args):
args = Tuple(*UnitaryOperator._eval_args(args))
_validate_targets_controls(args)
return args
@classmethod
def _eval_hilbert_space(cls, args):
"""This returns the smallest possible Hilbert space."""
return ComplexSpace(2)**(_max(args) + 1)
#-------------------------------------------------------------------------
# Properties
#-------------------------------------------------------------------------
@property
def nqubits(self):
"""The total number of qubits this gate acts on.
For controlled gate subclasses this includes both target and control
qubits, so that, for examples the CNOT gate acts on 2 qubits.
"""
return len(self.targets)
@property
def min_qubits(self):
"""The minimum number of qubits this gate needs to act on."""
return _max(self.targets) + 1
@property
def targets(self):
"""A tuple of target qubits."""
return self.label
@property
def gate_name_plot(self):
return r'$%s$' % self.gate_name_latex
#-------------------------------------------------------------------------
# Gate methods
#-------------------------------------------------------------------------
def get_target_matrix(self, format='sympy'):
"""The matrix rep. of the target part of the gate.
Parameters
----------
format : str
The format string ('sympy','numpy', etc.)
"""
raise NotImplementedError(
'get_target_matrix is not implemented in Gate.')
#-------------------------------------------------------------------------
# Apply
#-------------------------------------------------------------------------
def _apply_operator_IntQubit(self, qubits, **options):
"""Redirect an apply from IntQubit to Qubit"""
return self._apply_operator_Qubit(qubits, **options)
def _apply_operator_Qubit(self, qubits, **options):
"""Apply this gate to a Qubit."""
# Check number of qubits this gate acts on.
if qubits.nqubits < self.min_qubits:
raise QuantumError(
'Gate needs a minimum of %r qubits to act on, got: %r' %
(self.min_qubits, qubits.nqubits)
)
# If the controls are not met, just return
if isinstance(self, CGate):
if not self.eval_controls(qubits):
return qubits
targets = self.targets
target_matrix = self.get_target_matrix(format='sympy')
# Find which column of the target matrix this applies to.
column_index = 0
n = 1
for target in targets:
column_index += n*qubits[target]
n = n << 1
column = target_matrix[:, int(column_index)]
# Now apply each column element to the qubit.
result = 0
for index in range(column.rows):
# TODO: This can be optimized to reduce the number of Qubit
# creations. We should simply manipulate the raw list of qubit
# values and then build the new Qubit object once.
# Make a copy of the incoming qubits.
new_qubit = qubits.__class__(*qubits.args)
# Flip the bits that need to be flipped.
for bit in range(len(targets)):
if new_qubit[targets[bit]] != (index >> bit) & 1:
new_qubit = new_qubit.flip(targets[bit])
# The value in that row and column times the flipped-bit qubit
# is the result for that part.
result += column[index]*new_qubit
return result
#-------------------------------------------------------------------------
# Represent
#-------------------------------------------------------------------------
def _represent_default_basis(self, **options):
return self._represent_ZGate(None, **options)
def _represent_ZGate(self, basis, **options):
format = options.get('format', 'sympy')
nqubits = options.get('nqubits', 0)
if nqubits == 0:
raise QuantumError(
'The number of qubits must be given as nqubits.')
# Make sure we have enough qubits for the gate.
if nqubits < self.min_qubits:
raise QuantumError(
'The number of qubits %r is too small for the gate.' % nqubits
)
target_matrix = self.get_target_matrix(format)
targets = self.targets
if isinstance(self, CGate):
controls = self.controls
else:
controls = []
m = represent_zbasis(
controls, targets, target_matrix, nqubits, format
)
return m
#-------------------------------------------------------------------------
# Print methods
#-------------------------------------------------------------------------
def _sympystr(self, printer, *args):
label = self._print_label(printer, *args)
return '%s(%s)' % (self.gate_name, label)
def _pretty(self, printer, *args):
a = stringPict(unicode(self.gate_name))
b = self._print_label_pretty(printer, *args)
return self._print_subscript_pretty(a, b)
def _latex(self, printer, *args):
label = self._print_label(printer, *args)
return '%s_{%s}' % (self.gate_name_latex, label)
def plot_gate(self, axes, gate_idx, gate_grid, wire_grid):
raise NotImplementedError('plot_gate is not implemented.')
class CGate(Gate):
"""A general unitary gate with control qubits.
A general control gate applies a target gate to a set of targets if all
of the control qubits have a particular values (set by
``CGate.control_value``).
Parameters
----------
label : tuple
The label in this case has the form (controls, gate), where controls
is a tuple/list of control qubits (as ints) and gate is a ``Gate``
instance that is the target operator.
Examples
========
"""
gate_name = u'C'
gate_name_latex = u'C'
# The values this class controls for.
control_value = Integer(1)
simplify_cgate=False
#-------------------------------------------------------------------------
# Initialization
#-------------------------------------------------------------------------
@classmethod
def _eval_args(cls, args):
# _eval_args has the right logic for the controls argument.
controls = args[0]
gate = args[1]
if not is_sequence(controls):
controls = (controls,)
controls = UnitaryOperator._eval_args(controls)
_validate_targets_controls(chain(controls, gate.targets))
return (Tuple(*controls), gate)
@classmethod
def _eval_hilbert_space(cls, args):
"""This returns the smallest possible Hilbert space."""
return ComplexSpace(2)**_max(_max(args[0]) + 1, args[1].min_qubits)
#-------------------------------------------------------------------------
# Properties
#-------------------------------------------------------------------------
@property
def nqubits(self):
"""The total number of qubits this gate acts on.
For controlled gate subclasses this includes both target and control
qubits, so that, for examples the CNOT gate acts on 2 qubits.
"""
return len(self.targets) + len(self.controls)
@property
def min_qubits(self):
"""The minimum number of qubits this gate needs to act on."""
return _max(_max(self.controls), _max(self.targets)) + 1
@property
def targets(self):
"""A tuple of target qubits."""
return self.gate.targets
@property
def controls(self):
"""A tuple of control qubits."""
return tuple(self.label[0])
@property
def gate(self):
"""The non-controlled gate that will be applied to the targets."""
return self.label[1]
#-------------------------------------------------------------------------
# Gate methods
#-------------------------------------------------------------------------
def get_target_matrix(self, format='sympy'):
return self.gate.get_target_matrix(format)
def eval_controls(self, qubit):
"""Return True/False to indicate if the controls are satisfied."""
return all(qubit[bit] == self.control_value for bit in self.controls)
def decompose(self, **options):
"""Decompose the controlled gate into CNOT and single qubits gates."""
if len(self.controls) == 1:
c = self.controls[0]
t = self.gate.targets[0]
if isinstance(self.gate, YGate):
g1 = PhaseGate(t)
g2 = CNotGate(c, t)
g3 = PhaseGate(t)
g4 = ZGate(t)
return g1*g2*g3*g4
if isinstance(self.gate, ZGate):
g1 = HadamardGate(t)
g2 = CNotGate(c, t)
g3 = HadamardGate(t)
return g1*g2*g3
else:
return self
#-------------------------------------------------------------------------
# Print methods
#-------------------------------------------------------------------------
def _print_label(self, printer, *args):
controls = self._print_sequence(self.controls, ',', printer, *args)
gate = printer._print(self.gate, *args)
return '(%s),%s' % (controls, gate)
def _pretty(self, printer, *args):
controls = self._print_sequence_pretty(
self.controls, ',', printer, *args)
gate = printer._print(self.gate)
gate_name = stringPict(unicode(self.gate_name))
first = self._print_subscript_pretty(gate_name, controls)
gate = self._print_parens_pretty(gate)
final = prettyForm(*first.right((gate)))
return final
def _latex(self, printer, *args):
controls = self._print_sequence(self.controls, ',', printer, *args)
gate = printer._print(self.gate, *args)
return r'%s_{%s}{\left(%s\right)}' % \
(self.gate_name_latex, controls, gate)
def plot_gate(self, circ_plot, gate_idx):
"""
Plot the controlled gate. If *simplify_cgate* is true, simplify
C-X and C-Z gates into their more familiar forms.
"""
min_wire = int(_min(chain(self.controls, self.targets)))
max_wire = int(_max(chain(self.controls, self.targets)))
circ_plot.control_line(gate_idx, min_wire, max_wire)
for c in self.controls:
circ_plot.control_point(gate_idx, int(c))
if self.simplify_cgate:
if self.gate.gate_name == u'X':
self.gate.plot_gate_plus(circ_plot, gate_idx)
elif self.gate.gate_name == u'Z':
circ_plot.control_point(gate_idx, self.targets[0])
else:
self.gate.plot_gate(circ_plot, gate_idx)
else:
self.gate.plot_gate(circ_plot, gate_idx)
#-------------------------------------------------------------------------
# Miscellaneous
#-------------------------------------------------------------------------
def _eval_dagger(self):
if isinstance(self.gate, HermitianOperator):
return self
else:
return Gate._eval_dagger(self)
def _eval_inverse(self):
if isinstance(self.gate, HermitianOperator):
return self
else:
return Gate._eval_inverse(self)
def _eval_power(self, exp):
if isinstance(self.gate, HermitianOperator):
if exp == -1:
return Gate._eval_power(self, exp)
elif abs(exp) % 2 == 0:
return self*(Gate._eval_inverse(self))
else:
return self
else:
return Gate._eval_power(self, exp)
class CGateS(CGate):
"""Version of CGate that allows gate simplifications.
I.e. cnot looks like an oplus, cphase has dots, etc.
"""
simplify_cgate=True
class UGate(Gate):
"""General gate specified by a set of targets and a target matrix.
Parameters
----------
label : tuple
A tuple of the form (targets, U), where targets is a tuple of the
target qubits and U is a unitary matrix with dimension of
len(targets).
"""
gate_name = u'U'
gate_name_latex = u'U'
#-------------------------------------------------------------------------
# Initialization
#-------------------------------------------------------------------------
@classmethod
def _eval_args(cls, args):
targets = args[0]
if not is_sequence(targets):
targets = (targets,)
targets = Gate._eval_args(targets)
_validate_targets_controls(targets)
mat = args[1]
if not isinstance(mat, MatrixBase):
raise TypeError('Matrix expected, got: %r' % mat)
dim = 2**len(targets)
if not all(dim == shape for shape in mat.shape):
raise IndexError(
'Number of targets must match the matrix size: %r %r' %
(targets, mat)
)
return (targets, mat)
@classmethod
def _eval_hilbert_space(cls, args):
"""This returns the smallest possible Hilbert space."""
return ComplexSpace(2)**(_max(args[0]) + 1)
#-------------------------------------------------------------------------
# Properties
#-------------------------------------------------------------------------
@property
def targets(self):
"""A tuple of target qubits."""
return tuple(self.label[0])
#-------------------------------------------------------------------------
# Gate methods
#-------------------------------------------------------------------------
def get_target_matrix(self, format='sympy'):
"""The matrix rep. of the target part of the gate.
Parameters
----------
format : str
The format string ('sympy','numpy', etc.)
"""
return self.label[1]
#-------------------------------------------------------------------------
# Print methods
#-------------------------------------------------------------------------
def _pretty(self, printer, *args):
targets = self._print_sequence_pretty(
self.targets, ',', printer, *args)
gate_name = stringPict(unicode(self.gate_name))
return self._print_subscript_pretty(gate_name, targets)
def _latex(self, printer, *args):
targets = self._print_sequence(self.targets, ',', printer, *args)
return r'%s_{%s}' % (self.gate_name_latex, targets)
def plot_gate(self, circ_plot, gate_idx):
circ_plot.one_qubit_box(
self.gate_name_plot,
gate_idx, int(self.targets[0])
)
class OneQubitGate(Gate):
"""A single qubit unitary gate base class."""
nqubits = Integer(1)
def plot_gate(self, circ_plot, gate_idx):
circ_plot.one_qubit_box(
self.gate_name_plot,
gate_idx, int(self.targets[0])
)
def _eval_commutator(self, other, **hints):
if isinstance(other, OneQubitGate):
if self.targets != other.targets or self.__class__ == other.__class__:
return Integer(0)
return Operator._eval_commutator(self, other, **hints)
def _eval_anticommutator(self, other, **hints):
if isinstance(other, OneQubitGate):
if self.targets != other.targets or self.__class__ == other.__class__:
return Integer(2)*self*other
return Operator._eval_anticommutator(self, other, **hints)
class TwoQubitGate(Gate):
"""A two qubit unitary gate base class."""
nqubits = Integer(2)
#-----------------------------------------------------------------------------
# Single Qubit Gates
#-----------------------------------------------------------------------------
class IdentityGate(OneQubitGate):
"""The single qubit identity gate.
Parameters
----------
target : int
The target qubit this gate will apply to.
Examples
========
"""
gate_name = u'1'
gate_name_latex = u'1'
def get_target_matrix(self, format='sympy'):
return matrix_cache.get_matrix('eye2', format)
def _eval_commutator(self, other, **hints):
return Integer(0)
def _eval_anticommutator(self, other, **hints):
return Integer(2)*other
class HadamardGate(HermitianOperator, OneQubitGate):
"""The single qubit Hadamard gate.
Parameters
----------
target : int
The target qubit this gate will apply to.
Examples
========
>>> from sympy import sqrt
>>> from sympy.physics.quantum.qubit import Qubit
>>> from sympy.physics.quantum.gate import HadamardGate
>>> from sympy.physics.quantum.qapply import qapply
>>> qapply(HadamardGate(0)*Qubit('1'))
sqrt(2)*|0>/2 - sqrt(2)*|1>/2
>>> # Hadamard on bell state, applied on 2 qubits.
>>> psi = 1/sqrt(2)*(Qubit('00')+Qubit('11'))
>>> qapply(HadamardGate(0)*HadamardGate(1)*psi)
sqrt(2)*|00>/2 + sqrt(2)*|11>/2
"""
gate_name = u'H'
gate_name_latex = u'H'
def get_target_matrix(self, format='sympy'):
if _normalized:
return matrix_cache.get_matrix('H', format)
else:
return matrix_cache.get_matrix('Hsqrt2', format)
def _eval_commutator_XGate(self, other, **hints):
return I*sqrt(2)*YGate(self.targets[0])
def _eval_commutator_YGate(self, other, **hints):
return I*sqrt(2)*(ZGate(self.targets[0]) - XGate(self.targets[0]))
def _eval_commutator_ZGate(self, other, **hints):
return -I*sqrt(2)*YGate(self.targets[0])
def _eval_anticommutator_XGate(self, other, **hints):
return sqrt(2)*IdentityGate(self.targets[0])
def _eval_anticommutator_YGate(self, other, **hints):
return Integer(0)
def _eval_anticommutator_ZGate(self, other, **hints):
return sqrt(2)*IdentityGate(self.targets[0])
class XGate(HermitianOperator, OneQubitGate):
"""The single qubit X, or NOT, gate.
Parameters
----------
target : int
The target qubit this gate will apply to.
Examples
========
"""
gate_name = u'X'
gate_name_latex = u'X'
def get_target_matrix(self, format='sympy'):
return matrix_cache.get_matrix('X', format)
def plot_gate(self, circ_plot, gate_idx):
OneQubitGate.plot_gate(self,circ_plot,gate_idx)
def plot_gate_plus(self, circ_plot, gate_idx):
circ_plot.not_point(
gate_idx, int(self.label[0])
)
def _eval_commutator_YGate(self, other, **hints):
return Integer(2)*I*ZGate(self.targets[0])
def _eval_anticommutator_XGate(self, other, **hints):
return Integer(2)*IdentityGate(self.targets[0])
def _eval_anticommutator_YGate(self, other, **hints):
return Integer(0)
def _eval_anticommutator_ZGate(self, other, **hints):
return Integer(0)
class YGate(HermitianOperator, OneQubitGate):
"""The single qubit Y gate.
Parameters
----------
target : int
The target qubit this gate will apply to.
Examples
========
"""
gate_name = u'Y'
gate_name_latex = u'Y'
def get_target_matrix(self, format='sympy'):
return matrix_cache.get_matrix('Y', format)
def _eval_commutator_ZGate(self, other, **hints):
return Integer(2)*I*XGate(self.targets[0])
def _eval_anticommutator_YGate(self, other, **hints):
return Integer(2)*IdentityGate(self.targets[0])
def _eval_anticommutator_ZGate(self, other, **hints):
return Integer(0)
class ZGate(HermitianOperator, OneQubitGate):
"""The single qubit Z gate.
Parameters
----------
target : int
The target qubit this gate will apply to.
Examples
========
"""
gate_name = u'Z'
gate_name_latex = u'Z'
def get_target_matrix(self, format='sympy'):
return matrix_cache.get_matrix('Z', format)
def _eval_commutator_XGate(self, other, **hints):
return Integer(2)*I*YGate(self.targets[0])
def _eval_anticommutator_YGate(self, other, **hints):
return Integer(0)
class PhaseGate(OneQubitGate):
"""The single qubit phase, or S, gate.
This gate rotates the phase of the state by pi/2 if the state is ``|1>`` and
does nothing if the state is ``|0>``.
Parameters
----------
target : int
The target qubit this gate will apply to.
Examples
========
"""
gate_name = u'S'
gate_name_latex = u'S'
def get_target_matrix(self, format='sympy'):
return matrix_cache.get_matrix('S', format)
def _eval_commutator_ZGate(self, other, **hints):
return Integer(0)
def _eval_commutator_TGate(self, other, **hints):
return Integer(0)
class TGate(OneQubitGate):
"""The single qubit pi/8 gate.
This gate rotates the phase of the state by pi/4 if the state is ``|1>`` and
does nothing if the state is ``|0>``.
Parameters
----------
target : int
The target qubit this gate will apply to.
Examples
========
"""
gate_name = u'T'
gate_name_latex = u'T'
def get_target_matrix(self, format='sympy'):
return matrix_cache.get_matrix('T', format)
def _eval_commutator_ZGate(self, other, **hints):
return Integer(0)
def _eval_commutator_PhaseGate(self, other, **hints):
return Integer(0)
# Aliases for gate names.
H = HadamardGate
X = XGate
Y = YGate
Z = ZGate
T = TGate
Phase = S = PhaseGate
#-----------------------------------------------------------------------------
# 2 Qubit Gates
#-----------------------------------------------------------------------------
class CNotGate(HermitianOperator, CGate, TwoQubitGate):
"""Two qubit controlled-NOT.
This gate performs the NOT or X gate on the target qubit if the control
qubits all have the value 1.
Parameters
----------
label : tuple
A tuple of the form (control, target).
Examples
========
>>> from sympy.physics.quantum.gate import CNOT
>>> from sympy.physics.quantum.qapply import qapply
>>> from sympy.physics.quantum.qubit import Qubit
>>> c = CNOT(1,0)
>>> qapply(c*Qubit('10')) # note that qubits are indexed from right to left
|11>
"""
gate_name = 'CNOT'
gate_name_latex = u'CNOT'
simplify_cgate = True
#-------------------------------------------------------------------------
# Initialization
#-------------------------------------------------------------------------
@classmethod
def _eval_args(cls, args):
args = Gate._eval_args(args)
return args
@classmethod
def _eval_hilbert_space(cls, args):
"""This returns the smallest possible Hilbert space."""
return ComplexSpace(2)**(_max(args) + 1)
#-------------------------------------------------------------------------
# Properties
#-------------------------------------------------------------------------
@property
def min_qubits(self):
"""The minimum number of qubits this gate needs to act on."""
return _max(self.label) + 1
@property
def targets(self):
"""A tuple of target qubits."""
return (self.label[1],)
@property
def controls(self):
"""A tuple of control qubits."""
return (self.label[0],)
@property
def gate(self):
"""The non-controlled gate that will be applied to the targets."""
return XGate(self.label[1])
#-------------------------------------------------------------------------
# Properties
#-------------------------------------------------------------------------
# The default printing of Gate works better than those of CGate, so we
# go around the overridden methods in CGate.
def _print_label(self, printer, *args):
return Gate._print_label(self, printer, *args)
def _pretty(self, printer, *args):
return Gate._pretty(self, printer, *args)
def _latex(self, printer, *args):
return Gate._latex(self, printer, *args)
#-------------------------------------------------------------------------
# Commutator/AntiCommutator
#-------------------------------------------------------------------------
def _eval_commutator_ZGate(self, other, **hints):
"""[CNOT(i, j), Z(i)] == 0."""
if self.controls[0] == other.targets[0]:
return Integer(0)
else:
raise NotImplementedError('Commutator not implemented: %r' % other)
def _eval_commutator_TGate(self, other, **hints):
"""[CNOT(i, j), T(i)] == 0."""
return self._eval_commutator_ZGate(other, **hints)
def _eval_commutator_PhaseGate(self, other, **hints):
"""[CNOT(i, j), S(i)] == 0."""
return self._eval_commutator_ZGate(other, **hints)
def _eval_commutator_XGate(self, other, **hints):
"""[CNOT(i, j), X(j)] == 0."""
if self.targets[0] == other.targets[0]:
return Integer(0)
else:
raise NotImplementedError('Commutator not implemented: %r' % other)
def _eval_commutator_CNotGate(self, other, **hints):
"""[CNOT(i, j), CNOT(i,k)] == 0."""
if self.controls[0] == other.controls[0]:
return Integer(0)
else:
raise NotImplementedError('Commutator not implemented: %r' % other)
class SwapGate(TwoQubitGate):
"""Two qubit SWAP gate.
This gate swap the values of the two qubits.
Parameters
----------
label : tuple
A tuple of the form (target1, target2).
Examples
========
"""
gate_name = 'SWAP'
gate_name_latex = u'SWAP'
def get_target_matrix(self, format='sympy'):
return matrix_cache.get_matrix('SWAP', format)
def decompose(self, **options):
"""Decompose the SWAP gate into CNOT gates."""
i, j = self.targets[0], self.targets[1]
g1 = CNotGate(i, j)
g2 = CNotGate(j, i)
return g1*g2*g1
def plot_gate(self, circ_plot, gate_idx):
min_wire = int(_min(self.targets))
max_wire = int(_max(self.targets))
circ_plot.control_line(gate_idx, min_wire, max_wire)
circ_plot.swap_point(gate_idx, min_wire)
circ_plot.swap_point(gate_idx, max_wire)
def _represent_ZGate(self, basis, **options):
"""Represent the SWAP gate in the computational basis.
The following representation is used to compute this:
SWAP = |1><1|x|1><1| + |0><0|x|0><0| + |1><0|x|0><1| + |0><1|x|1><0|
"""
format = options.get('format', 'sympy')
targets = [int(t) for t in self.targets]
min_target = _min(targets)
max_target = _max(targets)
nqubits = options.get('nqubits', self.min_qubits)
op01 = matrix_cache.get_matrix('op01', format)
op10 = matrix_cache.get_matrix('op10', format)
op11 = matrix_cache.get_matrix('op11', format)
op00 = matrix_cache.get_matrix('op00', format)
eye2 = matrix_cache.get_matrix('eye2', format)
result = None
for i, j in ((op01, op10), (op10, op01), (op00, op00), (op11, op11)):
product = nqubits*[eye2]
product[nqubits - min_target - 1] = i
product[nqubits - max_target - 1] = j
new_result = matrix_tensor_product(*product)
if result is None:
result = new_result
else:
result = result + new_result
return result
# Aliases for gate names.
CNOT = CNotGate
SWAP = SwapGate
def CPHASE(a,b): return CGateS((a,),Z(b))
#-----------------------------------------------------------------------------
# Represent
#-----------------------------------------------------------------------------
def represent_zbasis(controls, targets, target_matrix, nqubits, format='sympy'):
"""Represent a gate with controls, targets and target_matrix.
This function does the low-level work of representing gates as matrices
in the standard computational basis (ZGate). Currently, we support two
main cases:
1. One target qubit and no control qubits.
2. One target qubits and multiple control qubits.
For the base of multiple controls, we use the following expression [1]:
1_{2**n} + (|1><1|)^{(n-1)} x (target-matrix - 1_{2})
Parameters
----------
controls : list, tuple
A sequence of control qubits.
targets : list, tuple
A sequence of target qubits.
target_matrix : sympy.Matrix, numpy.matrix, scipy.sparse
The matrix form of the transformation to be performed on the target
qubits. The format of this matrix must match that passed into
the `format` argument.
nqubits : int
The total number of qubits used for the representation.
format : str
The format of the final matrix ('sympy', 'numpy', 'scipy.sparse').
Examples
========
References
----------
[1] http://www.johnlapeyre.com/qinf/qinf_html/node6.html.
"""
controls = [int(x) for x in controls]
targets = [int(x) for x in targets]
nqubits = int(nqubits)
# This checks for the format as well.
op11 = matrix_cache.get_matrix('op11', format)
eye2 = matrix_cache.get_matrix('eye2', format)
# Plain single qubit case
if len(controls) == 0 and len(targets) == 1:
product = []
bit = targets[0]
# Fill product with [I1,Gate,I2] such that the unitaries,
# I, cause the gate to be applied to the correct Qubit
if bit != nqubits - 1:
product.append(matrix_eye(2**(nqubits - bit - 1), format=format))
product.append(target_matrix)
if bit != 0:
product.append(matrix_eye(2**bit, format=format))
return matrix_tensor_product(*product)
# Single target, multiple controls.
elif len(targets) == 1 and len(controls) >= 1:
target = targets[0]
# Build the non-trivial part.
product2 = []
for i in range(nqubits):
product2.append(matrix_eye(2, format=format))
for control in controls:
product2[nqubits - 1 - control] = op11
product2[nqubits - 1 - target] = target_matrix - eye2
return matrix_eye(2**nqubits, format=format) + \
matrix_tensor_product(*product2)
# Multi-target, multi-control is not yet implemented.
else:
raise NotImplementedError(
'The representation of multi-target, multi-control gates '
'is not implemented.'
)
#-----------------------------------------------------------------------------
# Gate manipulation functions.
#-----------------------------------------------------------------------------
def gate_simp(circuit):
"""Simplifies gates symbolically
It first sorts gates using gate_sort. It then applies basic
simplification rules to the circuit, e.g., XGate**2 = Identity
"""
# Bubble sort out gates that commute.
circuit = gate_sort(circuit)
# Do simplifications by subing a simplification into the first element
# which can be simplified. We recursively call gate_simp with new circuit
# as input more simplifications exist.
if isinstance(circuit, Add):
return sum(gate_simp(t) for t in circuit.args)
elif isinstance(circuit, Mul):
circuit_args = circuit.args
elif isinstance(circuit, Pow):
b, e = circuit.as_base_exp()
circuit_args = (gate_simp(b)**e,)
else:
return circuit
# Iterate through each element in circuit, simplify if possible.
for i in range(len(circuit_args)):
# H,X,Y or Z squared is 1.
# T**2 = S, S**2 = Z
if isinstance(circuit_args[i], Pow):
if isinstance(circuit_args[i].base,
(HadamardGate, XGate, YGate, ZGate)) \
and isinstance(circuit_args[i].exp, Number):
# Build a new circuit taking replacing the
# H,X,Y,Z squared with one.
newargs = (circuit_args[:i] +
(circuit_args[i].base**(circuit_args[i].exp % 2),) +
circuit_args[i + 1:])
# Recursively simplify the new circuit.
circuit = gate_simp(Mul(*newargs))
break
elif isinstance(circuit_args[i].base, PhaseGate):
# Build a new circuit taking old circuit but splicing
# in simplification.
newargs = circuit_args[:i]
# Replace PhaseGate**2 with ZGate.
newargs = newargs + (ZGate(circuit_args[i].base.args[0])**
(Integer(circuit_args[i].exp/2)), circuit_args[i].base**
(circuit_args[i].exp % 2))
# Append the last elements.
newargs = newargs + circuit_args[i + 1:]
# Recursively simplify the new circuit.
circuit = gate_simp(Mul(*newargs))
break
elif isinstance(circuit_args[i].base, TGate):
# Build a new circuit taking all the old elements.
newargs = circuit_args[:i]
# Put an Phasegate in place of any TGate**2.
newargs = newargs + (PhaseGate(circuit_args[i].base.args[0])**
Integer(circuit_args[i].exp/2), circuit_args[i].base**
(circuit_args[i].exp % 2))
# Append the last elements.
newargs = newargs + circuit_args[i + 1:]
# Recursively simplify the new circuit.
circuit = gate_simp(Mul(*newargs))
break
return circuit
def gate_sort(circuit):
"""Sorts the gates while keeping track of commutation relations
This function uses a bubble sort to rearrange the order of gate
application. Keeps track of Quantum computations special commutation
relations (e.g. things that apply to the same Qubit do not commute with
each other)
circuit is the Mul of gates that are to be sorted.
"""
# Make sure we have an Add or Mul.
if isinstance(circuit, Add):
return sum(gate_sort(t) for t in circuit.args)
if isinstance(circuit, Pow):
return gate_sort(circuit.base)**circuit.exp
elif isinstance(circuit, Gate):
return circuit
if not isinstance(circuit, Mul):
return circuit
changes = True
while changes:
changes = False
circ_array = circuit.args
for i in range(len(circ_array) - 1):
# Go through each element and switch ones that are in wrong order
if isinstance(circ_array[i], (Gate, Pow)) and \
isinstance(circ_array[i + 1], (Gate, Pow)):
# If we have a Pow object, look at only the base
first_base, first_exp = circ_array[i].as_base_exp()
second_base, second_exp = circ_array[i + 1].as_base_exp()
# Use sympy's hash based sorting. This is not mathematical
# sorting, but is rather based on comparing hashes of objects.
# See Basic.compare for details.
if first_base.compare(second_base) > 0:
if Commutator(first_base, second_base).doit() == 0:
new_args = (circuit.args[:i] + (circuit.args[i + 1],) +
(circuit.args[i],) + circuit.args[i + 2:])
circuit = Mul(*new_args)
changes = True
break
if AntiCommutator(first_base, second_base).doit() == 0:
new_args = (circuit.args[:i] + (circuit.args[i + 1],) +
(circuit.args[i],) + circuit.args[i + 2:])
sign = Integer(-1)**(first_exp*second_exp)
circuit = sign*Mul(*new_args)
changes = True
break
return circuit
#-----------------------------------------------------------------------------
# Utility functions
#-----------------------------------------------------------------------------
def random_circuit(ngates, nqubits, gate_space=(X, Y, Z, S, T, H, CNOT, SWAP)):
"""Return a random circuit of ngates and nqubits.
This uses an equally weighted sample of (X, Y, Z, S, T, H, CNOT, SWAP)
gates.
Parameters
----------
ngates : int
The number of gates in the circuit.
nqubits : int
The number of qubits in the circuit.
gate_space : tuple
A tuple of the gate classes that will be used in the circuit.
Repeating gate classes multiple times in this tuple will increase
the frequency they appear in the random circuit.
"""
qubit_space = range(nqubits)
result = []
for i in range(ngates):
g = random.choice(gate_space)
if g == CNotGate or g == SwapGate:
qubits = random.sample(qubit_space, 2)
g = g(*qubits)
else:
qubit = random.choice(qubit_space)
g = g(qubit)
result.append(g)
return Mul(*result)
def zx_basis_transform(self, format='sympy'):
"""Transformation matrix from Z to X basis."""
return matrix_cache.get_matrix('ZX', format)
def zy_basis_transform(self, format='sympy'):
"""Transformation matrix from Z to Y basis."""
return matrix_cache.get_matrix('ZY', format)
|
f155d88068de2d8c86aed789a024d462e054c779e718369398a8573f5fdedc9b
|
from __future__ import print_function, division
from itertools import product
from sympy import Tuple, Add, Mul, Matrix, log, expand, Rational
from sympy.core.trace import Tr
from sympy.printing.pretty.stringpict import prettyForm
from sympy.physics.quantum.dagger import Dagger
from sympy.physics.quantum.operator import HermitianOperator
from sympy.physics.quantum.represent import represent
from sympy.physics.quantum.matrixutils import numpy_ndarray, scipy_sparse_matrix, to_numpy
from sympy.physics.quantum.tensorproduct import TensorProduct, tensor_product_simp
class Density(HermitianOperator):
"""Density operator for representing mixed states.
TODO: Density operator support for Qubits
Parameters
==========
values : tuples/lists
Each tuple/list should be of form (state, prob) or [state,prob]
Examples
========
Create a density operator with 2 states represented by Kets.
>>> from sympy.physics.quantum.state import Ket
>>> from sympy.physics.quantum.density import Density
>>> d = Density([Ket(0), 0.5], [Ket(1),0.5])
>>> d
'Density'((|0>, 0.5),(|1>, 0.5))
"""
@classmethod
def _eval_args(cls, args):
# call this to qsympify the args
args = super(Density, cls)._eval_args(args)
for arg in args:
# Check if arg is a tuple
if not (isinstance(arg, Tuple) and
len(arg) == 2):
raise ValueError("Each argument should be of form [state,prob]"
" or ( state, prob )")
return args
def states(self):
"""Return list of all states.
Examples
========
>>> from sympy.physics.quantum.state import Ket
>>> from sympy.physics.quantum.density import Density
>>> d = Density([Ket(0), 0.5], [Ket(1),0.5])
>>> d.states()
(|0>, |1>)
"""
return Tuple(*[arg[0] for arg in self.args])
def probs(self):
"""Return list of all probabilities.
Examples
========
>>> from sympy.physics.quantum.state import Ket
>>> from sympy.physics.quantum.density import Density
>>> d = Density([Ket(0), 0.5], [Ket(1),0.5])
>>> d.probs()
(0.5, 0.5)
"""
return Tuple(*[arg[1] for arg in self.args])
def get_state(self, index):
"""Return specific state by index.
Parameters
==========
index : index of state to be returned
Examples
========
>>> from sympy.physics.quantum.state import Ket
>>> from sympy.physics.quantum.density import Density
>>> d = Density([Ket(0), 0.5], [Ket(1),0.5])
>>> d.states()[1]
|1>
"""
state = self.args[index][0]
return state
def get_prob(self, index):
"""Return probability of specific state by index.
Parameters
===========
index : index of states whose probability is returned.
Examples
========
>>> from sympy.physics.quantum.state import Ket
>>> from sympy.physics.quantum.density import Density
>>> d = Density([Ket(0), 0.5], [Ket(1),0.5])
>>> d.probs()[1]
0.500000000000000
"""
prob = self.args[index][1]
return prob
def apply_op(self, op):
"""op will operate on each individual state.
Parameters
==========
op : Operator
Examples
========
>>> from sympy.physics.quantum.state import Ket
>>> from sympy.physics.quantum.density import Density
>>> from sympy.physics.quantum.operator import Operator
>>> A = Operator('A')
>>> d = Density([Ket(0), 0.5], [Ket(1),0.5])
>>> d.apply_op(A)
'Density'((A*|0>, 0.5),(A*|1>, 0.5))
"""
new_args = [(op*state, prob) for (state, prob) in self.args]
return Density(*new_args)
def doit(self, **hints):
"""Expand the density operator into an outer product format.
Examples
========
>>> from sympy.physics.quantum.state import Ket
>>> from sympy.physics.quantum.density import Density
>>> from sympy.physics.quantum.operator import Operator
>>> A = Operator('A')
>>> d = Density([Ket(0), 0.5], [Ket(1),0.5])
>>> d.doit()
0.5*|0><0| + 0.5*|1><1|
"""
terms = []
for (state, prob) in self.args:
state = state.expand() # needed to break up (a+b)*c
if (isinstance(state, Add)):
for arg in product(state.args, repeat=2):
terms.append(prob *
self._generate_outer_prod(arg[0], arg[1]))
else:
terms.append(prob *
self._generate_outer_prod(state, state))
return Add(*terms)
def _generate_outer_prod(self, arg1, arg2):
c_part1, nc_part1 = arg1.args_cnc()
c_part2, nc_part2 = arg2.args_cnc()
if ( len(nc_part1) == 0 or
len(nc_part2) == 0 ):
raise ValueError('Atleast one-pair of'
' Non-commutative instance required'
' for outer product.')
# Muls of Tensor Products should be expanded
# before this function is called
if (isinstance(nc_part1[0], TensorProduct) and
len(nc_part1) == 1 and len(nc_part2) == 1):
op = tensor_product_simp(nc_part1[0] * Dagger(nc_part2[0]))
else:
op = Mul(*nc_part1) * Dagger(Mul(*nc_part2))
return Mul(*c_part1)*Mul(*c_part2)*op
def _represent(self, **options):
return represent(self.doit(), **options)
def _print_operator_name_latex(self, printer, *args):
return printer._print(r'\rho', *args)
def _print_operator_name_pretty(self, printer, *args):
return prettyForm(unichr('\N{GREEK SMALL LETTER RHO}'))
def _eval_trace(self, **kwargs):
indices = kwargs.get('indices', [])
return Tr(self.doit(), indices).doit()
def entropy(self):
""" Compute the entropy of a density matrix.
Refer to density.entropy() method for examples.
"""
return entropy(self)
def entropy(density):
"""Compute the entropy of a matrix/density object.
This computes -Tr(density*ln(density)) using the eigenvalue decomposition
of density, which is given as either a Density instance or a matrix
(numpy.ndarray, sympy.Matrix or scipy.sparse).
Parameters
==========
density : density matrix of type Density, sympy matrix,
scipy.sparse or numpy.ndarray
Examples
========
>>> from sympy.physics.quantum.density import Density, entropy
>>> from sympy.physics.quantum.represent import represent
>>> from sympy.physics.quantum.matrixutils import scipy_sparse_matrix
>>> from sympy.physics.quantum.spin import JzKet, Jz
>>> from sympy import S, log
>>> up = JzKet(S(1)/2,S(1)/2)
>>> down = JzKet(S(1)/2,-S(1)/2)
>>> d = Density((up,S(1)/2),(down,S(1)/2))
>>> entropy(d)
log(2)/2
"""
if isinstance(density, Density):
density = represent(density) # represent in Matrix
if isinstance(density, scipy_sparse_matrix):
density = to_numpy(density)
if isinstance(density, Matrix):
eigvals = density.eigenvals().keys()
return expand(-sum(e*log(e) for e in eigvals))
elif isinstance(density, numpy_ndarray):
import numpy as np
eigvals = np.linalg.eigvals(density)
return -np.sum(eigvals*np.log(eigvals))
else:
raise ValueError(
"numpy.ndarray, scipy.sparse or sympy matrix expected")
def fidelity(state1, state2):
""" Computes the fidelity [1]_ between two quantum states
The arguments provided to this function should be a square matrix or a
Density object. If it is a square matrix, it is assumed to be diagonalizable.
Parameters
==========
state1, state2 : a density matrix or Matrix
Examples
========
>>> from sympy import S, sqrt
>>> from sympy.physics.quantum.dagger import Dagger
>>> from sympy.physics.quantum.spin import JzKet
>>> from sympy.physics.quantum.density import Density, fidelity
>>> from sympy.physics.quantum.represent import represent
>>>
>>> up = JzKet(S(1)/2,S(1)/2)
>>> down = JzKet(S(1)/2,-S(1)/2)
>>> amp = 1/sqrt(2)
>>> updown = (amp * up) + (amp * down)
>>>
>>> # represent turns Kets into matrices
>>> up_dm = represent(up * Dagger(up))
>>> down_dm = represent(down * Dagger(down))
>>> updown_dm = represent(updown * Dagger(updown))
>>>
>>> fidelity(up_dm, up_dm)
1
>>> fidelity(up_dm, down_dm) #orthogonal states
0
>>> fidelity(up_dm, updown_dm).evalf().round(3)
0.707
References
==========
.. [1] https://en.wikipedia.org/wiki/Fidelity_of_quantum_states
"""
state1 = represent(state1) if isinstance(state1, Density) else state1
state2 = represent(state2) if isinstance(state2, Density) else state2
if (not isinstance(state1, Matrix) or
not isinstance(state2, Matrix)):
raise ValueError("state1 and state2 must be of type Density or Matrix "
"received type=%s for state1 and type=%s for state2" %
(type(state1), type(state2)))
if ( state1.shape != state2.shape and state1.is_square):
raise ValueError("The dimensions of both args should be equal and the "
"matrix obtained should be a square matrix")
sqrt_state1 = state1**Rational(1, 2)
return Tr((sqrt_state1 * state2 * sqrt_state1)**Rational(1, 2)).doit()
|
10b99f1782a286959528ab18712ae4418c9a4519e2e3d4c0ac1828d30b513742
|
"""Logic for applying operators to states.
Todo:
* Sometimes the final result needs to be expanded, we should do this by hand.
"""
from __future__ import print_function, division
from sympy import Add, Mul, Pow, sympify, S
from sympy.core.compatibility import range
from sympy.physics.quantum.anticommutator import AntiCommutator
from sympy.physics.quantum.commutator import Commutator
from sympy.physics.quantum.dagger import Dagger
from sympy.physics.quantum.innerproduct import InnerProduct
from sympy.physics.quantum.operator import OuterProduct, Operator
from sympy.physics.quantum.state import State, KetBase, BraBase, Wavefunction
from sympy.physics.quantum.tensorproduct import TensorProduct
__all__ = [
'qapply'
]
#-----------------------------------------------------------------------------
# Main code
#-----------------------------------------------------------------------------
def qapply(e, **options):
"""Apply operators to states in a quantum expression.
Parameters
==========
e : Expr
The expression containing operators and states. This expression tree
will be walked to find operators acting on states symbolically.
options : dict
A dict of key/value pairs that determine how the operator actions
are carried out.
The following options are valid:
* ``dagger``: try to apply Dagger operators to the left
(default: False).
* ``ip_doit``: call ``.doit()`` in inner products when they are
encountered (default: True).
Returns
=======
e : Expr
The original expression, but with the operators applied to states.
Examples
========
>>> from sympy.physics.quantum import qapply, Ket, Bra
>>> b = Bra('b')
>>> k = Ket('k')
>>> A = k * b
>>> A
|k><b|
>>> qapply(A * b.dual / (b * b.dual))
|k>
>>> qapply(k.dual * A / (k.dual * k), dagger=True)
<b|
>>> qapply(k.dual * A / (k.dual * k))
<k|*|k><b|/<k|k>
"""
from sympy.physics.quantum.density import Density
dagger = options.get('dagger', False)
if e == 0:
return S.Zero
# This may be a bit aggressive but ensures that everything gets expanded
# to its simplest form before trying to apply operators. This includes
# things like (A+B+C)*|a> and A*(|a>+|b>) and all Commutators and
# TensorProducts. The only problem with this is that if we can't apply
# all the Operators, we have just expanded everything.
# TODO: don't expand the scalars in front of each Mul.
e = e.expand(commutator=True, tensorproduct=True)
# If we just have a raw ket, return it.
if isinstance(e, KetBase):
return e
# We have an Add(a, b, c, ...) and compute
# Add(qapply(a), qapply(b), ...)
elif isinstance(e, Add):
result = 0
for arg in e.args:
result += qapply(arg, **options)
return result.expand()
# For a Density operator call qapply on its state
elif isinstance(e, Density):
new_args = [(qapply(state, **options), prob) for (state,
prob) in e.args]
return Density(*new_args)
# For a raw TensorProduct, call qapply on its args.
elif isinstance(e, TensorProduct):
return TensorProduct(*[qapply(t, **options) for t in e.args])
# For a Pow, call qapply on its base.
elif isinstance(e, Pow):
return qapply(e.base, **options)**e.exp
# We have a Mul where there might be actual operators to apply to kets.
elif isinstance(e, Mul):
c_part, nc_part = e.args_cnc()
c_mul = Mul(*c_part)
nc_mul = Mul(*nc_part)
if isinstance(nc_mul, Mul):
result = c_mul*qapply_Mul(nc_mul, **options)
else:
result = c_mul*qapply(nc_mul, **options)
if result == e and dagger:
return Dagger(qapply_Mul(Dagger(e), **options))
else:
return result
# In all other cases (State, Operator, Pow, Commutator, InnerProduct,
# OuterProduct) we won't ever have operators to apply to kets.
else:
return e
def qapply_Mul(e, **options):
ip_doit = options.get('ip_doit', True)
args = list(e.args)
# If we only have 0 or 1 args, we have nothing to do and return.
if len(args) <= 1 or not isinstance(e, Mul):
return e
rhs = args.pop()
lhs = args.pop()
# Make sure we have two non-commutative objects before proceeding.
if (sympify(rhs).is_commutative and not isinstance(rhs, Wavefunction)) or \
(sympify(lhs).is_commutative and not isinstance(lhs, Wavefunction)):
return e
# For a Pow with an integer exponent, apply one of them and reduce the
# exponent by one.
if isinstance(lhs, Pow) and lhs.exp.is_Integer:
args.append(lhs.base**(lhs.exp - 1))
lhs = lhs.base
# Pull OuterProduct apart
if isinstance(lhs, OuterProduct):
args.append(lhs.ket)
lhs = lhs.bra
# Call .doit() on Commutator/AntiCommutator.
if isinstance(lhs, (Commutator, AntiCommutator)):
comm = lhs.doit()
if isinstance(comm, Add):
return qapply(
e.func(*(args + [comm.args[0], rhs])) +
e.func(*(args + [comm.args[1], rhs])),
**options
)
else:
return qapply(e.func(*args)*comm*rhs, **options)
# Apply tensor products of operators to states
if isinstance(lhs, TensorProduct) and all([isinstance(arg, (Operator, State, Mul, Pow)) or arg == 1 for arg in lhs.args]) and \
isinstance(rhs, TensorProduct) and all([isinstance(arg, (Operator, State, Mul, Pow)) or arg == 1 for arg in rhs.args]) and \
len(lhs.args) == len(rhs.args):
result = TensorProduct(*[qapply(lhs.args[n]*rhs.args[n], **options) for n in range(len(lhs.args))]).expand(tensorproduct=True)
return qapply_Mul(e.func(*args), **options)*result
# Now try to actually apply the operator and build an inner product.
try:
result = lhs._apply_operator(rhs, **options)
except (NotImplementedError, AttributeError):
try:
result = rhs._apply_operator(lhs, **options)
except (NotImplementedError, AttributeError):
if isinstance(lhs, BraBase) and isinstance(rhs, KetBase):
result = InnerProduct(lhs, rhs)
if ip_doit:
result = result.doit()
else:
result = None
# TODO: I may need to expand before returning the final result.
if result == 0:
return S.Zero
elif result is None:
if len(args) == 0:
# We had two args to begin with so args=[].
return e
else:
return qapply_Mul(e.func(*(args + [lhs])), **options)*rhs
elif isinstance(result, InnerProduct):
return result*qapply_Mul(e.func(*args), **options)
else: # result is a scalar times a Mul, Add or TensorProduct
return qapply(e.func(*args)*result, **options)
|
a0ac9c7699afb0878b64fe16ed328dddba1525381c449fd995043e222558b486
|
from sympy.core.backend import eye, Matrix, zeros
from sympy.physics.mechanics import dynamicsymbols
from sympy.physics.mechanics.functions import find_dynamicsymbols
__all__ = ['SymbolicSystem']
class SymbolicSystem(object):
"""SymbolicSystem is a class that contains all the information about a
system in a symbolic format such as the equations of motions and the bodies
and loads in the system.
There are three ways that the equations of motion can be described for
Symbolic System:
[1] Explicit form where the kinematics and dynamics are combined
x' = F_1(x, t, r, p)
[2] Implicit form where the kinematics and dynamics are combined
M_2(x, p) x' = F_2(x, t, r, p)
[3] Implicit form where the kinematics and dynamics are separate
M_3(q, p) u' = F_3(q, u, t, r, p)
q' = G(q, u, t, r, p)
where
x : states, e.g. [q, u]
t : time
r : specified (exogenous) inputs
p : constants
q : generalized coordinates
u : generalized speeds
F_1 : right hand side of the combined equations in explicit form
F_2 : right hand side of the combined equations in implicit form
F_3 : right hand side of the dynamical equations in implicit form
M_2 : mass matrix of the combined equations in implicit form
M_3 : mass matrix of the dynamical equations in implicit form
G : right hand side of the kinematical differential equations
Parameters
==========
coord_states : ordered iterable of functions of time
This input will either be a collection of the coordinates or states
of the system depending on whether or not the speeds are also
given. If speeds are specified this input will be assumed to
be the coordinates otherwise this input will be assumed to
be the states.
right_hand_side : Matrix
This variable is the right hand side of the equations of motion in
any of the forms. The specific form will be assumed depending on
whether a mass matrix or coordinate derivatives are given.
speeds : ordered iterable of functions of time, optional
This is a collection of the generalized speeds of the system. If
given it will be assumed that the first argument (coord_states)
will represent the generalized coordinates of the system.
mass_matrix : Matrix, optional
The matrix of the implicit forms of the equations of motion (forms
[2] and [3]). The distinction between the forms is determined by
whether or not the coordinate derivatives are passed in. If
they are given form [3] will be assumed otherwise form [2] is
assumed.
coordinate_derivatives : Matrix, optional
The right hand side of the kinematical equations in explicit form.
If given it will be assumed that the equations of motion are being
entered in form [3].
alg_con : Iterable, optional
The indexes of the rows in the equations of motion that contain
algebraic constraints instead of differential equations. If the
equations are input in form [3], it will be assumed the indexes are
referencing the mass_matrix/right_hand_side combination and not the
coordinate_derivatives.
output_eqns : Dictionary, optional
Any output equations that are desired to be tracked are stored in a
dictionary where the key corresponds to the name given for the
specific equation and the value is the equation itself in symbolic
form
coord_idxs : Iterable, optional
If coord_states corresponds to the states rather than the
coordinates this variable will tell SymbolicSystem which indexes of
the states correspond to generalized coordinates.
speed_idxs : Iterable, optional
If coord_states corresponds to the states rather than the
coordinates this variable will tell SymbolicSystem which indexes of
the states correspond to generalized speeds.
bodies : iterable of Body/Rigidbody objects, optional
Iterable containing the bodies of the system
loads : iterable of load instances (described below), optional
Iterable containing the loads of the system where forces are given
by (point of application, force vector) and torques are given by
(reference frame acting upon, torque vector). Ex [(point, force),
(ref_frame, torque)]
Attributes
==========
coordinates : Matrix, shape(n, 1)
This is a matrix containing the generalized coordinates of the system
speeds : Matrix, shape(m, 1)
This is a matrix containing the generalized speeds of the system
states : Matrix, shape(o, 1)
This is a matrix containing the state variables of the system
alg_con : List
This list contains the indices of the algebraic constraints in the
combined equations of motion. The presence of these constraints
requires that a DAE solver be used instead of an ODE solver.
If the system is given in form [3] the alg_con variable will be
adjusted such that it is a representation of the combined kinematics
and dynamics thus make sure it always matches the mass matrix
entered.
dyn_implicit_mat : Matrix, shape(m, m)
This is the M matrix in form [3] of the equations of motion (the mass
matrix or generalized inertia matrix of the dynamical equations of
motion in implicit form).
dyn_implicit_rhs : Matrix, shape(m, 1)
This is the F vector in form [3] of the equations of motion (the right
hand side of the dynamical equations of motion in implicit form).
comb_implicit_mat : Matrix, shape(o, o)
This is the M matrix in form [2] of the equations of motion.
This matrix contains a block diagonal structure where the top
left block (the first rows) represent the matrix in the
implicit form of the kinematical equations and the bottom right
block (the last rows) represent the matrix in the implicit form
of the dynamical equations.
comb_implicit_rhs : Matrix, shape(o, 1)
This is the F vector in form [2] of the equations of motion. The top
part of the vector represents the right hand side of the implicit form
of the kinemaical equations and the bottom of the vector represents the
right hand side of the implicit form of the dynamical equations of
motion.
comb_explicit_rhs : Matrix, shape(o, 1)
This vector represents the right hand side of the combined equations of
motion in explicit form (form [1] from above).
kin_explicit_rhs : Matrix, shape(m, 1)
This is the right hand side of the explicit form of the kinematical
equations of motion as can be seen in form [3] (the G matrix).
output_eqns : Dictionary
If output equations were given they are stored in a dictionary where
the key corresponds to the name given for the specific equation and
the value is the equation itself in symbolic form
bodies : Tuple
If the bodies in the system were given they are stored in a tuple for
future access
loads : Tuple
If the loads in the system were given they are stored in a tuple for
future access. This includes forces and torques where forces are given
by (point of application, force vector) and torques are given by
(reference frame acted upon, torque vector).
Example
=======
As a simple example, the dynamics of a simple pendulum will be input into a
SymbolicSystem object manually. First some imports will be needed and then
symbols will be set up for the length of the pendulum (l), mass at the end
of the pendulum (m), and a constant for gravity (g). ::
>>> from sympy import Matrix, sin, symbols
>>> from sympy.physics.mechanics import dynamicsymbols, SymbolicSystem
>>> l, m, g = symbols('l m g')
The system will be defined by an angle of theta from the vertical and a
generalized speed of omega will be used where omega = theta_dot. ::
>>> theta, omega = dynamicsymbols('theta omega')
Now the equations of motion are ready to be formed and passed to the
SymbolicSystem object. ::
>>> kin_explicit_rhs = Matrix([omega])
>>> dyn_implicit_mat = Matrix([l**2 * m])
>>> dyn_implicit_rhs = Matrix([-g * l * m * sin(theta)])
>>> symsystem = SymbolicSystem([theta], dyn_implicit_rhs, [omega],
... dyn_implicit_mat)
Notes
=====
m : number of generalized speeds
n : number of generalized coordinates
o : number of states
"""
def __init__(self, coord_states, right_hand_side, speeds=None,
mass_matrix=None, coordinate_derivatives=None, alg_con=None,
output_eqns={}, coord_idxs=None, speed_idxs=None, bodies=None,
loads=None):
"""Initializes a SymbolicSystem object"""
# Extract information on speeds, coordinates and states
if speeds is None:
self._states = Matrix(coord_states)
if coord_idxs is None:
self._coordinates = None
else:
coords = [coord_states[i] for i in coord_idxs]
self._coordinates = Matrix(coords)
if speed_idxs is None:
self._speeds = None
else:
speeds_inter = [coord_states[i] for i in speed_idxs]
self._speeds = Matrix(speeds_inter)
else:
self._coordinates = Matrix(coord_states)
self._speeds = Matrix(speeds)
self._states = self._coordinates.col_join(self._speeds)
# Extract equations of motion form
if coordinate_derivatives is not None:
self._kin_explicit_rhs = coordinate_derivatives
self._dyn_implicit_rhs = right_hand_side
self._dyn_implicit_mat = mass_matrix
self._comb_implicit_rhs = None
self._comb_implicit_mat = None
self._comb_explicit_rhs = None
elif mass_matrix is not None:
self._kin_explicit_rhs = None
self._dyn_implicit_rhs = None
self._dyn_implicit_mat = None
self._comb_implicit_rhs = right_hand_side
self._comb_implicit_mat = mass_matrix
self._comb_explicit_rhs = None
else:
self._kin_explicit_rhs = None
self._dyn_implicit_rhs = None
self._dyn_implicit_mat = None
self._comb_implicit_rhs = None
self._comb_implicit_mat = None
self._comb_explicit_rhs = right_hand_side
# Set the remainder of the inputs as instance attributes
if alg_con is not None and coordinate_derivatives is not None:
alg_con = [i + len(coordinate_derivatives) for i in alg_con]
self._alg_con = alg_con
self.output_eqns = output_eqns
# Change the body and loads iterables to tuples if they are not tuples
# already
if type(bodies) != tuple and bodies is not None:
bodies = tuple(bodies)
if type(loads) != tuple and loads is not None:
loads = tuple(loads)
self._bodies = bodies
self._loads = loads
@property
def coordinates(self):
"""Returns the column matrix of the generalized coordinates"""
if self._coordinates is None:
raise AttributeError("The coordinates were not specified.")
else:
return self._coordinates
@property
def speeds(self):
"""Returns the column matrix of generalized speeds"""
if self._speeds is None:
raise AttributeError("The speeds were not specified.")
else:
return self._speeds
@property
def states(self):
"""Returns the column matrix of the state variables"""
return self._states
@property
def alg_con(self):
"""Returns a list with the indices of the rows containing algebraic
constraints in the combined form of the equations of motion"""
return self._alg_con
@property
def dyn_implicit_mat(self):
"""Returns the matrix, M, corresponding to the dynamic equations in
implicit form, M x' = F, where the kinematical equations are not
included"""
if self._dyn_implicit_mat is None:
raise AttributeError("dyn_implicit_mat is not specified for "
"equations of motion form [1] or [2].")
else:
return self._dyn_implicit_mat
@property
def dyn_implicit_rhs(self):
"""Returns the column matrix, F, corresponding to the dynamic equations
in implicit form, M x' = F, where the kinematical equations are not
included"""
if self._dyn_implicit_rhs is None:
raise AttributeError("dyn_implicit_rhs is not specified for "
"equations of motion form [1] or [2].")
else:
return self._dyn_implicit_rhs
@property
def comb_implicit_mat(self):
"""Returns the matrix, M, corresponding to the equations of motion in
implicit form (form [2]), M x' = F, where the kinematical equations are
included"""
if self._comb_implicit_mat is None:
if self._dyn_implicit_mat is not None:
num_kin_eqns = len(self._kin_explicit_rhs)
num_dyn_eqns = len(self._dyn_implicit_rhs)
zeros1 = zeros(num_kin_eqns, num_dyn_eqns)
zeros2 = zeros(num_dyn_eqns, num_kin_eqns)
inter1 = eye(num_kin_eqns).row_join(zeros1)
inter2 = zeros2.row_join(self._dyn_implicit_mat)
self._comb_implicit_mat = inter1.col_join(inter2)
return self._comb_implicit_mat
else:
raise AttributeError("comb_implicit_mat is not specified for "
"equations of motion form [1].")
else:
return self._comb_implicit_mat
@property
def comb_implicit_rhs(self):
"""Returns the column matrix, F, corresponding to the equations of
motion in implicit form (form [2]), M x' = F, where the kinematical
equations are included"""
if self._comb_implicit_rhs is None:
if self._dyn_implicit_rhs is not None:
kin_inter = self._kin_explicit_rhs
dyn_inter = self._dyn_implicit_rhs
self._comb_implicit_rhs = kin_inter.col_join(dyn_inter)
return self._comb_implicit_rhs
else:
raise AttributeError("comb_implicit_mat is not specified for "
"equations of motion in form [1].")
else:
return self._comb_implicit_rhs
def compute_explicit_form(self):
"""If the explicit right hand side of the combined equations of motion
is to provided upon initialization, this method will calculate it. This
calculation can potentially take awhile to compute."""
if self._comb_explicit_rhs is not None:
raise AttributeError("comb_explicit_rhs is already formed.")
inter1 = getattr(self, 'kin_explicit_rhs', None)
if inter1 is not None:
inter2 = self._dyn_implicit_mat.LUsolve(self._dyn_implicit_rhs)
out = inter1.col_join(inter2)
else:
out = self._comb_implicit_mat.LUsolve(self._comb_implicit_rhs)
self._comb_explicit_rhs = out
@property
def comb_explicit_rhs(self):
"""Returns the right hand side of the equations of motion in explicit
form, x' = F, where the kinematical equations are included"""
if self._comb_explicit_rhs is None:
raise AttributeError("Please run .combute_explicit_form before "
"attempting to access comb_explicit_rhs.")
else:
return self._comb_explicit_rhs
@property
def kin_explicit_rhs(self):
"""Returns the right hand side of the kinematical equations in explicit
form, q' = G"""
if self._kin_explicit_rhs is None:
raise AttributeError("kin_explicit_rhs is not specified for "
"equations of motion form [1] or [2].")
else:
return self._kin_explicit_rhs
def dynamic_symbols(self):
"""Returns a column matrix containing all of the symbols in the system
that depend on time"""
# Create a list of all of the expressions in the equations of motion
if self._comb_explicit_rhs is None:
eom_expressions = (self.comb_implicit_mat[:] +
self.comb_implicit_rhs[:])
else:
eom_expressions = (self._comb_explicit_rhs[:])
functions_of_time = set()
for expr in eom_expressions:
functions_of_time = functions_of_time.union(
find_dynamicsymbols(expr))
functions_of_time = functions_of_time.union(self._states)
return tuple(functions_of_time)
def constant_symbols(self):
"""Returns a column matrix containing all of the symbols in the system
that do not depend on time"""
# Create a list of all of the expressions in the equations of motion
if self._comb_explicit_rhs is None:
eom_expressions = (self.comb_implicit_mat[:] +
self.comb_implicit_rhs[:])
else:
eom_expressions = (self._comb_explicit_rhs[:])
constants = set()
for expr in eom_expressions:
constants = constants.union(expr.free_symbols)
constants.remove(dynamicsymbols._t)
return tuple(constants)
@property
def bodies(self):
"""Returns the bodies in the system"""
if self._bodies is None:
raise AttributeError("bodies were not specified for the system.")
else:
return self._bodies
@property
def loads(self):
"""Returns the loads in the system"""
if self._loads is None:
raise AttributeError("loads were not specified for the system.")
else:
return self._loads
|
9172c37e6377b1d9912c5affa50999efb91eb2e8ce0d54dd5612b56c8b778355
|
#!/usr/bin/env python
"""This module contains some sample symbolic models used for testing and
examples."""
# Internal imports
from sympy.core import backend as sm
import sympy.physics.mechanics as me
def multi_mass_spring_damper(n=1, apply_gravity=False,
apply_external_forces=False):
r"""Returns a system containing the symbolic equations of motion and
associated variables for a simple multi-degree of freedom point mass,
spring, damper system with optional gravitational and external
specified forces. For example, a two mass system under the influence of
gravity and external forces looks like:
::
----------------
| | | | g
\ | | | V
k0 / --- c0 |
| | | x0, v0
--------- V
| m0 | -----
--------- |
| | | |
\ v | | |
k1 / f0 --- c1 |
| | | x1, v1
--------- V
| m1 | -----
---------
| f1
V
Parameters
----------
n : integer
The number of masses in the serial chain.
apply_gravity : boolean
If true, gravity will be applied to each mass.
apply_external_forces : boolean
If true, a time varying external force will be applied to each mass.
Returns
-------
kane : sympy.physics.mechanics.kane.KanesMethod
A KanesMethod object.
"""
mass = sm.symbols('m:{}'.format(n))
stiffness = sm.symbols('k:{}'.format(n))
damping = sm.symbols('c:{}'.format(n))
acceleration_due_to_gravity = sm.symbols('g')
coordinates = me.dynamicsymbols('x:{}'.format(n))
speeds = me.dynamicsymbols('v:{}'.format(n))
specifieds = me.dynamicsymbols('f:{}'.format(n))
ceiling = me.ReferenceFrame('N')
origin = me.Point('origin')
origin.set_vel(ceiling, 0)
points = [origin]
kinematic_equations = []
particles = []
forces = []
for i in range(n):
center = points[-1].locatenew('center{}'.format(i),
coordinates[i] * ceiling.x)
center.set_vel(ceiling, points[-1].vel(ceiling) +
speeds[i] * ceiling.x)
points.append(center)
block = me.Particle('block{}'.format(i), center, mass[i])
kinematic_equations.append(speeds[i] - coordinates[i].diff())
total_force = (-stiffness[i] * coordinates[i] -
damping[i] * speeds[i])
try:
total_force += (stiffness[i + 1] * coordinates[i + 1] +
damping[i + 1] * speeds[i + 1])
except IndexError: # no force from below on last mass
pass
if apply_gravity:
total_force += mass[i] * acceleration_due_to_gravity
if apply_external_forces:
total_force += specifieds[i]
forces.append((center, total_force * ceiling.x))
particles.append(block)
kane = me.KanesMethod(ceiling, q_ind=coordinates, u_ind=speeds,
kd_eqs=kinematic_equations)
kane.kanes_equations(particles, forces)
return kane
def n_link_pendulum_on_cart(n=1, cart_force=True, joint_torques=False):
r"""Returns the system containing the symbolic first order equations of
motion for a 2D n-link pendulum on a sliding cart under the influence of
gravity.
::
|
o y v
\ 0 ^ g
\ |
--\-|----
| \| |
F-> | o --|---> x
| |
---------
o o
Parameters
----------
n : integer
The number of links in the pendulum.
cart_force : boolean, default=True
If true an external specified lateral force is applied to the cart.
joint_torques : boolean, default=False
If true joint torques will be added as specified inputs at each
joint.
Returns
-------
kane : sympy.physics.mechanics.kane.KanesMethod
A KanesMethod object.
Notes
-----
The degrees of freedom of the system are n + 1, i.e. one for each
pendulum link and one for the lateral motion of the cart.
M x' = F, where x = [u0, ..., un+1, q0, ..., qn+1]
The joint angles are all defined relative to the ground where the x axis
defines the ground line and the y axis points up. The joint torques are
applied between each adjacent link and the between the cart and the
lower link where a positive torque corresponds to positive angle.
"""
if n <= 0:
raise ValueError('The number of links must be a positive integer.')
q = me.dynamicsymbols('q:{}'.format(n + 1))
u = me.dynamicsymbols('u:{}'.format(n + 1))
if joint_torques is True:
T = me.dynamicsymbols('T1:{}'.format(n + 1))
m = sm.symbols('m:{}'.format(n + 1))
l = sm.symbols('l:{}'.format(n))
g, t = sm.symbols('g t')
I = me.ReferenceFrame('I')
O = me.Point('O')
O.set_vel(I, 0)
P0 = me.Point('P0')
P0.set_pos(O, q[0] * I.x)
P0.set_vel(I, u[0] * I.x)
Pa0 = me.Particle('Pa0', P0, m[0])
frames = [I]
points = [P0]
particles = [Pa0]
forces = [(P0, -m[0] * g * I.y)]
kindiffs = [q[0].diff(t) - u[0]]
if cart_force is True or joint_torques is True:
specified = []
else:
specified = None
for i in range(n):
Bi = I.orientnew('B{}'.format(i), 'Axis', [q[i + 1], I.z])
Bi.set_ang_vel(I, u[i + 1] * I.z)
frames.append(Bi)
Pi = points[-1].locatenew('P{}'.format(i + 1), l[i] * Bi.y)
Pi.v2pt_theory(points[-1], I, Bi)
points.append(Pi)
Pai = me.Particle('Pa' + str(i + 1), Pi, m[i + 1])
particles.append(Pai)
forces.append((Pi, -m[i + 1] * g * I.y))
if joint_torques is True:
specified.append(T[i])
if i == 0:
forces.append((I, -T[i] * I.z))
if i == n - 1:
forces.append((Bi, T[i] * I.z))
else:
forces.append((Bi, T[i] * I.z - T[i + 1] * I.z))
kindiffs.append(q[i + 1].diff(t) - u[i + 1])
if cart_force is True:
F = me.dynamicsymbols('F')
forces.append((P0, F * I.x))
specified.append(F)
kane = me.KanesMethod(I, q_ind=q, u_ind=u, kd_eqs=kindiffs)
kane.kanes_equations(particles, forces)
return kane
|
563cc958c0c3aa4703299f9ffa23dcca6843dcb9608ac766d82027042bc802e7
|
from __future__ import print_function, division
from sympy.core.backend import zeros, Matrix, diff, eye
from sympy import solve_linear_system_LU
from sympy.core.compatibility import range
from sympy.utilities import default_sort_key
from sympy.physics.vector import (ReferenceFrame, dynamicsymbols,
partial_velocity)
from sympy.physics.mechanics.particle import Particle
from sympy.physics.mechanics.rigidbody import RigidBody
from sympy.physics.mechanics.functions import (msubs, find_dynamicsymbols,
_f_list_parser)
from sympy.physics.mechanics.linearize import Linearizer
from sympy.utilities.exceptions import SymPyDeprecationWarning
from sympy.utilities.iterables import iterable
__all__ = ['KanesMethod']
class KanesMethod(object):
"""Kane's method object.
This object is used to do the "book-keeping" as you go through and form
equations of motion in the way Kane presents in:
Kane, T., Levinson, D. Dynamics Theory and Applications. 1985 McGraw-Hill
The attributes are for equations in the form [M] udot = forcing.
Attributes
==========
q, u : Matrix
Matrices of the generalized coordinates and speeds
bodylist : iterable
Iterable of Point and RigidBody objects in the system.
forcelist : iterable
Iterable of (Point, vector) or (ReferenceFrame, vector) tuples
describing the forces on the system.
auxiliary : Matrix
If applicable, the set of auxiliary Kane's
equations used to solve for non-contributing
forces.
mass_matrix : Matrix
The system's mass matrix
forcing : Matrix
The system's forcing vector
mass_matrix_full : Matrix
The "mass matrix" for the u's and q's
forcing_full : Matrix
The "forcing vector" for the u's and q's
Examples
========
This is a simple example for a one degree of freedom translational
spring-mass-damper.
In this example, we first need to do the kinematics.
This involves creating generalized speeds and coordinates and their
derivatives.
Then we create a point and set its velocity in a frame.
>>> from sympy import symbols
>>> from sympy.physics.mechanics import dynamicsymbols, ReferenceFrame
>>> from sympy.physics.mechanics import Point, Particle, KanesMethod
>>> q, u = dynamicsymbols('q u')
>>> qd, ud = dynamicsymbols('q u', 1)
>>> m, c, k = symbols('m c k')
>>> N = ReferenceFrame('N')
>>> P = Point('P')
>>> P.set_vel(N, u * N.x)
Next we need to arrange/store information in the way that KanesMethod
requires. The kinematic differential equations need to be stored in a
dict. A list of forces/torques must be constructed, where each entry in
the list is a (Point, Vector) or (ReferenceFrame, Vector) tuple, where the
Vectors represent the Force or Torque.
Next a particle needs to be created, and it needs to have a point and mass
assigned to it.
Finally, a list of all bodies and particles needs to be created.
>>> kd = [qd - u]
>>> FL = [(P, (-k * q - c * u) * N.x)]
>>> pa = Particle('pa', P, m)
>>> BL = [pa]
Finally we can generate the equations of motion.
First we create the KanesMethod object and supply an inertial frame,
coordinates, generalized speeds, and the kinematic differential equations.
Additional quantities such as configuration and motion constraints,
dependent coordinates and speeds, and auxiliary speeds are also supplied
here (see the online documentation).
Next we form FR* and FR to complete: Fr + Fr* = 0.
We have the equations of motion at this point.
It makes sense to rearrange them though, so we calculate the mass matrix and
the forcing terms, for E.o.M. in the form: [MM] udot = forcing, where MM is
the mass matrix, udot is a vector of the time derivatives of the
generalized speeds, and forcing is a vector representing "forcing" terms.
>>> KM = KanesMethod(N, q_ind=[q], u_ind=[u], kd_eqs=kd)
>>> (fr, frstar) = KM.kanes_equations(BL, FL)
>>> MM = KM.mass_matrix
>>> forcing = KM.forcing
>>> rhs = MM.inv() * forcing
>>> rhs
Matrix([[(-c*u(t) - k*q(t))/m]])
>>> KM.linearize(A_and_B=True)[0]
Matrix([
[ 0, 1],
[-k/m, -c/m]])
Please look at the documentation pages for more information on how to
perform linearization and how to deal with dependent coordinates & speeds,
and how do deal with bringing non-contributing forces into evidence.
"""
def __init__(self, frame, q_ind, u_ind, kd_eqs=None, q_dependent=None,
configuration_constraints=None, u_dependent=None,
velocity_constraints=None, acceleration_constraints=None,
u_auxiliary=None):
"""Please read the online documentation. """
if not q_ind:
q_ind = [dynamicsymbols('dummy_q')]
kd_eqs = [dynamicsymbols('dummy_kd')]
if not isinstance(frame, ReferenceFrame):
raise TypeError('An intertial ReferenceFrame must be supplied')
self._inertial = frame
self._fr = None
self._frstar = None
self._forcelist = None
self._bodylist = None
self._initialize_vectors(q_ind, q_dependent, u_ind, u_dependent,
u_auxiliary)
self._initialize_kindiffeq_matrices(kd_eqs)
self._initialize_constraint_matrices(configuration_constraints,
velocity_constraints, acceleration_constraints)
def _initialize_vectors(self, q_ind, q_dep, u_ind, u_dep, u_aux):
"""Initialize the coordinate and speed vectors."""
none_handler = lambda x: Matrix(x) if x else Matrix()
# Initialize generalized coordinates
q_dep = none_handler(q_dep)
if not iterable(q_ind):
raise TypeError('Generalized coordinates must be an iterable.')
if not iterable(q_dep):
raise TypeError('Dependent coordinates must be an iterable.')
q_ind = Matrix(q_ind)
self._qdep = q_dep
self._q = Matrix([q_ind, q_dep])
self._qdot = self.q.diff(dynamicsymbols._t)
# Initialize generalized speeds
u_dep = none_handler(u_dep)
if not iterable(u_ind):
raise TypeError('Generalized speeds must be an iterable.')
if not iterable(u_dep):
raise TypeError('Dependent speeds must be an iterable.')
u_ind = Matrix(u_ind)
self._udep = u_dep
self._u = Matrix([u_ind, u_dep])
self._udot = self.u.diff(dynamicsymbols._t)
self._uaux = none_handler(u_aux)
def _initialize_constraint_matrices(self, config, vel, acc):
"""Initializes constraint matrices."""
# Define vector dimensions
o = len(self.u)
m = len(self._udep)
p = o - m
none_handler = lambda x: Matrix(x) if x else Matrix()
# Initialize configuration constraints
config = none_handler(config)
if len(self._qdep) != len(config):
raise ValueError('There must be an equal number of dependent '
'coordinates and configuration constraints.')
self._f_h = none_handler(config)
# Initialize velocity and acceleration constraints
vel = none_handler(vel)
acc = none_handler(acc)
if len(vel) != m:
raise ValueError('There must be an equal number of dependent '
'speeds and velocity constraints.')
if acc and (len(acc) != m):
raise ValueError('There must be an equal number of dependent '
'speeds and acceleration constraints.')
if vel:
u_zero = dict((i, 0) for i in self.u)
udot_zero = dict((i, 0) for i in self._udot)
# When calling kanes_equations, another class instance will be
# created if auxiliary u's are present. In this case, the
# computation of kinetic differential equation matrices will be
# skipped as this was computed during the original KanesMethod
# object, and the qd_u_map will not be available.
if self._qdot_u_map is not None:
vel = msubs(vel, self._qdot_u_map)
self._f_nh = msubs(vel, u_zero)
self._k_nh = (vel - self._f_nh).jacobian(self.u)
# If no acceleration constraints given, calculate them.
if not acc:
self._f_dnh = (self._k_nh.diff(dynamicsymbols._t) * self.u +
self._f_nh.diff(dynamicsymbols._t))
self._k_dnh = self._k_nh
else:
if self._qdot_u_map is not None:
acc = msubs(acc, self._qdot_u_map)
self._f_dnh = msubs(acc, udot_zero)
self._k_dnh = (acc - self._f_dnh).jacobian(self._udot)
# Form of non-holonomic constraints is B*u + C = 0.
# We partition B into independent and dependent columns:
# Ars is then -B_dep.inv() * B_ind, and it relates dependent speeds
# to independent speeds as: udep = Ars*uind, neglecting the C term.
B_ind = self._k_nh[:, :p]
B_dep = self._k_nh[:, p:o]
self._Ars = -B_dep.LUsolve(B_ind)
else:
self._f_nh = Matrix()
self._k_nh = Matrix()
self._f_dnh = Matrix()
self._k_dnh = Matrix()
self._Ars = Matrix()
def _initialize_kindiffeq_matrices(self, kdeqs):
"""Initialize the kinematic differential equation matrices."""
if kdeqs:
if len(self.q) != len(kdeqs):
raise ValueError('There must be an equal number of kinematic '
'differential equations and coordinates.')
kdeqs = Matrix(kdeqs)
u = self.u
qdot = self._qdot
# Dictionaries setting things to zero
u_zero = dict((i, 0) for i in u)
uaux_zero = dict((i, 0) for i in self._uaux)
qdot_zero = dict((i, 0) for i in qdot)
f_k = msubs(kdeqs, u_zero, qdot_zero)
k_ku = (msubs(kdeqs, qdot_zero) - f_k).jacobian(u)
k_kqdot = (msubs(kdeqs, u_zero) - f_k).jacobian(qdot)
f_k = k_kqdot.LUsolve(f_k)
k_ku = k_kqdot.LUsolve(k_ku)
k_kqdot = eye(len(qdot))
self._qdot_u_map = solve_linear_system_LU(
Matrix([k_kqdot.T, -(k_ku * u + f_k).T]).T, qdot)
self._f_k = msubs(f_k, uaux_zero)
self._k_ku = msubs(k_ku, uaux_zero)
self._k_kqdot = k_kqdot
else:
self._qdot_u_map = None
self._f_k = Matrix()
self._k_ku = Matrix()
self._k_kqdot = Matrix()
def _form_fr(self, fl):
"""Form the generalized active force."""
if fl is not None and (len(fl) == 0 or not iterable(fl)):
raise ValueError('Force pairs must be supplied in an '
'non-empty iterable or None.')
N = self._inertial
# pull out relevant velocities for constructing partial velocities
vel_list, f_list = _f_list_parser(fl, N)
vel_list = [msubs(i, self._qdot_u_map) for i in vel_list]
# Fill Fr with dot product of partial velocities and forces
o = len(self.u)
b = len(f_list)
FR = zeros(o, 1)
partials = partial_velocity(vel_list, self.u, N)
for i in range(o):
FR[i] = sum(partials[j][i] & f_list[j] for j in range(b))
# In case there are dependent speeds
if self._udep:
p = o - len(self._udep)
FRtilde = FR[:p, 0]
FRold = FR[p:o, 0]
FRtilde += self._Ars.T * FRold
FR = FRtilde
self._forcelist = fl
self._fr = FR
return FR
def _form_frstar(self, bl):
"""Form the generalized inertia force."""
if not iterable(bl):
raise TypeError('Bodies must be supplied in an iterable.')
t = dynamicsymbols._t
N = self._inertial
# Dicts setting things to zero
udot_zero = dict((i, 0) for i in self._udot)
uaux_zero = dict((i, 0) for i in self._uaux)
uauxdot = [diff(i, t) for i in self._uaux]
uauxdot_zero = dict((i, 0) for i in uauxdot)
# Dictionary of q' and q'' to u and u'
q_ddot_u_map = dict((k.diff(t), v.diff(t)) for (k, v) in
self._qdot_u_map.items())
q_ddot_u_map.update(self._qdot_u_map)
# Fill up the list of partials: format is a list with num elements
# equal to number of entries in body list. Each of these elements is a
# list - either of length 1 for the translational components of
# particles or of length 2 for the translational and rotational
# components of rigid bodies. The inner most list is the list of
# partial velocities.
def get_partial_velocity(body):
if isinstance(body, RigidBody):
vlist = [body.masscenter.vel(N), body.frame.ang_vel_in(N)]
elif isinstance(body, Particle):
vlist = [body.point.vel(N),]
else:
raise TypeError('The body list may only contain either '
'RigidBody or Particle as list elements.')
v = [msubs(vel, self._qdot_u_map) for vel in vlist]
return partial_velocity(v, self.u, N)
partials = [get_partial_velocity(body) for body in bl]
# Compute fr_star in two components:
# fr_star = -(MM*u' + nonMM)
o = len(self.u)
MM = zeros(o, o)
nonMM = zeros(o, 1)
zero_uaux = lambda expr: msubs(expr, uaux_zero)
zero_udot_uaux = lambda expr: msubs(msubs(expr, udot_zero), uaux_zero)
for i, body in enumerate(bl):
if isinstance(body, RigidBody):
M = zero_uaux(body.mass)
I = zero_uaux(body.central_inertia)
vel = zero_uaux(body.masscenter.vel(N))
omega = zero_uaux(body.frame.ang_vel_in(N))
acc = zero_udot_uaux(body.masscenter.acc(N))
inertial_force = (M.diff(t) * vel + M * acc)
inertial_torque = zero_uaux((I.dt(body.frame) & omega) +
msubs(I & body.frame.ang_acc_in(N), udot_zero) +
(omega ^ (I & omega)))
for j in range(o):
tmp_vel = zero_uaux(partials[i][0][j])
tmp_ang = zero_uaux(I & partials[i][1][j])
for k in range(o):
# translational
MM[j, k] += M * (tmp_vel & partials[i][0][k])
# rotational
MM[j, k] += (tmp_ang & partials[i][1][k])
nonMM[j] += inertial_force & partials[i][0][j]
nonMM[j] += inertial_torque & partials[i][1][j]
else:
M = zero_uaux(body.mass)
vel = zero_uaux(body.point.vel(N))
acc = zero_udot_uaux(body.point.acc(N))
inertial_force = (M.diff(t) * vel + M * acc)
for j in range(o):
temp = zero_uaux(partials[i][0][j])
for k in range(o):
MM[j, k] += M * (temp & partials[i][0][k])
nonMM[j] += inertial_force & partials[i][0][j]
# Compose fr_star out of MM and nonMM
MM = zero_uaux(msubs(MM, q_ddot_u_map))
nonMM = msubs(msubs(nonMM, q_ddot_u_map),
udot_zero, uauxdot_zero, uaux_zero)
fr_star = -(MM * msubs(Matrix(self._udot), uauxdot_zero) + nonMM)
# If there are dependent speeds, we need to find fr_star_tilde
if self._udep:
p = o - len(self._udep)
fr_star_ind = fr_star[:p, 0]
fr_star_dep = fr_star[p:o, 0]
fr_star = fr_star_ind + (self._Ars.T * fr_star_dep)
# Apply the same to MM
MMi = MM[:p, :]
MMd = MM[p:o, :]
MM = MMi + (self._Ars.T * MMd)
self._bodylist = bl
self._frstar = fr_star
self._k_d = MM
self._f_d = -msubs(self._fr + self._frstar, udot_zero)
return fr_star
def to_linearizer(self):
"""Returns an instance of the Linearizer class, initiated from the
data in the KanesMethod class. This may be more desirable than using
the linearize class method, as the Linearizer object will allow more
efficient recalculation (i.e. about varying operating points)."""
if (self._fr is None) or (self._frstar is None):
raise ValueError('Need to compute Fr, Fr* first.')
# Get required equation components. The Kane's method class breaks
# these into pieces. Need to reassemble
f_c = self._f_h
if self._f_nh and self._k_nh:
f_v = self._f_nh + self._k_nh*Matrix(self.u)
else:
f_v = Matrix()
if self._f_dnh and self._k_dnh:
f_a = self._f_dnh + self._k_dnh*Matrix(self._udot)
else:
f_a = Matrix()
# Dicts to sub to zero, for splitting up expressions
u_zero = dict((i, 0) for i in self.u)
ud_zero = dict((i, 0) for i in self._udot)
qd_zero = dict((i, 0) for i in self._qdot)
qd_u_zero = dict((i, 0) for i in Matrix([self._qdot, self.u]))
# Break the kinematic differential eqs apart into f_0 and f_1
f_0 = msubs(self._f_k, u_zero) + self._k_kqdot*Matrix(self._qdot)
f_1 = msubs(self._f_k, qd_zero) + self._k_ku*Matrix(self.u)
# Break the dynamic differential eqs into f_2 and f_3
f_2 = msubs(self._frstar, qd_u_zero)
f_3 = msubs(self._frstar, ud_zero) + self._fr
f_4 = zeros(len(f_2), 1)
# Get the required vector components
q = self.q
u = self.u
if self._qdep:
q_i = q[:-len(self._qdep)]
else:
q_i = q
q_d = self._qdep
if self._udep:
u_i = u[:-len(self._udep)]
else:
u_i = u
u_d = self._udep
# Form dictionary to set auxiliary speeds & their derivatives to 0.
uaux = self._uaux
uauxdot = uaux.diff(dynamicsymbols._t)
uaux_zero = dict((i, 0) for i in Matrix([uaux, uauxdot]))
# Checking for dynamic symbols outside the dynamic differential
# equations; throws error if there is.
sym_list = set(Matrix([q, self._qdot, u, self._udot, uaux, uauxdot]))
if any(find_dynamicsymbols(i, sym_list) for i in [self._k_kqdot,
self._k_ku, self._f_k, self._k_dnh, self._f_dnh, self._k_d]):
raise ValueError('Cannot have dynamicsymbols outside dynamic \
forcing vector.')
# Find all other dynamic symbols, forming the forcing vector r.
# Sort r to make it canonical.
r = list(find_dynamicsymbols(msubs(self._f_d, uaux_zero), sym_list))
r.sort(key=default_sort_key)
# Check for any derivatives of variables in r that are also found in r.
for i in r:
if diff(i, dynamicsymbols._t) in r:
raise ValueError('Cannot have derivatives of specified \
quantities when linearizing forcing terms.')
return Linearizer(f_0, f_1, f_2, f_3, f_4, f_c, f_v, f_a, q, u, q_i,
q_d, u_i, u_d, r)
def linearize(self, **kwargs):
""" Linearize the equations of motion about a symbolic operating point.
If kwarg A_and_B is False (default), returns M, A, B, r for the
linearized form, M*[q', u']^T = A*[q_ind, u_ind]^T + B*r.
If kwarg A_and_B is True, returns A, B, r for the linearized form
dx = A*x + B*r, where x = [q_ind, u_ind]^T. Note that this is
computationally intensive if there are many symbolic parameters. For
this reason, it may be more desirable to use the default A_and_B=False,
returning M, A, and B. Values may then be substituted in to these
matrices, and the state space form found as
A = P.T*M.inv()*A, B = P.T*M.inv()*B, where P = Linearizer.perm_mat.
In both cases, r is found as all dynamicsymbols in the equations of
motion that are not part of q, u, q', or u'. They are sorted in
canonical form.
The operating points may be also entered using the ``op_point`` kwarg.
This takes a dictionary of {symbol: value}, or a an iterable of such
dictionaries. The values may be numeric or symbolic. The more values
you can specify beforehand, the faster this computation will run.
For more documentation, please see the ``Linearizer`` class."""
# TODO : Remove this after 1.1 has been released.
_ = kwargs.pop('new_method', None)
linearizer = self.to_linearizer()
result = linearizer.linearize(**kwargs)
return result + (linearizer.r,)
def kanes_equations(self, bodies, loads=None):
""" Method to form Kane's equations, Fr + Fr* = 0.
Returns (Fr, Fr*). In the case where auxiliary generalized speeds are
present (say, s auxiliary speeds, o generalized speeds, and m motion
constraints) the length of the returned vectors will be o - m + s in
length. The first o - m equations will be the constrained Kane's
equations, then the s auxiliary Kane's equations. These auxiliary
equations can be accessed with the auxiliary_eqs().
Parameters
==========
bodies : iterable
An iterable of all RigidBody's and Particle's in the system.
A system must have at least one body.
loads : iterable
Takes in an iterable of (Particle, Vector) or (ReferenceFrame, Vector)
tuples which represent the force at a point or torque on a frame.
Must be either a non-empty iterable of tuples or None which corresponds
to a system with no constraints.
"""
if (bodies is None and loads is not None) or isinstance(bodies[0], tuple):
# This switches the order if they use the old way.
bodies, loads = loads, bodies
SymPyDeprecationWarning(value='The API for kanes_equations() has changed such '
'that the loads (forces and torques) are now the second argument '
'and is optional with None being the default.',
feature='The kanes_equation() argument order',
useinstead='switched argument order to update your code, For example: '
'kanes_equations(loads, bodies) > kanes_equations(bodies, loads).',
issue=10945, deprecated_since_version="1.1").warn()
if not self._k_kqdot:
raise AttributeError('Create an instance of KanesMethod with '
'kinematic differential equations to use this method.')
fr = self._form_fr(loads)
frstar = self._form_frstar(bodies)
if self._uaux:
if not self._udep:
km = KanesMethod(self._inertial, self.q, self._uaux,
u_auxiliary=self._uaux)
else:
km = KanesMethod(self._inertial, self.q, self._uaux,
u_auxiliary=self._uaux, u_dependent=self._udep,
velocity_constraints=(self._k_nh * self.u +
self._f_nh))
km._qdot_u_map = self._qdot_u_map
self._km = km
fraux = km._form_fr(loads)
frstaraux = km._form_frstar(bodies)
self._aux_eq = fraux + frstaraux
self._fr = fr.col_join(fraux)
self._frstar = frstar.col_join(frstaraux)
return (self._fr, self._frstar)
def rhs(self, inv_method=None):
"""Returns the system's equations of motion in first order form. The
output is the right hand side of::
x' = |q'| =: f(q, u, r, p, t)
|u'|
The right hand side is what is needed by most numerical ODE
integrators.
Parameters
==========
inv_method : str
The specific sympy inverse matrix calculation method to use. For a
list of valid methods, see
:meth:`~sympy.matrices.matrices.MatrixBase.inv`
"""
rhs = zeros(len(self.q) + len(self.u), 1)
kdes = self.kindiffdict()
for i, q_i in enumerate(self.q):
rhs[i] = kdes[q_i.diff()]
if inv_method is None:
rhs[len(self.q):, 0] = self.mass_matrix.LUsolve(self.forcing)
else:
rhs[len(self.q):, 0] = (self.mass_matrix.inv(inv_method,
try_block_diag=True) *
self.forcing)
return rhs
def kindiffdict(self):
"""Returns a dictionary mapping q' to u."""
if not self._qdot_u_map:
raise AttributeError('Create an instance of KanesMethod with '
'kinematic differential equations to use this method.')
return self._qdot_u_map
@property
def auxiliary_eqs(self):
"""A matrix containing the auxiliary equations."""
if not self._fr or not self._frstar:
raise ValueError('Need to compute Fr, Fr* first.')
if not self._uaux:
raise ValueError('No auxiliary speeds have been declared.')
return self._aux_eq
@property
def mass_matrix(self):
"""The mass matrix of the system."""
if not self._fr or not self._frstar:
raise ValueError('Need to compute Fr, Fr* first.')
return Matrix([self._k_d, self._k_dnh])
@property
def mass_matrix_full(self):
"""The mass matrix of the system, augmented by the kinematic
differential equations."""
if not self._fr or not self._frstar:
raise ValueError('Need to compute Fr, Fr* first.')
o = len(self.u)
n = len(self.q)
return ((self._k_kqdot).row_join(zeros(n, o))).col_join((zeros(o,
n)).row_join(self.mass_matrix))
@property
def forcing(self):
"""The forcing vector of the system."""
if not self._fr or not self._frstar:
raise ValueError('Need to compute Fr, Fr* first.')
return -Matrix([self._f_d, self._f_dnh])
@property
def forcing_full(self):
"""The forcing vector of the system, augmented by the kinematic
differential equations."""
if not self._fr or not self._frstar:
raise ValueError('Need to compute Fr, Fr* first.')
f1 = self._k_ku * Matrix(self.u) + self._f_k
return -Matrix([f1, self._f_d, self._f_dnh])
@property
def q(self):
return self._q
@property
def u(self):
return self._u
@property
def bodylist(self):
return self._bodylist
@property
def forcelist(self):
return self._forcelist
|
6e7e67a6f639e164e860daa8570e1d4d068c4c364211bee0e878f03cc045b8d4
|
# -*- encoding: utf-8 -*-
from __future__ import print_function, division
from sympy.core.backend import sympify
from sympy.core.compatibility import string_types
from sympy.physics.vector import Point, ReferenceFrame, Dyadic
__all__ = ['RigidBody']
class RigidBody(object):
"""An idealized rigid body.
This is essentially a container which holds the various components which
describe a rigid body: a name, mass, center of mass, reference frame, and
inertia.
All of these need to be supplied on creation, but can be changed
afterwards.
Attributes
==========
name : string
The body's name.
masscenter : Point
The point which represents the center of mass of the rigid body.
frame : ReferenceFrame
The ReferenceFrame which the rigid body is fixed in.
mass : Sympifyable
The body's mass.
inertia : (Dyadic, Point)
The body's inertia about a point; stored in a tuple as shown above.
Examples
========
>>> from sympy import Symbol
>>> from sympy.physics.mechanics import ReferenceFrame, Point, RigidBody
>>> from sympy.physics.mechanics import outer
>>> m = Symbol('m')
>>> A = ReferenceFrame('A')
>>> P = Point('P')
>>> I = outer (A.x, A.x)
>>> inertia_tuple = (I, P)
>>> B = RigidBody('B', P, A, m, inertia_tuple)
>>> # Or you could change them afterwards
>>> m2 = Symbol('m2')
>>> B.mass = m2
"""
def __init__(self, name, masscenter, frame, mass, inertia):
if not isinstance(name, string_types):
raise TypeError('Supply a valid name.')
self._name = name
self.masscenter = masscenter
self.mass = mass
self.frame = frame
self.inertia = inertia
self.potential_energy = 0
def __str__(self):
return self._name
__repr__ = __str__
@property
def frame(self):
return self._frame
@frame.setter
def frame(self, F):
if not isinstance(F, ReferenceFrame):
raise TypeError("RigdBody frame must be a ReferenceFrame object.")
self._frame = F
@property
def masscenter(self):
return self._masscenter
@masscenter.setter
def masscenter(self, p):
if not isinstance(p, Point):
raise TypeError("RigidBody center of mass must be a Point object.")
self._masscenter = p
@property
def mass(self):
return self._mass
@mass.setter
def mass(self, m):
self._mass = sympify(m)
@property
def inertia(self):
return (self._inertia, self._inertia_point)
@inertia.setter
def inertia(self, I):
if not isinstance(I[0], Dyadic):
raise TypeError("RigidBody inertia must be a Dyadic object.")
if not isinstance(I[1], Point):
raise TypeError("RigidBody inertia must be about a Point.")
self._inertia = I[0]
self._inertia_point = I[1]
# have I S/O, want I S/S*
# I S/O = I S/S* + I S*/O; I S/S* = I S/O - I S*/O
# I_S/S* = I_S/O - I_S*/O
from sympy.physics.mechanics.functions import inertia_of_point_mass
I_Ss_O = inertia_of_point_mass(self.mass,
self.masscenter.pos_from(I[1]),
self.frame)
self._central_inertia = I[0] - I_Ss_O
@property
def central_inertia(self):
"""The body's central inertia dyadic."""
return self._central_inertia
def linear_momentum(self, frame):
""" Linear momentum of the rigid body.
The linear momentum L, of a rigid body B, with respect to frame N is
given by
L = M * v*
where M is the mass of the rigid body and v* is the velocity of
the mass center of B in the frame, N.
Parameters
==========
frame : ReferenceFrame
The frame in which linear momentum is desired.
Examples
========
>>> from sympy.physics.mechanics import Point, ReferenceFrame, outer
>>> from sympy.physics.mechanics import RigidBody, dynamicsymbols
>>> M, v = dynamicsymbols('M v')
>>> N = ReferenceFrame('N')
>>> P = Point('P')
>>> P.set_vel(N, v * N.x)
>>> I = outer (N.x, N.x)
>>> Inertia_tuple = (I, P)
>>> B = RigidBody('B', P, N, M, Inertia_tuple)
>>> B.linear_momentum(N)
M*v*N.x
"""
return self.mass * self.masscenter.vel(frame)
def angular_momentum(self, point, frame):
"""Returns the angular momentum of the rigid body about a point in the
given frame.
The angular momentum H of a rigid body B about some point O in a frame
N is given by:
H = I·w + r×Mv
where I is the central inertia dyadic of B, w is the angular velocity
of body B in the frame, N, r is the position vector from point O to the
mass center of B, and v is the velocity of the mass center in the
frame, N.
Parameters
==========
point : Point
The point about which angular momentum is desired.
frame : ReferenceFrame
The frame in which angular momentum is desired.
Examples
========
>>> from sympy.physics.mechanics import Point, ReferenceFrame, outer
>>> from sympy.physics.mechanics import RigidBody, dynamicsymbols
>>> M, v, r, omega = dynamicsymbols('M v r omega')
>>> N = ReferenceFrame('N')
>>> b = ReferenceFrame('b')
>>> b.set_ang_vel(N, omega * b.x)
>>> P = Point('P')
>>> P.set_vel(N, 1 * N.x)
>>> I = outer(b.x, b.x)
>>> B = RigidBody('B', P, b, M, (I, P))
>>> B.angular_momentum(P, N)
omega*b.x
"""
I = self.central_inertia
w = self.frame.ang_vel_in(frame)
m = self.mass
r = self.masscenter.pos_from(point)
v = self.masscenter.vel(frame)
return I.dot(w) + r.cross(m * v)
def kinetic_energy(self, frame):
"""Kinetic energy of the rigid body
The kinetic energy, T, of a rigid body, B, is given by
'T = 1/2 (I omega^2 + m v^2)'
where I and m are the central inertia dyadic and mass of rigid body B,
respectively, omega is the body's angular velocity and v is the
velocity of the body's mass center in the supplied ReferenceFrame.
Parameters
==========
frame : ReferenceFrame
The RigidBody's angular velocity and the velocity of it's mass
center are typically defined with respect to an inertial frame but
any relevant frame in which the velocities are known can be supplied.
Examples
========
>>> from sympy.physics.mechanics import Point, ReferenceFrame, outer
>>> from sympy.physics.mechanics import RigidBody
>>> from sympy import symbols
>>> M, v, r, omega = symbols('M v r omega')
>>> N = ReferenceFrame('N')
>>> b = ReferenceFrame('b')
>>> b.set_ang_vel(N, omega * b.x)
>>> P = Point('P')
>>> P.set_vel(N, v * N.x)
>>> I = outer (b.x, b.x)
>>> inertia_tuple = (I, P)
>>> B = RigidBody('B', P, b, M, inertia_tuple)
>>> B.kinetic_energy(N)
M*v**2/2 + omega**2/2
"""
rotational_KE = (self.frame.ang_vel_in(frame) & (self.central_inertia &
self.frame.ang_vel_in(frame)) / sympify(2))
translational_KE = (self.mass * (self.masscenter.vel(frame) &
self.masscenter.vel(frame)) / sympify(2))
return rotational_KE + translational_KE
@property
def potential_energy(self):
"""The potential energy of the RigidBody.
Examples
========
>>> from sympy.physics.mechanics import RigidBody, Point, outer, ReferenceFrame
>>> from sympy import symbols
>>> M, g, h = symbols('M g h')
>>> b = ReferenceFrame('b')
>>> P = Point('P')
>>> I = outer (b.x, b.x)
>>> Inertia_tuple = (I, P)
>>> B = RigidBody('B', P, b, M, Inertia_tuple)
>>> B.potential_energy = M * g * h
>>> B.potential_energy
M*g*h
"""
return self._pe
@potential_energy.setter
def potential_energy(self, scalar):
"""Used to set the potential energy of this RigidBody.
Parameters
==========
scalar: Sympifyable
The potential energy (a scalar) of the RigidBody.
Examples
========
>>> from sympy.physics.mechanics import Particle, Point, outer
>>> from sympy.physics.mechanics import RigidBody, ReferenceFrame
>>> from sympy import symbols
>>> b = ReferenceFrame('b')
>>> M, g, h = symbols('M g h')
>>> P = Point('P')
>>> I = outer (b.x, b.x)
>>> Inertia_tuple = (I, P)
>>> B = RigidBody('B', P, b, M, Inertia_tuple)
>>> B.potential_energy = M * g * h
"""
self._pe = sympify(scalar)
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71b9e8c9f4139859d09bb9aa56df0f7d70b1ae112e6c4dda3bae0ea2a1360ead
|
from __future__ import print_function, division
from sympy.utilities import dict_merge
from sympy.utilities.iterables import iterable
from sympy.physics.vector import (Dyadic, Vector, ReferenceFrame,
Point, dynamicsymbols)
from sympy.physics.vector.printing import (vprint, vsprint, vpprint, vlatex,
init_vprinting)
from sympy.physics.mechanics.particle import Particle
from sympy.physics.mechanics.rigidbody import RigidBody
from sympy import simplify
from sympy.core.backend import (Matrix, sympify, Mul, Derivative, sin, cos,
tan, AppliedUndef, S)
__all__ = ['inertia',
'inertia_of_point_mass',
'linear_momentum',
'angular_momentum',
'kinetic_energy',
'potential_energy',
'Lagrangian',
'mechanics_printing',
'mprint',
'msprint',
'mpprint',
'mlatex',
'msubs',
'find_dynamicsymbols']
# These are functions that we've moved and renamed during extracting the
# basic vector calculus code from the mechanics packages.
mprint = vprint
msprint = vsprint
mpprint = vpprint
mlatex = vlatex
def mechanics_printing(**kwargs):
"""
Initializes time derivative printing for all SymPy objects in
mechanics module.
"""
init_vprinting(**kwargs)
mechanics_printing.__doc__ = init_vprinting.__doc__
def inertia(frame, ixx, iyy, izz, ixy=0, iyz=0, izx=0):
"""Simple way to create inertia Dyadic object.
If you don't know what a Dyadic is, just treat this like the inertia
tensor. Then, do the easy thing and define it in a body-fixed frame.
Parameters
==========
frame : ReferenceFrame
The frame the inertia is defined in
ixx : Sympifyable
the xx element in the inertia dyadic
iyy : Sympifyable
the yy element in the inertia dyadic
izz : Sympifyable
the zz element in the inertia dyadic
ixy : Sympifyable
the xy element in the inertia dyadic
iyz : Sympifyable
the yz element in the inertia dyadic
izx : Sympifyable
the zx element in the inertia dyadic
Examples
========
>>> from sympy.physics.mechanics import ReferenceFrame, inertia
>>> N = ReferenceFrame('N')
>>> inertia(N, 1, 2, 3)
(N.x|N.x) + 2*(N.y|N.y) + 3*(N.z|N.z)
"""
if not isinstance(frame, ReferenceFrame):
raise TypeError('Need to define the inertia in a frame')
ol = sympify(ixx) * (frame.x | frame.x)
ol += sympify(ixy) * (frame.x | frame.y)
ol += sympify(izx) * (frame.x | frame.z)
ol += sympify(ixy) * (frame.y | frame.x)
ol += sympify(iyy) * (frame.y | frame.y)
ol += sympify(iyz) * (frame.y | frame.z)
ol += sympify(izx) * (frame.z | frame.x)
ol += sympify(iyz) * (frame.z | frame.y)
ol += sympify(izz) * (frame.z | frame.z)
return ol
def inertia_of_point_mass(mass, pos_vec, frame):
"""Inertia dyadic of a point mass relative to point O.
Parameters
==========
mass : Sympifyable
Mass of the point mass
pos_vec : Vector
Position from point O to point mass
frame : ReferenceFrame
Reference frame to express the dyadic in
Examples
========
>>> from sympy import symbols
>>> from sympy.physics.mechanics import ReferenceFrame, inertia_of_point_mass
>>> N = ReferenceFrame('N')
>>> r, m = symbols('r m')
>>> px = r * N.x
>>> inertia_of_point_mass(m, px, N)
m*r**2*(N.y|N.y) + m*r**2*(N.z|N.z)
"""
return mass * (((frame.x | frame.x) + (frame.y | frame.y) +
(frame.z | frame.z)) * (pos_vec & pos_vec) -
(pos_vec | pos_vec))
def linear_momentum(frame, *body):
"""Linear momentum of the system.
This function returns the linear momentum of a system of Particle's and/or
RigidBody's. The linear momentum of a system is equal to the vector sum of
the linear momentum of its constituents. Consider a system, S, comprised of
a rigid body, A, and a particle, P. The linear momentum of the system, L,
is equal to the vector sum of the linear momentum of the particle, L1, and
the linear momentum of the rigid body, L2, i.e.
L = L1 + L2
Parameters
==========
frame : ReferenceFrame
The frame in which linear momentum is desired.
body1, body2, body3... : Particle and/or RigidBody
The body (or bodies) whose linear momentum is required.
Examples
========
>>> from sympy.physics.mechanics import Point, Particle, ReferenceFrame
>>> from sympy.physics.mechanics import RigidBody, outer, linear_momentum
>>> N = ReferenceFrame('N')
>>> P = Point('P')
>>> P.set_vel(N, 10 * N.x)
>>> Pa = Particle('Pa', P, 1)
>>> Ac = Point('Ac')
>>> Ac.set_vel(N, 25 * N.y)
>>> I = outer(N.x, N.x)
>>> A = RigidBody('A', Ac, N, 20, (I, Ac))
>>> linear_momentum(N, A, Pa)
10*N.x + 500*N.y
"""
if not isinstance(frame, ReferenceFrame):
raise TypeError('Please specify a valid ReferenceFrame')
else:
linear_momentum_sys = Vector(0)
for e in body:
if isinstance(e, (RigidBody, Particle)):
linear_momentum_sys += e.linear_momentum(frame)
else:
raise TypeError('*body must have only Particle or RigidBody')
return linear_momentum_sys
def angular_momentum(point, frame, *body):
"""Angular momentum of a system
This function returns the angular momentum of a system of Particle's and/or
RigidBody's. The angular momentum of such a system is equal to the vector
sum of the angular momentum of its constituents. Consider a system, S,
comprised of a rigid body, A, and a particle, P. The angular momentum of
the system, H, is equal to the vector sum of the angular momentum of the
particle, H1, and the angular momentum of the rigid body, H2, i.e.
H = H1 + H2
Parameters
==========
point : Point
The point about which angular momentum of the system is desired.
frame : ReferenceFrame
The frame in which angular momentum is desired.
body1, body2, body3... : Particle and/or RigidBody
The body (or bodies) whose angular momentum is required.
Examples
========
>>> from sympy.physics.mechanics import Point, Particle, ReferenceFrame
>>> from sympy.physics.mechanics import RigidBody, outer, angular_momentum
>>> N = ReferenceFrame('N')
>>> O = Point('O')
>>> O.set_vel(N, 0 * N.x)
>>> P = O.locatenew('P', 1 * N.x)
>>> P.set_vel(N, 10 * N.x)
>>> Pa = Particle('Pa', P, 1)
>>> Ac = O.locatenew('Ac', 2 * N.y)
>>> Ac.set_vel(N, 5 * N.y)
>>> a = ReferenceFrame('a')
>>> a.set_ang_vel(N, 10 * N.z)
>>> I = outer(N.z, N.z)
>>> A = RigidBody('A', Ac, a, 20, (I, Ac))
>>> angular_momentum(O, N, Pa, A)
10*N.z
"""
if not isinstance(frame, ReferenceFrame):
raise TypeError('Please enter a valid ReferenceFrame')
if not isinstance(point, Point):
raise TypeError('Please specify a valid Point')
else:
angular_momentum_sys = Vector(0)
for e in body:
if isinstance(e, (RigidBody, Particle)):
angular_momentum_sys += e.angular_momentum(point, frame)
else:
raise TypeError('*body must have only Particle or RigidBody')
return angular_momentum_sys
def kinetic_energy(frame, *body):
"""Kinetic energy of a multibody system.
This function returns the kinetic energy of a system of Particle's and/or
RigidBody's. The kinetic energy of such a system is equal to the sum of
the kinetic energies of its constituents. Consider a system, S, comprising
a rigid body, A, and a particle, P. The kinetic energy of the system, T,
is equal to the vector sum of the kinetic energy of the particle, T1, and
the kinetic energy of the rigid body, T2, i.e.
T = T1 + T2
Kinetic energy is a scalar.
Parameters
==========
frame : ReferenceFrame
The frame in which the velocity or angular velocity of the body is
defined.
body1, body2, body3... : Particle and/or RigidBody
The body (or bodies) whose kinetic energy is required.
Examples
========
>>> from sympy.physics.mechanics import Point, Particle, ReferenceFrame
>>> from sympy.physics.mechanics import RigidBody, outer, kinetic_energy
>>> N = ReferenceFrame('N')
>>> O = Point('O')
>>> O.set_vel(N, 0 * N.x)
>>> P = O.locatenew('P', 1 * N.x)
>>> P.set_vel(N, 10 * N.x)
>>> Pa = Particle('Pa', P, 1)
>>> Ac = O.locatenew('Ac', 2 * N.y)
>>> Ac.set_vel(N, 5 * N.y)
>>> a = ReferenceFrame('a')
>>> a.set_ang_vel(N, 10 * N.z)
>>> I = outer(N.z, N.z)
>>> A = RigidBody('A', Ac, a, 20, (I, Ac))
>>> kinetic_energy(N, Pa, A)
350
"""
if not isinstance(frame, ReferenceFrame):
raise TypeError('Please enter a valid ReferenceFrame')
ke_sys = S(0)
for e in body:
if isinstance(e, (RigidBody, Particle)):
ke_sys += e.kinetic_energy(frame)
else:
raise TypeError('*body must have only Particle or RigidBody')
return ke_sys
def potential_energy(*body):
"""Potential energy of a multibody system.
This function returns the potential energy of a system of Particle's and/or
RigidBody's. The potential energy of such a system is equal to the sum of
the potential energy of its constituents. Consider a system, S, comprising
a rigid body, A, and a particle, P. The potential energy of the system, V,
is equal to the vector sum of the potential energy of the particle, V1, and
the potential energy of the rigid body, V2, i.e.
V = V1 + V2
Potential energy is a scalar.
Parameters
==========
body1, body2, body3... : Particle and/or RigidBody
The body (or bodies) whose potential energy is required.
Examples
========
>>> from sympy.physics.mechanics import Point, Particle, ReferenceFrame
>>> from sympy.physics.mechanics import RigidBody, outer, potential_energy
>>> from sympy import symbols
>>> M, m, g, h = symbols('M m g h')
>>> N = ReferenceFrame('N')
>>> O = Point('O')
>>> O.set_vel(N, 0 * N.x)
>>> P = O.locatenew('P', 1 * N.x)
>>> Pa = Particle('Pa', P, m)
>>> Ac = O.locatenew('Ac', 2 * N.y)
>>> a = ReferenceFrame('a')
>>> I = outer(N.z, N.z)
>>> A = RigidBody('A', Ac, a, M, (I, Ac))
>>> Pa.potential_energy = m * g * h
>>> A.potential_energy = M * g * h
>>> potential_energy(Pa, A)
M*g*h + g*h*m
"""
pe_sys = S(0)
for e in body:
if isinstance(e, (RigidBody, Particle)):
pe_sys += e.potential_energy
else:
raise TypeError('*body must have only Particle or RigidBody')
return pe_sys
def gravity(acceleration, *bodies):
"""
Returns a list of gravity forces given the acceleration
due to gravity and any number of particles or rigidbodies.
Example
=======
>>> from sympy.physics.mechanics import ReferenceFrame, Point, Particle, outer, RigidBody
>>> from sympy.physics.mechanics.functions import gravity
>>> from sympy import symbols
>>> N = ReferenceFrame('N')
>>> m, M, g = symbols('m M g')
>>> F1, F2 = symbols('F1 F2')
>>> po = Point('po')
>>> pa = Particle('pa', po, m)
>>> A = ReferenceFrame('A')
>>> P = Point('P')
>>> I = outer(A.x, A.x)
>>> B = RigidBody('B', P, A, M, (I, P))
>>> forceList = [(po, F1), (P, F2)]
>>> forceList.extend(gravity(g*N.y, pa, B))
>>> forceList
[(po, F1), (P, F2), (po, g*m*N.y), (P, M*g*N.y)]
"""
gravity_force = []
if not bodies:
raise TypeError("No bodies(instances of Particle or Rigidbody) were passed.")
for e in bodies:
point = getattr(e, 'masscenter', None)
if point is None:
point = e.point
gravity_force.append((point, e.mass*acceleration))
return gravity_force
def center_of_mass(point, *bodies):
"""
Returns the position vector from the given point to the center of mass
of the given bodies(particles or rigidbodies).
Example
=======
>>> from sympy import symbols, S
>>> from sympy.physics.vector import Point
>>> from sympy.physics.mechanics import Particle, ReferenceFrame, RigidBody, outer
>>> from sympy.physics.mechanics.functions import center_of_mass
>>> a = ReferenceFrame('a')
>>> m = symbols('m', real=True)
>>> p1 = Particle('p1', Point('p1_pt'), S(1))
>>> p2 = Particle('p2', Point('p2_pt'), S(2))
>>> p3 = Particle('p3', Point('p3_pt'), S(3))
>>> p4 = Particle('p4', Point('p4_pt'), m)
>>> b_f = ReferenceFrame('b_f')
>>> b_cm = Point('b_cm')
>>> mb = symbols('mb')
>>> b = RigidBody('b', b_cm, b_f, mb, (outer(b_f.x, b_f.x), b_cm))
>>> p2.point.set_pos(p1.point, a.x)
>>> p3.point.set_pos(p1.point, a.x + a.y)
>>> p4.point.set_pos(p1.point, a.y)
>>> b.masscenter.set_pos(p1.point, a.y + a.z)
>>> point_o=Point('o')
>>> point_o.set_pos(p1.point, center_of_mass(p1.point, p1, p2, p3, p4, b))
>>> expr = 5/(m + mb + 6)*a.x + (m + mb + 3)/(m + mb + 6)*a.y + mb/(m + mb + 6)*a.z
>>> point_o.pos_from(p1.point)
5/(m + mb + 6)*a.x + (m + mb + 3)/(m + mb + 6)*a.y + mb/(m + mb + 6)*a.z
"""
if not bodies:
raise TypeError("No bodies(instances of Particle or Rigidbody) were passed.")
total_mass = 0
vec = Vector(0)
for i in bodies:
total_mass += i.mass
masscenter = getattr(i, 'masscenter', None)
if masscenter is None:
masscenter = i.point
vec += i.mass*masscenter.pos_from(point)
return vec/total_mass
def Lagrangian(frame, *body):
"""Lagrangian of a multibody system.
This function returns the Lagrangian of a system of Particle's and/or
RigidBody's. The Lagrangian of such a system is equal to the difference
between the kinetic energies and potential energies of its constituents. If
T and V are the kinetic and potential energies of a system then it's
Lagrangian, L, is defined as
L = T - V
The Lagrangian is a scalar.
Parameters
==========
frame : ReferenceFrame
The frame in which the velocity or angular velocity of the body is
defined to determine the kinetic energy.
body1, body2, body3... : Particle and/or RigidBody
The body (or bodies) whose Lagrangian is required.
Examples
========
>>> from sympy.physics.mechanics import Point, Particle, ReferenceFrame
>>> from sympy.physics.mechanics import RigidBody, outer, Lagrangian
>>> from sympy import symbols
>>> M, m, g, h = symbols('M m g h')
>>> N = ReferenceFrame('N')
>>> O = Point('O')
>>> O.set_vel(N, 0 * N.x)
>>> P = O.locatenew('P', 1 * N.x)
>>> P.set_vel(N, 10 * N.x)
>>> Pa = Particle('Pa', P, 1)
>>> Ac = O.locatenew('Ac', 2 * N.y)
>>> Ac.set_vel(N, 5 * N.y)
>>> a = ReferenceFrame('a')
>>> a.set_ang_vel(N, 10 * N.z)
>>> I = outer(N.z, N.z)
>>> A = RigidBody('A', Ac, a, 20, (I, Ac))
>>> Pa.potential_energy = m * g * h
>>> A.potential_energy = M * g * h
>>> Lagrangian(N, Pa, A)
-M*g*h - g*h*m + 350
"""
if not isinstance(frame, ReferenceFrame):
raise TypeError('Please supply a valid ReferenceFrame')
for e in body:
if not isinstance(e, (RigidBody, Particle)):
raise TypeError('*body must have only Particle or RigidBody')
return kinetic_energy(frame, *body) - potential_energy(*body)
def find_dynamicsymbols(expression, exclude=None, reference_frame=None):
"""Find all dynamicsymbols in expression.
If the optional ``exclude`` kwarg is used, only dynamicsymbols
not in the iterable ``exclude`` are returned.
If we intend to apply this function on a vector, the optional
''reference_frame'' is also used to inform about the corresponding frame
with respect to which the dynamic symbols of the given vector is to be
determined.
Parameters
==========
expression : sympy expression
exclude : iterable of dynamicsymbols, optional
reference_frame : ReferenceFrame, optional
The frame with respect to which the dynamic symbols of the
given vector is to be determined.
Examples
========
>>> from sympy.physics.mechanics import dynamicsymbols, find_dynamicsymbols
>>> from sympy.physics.mechanics import ReferenceFrame
>>> x, y = dynamicsymbols('x, y')
>>> expr = x + x.diff()*y
>>> find_dynamicsymbols(expr)
{x(t), y(t), Derivative(x(t), t)}
>>> find_dynamicsymbols(expr, exclude=[x, y])
{Derivative(x(t), t)}
>>> a, b, c = dynamicsymbols('a, b, c')
>>> A = ReferenceFrame('A')
>>> v = a * A.x + b * A.y + c * A.z
>>> find_dynamicsymbols(v, reference_frame=A)
{a(t), b(t), c(t)}
"""
t_set = {dynamicsymbols._t}
if exclude:
if iterable(exclude):
exclude_set = set(exclude)
else:
raise TypeError("exclude kwarg must be iterable")
else:
exclude_set = set()
if isinstance(expression, Vector):
if reference_frame is None:
raise ValueError("You must provide reference_frame when passing a "
"vector expression, got %s." % reference_frame)
else:
expression = expression.to_matrix(reference_frame)
return set([i for i in expression.atoms(AppliedUndef, Derivative) if
i.free_symbols == t_set]) - exclude_set
def msubs(expr, *sub_dicts, **kwargs):
"""A custom subs for use on expressions derived in physics.mechanics.
Traverses the expression tree once, performing the subs found in sub_dicts.
Terms inside ``Derivative`` expressions are ignored:
>>> from sympy.physics.mechanics import dynamicsymbols, msubs
>>> x = dynamicsymbols('x')
>>> msubs(x.diff() + x, {x: 1})
Derivative(x(t), t) + 1
Note that sub_dicts can be a single dictionary, or several dictionaries:
>>> x, y, z = dynamicsymbols('x, y, z')
>>> sub1 = {x: 1, y: 2}
>>> sub2 = {z: 3, x.diff(): 4}
>>> msubs(x.diff() + x + y + z, sub1, sub2)
10
If smart=True (default False), also checks for conditions that may result
in ``nan``, but if simplified would yield a valid expression. For example:
>>> from sympy import sin, tan
>>> (sin(x)/tan(x)).subs(x, 0)
nan
>>> msubs(sin(x)/tan(x), {x: 0}, smart=True)
1
It does this by first replacing all ``tan`` with ``sin/cos``. Then each
node is traversed. If the node is a fraction, subs is first evaluated on
the denominator. If this results in 0, simplification of the entire
fraction is attempted. Using this selective simplification, only
subexpressions that result in 1/0 are targeted, resulting in faster
performance.
"""
sub_dict = dict_merge(*sub_dicts)
smart = kwargs.pop('smart', False)
if smart:
func = _smart_subs
elif hasattr(expr, 'msubs'):
return expr.msubs(sub_dict)
else:
func = lambda expr, sub_dict: _crawl(expr, _sub_func, sub_dict)
if isinstance(expr, (Matrix, Vector, Dyadic)):
return expr.applyfunc(lambda x: func(x, sub_dict))
else:
return func(expr, sub_dict)
def _crawl(expr, func, *args, **kwargs):
"""Crawl the expression tree, and apply func to every node."""
val = func(expr, *args, **kwargs)
if val is not None:
return val
new_args = (_crawl(arg, func, *args, **kwargs) for arg in expr.args)
return expr.func(*new_args)
def _sub_func(expr, sub_dict):
"""Perform direct matching substitution, ignoring derivatives."""
if expr in sub_dict:
return sub_dict[expr]
elif not expr.args or expr.is_Derivative:
return expr
def _tan_repl_func(expr):
"""Replace tan with sin/cos."""
if isinstance(expr, tan):
return sin(*expr.args) / cos(*expr.args)
elif not expr.args or expr.is_Derivative:
return expr
def _smart_subs(expr, sub_dict):
"""Performs subs, checking for conditions that may result in `nan` or
`oo`, and attempts to simplify them out.
The expression tree is traversed twice, and the following steps are
performed on each expression node:
- First traverse:
Replace all `tan` with `sin/cos`.
- Second traverse:
If node is a fraction, check if the denominator evaluates to 0.
If so, attempt to simplify it out. Then if node is in sub_dict,
sub in the corresponding value."""
expr = _crawl(expr, _tan_repl_func)
def _recurser(expr, sub_dict):
# Decompose the expression into num, den
num, den = _fraction_decomp(expr)
if den != 1:
# If there is a non trivial denominator, we need to handle it
denom_subbed = _recurser(den, sub_dict)
if denom_subbed.evalf() == 0:
# If denom is 0 after this, attempt to simplify the bad expr
expr = simplify(expr)
else:
# Expression won't result in nan, find numerator
num_subbed = _recurser(num, sub_dict)
return num_subbed / denom_subbed
# We have to crawl the tree manually, because `expr` may have been
# modified in the simplify step. First, perform subs as normal:
val = _sub_func(expr, sub_dict)
if val is not None:
return val
new_args = (_recurser(arg, sub_dict) for arg in expr.args)
return expr.func(*new_args)
return _recurser(expr, sub_dict)
def _fraction_decomp(expr):
"""Return num, den such that expr = num/den"""
if not isinstance(expr, Mul):
return expr, 1
num = []
den = []
for a in expr.args:
if a.is_Pow and a.args[1] < 0:
den.append(1 / a)
else:
num.append(a)
if not den:
return expr, 1
num = Mul(*num)
den = Mul(*den)
return num, den
def _f_list_parser(fl, ref_frame):
"""Parses the provided forcelist composed of items
of the form (obj, force).
Returns a tuple containing:
vel_list: The velocity (ang_vel for Frames, vel for Points) in
the provided reference frame.
f_list: The forces.
Used internally in the KanesMethod and LagrangesMethod classes.
"""
def flist_iter():
for pair in fl:
obj, force = pair
if isinstance(obj, ReferenceFrame):
yield obj.ang_vel_in(ref_frame), force
elif isinstance(obj, Point):
yield obj.vel(ref_frame), force
else:
raise TypeError('First entry in each forcelist pair must '
'be a point or frame.')
if not fl:
vel_list, f_list = (), ()
else:
unzip = lambda l: list(zip(*l)) if l[0] else [(), ()]
vel_list, f_list = unzip(list(flist_iter()))
return vel_list, f_list
|
0d668cd9f60c28fdf2bcd399b10f73a77640e648a1cb20cb4054337c75e80725
|
from __future__ import print_function, division
from sympy.core.backend import sympify
from sympy.core.compatibility import string_types
from sympy.physics.vector import Point
__all__ = ['Particle']
class Particle(object):
"""A particle.
Particles have a non-zero mass and lack spatial extension; they take up no
space.
Values need to be supplied on initialization, but can be changed later.
Parameters
==========
name : str
Name of particle
point : Point
A physics/mechanics Point which represents the position, velocity, and
acceleration of this Particle
mass : sympifyable
A SymPy expression representing the Particle's mass
Examples
========
>>> from sympy.physics.mechanics import Particle, Point
>>> from sympy import Symbol
>>> po = Point('po')
>>> m = Symbol('m')
>>> pa = Particle('pa', po, m)
>>> # Or you could change these later
>>> pa.mass = m
>>> pa.point = po
"""
def __init__(self, name, point, mass):
if not isinstance(name, string_types):
raise TypeError('Supply a valid name.')
self._name = name
self.mass = mass
self.point = point
self.potential_energy = 0
def __str__(self):
return self._name
__repr__ = __str__
@property
def mass(self):
"""Mass of the particle."""
return self._mass
@mass.setter
def mass(self, value):
self._mass = sympify(value)
@property
def point(self):
"""Point of the particle."""
return self._point
@point.setter
def point(self, p):
if not isinstance(p, Point):
raise TypeError("Particle point attribute must be a Point object.")
self._point = p
def linear_momentum(self, frame):
"""Linear momentum of the particle.
The linear momentum L, of a particle P, with respect to frame N is
given by
L = m * v
where m is the mass of the particle, and v is the velocity of the
particle in the frame N.
Parameters
==========
frame : ReferenceFrame
The frame in which linear momentum is desired.
Examples
========
>>> from sympy.physics.mechanics import Particle, Point, ReferenceFrame
>>> from sympy.physics.mechanics import dynamicsymbols
>>> m, v = dynamicsymbols('m v')
>>> N = ReferenceFrame('N')
>>> P = Point('P')
>>> A = Particle('A', P, m)
>>> P.set_vel(N, v * N.x)
>>> A.linear_momentum(N)
m*v*N.x
"""
return self.mass * self.point.vel(frame)
def angular_momentum(self, point, frame):
"""Angular momentum of the particle about the point.
The angular momentum H, about some point O of a particle, P, is given
by:
H = r x m * v
where r is the position vector from point O to the particle P, m is
the mass of the particle, and v is the velocity of the particle in
the inertial frame, N.
Parameters
==========
point : Point
The point about which angular momentum of the particle is desired.
frame : ReferenceFrame
The frame in which angular momentum is desired.
Examples
========
>>> from sympy.physics.mechanics import Particle, Point, ReferenceFrame
>>> from sympy.physics.mechanics import dynamicsymbols
>>> m, v, r = dynamicsymbols('m v r')
>>> N = ReferenceFrame('N')
>>> O = Point('O')
>>> A = O.locatenew('A', r * N.x)
>>> P = Particle('P', A, m)
>>> P.point.set_vel(N, v * N.y)
>>> P.angular_momentum(O, N)
m*r*v*N.z
"""
return self.point.pos_from(point) ^ (self.mass * self.point.vel(frame))
def kinetic_energy(self, frame):
"""Kinetic energy of the particle
The kinetic energy, T, of a particle, P, is given by
'T = 1/2 m v^2'
where m is the mass of particle P, and v is the velocity of the
particle in the supplied ReferenceFrame.
Parameters
==========
frame : ReferenceFrame
The Particle's velocity is typically defined with respect to
an inertial frame but any relevant frame in which the velocity is
known can be supplied.
Examples
========
>>> from sympy.physics.mechanics import Particle, Point, ReferenceFrame
>>> from sympy import symbols
>>> m, v, r = symbols('m v r')
>>> N = ReferenceFrame('N')
>>> O = Point('O')
>>> P = Particle('P', O, m)
>>> P.point.set_vel(N, v * N.y)
>>> P.kinetic_energy(N)
m*v**2/2
"""
return (self.mass / sympify(2) * self.point.vel(frame) &
self.point.vel(frame))
@property
def potential_energy(self):
"""The potential energy of the Particle.
Examples
========
>>> from sympy.physics.mechanics import Particle, Point
>>> from sympy import symbols
>>> m, g, h = symbols('m g h')
>>> O = Point('O')
>>> P = Particle('P', O, m)
>>> P.potential_energy = m * g * h
>>> P.potential_energy
g*h*m
"""
return self._pe
@potential_energy.setter
def potential_energy(self, scalar):
"""Used to set the potential energy of the Particle.
Parameters
==========
scalar : Sympifyable
The potential energy (a scalar) of the Particle.
Examples
========
>>> from sympy.physics.mechanics import Particle, Point
>>> from sympy import symbols
>>> m, g, h = symbols('m g h')
>>> O = Point('O')
>>> P = Particle('P', O, m)
>>> P.potential_energy = m * g * h
"""
self._pe = sympify(scalar)
|
cb728d25de742c3e47a1b03474856f9ecb0fcbb8c980eadf77724fc1ea38f623
|
# isort:skip_file
"""
Dimensional analysis and unit systems.
This module defines dimension/unit systems and physical quantities. It is
based on a group-theoretical construction where dimensions are represented as
vectors (coefficients being the exponents), and units are defined as a dimension
to which we added a scale.
Quantities are built from a factor and a unit, and are the basic objects that
one will use when doing computations.
All objects except systems and prefixes can be used in sympy expressions.
Note that as part of a CAS, various objects do not combine automatically
under operations.
Details about the implementation can be found in the documentation, and we
will not repeat all the explanations we gave there concerning our approach.
Ideas about future developments can be found on the `Github wiki
<https://github.com/sympy/sympy/wiki/Unit-systems>`_, and you should consult
this page if you are willing to help.
Useful functions:
- ``find_unit``: easily lookup pre-defined units.
- ``convert_to(expr, newunit)``: converts an expression into the same
expression expressed in another unit.
"""
from sympy.core.compatibility import string_types
from .dimensions import Dimension, DimensionSystem
from .unitsystem import UnitSystem
from .util import convert_to
from .quantities import Quantity
from .dimensions import (
amount_of_substance, acceleration, action,
capacitance, charge, conductance, current, energy,
force, frequency, impedance, inductance, length,
luminous_intensity, magnetic_density,
magnetic_flux, mass, momentum, power, pressure, temperature, time,
velocity, voltage, volume
)
Unit = Quantity
speed = velocity
luminosity = luminous_intensity
magnetic_flux_density = magnetic_density
amount = amount_of_substance
from .prefixes import (
# 10-power based:
yotta,
zetta,
exa,
peta,
tera,
giga,
mega,
kilo,
hecto,
deca,
deci,
centi,
milli,
micro,
nano,
pico,
femto,
atto,
zepto,
yocto,
# 2-power based:
kibi,
mebi,
gibi,
tebi,
pebi,
exbi,
)
from .definitions import (
percent, percents,
permille,
rad, radian, radians,
deg, degree, degrees,
sr, steradian, steradians,
mil, angular_mil, angular_mils,
m, meter, meters,
kg, kilogram, kilograms,
s, second, seconds,
A, ampere, amperes,
K, kelvin, kelvins,
mol, mole, moles,
cd, candela, candelas,
g, gram, grams,
mg, milligram, milligrams,
ug, microgram, micrograms,
newton, newtons, N,
joule, joules, J,
watt, watts, W,
pascal, pascals, Pa, pa,
hertz, hz, Hz,
coulomb, coulombs, C,
volt, volts, v, V,
ohm, ohms,
siemens, S, mho, mhos,
farad, farads, F,
henry, henrys, H,
tesla, teslas, T,
weber, webers, Wb, wb,
optical_power, dioptre, D,
lux, lx,
katal, kat,
gray, Gy,
becquerel, Bq,
km, kilometer, kilometers,
dm, decimeter, decimeters,
cm, centimeter, centimeters,
mm, millimeter, millimeters,
um, micrometer, micrometers, micron, microns,
nm, nanometer, nanometers,
pm, picometer, picometers,
ft, foot, feet,
inch, inches,
yd, yard, yards,
mi, mile, miles,
nmi, nautical_mile, nautical_miles,
l, liter, liters,
dl, deciliter, deciliters,
cl, centiliter, centiliters,
ml, milliliter, milliliters,
ms, millisecond, milliseconds,
us, microsecond, microseconds,
ns, nanosecond, nanoseconds,
ps, picosecond, picoseconds,
minute, minutes,
h, hour, hours,
day, days,
anomalistic_year, anomalistic_years,
sidereal_year, sidereal_years,
tropical_year, tropical_years,
common_year, common_years,
julian_year, julian_years,
draconic_year, draconic_years,
gaussian_year, gaussian_years,
full_moon_cycle, full_moon_cycles,
year, years, tropical_year,
G, gravitational_constant,
c, speed_of_light,
Z0,
hbar,
planck,
eV, electronvolt, electronvolts,
avogadro_number,
avogadro, avogadro_constant,
boltzmann, boltzmann_constant,
stefan, stefan_boltzmann_constant,
R, molar_gas_constant,
faraday_constant,
josephson_constant,
von_klitzing_constant,
amu, amus, atomic_mass_unit, atomic_mass_constant,
gee, gees, acceleration_due_to_gravity,
u0, magnetic_constant, vacuum_permeability,
e0, electric_constant, vacuum_permittivity,
Z0, vacuum_impedance,
coulomb_constant, electric_force_constant,
atmosphere, atmospheres, atm,
kPa,
bar, bars,
pound, pounds,
psi,
dHg0,
mmHg, torr,
mmu, mmus, milli_mass_unit,
quart, quarts,
ly, lightyear, lightyears,
au, astronomical_unit, astronomical_units,
planck_mass,
planck_time,
planck_temperature,
planck_length,
planck_charge,
planck_area,
planck_volume,
planck_momentum,
planck_energy,
planck_force,
planck_power,
planck_density,
planck_energy_density,
planck_intensity,
planck_angular_frequency,
planck_pressure,
planck_current,
planck_voltage,
planck_impedance,
planck_acceleration,
bit, bits,
byte,
kibibyte, kibibytes,
mebibyte, mebibytes,
gibibyte, gibibytes,
tebibyte, tebibytes,
pebibyte, pebibytes,
exbibyte, exbibytes,
)
def find_unit(quantity):
"""
Return a list of matching units or dimension names.
- If ``quantity`` is a string -- units/dimensions containing the string
`quantity`.
- If ``quantity`` is a unit or dimension -- units having matching base
units or dimensions.
Examples
========
>>> from sympy.physics import units as u
>>> u.find_unit('charge')
['C', 'coulomb', 'coulombs', 'planck_charge']
>>> u.find_unit(u.charge)
['C', 'coulomb', 'coulombs', 'planck_charge']
>>> u.find_unit("ampere")
['ampere', 'amperes']
>>> u.find_unit('volt')
['volt', 'volts', 'electronvolt', 'electronvolts', 'planck_voltage']
>>> u.find_unit(u.inch**3)[:5]
['l', 'cl', 'dl', 'ml', 'liter']
"""
import sympy.physics.units as u
rv = []
if isinstance(quantity, string_types):
rv = [i for i in dir(u) if quantity in i and isinstance(getattr(u, i), Quantity)]
dim = getattr(u, quantity)
if isinstance(dim, Dimension):
rv.extend(find_unit(dim))
else:
for i in sorted(dir(u)):
other = getattr(u, i)
if not isinstance(other, Quantity):
continue
if isinstance(quantity, Quantity):
if quantity.dimension == other.dimension:
rv.append(str(i))
elif isinstance(quantity, Dimension):
if other.dimension == quantity:
rv.append(str(i))
elif other.dimension == Dimension(Quantity.get_dimensional_expr(quantity)):
rv.append(str(i))
return sorted(set(rv), key=lambda x: (len(x), x))
# NOTE: the old units module had additional variables:
# 'density', 'illuminance', 'resistance'.
# They were not dimensions, but units (old Unit class).
|
52adf78f279f85a02769786b1c56e75839d75c28e6ec94f7d444a1573fd3ebfd
|
"""
Unit system for physical quantities; include definition of constants.
"""
from __future__ import division
from sympy import S
from sympy.core.decorators import deprecated
from sympy.physics.units.quantities import Quantity
from sympy.utilities.exceptions import SymPyDeprecationWarning
from .dimensions import DimensionSystem
class UnitSystem(object):
"""
UnitSystem represents a coherent set of units.
A unit system is basically a dimension system with notions of scales. Many
of the methods are defined in the same way.
It is much better if all base units have a symbol.
"""
def __init__(self, base, units=(), name="", descr=""):
self.name = name
self.descr = descr
# construct the associated dimension system
base_dims = [u.dimension for u in base]
derived_dims = [u.dimension for u in units if u.dimension not in base_dims]
self._system = DimensionSystem(base_dims, derived_dims)
if not self.is_consistent:
raise ValueError("UnitSystem is not consistent")
self._units = tuple(set(base) | set(units))
# create a dict linkin
# this is possible since we have already verified that the base units
# form a coherent system
base_dict = dict((u.dimension, u) for u in base)
# order the base units in the same order than the dimensions in the
# associated system, in order to ensure that we get always the same
self._base_units = tuple(base_dict[d] for d in self._system.base_dims)
def __str__(self):
"""
Return the name of the system.
If it does not exist, then it makes a list of symbols (or names) of
the base dimensions.
"""
if self.name != "":
return self.name
else:
return "UnitSystem((%s))" % ", ".join(
str(d) for d in self._base_units)
def __repr__(self):
return '<UnitSystem: %s>' % repr(self._base_units)
def extend(self, base, units=(), name="", description=""):
"""Extend the current system into a new one.
Take the base and normal units of the current system to merge
them to the base and normal units given in argument.
If not provided, name and description are overridden by empty strings.
"""
base = self._base_units + tuple(base)
units = self._units + tuple(units)
return UnitSystem(base, units, name, description)
def print_unit_base(self, unit):
"""
Useless method.
DO NOT USE, use instead ``convert_to``.
Give the string expression of a unit in term of the basis.
Units are displayed by decreasing power.
"""
SymPyDeprecationWarning(
deprecated_since_version="1.2",
issue=13336,
feature="print_unit_base",
useinstead="convert_to",
).warn()
from sympy.physics.units import convert_to
return convert_to(unit, self._base_units)
@property
def dim(self):
"""
Give the dimension of the system.
That is return the number of units forming the basis.
"""
return self._system.dim
@property
def is_consistent(self):
"""
Check if the underlying dimension system is consistent.
"""
# test is performed in DimensionSystem
return self._system.is_consistent
|
7d1ea2adcf0f933e9845d3afeb8bf506e98bb15ea55eeb090b3aa4cb8669b6fc
|
"""
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 __future__ import division
import collections
from sympy import Integer, Matrix, S, Symbol, sympify, Basic, Tuple, Dict, default_sort_key
from sympy.core.compatibility import reduce, string_types
from sympy.core.basic import Basic
from sympy.core.expr import Expr
from sympy.core.power import Pow
from sympy.utilities.exceptions import SymPyDeprecationWarning
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.dimensions 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
_dimensional_dependencies = dict()
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, string_types):
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, string_types):
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 _register_as_base_dim(self):
SymPyDeprecationWarning(
deprecated_since_version="1.2",
issue=13336,
feature="do not call ._register_as_base_dim()",
useinstead="DimensionSystem"
).warn()
if not self.name.is_Symbol:
raise TypeError("Base dimensions need to have symbolic name")
name = self.name
if name in dimsys_default.dimensional_dependencies:
raise IndexError("already in dependencies dict")
# Horrible code:
d = dict(dimsys_default.dimensional_dependencies)
d[name] = Dict({name: 1})
dimsys_default._args = (dimsys_default.args[:2] + (Dict(d),))
def __add__(self, other):
from sympy.physics.units.quantities import Quantity
other = sympify(other)
if isinstance(other, Basic):
if other.has(Quantity):
other = Dimension(Quantity.get_dimensional_expr(other))
if isinstance(other, Dimension) and self == other:
return self
return super(Dimension, self).__add__(other)
return self
def __radd__(self, other):
return self + 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):
other = Dimension(Quantity.get_dimensional_expr(other))
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(Dimension, self).__mul__(other)
return self
def __rmul__(self, other):
return self*other
def __div__(self, other):
return self*Pow(other, -1)
def __rdiv__(self, other):
return other * pow(self, -1)
__truediv__ = __div__
__rtruediv__ = __rdiv__
def get_dimensional_dependencies(self, mark_dimensionless=False):
SymPyDeprecationWarning(
deprecated_since_version="1.2",
issue=13336,
feature="do not call",
useinstead="DimensionSystem"
).warn()
name = self.name
dimdep = dimsys_default.get_dimensional_dependencies(name)
if mark_dimensionless and dimdep == {}:
return {'dimensionless': 1}
return {str(i): j for i, j in dimdep.items()}
@classmethod
def _from_dimensional_dependencies(cls, dependencies):
return reduce(lambda x, y: x * y, (
Dimension(d)**e for d, e in dependencies.items()
))
@classmethod
def _get_dimensional_dependencies_for_name(cls, name):
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, dimensional_dependencies=None):
"""
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
if dimensional_dependencies is None:
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
else:
return True
# base dimensions (MKS)
length = Dimension(name="length", symbol="L")
mass = Dimension(name="mass", symbol="M")
time = Dimension(name="time", symbol="T")
# base dimensions (MKSA not in MKS)
current = Dimension(name='current', symbol='I')
# other base dimensions:
temperature = Dimension("temperature", "T")
amount_of_substance = Dimension("amount_of_substance")
luminous_intensity = Dimension("luminous_intensity")
# derived dimensions (MKS)
velocity = Dimension(name="velocity")
acceleration = Dimension(name="acceleration")
momentum = Dimension(name="momentum")
force = Dimension(name="force", symbol="F")
energy = Dimension(name="energy", symbol="E")
power = Dimension(name="power")
pressure = Dimension(name="pressure")
frequency = Dimension(name="frequency", symbol="f")
action = Dimension(name="action", symbol="A")
volume = Dimension("volume")
# derived dimensions (MKSA not in MKS)
voltage = Dimension(name='voltage', symbol='U')
impedance = Dimension(name='impedance', symbol='Z')
conductance = Dimension(name='conductance', symbol='G')
capacitance = Dimension(name='capacitance')
inductance = Dimension(name='inductance')
charge = Dimension(name='charge', symbol='Q')
magnetic_density = Dimension(name='magnetic_density', symbol='B')
magnetic_flux = Dimension(name='magnetic_flux')
# Dimensions in information theory:
information = Dimension(name='information')
# 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):
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, string_types):
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, string_types):
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] = 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 name.is_Symbol:
return dict(self.dimensional_dependencies.get(name, {}))
if name.is_Number:
return {}
get_for_name = dimsys_default._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_Pow:
dim = get_for_name(name.base)
return {k: v*name.exp for (k, v) in dim.items()}
if name.is_Function:
args = (Dimension._from_dimensional_dependencies(
get_for_name(arg)) for arg in name.args)
result = name.func(*args)
if isinstance(result, Dimension):
return dimsys_default.get_dimensional_dependencies(result)
elif result.func == name.func:
return {}
else:
return get_for_name(result)
def get_dimensional_dependencies(self, name, mark_dimensionless=False):
if isinstance(name, Dimension):
name = name.name
if isinstance(name, string_types):
name = Symbol(name)
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)
return DimensionSystem(
tuple(self.base_dims) + tuple(new_base_dims),
tuple(self.derived_dims) + tuple(new_derived_dims),
deps
)
@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(dimsys_default.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(dimsys_default.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
dimsys_MKS = DimensionSystem([
# Dimensional dependencies for MKS base dimensions
length,
mass,
time,
], dimensional_dependencies=dict(
# Dimensional dependencies for derived dimensions
velocity=dict(length=1, time=-1),
acceleration=dict(length=1, time=-2),
momentum=dict(mass=1, length=1, time=-1),
force=dict(mass=1, length=1, time=-2),
energy=dict(mass=1, length=2, time=-2),
power=dict(length=2, mass=1, time=-3),
pressure=dict(mass=1, length=-1, time=-2),
frequency=dict(time=-1),
action=dict(length=2, mass=1, time=-1),
volume=dict(length=3),
))
dimsys_MKSA = dimsys_MKS.extend([
# Dimensional dependencies for base dimensions (MKSA not in MKS)
current,
], new_dim_deps=dict(
# Dimensional dependencies for derived dimensions
voltage=dict(mass=1, length=2, current=-1, time=-3),
impedance=dict(mass=1, length=2, current=-2, time=-3),
conductance=dict(mass=-1, length=-2, current=2, time=3),
capacitance=dict(mass=-1, length=-2, current=2, time=4),
inductance=dict(mass=1, length=2, current=-2, time=-2),
charge=dict(current=1, time=1),
magnetic_density=dict(mass=1, current=-1, time=-2),
magnetic_flux=dict(length=2, mass=1, current=-1, time=-2),
))
dimsys_SI = dimsys_MKSA.extend(
[
# Dimensional dependencies for other base dimensions:
temperature,
amount_of_substance,
luminous_intensity,
])
dimsys_default = dimsys_SI.extend(
[information],
)
|
230e6d35ea1bdbd8d8d6b5a370f0f95b55ed1311e991ab995aa76e012fad80f3
|
"""
Module defining unit prefixe class and some constants.
Constant dict for SI and binary prefixes are defined as PREFIXES and
BIN_PREFIXES.
"""
from sympy import Expr, sympify
class Prefix(Expr):
"""
This class represent prefixes, with their name, symbol and factor.
Prefixes are used to create derived units from a given unit. They should
always be encapsulated into units.
The factor is constructed from a base (default is 10) to some power, and
it gives the total multiple or fraction. For example the kilometer km
is constructed from the meter (factor 1) and the kilo (10 to the power 3,
i.e. 1000). The base can be changed to allow e.g. binary prefixes.
A prefix multiplied by something will always return the product of this
other object times the factor, except if the other object:
- is a prefix and they can be combined into a new prefix;
- defines multiplication with prefixes (which is the case for the Unit
class).
"""
_op_priority = 13.0
is_commutative = True
def __new__(cls, name, abbrev, exponent, base=sympify(10)):
name = sympify(name)
abbrev = sympify(abbrev)
exponent = sympify(exponent)
base = sympify(base)
obj = Expr.__new__(cls, name, abbrev, exponent, base)
obj._name = name
obj._abbrev = abbrev
obj._scale_factor = base**exponent
obj._exponent = exponent
obj._base = base
return obj
@property
def name(self):
return self._name
@property
def abbrev(self):
return self._abbrev
@property
def scale_factor(self):
return self._scale_factor
@property
def base(self):
return self._base
def __str__(self):
# TODO: add proper printers and tests:
if self.base == 10:
return "Prefix(%r, %r, %r)" % (
str(self.name), str(self.abbrev), self._exponent)
else:
return "Prefix(%r, %r, %r, %r)" % (
str(self.name), str(self.abbrev), self._exponent, self.base)
__repr__ = __str__
def __mul__(self, other):
if not hasattr(other, "scale_factor"):
return super(Prefix, self).__mul__(other)
fact = self.scale_factor * other.scale_factor
if fact == 1:
return 1
elif isinstance(other, Prefix):
# simplify prefix
for p in PREFIXES:
if PREFIXES[p].scale_factor == fact:
return PREFIXES[p]
return fact
return self.scale_factor * other
def __div__(self, other):
if not hasattr(other, "scale_factor"):
return super(Prefix, self).__div__(other)
fact = self.scale_factor / other.scale_factor
if fact == 1:
return 1
elif isinstance(other, Prefix):
for p in PREFIXES:
if PREFIXES[p].scale_factor == fact:
return PREFIXES[p]
return fact
return self.scale_factor / other
__truediv__ = __div__
def __rdiv__(self, other):
if other == 1:
for p in PREFIXES:
if PREFIXES[p].scale_factor == 1 / self.scale_factor:
return PREFIXES[p]
return other / self.scale_factor
__rtruediv__ = __rdiv__
def prefix_unit(unit, prefixes):
"""
Return a list of all units formed by unit and the given prefixes.
You can use the predefined PREFIXES or BIN_PREFIXES, but you can also
pass as argument a subdict of them if you don't want all prefixed units.
>>> from sympy.physics.units.prefixes import (PREFIXES,
... prefix_unit)
>>> from sympy.physics.units.systems import MKS
>>> from sympy.physics.units import m
>>> pref = {"m": PREFIXES["m"], "c": PREFIXES["c"], "d": PREFIXES["d"]}
>>> prefix_unit(m, pref) #doctest: +SKIP
[cm, dm, mm]
"""
from sympy.physics.units.quantities import Quantity
prefixed_units = []
for prefix_abbr, prefix in prefixes.items():
quantity = Quantity(
"%s%s" % (prefix.name, unit.name),
abbrev=("%s%s" % (prefix.abbrev, unit.abbrev))
)
quantity.set_dimension(unit.dimension)
quantity.set_scale_factor(unit.scale_factor*prefix)
prefixed_units.append(quantity)
return prefixed_units
yotta = Prefix('yotta', 'Y', 24)
zetta = Prefix('zetta', 'Z', 21)
exa = Prefix('exa', 'E', 18)
peta = Prefix('peta', 'P', 15)
tera = Prefix('tera', 'T', 12)
giga = Prefix('giga', 'G', 9)
mega = Prefix('mega', 'M', 6)
kilo = Prefix('kilo', 'k', 3)
hecto = Prefix('hecto', 'h', 2)
deca = Prefix('deca', 'da', 1)
deci = Prefix('deci', 'd', -1)
centi = Prefix('centi', 'c', -2)
milli = Prefix('milli', 'm', -3)
micro = Prefix('micro', 'mu', -6)
nano = Prefix('nano', 'n', -9)
pico = Prefix('pico', 'p', -12)
femto = Prefix('femto', 'f', -15)
atto = Prefix('atto', 'a', -18)
zepto = Prefix('zepto', 'z', -21)
yocto = Prefix('yocto', 'y', -24)
# http://physics.nist.gov/cuu/Units/prefixes.html
PREFIXES = {
'Y': yotta,
'Z': zetta,
'E': exa,
'P': peta,
'T': tera,
'G': giga,
'M': mega,
'k': kilo,
'h': hecto,
'da': deca,
'd': deci,
'c': centi,
'm': milli,
'mu': micro,
'n': nano,
'p': pico,
'f': femto,
'a': atto,
'z': zepto,
'y': yocto,
}
kibi = Prefix('kibi', 'Y', 10, 2)
mebi = Prefix('mebi', 'Y', 20, 2)
gibi = Prefix('gibi', 'Y', 30, 2)
tebi = Prefix('tebi', 'Y', 40, 2)
pebi = Prefix('pebi', 'Y', 50, 2)
exbi = Prefix('exbi', 'Y', 60, 2)
# http://physics.nist.gov/cuu/Units/binary.html
BIN_PREFIXES = {
'Ki': kibi,
'Mi': mebi,
'Gi': gibi,
'Ti': tebi,
'Pi': pebi,
'Ei': exbi,
}
|
92952b1d5fc6d32204ac5137c026ffddbaa8ad694099561ab2e2f57489a2424a
|
"""
Several methods to simplify expressions involving unit objects.
"""
from __future__ import division
from sympy.utilities.exceptions import SymPyDeprecationWarning
from sympy import Add, Function, Mul, Pow, Rational, Tuple, sympify
from sympy.core.compatibility import reduce, Iterable, ordered
from sympy.physics.units.dimensions import Dimension, dimsys_default
from sympy.physics.units.quantities import Quantity
from sympy.physics.units.prefixes import Prefix
from sympy.utilities.iterables import sift
def dim_simplify(expr):
"""
NOTE: this function could be deprecated in the future.
Simplify expression by recursively evaluating the dimension arguments.
This function proceeds to a very rough dimensional analysis. It tries to
simplify expression with dimensions, and it deletes all what multiplies a
dimension without being a dimension. This is necessary to avoid strange
behavior when Add(L, L) be transformed into Mul(2, L).
"""
SymPyDeprecationWarning(
deprecated_since_version="1.2",
feature="dimensional simplification function",
issue=13336,
useinstead="don't use",
).warn()
_, expr = Quantity._collect_factor_and_dimension(expr)
return expr
def _get_conversion_matrix_for_expr(expr, target_units):
from sympy import Matrix
expr_dim = Dimension(Quantity.get_dimensional_expr(expr))
dim_dependencies = dimsys_default.get_dimensional_dependencies(expr_dim, mark_dimensionless=True)
target_dims = [Dimension(Quantity.get_dimensional_expr(x)) for x in target_units]
canon_dim_units = {i for x in target_dims for i in dimsys_default.get_dimensional_dependencies(x, mark_dimensionless=True)}
canon_expr_units = {i for i in dim_dependencies}
if not canon_expr_units.issubset(canon_dim_units):
return None
canon_dim_units = sorted(canon_dim_units)
camat = Matrix([[dimsys_default.get_dimensional_dependencies(i, mark_dimensionless=True).get(j, 0) for i in target_dims] for j in canon_dim_units])
exprmat = Matrix([dim_dependencies.get(k, 0) for k in canon_dim_units])
res_exponents = camat.solve_least_squares(exprmat, method=None)
return res_exponents
def convert_to(expr, target_units):
"""
Convert ``expr`` to the same expression with all of its units and quantities
represented as factors of ``target_units``, whenever the dimension is compatible.
``target_units`` may be a single unit/quantity, or a collection of
units/quantities.
Examples
========
>>> from sympy.physics.units import speed_of_light, meter, gram, second, day
>>> from sympy.physics.units import mile, newton, kilogram, atomic_mass_constant
>>> from sympy.physics.units import kilometer, centimeter
>>> from sympy.physics.units import convert_to
>>> convert_to(mile, kilometer)
25146*kilometer/15625
>>> convert_to(mile, kilometer).n()
1.609344*kilometer
>>> convert_to(speed_of_light, meter/second)
299792458*meter/second
>>> convert_to(day, second)
86400*second
>>> 3*newton
3*newton
>>> convert_to(3*newton, kilogram*meter/second**2)
3*kilogram*meter/second**2
>>> convert_to(atomic_mass_constant, gram)
1.66053904e-24*gram
Conversion to multiple units:
>>> convert_to(speed_of_light, [meter, second])
299792458*meter/second
>>> convert_to(3*newton, [centimeter, gram, second])
300000*centimeter*gram/second**2
Conversion to Planck units:
>>> from sympy.physics.units import gravitational_constant, hbar
>>> convert_to(atomic_mass_constant, [gravitational_constant, speed_of_light, hbar]).n()
7.62950196312651e-20*gravitational_constant**(-0.5)*hbar**0.5*speed_of_light**0.5
"""
if not isinstance(target_units, (Iterable, Tuple)):
target_units = [target_units]
if isinstance(expr, Add):
return Add.fromiter(convert_to(i, target_units) for i in expr.args)
expr = sympify(expr)
if not isinstance(expr, Quantity) and expr.has(Quantity):
expr = expr.replace(lambda x: isinstance(x, Quantity), lambda x: x.convert_to(target_units))
def get_total_scale_factor(expr):
if isinstance(expr, Mul):
return reduce(lambda x, y: x * y, [get_total_scale_factor(i) for i in expr.args])
elif isinstance(expr, Pow):
return get_total_scale_factor(expr.base) ** expr.exp
elif isinstance(expr, Quantity):
return expr.scale_factor
return expr
depmat = _get_conversion_matrix_for_expr(expr, target_units)
if depmat is None:
return expr
expr_scale_factor = get_total_scale_factor(expr)
return expr_scale_factor * Mul.fromiter((1/get_total_scale_factor(u) * u) ** p for u, p in zip(target_units, depmat))
def quantity_simplify(expr):
"""Return an equivalent expression in which prefixes are replaced
with numerical values and all units of a given dimension are the
unified in a canonical manner.
Examples
========
>>> from sympy.physics.units.util import quantity_simplify
>>> from sympy.physics.units.prefixes import kilo
>>> from sympy.physics.units import foot, inch
>>> quantity_simplify(kilo*foot*inch)
250*foot**2/3
>>> quantity_simplify(foot - 6*inch)
foot/2
"""
if expr.is_Atom or not expr.has(Prefix, Quantity):
return expr
# replace all prefixes with numerical values
p = expr.atoms(Prefix)
expr = expr.xreplace({p: p.scale_factor for p in p})
# replace all quantities of given dimension with a canonical
# quantity, chosen from those in the expression
d = sift(expr.atoms(Quantity), lambda i: i.dimension)
for k in d:
if len(d[k]) == 1:
continue
v = list(ordered(d[k]))
ref = v[0]/v[0].scale_factor
expr = expr.xreplace({vi: ref*vi.scale_factor for vi in v[1:]})
return expr
def check_dimensions(expr):
"""Return expr if there are not unitless values added to
dimensional quantities, else raise a ValueError."""
from sympy.solvers.solveset import _term_factors
# the case of adding a number to a dimensional quantity
# is ignored for the sake of SymPy core routines, so this
# function will raise an error now if such an addend is
# found.
# Also, when doing substitutions, multiplicative constants
# might be introduced, so remove those now
adds = expr.atoms(Add)
DIM_OF = dimsys_default.get_dimensional_dependencies
for a in adds:
deset = set()
for ai in a.args:
if ai.is_number:
deset.add(())
continue
dims = []
skip = False
for i in Mul.make_args(ai):
if i.has(Quantity):
i = Dimension(Quantity.get_dimensional_expr(i))
if i.has(Dimension):
dims.extend(DIM_OF(i).items())
elif i.free_symbols:
skip = True
break
if not skip:
deset.add(tuple(sorted(dims)))
if len(deset) > 1:
raise ValueError(
"addends have incompatible dimensions")
# clear multiplicative constants on Dimensions which may be
# left after substitution
reps = {}
for m in expr.atoms(Mul):
if any(isinstance(i, Dimension) for i in m.args):
reps[m] = m.func(*[
i for i in m.args if not i.is_number])
return expr.xreplace(reps)
|
1878f2aebd9be98c985725162b764340519e6170c7eeb498b1a1b8b8ecbfd18d
|
"""
Physical quantities.
"""
from __future__ import division
from sympy import (Abs, Add, AtomicExpr, Basic, Derivative, Function, Mul,
Pow, S, Symbol, sympify)
from sympy.core.compatibility import string_types
from sympy.physics.units import Dimension, dimensions
from sympy.physics.units.dimensions import dimsys_default, DimensionSystem
from sympy.physics.units.prefixes import Prefix
from sympy.utilities.exceptions import SymPyDeprecationWarning
class Quantity(AtomicExpr):
"""
Physical quantity: can be a unit of measure, a constant or a generic quantity.
"""
is_commutative = True
is_real = True
is_number = False
is_nonzero = True
_diff_wrt = True
def __new__(cls, name, abbrev=None, dimension=None, scale_factor=None, **assumptions):
if not isinstance(name, Symbol):
name = Symbol(name)
# For Quantity(name, dim, scale, abbrev) to work like in the
# old version of Sympy:
if not isinstance(abbrev, string_types) and not \
isinstance(abbrev, Symbol):
dimension, scale_factor, abbrev = abbrev, dimension, scale_factor
if dimension is not None:
SymPyDeprecationWarning(
deprecated_since_version="1.3",
issue=14319,
feature="Quantity arguments",
useinstead="SI_quantity_dimension_map",
).warn()
if scale_factor is not None:
SymPyDeprecationWarning(
deprecated_since_version="1.3",
issue=14319,
feature="Quantity arguments",
useinstead="SI_quantity_scale_factors",
).warn()
if abbrev is None:
abbrev = name
elif isinstance(abbrev, string_types):
abbrev = Symbol(abbrev)
obj = AtomicExpr.__new__(cls, name, abbrev)
obj._name = name
obj._abbrev = abbrev
if dimension is not None:
# TODO: remove after deprecation:
obj.set_dimension(dimension)
if scale_factor is not None:
# TODO: remove after deprecation:
obj.set_scale_factor(scale_factor)
return obj
### Currently only SI is supported: ###
# Dimensional representations for the SI units:
SI_quantity_dimension_map = {}
# Scale factors in SI units:
SI_quantity_scale_factors = {}
def set_dimension(self, dimension, unit_system="SI"):
from sympy.physics.units.dimensions import dimsys_default, DimensionSystem
if unit_system != "SI":
# TODO: add support for more units and dimension systems:
raise NotImplementedError("Currently only SI is supported")
dim_sys = dimsys_default
if not isinstance(dimension, dimensions.Dimension):
if dimension == 1:
dimension = Dimension(1)
else:
raise ValueError("expected dimension or 1")
else:
for dim_sym in dimension.name.atoms(Dimension):
if dim_sym not in [i.name for i in dim_sys._dimensional_dependencies]:
raise ValueError("Dimension %s is not registered in the "
"dimensional dependency tree." % dim_sym)
Quantity.SI_quantity_dimension_map[self] = dimension
def set_scale_factor(self, scale_factor, unit_system="SI"):
if unit_system != "SI":
# TODO: add support for more units and dimension systems:
raise NotImplementedError("Currently only SI is supported")
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: x.scale_factor)
Quantity.SI_quantity_scale_factors[self] = scale_factor
@property
def name(self):
return self._name
@property
def dimension(self):
# TODO: add support for units other than SI:
return Quantity.SI_quantity_dimension_map[self]
@property
def abbrev(self):
"""
Symbol representing the unit name.
Prepend the abbreviation with the prefix symbol if it is defines.
"""
return self._abbrev
@property
def scale_factor(self):
"""
Overall magnitude of the quantity as compared to the canonical units.
"""
return Quantity.SI_quantity_scale_factors.get(self, S.One)
def _eval_is_positive(self):
return self.scale_factor.is_positive
def _eval_is_constant(self):
return self.scale_factor.is_constant()
def _eval_Abs(self):
scale_factor = Abs(self.scale_factor)
if scale_factor == self.scale_factor:
return self
return None
q = self.func(self.name, self.abbrev)
def _eval_subs(self, old, new):
if isinstance(new, Quantity) and self != old:
return self
@staticmethod
def get_dimensional_expr(expr):
if isinstance(expr, Mul):
return Mul(*[Quantity.get_dimensional_expr(i) for i in expr.args])
elif isinstance(expr, Pow):
return Quantity.get_dimensional_expr(expr.base) ** expr.exp
elif isinstance(expr, Add):
return Quantity.get_dimensional_expr(expr.args[0])
elif isinstance(expr, Derivative):
dim = Quantity.get_dimensional_expr(expr.expr)
for independent, count in expr.variable_count:
dim /= Quantity.get_dimensional_expr(independent)**count
return dim
elif isinstance(expr, Function):
args = [Quantity.get_dimensional_expr(arg) for arg in expr.args]
if all(i == 1 for i in args):
return S.One
return expr.func(*args)
elif isinstance(expr, Quantity):
return expr.dimension.name
return S.One
@staticmethod
def _collect_factor_and_dimension(expr):
"""Return tuple with factor expression and dimension expression."""
if isinstance(expr, Quantity):
return expr.scale_factor, expr.dimension
elif isinstance(expr, Mul):
factor = 1
dimension = Dimension(1)
for arg in expr.args:
arg_factor, arg_dim = Quantity._collect_factor_and_dimension(arg)
factor *= arg_factor
dimension *= arg_dim
return factor, dimension
elif isinstance(expr, Pow):
factor, dim = Quantity._collect_factor_and_dimension(expr.base)
exp_factor, exp_dim = Quantity._collect_factor_and_dimension(expr.exp)
if exp_dim.is_dimensionless:
exp_dim = 1
return factor ** exp_factor, dim ** (exp_factor * exp_dim)
elif isinstance(expr, Add):
factor, dim = Quantity._collect_factor_and_dimension(expr.args[0])
for addend in expr.args[1:]:
addend_factor, addend_dim = \
Quantity._collect_factor_and_dimension(addend)
if dim != addend_dim:
raise ValueError(
'Dimension of "{0}" is {1}, '
'but it should be {2}'.format(
addend, addend_dim.name, dim.name))
factor += addend_factor
return factor, dim
elif isinstance(expr, Derivative):
factor, dim = Quantity._collect_factor_and_dimension(expr.args[0])
for independent, count in expr.variable_count:
ifactor, idim = Quantity._collect_factor_and_dimension(independent)
factor /= ifactor**count
dim /= idim**count
return factor, dim
elif isinstance(expr, Function):
fds = [Quantity._collect_factor_and_dimension(
arg) for arg in expr.args]
return (expr.func(*(f[0] for f in fds)),
expr.func(*(d[1] for d in fds)))
elif isinstance(expr, Dimension):
return 1, expr
else:
return expr, Dimension(1)
def convert_to(self, other):
"""
Convert the quantity to another quantity of same dimensions.
Examples
========
>>> from sympy.physics.units import speed_of_light, meter, second
>>> speed_of_light
speed_of_light
>>> speed_of_light.convert_to(meter/second)
299792458*meter/second
>>> from sympy.physics.units import liter
>>> liter.convert_to(meter**3)
meter**3/1000
"""
from .util import convert_to
return convert_to(self, other)
@property
def free_symbols(self):
"""Return free symbols from quantity."""
return self.scale_factor.free_symbols
|
523d6f085b7a35138ffbb76312a15cb54709c4d862f52b4f0b70a6e32fbff9c2
|
from sympy import Derivative
from sympy.core.function import UndefinedFunction, AppliedUndef
from sympy.core.symbol import Symbol
from sympy.interactive.printing import init_printing
from sympy.printing.conventions import split_super_sub
from sympy.printing.latex import LatexPrinter, translate
from sympy.printing.pretty.pretty import PrettyPrinter
from sympy.printing.pretty.pretty_symbology import center_accent
from sympy.printing.str import StrPrinter
__all__ = ['vprint', 'vsstrrepr', 'vsprint', 'vpprint', 'vlatex',
'init_vprinting']
class VectorStrPrinter(StrPrinter):
"""String Printer for vector expressions. """
def _print_Derivative(self, e):
from sympy.physics.vector.functions import dynamicsymbols
t = dynamicsymbols._t
if (bool(sum([i == t for i in e.variables])) &
isinstance(type(e.args[0]), UndefinedFunction)):
ol = str(e.args[0].func)
for i, v in enumerate(e.variables):
ol += dynamicsymbols._str
return ol
else:
return StrPrinter().doprint(e)
def _print_Function(self, e):
from sympy.physics.vector.functions import dynamicsymbols
t = dynamicsymbols._t
if isinstance(type(e), UndefinedFunction):
return StrPrinter().doprint(e).replace("(%s)" % t, '')
return e.func.__name__ + "(%s)" % self.stringify(e.args, ", ")
class VectorStrReprPrinter(VectorStrPrinter):
"""String repr printer for vector expressions."""
def _print_str(self, s):
return repr(s)
class VectorLatexPrinter(LatexPrinter):
"""Latex Printer for vector expressions. """
def _print_Function(self, expr, exp=None):
from sympy.physics.vector.functions import dynamicsymbols
func = expr.func.__name__
t = dynamicsymbols._t
if hasattr(self, '_print_' + func) and \
not isinstance(type(expr), UndefinedFunction):
return getattr(self, '_print_' + func)(expr, exp)
elif isinstance(type(expr), UndefinedFunction) and (expr.args == (t,)):
name, supers, subs = split_super_sub(func)
name = translate(name)
supers = [translate(sup) for sup in supers]
subs = [translate(sub) for sub in subs]
if len(supers) != 0:
supers = r"^{%s}" % "".join(supers)
else:
supers = r""
if len(subs) != 0:
subs = r"_{%s}" % "".join(subs)
else:
subs = r""
if exp:
supers += r"^{%s}" % self._print(exp)
return r"%s" % (name + supers + subs)
else:
args = [str(self._print(arg)) for arg in expr.args]
# How inverse trig functions should be displayed, formats are:
# abbreviated: asin, full: arcsin, power: sin^-1
inv_trig_style = self._settings['inv_trig_style']
# If we are dealing with a power-style inverse trig function
inv_trig_power_case = False
# If it is applicable to fold the argument brackets
can_fold_brackets = self._settings['fold_func_brackets'] and \
len(args) == 1 and \
not self._needs_function_brackets(expr.args[0])
inv_trig_table = ["asin", "acos", "atan", "acot"]
# If the function is an inverse trig function, handle the style
if func in inv_trig_table:
if inv_trig_style == "abbreviated":
pass
elif inv_trig_style == "full":
func = "arc" + func[1:]
elif inv_trig_style == "power":
func = func[1:]
inv_trig_power_case = True
# Can never fold brackets if we're raised to a power
if exp is not None:
can_fold_brackets = False
if inv_trig_power_case:
name = r"\operatorname{%s}^{-1}" % func
elif exp is not None:
name = r"\operatorname{%s}^{%s}" % (func, exp)
else:
name = r"\operatorname{%s}" % func
if can_fold_brackets:
name += r"%s"
else:
name += r"\left(%s\right)"
if inv_trig_power_case and exp is not None:
name += r"^{%s}" % exp
return name % ",".join(args)
def _print_Derivative(self, der_expr):
from sympy.physics.vector.functions import dynamicsymbols
# make sure it is in the right form
der_expr = der_expr.doit()
if not isinstance(der_expr, Derivative):
return r"\left(%s\right)" % self.doprint(der_expr)
# check if expr is a dynamicsymbol
t = dynamicsymbols._t
expr = der_expr.expr
red = expr.atoms(AppliedUndef)
syms = der_expr.variables
test1 = not all([True for i in red if i.free_symbols == {t}])
test2 = not all([(t == i) for i in syms])
if test1 or test2:
return LatexPrinter().doprint(der_expr)
# done checking
dots = len(syms)
base = self._print_Function(expr)
base_split = base.split('_', 1)
base = base_split[0]
if dots == 1:
base = r"\dot{%s}" % base
elif dots == 2:
base = r"\ddot{%s}" % base
elif dots == 3:
base = r"\dddot{%s}" % base
elif dots == 4:
base = r"\ddddot{%s}" % base
else: # Fallback to standard printing
return LatexPrinter().doprint(der_expr)
if len(base_split) is not 1:
base += '_' + base_split[1]
return base
class VectorPrettyPrinter(PrettyPrinter):
"""Pretty Printer for vectorialexpressions. """
def _print_Derivative(self, deriv):
from sympy.physics.vector.functions import dynamicsymbols
# XXX use U('PARTIAL DIFFERENTIAL') here ?
t = dynamicsymbols._t
dot_i = 0
syms = list(reversed(deriv.variables))
while len(syms) > 0:
if syms[-1] == t:
syms.pop()
dot_i += 1
else:
return super(VectorPrettyPrinter, self)._print_Derivative(deriv)
if not (isinstance(type(deriv.expr), UndefinedFunction)
and (deriv.expr.args == (t,))):
return super(VectorPrettyPrinter, self)._print_Derivative(deriv)
else:
pform = self._print_Function(deriv.expr)
# the following condition would happen with some sort of non-standard
# dynamic symbol I guess, so we'll just print the SymPy way
if len(pform.picture) > 1:
return super(VectorPrettyPrinter, self)._print_Derivative(deriv)
# There are only special symbols up to fourth-order derivatives
if dot_i >= 5:
return super(VectorPrettyPrinter, self)._print_Derivative(deriv)
# Deal with special symbols
dots = {0 : u"",
1 : u"\N{COMBINING DOT ABOVE}",
2 : u"\N{COMBINING DIAERESIS}",
3 : u"\N{COMBINING THREE DOTS ABOVE}",
4 : u"\N{COMBINING FOUR DOTS ABOVE}"}
d = pform.__dict__
#if unicode is false then calculate number of apostrophes needed and add to output
if not self._use_unicode:
apostrophes = ""
for i in range(0, dot_i):
apostrophes += "'"
d['picture'][0] += apostrophes + "(t)"
else:
d['picture'] = [center_accent(d['picture'][0], dots[dot_i])]
d['unicode'] = center_accent(d['unicode'], dots[dot_i])
return pform
def _print_Function(self, e):
from sympy.physics.vector.functions import dynamicsymbols
t = dynamicsymbols._t
# XXX works only for applied functions
func = e.func
args = e.args
func_name = func.__name__
pform = self._print_Symbol(Symbol(func_name))
# If this function is an Undefined function of t, it is probably a
# dynamic symbol, so we'll skip the (t). The rest of the code is
# identical to the normal PrettyPrinter code
if not (isinstance(func, UndefinedFunction) and (args == (t,))):
return super(VectorPrettyPrinter, self)._print_Function(e)
return pform
def vprint(expr, **settings):
r"""Function for printing of expressions generated in the
sympy.physics vector package.
Extends SymPy's StrPrinter, takes the same setting accepted by SymPy's
`sstr()`, and is equivalent to `print(sstr(foo))`.
Parameters
==========
expr : valid SymPy object
SymPy expression to print.
settings : args
Same as the settings accepted by SymPy's sstr().
Examples
========
>>> from sympy.physics.vector import vprint, dynamicsymbols
>>> u1 = dynamicsymbols('u1')
>>> print(u1)
u1(t)
>>> vprint(u1)
u1
"""
outstr = vsprint(expr, **settings)
from sympy.core.compatibility import builtins
if (outstr != 'None'):
builtins._ = outstr
print(outstr)
def vsstrrepr(expr, **settings):
"""Function for displaying expression representation's with vector
printing enabled.
Parameters
==========
expr : valid SymPy object
SymPy expression to print.
settings : args
Same as the settings accepted by SymPy's sstrrepr().
"""
p = VectorStrReprPrinter(settings)
return p.doprint(expr)
def vsprint(expr, **settings):
r"""Function for displaying expressions generated in the
sympy.physics vector package.
Returns the output of vprint() as a string.
Parameters
==========
expr : valid SymPy object
SymPy expression to print
settings : args
Same as the settings accepted by SymPy's sstr().
Examples
========
>>> from sympy.physics.vector import vsprint, dynamicsymbols
>>> u1, u2 = dynamicsymbols('u1 u2')
>>> u2d = dynamicsymbols('u2', level=1)
>>> print("%s = %s" % (u1, u2 + u2d))
u1(t) = u2(t) + Derivative(u2(t), t)
>>> print("%s = %s" % (vsprint(u1), vsprint(u2 + u2d)))
u1 = u2 + u2'
"""
string_printer = VectorStrPrinter(settings)
return string_printer.doprint(expr)
def vpprint(expr, **settings):
r"""Function for pretty printing of expressions generated in the
sympy.physics vector package.
Mainly used for expressions not inside a vector; the output of running
scripts and generating equations of motion. Takes the same options as
SymPy's pretty_print(); see that function for more information.
Parameters
==========
expr : valid SymPy object
SymPy expression to pretty print
settings : args
Same as those accepted by SymPy's pretty_print.
"""
pp = VectorPrettyPrinter(settings)
# Note that this is copied from sympy.printing.pretty.pretty_print:
# XXX: this is an ugly hack, but at least it works
use_unicode = pp._settings['use_unicode']
from sympy.printing.pretty.pretty_symbology import pretty_use_unicode
uflag = pretty_use_unicode(use_unicode)
try:
return pp.doprint(expr)
finally:
pretty_use_unicode(uflag)
def vlatex(expr, **settings):
r"""Function for printing latex representation of sympy.physics.vector
objects.
For latex representation of Vectors, Dyadics, and dynamicsymbols. Takes the
same options as SymPy's latex(); see that function for more information;
Parameters
==========
expr : valid SymPy object
SymPy expression to represent in LaTeX form
settings : args
Same as latex()
Examples
========
>>> from sympy.physics.vector import vlatex, ReferenceFrame, dynamicsymbols
>>> N = ReferenceFrame('N')
>>> q1, q2 = dynamicsymbols('q1 q2')
>>> q1d, q2d = dynamicsymbols('q1 q2', 1)
>>> q1dd, q2dd = dynamicsymbols('q1 q2', 2)
>>> vlatex(N.x + N.y)
'\\mathbf{\\hat{n}_x} + \\mathbf{\\hat{n}_y}'
>>> vlatex(q1 + q2)
'q_{1} + q_{2}'
>>> vlatex(q1d)
'\\dot{q}_{1}'
>>> vlatex(q1 * q2d)
'q_{1} \\dot{q}_{2}'
>>> vlatex(q1dd * q1 / q1d)
'\\frac{q_{1} \\ddot{q}_{1}}{\\dot{q}_{1}}'
"""
latex_printer = VectorLatexPrinter(settings)
return latex_printer.doprint(expr)
def init_vprinting(**kwargs):
"""Initializes time derivative printing for all SymPy objects, i.e. any
functions of time will be displayed in a more compact notation. The main
benefit of this is for printing of time derivatives; instead of
displaying as ``Derivative(f(t),t)``, it will display ``f'``. This is
only actually needed for when derivatives are present and are not in a
physics.vector.Vector or physics.vector.Dyadic object. This function is a
light wrapper to `sympy.interactive.init_printing`. Any keyword
arguments for it are valid here.
{0}
Examples
========
>>> from sympy import Function, symbols
>>> from sympy.physics.vector import init_vprinting
>>> t, x = symbols('t, x')
>>> omega = Function('omega')
>>> omega(x).diff()
Derivative(omega(x), x)
>>> omega(t).diff()
Derivative(omega(t), t)
Now use the string printer:
>>> init_vprinting(pretty_print=False)
>>> omega(x).diff()
Derivative(omega(x), x)
>>> omega(t).diff()
omega'
"""
kwargs['str_printer'] = vsstrrepr
kwargs['pretty_printer'] = vpprint
kwargs['latex_printer'] = vlatex
init_printing(**kwargs)
params = init_printing.__doc__.split('Examples\n ========')[0]
init_vprinting.__doc__ = init_vprinting.__doc__.format(params)
|
a6da48c3ec457ac2b19cf45a725dace00e29070351658e04e349d7c47ed204ec
|
from __future__ import print_function, division
from sympy.core.compatibility import range, string_types
from .vector import Vector, _check_vector
from .frame import _check_frame
__all__ = ['Point']
class Point(object):
"""This object represents a point in a dynamic system.
It stores the: position, velocity, and acceleration of a point.
The position is a vector defined as the vector distance from a parent
point to this point.
"""
def __init__(self, name):
"""Initialization of a Point object. """
self.name = name
self._pos_dict = {}
self._vel_dict = {}
self._acc_dict = {}
self._pdlist = [self._pos_dict, self._vel_dict, self._acc_dict]
def __str__(self):
return self.name
__repr__ = __str__
def _check_point(self, other):
if not isinstance(other, Point):
raise TypeError('A Point must be supplied')
def _pdict_list(self, other, num):
"""Creates a list from self to other using _dcm_dict. """
outlist = [[self]]
oldlist = [[]]
while outlist != oldlist:
oldlist = outlist[:]
for i, v in enumerate(outlist):
templist = v[-1]._pdlist[num].keys()
for i2, v2 in enumerate(templist):
if not v.__contains__(v2):
littletemplist = v + [v2]
if not outlist.__contains__(littletemplist):
outlist.append(littletemplist)
for i, v in enumerate(oldlist):
if v[-1] != other:
outlist.remove(v)
outlist.sort(key=len)
if len(outlist) != 0:
return outlist[0]
raise ValueError('No Connecting Path found between ' + other.name +
' and ' + self.name)
def a1pt_theory(self, otherpoint, outframe, interframe):
"""Sets the acceleration of this point with the 1-point theory.
The 1-point theory for point acceleration looks like this:
^N a^P = ^B a^P + ^N a^O + ^N alpha^B x r^OP + ^N omega^B x (^N omega^B
x r^OP) + 2 ^N omega^B x ^B v^P
where O is a point fixed in B, P is a point moving in B, and B is
rotating in frame N.
Parameters
==========
otherpoint : Point
The first point of the 1-point theory (O)
outframe : ReferenceFrame
The frame we want this point's acceleration defined in (N)
fixedframe : ReferenceFrame
The intermediate frame in this calculation (B)
Examples
========
>>> from sympy.physics.vector import Point, ReferenceFrame
>>> from sympy.physics.vector import Vector, dynamicsymbols
>>> q = dynamicsymbols('q')
>>> q2 = dynamicsymbols('q2')
>>> qd = dynamicsymbols('q', 1)
>>> q2d = dynamicsymbols('q2', 1)
>>> N = ReferenceFrame('N')
>>> B = ReferenceFrame('B')
>>> B.set_ang_vel(N, 5 * B.y)
>>> O = Point('O')
>>> P = O.locatenew('P', q * B.x)
>>> P.set_vel(B, qd * B.x + q2d * B.y)
>>> O.set_vel(N, 0)
>>> P.a1pt_theory(O, N, B)
(-25*q + q'')*B.x + q2''*B.y - 10*q'*B.z
"""
_check_frame(outframe)
_check_frame(interframe)
self._check_point(otherpoint)
dist = self.pos_from(otherpoint)
v = self.vel(interframe)
a1 = otherpoint.acc(outframe)
a2 = self.acc(interframe)
omega = interframe.ang_vel_in(outframe)
alpha = interframe.ang_acc_in(outframe)
self.set_acc(outframe, a2 + 2 * (omega ^ v) + a1 + (alpha ^ dist) +
(omega ^ (omega ^ dist)))
return self.acc(outframe)
def a2pt_theory(self, otherpoint, outframe, fixedframe):
"""Sets the acceleration of this point with the 2-point theory.
The 2-point theory for point acceleration looks like this:
^N a^P = ^N a^O + ^N alpha^B x r^OP + ^N omega^B x (^N omega^B x r^OP)
where O and P are both points fixed in frame B, which is rotating in
frame N.
Parameters
==========
otherpoint : Point
The first point of the 2-point theory (O)
outframe : ReferenceFrame
The frame we want this point's acceleration defined in (N)
fixedframe : ReferenceFrame
The frame in which both points are fixed (B)
Examples
========
>>> from sympy.physics.vector import Point, ReferenceFrame, dynamicsymbols
>>> q = dynamicsymbols('q')
>>> qd = dynamicsymbols('q', 1)
>>> N = ReferenceFrame('N')
>>> B = N.orientnew('B', 'Axis', [q, N.z])
>>> O = Point('O')
>>> P = O.locatenew('P', 10 * B.x)
>>> O.set_vel(N, 5 * N.x)
>>> P.a2pt_theory(O, N, B)
- 10*q'**2*B.x + 10*q''*B.y
"""
_check_frame(outframe)
_check_frame(fixedframe)
self._check_point(otherpoint)
dist = self.pos_from(otherpoint)
a = otherpoint.acc(outframe)
omega = fixedframe.ang_vel_in(outframe)
alpha = fixedframe.ang_acc_in(outframe)
self.set_acc(outframe, a + (alpha ^ dist) + (omega ^ (omega ^ dist)))
return self.acc(outframe)
def acc(self, frame):
"""The acceleration Vector of this Point in a ReferenceFrame.
Parameters
==========
frame : ReferenceFrame
The frame in which the returned acceleration vector will be defined in
Examples
========
>>> from sympy.physics.vector import Point, ReferenceFrame
>>> N = ReferenceFrame('N')
>>> p1 = Point('p1')
>>> p1.set_acc(N, 10 * N.x)
>>> p1.acc(N)
10*N.x
"""
_check_frame(frame)
if not (frame in self._acc_dict):
if self._vel_dict[frame] != 0:
return (self._vel_dict[frame]).dt(frame)
else:
return Vector(0)
return self._acc_dict[frame]
def locatenew(self, name, value):
"""Creates a new point with a position defined from this point.
Parameters
==========
name : str
The name for the new point
value : Vector
The position of the new point relative to this point
Examples
========
>>> from sympy.physics.vector import ReferenceFrame, Point
>>> N = ReferenceFrame('N')
>>> P1 = Point('P1')
>>> P2 = P1.locatenew('P2', 10 * N.x)
"""
if not isinstance(name, string_types):
raise TypeError('Must supply a valid name')
if value == 0:
value = Vector(0)
value = _check_vector(value)
p = Point(name)
p.set_pos(self, value)
self.set_pos(p, -value)
return p
def pos_from(self, otherpoint):
"""Returns a Vector distance between this Point and the other Point.
Parameters
==========
otherpoint : Point
The otherpoint we are locating this one relative to
Examples
========
>>> from sympy.physics.vector import Point, ReferenceFrame
>>> N = ReferenceFrame('N')
>>> p1 = Point('p1')
>>> p2 = Point('p2')
>>> p1.set_pos(p2, 10 * N.x)
>>> p1.pos_from(p2)
10*N.x
"""
outvec = Vector(0)
plist = self._pdict_list(otherpoint, 0)
for i in range(len(plist) - 1):
outvec += plist[i]._pos_dict[plist[i + 1]]
return outvec
def set_acc(self, frame, value):
"""Used to set the acceleration of this Point in a ReferenceFrame.
Parameters
==========
frame : ReferenceFrame
The frame in which this point's acceleration is defined
value : Vector
The vector value of this point's acceleration in the frame
Examples
========
>>> from sympy.physics.vector import Point, ReferenceFrame
>>> N = ReferenceFrame('N')
>>> p1 = Point('p1')
>>> p1.set_acc(N, 10 * N.x)
>>> p1.acc(N)
10*N.x
"""
if value == 0:
value = Vector(0)
value = _check_vector(value)
_check_frame(frame)
self._acc_dict.update({frame: value})
def set_pos(self, otherpoint, value):
"""Used to set the position of this point w.r.t. another point.
Parameters
==========
otherpoint : Point
The other point which this point's location is defined relative to
value : Vector
The vector which defines the location of this point
Examples
========
>>> from sympy.physics.vector import Point, ReferenceFrame
>>> N = ReferenceFrame('N')
>>> p1 = Point('p1')
>>> p2 = Point('p2')
>>> p1.set_pos(p2, 10 * N.x)
>>> p1.pos_from(p2)
10*N.x
"""
if value == 0:
value = Vector(0)
value = _check_vector(value)
self._check_point(otherpoint)
self._pos_dict.update({otherpoint: value})
otherpoint._pos_dict.update({self: -value})
def set_vel(self, frame, value):
"""Sets the velocity Vector of this Point in a ReferenceFrame.
Parameters
==========
frame : ReferenceFrame
The frame in which this point's velocity is defined
value : Vector
The vector value of this point's velocity in the frame
Examples
========
>>> from sympy.physics.vector import Point, ReferenceFrame
>>> N = ReferenceFrame('N')
>>> p1 = Point('p1')
>>> p1.set_vel(N, 10 * N.x)
>>> p1.vel(N)
10*N.x
"""
if value == 0:
value = Vector(0)
value = _check_vector(value)
_check_frame(frame)
self._vel_dict.update({frame: value})
def v1pt_theory(self, otherpoint, outframe, interframe):
"""Sets the velocity of this point with the 1-point theory.
The 1-point theory for point velocity looks like this:
^N v^P = ^B v^P + ^N v^O + ^N omega^B x r^OP
where O is a point fixed in B, P is a point moving in B, and B is
rotating in frame N.
Parameters
==========
otherpoint : Point
The first point of the 2-point theory (O)
outframe : ReferenceFrame
The frame we want this point's velocity defined in (N)
interframe : ReferenceFrame
The intermediate frame in this calculation (B)
Examples
========
>>> from sympy.physics.vector import Point, ReferenceFrame
>>> from sympy.physics.vector import Vector, dynamicsymbols
>>> q = dynamicsymbols('q')
>>> q2 = dynamicsymbols('q2')
>>> qd = dynamicsymbols('q', 1)
>>> q2d = dynamicsymbols('q2', 1)
>>> N = ReferenceFrame('N')
>>> B = ReferenceFrame('B')
>>> B.set_ang_vel(N, 5 * B.y)
>>> O = Point('O')
>>> P = O.locatenew('P', q * B.x)
>>> P.set_vel(B, qd * B.x + q2d * B.y)
>>> O.set_vel(N, 0)
>>> P.v1pt_theory(O, N, B)
q'*B.x + q2'*B.y - 5*q*B.z
"""
_check_frame(outframe)
_check_frame(interframe)
self._check_point(otherpoint)
dist = self.pos_from(otherpoint)
v1 = self.vel(interframe)
v2 = otherpoint.vel(outframe)
omega = interframe.ang_vel_in(outframe)
self.set_vel(outframe, v1 + v2 + (omega ^ dist))
return self.vel(outframe)
def v2pt_theory(self, otherpoint, outframe, fixedframe):
"""Sets the velocity of this point with the 2-point theory.
The 2-point theory for point velocity looks like this:
^N v^P = ^N v^O + ^N omega^B x r^OP
where O and P are both points fixed in frame B, which is rotating in
frame N.
Parameters
==========
otherpoint : Point
The first point of the 2-point theory (O)
outframe : ReferenceFrame
The frame we want this point's velocity defined in (N)
fixedframe : ReferenceFrame
The frame in which both points are fixed (B)
Examples
========
>>> from sympy.physics.vector import Point, ReferenceFrame, dynamicsymbols
>>> q = dynamicsymbols('q')
>>> qd = dynamicsymbols('q', 1)
>>> N = ReferenceFrame('N')
>>> B = N.orientnew('B', 'Axis', [q, N.z])
>>> O = Point('O')
>>> P = O.locatenew('P', 10 * B.x)
>>> O.set_vel(N, 5 * N.x)
>>> P.v2pt_theory(O, N, B)
5*N.x + 10*q'*B.y
"""
_check_frame(outframe)
_check_frame(fixedframe)
self._check_point(otherpoint)
dist = self.pos_from(otherpoint)
v = otherpoint.vel(outframe)
omega = fixedframe.ang_vel_in(outframe)
self.set_vel(outframe, v + (omega ^ dist))
return self.vel(outframe)
def vel(self, frame):
"""The velocity Vector of this Point in the ReferenceFrame.
Parameters
==========
frame : ReferenceFrame
The frame in which the returned velocity vector will be defined in
Examples
========
>>> from sympy.physics.vector import Point, ReferenceFrame
>>> N = ReferenceFrame('N')
>>> p1 = Point('p1')
>>> p1.set_vel(N, 10 * N.x)
>>> p1.vel(N)
10*N.x
"""
_check_frame(frame)
if not (frame in self._vel_dict):
raise ValueError('Velocity of point ' + self.name + ' has not been'
' defined in ReferenceFrame ' + frame.name)
return self._vel_dict[frame]
def partial_velocity(self, frame, *gen_speeds):
"""Returns the partial velocities of the linear velocity vector of this
point in the given frame with respect to one or more provided
generalized speeds.
Parameters
==========
frame : ReferenceFrame
The frame with which the velocity is defined in.
gen_speeds : functions of time
The generalized speeds.
Returns
=======
partial_velocities : tuple of Vector
The partial velocity vectors corresponding to the provided
generalized speeds.
Examples
========
>>> from sympy.physics.vector import ReferenceFrame, Point
>>> from sympy.physics.vector import dynamicsymbols
>>> N = ReferenceFrame('N')
>>> A = ReferenceFrame('A')
>>> p = Point('p')
>>> u1, u2 = dynamicsymbols('u1, u2')
>>> p.set_vel(N, u1 * N.x + u2 * A.y)
>>> p.partial_velocity(N, u1)
N.x
>>> p.partial_velocity(N, u1, u2)
(N.x, A.y)
"""
partials = [self.vel(frame).diff(speed, frame, var_in_dcm=False) for
speed in gen_speeds]
if len(partials) == 1:
return partials[0]
else:
return tuple(partials)
|
7d5943c57cfa00528d4d3061a30227e14552c9181bf8668cbb5bc8d210be58ca
|
from __future__ import print_function, division
from sympy.core.backend import (sympify, diff, sin, cos, Matrix, symbols,
Function, S, Symbol)
from sympy import integrate, trigsimp
from sympy.core.compatibility import reduce
from .vector import Vector, _check_vector
from .frame import CoordinateSym, _check_frame
from .dyadic import Dyadic
from .printing import vprint, vsprint, vpprint, vlatex, init_vprinting
from sympy.utilities.iterables import iterable
from sympy.utilities.misc import translate
__all__ = ['cross', 'dot', 'express', 'time_derivative', 'outer',
'kinematic_equations', 'get_motion_params', 'partial_velocity',
'dynamicsymbols', 'vprint', 'vsprint', 'vpprint', 'vlatex',
'init_vprinting']
def cross(vec1, vec2):
"""Cross product convenience wrapper for Vector.cross(): \n"""
if not isinstance(vec1, (Vector, Dyadic)):
raise TypeError('Cross product is between two vectors')
return vec1 ^ vec2
cross.__doc__ += Vector.cross.__doc__
def dot(vec1, vec2):
"""Dot product convenience wrapper for Vector.dot(): \n"""
if not isinstance(vec1, (Vector, Dyadic)):
raise TypeError('Dot product is between two vectors')
return vec1 & vec2
dot.__doc__ += Vector.dot.__doc__
def express(expr, frame, frame2=None, variables=False):
"""
Global function for 'express' functionality.
Re-expresses a Vector, scalar(sympyfiable) or Dyadic in given frame.
Refer to the local methods of Vector and Dyadic for details.
If 'variables' is True, then the coordinate variables (CoordinateSym
instances) of other frames present in the vector/scalar field or
dyadic expression are also substituted in terms of the base scalars of
this frame.
Parameters
==========
expr : Vector/Dyadic/scalar(sympyfiable)
The expression to re-express in ReferenceFrame 'frame'
frame: ReferenceFrame
The reference frame to express expr in
frame2 : ReferenceFrame
The other frame required for re-expression(only for Dyadic expr)
variables : boolean
Specifies whether to substitute the coordinate variables present
in expr, in terms of those of frame
Examples
========
>>> from sympy.physics.vector import ReferenceFrame, outer, dynamicsymbols
>>> N = ReferenceFrame('N')
>>> q = dynamicsymbols('q')
>>> B = N.orientnew('B', 'Axis', [q, N.z])
>>> d = outer(N.x, N.x)
>>> from sympy.physics.vector import express
>>> express(d, B, N)
cos(q)*(B.x|N.x) - sin(q)*(B.y|N.x)
>>> express(B.x, N)
cos(q)*N.x + sin(q)*N.y
>>> express(N[0], B, variables=True)
B_x*cos(q(t)) - B_y*sin(q(t))
"""
_check_frame(frame)
if expr == 0:
return expr
if isinstance(expr, Vector):
#Given expr is a Vector
if variables:
#If variables attribute is True, substitute
#the coordinate variables in the Vector
frame_list = [x[-1] for x in expr.args]
subs_dict = {}
for f in frame_list:
subs_dict.update(f.variable_map(frame))
expr = expr.subs(subs_dict)
#Re-express in this frame
outvec = Vector([])
for i, v in enumerate(expr.args):
if v[1] != frame:
temp = frame.dcm(v[1]) * v[0]
if Vector.simp:
temp = temp.applyfunc(lambda x:
trigsimp(x, method='fu'))
outvec += Vector([(temp, frame)])
else:
outvec += Vector([v])
return outvec
if isinstance(expr, Dyadic):
if frame2 is None:
frame2 = frame
_check_frame(frame2)
ol = Dyadic(0)
for i, v in enumerate(expr.args):
ol += express(v[0], frame, variables=variables) * \
(express(v[1], frame, variables=variables) |
express(v[2], frame2, variables=variables))
return ol
else:
if variables:
#Given expr is a scalar field
frame_set = set([])
expr = sympify(expr)
#Subsitute all the coordinate variables
for x in expr.free_symbols:
if isinstance(x, CoordinateSym)and x.frame != frame:
frame_set.add(x.frame)
subs_dict = {}
for f in frame_set:
subs_dict.update(f.variable_map(frame))
return expr.subs(subs_dict)
return expr
def time_derivative(expr, frame, order=1):
"""
Calculate the time derivative of a vector/scalar field function
or dyadic expression in given frame.
References
==========
https://en.wikipedia.org/wiki/Rotating_reference_frame#Time_derivatives_in_the_two_frames
Parameters
==========
expr : Vector/Dyadic/sympifyable
The expression whose time derivative is to be calculated
frame : ReferenceFrame
The reference frame to calculate the time derivative in
order : integer
The order of the derivative to be calculated
Examples
========
>>> from sympy.physics.vector import ReferenceFrame, dynamicsymbols
>>> from sympy import Symbol
>>> q1 = Symbol('q1')
>>> u1 = dynamicsymbols('u1')
>>> N = ReferenceFrame('N')
>>> A = N.orientnew('A', 'Axis', [q1, N.x])
>>> v = u1 * N.x
>>> A.set_ang_vel(N, 10*A.x)
>>> from sympy.physics.vector import time_derivative
>>> time_derivative(v, N)
u1'*N.x
>>> time_derivative(u1*A[0], N)
N_x*Derivative(u1(t), t)
>>> B = N.orientnew('B', 'Axis', [u1, N.z])
>>> from sympy.physics.vector import outer
>>> d = outer(N.x, N.x)
>>> time_derivative(d, B)
- u1'*(N.y|N.x) - u1'*(N.x|N.y)
"""
t = dynamicsymbols._t
_check_frame(frame)
if order == 0:
return expr
if order % 1 != 0 or order < 0:
raise ValueError("Unsupported value of order entered")
if isinstance(expr, Vector):
outlist = []
for i, v in enumerate(expr.args):
if v[1] == frame:
outlist += [(express(v[0], frame,
variables=True).diff(t), frame)]
else:
outlist += (time_derivative(Vector([v]), v[1]) + \
(v[1].ang_vel_in(frame) ^ Vector([v]))).args
outvec = Vector(outlist)
return time_derivative(outvec, frame, order - 1)
if isinstance(expr, Dyadic):
ol = Dyadic(0)
for i, v in enumerate(expr.args):
ol += (v[0].diff(t) * (v[1] | v[2]))
ol += (v[0] * (time_derivative(v[1], frame) | v[2]))
ol += (v[0] * (v[1] | time_derivative(v[2], frame)))
return time_derivative(ol, frame, order - 1)
else:
return diff(express(expr, frame, variables=True), t, order)
def outer(vec1, vec2):
"""Outer product convenience wrapper for Vector.outer():\n"""
if not isinstance(vec1, Vector):
raise TypeError('Outer product is between two Vectors')
return vec1 | vec2
outer.__doc__ += Vector.outer.__doc__
def kinematic_equations(speeds, coords, rot_type, rot_order=''):
"""Gives equations relating the qdot's to u's for a rotation type.
Supply rotation type and order as in orient. Speeds are assumed to be
body-fixed; if we are defining the orientation of B in A using by rot_type,
the angular velocity of B in A is assumed to be in the form: speed[0]*B.x +
speed[1]*B.y + speed[2]*B.z
Parameters
==========
speeds : list of length 3
The body fixed angular velocity measure numbers.
coords : list of length 3 or 4
The coordinates used to define the orientation of the two frames.
rot_type : str
The type of rotation used to create the equations. Body, Space, or
Quaternion only
rot_order : str or int
If applicable, the order of a series of rotations.
Examples
========
>>> from sympy.physics.vector import dynamicsymbols
>>> from sympy.physics.vector import kinematic_equations, vprint
>>> u1, u2, u3 = dynamicsymbols('u1 u2 u3')
>>> q1, q2, q3 = dynamicsymbols('q1 q2 q3')
>>> vprint(kinematic_equations([u1,u2,u3], [q1,q2,q3], 'body', '313'),
... order=None)
[-(u1*sin(q3) + u2*cos(q3))/sin(q2) + q1', -u1*cos(q3) + u2*sin(q3) + q2', (u1*sin(q3) + u2*cos(q3))*cos(q2)/sin(q2) - u3 + q3']
"""
# Code below is checking and sanitizing input
approved_orders = ('123', '231', '312', '132', '213', '321', '121', '131',
'212', '232', '313', '323', '1', '2', '3', '')
# make sure XYZ => 123 and rot_type is in lower case
rot_order = translate(str(rot_order), 'XYZxyz', '123123')
rot_type = rot_type.lower()
if not isinstance(speeds, (list, tuple)):
raise TypeError('Need to supply speeds in a list')
if len(speeds) != 3:
raise TypeError('Need to supply 3 body-fixed speeds')
if not isinstance(coords, (list, tuple)):
raise TypeError('Need to supply coordinates in a list')
if rot_type in ['body', 'space']:
if rot_order not in approved_orders:
raise ValueError('Not an acceptable rotation order')
if len(coords) != 3:
raise ValueError('Need 3 coordinates for body or space')
# Actual hard-coded kinematic differential equations
w1, w2, w3 = speeds
if w1 == w2 == w3 == 0:
return [S.Zero]*3
q1, q2, q3 = coords
q1d, q2d, q3d = [diff(i, dynamicsymbols._t) for i in coords]
s1, s2, s3 = [sin(q1), sin(q2), sin(q3)]
c1, c2, c3 = [cos(q1), cos(q2), cos(q3)]
if rot_type == 'body':
if rot_order == '123':
return [q1d - (w1 * c3 - w2 * s3) / c2, q2d - w1 * s3 - w2 *
c3, q3d - (-w1 * c3 + w2 * s3) * s2 / c2 - w3]
if rot_order == '231':
return [q1d - (w2 * c3 - w3 * s3) / c2, q2d - w2 * s3 - w3 *
c3, q3d - w1 - (- w2 * c3 + w3 * s3) * s2 / c2]
if rot_order == '312':
return [q1d - (-w1 * s3 + w3 * c3) / c2, q2d - w1 * c3 - w3 *
s3, q3d - (w1 * s3 - w3 * c3) * s2 / c2 - w2]
if rot_order == '132':
return [q1d - (w1 * c3 + w3 * s3) / c2, q2d + w1 * s3 - w3 *
c3, q3d - (w1 * c3 + w3 * s3) * s2 / c2 - w2]
if rot_order == '213':
return [q1d - (w1 * s3 + w2 * c3) / c2, q2d - w1 * c3 + w2 *
s3, q3d - (w1 * s3 + w2 * c3) * s2 / c2 - w3]
if rot_order == '321':
return [q1d - (w2 * s3 + w3 * c3) / c2, q2d - w2 * c3 + w3 *
s3, q3d - w1 - (w2 * s3 + w3 * c3) * s2 / c2]
if rot_order == '121':
return [q1d - (w2 * s3 + w3 * c3) / s2, q2d - w2 * c3 + w3 *
s3, q3d - w1 + (w2 * s3 + w3 * c3) * c2 / s2]
if rot_order == '131':
return [q1d - (-w2 * c3 + w3 * s3) / s2, q2d - w2 * s3 - w3 *
c3, q3d - w1 - (w2 * c3 - w3 * s3) * c2 / s2]
if rot_order == '212':
return [q1d - (w1 * s3 - w3 * c3) / s2, q2d - w1 * c3 - w3 *
s3, q3d - (-w1 * s3 + w3 * c3) * c2 / s2 - w2]
if rot_order == '232':
return [q1d - (w1 * c3 + w3 * s3) / s2, q2d + w1 * s3 - w3 *
c3, q3d + (w1 * c3 + w3 * s3) * c2 / s2 - w2]
if rot_order == '313':
return [q1d - (w1 * s3 + w2 * c3) / s2, q2d - w1 * c3 + w2 *
s3, q3d + (w1 * s3 + w2 * c3) * c2 / s2 - w3]
if rot_order == '323':
return [q1d - (-w1 * c3 + w2 * s3) / s2, q2d - w1 * s3 - w2 *
c3, q3d - (w1 * c3 - w2 * s3) * c2 / s2 - w3]
if rot_type == 'space':
if rot_order == '123':
return [q1d - w1 - (w2 * s1 + w3 * c1) * s2 / c2, q2d - w2 *
c1 + w3 * s1, q3d - (w2 * s1 + w3 * c1) / c2]
if rot_order == '231':
return [q1d - (w1 * c1 + w3 * s1) * s2 / c2 - w2, q2d + w1 *
s1 - w3 * c1, q3d - (w1 * c1 + w3 * s1) / c2]
if rot_order == '312':
return [q1d - (w1 * s1 + w2 * c1) * s2 / c2 - w3, q2d - w1 *
c1 + w2 * s1, q3d - (w1 * s1 + w2 * c1) / c2]
if rot_order == '132':
return [q1d - w1 - (-w2 * c1 + w3 * s1) * s2 / c2, q2d - w2 *
s1 - w3 * c1, q3d - (w2 * c1 - w3 * s1) / c2]
if rot_order == '213':
return [q1d - (w1 * s1 - w3 * c1) * s2 / c2 - w2, q2d - w1 *
c1 - w3 * s1, q3d - (-w1 * s1 + w3 * c1) / c2]
if rot_order == '321':
return [q1d - (-w1 * c1 + w2 * s1) * s2 / c2 - w3, q2d - w1 *
s1 - w2 * c1, q3d - (w1 * c1 - w2 * s1) / c2]
if rot_order == '121':
return [q1d - w1 + (w2 * s1 + w3 * c1) * c2 / s2, q2d - w2 *
c1 + w3 * s1, q3d - (w2 * s1 + w3 * c1) / s2]
if rot_order == '131':
return [q1d - w1 - (w2 * c1 - w3 * s1) * c2 / s2, q2d - w2 *
s1 - w3 * c1, q3d - (-w2 * c1 + w3 * s1) / s2]
if rot_order == '212':
return [q1d - (-w1 * s1 + w3 * c1) * c2 / s2 - w2, q2d - w1 *
c1 - w3 * s1, q3d - (w1 * s1 - w3 * c1) / s2]
if rot_order == '232':
return [q1d + (w1 * c1 + w3 * s1) * c2 / s2 - w2, q2d + w1 *
s1 - w3 * c1, q3d - (w1 * c1 + w3 * s1) / s2]
if rot_order == '313':
return [q1d + (w1 * s1 + w2 * c1) * c2 / s2 - w3, q2d - w1 *
c1 + w2 * s1, q3d - (w1 * s1 + w2 * c1) / s2]
if rot_order == '323':
return [q1d - (w1 * c1 - w2 * s1) * c2 / s2 - w3, q2d - w1 *
s1 - w2 * c1, q3d - (-w1 * c1 + w2 * s1) / s2]
elif rot_type == 'quaternion':
if rot_order != '':
raise ValueError('Cannot have rotation order for quaternion')
if len(coords) != 4:
raise ValueError('Need 4 coordinates for quaternion')
# Actual hard-coded kinematic differential equations
e0, e1, e2, e3 = coords
w = Matrix(speeds + [0])
E = Matrix([[e0, -e3, e2, e1], [e3, e0, -e1, e2], [-e2, e1, e0, e3],
[-e1, -e2, -e3, e0]])
edots = Matrix([diff(i, dynamicsymbols._t) for i in [e1, e2, e3, e0]])
return list(edots.T - 0.5 * w.T * E.T)
else:
raise ValueError('Not an approved rotation type for this function')
def get_motion_params(frame, **kwargs):
"""
Returns the three motion parameters - (acceleration, velocity, and
position) as vectorial functions of time in the given frame.
If a higher order differential function is provided, the lower order
functions are used as boundary conditions. For example, given the
acceleration, the velocity and position parameters are taken as
boundary conditions.
The values of time at which the boundary conditions are specified
are taken from timevalue1(for position boundary condition) and
timevalue2(for velocity boundary condition).
If any of the boundary conditions are not provided, they are taken
to be zero by default (zero vectors, in case of vectorial inputs). If
the boundary conditions are also functions of time, they are converted
to constants by substituting the time values in the dynamicsymbols._t
time Symbol.
This function can also be used for calculating rotational motion
parameters. Have a look at the Parameters and Examples for more clarity.
Parameters
==========
frame : ReferenceFrame
The frame to express the motion parameters in
acceleration : Vector
Acceleration of the object/frame as a function of time
velocity : Vector
Velocity as function of time or as boundary condition
of velocity at time = timevalue1
position : Vector
Velocity as function of time or as boundary condition
of velocity at time = timevalue1
timevalue1 : sympyfiable
Value of time for position boundary condition
timevalue2 : sympyfiable
Value of time for velocity boundary condition
Examples
========
>>> from sympy.physics.vector import ReferenceFrame, get_motion_params, dynamicsymbols
>>> from sympy import symbols
>>> R = ReferenceFrame('R')
>>> v1, v2, v3 = dynamicsymbols('v1 v2 v3')
>>> v = v1*R.x + v2*R.y + v3*R.z
>>> get_motion_params(R, position = v)
(v1''*R.x + v2''*R.y + v3''*R.z, v1'*R.x + v2'*R.y + v3'*R.z, v1*R.x + v2*R.y + v3*R.z)
>>> a, b, c = symbols('a b c')
>>> v = a*R.x + b*R.y + c*R.z
>>> get_motion_params(R, velocity = v)
(0, a*R.x + b*R.y + c*R.z, a*t*R.x + b*t*R.y + c*t*R.z)
>>> parameters = get_motion_params(R, acceleration = v)
>>> parameters[1]
a*t*R.x + b*t*R.y + c*t*R.z
>>> parameters[2]
a*t**2/2*R.x + b*t**2/2*R.y + c*t**2/2*R.z
"""
##Helper functions
def _process_vector_differential(vectdiff, condition, \
variable, ordinate, frame):
"""
Helper function for get_motion methods. Finds derivative of vectdiff wrt
variable, and its integral using the specified boundary condition at
value of variable = ordinate.
Returns a tuple of - (derivative, function and integral) wrt vectdiff
"""
#Make sure boundary condition is independent of 'variable'
if condition != 0:
condition = express(condition, frame, variables=True)
#Special case of vectdiff == 0
if vectdiff == Vector(0):
return (0, 0, condition)
#Express vectdiff completely in condition's frame to give vectdiff1
vectdiff1 = express(vectdiff, frame)
#Find derivative of vectdiff
vectdiff2 = time_derivative(vectdiff, frame)
#Integrate and use boundary condition
vectdiff0 = Vector(0)
lims = (variable, ordinate, variable)
for dim in frame:
function1 = vectdiff1.dot(dim)
abscissa = dim.dot(condition).subs({variable : ordinate})
# Indefinite integral of 'function1' wrt 'variable', using
# the given initial condition (ordinate, abscissa).
vectdiff0 += (integrate(function1, lims) + abscissa) * dim
#Return tuple
return (vectdiff2, vectdiff, vectdiff0)
##Function body
_check_frame(frame)
#Decide mode of operation based on user's input
if 'acceleration' in kwargs:
mode = 2
elif 'velocity' in kwargs:
mode = 1
else:
mode = 0
#All the possible parameters in kwargs
#Not all are required for every case
#If not specified, set to default values(may or may not be used in
#calculations)
conditions = ['acceleration', 'velocity', 'position',
'timevalue', 'timevalue1', 'timevalue2']
for i, x in enumerate(conditions):
if x not in kwargs:
if i < 3:
kwargs[x] = Vector(0)
else:
kwargs[x] = S(0)
elif i < 3:
_check_vector(kwargs[x])
else:
kwargs[x] = sympify(kwargs[x])
if mode == 2:
vel = _process_vector_differential(kwargs['acceleration'],
kwargs['velocity'],
dynamicsymbols._t,
kwargs['timevalue2'], frame)[2]
pos = _process_vector_differential(vel, kwargs['position'],
dynamicsymbols._t,
kwargs['timevalue1'], frame)[2]
return (kwargs['acceleration'], vel, pos)
elif mode == 1:
return _process_vector_differential(kwargs['velocity'],
kwargs['position'],
dynamicsymbols._t,
kwargs['timevalue1'], frame)
else:
vel = time_derivative(kwargs['position'], frame)
acc = time_derivative(vel, frame)
return (acc, vel, kwargs['position'])
def partial_velocity(vel_vecs, gen_speeds, frame):
"""Returns a list of partial velocities with respect to the provided
generalized speeds in the given reference frame for each of the supplied
velocity vectors.
The output is a list of lists. The outer list has a number of elements
equal to the number of supplied velocity vectors. The inner lists are, for
each velocity vector, the partial derivatives of that velocity vector with
respect to the generalized speeds supplied.
Parameters
==========
vel_vecs : iterable
An iterable of velocity vectors (angular or linear).
gen_speeds : iterable
An iterable of generalized speeds.
frame : ReferenceFrame
The reference frame that the partial derivatives are going to be taken
in.
Examples
========
>>> from sympy.physics.vector import Point, ReferenceFrame
>>> from sympy.physics.vector import dynamicsymbols
>>> from sympy.physics.vector import partial_velocity
>>> u = dynamicsymbols('u')
>>> N = ReferenceFrame('N')
>>> P = Point('P')
>>> P.set_vel(N, u * N.x)
>>> vel_vecs = [P.vel(N)]
>>> gen_speeds = [u]
>>> partial_velocity(vel_vecs, gen_speeds, N)
[[N.x]]
"""
if not iterable(vel_vecs):
raise TypeError('Velocity vectors must be contained in an iterable.')
if not iterable(gen_speeds):
raise TypeError('Generalized speeds must be contained in an iterable')
vec_partials = []
for vec in vel_vecs:
partials = []
for speed in gen_speeds:
partials.append(vec.diff(speed, frame, var_in_dcm=False))
vec_partials.append(partials)
return vec_partials
def dynamicsymbols(names, level=0):
"""Uses symbols and Function for functions of time.
Creates a SymPy UndefinedFunction, which is then initialized as a function
of a variable, the default being Symbol('t').
Parameters
==========
names : str
Names of the dynamic symbols you want to create; works the same way as
inputs to symbols
level : int
Level of differentiation of the returned function; d/dt once of t,
twice of t, etc.
Examples
========
>>> from sympy.physics.vector import dynamicsymbols
>>> from sympy import diff, Symbol
>>> q1 = dynamicsymbols('q1')
>>> q1
q1(t)
>>> diff(q1, Symbol('t'))
Derivative(q1(t), t)
"""
esses = symbols(names, cls=Function)
t = dynamicsymbols._t
if iterable(esses):
esses = [reduce(diff, [t] * level, e(t)) for e in esses]
return esses
else:
return reduce(diff, [t] * level, esses(t))
dynamicsymbols._t = Symbol('t')
dynamicsymbols._str = '\''
|
8f0e3536dd62b73c856d1ddd3b7dcc70366638be47ea58be6dabb7f990770e15
|
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.core.compatibility import string_types, range
from sympy.physics.vector.vector import Vector, _check_vector
from sympy.utilities.misc import translate
__all__ = ['CoordinateSym', 'ReferenceFrame']
class CoordinateSym(Symbol):
"""
A coordinate symbol/base scalar associated wrt a Reference Frame.
Ideally, users should not instantiate this class. Instances of
this class must only be accessed through the corresponding frame
as 'frame[index]'.
CoordinateSyms having the same frame and index parameters are equal
(even though they may be instantiated separately).
Parameters
==========
name : string
The display name of the CoordinateSym
frame : ReferenceFrame
The reference frame this base scalar belongs to
index : 0, 1 or 2
The index of the dimension denoted by this coordinate variable
Examples
========
>>> from sympy.physics.vector import ReferenceFrame, CoordinateSym
>>> A = ReferenceFrame('A')
>>> A[1]
A_y
>>> type(A[0])
<class 'sympy.physics.vector.frame.CoordinateSym'>
>>> a_y = CoordinateSym('a_y', A, 1)
>>> a_y == A[1]
True
"""
def __new__(cls, name, frame, index):
# We can't use the cached Symbol.__new__ because this class depends on
# frame and index, which are not passed to Symbol.__xnew__.
assumptions = {}
super(CoordinateSym, cls)._sanitize(assumptions, cls)
obj = super(CoordinateSym, cls).__xnew__(cls, name, **assumptions)
_check_frame(frame)
if index not in range(0, 3):
raise ValueError("Invalid index specified")
obj._id = (frame, index)
return obj
@property
def frame(self):
return self._id[0]
def __eq__(self, other):
#Check if the other object is a CoordinateSym of the same frame
#and same index
if isinstance(other, CoordinateSym):
if other._id == self._id:
return True
return False
def __ne__(self, other):
return not self == other
def __hash__(self):
return tuple((self._id[0].__hash__(), self._id[1])).__hash__()
class ReferenceFrame(object):
"""A reference frame in classical mechanics.
ReferenceFrame is a class used to represent a reference frame in classical
mechanics. It has a standard basis of three unit vectors in the frame's
x, y, and z directions.
It also can have a rotation relative to a parent frame; this rotation is
defined by a direction cosine matrix relating this frame's basis vectors to
the parent frame's basis vectors. It can also have an angular velocity
vector, defined in another frame.
"""
_count = 0
def __init__(self, name, indices=None, latexs=None, variables=None):
"""ReferenceFrame initialization method.
A ReferenceFrame has a set of orthonormal basis vectors, along with
orientations relative to other ReferenceFrames and angular velocities
relative to other ReferenceFrames.
Parameters
==========
indices : list (of strings)
If custom indices are desired for console, pretty, and LaTeX
printing, supply three as a list. The basis vectors can then be
accessed with the get_item method.
latexs : list (of strings)
If custom names are desired for LaTeX printing of each basis
vector, supply the names here in a list.
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'
"""
if not isinstance(name, string_types):
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, string_types):
raise TypeError('Indices must be strings')
self.str_vecs = [(name + '[\'' + indices[0] + '\']'),
(name + '[\'' + indices[1] + '\']'),
(name + '[\'' + indices[2] + '\']')]
self.pretty_vecs = [(name.lower() + u"_" + indices[0]),
(name.lower() + u"_" + indices[1]),
(name.lower() + u"_" + indices[2])]
self.latex_vecs = [(r"\mathbf{\hat{%s}_{%s}}" % (name.lower(),
indices[0])), (r"\mathbf{\hat{%s}_{%s}}" %
(name.lower(), indices[1])),
(r"\mathbf{\hat{%s}_{%s}}" % (name.lower(),
indices[2]))]
self.indices = indices
# Second case, when no custom indices are supplied
else:
self.str_vecs = [(name + '.x'), (name + '.y'), (name + '.z')]
self.pretty_vecs = [name.lower() + u"_x",
name.lower() + u"_y",
name.lower() + u"_z"]
self.latex_vecs = [(r"\mathbf{\hat{%s}_x}" % name.lower()),
(r"\mathbf{\hat{%s}_y}" % name.lower()),
(r"\mathbf{\hat{%s}_z}" % name.lower())]
self.indices = ['x', 'y', 'z']
# Different step, for custom latex basis vectors
if latexs is not None:
if not isinstance(latexs, (tuple, list)):
raise TypeError('Supply the indices as a list')
if len(latexs) != 3:
raise ValueError('Supply 3 indices')
for i in latexs:
if not isinstance(i, string_types):
raise TypeError('Latex entries must be strings')
self.latex_vecs = latexs
self.name = name
self._var_dict = {}
#The _dcm_dict dictionary will only store the dcms of parent-child
#relationships. The _dcm_cache dictionary will work as the dcm
#cache.
self._dcm_dict = {}
self._dcm_cache = {}
self._ang_vel_dict = {}
self._ang_acc_dict = {}
self._dlist = [self._dcm_dict, self._ang_vel_dict, self._ang_acc_dict]
self._cur = 0
self._x = Vector([(Matrix([1, 0, 0]), self)])
self._y = Vector([(Matrix([0, 1, 0]), self)])
self._z = Vector([(Matrix([0, 0, 1]), self)])
#Associate coordinate symbols wrt this frame
if variables is not None:
if not isinstance(variables, (tuple, list)):
raise TypeError('Supply the variable names as a list/tuple')
if len(variables) != 3:
raise ValueError('Supply 3 variable names')
for i in variables:
if not isinstance(i, string_types):
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, string_types):
if ind < 3:
return self.varlist[ind]
else:
raise ValueError("Invalid index provided")
if self.indices[0] == ind:
return self.x
if self.indices[1] == ind:
return self.y
if self.indices[2] == ind:
return self.z
else:
raise ValueError('Not a defined index')
def __iter__(self):
return iter([self.x, self.y, self.z])
def __str__(self):
"""Returns the name of the frame. """
return self.name
__repr__ = __str__
def _dict_list(self, other, num):
"""Creates a list from self to other using _dcm_dict. """
outlist = [[self]]
oldlist = [[]]
while outlist != oldlist:
oldlist = outlist[:]
for i, v in enumerate(outlist):
templist = v[-1]._dlist[num].keys()
for i2, v2 in enumerate(templist):
if not v.__contains__(v2):
littletemplist = v + [v2]
if not outlist.__contains__(littletemplist):
outlist.append(littletemplist)
for i, v in enumerate(oldlist):
if v[-1] != other:
outlist.remove(v)
outlist.sort(key=len)
if len(outlist) != 0:
return outlist[0]
raise ValueError('No Connecting Path found between ' + self.name +
' and ' + other.name)
def _w_diff_dcm(self, otherframe):
"""Angular velocity from time differentiating the DCM. """
from sympy.physics.vector.functions import dynamicsymbols
dcm2diff = self.dcm(otherframe)
diffed = dcm2diff.diff(dynamicsymbols._t)
# angvelmat = diffed * dcm2diff.T
# This one seems to produce the correct result when I checked using Autolev.
angvelmat = dcm2diff*diffed.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]), self)])
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, Vector
>>> 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, Vector
>>> 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):
"""The direction cosine matrix between frames.
This gives the DCM between this frame and the otherframe.
The format is N.xyz = N.dcm(B) * B.xyz
A SymPy Matrix is returned.
Parameters
==========
otherframe : ReferenceFrame
The otherframe which the DCM is generated to.
Examples
========
>>> from sympy.physics.vector import ReferenceFrame, Vector
>>> from sympy import symbols
>>> 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)]])
"""
_check_frame(otherframe)
#Check if the dcm wrt that frame has already been calculated
if otherframe in self._dcm_cache:
return self._dcm_cache[otherframe]
flist = self._dict_list(otherframe, 0)
outdcm = eye(3)
for i in range(len(flist) - 1):
outdcm = outdcm * flist[i]._dcm_dict[flist[i + 1]]
#After calculation, store the dcm in dcm cache for faster
#future retrieval
self._dcm_cache[otherframe] = outdcm
otherframe._dcm_cache[self] = outdcm.T
return outdcm
def orient(self, parent, rot_type, amounts, rot_order=''):
"""Defines the orientation of this frame relative to a parent frame.
Parameters
==========
parent : ReferenceFrame
The frame that this ReferenceFrame will have its orientation matrix
defined in relation to.
rot_type : str
The type of orientation matrix that is being created. Supported
types are 'Body', 'Space', 'Quaternion', 'Axis', and 'DCM'.
See examples for correct usage.
amounts : list OR value
The quantities that the orientation matrix will be defined by.
In case of rot_type='DCM', value must be a
sympy.matrices.MatrixBase object (or subclasses of it).
rot_order : str or int
If applicable, the order of a series of rotations.
Examples
========
>>> from sympy.physics.vector import ReferenceFrame, Vector
>>> from sympy import symbols, eye, ImmutableMatrix
>>> q0, q1, q2, q3 = symbols('q0 q1 q2 q3')
>>> N = ReferenceFrame('N')
>>> B = ReferenceFrame('B')
Now we have a choice of how to implement the orientation. First is
Body. Body orientation takes this reference frame through three
successive simple rotations. Acceptable rotation orders are of length
3, expressed in XYZ or 123, and cannot have a rotation about about an
axis twice in a row.
>>> B.orient(N, 'Body', [q1, q2, q3], 123)
>>> B.orient(N, 'Body', [q1, q2, 0], 'ZXZ')
>>> B.orient(N, 'Body', [0, 0, 0], 'XYX')
Next is Space. Space is like Body, but the rotations are applied in the
opposite order.
>>> B.orient(N, 'Space', [q1, q2, q3], '312')
Next is Quaternion. This orients the new ReferenceFrame with
Quaternions, defined as a finite rotation about lambda, a unit vector,
by some amount theta.
This orientation is described by four parameters:
q0 = cos(theta/2)
q1 = lambda_x sin(theta/2)
q2 = lambda_y sin(theta/2)
q3 = lambda_z sin(theta/2)
Quaternion does not take in a rotation order.
>>> B.orient(N, 'Quaternion', [q0, q1, q2, q3])
Next is Axis. This is a rotation about an arbitrary, non-time-varying
axis by some angle. The axis is supplied as a Vector. This is how
simple rotations are defined.
>>> B.orient(N, 'Axis', [q1, N.x + 2 * N.y])
Last is DCM (Direction Cosine Matrix). This is a rotation matrix
given manually.
>>> B.orient(N, 'DCM', eye(3))
>>> B.orient(N, 'DCM', ImmutableMatrix([[0, 1, 0], [0, 0, -1], [-1, 0, 0]]))
"""
from sympy.physics.vector.functions import dynamicsymbols
_check_frame(parent)
# Allow passing a rotation matrix manually.
if rot_type == 'DCM':
# When rot_type == 'DCM', then amounts must be a Matrix type object
# (e.g. sympy.matrices.dense.MutableDenseMatrix).
if not isinstance(amounts, MatrixBase):
raise TypeError("Amounts must be a sympy Matrix type object.")
else:
amounts = list(amounts)
for i, v in enumerate(amounts):
if not isinstance(v, Vector):
amounts[i] = sympify(v)
def _rot(axis, angle):
"""DCM for simple axis 1,2,or 3 rotations. """
if axis == 1:
return Matrix([[1, 0, 0],
[0, cos(angle), -sin(angle)],
[0, sin(angle), cos(angle)]])
elif axis == 2:
return Matrix([[cos(angle), 0, sin(angle)],
[0, 1, 0],
[-sin(angle), 0, cos(angle)]])
elif axis == 3:
return Matrix([[cos(angle), -sin(angle), 0],
[sin(angle), cos(angle), 0],
[0, 0, 1]])
approved_orders = ('123', '231', '312', '132', '213', '321', '121',
'131', '212', '232', '313', '323', '')
# make sure XYZ => 123 and rot_type is in upper case
rot_order = translate(str(rot_order), 'XYZxyz', '123123')
rot_type = rot_type.upper()
if not rot_order in approved_orders:
raise TypeError('The supplied order is not an approved type')
parent_orient = []
if rot_type == 'AXIS':
if not rot_order == '':
raise TypeError('Axis orientation takes no rotation order')
if not (isinstance(amounts, (list, tuple)) & (len(amounts) == 2)):
raise TypeError('Amounts are a list or tuple of length 2')
theta = amounts[0]
axis = amounts[1]
axis = _check_vector(axis)
if not axis.dt(parent) == 0:
raise ValueError('Axis cannot be time-varying')
axis = axis.express(parent).normalize()
axis = axis.args[0][0]
parent_orient = ((eye(3) - axis * axis.T) * cos(theta) +
Matrix([[0, -axis[2], axis[1]], [axis[2], 0, -axis[0]],
[-axis[1], axis[0], 0]]) * sin(theta) + axis * axis.T)
elif rot_type == 'QUATERNION':
if not rot_order == '':
raise TypeError(
'Quaternion orientation takes no rotation order')
if not (isinstance(amounts, (list, tuple)) & (len(amounts) == 4)):
raise TypeError('Amounts are a list or tuple of length 4')
q0, q1, q2, q3 = amounts
parent_orient = (Matrix([[q0 ** 2 + q1 ** 2 - q2 ** 2 - q3 **
2, 2 * (q1 * q2 - q0 * q3), 2 * (q0 * q2 + q1 * q3)],
[2 * (q1 * q2 + q0 * q3), q0 ** 2 - q1 ** 2 + q2 ** 2 - q3 ** 2,
2 * (q2 * q3 - q0 * q1)], [2 * (q1 * q3 - q0 * q2), 2 * (q0 *
q1 + q2 * q3), q0 ** 2 - q1 ** 2 - q2 ** 2 + q3 ** 2]]))
elif rot_type == 'BODY':
if not (len(amounts) == 3 & len(rot_order) == 3):
raise TypeError('Body orientation takes 3 values & 3 orders')
a1 = int(rot_order[0])
a2 = int(rot_order[1])
a3 = int(rot_order[2])
parent_orient = (_rot(a1, amounts[0]) * _rot(a2, amounts[1])
* _rot(a3, amounts[2]))
elif rot_type == 'SPACE':
if not (len(amounts) == 3 & len(rot_order) == 3):
raise TypeError('Space orientation takes 3 values & 3 orders')
a1 = int(rot_order[0])
a2 = int(rot_order[1])
a3 = int(rot_order[2])
parent_orient = (_rot(a3, amounts[2]) * _rot(a2, amounts[1])
* _rot(a1, amounts[0]))
elif rot_type == 'DCM':
parent_orient = amounts
else:
raise NotImplementedError('That is not an implemented rotation')
#Reset the _dcm_cache of this frame, and remove it from the _dcm_caches
#of the frames it is linked to. Also remove it from the _dcm_dict of
#its parent
frames = self._dcm_cache.keys()
dcm_dict_del = []
dcm_cache_del = []
for frame in frames:
if frame in self._dcm_dict:
dcm_dict_del += [frame]
dcm_cache_del += [frame]
for frame in dcm_dict_del:
del frame._dcm_dict[self]
for frame in dcm_cache_del:
del frame._dcm_cache[self]
#Add the dcm relationship to _dcm_dict
self._dcm_dict = self._dlist[0] = {}
self._dcm_dict.update({parent: parent_orient.T})
parent._dcm_dict.update({self: parent_orient})
#Also update the dcm cache after resetting it
self._dcm_cache = {}
self._dcm_cache.update({parent: parent_orient.T})
parent._dcm_cache.update({self: parent_orient})
if rot_type == 'QUATERNION':
t = dynamicsymbols._t
q0, q1, q2, q3 = amounts
q0d = diff(q0, t)
q1d = diff(q1, t)
q2d = diff(q2, t)
q3d = diff(q3, t)
w1 = 2 * (q1d * q0 + q2d * q3 - q3d * q2 - q0d * q1)
w2 = 2 * (q2d * q0 + q3d * q1 - q1d * q3 - q0d * q2)
w3 = 2 * (q3d * q0 + q1d * q2 - q2d * q1 - q0d * q3)
wvec = Vector([(Matrix([w1, w2, w3]), self)])
elif rot_type == 'AXIS':
thetad = (amounts[0]).diff(dynamicsymbols._t)
wvec = thetad * amounts[1].express(parent).normalize()
elif rot_type == 'DCM':
wvec = self._w_diff_dcm(parent)
else:
try:
from sympy.polys.polyerrors import CoercionFailed
from sympy.physics.vector.functions import kinematic_equations
q1, q2, q3 = amounts
u1, u2, u3 = symbols('u1, u2, u3', cls=Dummy)
templist = kinematic_equations([u1, u2, u3], [q1, q2, q3],
rot_type, rot_order)
templist = [expand(i) for i in templist]
td = solve(templist, [u1, u2, u3])
u1 = expand(td[u1])
u2 = expand(td[u2])
u3 = expand(td[u3])
wvec = u1 * self.x + u2 * self.y + u3 * self.z
except (CoercionFailed, AssertionError):
wvec = self._w_diff_dcm(parent)
self._ang_vel_dict.update({parent: wvec})
parent._ang_vel_dict.update({self: -wvec})
self._var_dict = {}
def orientnew(self, newname, rot_type, amounts, rot_order='',
variables=None, indices=None, latexs=None):
"""Creates a new ReferenceFrame oriented with respect to this Frame.
See ReferenceFrame.orient() for acceptable rotation types, amounts,
and orders. Parent is going to be self.
Parameters
==========
newname : str
The name for the new ReferenceFrame
rot_type : str
The type of orientation matrix that is being created.
amounts : list OR value
The quantities that the orientation matrix will be defined by.
rot_order : str
If applicable, the order of a series of rotations.
Examples
========
>>> from sympy.physics.vector import ReferenceFrame, Vector
>>> from sympy import symbols
>>> q0, q1, q2, q3 = symbols('q0 q1 q2 q3')
>>> N = ReferenceFrame('N')
Now we have a choice of how to implement the orientation. First is
Body. Body orientation takes this reference frame through three
successive simple rotations. Acceptable rotation orders are of length
3, expressed in XYZ or 123, and cannot have a rotation about about an
axis twice in a row.
>>> A = N.orientnew('A', 'Body', [q1, q2, q3], '123')
>>> A = N.orientnew('A', 'Body', [q1, q2, 0], 'ZXZ')
>>> A = N.orientnew('A', 'Body', [0, 0, 0], 'XYX')
Next is Space. Space is like Body, but the rotations are applied in the
opposite order.
>>> A = N.orientnew('A', 'Space', [q1, q2, q3], '312')
Next is Quaternion. This orients the new ReferenceFrame with
Quaternions, defined as a finite rotation about lambda, a unit vector,
by some amount theta.
This orientation is described by four parameters:
q0 = cos(theta/2)
q1 = lambda_x sin(theta/2)
q2 = lambda_y sin(theta/2)
q3 = lambda_z sin(theta/2)
Quaternion does not take in a rotation order.
>>> A = N.orientnew('A', 'Quaternion', [q0, q1, q2, q3])
Last is Axis. This is a rotation about an arbitrary, non-time-varying
axis by some angle. The axis is supplied as a Vector. This is how
simple rotations are defined.
>>> A = N.orientnew('A', 'Axis', [q1, N.x])
"""
newframe = self.__class__(newname, variables=variables,
indices=indices, latexs=latexs)
newframe.orient(self, rot_type, amounts, rot_order)
return newframe
def set_ang_acc(self, otherframe, value):
"""Define the angular acceleration Vector in a ReferenceFrame.
Defines the angular acceleration of this ReferenceFrame, in another.
Angular acceleration can be defined with respect to multiple different
ReferenceFrames. Care must be taken to not create loops which are
inconsistent.
Parameters
==========
otherframe : ReferenceFrame
A ReferenceFrame to define the angular acceleration in
value : Vector
The Vector representing angular acceleration
Examples
========
>>> from sympy.physics.vector import ReferenceFrame, Vector
>>> 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, Vector
>>> 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'))
|
2110920bdc1502261907e1eab959811b8a5bb0f8e0771cfa7f4a873fc23bc8ca
|
from sympy.core.backend import sympify, Add, ImmutableMatrix as Matrix
from sympy.core.compatibility import unicode
from .printing import (VectorLatexPrinter, VectorPrettyPrinter,
VectorStrPrinter)
__all__ = ['Dyadic']
class Dyadic(object):
"""A Dyadic object.
See:
https://en.wikipedia.org/wiki/Dyadic_tensor
Kane, T., Levinson, D. Dynamics Theory and Applications. 1985 McGraw-Hill
A more powerful way to represent a rigid body's inertia. While it is more
complex, by choosing Dyadic components to be in body fixed basis vectors,
the resulting matrix is equivalent to the inertia tensor.
"""
def __init__(self, inlist):
"""
Just like Vector's init, you shouldn't call this unless creating a
zero dyadic.
zd = Dyadic(0)
Stores a Dyadic as a list of lists; the inner list has the measure
number and the two unit vectors; the outerlist holds each unique
unit vector pair.
"""
self.args = []
if inlist == 0:
inlist = []
while len(inlist) != 0:
added = 0
for i, v in enumerate(self.args):
if ((str(inlist[0][1]) == str(self.args[i][1])) and
(str(inlist[0][2]) == str(self.args[i][2]))):
self.args[i] = (self.args[i][0] + inlist[0][0],
inlist[0][1], inlist[0][2])
inlist.remove(inlist[0])
added = 1
break
if added != 1:
self.args.append(inlist[0])
inlist.remove(inlist[0])
i = 0
# This code is to remove empty parts from the list
while i < len(self.args):
if ((self.args[i][0] == 0) | (self.args[i][1] == 0) |
(self.args[i][2] == 0)):
self.args.remove(self.args[i])
i -= 1
i += 1
def __add__(self, other):
"""The add operator for Dyadic. """
other = _check_dyadic(other)
return Dyadic(self.args + other.args)
def __and__(self, other):
"""The inner product operator for a Dyadic and a Dyadic or Vector.
Parameters
==========
other : Dyadic or Vector
The other Dyadic or Vector to take the inner product with
Examples
========
>>> from sympy.physics.vector import ReferenceFrame, outer
>>> N = ReferenceFrame('N')
>>> D1 = outer(N.x, N.y)
>>> D2 = outer(N.y, N.y)
>>> D1.dot(D2)
(N.x|N.y)
>>> D1.dot(N.y)
N.x
"""
from sympy.physics.vector.vector import Vector, _check_vector
if isinstance(other, Dyadic):
other = _check_dyadic(other)
ol = Dyadic(0)
for i, v in enumerate(self.args):
for i2, v2 in enumerate(other.args):
ol += v[0] * v2[0] * (v[2] & v2[1]) * (v[1] | v2[2])
else:
other = _check_vector(other)
ol = Vector(0)
for i, v in enumerate(self.args):
ol += v[0] * v[1] * (v[2] & other)
return ol
def __div__(self, other):
"""Divides the Dyadic by a sympifyable expression. """
return self.__mul__(1 / other)
__truediv__ = __div__
def __eq__(self, other):
"""Tests for equality.
Is currently weak; needs stronger comparison testing
"""
if other == 0:
other = Dyadic(0)
other = _check_dyadic(other)
if (self.args == []) and (other.args == []):
return True
elif (self.args == []) or (other.args == []):
return False
return set(self.args) == set(other.args)
def __mul__(self, other):
"""Multiplies the Dyadic by a sympifyable expression.
Parameters
==========
other : Sympafiable
The scalar to multiply this Dyadic with
Examples
========
>>> from sympy.physics.vector import ReferenceFrame, outer
>>> N = ReferenceFrame('N')
>>> d = outer(N.x, N.x)
>>> 5 * d
5*(N.x|N.x)
"""
newlist = [v for v in self.args]
for i, v in enumerate(newlist):
newlist[i] = (sympify(other) * newlist[i][0], newlist[i][1],
newlist[i][2])
return Dyadic(newlist)
def __ne__(self, other):
return not self == other
def __neg__(self):
return self * -1
def _latex(self, printer=None):
ar = self.args # just to shorten things
if len(ar) == 0:
return str(0)
ol = [] # output list, to be concatenated to a string
mlp = VectorLatexPrinter()
for i, v in enumerate(ar):
# if the coef of the dyadic is 1, we skip the 1
if ar[i][0] == 1:
ol.append(' + ' + mlp.doprint(ar[i][1]) + r"\otimes " +
mlp.doprint(ar[i][2]))
# if the coef of the dyadic is -1, we skip the 1
elif ar[i][0] == -1:
ol.append(' - ' +
mlp.doprint(ar[i][1]) +
r"\otimes " +
mlp.doprint(ar[i][2]))
# If the coefficient of the dyadic is not 1 or -1,
# we might wrap it in parentheses, for readability.
elif ar[i][0] != 0:
arg_str = mlp.doprint(ar[i][0])
if isinstance(ar[i][0], Add):
arg_str = '(%s)' % arg_str
if arg_str.startswith('-'):
arg_str = arg_str[1:]
str_start = ' - '
else:
str_start = ' + '
ol.append(str_start + arg_str + mlp.doprint(ar[i][1]) +
r"\otimes " + mlp.doprint(ar[i][2]))
outstr = ''.join(ol)
if outstr.startswith(' + '):
outstr = outstr[3:]
elif outstr.startswith(' '):
outstr = outstr[1:]
return outstr
def _pretty(self, printer=None):
e = self
class Fake(object):
baseline = 0
def render(self, *args, **kwargs):
ar = e.args # just to shorten things
settings = printer._settings if printer else {}
if printer:
use_unicode = printer._use_unicode
else:
from sympy.printing.pretty.pretty_symbology import (
pretty_use_unicode)
use_unicode = pretty_use_unicode()
mpp = printer if printer else VectorPrettyPrinter(settings)
if len(ar) == 0:
return unicode(0)
bar = u"\N{CIRCLED TIMES}" if use_unicode else "|"
ol = [] # output list, to be concatenated to a string
for i, v in enumerate(ar):
# if the coef of the dyadic is 1, we skip the 1
if ar[i][0] == 1:
ol.extend([u" + ",
mpp.doprint(ar[i][1]),
bar,
mpp.doprint(ar[i][2])])
# if the coef of the dyadic is -1, we skip the 1
elif ar[i][0] == -1:
ol.extend([u" - ",
mpp.doprint(ar[i][1]),
bar,
mpp.doprint(ar[i][2])])
# If the coefficient of the dyadic is not 1 or -1,
# we might wrap it in parentheses, for readability.
elif ar[i][0] != 0:
if isinstance(ar[i][0], Add):
arg_str = mpp._print(
ar[i][0]).parens()[0]
else:
arg_str = mpp.doprint(ar[i][0])
if arg_str.startswith(u"-"):
arg_str = arg_str[1:]
str_start = u" - "
else:
str_start = u" + "
ol.extend([str_start, arg_str, u" ",
mpp.doprint(ar[i][1]),
bar,
mpp.doprint(ar[i][2])])
outstr = u"".join(ol)
if outstr.startswith(u" + "):
outstr = outstr[3:]
elif outstr.startswith(" "):
outstr = outstr[1:]
return outstr
return Fake()
def __rand__(self, other):
"""The inner product operator for a Vector or Dyadic, and a Dyadic
This is for: Vector dot Dyadic
Parameters
==========
other : Vector
The vector we are dotting with
Examples
========
>>> from sympy.physics.vector import ReferenceFrame, dot, outer
>>> N = ReferenceFrame('N')
>>> d = outer(N.x, N.x)
>>> dot(N.x, d)
N.x
"""
from sympy.physics.vector.vector import Vector, _check_vector
other = _check_vector(other)
ol = Vector(0)
for i, v in enumerate(self.args):
ol += v[0] * v[2] * (v[1] & other)
return ol
def __rsub__(self, other):
return (-1 * self) + other
def __rxor__(self, other):
"""For a cross product in the form: Vector x Dyadic
Parameters
==========
other : Vector
The Vector that we are crossing this Dyadic with
Examples
========
>>> from sympy.physics.vector import ReferenceFrame, outer, cross
>>> N = ReferenceFrame('N')
>>> d = outer(N.x, N.x)
>>> cross(N.y, d)
- (N.z|N.x)
"""
from sympy.physics.vector.vector import _check_vector
other = _check_vector(other)
ol = Dyadic(0)
for i, v in enumerate(self.args):
ol += v[0] * ((other ^ v[1]) | v[2])
return ol
def __str__(self, printer=None):
"""Printing method. """
ar = self.args # just to shorten things
if len(ar) == 0:
return str(0)
ol = [] # output list, to be concatenated to a string
for i, v in enumerate(ar):
# if the coef of the dyadic is 1, we skip the 1
if ar[i][0] == 1:
ol.append(' + (' + str(ar[i][1]) + '|' + str(ar[i][2]) + ')')
# if the coef of the dyadic is -1, we skip the 1
elif ar[i][0] == -1:
ol.append(' - (' + str(ar[i][1]) + '|' + str(ar[i][2]) + ')')
# If the coefficient of the dyadic is not 1 or -1,
# we might wrap it in parentheses, for readability.
elif ar[i][0] != 0:
arg_str = VectorStrPrinter().doprint(ar[i][0])
if isinstance(ar[i][0], Add):
arg_str = "(%s)" % arg_str
if arg_str[0] == '-':
arg_str = arg_str[1:]
str_start = ' - '
else:
str_start = ' + '
ol.append(str_start + arg_str + '*(' + str(ar[i][1]) +
'|' + str(ar[i][2]) + ')')
outstr = ''.join(ol)
if outstr.startswith(' + '):
outstr = outstr[3:]
elif outstr.startswith(' '):
outstr = outstr[1:]
return outstr
def __sub__(self, other):
"""The subtraction operator. """
return self.__add__(other * -1)
def __xor__(self, other):
"""For a cross product in the form: Dyadic x Vector.
Parameters
==========
other : Vector
The Vector that we are crossing this Dyadic with
Examples
========
>>> from sympy.physics.vector import ReferenceFrame, outer, cross
>>> N = ReferenceFrame('N')
>>> d = outer(N.x, N.x)
>>> cross(d, N.y)
(N.x|N.z)
"""
from sympy.physics.vector.vector import _check_vector
other = _check_vector(other)
ol = Dyadic(0)
for i, v in enumerate(self.args):
ol += v[0] * (v[1] | (v[2] ^ other))
return ol
# We don't define _repr_png_ here because it would add a large amount of
# data to any notebook containing SymPy expressions, without adding
# anything useful to the notebook. It can still enabled manually, e.g.,
# for the qtconsole, with init_printing().
def _repr_latex_(self):
"""
IPython/Jupyter LaTeX printing
To change the behavior of this (e.g., pass in some settings to LaTeX),
use init_printing(). init_printing() will also enable LaTeX printing
for built in numeric types like ints and container types that contain
SymPy objects, like lists and dictionaries of expressions.
"""
from sympy.printing.latex import latex
s = latex(self, mode='plain')
return "$\\displaystyle %s$" % s
_repr_latex_orig = _repr_latex_
_sympystr = __str__
_sympyrepr = _sympystr
__repr__ = __str__
__radd__ = __add__
__rmul__ = __mul__
def express(self, frame1, frame2=None):
"""Expresses this Dyadic in alternate frame(s)
The first frame is the list side expression, the second frame is the
right side; if Dyadic is in form A.x|B.y, you can express it in two
different frames. If no second frame is given, the Dyadic is
expressed in only one frame.
Calls the global express function
Parameters
==========
frame1 : ReferenceFrame
The frame to express the left side of the Dyadic in
frame2 : ReferenceFrame
If provided, the frame to express the right side of the Dyadic in
Examples
========
>>> from sympy.physics.vector import ReferenceFrame, outer, dynamicsymbols
>>> N = ReferenceFrame('N')
>>> q = dynamicsymbols('q')
>>> B = N.orientnew('B', 'Axis', [q, N.z])
>>> d = outer(N.x, N.x)
>>> d.express(B, N)
cos(q)*(B.x|N.x) - sin(q)*(B.y|N.x)
"""
from sympy.physics.vector.functions import express
return express(self, frame1, frame2)
def to_matrix(self, reference_frame, second_reference_frame=None):
"""Returns the matrix form of the dyadic with respect to one or two
reference frames.
Parameters
----------
reference_frame : ReferenceFrame
The reference frame that the rows and columns of the matrix
correspond to. If a second reference frame is provided, this
only corresponds to the rows of the matrix.
second_reference_frame : ReferenceFrame, optional, default=None
The reference frame that the columns of the matrix correspond
to.
Returns
-------
matrix : ImmutableMatrix, shape(3,3)
The matrix that gives the 2D tensor form.
Examples
========
>>> from sympy import symbols
>>> from sympy.physics.vector import ReferenceFrame, Vector
>>> Vector.simp = True
>>> from sympy.physics.mechanics import inertia
>>> Ixx, Iyy, Izz, Ixy, Iyz, Ixz = symbols('Ixx, Iyy, Izz, Ixy, Iyz, Ixz')
>>> N = ReferenceFrame('N')
>>> inertia_dyadic = inertia(N, Ixx, Iyy, Izz, Ixy, Iyz, Ixz)
>>> inertia_dyadic.to_matrix(N)
Matrix([
[Ixx, Ixy, Ixz],
[Ixy, Iyy, Iyz],
[Ixz, Iyz, Izz]])
>>> beta = symbols('beta')
>>> A = N.orientnew('A', 'Axis', (beta, N.x))
>>> inertia_dyadic.to_matrix(A)
Matrix([
[ Ixx, Ixy*cos(beta) + Ixz*sin(beta), -Ixy*sin(beta) + Ixz*cos(beta)],
[ Ixy*cos(beta) + Ixz*sin(beta), Iyy*cos(2*beta)/2 + Iyy/2 + Iyz*sin(2*beta) - Izz*cos(2*beta)/2 + Izz/2, -Iyy*sin(2*beta)/2 + Iyz*cos(2*beta) + Izz*sin(2*beta)/2],
[-Ixy*sin(beta) + Ixz*cos(beta), -Iyy*sin(2*beta)/2 + Iyz*cos(2*beta) + Izz*sin(2*beta)/2, -Iyy*cos(2*beta)/2 + Iyy/2 - Iyz*sin(2*beta) + Izz*cos(2*beta)/2 + Izz/2]])
"""
if second_reference_frame is None:
second_reference_frame = reference_frame
return Matrix([i.dot(self).dot(j) for i in reference_frame for j in
second_reference_frame]).reshape(3, 3)
def doit(self, **hints):
"""Calls .doit() on each term in the Dyadic"""
return sum([Dyadic([(v[0].doit(**hints), v[1], v[2])])
for v in self.args], Dyadic(0))
def dt(self, frame):
"""Take the time derivative of this Dyadic in a frame.
This function calls the global time_derivative method
Parameters
==========
frame : ReferenceFrame
The frame to take the time derivative in
Examples
========
>>> from sympy.physics.vector import ReferenceFrame, outer, dynamicsymbols
>>> N = ReferenceFrame('N')
>>> q = dynamicsymbols('q')
>>> B = N.orientnew('B', 'Axis', [q, N.z])
>>> d = outer(N.x, N.x)
>>> d.dt(B)
- q'*(N.y|N.x) - q'*(N.x|N.y)
"""
from sympy.physics.vector.functions import time_derivative
return time_derivative(self, frame)
def simplify(self):
"""Returns a simplified Dyadic."""
out = Dyadic(0)
for v in self.args:
out += Dyadic([(v[0].simplify(), v[1], v[2])])
return out
def subs(self, *args, **kwargs):
"""Substitution on the Dyadic.
Examples
========
>>> from sympy.physics.vector import ReferenceFrame
>>> from sympy import Symbol
>>> N = ReferenceFrame('N')
>>> s = Symbol('s')
>>> a = s*(N.x|N.x)
>>> a.subs({s: 2})
2*(N.x|N.x)
"""
return sum([Dyadic([(v[0].subs(*args, **kwargs), v[1], v[2])])
for v in self.args], Dyadic(0))
def applyfunc(self, f):
"""Apply a function to each component of a Dyadic."""
if not callable(f):
raise TypeError("`f` must be callable.")
out = Dyadic(0)
for a, b, c in self.args:
out += f(a) * (b|c)
return out
dot = __and__
cross = __xor__
def _check_dyadic(other):
if not isinstance(other, Dyadic):
raise TypeError('A Dyadic must be supplied')
return other
|
c3f9f64eac9d9c32c93238ae0d45184d0df7d3bbdd85da447d87c7a01ca5cb76
|
"""
This module can be used to solve 2D beam bending problems with
singularity functions in mechanics.
"""
from __future__ import print_function, division
from sympy.core import S, Symbol, diff, symbols
from sympy.solvers import linsolve
from sympy.printing import sstr
from sympy.functions import SingularityFunction, Piecewise, factorial
from sympy.core import sympify
from sympy.integrals import integrate
from sympy.series import limit
from sympy.plotting import plot
from sympy.external import import_module
from sympy.utilities.decorator import doctest_depends_on
from sympy import lambdify
matplotlib = import_module('matplotlib', __import__kwargs={'fromlist':['pyplot']})
numpy = import_module('numpy', __import__kwargs={'fromlist':['linspace']})
__doctest_requires__ = {('Beam.plot_loading_results',): ['matplotlib']}
class Beam(object):
"""
A Beam is a structural element that is capable of withstanding load
primarily by resisting against bending. Beams are characterized by
their cross sectional profile(Second moment of area), their length
and their material.
.. note::
While solving a beam bending problem, a user should choose its
own sign convention and should stick to it. The results will
automatically follow the chosen sign convention.
Examples
========
There is a beam of length 4 meters. A constant distributed load of 6 N/m
is applied from half of the beam till the end. There are two simple supports
below the beam, one at the starting point and another at the ending point
of the beam. The deflection of the beam at the end is restricted.
Using the sign convention of downwards forces being positive.
>>> from sympy.physics.continuum_mechanics.beam import Beam
>>> from sympy import symbols, Piecewise
>>> E, I = symbols('E, I')
>>> R1, R2 = symbols('R1, R2')
>>> b = Beam(4, E, I)
>>> b.apply_load(R1, 0, -1)
>>> b.apply_load(6, 2, 0)
>>> b.apply_load(R2, 4, -1)
>>> b.bc_deflection = [(0, 0), (4, 0)]
>>> b.boundary_conditions
{'deflection': [(0, 0), (4, 0)], 'slope': []}
>>> b.load
R1*SingularityFunction(x, 0, -1) + R2*SingularityFunction(x, 4, -1) + 6*SingularityFunction(x, 2, 0)
>>> b.solve_for_reaction_loads(R1, R2)
>>> b.load
-3*SingularityFunction(x, 0, -1) + 6*SingularityFunction(x, 2, 0) - 9*SingularityFunction(x, 4, -1)
>>> b.shear_force()
-3*SingularityFunction(x, 0, 0) + 6*SingularityFunction(x, 2, 1) - 9*SingularityFunction(x, 4, 0)
>>> b.bending_moment()
-3*SingularityFunction(x, 0, 1) + 3*SingularityFunction(x, 2, 2) - 9*SingularityFunction(x, 4, 1)
>>> b.slope()
(-3*SingularityFunction(x, 0, 2)/2 + SingularityFunction(x, 2, 3) - 9*SingularityFunction(x, 4, 2)/2 + 7)/(E*I)
>>> b.deflection()
(7*x - SingularityFunction(x, 0, 3)/2 + SingularityFunction(x, 2, 4)/4 - 3*SingularityFunction(x, 4, 3)/2)/(E*I)
>>> b.deflection().rewrite(Piecewise)
(7*x - Piecewise((x**3, x > 0), (0, True))/2
- 3*Piecewise(((x - 4)**3, x - 4 > 0), (0, True))/2
+ Piecewise(((x - 2)**4, x - 2 > 0), (0, True))/4)/(E*I)
"""
def __init__(self, length, elastic_modulus, second_moment, 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
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.
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
self.second_moment = second_moment
self.variable = variable
self._base_char = base_char
self._boundary_conditions = {'deflection': [], 'slope': []}
self._load = 0
self._applied_loads = []
self._reaction_loads = {}
self._composite_type = None
self._hinge_position = None
def __str__(self):
str_sol = 'Beam({}, {}, {})'.format(sstr(self._length), sstr(self._elastic_modulus), sstr(self._second_moment))
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 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 = symbols('E, I')
>>> 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, 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._second_moment = sympify(i)
@property
def boundary_conditions(self):
"""
Returns a dictionary of boundary conditions applied on the beam.
The dictionary has three kewwords namely moment, slope and deflection.
The value of each keyword is a list of tuple, where each tuple
contains loaction 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)
"""
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))
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 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 point load of magnitude 4 N is applied from the top of the beam at
2 meters from the starting point and a parabolic ramp load of magnitude
2 N/m is applied below the beam starting from 2 meters to 3 meters
away from the starting point of the beam.
>>> from sympy.physics.continuum_mechanics.beam import Beam
>>> from sympy import symbols
>>> E, I = symbols('E, I')
>>> b = Beam(4, E, I)
>>> b.apply_load(-3, 0, -2)
>>> b.apply_load(4, 2, -1)
>>> b.apply_load(-2, 2, 2, end=3)
>>> b.load
-3*SingularityFunction(x, 0, -2) + 4*SingularityFunction(x, 2, -1) - 2*SingularityFunction(x, 2, 2) + 2*SingularityFunction(x, 3, 0) + 4*SingularityFunction(x, 3, 1) + 2*SingularityFunction(x, 3, 2)
"""
x = self.variable
value = sympify(value)
start = sympify(start)
order = sympify(order)
self._applied_loads.append((value, start, order, end))
self._load += value*SingularityFunction(x, start, order)
if end:
if order.is_negative:
msg = ("If 'end' is provided the 'order' of the load cannot "
"be negative, i.e. 'end' is only valid for distributed "
"loads.")
raise ValueError(msg)
# NOTE : A Taylor series can be used to define the summation of
# singularity functions that subtract from the load past the end
# point such that it evaluates to zero past 'end'.
f = value * x**order
for i in range(0, order + 1):
self._load -= (f.diff(x, i).subs(x, end - start) *
SingularityFunction(x, end, i) / factorial(i))
def remove_load(self, value, start, order, end=None):
"""
This method removes a particular load present on the beam object.
Returns a ValueError if the load passed as an argument is not
present on the beam.
Parameters
==========
value : Sympifyable
The magnitude of an applied load.
start : Sympifyable
The starting point of the applied load. For point moments and
point forces this is the location of application.
order : Integer
The order of the applied load.
- For moments, order= -2
- For point loads, order=-1
- For constant distributed load, order=0
- For ramp loads, order=1
- For parabolic ramp loads, order=2
- ... so on.
end : Sympifyable, optional
An optional argument that can be used if the load has an end point
within the length of the beam.
Examples
========
There is a beam of length 4 meters. A moment of magnitude 3 Nm is
applied in the clockwise direction at the starting point of the beam.
A pointload of magnitude 4 N is applied from the top of the beam at
2 meters from the starting point and a parabolic ramp load of magnitude
2 N/m is applied below the beam starting from 2 meters to 3 meters
away from the starting point of the beam.
>>> from sympy.physics.continuum_mechanics.beam import Beam
>>> from sympy import symbols
>>> E, I = symbols('E, I')
>>> b = Beam(4, E, I)
>>> b.apply_load(-3, 0, -2)
>>> b.apply_load(4, 2, -1)
>>> b.apply_load(-2, 2, 2, end=3)
>>> b.load
-3*SingularityFunction(x, 0, -2) + 4*SingularityFunction(x, 2, -1) - 2*SingularityFunction(x, 2, 2) + 2*SingularityFunction(x, 3, 0) + 4*SingularityFunction(x, 3, 1) + 2*SingularityFunction(x, 3, 2)
>>> b.remove_load(-2, 2, 2, end = 3)
>>> b.load
-3*SingularityFunction(x, 0, -2) + 4*SingularityFunction(x, 2, -1)
"""
x = self.variable
value = sympify(value)
start = sympify(start)
order = sympify(order)
if (value, start, order, end) in self._applied_loads:
self._load -= value*SingularityFunction(x, start, order)
self._applied_loads.remove((value, start, order, end))
else:
msg = "No such load distribution exists on the beam object."
raise ValueError(msg)
if end:
# TODO : This is essentially duplicate code wrt to apply_load,
# would be better to move it to one location and both methods use
# it.
if order.is_negative:
msg = ("If 'end' is provided the 'order' of the load cannot "
"be negative, i.e. 'end' is only valid for distributed "
"loads.")
raise ValueError(msg)
# NOTE : A Taylor series can be used to define the summation of
# singularity functions that subtract from the load past the end
# point such that it evaluates to zero past 'end'.
f = value * x**order
for i in range(0, order + 1):
self._load += (f.diff(x, i).subs(x, end - start) *
SingularityFunction(x, end, i) / factorial(i))
@property
def load(self):
"""
Returns a Singularity Function expression which represents
the load distribution curve of the Beam object.
Examples
========
There is a beam of length 4 meters. A moment of magnitude 3 Nm is
applied in the clockwise direction at the starting point of the beam.
A point load of magnitude 4 N is applied from the top of the beam at
2 meters from the starting point and a parabolic ramp load of magnitude
2 N/m is applied below the beam starting from 3 meters away from the
starting point of the beam.
>>> from sympy.physics.continuum_mechanics.beam import Beam
>>> from sympy import symbols
>>> E, I = symbols('E, I')
>>> b = Beam(4, E, I)
>>> b.apply_load(-3, 0, -2)
>>> b.apply_load(4, 2, -1)
>>> b.apply_load(-2, 3, 2)
>>> b.load
-3*SingularityFunction(x, 0, -2) + 4*SingularityFunction(x, 2, -1) - 2*SingularityFunction(x, 3, 2)
"""
return self._load
@property
def applied_loads(self):
"""
Returns a list of all loads applied on the beam object.
Each load in the list is a tuple of form (value, start, order, end).
Examples
========
There is a beam of length 4 meters. A moment of magnitude 3 Nm is
applied in the clockwise direction at the starting point of the beam.
A pointload of magnitude 4 N is applied from the top of the beam at
2 meters from the starting point. Another pointload of magnitude 5 N
is applied at same position.
>>> from sympy.physics.continuum_mechanics.beam import Beam
>>> from sympy import symbols
>>> E, I = symbols('E, I')
>>> b = Beam(4, E, I)
>>> b.apply_load(-3, 0, -2)
>>> b.apply_load(4, 2, -1)
>>> b.apply_load(5, 2, -1)
>>> b.load
-3*SingularityFunction(x, 0, -2) + 9*SingularityFunction(x, 2, -1)
>>> b.applied_loads
[(-3, 0, -2, None), (4, 2, -1, None), (5, 2, -1, None)]
"""
return self._applied_loads
def _solve_hinge_beams(self, *reactions):
"""Method to find integration constants and reactional variables in a
composite beam connected via hinge.
This method resolves the composite Beam into its sub-beams and then
equations of shear force, bending moment, slope and deflection are
evaluated for both of them separately. These equations are then solved
for unknown reactions and integration constants using the boundary
conditions applied on the Beam. Equal deflection of both sub-beams
at the hinge joint gives us another equation to solve the system.
Examples
========
A combined beam, with constant fkexural rigidity E*I, is formed by joining
a Beam of length 2*l to the right of another Beam of length l. The whole beam
is fixed at both of its both end. A point load of magnitude P is also applied
from the top at a distance of 2*l from starting point.
>>> from sympy.physics.continuum_mechanics.beam import Beam
>>> from sympy import symbols
>>> E, I = symbols('E, I')
>>> l=symbols('l', positive=True)
>>> b1=Beam(l ,E,I)
>>> b2=Beam(2*l ,E,I)
>>> b=b1.join(b2,"hinge")
>>> M1, A1, M2, A2, P = symbols('M1 A1 M2 A2 P')
>>> b.apply_load(A1,0,-1)
>>> b.apply_load(M1,0,-2)
>>> b.apply_load(P,2*l,-1)
>>> b.apply_load(A2,3*l,-1)
>>> b.apply_load(M2,3*l,-2)
>>> b.bc_slope=[(0,0), (3*l, 0)]
>>> b.bc_deflection=[(0,0), (3*l, 0)]
>>> b.solve_for_reaction_loads(M1, A1, M2, A2)
>>> b.reaction_loads
{A1: -5*P/18, A2: -13*P/18, M1: 5*P*l/18, M2: -4*P*l/9}
>>> b.slope()
(5*P*l*SingularityFunction(x, 0, 1)/18 - 5*P*SingularityFunction(x, 0, 2)/36 + 5*P*SingularityFunction(x, l, 2)/36)*SingularityFunction(x, 0, 0)/(E*I)
- (5*P*l*SingularityFunction(x, 0, 1)/18 - 5*P*SingularityFunction(x, 0, 2)/36 + 5*P*SingularityFunction(x, l, 2)/36)*SingularityFunction(x, l, 0)/(E*I)
+ (P*l**2/18 - 4*P*l*SingularityFunction(-l + x, 2*l, 1)/9 - 5*P*SingularityFunction(-l + x, 0, 2)/36 + P*SingularityFunction(-l + x, l, 2)/2
- 13*P*SingularityFunction(-l + x, 2*l, 2)/36)*SingularityFunction(x, l, 0)/(E*I)
>>> b.deflection()
(5*P*l*SingularityFunction(x, 0, 2)/36 - 5*P*SingularityFunction(x, 0, 3)/108 + 5*P*SingularityFunction(x, l, 3)/108)*SingularityFunction(x, 0, 0)/(E*I)
- (5*P*l*SingularityFunction(x, 0, 2)/36 - 5*P*SingularityFunction(x, 0, 3)/108 + 5*P*SingularityFunction(x, l, 3)/108)*SingularityFunction(x, l, 0)/(E*I)
+ (5*P*l**3/54 + P*l**2*(-l + x)/18 - 2*P*l*SingularityFunction(-l + x, 2*l, 2)/9 - 5*P*SingularityFunction(-l + x, 0, 3)/108 + P*SingularityFunction(-l + x, l, 3)/6
- 13*P*SingularityFunction(-l + x, 2*l, 3)/108)*SingularityFunction(x, l, 0)/(E*I)
"""
x = self.variable
l = self._hinge_position
E = self._elastic_modulus
I = self._second_moment
if isinstance(I, Piecewise):
I1 = I.args[0][0]
I2 = I.args[1][0]
else:
I1 = I2 = I
load_1 = 0 # Load equation on first segment of composite beam
load_2 = 0 # Load equation on second segment of composite beam
# Distributing load on both segments
for load in self.applied_loads:
if load[1] < l:
load_1 += load[0]*SingularityFunction(x, load[1], load[2])
if load[2] == 0:
load_1 -= load[0]*SingularityFunction(x, load[3], load[2])
elif load[2] > 0:
load_1 -= load[0]*SingularityFunction(x, load[3], load[2]) + load[0]*SingularityFunction(x, load[3], 0)
elif load[1] == l:
load_1 += load[0]*SingularityFunction(x, load[1], load[2])
load_2 += load[0]*SingularityFunction(x, load[1] - l, load[2])
elif load[1] > l:
load_2 += load[0]*SingularityFunction(x, load[1] - l, load[2])
if load[2] == 0:
load_2 -= load[0]*SingularityFunction(x, load[3] - l, load[2])
elif load[2] > 0:
load_2 -= load[0]*SingularityFunction(x, load[3] - l, load[2]) + load[0]*SingularityFunction(x, load[3] - l, 0)
h = Symbol('h') # Force due to hinge
load_1 += h*SingularityFunction(x, l, -1)
load_2 -= h*SingularityFunction(x, 0, -1)
eq = []
shear_1 = integrate(load_1, x)
shear_curve_1 = limit(shear_1, x, l)
eq.append(shear_curve_1)
bending_1 = integrate(shear_1, x)
moment_curve_1 = limit(bending_1, x, l)
eq.append(moment_curve_1)
shear_2 = integrate(load_2, x)
shear_curve_2 = limit(shear_2, x, self.length - l)
eq.append(shear_curve_2)
bending_2 = integrate(shear_2, x)
moment_curve_2 = limit(bending_2, x, self.length - l)
eq.append(moment_curve_2)
C1 = Symbol('C1')
C2 = Symbol('C2')
C3 = Symbol('C3')
C4 = Symbol('C4')
slope_1 = S(1)/(E*I1)*(integrate(bending_1, x) + C1)
def_1 = S(1)/(E*I1)*(integrate((E*I)*slope_1, x) + C1*x + C2)
slope_2 = S(1)/(E*I2)*(integrate(integrate(integrate(load_2, x), x), x) + C3)
def_2 = S(1)/(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, linsolve, limit
>>> 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(1)/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(1)/(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(1)/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(1)/(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(1)/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(1)/(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(1)/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 coresponding 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 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.
>>> 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='position', ylabel='Value', 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.
>>> 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='position', ylabel='Value', 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.
>>> 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='position', ylabel='Value', 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.
>>> 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='position', ylabel='Value',
line_color='r')
@doctest_depends_on(modules=('numpy', 'matplotlib',))
def plot_loading_results(self, subs=None):
"""
Returns Axes object containing subplots 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.
.. note::
This method only works if numpy and matplotlib libraries
are installed on the system.
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.
>>> 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()
"""
if matplotlib is None:
raise ImportError('Install matplotlib to use this method.')
else:
plt = matplotlib.pyplot
if numpy is None:
raise ImportError('Install numpy to use this method.')
else:
linspace = numpy.linspace
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
# As we are using matplotlib directly in this method, we need to change
# SymPy methods to numpy functions.
shear = lambdify(variable,
self.shear_force().subs(subs).rewrite(Piecewise),
'numpy')
moment = lambdify(variable,
self.bending_moment().subs(subs).rewrite(Piecewise),
'numpy')
slope = lambdify(variable, self.slope().subs(subs).rewrite(Piecewise),
'numpy')
deflection = lambdify(variable,
self.deflection().subs(subs).rewrite(Piecewise),
'numpy')
points = linspace(0, float(length), num=100*length)
# Creating a grid for subplots with 2 rows and 2 columns
fig, axs = plt.subplots(4, 1)
# axs is a 2D-numpy array containing axes
axs[0].plot(points, shear(points))
axs[0].set_title("Shear Force")
axs[1].plot(points, moment(points))
axs[1].set_title("Bending Moment")
axs[2].plot(points, slope(points))
axs[2].set_title("Slope")
axs[3].plot(points, deflection(points))
axs[3].set_title("Deflection")
fig.tight_layout() # For better spacing between subplots
return axs
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
>>> l, E, G, I, A = symbols('l, E, G, I, A')
>>> b = Beam3D(l, E, G, I, A)
>>> x, q, m = symbols('x, q, m')
>>> b.apply_load(q, 0, 0, dir="y")
>>> b.apply_moment_load(m, 0, -1, dir="z")
>>> b.shear_force()
[0, -q*x, 0]
>>> b.bending_moment()
[0, 0, -m*x + q*x**2/2]
>>> b.bc_slope = [(0, [0, 0, 0]), (l, [0, 0, 0])]
>>> b.bc_deflection = [(0, [0, 0, 0]), (l, [0, 0, 0])]
>>> b.solve_slope_deflection()
>>> b.slope()
[0, 0, l*x*(-l*q + 3*l*(A*G*l*(l*q - 2*m) + 12*E*I*q)/(2*(A*G*l**2 + 12*E*I)) + 3*m)/(6*E*I)
+ q*x**3/(6*E*I) + x**2*(-l*(A*G*l*(l*q - 2*m) + 12*E*I*q)/(2*(A*G*l**2 + 12*E*I))
- m)/(2*E*I)]
>>> dx, dy, dz = b.deflection()
>>> dx
0
>>> dz
0
>>> expectedy = (
... -l**2*q*x**2/(12*E*I) + l**2*x**2*(A*G*l*(l*q - 2*m) + 12*E*I*q)/(8*E*I*(A*G*l**2 + 12*E*I))
... + l*m*x**2/(4*E*I) - l*x**3*(A*G*l*(l*q - 2*m) + 12*E*I*q)/(12*E*I*(A*G*l**2 + 12*E*I)) - m*x**3/(6*E*I)
... + q*x**4/(24*E*I) + l*x*(A*G*l*(l*q - 2*m) + 12*E*I*q)/(2*A*G*(A*G*l**2 + 12*E*I)) - q*x**2/(2*A*G)
... )
>>> simplify(dy - expectedy)
0
References
==========
.. [1] http://homes.civil.aau.dk/jc/FemteSemester/Beams3D.pdf
"""
def __init__(self, length, elastic_modulus, shear_modulus , second_moment, area, variable=Symbol('x')):
"""Initializes the class.
Parameters
==========
length : Sympifyable
A Symbol or value representing the Beam's length.
elastic_modulus : Sympifyable
A SymPy expression representing the Beam's Modulus of Elasticity.
It is a measure of the stiffness of the Beam material.
shear_modulus : Sympifyable
A SymPy expression representing the Beam's Modulus of rigidity.
It is a measure of rigidity of the Beam material.
second_moment : Sympifyable or list
A list of two elements having SymPy expression representing the
Beam's Second moment of area. First value represent Second moment
across y-axis and second across z-axis.
Single SymPy expression can be passed if both values are same
area : Sympifyable
A SymPy expression representing the Beam's cross-sectional area
in a plane prependicular to length of the Beam.
variable : Symbol, optional
A Symbol object that will be used as the variable along the beam
while representing the load, shear, moment, slope and deflection
curve. By default, it is set to ``Symbol('x')``.
"""
super(Beam3D, self).__init__(length, elastic_modulus, second_moment, variable)
self.shear_modulus = shear_modulus
self.area = area
self._load_vector = [0, 0, 0]
self._moment_load_vector = [0, 0, 0]
self._load_Singularity = [0, 0, 0]
self._slope = [0, 0, 0]
self._deflection = [0, 0, 0]
@property
def shear_modulus(self):
"""Young's Modulus of the Beam. """
return self._shear_modulus
@shear_modulus.setter
def shear_modulus(self, e):
self._shear_modulus = sympify(e)
@property
def second_moment(self):
"""Second moment of area of the Beam. """
return self._second_moment
@second_moment.setter
def second_moment(self, i):
if isinstance(i, list):
i = [sympify(x) for x in i]
self._second_moment = i
else:
self._second_moment = sympify(i)
@property
def area(self):
"""Cross-sectional area of the Beam. """
return self._area
@area.setter
def area(self, a):
self._area = sympify(a)
@property
def load_vector(self):
"""
Returns a three element list representing the load vector.
"""
return self._load_vector
@property
def moment_load_vector(self):
"""
Returns a three element list representing moment loads on Beam.
"""
return self._moment_load_vector
@property
def boundary_conditions(self):
"""
Returns a dictionary of boundary conditions applied on the beam.
The dictionary has two keywords namely slope and deflection.
The value of each keyword is a list of tuple, where each tuple
contains loaction 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 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 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
|
9fe7ea6796e61c0e30df9a4897a6bb151513a03d86b604a3898bb426037b56ed
|
__all__ = []
# The following pattern is used below for importing sub-modules:
#
# 1. "from foo import *". This imports all the names from foo.__all__ into
# this module. But, this does not put those names into the __all__ of
# this module. This enables "from sympy.physics.optics import TWave" to
# work.
# 2. "import foo; __all__.extend(foo.__all__)". This adds all the names in
# foo.__all__ to the __all__ of this module. The names in __all__
# determine which names are imported when
# "from sympy.physics.optics import *" is done.
from . import waves
from .waves import TWave
__all__.extend(waves.__all__)
from . import gaussopt
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)
__all__.extend(gaussopt.__all__)
from . import medium
from .medium import Medium
__all__.extend(medium.__all__)
from . import utils
from .utils import (refraction_angle, fresnel_coefficients,
deviation, brewster_angle, critical_angle, lens_makers_formula,
mirror_formula, lens_formula, hyperfocal_distance, transverse_magnification)
__all__.extend(utils.__all__)
|
56236e3f6eb0e495153bd143a37e027fee2be2a74d0f8b61197ba3e1f797694e
|
"""
**Contains**
* Medium
"""
from __future__ import division
from sympy.physics.units import second, meter, kilogram, ampere
__all__ = ['Medium']
from sympy import Symbol, sympify, sqrt
from sympy.physics.units import speed_of_light, u0, e0
c = speed_of_light.convert_to(meter/second)
_e0mksa = e0.convert_to(ampere**2*second**4/(kilogram*meter**3))
_u0mksa = u0.convert_to(meter*kilogram/(ampere**2*second**2))
class Medium(Symbol):
"""
This class represents an optical medium. The prime reason to implement this is
to facilitate refraction, Fermat's principle, etc.
An optical medium is a material through which electromagnetic waves propagate.
The permittivity and permeability of the medium define how electromagnetic
waves propagate in it.
Parameters
==========
name: string
The display name of the Medium.
permittivity: Sympifyable
Electric permittivity of the space.
permeability: Sympifyable
Magnetic permeability of the space.
n: Sympifyable
Index of refraction of the medium.
Examples
========
>>> from sympy.abc import epsilon, mu
>>> from sympy.physics.optics import Medium
>>> m1 = Medium('m1')
>>> m2 = Medium('m2', epsilon, mu)
>>> m1.intrinsic_impedance
149896229*pi*kilogram*meter**2/(1250000*ampere**2*second**3)
>>> m2.refractive_index
299792458*meter*sqrt(epsilon*mu)/second
References
==========
.. [1] https://en.wikipedia.org/wiki/Optical_medium
"""
def __new__(cls, name, permittivity=None, permeability=None, n=None):
obj = super(Medium, cls).__new__(cls, name)
obj._permittivity = sympify(permittivity)
obj._permeability = sympify(permeability)
obj._n = sympify(n)
if n is not None:
if permittivity is not None and permeability is None:
obj._permeability = n**2/(c**2*obj._permittivity)
if permeability is not None and permittivity is None:
obj._permittivity = n**2/(c**2*obj._permeability)
if permittivity is not None and permittivity is not None:
if abs(n - c*sqrt(obj._permittivity*obj._permeability)) > 1e-6:
raise ValueError("Values are not consistent.")
elif permittivity is not None and permeability is not None:
obj._n = c*sqrt(permittivity*permeability)
elif permittivity is None and permeability is None:
obj._permittivity = _e0mksa
obj._permeability = _u0mksa
return obj
@property
def intrinsic_impedance(self):
"""
Returns intrinsic impedance of the medium.
The intrinsic impedance of a medium is the ratio of the
transverse components of the electric and magnetic fields
of the electromagnetic wave travelling in the medium.
In a region with no electrical conductivity it simplifies
to the square root of ratio of magnetic permeability to
electric permittivity.
Examples
========
>>> from sympy.physics.optics import Medium
>>> m = Medium('m')
>>> m.intrinsic_impedance
149896229*pi*kilogram*meter**2/(1250000*ampere**2*second**3)
"""
return sqrt(self._permeability/self._permittivity)
@property
def speed(self):
"""
Returns speed of the electromagnetic wave travelling in the medium.
Examples
========
>>> from sympy.physics.optics import Medium
>>> m = Medium('m')
>>> m.speed
299792458*meter/second
>>> m2 = Medium('m2', n=1)
>>> m.speed == m2.speed
True
"""
if self._permittivity is not None and self._permeability is not None:
return 1/sqrt(self._permittivity*self._permeability)
else:
return c/self._n
@property
def refractive_index(self):
"""
Returns refractive index of the medium.
Examples
========
>>> from sympy.physics.optics import Medium
>>> m = Medium('m')
>>> m.refractive_index
1
"""
return (c/self.speed)
@property
def permittivity(self):
"""
Returns electric permittivity of the medium.
Examples
========
>>> from sympy.physics.optics import Medium
>>> m = Medium('m')
>>> m.permittivity
625000*ampere**2*second**4/(22468879468420441*pi*kilogram*meter**3)
"""
return self._permittivity
@property
def permeability(self):
"""
Returns magnetic permeability of the medium.
Examples
========
>>> from sympy.physics.optics import Medium
>>> m = Medium('m')
>>> m.permeability
pi*kilogram*meter/(2500000*ampere**2*second**2)
"""
return self._permeability
def __str__(self):
from sympy.printing import sstr
return type(self).__name__ + ': ' + sstr([self._permittivity,
self._permeability, self._n])
def __lt__(self, other):
"""
Compares based on refractive index of the medium.
"""
return self.refractive_index < other.refractive_index
def __gt__(self, other):
return not self < other
def __eq__(self, other):
return self.refractive_index == other.refractive_index
def __ne__(self, other):
return not self == other
|
947bcb89cdd4daf3190170615cffbae46ab7fa5bfb6768a8370d46a0198e0ead
|
"""
**Contains**
* refraction_angle
* fresnel_coefficients
* deviation
* brewster_angle
* critical_angle
* lens_makers_formula
* mirror_formula
* lens_formula
* hyperfocal_distance
* transverse_magnification
"""
from __future__ import division
__all__ = ['refraction_angle',
'deviation',
'fresnel_coefficients',
'brewster_angle',
'critical_angle',
'lens_makers_formula',
'mirror_formula',
'lens_formula',
'hyperfocal_distance',
'transverse_magnification'
]
from sympy import Symbol, sympify, sqrt, Matrix, acos, oo, Limit, atan2, asin,\
cos, sin, tan, I, cancel
from sympy.core.compatibility import is_sequence
from sympy.geometry.line import Ray3D, Point3D
from sympy.geometry.util import intersection
from sympy.geometry.plane import Plane
from .medium import Medium
def refraction_angle(incident, medium1, medium2, normal=None, plane=None):
"""
This function calculates transmitted vector after refraction at planar
surface. `medium1` and `medium2` can be `Medium` or any sympifiable object.
If `incident` is an object of `Ray3D`, `normal` also has to be an instance
of `Ray3D` in order to get the output as a `Ray3D`. Please note that if
plane of separation is not provided and normal is an instance of `Ray3D`,
normal will be assumed to be intersecting incident ray at the plane of
separation. This will not be the case when `normal` is a `Matrix` or
any other sequence.
If `incident` is an instance of `Ray3D` and `plane` has not been provided
and `normal` is not `Ray3D`, output will be a `Matrix`.
Parameters
==========
incident : Matrix, Ray3D, or sequence
Incident vector
medium1 : sympy.physics.optics.medium.Medium or sympifiable
Medium 1 or its refractive index
medium2 : sympy.physics.optics.medium.Medium or sympifiable
Medium 2 or its refractive index
normal : Matrix, Ray3D, or sequence
Normal vector
plane : Plane
Plane of separation of the two media.
Examples
========
>>> from sympy.physics.optics import refraction_angle
>>> from sympy.geometry import Point3D, Ray3D, Plane
>>> from sympy.matrices import Matrix
>>> from sympy import symbols
>>> n = Matrix([0, 0, 1])
>>> P = Plane(Point3D(0, 0, 0), normal_vector=[0, 0, 1])
>>> r1 = Ray3D(Point3D(-1, -1, 1), Point3D(0, 0, 0))
>>> refraction_angle(r1, 1, 1, n)
Matrix([
[ 1],
[ 1],
[-1]])
>>> refraction_angle(r1, 1, 1, plane=P)
Ray3D(Point3D(0, 0, 0), Point3D(1, 1, -1))
With different index of refraction of the two media
>>> n1, n2 = symbols('n1, n2')
>>> refraction_angle(r1, n1, n2, n)
Matrix([
[ n1/n2],
[ n1/n2],
[-sqrt(3)*sqrt(-2*n1**2/(3*n2**2) + 1)]])
>>> refraction_angle(r1, n1, n2, plane=P)
Ray3D(Point3D(0, 0, 0), Point3D(n1/n2, n1/n2, -sqrt(3)*sqrt(-2*n1**2/(3*n2**2) + 1)))
"""
# A flag to check whether to return Ray3D or not
return_ray = False
if plane is not None and normal is not None:
raise ValueError("Either plane or normal is acceptable.")
if not isinstance(incident, Matrix):
if is_sequence(incident):
_incident = Matrix(incident)
elif isinstance(incident, Ray3D):
_incident = Matrix(incident.direction_ratio)
else:
raise TypeError(
"incident should be a Matrix, Ray3D, or sequence")
else:
_incident = incident
# If plane is provided, get direction ratios of the normal
# to the plane from the plane else go with `normal` param.
if plane is not None:
if not isinstance(plane, Plane):
raise TypeError("plane should be an instance of geometry.plane.Plane")
# If we have the plane, we can get the intersection
# point of incident ray and the plane and thus return
# an instance of Ray3D.
if isinstance(incident, Ray3D):
return_ray = True
intersection_pt = plane.intersection(incident)[0]
_normal = Matrix(plane.normal_vector)
else:
if not isinstance(normal, Matrix):
if is_sequence(normal):
_normal = Matrix(normal)
elif isinstance(normal, Ray3D):
_normal = Matrix(normal.direction_ratio)
if isinstance(incident, Ray3D):
intersection_pt = intersection(incident, normal)
if len(intersection_pt) == 0:
raise ValueError(
"Normal isn't concurrent with the incident ray.")
else:
return_ray = True
intersection_pt = intersection_pt[0]
else:
raise TypeError(
"Normal should be a Matrix, Ray3D, or sequence")
else:
_normal = normal
n1, n2 = None, None
if isinstance(medium1, Medium):
n1 = medium1.refractive_index
else:
n1 = sympify(medium1)
if isinstance(medium2, Medium):
n2 = medium2.refractive_index
else:
n2 = sympify(medium2)
eta = n1/n2 # Relative index of refraction
# Calculating magnitude of the vectors
mag_incident = sqrt(sum([i**2 for i in _incident]))
mag_normal = sqrt(sum([i**2 for i in _normal]))
# Converting vectors to unit vectors by dividing
# them with their magnitudes
_incident /= mag_incident
_normal /= mag_normal
c1 = -_incident.dot(_normal) # cos(angle_of_incidence)
cs2 = 1 - eta**2*(1 - c1**2) # cos(angle_of_refraction)**2
if cs2.is_negative: # This is the case of total internal reflection(TIR).
return 0
drs = eta*_incident + (eta*c1 - sqrt(cs2))*_normal
# Multiplying unit vector by its magnitude
drs = drs*mag_incident
if not return_ray:
return drs
else:
return Ray3D(intersection_pt, direction_ratio=drs)
def fresnel_coefficients(angle_of_incidence, medium1, medium2):
"""
This function uses Fresnel equations to calculate reflection and
transmission coefficients. Those are obtained for both polarisations
when the electric field vector is in the plane of incidence (labelled 'p')
and when the electric field vector is perpendicular to the plane of
incidence (labelled 's'). There are four real coefficients unless the
incident ray reflects in total internal in which case there are two complex
ones. Angle of incidence is the angle between the incident ray and the
surface normal. ``medium1`` and ``medium2`` can be ``Medium`` or any
sympifiable object.
Parameters
==========
angle_of_incidence : sympifiable
medium1 : Medium or sympifiable
Medium 1 or its refractive index
medium2 : Medium or sympifiable
Medium 2 or its refractive index
Returns a list with four real Fresnel coefficients:
[reflection p (TM), reflection s (TE),
transmission p (TM), transmission s (TE)]
If the ray is undergoes total internal reflection then returns a
list of two complex Fresnel coefficients:
[reflection p (TM), reflection s (TE)]
Examples
========
>>> from sympy.physics.optics import fresnel_coefficients
>>> fresnel_coefficients(0.3, 1, 2)
[0.317843553417859, -0.348645229818821,
0.658921776708929, 0.651354770181179]
>>> fresnel_coefficients(0.6, 2, 1)
[-0.235625382192159 - 0.971843958291041*I,
0.816477005968898 - 0.577377951366403*I]
References
==========
https://en.wikipedia.org/wiki/Fresnel_equations
"""
if isinstance(medium1, Medium):
n1 = medium1.refractive_index
else:
n1 = sympify(medium1)
if isinstance(medium2, Medium):
n2 = medium2.refractive_index
else:
n2 = sympify(medium2)
angle_of_refraction = asin(n1*sin(angle_of_incidence)/n2)
try:
angle_of_total_internal_reflection_onset = critical_angle(n1, n2)
except ValueError:
angle_of_total_internal_reflection_onset = None
if angle_of_total_internal_reflection_onset == None or\
angle_of_total_internal_reflection_onset > angle_of_incidence:
R_s = -sin(angle_of_incidence - angle_of_refraction)\
/sin(angle_of_incidence + angle_of_refraction)
R_p = tan(angle_of_incidence - angle_of_refraction)\
/tan(angle_of_incidence + angle_of_refraction)
T_s = 2*sin(angle_of_refraction)*cos(angle_of_incidence)\
/sin(angle_of_incidence + angle_of_refraction)
T_p = 2*sin(angle_of_refraction)*cos(angle_of_incidence)\
/(sin(angle_of_incidence + angle_of_refraction)\
*cos(angle_of_incidence - angle_of_refraction))
return [R_p, R_s, T_p, T_s]
else:
n = n2/n1
R_s = cancel((cos(angle_of_incidence)-\
I*sqrt(sin(angle_of_incidence)**2 - n**2))\
/(cos(angle_of_incidence)+\
I*sqrt(sin(angle_of_incidence)**2 - n**2)))
R_p = cancel((n**2*cos(angle_of_incidence)-\
I*sqrt(sin(angle_of_incidence)**2 - n**2))\
/(n**2*cos(angle_of_incidence)+\
I*sqrt(sin(angle_of_incidence)**2 - n**2)))
return [R_p, R_s]
def deviation(incident, medium1, medium2, normal=None, plane=None):
"""
This function calculates the angle of deviation of a ray
due to refraction at planar surface.
Parameters
==========
incident : Matrix, Ray3D, or sequence
Incident vector
medium1 : sympy.physics.optics.medium.Medium or sympifiable
Medium 1 or its refractive index
medium2 : sympy.physics.optics.medium.Medium or sympifiable
Medium 2 or its refractive index
normal : Matrix, Ray3D, or sequence
Normal vector
plane : Plane
Plane of separation of the two media.
Examples
========
>>> from sympy.physics.optics import deviation
>>> from sympy.geometry import Point3D, Ray3D, Plane
>>> from sympy.matrices import Matrix
>>> from sympy import symbols
>>> n1, n2 = symbols('n1, n2')
>>> n = Matrix([0, 0, 1])
>>> P = Plane(Point3D(0, 0, 0), normal_vector=[0, 0, 1])
>>> r1 = Ray3D(Point3D(-1, -1, 1), Point3D(0, 0, 0))
>>> deviation(r1, 1, 1, n)
0
>>> deviation(r1, n1, n2, plane=P)
-acos(-sqrt(-2*n1**2/(3*n2**2) + 1)) + acos(-sqrt(3)/3)
"""
refracted = refraction_angle(incident,
medium1,
medium2,
normal=normal,
plane=plane)
if refracted != 0:
if isinstance(refracted, Ray3D):
refracted = Matrix(refracted.direction_ratio)
if not isinstance(incident, Matrix):
if is_sequence(incident):
_incident = Matrix(incident)
elif isinstance(incident, Ray3D):
_incident = Matrix(incident.direction_ratio)
else:
raise TypeError(
"incident should be a Matrix, Ray3D, or sequence")
else:
_incident = incident
if plane is None:
if not isinstance(normal, Matrix):
if is_sequence(normal):
_normal = Matrix(normal)
elif isinstance(normal, Ray3D):
_normal = Matrix(normal.direction_ratio)
else:
raise TypeError(
"normal should be a Matrix, Ray3D, or sequence")
else:
_normal = normal
else:
_normal = Matrix(plane.normal_vector)
mag_incident = sqrt(sum([i**2 for i in _incident]))
mag_normal = sqrt(sum([i**2 for i in _normal]))
mag_refracted = sqrt(sum([i**2 for i in refracted]))
_incident /= mag_incident
_normal /= mag_normal
refracted /= mag_refracted
i = acos(_incident.dot(_normal))
r = acos(refracted.dot(_normal))
return i - r
def brewster_angle(medium1, medium2):
"""
This function calculates the Brewster's angle of incidence to Medium 2 from
Medium 1 in radians.
Parameters
==========
medium 1 : Medium or sympifiable
Refractive index of Medium 1
medium 2 : Medium or sympifiable
Refractive index of Medium 1
Examples
========
>>> from sympy.physics.optics import brewster_angle
>>> brewster_angle(1, 1.33)
0.926093295503462
"""
n1, n2 = None, None
if isinstance(medium1, Medium):
n1 = medium1.refractive_index
else:
n1 = sympify(medium1)
if isinstance(medium2, Medium):
n2 = medium2.refractive_index
else:
n2 = sympify(medium2)
return atan2(n2, n1)
def critical_angle(medium1, medium2):
"""
This function calculates the critical angle of incidence (marking the onset
of total internal) to Medium 2 from Medium 1 in radians.
Parameters
==========
medium 1 : Medium or sympifiable
Refractive index of Medium 1
medium 2 : Medium or sympifiable
Refractive index of Medium 1
Examples
========
>>> from sympy.physics.optics import critical_angle
>>> critical_angle(1.33, 1)
0.850908514477849
"""
n1, n2 = None, None
if isinstance(medium1, Medium):
n1 = medium1.refractive_index
else:
n1 = sympify(medium1)
if isinstance(medium2, Medium):
n2 = medium2.refractive_index
else:
n2 = sympify(medium2)
if n2 > n1:
raise ValueError('Total internal reflection impossible for n1 < n2')
else:
return asin(n2/n1)
def lens_makers_formula(n_lens, n_surr, r1, r2):
"""
This function calculates focal length of a thin lens.
It follows cartesian sign convention.
Parameters
==========
n_lens : Medium or sympifiable
Index of refraction of lens.
n_surr : Medium or sympifiable
Index of reflection of surrounding.
r1 : sympifiable
Radius of curvature of first surface.
r2 : sympifiable
Radius of curvature of second surface.
Examples
========
>>> from sympy.physics.optics import lens_makers_formula
>>> lens_makers_formula(1.33, 1, 10, -10)
15.1515151515151
"""
if isinstance(n_lens, Medium):
n_lens = n_lens.refractive_index
else:
n_lens = sympify(n_lens)
if isinstance(n_surr, Medium):
n_surr = n_surr.refractive_index
else:
n_surr = sympify(n_surr)
r1 = sympify(r1)
r2 = sympify(r2)
return 1/((n_lens - n_surr)/n_surr*(1/r1 - 1/r2))
def mirror_formula(focal_length=None, u=None, v=None):
"""
This function provides one of the three parameters
when two of them are supplied.
This is valid only for paraxial rays.
Parameters
==========
focal_length : sympifiable
Focal length of the mirror.
u : sympifiable
Distance of object from the pole on
the principal axis.
v : sympifiable
Distance of the image from the pole
on the principal axis.
Examples
========
>>> from sympy.physics.optics import mirror_formula
>>> from sympy.abc import f, u, v
>>> mirror_formula(focal_length=f, u=u)
f*u/(-f + u)
>>> mirror_formula(focal_length=f, v=v)
f*v/(-f + v)
>>> mirror_formula(u=u, v=v)
u*v/(u + v)
"""
if focal_length and u and v:
raise ValueError("Please provide only two parameters")
focal_length = sympify(focal_length)
u = sympify(u)
v = sympify(v)
if u == oo:
_u = Symbol('u')
if v == oo:
_v = Symbol('v')
if focal_length == oo:
_f = Symbol('f')
if focal_length is None:
if u == oo and v == oo:
return Limit(Limit(_v*_u/(_v + _u), _u, oo), _v, oo).doit()
if u == oo:
return Limit(v*_u/(v + _u), _u, oo).doit()
if v == oo:
return Limit(_v*u/(_v + u), _v, oo).doit()
return v*u/(v + u)
if u is None:
if v == oo and focal_length == oo:
return Limit(Limit(_v*_f/(_v - _f), _v, oo), _f, oo).doit()
if v == oo:
return Limit(_v*focal_length/(_v - focal_length), _v, oo).doit()
if focal_length == oo:
return Limit(v*_f/(v - _f), _f, oo).doit()
return v*focal_length/(v - focal_length)
if v is None:
if u == oo and focal_length == oo:
return Limit(Limit(_u*_f/(_u - _f), _u, oo), _f, oo).doit()
if u == oo:
return Limit(_u*focal_length/(_u - focal_length), _u, oo).doit()
if focal_length == oo:
return Limit(u*_f/(u - _f), _f, oo).doit()
return u*focal_length/(u - focal_length)
def lens_formula(focal_length=None, u=None, v=None):
"""
This function provides one of the three parameters
when two of them are supplied.
This is valid only for paraxial rays.
Parameters
==========
focal_length : sympifiable
Focal length of the mirror.
u : sympifiable
Distance of object from the optical center on
the principal axis.
v : sympifiable
Distance of the image from the optical center
on the principal axis.
Examples
========
>>> from sympy.physics.optics import lens_formula
>>> from sympy.abc import f, u, v
>>> lens_formula(focal_length=f, u=u)
f*u/(f + u)
>>> lens_formula(focal_length=f, v=v)
f*v/(f - v)
>>> lens_formula(u=u, v=v)
u*v/(u - v)
"""
if focal_length and u and v:
raise ValueError("Please provide only two parameters")
focal_length = sympify(focal_length)
u = sympify(u)
v = sympify(v)
if u == oo:
_u = Symbol('u')
if v == oo:
_v = Symbol('v')
if focal_length == oo:
_f = Symbol('f')
if focal_length is None:
if u == oo and v == oo:
return Limit(Limit(_v*_u/(_u - _v), _u, oo), _v, oo).doit()
if u == oo:
return Limit(v*_u/(_u - v), _u, oo).doit()
if v == oo:
return Limit(_v*u/(u - _v), _v, oo).doit()
return v*u/(u - v)
if u is None:
if v == oo and focal_length == oo:
return Limit(Limit(_v*_f/(_f - _v), _v, oo), _f, oo).doit()
if v == oo:
return Limit(_v*focal_length/(focal_length - _v), _v, oo).doit()
if focal_length == oo:
return Limit(v*_f/(_f - v), _f, oo).doit()
return v*focal_length/(focal_length - v)
if v is None:
if u == oo and focal_length == oo:
return Limit(Limit(_u*_f/(_u + _f), _u, oo), _f, oo).doit()
if u == oo:
return Limit(_u*focal_length/(_u + focal_length), _u, oo).doit()
if focal_length == oo:
return Limit(u*_f/(u + _f), _f, oo).doit()
return u*focal_length/(u + focal_length)
def hyperfocal_distance(f, N, c):
"""
Parameters
==========
f: sympifiable
Focal length of a given lens
N: sympifiable
F-number of a given lens
c: sympifiable
Circle of Confusion (CoC) of a given image format
Example
=======
>>> from sympy.physics.optics import hyperfocal_distance
>>> from sympy.abc import f, N, c
>>> round(hyperfocal_distance(f = 0.5, N = 8, c = 0.0033), 2)
9.47
"""
f = sympify(f)
N = sympify(N)
c = sympify(c)
return (1/(N * c))*(f**2)
def transverse_magnification(si, so):
"""
Calculates the transverse magnification, which is the ratio of the
image size to the object size.
Parameters
==========
so: sympifiable
Lens-object distance
si: sympifiable
Lens-image distance
Example
=======
>>> from sympy.physics.optics import transverse_magnification
>>> transverse_magnification(30, 15)
-2
"""
si = sympify(si)
so = sympify(so)
return (-(si/so))
|
3ab960636268e28990f014e8733e4343aff5465599b4cd4792f654e16531c827
|
"""
Gaussian optics.
The module implements:
- Ray transfer matrices for geometrical and gaussian optics.
See RayTransferMatrix, GeometricRay and BeamParameter
- Conjugation relations for geometrical and gaussian optics.
See geometric_conj*, gauss_conj and conjugate_gauss_beams
The conventions for the distances are as follows:
focal distance
positive for convergent lenses
object distance
positive for real objects
image distance
positive for real images
"""
from __future__ import print_function, division
__all__ = [
'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 sympy import (atan2, Expr, I, im, Matrix, oo, pi, re, sqrt, sympify,
together)
from sympy.utilities.misc import filldedent
###
# A, B, C, D matrices
###
class RayTransferMatrix(Matrix):
"""
Base class for a Ray Transfer Matrix.
It should be used if there isn't already a more specific subclass mentioned
in See Also.
Parameters
==========
parameters : A, B, C and D or 2x2 matrix (Matrix(2, 2, [A, B, C, D]))
Examples
========
>>> from sympy.physics.optics import RayTransferMatrix, ThinLens
>>> from sympy import Symbol, Matrix
>>> mat = RayTransferMatrix(1, 2, 3, 4)
>>> mat
Matrix([
[1, 2],
[3, 4]])
>>> RayTransferMatrix(Matrix([[1, 2], [3, 4]]))
Matrix([
[1, 2],
[3, 4]])
>>> mat.A
1
>>> f = Symbol('f')
>>> lens = ThinLens(f)
>>> lens
Matrix([
[ 1, 0],
[-1/f, 1]])
>>> lens.C
-1/f
See Also
========
GeometricRay, BeamParameter,
FreeSpace, FlatRefraction, CurvedRefraction,
FlatMirror, CurvedMirror, ThinLens
References
==========
.. [1] https://en.wikipedia.org/wiki/Ray_transfer_matrix_analysis
"""
def __new__(cls, *args):
if len(args) == 4:
temp = ((args[0], args[1]), (args[2], args[3]))
elif len(args) == 1 \
and isinstance(args[0], Matrix) \
and args[0].shape == (2, 2):
temp = args[0]
else:
raise ValueError(filldedent('''
Expecting 2x2 Matrix or the 4 elements of
the Matrix but got %s''' % str(args)))
return Matrix.__new__(cls, temp)
def __mul__(self, other):
if isinstance(other, RayTransferMatrix):
return RayTransferMatrix(Matrix.__mul__(self, other))
elif isinstance(other, GeometricRay):
return GeometricRay(Matrix.__mul__(self, other))
elif isinstance(other, BeamParameter):
temp = self*Matrix(((other.q,), (1,)))
q = (temp[0]/temp[1]).expand(complex=True)
return BeamParameter(other.wavelen,
together(re(q)),
z_r=together(im(q)))
else:
return Matrix.__mul__(self, other)
@property
def A(self):
"""
The A parameter of the Matrix.
Examples
========
>>> from sympy.physics.optics import RayTransferMatrix
>>> mat = RayTransferMatrix(1, 2, 3, 4)
>>> mat.A
1
"""
return self[0, 0]
@property
def B(self):
"""
The B parameter of the Matrix.
Examples
========
>>> from sympy.physics.optics import RayTransferMatrix
>>> mat = RayTransferMatrix(1, 2, 3, 4)
>>> mat.B
2
"""
return self[0, 1]
@property
def C(self):
"""
The C parameter of the Matrix.
Examples
========
>>> from sympy.physics.optics import RayTransferMatrix
>>> mat = RayTransferMatrix(1, 2, 3, 4)
>>> mat.C
3
"""
return self[1, 0]
@property
def D(self):
"""
The D parameter of the Matrix.
Examples
========
>>> from sympy.physics.optics import RayTransferMatrix
>>> mat = RayTransferMatrix(1, 2, 3, 4)
>>> mat.D
4
"""
return self[1, 1]
class FreeSpace(RayTransferMatrix):
"""
Ray Transfer Matrix for free space.
Parameters
==========
distance
See Also
========
RayTransferMatrix
Examples
========
>>> from sympy.physics.optics import FreeSpace
>>> from sympy import symbols
>>> d = symbols('d')
>>> FreeSpace(d)
Matrix([
[1, d],
[0, 1]])
"""
def __new__(cls, d):
return RayTransferMatrix.__new__(cls, 1, d, 0, 1)
class FlatRefraction(RayTransferMatrix):
"""
Ray Transfer Matrix for refraction.
Parameters
==========
n1 : refractive index of one medium
n2 : refractive index of other medium
See Also
========
RayTransferMatrix
Examples
========
>>> from sympy.physics.optics import FlatRefraction
>>> from sympy import symbols
>>> n1, n2 = symbols('n1 n2')
>>> FlatRefraction(n1, n2)
Matrix([
[1, 0],
[0, n1/n2]])
"""
def __new__(cls, n1, n2):
n1, n2 = map(sympify, (n1, n2))
return RayTransferMatrix.__new__(cls, 1, 0, 0, n1/n2)
class CurvedRefraction(RayTransferMatrix):
"""
Ray Transfer Matrix for refraction on curved interface.
Parameters
==========
R : radius of curvature (positive for concave)
n1 : refractive index of one medium
n2 : refractive index of other medium
See Also
========
RayTransferMatrix
Examples
========
>>> from sympy.physics.optics import CurvedRefraction
>>> from sympy import symbols
>>> R, n1, n2 = symbols('R n1 n2')
>>> CurvedRefraction(R, n1, n2)
Matrix([
[ 1, 0],
[(n1 - n2)/(R*n2), n1/n2]])
"""
def __new__(cls, R, n1, n2):
R, n1, n2 = map(sympify, (R, n1, n2))
return RayTransferMatrix.__new__(cls, 1, 0, (n1 - n2)/R/n2, n1/n2)
class FlatMirror(RayTransferMatrix):
"""
Ray Transfer Matrix for reflection.
See Also
========
RayTransferMatrix
Examples
========
>>> from sympy.physics.optics import FlatMirror
>>> FlatMirror()
Matrix([
[1, 0],
[0, 1]])
"""
def __new__(cls):
return RayTransferMatrix.__new__(cls, 1, 0, 0, 1)
class CurvedMirror(RayTransferMatrix):
"""
Ray Transfer Matrix for reflection from curved surface.
Parameters
==========
R : radius of curvature (positive for concave)
See Also
========
RayTransferMatrix
Examples
========
>>> from sympy.physics.optics import CurvedMirror
>>> from sympy import symbols
>>> R = symbols('R')
>>> CurvedMirror(R)
Matrix([
[ 1, 0],
[-2/R, 1]])
"""
def __new__(cls, R):
R = sympify(R)
return RayTransferMatrix.__new__(cls, 1, 0, -2/R, 1)
class ThinLens(RayTransferMatrix):
"""
Ray Transfer Matrix for a thin lens.
Parameters
==========
f : the focal distance
See Also
========
RayTransferMatrix
Examples
========
>>> from sympy.physics.optics import ThinLens
>>> from sympy import symbols
>>> f = symbols('f')
>>> ThinLens(f)
Matrix([
[ 1, 0],
[-1/f, 1]])
"""
def __new__(cls, f):
f = sympify(f)
return RayTransferMatrix.__new__(cls, 1, 0, -1/f, 1)
###
# Representation for geometric ray
###
class GeometricRay(Matrix):
"""
Representation for a geometric ray in the Ray Transfer Matrix formalism.
Parameters
==========
h : height, and
angle : angle, or
matrix : a 2x1 matrix (Matrix(2, 1, [height, angle]))
Examples
========
>>> from sympy.physics.optics import GeometricRay, FreeSpace
>>> from sympy import symbols, Matrix
>>> d, h, angle = symbols('d, h, angle')
>>> GeometricRay(h, angle)
Matrix([
[ h],
[angle]])
>>> FreeSpace(d)*GeometricRay(h, angle)
Matrix([
[angle*d + h],
[ angle]])
>>> GeometricRay( Matrix( ((h,), (angle,)) ) )
Matrix([
[ h],
[angle]])
See Also
========
RayTransferMatrix
"""
def __new__(cls, *args):
if len(args) == 1 and isinstance(args[0], Matrix) \
and args[0].shape == (2, 1):
temp = args[0]
elif len(args) == 2:
temp = ((args[0],), (args[1],))
else:
raise ValueError(filldedent('''
Expecting 2x1 Matrix or the 2 elements of
the Matrix but got %s''' % str(args)))
return Matrix.__new__(cls, temp)
@property
def height(self):
"""
The distance from the optical axis.
Examples
========
>>> from sympy.physics.optics import GeometricRay
>>> from sympy import symbols
>>> h, angle = symbols('h, angle')
>>> gRay = GeometricRay(h, angle)
>>> gRay.height
h
"""
return self[0]
@property
def angle(self):
"""
The angle with the optical axis.
Examples
========
>>> from sympy.physics.optics import GeometricRay
>>> from sympy import symbols
>>> h, angle = symbols('h, angle')
>>> gRay = GeometricRay(h, angle)
>>> gRay.angle
angle
"""
return self[1]
###
# Representation for gauss beam
###
class BeamParameter(Expr):
"""
Representation for a gaussian ray in the Ray Transfer Matrix formalism.
Parameters
==========
wavelen : the wavelength,
z : the distance to waist, and
w : the waist, or
z_r : the rayleigh range
Examples
========
>>> from sympy.physics.optics import BeamParameter
>>> p = BeamParameter(530e-9, 1, w=1e-3)
>>> p.q
1 + 1.88679245283019*I*pi
>>> p.q.n()
1.0 + 5.92753330865999*I
>>> p.w_0.n()
0.00100000000000000
>>> p.z_r.n()
5.92753330865999
>>> from sympy.physics.optics import FreeSpace
>>> fs = FreeSpace(10)
>>> p1 = fs*p
>>> p.w.n()
0.00101413072159615
>>> p1.w.n()
0.00210803120913829
See Also
========
RayTransferMatrix
References
==========
.. [1] https://en.wikipedia.org/wiki/Complex_beam_parameter
.. [2] https://en.wikipedia.org/wiki/Gaussian_beam
"""
#TODO A class Complex may be implemented. The BeamParameter may
# subclass it. See:
# https://groups.google.com/d/topic/sympy/7XkU07NRBEs/discussion
__slots__ = ['z', 'z_r', 'wavelen']
def __new__(cls, wavelen, z, **kwargs):
wavelen, z = map(sympify, (wavelen, z))
inst = Expr.__new__(cls, wavelen, z)
inst.wavelen = wavelen
inst.z = z
if len(kwargs) != 1:
raise ValueError('Constructor expects exactly one named argument.')
elif 'z_r' in kwargs:
inst.z_r = sympify(kwargs['z_r'])
elif 'w' in kwargs:
inst.z_r = waist2rayleigh(sympify(kwargs['w']), wavelen)
else:
raise ValueError('The constructor needs named argument w or z_r')
return inst
@property
def q(self):
"""
The complex parameter representing the beam.
Examples
========
>>> from sympy.physics.optics import BeamParameter
>>> p = BeamParameter(530e-9, 1, w=1e-3)
>>> p.q
1 + 1.88679245283019*I*pi
"""
return self.z + I*self.z_r
@property
def radius(self):
"""
The radius of curvature of the phase front.
Examples
========
>>> from sympy.physics.optics import BeamParameter
>>> p = BeamParameter(530e-9, 1, w=1e-3)
>>> p.radius
1 + 3.55998576005696*pi**2
"""
return self.z*(1 + (self.z_r/self.z)**2)
@property
def w(self):
"""
The beam radius at `1/e^2` intensity.
See Also
========
w_0 : the minimal radius of beam
Examples
========
>>> from sympy.physics.optics import BeamParameter
>>> p = BeamParameter(530e-9, 1, w=1e-3)
>>> p.w
0.001*sqrt(0.2809/pi**2 + 1)
"""
return self.w_0*sqrt(1 + (self.z/self.z_r)**2)
@property
def w_0(self):
"""
The beam waist (minimal radius).
See Also
========
w : the beam radius at `1/e^2` intensity
Examples
========
>>> from sympy.physics.optics import BeamParameter
>>> p = BeamParameter(530e-9, 1, w=1e-3)
>>> p.w_0
0.00100000000000000
"""
return sqrt(self.z_r/pi*self.wavelen)
@property
def divergence(self):
"""
Half of the total angular spread.
Examples
========
>>> from sympy.physics.optics import BeamParameter
>>> p = BeamParameter(530e-9, 1, w=1e-3)
>>> p.divergence
0.00053/pi
"""
return self.wavelen/pi/self.w_0
@property
def gouy(self):
"""
The Gouy phase.
Examples
========
>>> from sympy.physics.optics import BeamParameter
>>> p = BeamParameter(530e-9, 1, w=1e-3)
>>> p.gouy
atan(0.53/pi)
"""
return atan2(self.z, self.z_r)
@property
def waist_approximation_limit(self):
"""
The minimal waist for which the gauss beam approximation is valid.
The gauss beam is a solution to the paraxial equation. For curvatures
that are too great it is not a valid approximation.
Examples
========
>>> from sympy.physics.optics import BeamParameter
>>> p = BeamParameter(530e-9, 1, w=1e-3)
>>> p.waist_approximation_limit
1.06e-6/pi
"""
return 2*self.wavelen/pi
###
# Utilities
###
def waist2rayleigh(w, wavelen):
"""
Calculate the rayleigh range from the waist of a gaussian beam.
See Also
========
rayleigh2waist, BeamParameter
Examples
========
>>> from sympy.physics.optics import waist2rayleigh
>>> from sympy import symbols
>>> w, wavelen = symbols('w wavelen')
>>> waist2rayleigh(w, wavelen)
pi*w**2/wavelen
"""
w, wavelen = map(sympify, (w, wavelen))
return w**2*pi/wavelen
def rayleigh2waist(z_r, wavelen):
"""Calculate the waist from the rayleigh range of a gaussian beam.
See Also
========
waist2rayleigh, BeamParameter
Examples
========
>>> from sympy.physics.optics import rayleigh2waist
>>> from sympy import symbols
>>> z_r, wavelen = symbols('z_r wavelen')
>>> rayleigh2waist(z_r, wavelen)
sqrt(wavelen*z_r)/sqrt(pi)
"""
z_r, wavelen = map(sympify, (z_r, wavelen))
return sqrt(z_r/pi*wavelen)
def geometric_conj_ab(a, b):
"""
Conjugation relation for geometrical beams under paraxial conditions.
Takes the distances to the optical element and returns the needed
focal distance.
See Also
========
geometric_conj_af, geometric_conj_bf
Examples
========
>>> from sympy.physics.optics import geometric_conj_ab
>>> from sympy import symbols
>>> a, b = symbols('a b')
>>> geometric_conj_ab(a, b)
a*b/(a + b)
"""
a, b = map(sympify, (a, b))
if abs(a) == oo or abs(b) == oo:
return a if abs(b) == oo else b
else:
return a*b/(a + b)
def geometric_conj_af(a, f):
"""
Conjugation relation for geometrical beams under paraxial conditions.
Takes the object distance (for geometric_conj_af) or the image distance
(for geometric_conj_bf) to the optical element and the focal distance.
Then it returns the other distance needed for conjugation.
See Also
========
geometric_conj_ab
Examples
========
>>> from sympy.physics.optics.gaussopt import geometric_conj_af, geometric_conj_bf
>>> from sympy import symbols
>>> a, b, f = symbols('a b f')
>>> geometric_conj_af(a, f)
a*f/(a - f)
>>> geometric_conj_bf(b, f)
b*f/(b - f)
"""
a, f = map(sympify, (a, f))
return -geometric_conj_ab(a, -f)
geometric_conj_bf = geometric_conj_af
def gaussian_conj(s_in, z_r_in, f):
"""
Conjugation relation for gaussian beams.
Parameters
==========
s_in : the distance to optical element from the waist
z_r_in : the rayleigh range of the incident beam
f : the focal length of the optical element
Returns
=======
a tuple containing (s_out, z_r_out, m)
s_out : the distance between the new waist and the optical element
z_r_out : the rayleigh range of the emergent beam
m : the ration between the new and the old waists
Examples
========
>>> from sympy.physics.optics import gaussian_conj
>>> from sympy import symbols
>>> s_in, z_r_in, f = symbols('s_in z_r_in f')
>>> gaussian_conj(s_in, z_r_in, f)[0]
1/(-1/(s_in + z_r_in**2/(-f + s_in)) + 1/f)
>>> gaussian_conj(s_in, z_r_in, f)[1]
z_r_in/(1 - s_in**2/f**2 + z_r_in**2/f**2)
>>> gaussian_conj(s_in, z_r_in, f)[2]
1/sqrt(1 - s_in**2/f**2 + z_r_in**2/f**2)
"""
s_in, z_r_in, f = map(sympify, (s_in, z_r_in, f))
s_out = 1 / ( -1/(s_in + z_r_in**2/(s_in - f)) + 1/f )
m = 1/sqrt((1 - (s_in/f)**2) + (z_r_in/f)**2)
z_r_out = z_r_in / ((1 - (s_in/f)**2) + (z_r_in/f)**2)
return (s_out, z_r_out, m)
def conjugate_gauss_beams(wavelen, waist_in, waist_out, **kwargs):
"""
Find the optical setup conjugating the object/image waists.
Parameters
==========
wavelen : the wavelength of the beam
waist_in and waist_out : the waists to be conjugated
f : the focal distance of the element used in the conjugation
Returns
=======
a tuple containing (s_in, s_out, f)
s_in : the distance before the optical element
s_out : the distance after the optical element
f : the focal distance of the optical element
Examples
========
>>> from sympy.physics.optics import conjugate_gauss_beams
>>> from sympy import symbols, factor
>>> l, w_i, w_o, f = symbols('l w_i w_o f')
>>> conjugate_gauss_beams(l, w_i, w_o, f=f)[0]
f*(1 - sqrt(w_i**2/w_o**2 - pi**2*w_i**4/(f**2*l**2)))
>>> factor(conjugate_gauss_beams(l, w_i, w_o, f=f)[1])
f*w_o**2*(w_i**2/w_o**2 - sqrt(w_i**2/w_o**2 -
pi**2*w_i**4/(f**2*l**2)))/w_i**2
>>> conjugate_gauss_beams(l, w_i, w_o, f=f)[2]
f
"""
#TODO add the other possible arguments
wavelen, waist_in, waist_out = map(sympify, (wavelen, waist_in, waist_out))
m = waist_out / waist_in
z = waist2rayleigh(waist_in, wavelen)
if len(kwargs) != 1:
raise ValueError("The function expects only one named argument")
elif 'dist' in kwargs:
raise NotImplementedError(filldedent('''
Currently only focal length is supported as a parameter'''))
elif 'f' in kwargs:
f = sympify(kwargs['f'])
s_in = f * (1 - sqrt(1/m**2 - z**2/f**2))
s_out = gaussian_conj(s_in, z, f)[0]
elif 's_in' in kwargs:
raise NotImplementedError(filldedent('''
Currently only focal length is supported as a parameter'''))
else:
raise ValueError(filldedent('''
The functions expects the focal length as a named argument'''))
return (s_in, s_out, f)
#TODO
#def plot_beam():
# """Plot the beam radius as it propagates in space."""
# pass
#TODO
#def plot_beam_conjugation():
# """
# Plot the intersection of two beams.
#
# Represents the conjugation relation.
#
# See Also
# ========
#
# conjugate_gauss_beams
# """
# pass
|
c075d2fc60221bd50c2fd2a1a56d6a9d4868be55256dc4366f9fb72648e54b1b
|
from sympy import symbols, S, log
from sympy.core.trace import Tr
from sympy.external import import_module
from sympy.physics.quantum.density import Density, entropy, fidelity
from sympy.physics.quantum.state import Ket, TimeDepKet
from sympy.physics.quantum.qubit import Qubit
from sympy.physics.quantum.represent import represent
from sympy.physics.quantum.dagger import Dagger
from sympy.physics.quantum.cartesian import XKet, PxKet, PxOp, XOp
from sympy.physics.quantum.spin import JzKet
from sympy.physics.quantum.operator import OuterProduct
from sympy.functions import sqrt
from sympy.utilities.pytest import raises, slow
from sympy.physics.quantum.matrixutils import scipy_sparse_matrix
from sympy.physics.quantum.tensorproduct import TensorProduct
def test_eval_args():
# check instance created
assert isinstance(Density([Ket(0), 0.5], [Ket(1), 0.5]), Density)
assert isinstance(Density([Qubit('00'), 1/sqrt(2)],
[Qubit('11'), 1/sqrt(2)]), Density)
#test if Qubit object type preserved
d = Density([Qubit('00'), 1/sqrt(2)], [Qubit('11'), 1/sqrt(2)])
for (state, prob) in d.args:
assert isinstance(state, Qubit)
# check for value error, when prob is not provided
raises(ValueError, lambda: Density([Ket(0)], [Ket(1)]))
def test_doit():
x, y = symbols('x y')
A, B, C, D, E, F = symbols('A B C D E F', commutative=False)
d = Density([XKet(), 0.5], [PxKet(), 0.5])
assert (0.5*(PxKet()*Dagger(PxKet())) +
0.5*(XKet()*Dagger(XKet()))) == d.doit()
# check for kets with expr in them
d_with_sym = Density([XKet(x*y), 0.5], [PxKet(x*y), 0.5])
assert (0.5*(PxKet(x*y)*Dagger(PxKet(x*y))) +
0.5*(XKet(x*y)*Dagger(XKet(x*y)))) == d_with_sym.doit()
d = Density([(A + B)*C, 1.0])
assert d.doit() == (1.0*A*C*Dagger(C)*Dagger(A) +
1.0*A*C*Dagger(C)*Dagger(B) +
1.0*B*C*Dagger(C)*Dagger(A) +
1.0*B*C*Dagger(C)*Dagger(B))
# With TensorProducts as args
# Density with simple tensor products as args
t = TensorProduct(A, B, C)
d = Density([t, 1.0])
assert d.doit() == \
1.0 * TensorProduct(A*Dagger(A), B*Dagger(B), C*Dagger(C))
# Density with multiple Tensorproducts as states
t2 = TensorProduct(A, B)
t3 = TensorProduct(C, D)
d = Density([t2, 0.5], [t3, 0.5])
assert d.doit() == (0.5 * TensorProduct(A*Dagger(A), B*Dagger(B)) +
0.5 * TensorProduct(C*Dagger(C), D*Dagger(D)))
#Density with mixed states
d = Density([t2 + t3, 1.0])
assert d.doit() == (1.0 * TensorProduct(A*Dagger(A), B*Dagger(B)) +
1.0 * TensorProduct(A*Dagger(C), B*Dagger(D)) +
1.0 * TensorProduct(C*Dagger(A), D*Dagger(B)) +
1.0 * TensorProduct(C*Dagger(C), D*Dagger(D)))
#Density operators with spin states
tp1 = TensorProduct(JzKet(1, 1), JzKet(1, -1))
d = Density([tp1, 1])
# full trace
t = Tr(d)
assert t.doit() == 1
#Partial trace on density operators with spin states
t = Tr(d, [0])
assert t.doit() == JzKet(1, -1) * Dagger(JzKet(1, -1))
t = Tr(d, [1])
assert t.doit() == JzKet(1, 1) * Dagger(JzKet(1, 1))
# with another spin state
tp2 = TensorProduct(JzKet(S(1)/2, S(1)/2), JzKet(S(1)/2, -S(1)/2))
d = Density([tp2, 1])
#full trace
t = Tr(d)
assert t.doit() == 1
#Partial trace on density operators with spin states
t = Tr(d, [0])
assert t.doit() == JzKet(S(1)/2, -S(1)/2) * Dagger(JzKet(S(1)/2, -S(1)/2))
t = Tr(d, [1])
assert t.doit() == JzKet(S(1)/2, S(1)/2) * Dagger(JzKet(S(1)/2, S(1)/2))
def test_apply_op():
d = Density([Ket(0), 0.5], [Ket(1), 0.5])
assert d.apply_op(XOp()) == Density([XOp()*Ket(0), 0.5],
[XOp()*Ket(1), 0.5])
def test_represent():
x, y = symbols('x y')
d = Density([XKet(), 0.5], [PxKet(), 0.5])
assert (represent(0.5*(PxKet()*Dagger(PxKet()))) +
represent(0.5*(XKet()*Dagger(XKet())))) == represent(d)
# check for kets with expr in them
d_with_sym = Density([XKet(x*y), 0.5], [PxKet(x*y), 0.5])
assert (represent(0.5*(PxKet(x*y)*Dagger(PxKet(x*y)))) +
represent(0.5*(XKet(x*y)*Dagger(XKet(x*y))))) == \
represent(d_with_sym)
# check when given explicit basis
assert (represent(0.5*(XKet()*Dagger(XKet())), basis=PxOp()) +
represent(0.5*(PxKet()*Dagger(PxKet())), basis=PxOp())) == \
represent(d, basis=PxOp())
def test_states():
d = Density([Ket(0), 0.5], [Ket(1), 0.5])
states = d.states()
assert states[0] == Ket(0) and states[1] == Ket(1)
def test_probs():
d = Density([Ket(0), .75], [Ket(1), 0.25])
probs = d.probs()
assert probs[0] == 0.75 and probs[1] == 0.25
#probs can be symbols
x, y = symbols('x y')
d = Density([Ket(0), x], [Ket(1), y])
probs = d.probs()
assert probs[0] == x and probs[1] == y
def test_get_state():
x, y = symbols('x y')
d = Density([Ket(0), x], [Ket(1), y])
states = (d.get_state(0), d.get_state(1))
assert states[0] == Ket(0) and states[1] == Ket(1)
def test_get_prob():
x, y = symbols('x y')
d = Density([Ket(0), x], [Ket(1), y])
probs = (d.get_prob(0), d.get_prob(1))
assert probs[0] == x and probs[1] == y
def test_entropy():
up = JzKet(S(1)/2, S(1)/2)
down = JzKet(S(1)/2, -S(1)/2)
d = Density((up, S(1)/2), (down, S(1)/2))
# test for density object
ent = entropy(d)
assert entropy(d) == 0.5*log(2)
assert d.entropy() == 0.5*log(2)
np = import_module('numpy', min_module_version='1.4.0')
if np:
#do this test only if 'numpy' is available on test machine
np_mat = represent(d, format='numpy')
ent = entropy(np_mat)
assert isinstance(np_mat, np.matrixlib.defmatrix.matrix)
assert ent.real == 0.69314718055994529
assert ent.imag == 0
scipy = import_module('scipy', __import__kwargs={'fromlist': ['sparse']})
if scipy and np:
#do this test only if numpy and scipy are available
mat = represent(d, format="scipy.sparse")
assert isinstance(mat, scipy_sparse_matrix)
assert ent.real == 0.69314718055994529
assert ent.imag == 0
def test_eval_trace():
up = JzKet(S(1)/2, S(1)/2)
down = JzKet(S(1)/2, -S(1)/2)
d = Density((up, 0.5), (down, 0.5))
t = Tr(d)
assert t.doit() == 1
#test dummy time dependent states
class TestTimeDepKet(TimeDepKet):
def _eval_trace(self, bra, **options):
return 1
x, t = symbols('x t')
k1 = TestTimeDepKet(0, 0.5)
k2 = TestTimeDepKet(0, 1)
d = Density([k1, 0.5], [k2, 0.5])
assert d.doit() == (0.5 * OuterProduct(k1, k1.dual) +
0.5 * OuterProduct(k2, k2.dual))
t = Tr(d)
assert t.doit() == 1
@slow
def test_fidelity():
#test with kets
up = JzKet(S(1)/2, S(1)/2)
down = JzKet(S(1)/2, -S(1)/2)
updown = (S(1)/sqrt(2))*up + (S(1)/sqrt(2))*down
#check with matrices
up_dm = represent(up * Dagger(up))
down_dm = represent(down * Dagger(down))
updown_dm = represent(updown * Dagger(updown))
assert abs(fidelity(up_dm, up_dm) - 1) < 1e-3
assert fidelity(up_dm, down_dm) < 1e-3
assert abs(fidelity(up_dm, updown_dm) - (S(1)/sqrt(2))) < 1e-3
assert abs(fidelity(updown_dm, down_dm) - (S(1)/sqrt(2))) < 1e-3
#check with density
up_dm = Density([up, 1.0])
down_dm = Density([down, 1.0])
updown_dm = Density([updown, 1.0])
assert abs(fidelity(up_dm, up_dm) - 1) < 1e-3
assert abs(fidelity(up_dm, down_dm)) < 1e-3
assert abs(fidelity(up_dm, updown_dm) - (S(1)/sqrt(2))) < 1e-3
assert abs(fidelity(updown_dm, down_dm) - (S(1)/sqrt(2))) < 1e-3
#check mixed states with density
updown2 = (sqrt(3)/2)*up + (S(1)/2)*down
d1 = Density([updown, 0.25], [updown2, 0.75])
d2 = Density([updown, 0.75], [updown2, 0.25])
assert abs(fidelity(d1, d2) - 0.991) < 1e-3
assert abs(fidelity(d2, d1) - fidelity(d1, d2)) < 1e-3
#using qubits/density(pure states)
state1 = Qubit('0')
state2 = Qubit('1')
state3 = (S(1)/sqrt(2))*state1 + (S(1)/sqrt(2))*state2
state4 = (sqrt(S(2)/3))*state1 + (S(1)/sqrt(3))*state2
state1_dm = Density([state1, 1])
state2_dm = Density([state2, 1])
state3_dm = Density([state3, 1])
assert fidelity(state1_dm, state1_dm) == 1
assert fidelity(state1_dm, state2_dm) == 0
assert abs(fidelity(state1_dm, state3_dm) - 1/sqrt(2)) < 1e-3
assert abs(fidelity(state3_dm, state2_dm) - 1/sqrt(2)) < 1e-3
#using qubits/density(mixed states)
d1 = Density([state3, 0.70], [state4, 0.30])
d2 = Density([state3, 0.20], [state4, 0.80])
assert abs(fidelity(d1, d1) - 1) < 1e-3
assert abs(fidelity(d1, d2) - 0.996) < 1e-3
assert abs(fidelity(d1, d2) - fidelity(d2, d1)) < 1e-3
#TODO: test for invalid arguments
# non-square matrix
mat1 = [[0, 0],
[0, 0],
[0, 0]]
mat2 = [[0, 0],
[0, 0]]
raises(ValueError, lambda: fidelity(mat1, mat2))
# unequal dimensions
mat1 = [[0, 0],
[0, 0]]
mat2 = [[0, 0, 0],
[0, 0, 0],
[0, 0, 0]]
raises(ValueError, lambda: fidelity(mat1, mat2))
# unsupported data-type
x, y = 1, 2 # random values that is not a matrix
raises(ValueError, lambda: fidelity(x, y))
|
53715c3b689613f54c14ddee49d57fe8f87df94fca529a8298eed3176a9dd738
|
from sympy.external import import_module
from sympy import Mul, Integer
from sympy.core.compatibility import PY3
from sympy.physics.quantum.dagger import Dagger
from sympy.physics.quantum.gate import (X, Y, Z, H, CNOT,
IdentityGate, CGate, PhaseGate, TGate)
from sympy.physics.quantum.identitysearch import (generate_gate_rules,
generate_equivalent_ids, GateIdentity, bfs_identity_search,
is_scalar_sparse_matrix,
is_scalar_nonsparse_matrix, is_degenerate, is_reducible)
from sympy.utilities.pytest import skip, XFAIL
def create_gate_sequence(qubit=0):
gates = (X(qubit), Y(qubit), Z(qubit), H(qubit))
return gates
def test_generate_gate_rules_1():
# Test with tuples
(x, y, z, h) = create_gate_sequence()
ph = PhaseGate(0)
cgate_t = CGate(0, TGate(1))
assert generate_gate_rules((x,)) == {((x,), ())}
gate_rules = set([((x, x), ()),
((x,), (x,))])
assert generate_gate_rules((x, x)) == gate_rules
gate_rules = set([((x, y, x), ()),
((y, x, x), ()),
((x, x, y), ()),
((y, x), (x,)),
((x, y), (x,)),
((y,), (x, x))])
assert generate_gate_rules((x, y, x)) == gate_rules
gate_rules = set([((x, y, z), ()), ((y, z, x), ()), ((z, x, y), ()),
((), (x, z, y)), ((), (y, x, z)), ((), (z, y, x)),
((x,), (z, y)), ((y, z), (x,)), ((y,), (x, z)),
((z, x), (y,)), ((z,), (y, x)), ((x, y), (z,))])
actual = generate_gate_rules((x, y, z))
assert actual == gate_rules
gate_rules = set(
[((), (h, z, y, x)), ((), (x, h, z, y)), ((), (y, x, h, z)),
((), (z, y, x, h)), ((h,), (z, y, x)), ((x,), (h, z, y)),
((y,), (x, h, z)), ((z,), (y, x, h)), ((h, x), (z, y)),
((x, y), (h, z)), ((y, z), (x, h)), ((z, h), (y, x)),
((h, x, y), (z,)), ((x, y, z), (h,)), ((y, z, h), (x,)),
((z, h, x), (y,)), ((h, x, y, z), ()), ((x, y, z, h), ()),
((y, z, h, x), ()), ((z, h, x, y), ())])
actual = generate_gate_rules((x, y, z, h))
assert actual == gate_rules
gate_rules = set([((), (cgate_t**(-1), ph**(-1), x)),
((), (ph**(-1), x, cgate_t**(-1))),
((), (x, cgate_t**(-1), ph**(-1))),
((cgate_t,), (ph**(-1), x)),
((ph,), (x, cgate_t**(-1))),
((x,), (cgate_t**(-1), ph**(-1))),
((cgate_t, x), (ph**(-1),)),
((ph, cgate_t), (x,)),
((x, ph), (cgate_t**(-1),)),
((cgate_t, x, ph), ()),
((ph, cgate_t, x), ()),
((x, ph, cgate_t), ())])
actual = generate_gate_rules((x, ph, cgate_t))
assert actual == gate_rules
gate_rules = set([(Integer(1), cgate_t**(-1)*ph**(-1)*x),
(Integer(1), ph**(-1)*x*cgate_t**(-1)),
(Integer(1), x*cgate_t**(-1)*ph**(-1)),
(cgate_t, ph**(-1)*x),
(ph, x*cgate_t**(-1)),
(x, cgate_t**(-1)*ph**(-1)),
(cgate_t*x, ph**(-1)),
(ph*cgate_t, x),
(x*ph, cgate_t**(-1)),
(cgate_t*x*ph, Integer(1)),
(ph*cgate_t*x, Integer(1)),
(x*ph*cgate_t, Integer(1))])
actual = generate_gate_rules((x, ph, cgate_t), return_as_muls=True)
assert actual == gate_rules
def test_generate_gate_rules_2():
# Test with Muls
(x, y, z, h) = create_gate_sequence()
ph = PhaseGate(0)
cgate_t = CGate(0, TGate(1))
# Note: 1 (type int) is not the same as 1 (type One)
expected = {(x, Integer(1))}
assert generate_gate_rules((x,), return_as_muls=True) == expected
expected = {(Integer(1), Integer(1))}
assert generate_gate_rules(x*x, return_as_muls=True) == expected
expected = {((), ())}
assert generate_gate_rules(x*x, return_as_muls=False) == expected
gate_rules = set([(x*y*x, Integer(1)),
(y, Integer(1)),
(y*x, x),
(x*y, x)])
assert generate_gate_rules(x*y*x, return_as_muls=True) == gate_rules
gate_rules = set([(x*y*z, Integer(1)),
(y*z*x, Integer(1)),
(z*x*y, Integer(1)),
(Integer(1), x*z*y),
(Integer(1), y*x*z),
(Integer(1), z*y*x),
(x, z*y),
(y*z, x),
(y, x*z),
(z*x, y),
(z, y*x),
(x*y, z)])
actual = generate_gate_rules(x*y*z, return_as_muls=True)
assert actual == gate_rules
gate_rules = set([(Integer(1), h*z*y*x),
(Integer(1), x*h*z*y),
(Integer(1), y*x*h*z),
(Integer(1), z*y*x*h),
(h, z*y*x), (x, h*z*y),
(y, x*h*z), (z, y*x*h),
(h*x, z*y), (z*h, y*x),
(x*y, h*z), (y*z, x*h),
(h*x*y, z), (x*y*z, h),
(y*z*h, x), (z*h*x, y),
(h*x*y*z, Integer(1)),
(x*y*z*h, Integer(1)),
(y*z*h*x, Integer(1)),
(z*h*x*y, Integer(1))])
actual = generate_gate_rules(x*y*z*h, return_as_muls=True)
assert actual == gate_rules
gate_rules = set([(Integer(1), cgate_t**(-1)*ph**(-1)*x),
(Integer(1), ph**(-1)*x*cgate_t**(-1)),
(Integer(1), x*cgate_t**(-1)*ph**(-1)),
(cgate_t, ph**(-1)*x),
(ph, x*cgate_t**(-1)),
(x, cgate_t**(-1)*ph**(-1)),
(cgate_t*x, ph**(-1)),
(ph*cgate_t, x),
(x*ph, cgate_t**(-1)),
(cgate_t*x*ph, Integer(1)),
(ph*cgate_t*x, Integer(1)),
(x*ph*cgate_t, Integer(1))])
actual = generate_gate_rules(x*ph*cgate_t, return_as_muls=True)
assert actual == gate_rules
gate_rules = set([((), (cgate_t**(-1), ph**(-1), x)),
((), (ph**(-1), x, cgate_t**(-1))),
((), (x, cgate_t**(-1), ph**(-1))),
((cgate_t,), (ph**(-1), x)),
((ph,), (x, cgate_t**(-1))),
((x,), (cgate_t**(-1), ph**(-1))),
((cgate_t, x), (ph**(-1),)),
((ph, cgate_t), (x,)),
((x, ph), (cgate_t**(-1),)),
((cgate_t, x, ph), ()),
((ph, cgate_t, x), ()),
((x, ph, cgate_t), ())])
actual = generate_gate_rules(x*ph*cgate_t)
assert actual == gate_rules
def test_generate_equivalent_ids_1():
# Test with tuples
(x, y, z, h) = create_gate_sequence()
assert generate_equivalent_ids((x,)) == {(x,)}
assert generate_equivalent_ids((x, x)) == {(x, x)}
assert generate_equivalent_ids((x, y)) == {(x, y), (y, x)}
gate_seq = (x, y, z)
gate_ids = set([(x, y, z), (y, z, x), (z, x, y), (z, y, x),
(y, x, z), (x, z, y)])
assert generate_equivalent_ids(gate_seq) == gate_ids
gate_ids = set([Mul(x, y, z), Mul(y, z, x), Mul(z, x, y),
Mul(z, y, x), Mul(y, x, z), Mul(x, z, y)])
assert generate_equivalent_ids(gate_seq, return_as_muls=True) == gate_ids
gate_seq = (x, y, z, h)
gate_ids = set([(x, y, z, h), (y, z, h, x),
(h, x, y, z), (h, z, y, x),
(z, y, x, h), (y, x, h, z),
(z, h, x, y), (x, h, z, y)])
assert generate_equivalent_ids(gate_seq) == gate_ids
gate_seq = (x, y, x, y)
gate_ids = {(x, y, x, y), (y, x, y, x)}
assert generate_equivalent_ids(gate_seq) == gate_ids
cgate_y = CGate((1,), y)
gate_seq = (y, cgate_y, y, cgate_y)
gate_ids = {(y, cgate_y, y, cgate_y), (cgate_y, y, cgate_y, y)}
assert generate_equivalent_ids(gate_seq) == gate_ids
cnot = CNOT(1, 0)
cgate_z = CGate((0,), Z(1))
gate_seq = (cnot, h, cgate_z, h)
gate_ids = set([(cnot, h, cgate_z, h), (h, cgate_z, h, cnot),
(h, cnot, h, cgate_z), (cgate_z, h, cnot, h)])
assert generate_equivalent_ids(gate_seq) == gate_ids
def test_generate_equivalent_ids_2():
# Test with Muls
(x, y, z, h) = create_gate_sequence()
assert generate_equivalent_ids((x,), return_as_muls=True) == {x}
gate_ids = {Integer(1)}
assert generate_equivalent_ids(x*x, return_as_muls=True) == gate_ids
gate_ids = {x*y, y*x}
assert generate_equivalent_ids(x*y, return_as_muls=True) == gate_ids
gate_ids = {(x, y), (y, x)}
assert generate_equivalent_ids(x*y) == gate_ids
circuit = Mul(*(x, y, z))
gate_ids = set([x*y*z, y*z*x, z*x*y, z*y*x,
y*x*z, x*z*y])
assert generate_equivalent_ids(circuit, return_as_muls=True) == gate_ids
circuit = Mul(*(x, y, z, h))
gate_ids = set([x*y*z*h, y*z*h*x,
h*x*y*z, h*z*y*x,
z*y*x*h, y*x*h*z,
z*h*x*y, x*h*z*y])
assert generate_equivalent_ids(circuit, return_as_muls=True) == gate_ids
circuit = Mul(*(x, y, x, y))
gate_ids = {x*y*x*y, y*x*y*x}
assert generate_equivalent_ids(circuit, return_as_muls=True) == gate_ids
cgate_y = CGate((1,), y)
circuit = Mul(*(y, cgate_y, y, cgate_y))
gate_ids = {y*cgate_y*y*cgate_y, cgate_y*y*cgate_y*y}
assert generate_equivalent_ids(circuit, return_as_muls=True) == gate_ids
cnot = CNOT(1, 0)
cgate_z = CGate((0,), Z(1))
circuit = Mul(*(cnot, h, cgate_z, h))
gate_ids = set([cnot*h*cgate_z*h, h*cgate_z*h*cnot,
h*cnot*h*cgate_z, cgate_z*h*cnot*h])
assert generate_equivalent_ids(circuit, return_as_muls=True) == gate_ids
def test_is_scalar_nonsparse_matrix():
numqubits = 2
id_only = False
id_gate = (IdentityGate(1),)
actual = is_scalar_nonsparse_matrix(id_gate, numqubits, id_only)
assert actual is True
x0 = X(0)
xx_circuit = (x0, x0)
actual = is_scalar_nonsparse_matrix(xx_circuit, numqubits, id_only)
assert actual is True
x1 = X(1)
y1 = Y(1)
xy_circuit = (x1, y1)
actual = is_scalar_nonsparse_matrix(xy_circuit, numqubits, id_only)
assert actual is False
z1 = Z(1)
xyz_circuit = (x1, y1, z1)
actual = is_scalar_nonsparse_matrix(xyz_circuit, numqubits, id_only)
assert actual is True
cnot = CNOT(1, 0)
cnot_circuit = (cnot, cnot)
actual = is_scalar_nonsparse_matrix(cnot_circuit, numqubits, id_only)
assert actual is True
h = H(0)
hh_circuit = (h, h)
actual = is_scalar_nonsparse_matrix(hh_circuit, numqubits, id_only)
assert actual is True
h1 = H(1)
xhzh_circuit = (x1, h1, z1, h1)
actual = is_scalar_nonsparse_matrix(xhzh_circuit, numqubits, id_only)
assert actual is True
id_only = True
actual = is_scalar_nonsparse_matrix(xhzh_circuit, numqubits, id_only)
assert actual is True
actual = is_scalar_nonsparse_matrix(xyz_circuit, numqubits, id_only)
assert actual is False
actual = is_scalar_nonsparse_matrix(cnot_circuit, numqubits, id_only)
assert actual is True
actual = is_scalar_nonsparse_matrix(hh_circuit, numqubits, id_only)
assert actual is True
def test_is_scalar_sparse_matrix():
np = import_module('numpy')
if not np:
skip("numpy not installed.")
scipy = import_module('scipy', __import__kwargs={'fromlist': ['sparse']})
if not scipy:
skip("scipy not installed.")
numqubits = 2
id_only = False
id_gate = (IdentityGate(1),)
assert is_scalar_sparse_matrix(id_gate, numqubits, id_only) is True
x0 = X(0)
xx_circuit = (x0, x0)
assert is_scalar_sparse_matrix(xx_circuit, numqubits, id_only) is True
x1 = X(1)
y1 = Y(1)
xy_circuit = (x1, y1)
assert is_scalar_sparse_matrix(xy_circuit, numqubits, id_only) is False
z1 = Z(1)
xyz_circuit = (x1, y1, z1)
assert is_scalar_sparse_matrix(xyz_circuit, numqubits, id_only) is True
cnot = CNOT(1, 0)
cnot_circuit = (cnot, cnot)
assert is_scalar_sparse_matrix(cnot_circuit, numqubits, id_only) is True
h = H(0)
hh_circuit = (h, h)
assert is_scalar_sparse_matrix(hh_circuit, numqubits, id_only) is True
# NOTE:
# The elements of the sparse matrix for the following circuit
# is actually 1.0000000000000002+0.0j.
h1 = H(1)
xhzh_circuit = (x1, h1, z1, h1)
assert is_scalar_sparse_matrix(xhzh_circuit, numqubits, id_only) is True
id_only = True
assert is_scalar_sparse_matrix(xhzh_circuit, numqubits, id_only) is True
assert is_scalar_sparse_matrix(xyz_circuit, numqubits, id_only) is False
assert is_scalar_sparse_matrix(cnot_circuit, numqubits, id_only) is True
assert is_scalar_sparse_matrix(hh_circuit, numqubits, id_only) is True
def test_is_degenerate():
(x, y, z, h) = create_gate_sequence()
gate_id = GateIdentity(x, y, z)
ids = {gate_id}
another_id = (z, y, x)
assert is_degenerate(ids, another_id) is True
def test_is_reducible():
nqubits = 2
(x, y, z, h) = create_gate_sequence()
circuit = (x, y, y)
assert is_reducible(circuit, nqubits, 1, 3) is True
circuit = (x, y, x)
assert is_reducible(circuit, nqubits, 1, 3) is False
circuit = (x, y, y, x)
assert is_reducible(circuit, nqubits, 0, 4) is True
circuit = (x, y, y, x)
assert is_reducible(circuit, nqubits, 1, 3) is True
circuit = (x, y, z, y, y)
assert is_reducible(circuit, nqubits, 1, 5) is True
def test_bfs_identity_search():
assert bfs_identity_search([], 1) == set()
(x, y, z, h) = create_gate_sequence()
gate_list = [x]
id_set = {GateIdentity(x, x)}
assert bfs_identity_search(gate_list, 1, max_depth=2) == id_set
# Set should not contain degenerate quantum circuits
gate_list = [x, y, z]
id_set = set([GateIdentity(x, x),
GateIdentity(y, y),
GateIdentity(z, z),
GateIdentity(x, y, z)])
assert bfs_identity_search(gate_list, 1) == id_set
id_set = set([GateIdentity(x, x),
GateIdentity(y, y),
GateIdentity(z, z),
GateIdentity(x, y, z),
GateIdentity(x, y, x, y),
GateIdentity(x, z, x, z),
GateIdentity(y, z, y, z)])
assert bfs_identity_search(gate_list, 1, max_depth=4) == id_set
assert bfs_identity_search(gate_list, 1, max_depth=5) == id_set
gate_list = [x, y, z, h]
id_set = set([GateIdentity(x, x),
GateIdentity(y, y),
GateIdentity(z, z),
GateIdentity(h, h),
GateIdentity(x, y, z),
GateIdentity(x, y, x, y),
GateIdentity(x, z, x, z),
GateIdentity(x, h, z, h),
GateIdentity(y, z, y, z),
GateIdentity(y, h, y, h)])
assert bfs_identity_search(gate_list, 1) == id_set
id_set = set([GateIdentity(x, x),
GateIdentity(y, y),
GateIdentity(z, z),
GateIdentity(h, h)])
assert id_set == bfs_identity_search(gate_list, 1, max_depth=3,
identity_only=True)
id_set = set([GateIdentity(x, x),
GateIdentity(y, y),
GateIdentity(z, z),
GateIdentity(h, h),
GateIdentity(x, y, z),
GateIdentity(x, y, x, y),
GateIdentity(x, z, x, z),
GateIdentity(x, h, z, h),
GateIdentity(y, z, y, z),
GateIdentity(y, h, y, h),
GateIdentity(x, y, h, x, h),
GateIdentity(x, z, h, y, h),
GateIdentity(y, z, h, z, h)])
assert bfs_identity_search(gate_list, 1, max_depth=5) == id_set
id_set = set([GateIdentity(x, x),
GateIdentity(y, y),
GateIdentity(z, z),
GateIdentity(h, h),
GateIdentity(x, h, z, h)])
assert id_set == bfs_identity_search(gate_list, 1, max_depth=4,
identity_only=True)
cnot = CNOT(1, 0)
gate_list = [x, cnot]
id_set = set([GateIdentity(x, x),
GateIdentity(cnot, cnot),
GateIdentity(x, cnot, x, cnot)])
assert bfs_identity_search(gate_list, 2, max_depth=4) == id_set
cgate_x = CGate((1,), x)
gate_list = [x, cgate_x]
id_set = set([GateIdentity(x, x),
GateIdentity(cgate_x, cgate_x),
GateIdentity(x, cgate_x, x, cgate_x)])
assert bfs_identity_search(gate_list, 2, max_depth=4) == id_set
cgate_z = CGate((0,), Z(1))
gate_list = [cnot, cgate_z, h]
id_set = set([GateIdentity(h, h),
GateIdentity(cgate_z, cgate_z),
GateIdentity(cnot, cnot),
GateIdentity(cnot, h, cgate_z, h)])
assert bfs_identity_search(gate_list, 2, max_depth=4) == id_set
s = PhaseGate(0)
t = TGate(0)
gate_list = [s, t]
id_set = {GateIdentity(s, s, s, s)}
assert bfs_identity_search(gate_list, 1, max_depth=4) == id_set
# @XFAIL
# Seems to fail on Python 2.7, but not 3.X, unless scipy is installed
def test_bfs_identity_search_xfail():
scipy = import_module('scipy', __import__kwargs={'fromlist': ['sparse']})
if scipy:
skip("scipy installed.")
s = PhaseGate(0)
t = TGate(0)
gate_list = [Dagger(s), t]
id_set = {GateIdentity(Dagger(s), t, t)}
assert bfs_identity_search(gate_list, 1, max_depth=3) == id_set
if not PY3:
test_bfs_identity_search_xfail = XFAIL(test_bfs_identity_search_xfail)
|
d609ef463921a054d885c23892d1da953cefe078fd554c87f333078232265cf3
|
from sympy import I, Integer, sqrt, symbols, S, Mul
from sympy.physics.quantum.anticommutator import AntiCommutator
from sympy.physics.quantum.commutator import Commutator
from sympy.physics.quantum.constants import hbar
from sympy.physics.quantum.dagger import Dagger
from sympy.physics.quantum.gate import H
from sympy.physics.quantum.operator import Operator
from sympy.physics.quantum.qapply import qapply
from sympy.physics.quantum.spin import Jx, Jy, Jz, Jplus, Jminus, J2, JzKet
from sympy.physics.quantum.tensorproduct import TensorProduct
from sympy.physics.quantum.state import Ket
from sympy.physics.quantum.density import Density
from sympy.physics.quantum.qubit import Qubit
from sympy.physics.quantum.boson import BosonOp, BosonFockKet, BosonFockBra
from sympy.physics.quantum.tensorproduct import TensorProduct
j, jp, m, mp = symbols("j j' m m'")
z = JzKet(1, 0)
po = JzKet(1, 1)
mo = JzKet(1, -1)
A = Operator('A')
class Foo(Operator):
def _apply_operator_JzKet(self, ket, **options):
return ket
def test_basic():
assert qapply(Jz*po) == hbar*po
assert qapply(Jx*z) == hbar*po/sqrt(2) + hbar*mo/sqrt(2)
assert qapply((Jplus + Jminus)*z/sqrt(2)) == hbar*po + hbar*mo
assert qapply(Jz*(po + mo)) == hbar*po - hbar*mo
assert qapply(Jz*po + Jz*mo) == hbar*po - hbar*mo
assert qapply(Jminus*Jminus*po) == 2*hbar**2*mo
assert qapply(Jplus**2*mo) == 2*hbar**2*po
assert qapply(Jplus**2*Jminus**2*po) == 4*hbar**4*po
def test_extra():
extra = z.dual*A*z
assert qapply(Jz*po*extra) == hbar*po*extra
assert qapply(Jx*z*extra) == (hbar*po/sqrt(2) + hbar*mo/sqrt(2))*extra
assert qapply(
(Jplus + Jminus)*z/sqrt(2)*extra) == hbar*po*extra + hbar*mo*extra
assert qapply(Jz*(po + mo)*extra) == hbar*po*extra - hbar*mo*extra
assert qapply(Jz*po*extra + Jz*mo*extra) == hbar*po*extra - hbar*mo*extra
assert qapply(Jminus*Jminus*po*extra) == 2*hbar**2*mo*extra
assert qapply(Jplus**2*mo*extra) == 2*hbar**2*po*extra
assert qapply(Jplus**2*Jminus**2*po*extra) == 4*hbar**4*po*extra
def test_innerproduct():
assert qapply(po.dual*Jz*po, ip_doit=False) == hbar*(po.dual*po)
assert qapply(po.dual*Jz*po) == hbar
def test_zero():
assert qapply(0) == 0
assert qapply(Integer(0)) == 0
def test_commutator():
assert qapply(Commutator(Jx, Jy)*Jz*po) == I*hbar**3*po
assert qapply(Commutator(J2, Jz)*Jz*po) == 0
assert qapply(Commutator(Jz, Foo('F'))*po) == 0
assert qapply(Commutator(Foo('F'), Jz)*po) == 0
def test_anticommutator():
assert qapply(AntiCommutator(Jz, Foo('F'))*po) == 2*hbar*po
assert qapply(AntiCommutator(Foo('F'), Jz)*po) == 2*hbar*po
def test_outerproduct():
e = Jz*(mo*po.dual)*Jz*po
assert qapply(e) == -hbar**2*mo
assert qapply(e, ip_doit=False) == -hbar**2*(po.dual*po)*mo
assert qapply(e).doit() == -hbar**2*mo
def test_tensorproduct():
a = BosonOp("a")
b = BosonOp("b")
ket1 = TensorProduct(BosonFockKet(1), BosonFockKet(2))
ket2 = TensorProduct(BosonFockKet(0), BosonFockKet(0))
ket3 = TensorProduct(BosonFockKet(0), BosonFockKet(2))
bra1 = TensorProduct(BosonFockBra(0), BosonFockBra(0))
bra2 = TensorProduct(BosonFockBra(1), BosonFockBra(2))
assert qapply(TensorProduct(a, b ** 2) * ket1) == sqrt(2) * ket2
assert qapply(TensorProduct(a, Dagger(b) * b) * ket1) == 2 * ket3
assert qapply(bra1 * TensorProduct(a, b * b),
dagger=True) == sqrt(2) * bra2
assert qapply(bra2 * ket1).doit() == TensorProduct(1, 1)
assert qapply(TensorProduct(a, b * b) * ket1) == sqrt(2) * ket2
assert qapply(Dagger(TensorProduct(a, b * b) * ket1),
dagger=True) == sqrt(2) * Dagger(ket2)
def test_dagger():
lhs = Dagger(Qubit(0))*Dagger(H(0))
rhs = Dagger(Qubit(1))/sqrt(2) + Dagger(Qubit(0))/sqrt(2)
assert qapply(lhs, dagger=True) == rhs
def test_issue_6073():
x, y = symbols('x y', commutative=False)
A = Ket(x, y)
B = Operator('B')
assert qapply(A) == A
assert qapply(A.dual*B) == A.dual*B
def test_density():
d = Density([Jz*mo, 0.5], [Jz*po, 0.5])
assert qapply(d) == Density([-hbar*mo, 0.5], [hbar*po, 0.5])
def test_issue3044():
expr1 = TensorProduct(Jz*JzKet(S(2),S(-1))/sqrt(2), Jz*JzKet(S(1)/2,S(1)/2))
result = Mul(S(-1), S(1)/4, (2**(S(1)/2)), hbar**2)
result *= TensorProduct(JzKet(2,-1), JzKet(S(1)/2,S(1)/2))
assert qapply(expr1) == result
|
1dddb941c1088175f1b6946f1646d7956aee8124b3a59176ffcede16f014bbb7
|
from sympy import S, sqrt, Sum, symbols
from sympy.physics.quantum.cg import Wigner3j, Wigner6j, Wigner9j, CG, cg_simp
from sympy.functions.special.tensor_functions import KroneckerDelta
from sympy.utilities.pytest import slow
def test_cg_simp_add():
j, m1, m1p, m2, m2p = symbols('j m1 m1p m2 m2p')
# Test Varshalovich 8.7.1 Eq 1
a = CG(S(1)/2, S(1)/2, 0, 0, S(1)/2, S(1)/2)
b = CG(S(1)/2, -S(1)/2, 0, 0, S(1)/2, -S(1)/2)
c = CG(1, 1, 0, 0, 1, 1)
d = CG(1, 0, 0, 0, 1, 0)
e = CG(1, -1, 0, 0, 1, -1)
assert cg_simp(a + b) == 2
assert cg_simp(c + d + e) == 3
assert cg_simp(a + b + c + d + e) == 5
assert cg_simp(a + b + c) == 2 + c
assert cg_simp(2*a + b) == 2 + a
assert cg_simp(2*c + d + e) == 3 + c
assert cg_simp(5*a + 5*b) == 10
assert cg_simp(5*c + 5*d + 5*e) == 15
assert cg_simp(-a - b) == -2
assert cg_simp(-c - d - e) == -3
assert cg_simp(-6*a - 6*b) == -12
assert cg_simp(-4*c - 4*d - 4*e) == -12
a = CG(S(1)/2, S(1)/2, j, 0, S(1)/2, S(1)/2)
b = CG(S(1)/2, -S(1)/2, j, 0, S(1)/2, -S(1)/2)
c = CG(1, 1, j, 0, 1, 1)
d = CG(1, 0, j, 0, 1, 0)
e = CG(1, -1, j, 0, 1, -1)
assert cg_simp(a + b) == 2*KroneckerDelta(j, 0)
assert cg_simp(c + d + e) == 3*KroneckerDelta(j, 0)
assert cg_simp(a + b + c + d + e) == 5*KroneckerDelta(j, 0)
assert cg_simp(a + b + c) == 2*KroneckerDelta(j, 0) + c
assert cg_simp(2*a + b) == 2*KroneckerDelta(j, 0) + a
assert cg_simp(2*c + d + e) == 3*KroneckerDelta(j, 0) + c
assert cg_simp(5*a + 5*b) == 10*KroneckerDelta(j, 0)
assert cg_simp(5*c + 5*d + 5*e) == 15*KroneckerDelta(j, 0)
assert cg_simp(-a - b) == -2*KroneckerDelta(j, 0)
assert cg_simp(-c - d - e) == -3*KroneckerDelta(j, 0)
assert cg_simp(-6*a - 6*b) == -12*KroneckerDelta(j, 0)
assert cg_simp(-4*c - 4*d - 4*e) == -12*KroneckerDelta(j, 0)
# Test Varshalovich 8.7.1 Eq 2
a = CG(S(1)/2, S(1)/2, S(1)/2, -S(1)/2, 0, 0)
b = CG(S(1)/2, -S(1)/2, S(1)/2, S(1)/2, 0, 0)
c = CG(1, 1, 1, -1, 0, 0)
d = CG(1, 0, 1, 0, 0, 0)
e = CG(1, -1, 1, 1, 0, 0)
assert cg_simp(a - b) == sqrt(2)
assert cg_simp(c - d + e) == sqrt(3)
assert cg_simp(a - b + c - d + e) == sqrt(2) + sqrt(3)
assert cg_simp(a - b + c) == sqrt(2) + c
assert cg_simp(2*a - b) == sqrt(2) + a
assert cg_simp(2*c - d + e) == sqrt(3) + c
assert cg_simp(5*a - 5*b) == 5*sqrt(2)
assert cg_simp(5*c - 5*d + 5*e) == 5*sqrt(3)
assert cg_simp(-a + b) == -sqrt(2)
assert cg_simp(-c + d - e) == -sqrt(3)
assert cg_simp(-6*a + 6*b) == -6*sqrt(2)
assert cg_simp(-4*c + 4*d - 4*e) == -4*sqrt(3)
a = CG(S(1)/2, S(1)/2, S(1)/2, -S(1)/2, j, 0)
b = CG(S(1)/2, -S(1)/2, S(1)/2, S(1)/2, j, 0)
c = CG(1, 1, 1, -1, j, 0)
d = CG(1, 0, 1, 0, j, 0)
e = CG(1, -1, 1, 1, j, 0)
assert cg_simp(a - b) == sqrt(2)*KroneckerDelta(j, 0)
assert cg_simp(c - d + e) == sqrt(3)*KroneckerDelta(j, 0)
assert cg_simp(a - b + c - d + e) == sqrt(
2)*KroneckerDelta(j, 0) + sqrt(3)*KroneckerDelta(j, 0)
assert cg_simp(a - b + c) == sqrt(2)*KroneckerDelta(j, 0) + c
assert cg_simp(2*a - b) == sqrt(2)*KroneckerDelta(j, 0) + a
assert cg_simp(2*c - d + e) == sqrt(3)*KroneckerDelta(j, 0) + c
assert cg_simp(5*a - 5*b) == 5*sqrt(2)*KroneckerDelta(j, 0)
assert cg_simp(5*c - 5*d + 5*e) == 5*sqrt(3)*KroneckerDelta(j, 0)
assert cg_simp(-a + b) == -sqrt(2)*KroneckerDelta(j, 0)
assert cg_simp(-c + d - e) == -sqrt(3)*KroneckerDelta(j, 0)
assert cg_simp(-6*a + 6*b) == -6*sqrt(2)*KroneckerDelta(j, 0)
assert cg_simp(-4*c + 4*d - 4*e) == -4*sqrt(3)*KroneckerDelta(j, 0)
# Test Varshalovich 8.7.2 Eq 9
# alpha=alphap,beta=betap case
# numerical
a = CG(S(1)/2, S(1)/2, S(1)/2, -S(1)/2, 1, 0)**2
b = CG(S(1)/2, S(1)/2, S(1)/2, -S(1)/2, 0, 0)**2
c = CG(1, 0, 1, 1, 1, 1)**2
d = CG(1, 0, 1, 1, 2, 1)**2
assert cg_simp(a + b) == 1
assert cg_simp(c + d) == 1
assert cg_simp(a + b + c + d) == 2
assert cg_simp(4*a + 4*b) == 4
assert cg_simp(4*c + 4*d) == 4
assert cg_simp(5*a + 3*b) == 3 + 2*a
assert cg_simp(5*c + 3*d) == 3 + 2*c
assert cg_simp(-a - b) == -1
assert cg_simp(-c - d) == -1
# symbolic
a = CG(S(1)/2, m1, S(1)/2, m2, 1, 1)**2
b = CG(S(1)/2, m1, S(1)/2, m2, 1, 0)**2
c = CG(S(1)/2, m1, S(1)/2, m2, 1, -1)**2
d = CG(S(1)/2, m1, S(1)/2, m2, 0, 0)**2
assert cg_simp(a + b + c + d) == 1
assert cg_simp(4*a + 4*b + 4*c + 4*d) == 4
assert cg_simp(3*a + 5*b + 3*c + 4*d) == 3 + 2*b + d
assert cg_simp(-a - b - c - d) == -1
a = CG(1, m1, 1, m2, 2, 2)**2
b = CG(1, m1, 1, m2, 2, 1)**2
c = CG(1, m1, 1, m2, 2, 0)**2
d = CG(1, m1, 1, m2, 2, -1)**2
e = CG(1, m1, 1, m2, 2, -2)**2
f = CG(1, m1, 1, m2, 1, 1)**2
g = CG(1, m1, 1, m2, 1, 0)**2
h = CG(1, m1, 1, m2, 1, -1)**2
i = CG(1, m1, 1, m2, 0, 0)**2
assert cg_simp(a + b + c + d + e + f + g + h + i) == 1
assert cg_simp(4*(a + b + c + d + e + f + g + h + i)) == 4
assert cg_simp(a + b + 2*c + d + 4*e + f + g + h + i) == 1 + c + 3*e
assert cg_simp(-a - b - c - d - e - f - g - h - i) == -1
# alpha!=alphap or beta!=betap case
# numerical
a = CG(S(1)/2, S(
1)/2, S(1)/2, -S(1)/2, 1, 0)*CG(S(1)/2, -S(1)/2, S(1)/2, S(1)/2, 1, 0)
b = CG(S(1)/2, S(
1)/2, S(1)/2, -S(1)/2, 0, 0)*CG(S(1)/2, -S(1)/2, S(1)/2, S(1)/2, 0, 0)
c = CG(1, 1, 1, 0, 2, 1)*CG(1, 0, 1, 1, 2, 1)
d = CG(1, 1, 1, 0, 1, 1)*CG(1, 0, 1, 1, 1, 1)
assert cg_simp(a + b) == 0
assert cg_simp(c + d) == 0
# symbolic
a = CG(S(1)/2, m1, S(1)/2, m2, 1, 1)*CG(S(1)/2, m1p, S(1)/2, m2p, 1, 1)
b = CG(S(1)/2, m1, S(1)/2, m2, 1, 0)*CG(S(1)/2, m1p, S(1)/2, m2p, 1, 0)
c = CG(S(1)/2, m1, S(1)/2, m2, 1, -1)*CG(S(1)/2, m1p, S(1)/2, m2p, 1, -1)
d = CG(S(1)/2, m1, S(1)/2, m2, 0, 0)*CG(S(1)/2, m1p, S(1)/2, m2p, 0, 0)
assert cg_simp(a + b + c + d) == KroneckerDelta(m1, m1p)*KroneckerDelta(m2, m2p)
a = CG(1, m1, 1, m2, 2, 2)*CG(1, m1p, 1, m2p, 2, 2)
b = CG(1, m1, 1, m2, 2, 1)*CG(1, m1p, 1, m2p, 2, 1)
c = CG(1, m1, 1, m2, 2, 0)*CG(1, m1p, 1, m2p, 2, 0)
d = CG(1, m1, 1, m2, 2, -1)*CG(1, m1p, 1, m2p, 2, -1)
e = CG(1, m1, 1, m2, 2, -2)*CG(1, m1p, 1, m2p, 2, -2)
f = CG(1, m1, 1, m2, 1, 1)*CG(1, m1p, 1, m2p, 1, 1)
g = CG(1, m1, 1, m2, 1, 0)*CG(1, m1p, 1, m2p, 1, 0)
h = CG(1, m1, 1, m2, 1, -1)*CG(1, m1p, 1, m2p, 1, -1)
i = CG(1, m1, 1, m2, 0, 0)*CG(1, m1p, 1, m2p, 0, 0)
assert cg_simp(
a + b + c + d + e + f + g + h + i) == KroneckerDelta(m1, m1p)*KroneckerDelta(m2, m2p)
def test_cg_simp_sum():
x, a, b, c, cp, alpha, beta, gamma, gammap = symbols(
'x a b c cp alpha beta gamma gammap')
# Varshalovich 8.7.1 Eq 1
assert cg_simp(x * Sum(CG(a, alpha, b, 0, a, alpha), (alpha, -a, a)
)) == x*(2*a + 1)*KroneckerDelta(b, 0)
assert cg_simp(x * Sum(CG(a, alpha, b, 0, a, alpha), (alpha, -a, a)) + CG(1, 0, 1, 0, 1, 0)) == x*(2*a + 1)*KroneckerDelta(b, 0) + CG(1, 0, 1, 0, 1, 0)
assert cg_simp(2 * Sum(CG(1, alpha, 0, 0, 1, alpha), (alpha, -1, 1))) == 6
# Varshalovich 8.7.1 Eq 2
assert cg_simp(x*Sum((-1)**(a - alpha) * CG(a, alpha, a, -alpha, c,
0), (alpha, -a, a))) == x*sqrt(2*a + 1)*KroneckerDelta(c, 0)
assert cg_simp(3*Sum((-1)**(2 - alpha) * CG(
2, alpha, 2, -alpha, 0, 0), (alpha, -2, 2))) == 3*sqrt(5)
# Varshalovich 8.7.2 Eq 4
assert cg_simp(Sum(CG(a, alpha, b, beta, c, gamma)*CG(a, alpha, b, beta, cp, gammap), (alpha, -a, a), (beta, -b, b))) == KroneckerDelta(c, cp)*KroneckerDelta(gamma, gammap)
assert cg_simp(Sum(CG(a, alpha, b, beta, c, gamma)*CG(a, alpha, b, beta, c, gammap), (alpha, -a, a), (beta, -b, b))) == KroneckerDelta(gamma, gammap)
assert cg_simp(Sum(CG(a, alpha, b, beta, c, gamma)*CG(a, alpha, b, beta, cp, gamma), (alpha, -a, a), (beta, -b, b))) == KroneckerDelta(c, cp)
assert cg_simp(Sum(CG(
a, alpha, b, beta, c, gamma)**2, (alpha, -a, a), (beta, -b, b))) == 1
assert cg_simp(Sum(CG(2, alpha, 1, beta, 2, gamma)*CG(2, alpha, 1, beta, 2, gammap), (alpha, -2, 2), (beta, -1, 1))) == KroneckerDelta(gamma, gammap)
def test_doit():
assert Wigner3j(S(1)/2, -S(1)/2, S(1)/2, S(1)/2, 0, 0).doit() == -sqrt(2)/2
assert Wigner6j(1, 2, 3, 2, 1, 2).doit() == sqrt(21)/105
assert Wigner6j(3, 1, 2, 2, 2, 1).doit() == sqrt(21) / 105
assert Wigner9j(
2, 1, 1, S(3)/2, S(1)/2, 1, S(1)/2, S(1)/2, 0).doit() == sqrt(2)/12
assert CG(S(1)/2, S(1)/2, S(1)/2, -S(1)/2, 1, 0).doit() == sqrt(2)/2
|
7063cec66c0b9176bb93d6398f01d10bdbdee5d5deb847952c99c1287a5e8e3a
|
from sympy import symbols
from sympy.physics.mechanics import Point, Particle, ReferenceFrame
from sympy.utilities.pytest import raises
def test_particle():
m, m2, v1, v2, v3, r, g, h = symbols('m m2 v1 v2 v3 r g h')
P = Point('P')
P2 = Point('P2')
p = Particle('pa', P, m)
assert p.__str__() == 'pa'
assert p.mass == m
assert p.point == P
# Test the mass setter
p.mass = m2
assert p.mass == m2
# Test the point setter
p.point = P2
assert p.point == P2
# Test the linear momentum function
N = ReferenceFrame('N')
O = Point('O')
P2.set_pos(O, r * N.y)
P2.set_vel(N, v1 * N.x)
raises(TypeError, lambda: Particle(P, P, m))
raises(TypeError, lambda: Particle('pa', m, m))
assert p.linear_momentum(N) == m2 * v1 * N.x
assert p.angular_momentum(O, N) == -m2 * r *v1 * N.z
P2.set_vel(N, v2 * N.y)
assert p.linear_momentum(N) == m2 * v2 * N.y
assert p.angular_momentum(O, N) == 0
P2.set_vel(N, v3 * N.z)
assert p.linear_momentum(N) == m2 * v3 * N.z
assert p.angular_momentum(O, N) == m2 * r * v3 * N.x
P2.set_vel(N, v1 * N.x + v2 * N.y + v3 * N.z)
assert p.linear_momentum(N) == m2 * (v1 * N.x + v2 * N.y + v3 * N.z)
assert p.angular_momentum(O, N) == m2 * r * (v3 * N.x - v1 * N.z)
p.potential_energy = m * g * h
assert p.potential_energy == m * g * h
# TODO make the result not be system-dependent
assert p.kinetic_energy(
N) in [m2*(v1**2 + v2**2 + v3**2)/2,
m2 * v1**2 / 2 + m2 * v2**2 / 2 + m2 * v3**2 / 2]
|
563894e9ce2093340a128bf0b872029923c65f544ad372beeb0773bf84252f45
|
from sympy.core.backend import sin, cos, tan, pi, symbols, Matrix, zeros, S
from sympy.physics.mechanics import (Particle, Point, ReferenceFrame,
RigidBody, Vector)
from sympy.physics.mechanics import (angular_momentum, dynamicsymbols,
inertia, inertia_of_point_mass,
kinetic_energy, linear_momentum,
outer, potential_energy, msubs,
find_dynamicsymbols, Lagrangian)
from sympy.physics.mechanics.functions import gravity, center_of_mass
from sympy.physics.vector.vector import Vector
from sympy.utilities.pytest import raises
Vector.simp = True
q1, q2, q3, q4, q5 = symbols('q1 q2 q3 q4 q5')
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])
def test_inertia():
N = ReferenceFrame('N')
ixx, iyy, izz = symbols('ixx iyy izz')
ixy, iyz, izx = symbols('ixy iyz izx')
assert inertia(N, ixx, iyy, izz) == (ixx * (N.x | N.x) + iyy *
(N.y | N.y) + izz * (N.z | N.z))
assert inertia(N, 0, 0, 0) == 0 * (N.x | N.x)
raises(TypeError, lambda: inertia(0, 0, 0, 0))
assert inertia(N, ixx, iyy, izz, ixy, iyz, izx) == (ixx * (N.x | N.x) +
ixy * (N.x | N.y) + izx * (N.x | N.z) + ixy * (N.y | N.x) + iyy *
(N.y | N.y) + iyz * (N.y | N.z) + izx * (N.z | N.x) + iyz * (N.z |
N.y) + izz * (N.z | N.z))
def test_inertia_of_point_mass():
r, s, t, m = symbols('r s t m')
N = ReferenceFrame('N')
px = r * N.x
I = inertia_of_point_mass(m, px, N)
assert I == m * r**2 * (N.y | N.y) + m * r**2 * (N.z | N.z)
py = s * N.y
I = inertia_of_point_mass(m, py, N)
assert I == m * s**2 * (N.x | N.x) + m * s**2 * (N.z | N.z)
pz = t * N.z
I = inertia_of_point_mass(m, pz, N)
assert I == m * t**2 * (N.x | N.x) + m * t**2 * (N.y | N.y)
p = px + py + pz
I = inertia_of_point_mass(m, p, N)
assert I == (m * (s**2 + t**2) * (N.x | N.x) -
m * r * s * (N.x | N.y) -
m * r * t * (N.x | N.z) -
m * r * s * (N.y | N.x) +
m * (r**2 + t**2) * (N.y | N.y) -
m * s * t * (N.y | N.z) -
m * r * t * (N.z | N.x) -
m * s * t * (N.z | N.y) +
m * (r**2 + s**2) * (N.z | N.z))
def test_linear_momentum():
N = ReferenceFrame('N')
Ac = Point('Ac')
Ac.set_vel(N, 25 * N.y)
I = outer(N.x, N.x)
A = RigidBody('A', Ac, N, 20, (I, Ac))
P = Point('P')
Pa = Particle('Pa', P, 1)
Pa.point.set_vel(N, 10 * N.x)
raises(TypeError, lambda: linear_momentum(A, A, Pa))
raises(TypeError, lambda: linear_momentum(N, N, Pa))
assert linear_momentum(N, A, Pa) == 10 * N.x + 500 * N.y
def test_angular_momentum_and_linear_momentum():
"""A rod with length 2l, centroidal inertia I, and mass M along with a
particle of mass m fixed to the end of the rod rotate with an angular rate
of omega about point O which is fixed to the non-particle end of the rod.
The rod's reference frame is A and the inertial frame is N."""
m, M, l, I = symbols('m, M, l, I')
omega = dynamicsymbols('omega')
N = ReferenceFrame('N')
a = ReferenceFrame('a')
O = Point('O')
Ac = O.locatenew('Ac', l * N.x)
P = Ac.locatenew('P', l * N.x)
O.set_vel(N, 0 * N.x)
a.set_ang_vel(N, omega * N.z)
Ac.v2pt_theory(O, N, a)
P.v2pt_theory(O, N, a)
Pa = Particle('Pa', P, m)
A = RigidBody('A', Ac, a, M, (I * outer(N.z, N.z), Ac))
expected = 2 * m * omega * l * N.y + M * l * omega * N.y
assert linear_momentum(N, A, Pa) == expected
raises(TypeError, lambda: angular_momentum(N, N, A, Pa))
raises(TypeError, lambda: angular_momentum(O, O, A, Pa))
raises(TypeError, lambda: angular_momentum(O, N, O, Pa))
expected = (I + M * l**2 + 4 * m * l**2) * omega * N.z
assert angular_momentum(O, N, A, Pa) == expected
def test_kinetic_energy():
m, M, l1 = symbols('m M l1')
omega = dynamicsymbols('omega')
N = ReferenceFrame('N')
O = Point('O')
O.set_vel(N, 0 * N.x)
Ac = O.locatenew('Ac', l1 * N.x)
P = Ac.locatenew('P', l1 * N.x)
a = ReferenceFrame('a')
a.set_ang_vel(N, omega * N.z)
Ac.v2pt_theory(O, N, a)
P.v2pt_theory(O, N, a)
Pa = Particle('Pa', P, m)
I = outer(N.z, N.z)
A = RigidBody('A', Ac, a, M, (I, Ac))
raises(TypeError, lambda: kinetic_energy(Pa, Pa, A))
raises(TypeError, lambda: kinetic_energy(N, N, A))
assert 0 == (kinetic_energy(N, Pa, A) - (M*l1**2*omega**2/2
+ 2*l1**2*m*omega**2 + omega**2/2)).expand()
def test_potential_energy():
m, M, l1, g, h, H = symbols('m M l1 g h H')
omega = dynamicsymbols('omega')
N = ReferenceFrame('N')
O = Point('O')
O.set_vel(N, 0 * N.x)
Ac = O.locatenew('Ac', l1 * N.x)
P = Ac.locatenew('P', l1 * N.x)
a = ReferenceFrame('a')
a.set_ang_vel(N, omega * N.z)
Ac.v2pt_theory(O, N, a)
P.v2pt_theory(O, N, a)
Pa = Particle('Pa', P, m)
I = outer(N.z, N.z)
A = RigidBody('A', Ac, a, M, (I, Ac))
Pa.potential_energy = m * g * h
A.potential_energy = M * g * H
assert potential_energy(A, Pa) == m * g * h + M * g * H
def test_Lagrangian():
M, m, g, h = symbols('M m g h')
N = ReferenceFrame('N')
O = Point('O')
O.set_vel(N, 0 * N.x)
P = O.locatenew('P', 1 * N.x)
P.set_vel(N, 10 * N.x)
Pa = Particle('Pa', P, 1)
Ac = O.locatenew('Ac', 2 * N.y)
Ac.set_vel(N, 5 * N.y)
a = ReferenceFrame('a')
a.set_ang_vel(N, 10 * N.z)
I = outer(N.z, N.z)
A = RigidBody('A', Ac, a, 20, (I, Ac))
Pa.potential_energy = m * g * h
A.potential_energy = M * g * h
raises(TypeError, lambda: Lagrangian(A, A, Pa))
raises(TypeError, lambda: Lagrangian(N, N, Pa))
def test_msubs():
a, b = symbols('a, b')
x, y, z = dynamicsymbols('x, y, z')
# Test simple substitution
expr = Matrix([[a*x + b, x*y.diff() + y],
[x.diff().diff(), z + sin(z.diff())]])
sol = Matrix([[a + b, y],
[x.diff().diff(), 1]])
sd = {x: 1, z: 1, z.diff(): 0, y.diff(): 0}
assert msubs(expr, sd) == sol
# Test smart substitution
expr = cos(x + y)*tan(x + y) + b*x.diff()
sd = {x: 0, y: pi/2, x.diff(): 1}
assert msubs(expr, sd, smart=True) == b + 1
N = ReferenceFrame('N')
v = x*N.x + y*N.y
d = x*(N.x|N.x) + y*(N.y|N.y)
v_sol = 1*N.y
d_sol = 1*(N.y|N.y)
sd = {x: 0, y: 1}
assert msubs(v, sd) == v_sol
assert msubs(d, sd) == d_sol
def test_find_dynamicsymbols():
a, b = symbols('a, b')
x, y, z = dynamicsymbols('x, y, z')
expr = Matrix([[a*x + b, x*y.diff() + y],
[x.diff().diff(), z + sin(z.diff())]])
# Test finding all dynamicsymbols
sol = {x, y.diff(), y, x.diff().diff(), z, z.diff()}
assert find_dynamicsymbols(expr) == sol
# Test finding all but those in sym_list
exclude_list = [x, y, z]
sol = {y.diff(), x.diff().diff(), z.diff()}
assert find_dynamicsymbols(expr, exclude=exclude_list) == sol
# Test finding all dynamicsymbols in a vector with a given reference frame
d, e, f = dynamicsymbols('d, e, f')
A = ReferenceFrame('A')
v = d * A.x + e * A.y + f * A.z
sol = {d, e, f}
assert find_dynamicsymbols(v, reference_frame=A) == sol
# Test if a ValueError is raised on supplying only a vector as input
raises(ValueError, lambda: find_dynamicsymbols(v))
def test_gravity():
N = ReferenceFrame('N')
m, M, g = symbols('m M g')
F1, F2 = dynamicsymbols('F1 F2')
po = Point('po')
pa = Particle('pa', po, m)
A = ReferenceFrame('A')
P = Point('P')
I = outer(A.x, A.x)
B = RigidBody('B', P, A, M, (I, P))
forceList = [(po, F1), (P, F2)]
forceList.extend(gravity(g*N.y, pa, B))
l = [(po, F1), (P, F2), (po, g*m*N.y), (P, g*M*N.y)]
for i in range(len(l)):
for j in range(len(l[i])):
assert forceList[i][j] == l[i][j]
# This function tests the center_of_mass() function
# that was added in PR #14758 to compute the center of
# mass of a system of bodies.
def test_center_of_mass():
a = ReferenceFrame('a')
m = symbols('m', real=True)
p1 = Particle('p1', Point('p1_pt'), S(1))
p2 = Particle('p2', Point('p2_pt'), S(2))
p3 = Particle('p3', Point('p3_pt'), S(3))
p4 = Particle('p4', Point('p4_pt'), m)
b_f = ReferenceFrame('b_f')
b_cm = Point('b_cm')
mb = symbols('mb')
b = RigidBody('b', b_cm, b_f, mb, (outer(b_f.x, b_f.x), b_cm))
p2.point.set_pos(p1.point, a.x)
p3.point.set_pos(p1.point, a.x + a.y)
p4.point.set_pos(p1.point, a.y)
b.masscenter.set_pos(p1.point, a.y + a.z)
point_o=Point('o')
point_o.set_pos(p1.point, center_of_mass(p1.point, p1, p2, p3, p4, b))
expr = 5/(m + mb + 6)*a.x + (m + mb + 3)/(m + mb + 6)*a.y + mb/(m + mb + 6)*a.z
assert point_o.pos_from(p1.point)-expr == 0
|
c823baafb03b79a9194c5bf56912319df5e086f776b2664da72c6745bb5802c6
|
from sympy.core.backend import symbols, Matrix, cos, sin, atan, sqrt, S
from sympy import solve, simplify
from sympy.physics.mechanics import dynamicsymbols, ReferenceFrame, Point,\
dot, cross, inertia, KanesMethod, Particle, RigidBody, Lagrangian,\
LagrangesMethod
from sympy.utilities.pytest import slow, warns_deprecated_sympy
@slow
def test_linearize_rolling_disc_kane():
# Symbols for time and constant parameters
t, r, m, g, v = symbols('t r m g v')
# Configuration variables and their time derivatives
q1, q2, q3, q4, q5, q6 = q = dynamicsymbols('q1:7')
q1d, q2d, q3d, q4d, q5d, q6d = qd = [qi.diff(t) for qi in q]
# Generalized speeds and their time derivatives
u = dynamicsymbols('u:6')
u1, u2, u3, u4, u5, u6 = u = dynamicsymbols('u1:7')
u1d, u2d, u3d, u4d, u5d, u6d = [ui.diff(t) for ui in u]
# Reference frames
N = ReferenceFrame('N') # Inertial frame
NO = Point('NO') # Inertial origin
A = N.orientnew('A', 'Axis', [q1, N.z]) # Yaw intermediate frame
B = A.orientnew('B', 'Axis', [q2, A.x]) # Lean intermediate frame
C = B.orientnew('C', 'Axis', [q3, B.y]) # Disc fixed frame
CO = NO.locatenew('CO', q4*N.x + q5*N.y + q6*N.z) # Disc center
# Disc angular velocity in N expressed using time derivatives of coordinates
w_c_n_qd = C.ang_vel_in(N)
w_b_n_qd = B.ang_vel_in(N)
# Inertial angular velocity and angular acceleration of disc fixed frame
C.set_ang_vel(N, u1*B.x + u2*B.y + u3*B.z)
# Disc center velocity in N expressed using time derivatives of coordinates
v_co_n_qd = CO.pos_from(NO).dt(N)
# Disc center velocity in N expressed using generalized speeds
CO.set_vel(N, u4*C.x + u5*C.y + u6*C.z)
# Disc Ground Contact Point
P = CO.locatenew('P', r*B.z)
P.v2pt_theory(CO, N, C)
# Configuration constraint
f_c = Matrix([q6 - dot(CO.pos_from(P), N.z)])
# Velocity level constraints
f_v = Matrix([dot(P.vel(N), uv) for uv in C])
# Kinematic differential equations
kindiffs = Matrix([dot(w_c_n_qd - C.ang_vel_in(N), uv) for uv in B] +
[dot(v_co_n_qd - CO.vel(N), uv) for uv in N])
qdots = solve(kindiffs, qd)
# Set angular velocity of remaining frames
B.set_ang_vel(N, w_b_n_qd.subs(qdots))
C.set_ang_acc(N, C.ang_vel_in(N).dt(B) + cross(B.ang_vel_in(N), C.ang_vel_in(N)))
# Active forces
F_CO = m*g*A.z
# Create inertia dyadic of disc C about point CO
I = (m * r**2) / 4
J = (m * r**2) / 2
I_C_CO = inertia(C, I, J, I)
Disc = RigidBody('Disc', CO, C, m, (I_C_CO, CO))
BL = [Disc]
FL = [(CO, F_CO)]
KM = KanesMethod(N, [q1, q2, q3, q4, q5], [u1, u2, u3], kd_eqs=kindiffs,
q_dependent=[q6], configuration_constraints=f_c,
u_dependent=[u4, u5, u6], velocity_constraints=f_v)
with warns_deprecated_sympy():
(fr, fr_star) = KM.kanes_equations(FL, BL)
# Test generalized form equations
linearizer = KM.to_linearizer()
assert linearizer.f_c == f_c
assert linearizer.f_v == f_v
assert linearizer.f_a == f_v.diff(t)
sol = solve(linearizer.f_0 + linearizer.f_1, qd)
for qi in qd:
assert sol[qi] == qdots[qi]
assert simplify(linearizer.f_2 + linearizer.f_3 - fr - fr_star) == Matrix([0, 0, 0])
# Perform the linearization
# Precomputed operating point
q_op = {q6: -r*cos(q2)}
u_op = {u1: 0,
u2: sin(q2)*q1d + q3d,
u3: cos(q2)*q1d,
u4: -r*(sin(q2)*q1d + q3d)*cos(q3),
u5: 0,
u6: -r*(sin(q2)*q1d + q3d)*sin(q3)}
qd_op = {q2d: 0,
q4d: -r*(sin(q2)*q1d + q3d)*cos(q1),
q5d: -r*(sin(q2)*q1d + q3d)*sin(q1),
q6d: 0}
ud_op = {u1d: 4*g*sin(q2)/(5*r) + sin(2*q2)*q1d**2/2 + 6*cos(q2)*q1d*q3d/5,
u2d: 0,
u3d: 0,
u4d: r*(sin(q2)*sin(q3)*q1d*q3d + sin(q3)*q3d**2),
u5d: r*(4*g*sin(q2)/(5*r) + sin(2*q2)*q1d**2/2 + 6*cos(q2)*q1d*q3d/5),
u6d: -r*(sin(q2)*cos(q3)*q1d*q3d + cos(q3)*q3d**2)}
A, B = linearizer.linearize(op_point=[q_op, u_op, qd_op, ud_op], A_and_B=True, simplify=True)
upright_nominal = {q1d: 0, q2: 0, m: 1, r: 1, g: 1}
# Precomputed solution
A_sol = Matrix([[0, 0, 0, 0, 0, 0, 0, 1],
[0, 0, 0, 0, 0, 1, 0, 0],
[0, 0, 0, 0, 0, 0, 1, 0],
[sin(q1)*q3d, 0, 0, 0, 0, -sin(q1), -cos(q1), 0],
[-cos(q1)*q3d, 0, 0, 0, 0, cos(q1), -sin(q1), 0],
[0, S(4)/5, 0, 0, 0, 0, 0, 6*q3d/5],
[0, 0, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, -2*q3d, 0, 0]])
B_sol = Matrix([])
# Check that linearization is correct
assert A.subs(upright_nominal) == A_sol
assert B.subs(upright_nominal) == B_sol
# Check eigenvalues at critical speed are all zero:
assert A.subs(upright_nominal).subs(q3d, 1/sqrt(3)).eigenvals() == {0: 8}
def test_linearize_pendulum_kane_minimal():
q1 = dynamicsymbols('q1') # angle of pendulum
u1 = dynamicsymbols('u1') # Angular velocity
q1d = dynamicsymbols('q1', 1) # Angular velocity
L, m, t = symbols('L, m, t')
g = 9.8
# Compose world frame
N = ReferenceFrame('N')
pN = Point('N*')
pN.set_vel(N, 0)
# A.x is along the pendulum
A = N.orientnew('A', 'axis', [q1, N.z])
A.set_ang_vel(N, u1*N.z)
# Locate point P relative to the origin N*
P = pN.locatenew('P', L*A.x)
P.v2pt_theory(pN, N, A)
pP = Particle('pP', P, m)
# Create Kinematic Differential Equations
kde = Matrix([q1d - u1])
# Input the force resultant at P
R = m*g*N.x
# Solve for eom with kanes method
KM = KanesMethod(N, q_ind=[q1], u_ind=[u1], kd_eqs=kde)
with warns_deprecated_sympy():
(fr, frstar) = KM.kanes_equations([(P, R)], [pP])
# Linearize
A, B, inp_vec = KM.linearize(A_and_B=True, simplify=True)
assert A == Matrix([[0, 1], [-9.8*cos(q1)/L, 0]])
assert B == Matrix([])
def test_linearize_pendulum_kane_nonminimal():
# Create generalized coordinates and speeds for this non-minimal realization
# q1, q2 = N.x and N.y coordinates of pendulum
# u1, u2 = N.x and N.y velocities of pendulum
q1, q2 = dynamicsymbols('q1:3')
q1d, q2d = dynamicsymbols('q1:3', level=1)
u1, u2 = dynamicsymbols('u1:3')
u1d, u2d = dynamicsymbols('u1:3', level=1)
L, m, t = symbols('L, m, t')
g = 9.8
# Compose world frame
N = ReferenceFrame('N')
pN = Point('N*')
pN.set_vel(N, 0)
# A.x is along the pendulum
theta1 = atan(q2/q1)
A = N.orientnew('A', 'axis', [theta1, N.z])
# Locate the pendulum mass
P = pN.locatenew('P1', q1*N.x + q2*N.y)
pP = Particle('pP', P, m)
# Calculate the kinematic differential equations
kde = Matrix([q1d - u1,
q2d - u2])
dq_dict = solve(kde, [q1d, q2d])
# Set velocity of point P
P.set_vel(N, P.pos_from(pN).dt(N).subs(dq_dict))
# Configuration constraint is length of pendulum
f_c = Matrix([P.pos_from(pN).magnitude() - L])
# Velocity constraint is that the velocity in the A.x direction is
# always zero (the pendulum is never getting longer).
f_v = Matrix([P.vel(N).express(A).dot(A.x)])
f_v.simplify()
# Acceleration constraints is the time derivative of the velocity constraint
f_a = f_v.diff(t)
f_a.simplify()
# Input the force resultant at P
R = m*g*N.x
# Derive the equations of motion using the KanesMethod class.
KM = KanesMethod(N, q_ind=[q2], u_ind=[u2], q_dependent=[q1],
u_dependent=[u1], configuration_constraints=f_c,
velocity_constraints=f_v, acceleration_constraints=f_a, kd_eqs=kde)
with warns_deprecated_sympy():
(fr, frstar) = KM.kanes_equations([(P, R)], [pP])
# Set the operating point to be straight down, and non-moving
q_op = {q1: L, q2: 0}
u_op = {u1: 0, u2: 0}
ud_op = {u1d: 0, u2d: 0}
A, B, inp_vec = KM.linearize(op_point=[q_op, u_op, ud_op], A_and_B=True,
simplify=True)
assert A.expand() == Matrix([[0, 1], [-9.8/L, 0]])
assert B == Matrix([])
def test_linearize_pendulum_lagrange_minimal():
q1 = dynamicsymbols('q1') # angle of pendulum
q1d = dynamicsymbols('q1', 1) # Angular velocity
L, m, t = symbols('L, m, t')
g = 9.8
# Compose world frame
N = ReferenceFrame('N')
pN = Point('N*')
pN.set_vel(N, 0)
# A.x is along the pendulum
A = N.orientnew('A', 'axis', [q1, N.z])
A.set_ang_vel(N, q1d*N.z)
# Locate point P relative to the origin N*
P = pN.locatenew('P', L*A.x)
P.v2pt_theory(pN, N, A)
pP = Particle('pP', P, m)
# Solve for eom with Lagranges method
Lag = Lagrangian(N, pP)
LM = LagrangesMethod(Lag, [q1], forcelist=[(P, m*g*N.x)], frame=N)
LM.form_lagranges_equations()
# Linearize
A, B, inp_vec = LM.linearize([q1], [q1d], A_and_B=True)
assert A == Matrix([[0, 1], [-9.8*cos(q1)/L, 0]])
assert B == Matrix([])
def test_linearize_pendulum_lagrange_nonminimal():
q1, q2 = dynamicsymbols('q1:3')
q1d, q2d = dynamicsymbols('q1:3', level=1)
L, m, t = symbols('L, m, t')
g = 9.8
# Compose World Frame
N = ReferenceFrame('N')
pN = Point('N*')
pN.set_vel(N, 0)
# A.x is along the pendulum
theta1 = atan(q2/q1)
A = N.orientnew('A', 'axis', [theta1, N.z])
# Create point P, the pendulum mass
P = pN.locatenew('P1', q1*N.x + q2*N.y)
P.set_vel(N, P.pos_from(pN).dt(N))
pP = Particle('pP', P, m)
# Constraint Equations
f_c = Matrix([q1**2 + q2**2 - L**2])
# Calculate the lagrangian, and form the equations of motion
Lag = Lagrangian(N, pP)
LM = LagrangesMethod(Lag, [q1, q2], hol_coneqs=f_c, forcelist=[(P, m*g*N.x)], frame=N)
LM.form_lagranges_equations()
# Compose operating point
op_point = {q1: L, q2: 0, q1d: 0, q2d: 0, q1d.diff(t): 0, q2d.diff(t): 0}
# Solve for multiplier operating point
lam_op = LM.solve_multipliers(op_point=op_point)
op_point.update(lam_op)
# Perform the Linearization
A, B, inp_vec = LM.linearize([q2], [q2d], [q1], [q1d],
op_point=op_point, A_and_B=True)
assert A == Matrix([[0, 1], [-9.8/L, 0]])
assert B == Matrix([])
def test_linearize_rolling_disc_lagrange():
q1, q2, q3 = q = dynamicsymbols('q1 q2 q3')
q1d, q2d, q3d = qd = dynamicsymbols('q1 q2 q3', 1)
r, m, g = symbols('r m g')
N = ReferenceFrame('N')
Y = N.orientnew('Y', 'Axis', [q1, N.z])
L = Y.orientnew('L', 'Axis', [q2, Y.x])
R = L.orientnew('R', 'Axis', [q3, L.y])
C = Point('C')
C.set_vel(N, 0)
Dmc = C.locatenew('Dmc', r * L.z)
Dmc.v2pt_theory(C, N, R)
I = inertia(L, m / 4 * r**2, m / 2 * r**2, m / 4 * r**2)
BodyD = RigidBody('BodyD', Dmc, R, m, (I, Dmc))
BodyD.potential_energy = - m * g * r * cos(q2)
Lag = Lagrangian(N, BodyD)
l = LagrangesMethod(Lag, q)
l.form_lagranges_equations()
# Linearize about steady-state upright rolling
op_point = {q1: 0, q2: 0, q3: 0,
q1d: 0, q2d: 0,
q1d.diff(): 0, q2d.diff(): 0, q3d.diff(): 0}
A = l.linearize(q_ind=q, qd_ind=qd, op_point=op_point, A_and_B=True)[0]
sol = Matrix([[0, 0, 0, 1, 0, 0],
[0, 0, 0, 0, 1, 0],
[0, 0, 0, 0, 0, 1],
[0, 0, 0, 0, -6*q3d, 0],
[0, -4*g/(5*r), 0, 6*q3d/5, 0, 0],
[0, 0, 0, 0, 0, 0]])
assert A == sol
|
46543f278ab2e6f1926314b53f316e1867b34e654198d8c3e0784571fd772561
|
from sympy import symbols
from sympy.physics.mechanics import Point, ReferenceFrame, Dyadic, RigidBody
from sympy.physics.mechanics import dynamicsymbols, outer, inertia
from sympy.physics.mechanics import inertia_of_point_mass
from sympy.core.backend import expand
from sympy.utilities.pytest import raises
def test_rigidbody():
m, m2, v1, v2, v3, omega = symbols('m m2 v1 v2 v3 omega')
A = ReferenceFrame('A')
A2 = ReferenceFrame('A2')
P = Point('P')
P2 = Point('P2')
I = Dyadic(0)
I2 = Dyadic(0)
B = RigidBody('B', P, A, m, (I, P))
assert B.mass == m
assert B.frame == A
assert B.masscenter == P
assert B.inertia == (I, B.masscenter)
B.mass = m2
B.frame = A2
B.masscenter = P2
B.inertia = (I2, B.masscenter)
raises(TypeError, lambda: RigidBody(P, P, A, m, (I, P)))
raises(TypeError, lambda: RigidBody('B', P, P, m, (I, P)))
raises(TypeError, lambda: RigidBody('B', P, A, m, (P, P)))
raises(TypeError, lambda: RigidBody('B', P, A, m, (I, I)))
assert B.__str__() == 'B'
assert B.mass == m2
assert B.frame == A2
assert B.masscenter == P2
assert B.inertia == (I2, B.masscenter)
assert B.masscenter == P2
assert B.inertia == (I2, B.masscenter)
# Testing linear momentum function assuming A2 is the inertial frame
N = ReferenceFrame('N')
P2.set_vel(N, v1 * N.x + v2 * N.y + v3 * N.z)
assert B.linear_momentum(N) == m2 * (v1 * N.x + v2 * N.y + v3 * N.z)
def test_rigidbody2():
M, v, r, omega, g, h = dynamicsymbols('M v r omega g h')
N = ReferenceFrame('N')
b = ReferenceFrame('b')
b.set_ang_vel(N, omega * b.x)
P = Point('P')
I = outer(b.x, b.x)
Inertia_tuple = (I, P)
B = RigidBody('B', P, b, M, Inertia_tuple)
P.set_vel(N, v * b.x)
assert B.angular_momentum(P, N) == omega * b.x
O = Point('O')
O.set_vel(N, v * b.x)
P.set_pos(O, r * b.y)
assert B.angular_momentum(O, N) == omega * b.x - M*v*r*b.z
B.potential_energy = M * g * h
assert B.potential_energy == M * g * h
assert expand(2 * B.kinetic_energy(N)) == omega**2 + M * v**2
def test_rigidbody3():
q1, q2, q3, q4 = dynamicsymbols('q1:5')
p1, p2, p3 = symbols('p1:4')
m = symbols('m')
A = ReferenceFrame('A')
B = A.orientnew('B', 'axis', [q1, A.x])
O = Point('O')
O.set_vel(A, q2*A.x + q3*A.y + q4*A.z)
P = O.locatenew('P', p1*B.x + p2*B.y + p3*B.z)
P.v2pt_theory(O, A, B)
I = outer(B.x, B.x)
rb1 = RigidBody('rb1', P, B, m, (I, P))
# I_S/O = I_S/S* + I_S*/O
rb2 = RigidBody('rb2', P, B, m,
(I + inertia_of_point_mass(m, P.pos_from(O), B), O))
assert rb1.central_inertia == rb2.central_inertia
assert rb1.angular_momentum(O, A) == rb2.angular_momentum(O, A)
def test_pendulum_angular_momentum():
"""Consider a pendulum of length OA = 2a, of mass m as a rigid body of
center of mass G (OG = a) which turn around (O,z). The angle between the
reference frame R and the rod is q. The inertia of the body is I =
(G,0,ma^2/3,ma^2/3). """
m, a = symbols('m, a')
q = dynamicsymbols('q')
R = ReferenceFrame('R')
R1 = R.orientnew('R1', 'Axis', [q, R.z])
R1.set_ang_vel(R, q.diff() * R.z)
I = inertia(R1, 0, m * a**2 / 3, m * a**2 / 3)
O = Point('O')
A = O.locatenew('A', 2*a * R1.x)
G = O.locatenew('G', a * R1.x)
S = RigidBody('S', G, R1, m, (I, G))
O.set_vel(R, 0)
A.v2pt_theory(O, R, R1)
G.v2pt_theory(O, R, R1)
assert (4 * m * a**2 / 3 * q.diff() * R.z -
S.angular_momentum(O, R).express(R)) == 0
|
0c86d2441a6b7e0ebf6ece66978278ce5697175494584150f72734c2cd75ab6c
|
"""
MKS unit system.
MKS stands for "meter, kilogram, second".
"""
from __future__ import division
from sympy.physics.units import DimensionSystem, UnitSystem
from sympy.physics.units.definitions import G, Hz, J, N, Pa, W, c, g, kg, m, s
from sympy.physics.units.dimensions import (
acceleration, action, energy, force, frequency, length, mass, momentum,
power, pressure, time, velocity, dimsys_MKS)
from sympy.physics.units.prefixes import PREFIXES, prefix_unit
dims = (velocity, acceleration, momentum, force, energy, power, pressure,
frequency, action)
# dimension system
_mks_dim = dimsys_MKS
units = [m, g, s, J, N, W, Pa, Hz]
all_units = []
# Prefixes of units like g, J, N etc get added using `prefix_unit`
# in the for loop, but the actual units have to be added manually.
all_units.extend([g, J, N, W, Pa, Hz])
for u in units:
all_units.extend(prefix_unit(u, PREFIXES))
all_units.extend([G, c])
# unit system
MKS = UnitSystem(base=(m, kg, s), units=all_units, name="MKS")
|
fefea9eb1f4ad86890d8e0f514b3723a73d5b427b2cdd18ec5c7351ec53a063b
|
"""
MKS unit system.
MKS stands for "meter, kilogram, second, ampere".
"""
from __future__ import division
from sympy.physics.units.definitions import Z0, A, C, F, H, S, T, V, Wb, ohm
from sympy.physics.units.dimensions import (
capacitance, charge, conductance, current, impedance, inductance,
magnetic_density, magnetic_flux, voltage, dimsys_MKSA)
from sympy.physics.units.prefixes import PREFIXES, prefix_unit
from sympy.physics.units.systems.mks import MKS, _mks_dim
dims = (voltage, impedance, conductance, capacitance, inductance, charge,
magnetic_density, magnetic_flux)
# dimension system
_mksa_dim = dimsys_MKSA
units = [A, V, ohm, S, F, H, C, T, Wb]
all_units = []
for u in units:
all_units.extend(prefix_unit(u, PREFIXES))
all_units.extend([Z0])
MKSA = MKS.extend(base=(A,), units=all_units, name='MKSA')
|
94da63d8cb310bb5e262585f411682ebba9f264a0a9d3b83bbcf174f290f7abc
|
from sympy.physics.units.systems.mks import MKS, _mks_dim
from sympy.physics.units.systems.mksa import MKSA, _mksa_dim
from sympy.physics.units.systems.natural import _natural_dim, natural
from sympy.physics.units.systems.si import SI, _si_dim
|
4e7e67566af98936b1482031ef1380c0dc6e39d192ee1889d59f340547f23264
|
"""
Naturalunit system.
The natural system comes from "setting c = 1, hbar = 1". From the computer
point of view it means that we use velocity and action instead of length and
time. Moreover instead of mass we use energy.
"""
from __future__ import division
from sympy.physics.units.definitions import c, eV, hbar
from sympy.physics.units.dimensions import (
DimensionSystem, action, energy, force, frequency, length, mass, momentum,
power, time, velocity)
from sympy.physics.units.prefixes import PREFIXES, prefix_unit
from sympy.physics.units.unitsystem import UnitSystem
# dimension system
_natural_dim = DimensionSystem(
base_dims=(action, energy, velocity),
derived_dims=(length, mass, time, momentum, force, power, frequency)
)
units = prefix_unit(eV, PREFIXES)
# unit system
natural = UnitSystem(base=(hbar, eV, c), units=units, name="Natural system")
|
e37ebe0f87c7f3473c8f7964a054de470927a554d04cde632fc0ea711c90b6a8
|
"""
SI unit system.
Based on MKSA, which stands for "meter, kilogram, second, ampere".
Added kelvin, candela and mole.
"""
from __future__ import division
from sympy.physics.units.definitions import (
K, cd, lux, mol,hertz, newton, pascal, joule, watt, coulomb, volt, farad,
ohm, siemens, weber, tesla, henry, candela, lux, becquerel, gray, katal)
from sympy.physics.units.dimensions import (
amount_of_substance, luminous_intensity, temperature, dimsys_SI,
frequency, force, pressure, energy, power, charge, voltage, capacitance,
conductance, magnetic_flux, magnetic_density, inductance,
luminous_intensity)
from sympy.physics.units.prefixes import PREFIXES, prefix_unit
from sympy.physics.units.systems.mksa import MKSA, _mksa_dim
derived_dims = (frequency, force, pressure, energy, power, charge, voltage,
capacitance, conductance, magnetic_flux,
magnetic_density, inductance, luminous_intensity)
base_dims = (amount_of_substance, luminous_intensity, temperature)
# dimension system
_si_dim = dimsys_SI
units = [mol, cd, K, lux, hertz, newton, pascal, joule, watt, coulomb, volt,
farad, ohm, siemens, weber, tesla, henry, candela, lux, becquerel,
gray, katal]
all_units = []
for u in units:
all_units.extend(prefix_unit(u, PREFIXES))
all_units.extend([mol, cd, K, lux])
SI = MKSA.extend(base=(mol, cd, K), units=all_units, name='SI')
|
b4791b1afd7070abd003c90317e64088943ce2dcbb5544460112fb3049f2a3ad
|
from sympy import (Abs, Add, Basic, Function, Number, Rational, S, Symbol,
diff, exp, integrate, log, sin, sqrt, symbols)
from sympy.physics.units import (amount_of_substance, convert_to, find_unit,
volume)
from sympy.physics.units.definitions import (amu, au, centimeter, coulomb,
day, energy, foot, grams, hour, inch, kg, km, m, meter, mile, millimeter,
minute, pressure, quart, s, second, speed_of_light, temperature, bit,
byte, kibibyte, mebibyte, gibibyte, tebibyte, pebibyte, exbibyte,
kilogram, gravitational_constant)
from sympy.physics.units.dimensions import Dimension, charge, length, time, dimsys_default
from sympy.physics.units.prefixes import PREFIXES, kilo
from sympy.physics.units.quantities import Quantity
from sympy.utilities.pytest import XFAIL, raises, warns_deprecated_sympy
k = PREFIXES["k"]
def test_str_repr():
assert str(kg) == "kilogram"
def test_eq():
# simple test
assert 10*m == 10*m
assert 10*m != 10*s
def test_convert_to():
q = Quantity("q1")
q.set_dimension(length)
q.set_scale_factor(S(5000))
assert q.convert_to(m) == 5000*m
assert speed_of_light.convert_to(m / s) == 299792458 * m / s
# TODO: eventually support this kind of conversion:
# assert (2*speed_of_light).convert_to(m / s) == 2 * 299792458 * m / s
assert day.convert_to(s) == 86400*s
# Wrong dimension to convert:
assert q.convert_to(s) == q
assert speed_of_light.convert_to(m) == speed_of_light
def test_Quantity_definition():
q = Quantity("s10", abbrev="sabbr")
q.set_dimension(time)
q.set_scale_factor(10)
u = Quantity("u", abbrev="dam")
u.set_dimension(length)
u.set_scale_factor(10)
km = Quantity("km")
km.set_dimension(length)
km.set_scale_factor(kilo)
v = Quantity("u")
v.set_dimension(length)
v.set_scale_factor(5*kilo)
assert q.scale_factor == 10
assert q.dimension == time
assert q.abbrev == Symbol("sabbr")
assert u.dimension == length
assert u.scale_factor == 10
assert u.abbrev == Symbol("dam")
assert km.scale_factor == 1000
assert km.func(*km.args) == km
assert km.func(*km.args).args == km.args
assert v.dimension == length
assert v.scale_factor == 5000
with warns_deprecated_sympy():
Quantity('invalid', 'dimension', 1)
with warns_deprecated_sympy():
Quantity('mismatch', dimension=length, scale_factor=kg)
def test_abbrev():
u = Quantity("u")
u.set_dimension(length)
u.set_scale_factor(S.One)
assert u.name == Symbol("u")
assert u.abbrev == Symbol("u")
u = Quantity("u", abbrev="om")
u.set_dimension(length)
u.set_scale_factor(S(2))
assert u.name == Symbol("u")
assert u.abbrev == Symbol("om")
assert u.scale_factor == 2
assert isinstance(u.scale_factor, Number)
u = Quantity("u", abbrev="ikm")
u.set_dimension(length)
u.set_scale_factor(3*kilo)
assert u.abbrev == Symbol("ikm")
assert u.scale_factor == 3000
def test_print():
u = Quantity("unitname", abbrev="dam")
assert repr(u) == "unitname"
assert str(u) == "unitname"
def test_Quantity_eq():
u = Quantity("u", abbrev="dam")
v = Quantity("v1")
assert u != v
v = Quantity("v2", abbrev="ds")
assert u != v
v = Quantity("v3", abbrev="dm")
assert u != v
def test_add_sub():
u = Quantity("u")
v = Quantity("v")
w = Quantity("w")
u.set_dimension(length)
v.set_dimension(length)
w.set_dimension(time)
u.set_scale_factor(S(10))
v.set_scale_factor(S(5))
w.set_scale_factor(S(2))
assert isinstance(u + v, Add)
assert (u + v.convert_to(u)) == (1 + S.Half)*u
# TODO: eventually add this:
# assert (u + v).convert_to(u) == (1 + S.Half)*u
assert isinstance(u - v, Add)
assert (u - v.convert_to(u)) == S.Half*u
# TODO: eventually add this:
# assert (u - v).convert_to(u) == S.Half*u
def test_quantity_abs():
v_w1 = Quantity('v_w1')
v_w2 = Quantity('v_w2')
v_w3 = Quantity('v_w3')
v_w1.set_dimension(length/time)
v_w2.set_dimension(length/time)
v_w3.set_dimension(length/time)
v_w1.set_scale_factor(meter/second)
v_w2.set_scale_factor(meter/second)
v_w3.set_scale_factor(meter/second)
expr = v_w3 - Abs(v_w1 - v_w2)
Dq = Dimension(Quantity.get_dimensional_expr(expr))
assert dimsys_default.get_dimensional_dependencies(Dq) == {
'length': 1,
'time': -1,
}
assert meter == sqrt(meter**2)
def test_check_unit_consistency():
u = Quantity("u")
v = Quantity("v")
w = Quantity("w")
u.set_dimension(length)
v.set_dimension(length)
w.set_dimension(time)
u.set_scale_factor(S(10))
v.set_scale_factor(S(5))
w.set_scale_factor(S(2))
def check_unit_consistency(expr):
Quantity._collect_factor_and_dimension(expr)
raises(ValueError, lambda: check_unit_consistency(u + w))
raises(ValueError, lambda: check_unit_consistency(u - w))
raises(ValueError, lambda: check_unit_consistency(u + 1))
raises(ValueError, lambda: check_unit_consistency(u - 1))
def test_mul_div():
u = Quantity("u")
v = Quantity("v")
t = Quantity("t")
ut = Quantity("ut")
v2 = Quantity("v")
u.set_dimension(length)
v.set_dimension(length)
t.set_dimension(time)
ut.set_dimension(length*time)
v2.set_dimension(length/time)
u.set_scale_factor(S(10))
v.set_scale_factor(S(5))
t.set_scale_factor(S(2))
ut.set_scale_factor(S(20))
v2.set_scale_factor(S(5))
assert 1 / u == u**(-1)
assert u / 1 == u
v1 = u / t
v2 = v
# Pow only supports structural equality:
assert v1 != v2
assert v1 == v2.convert_to(v1)
# TODO: decide whether to allow such expression in the future
# (requires somehow manipulating the core).
# assert u / Quantity('l2', dimension=length, scale_factor=2) == 5
assert u * 1 == u
ut1 = u * t
ut2 = ut
# Mul only supports structural equality:
assert ut1 != ut2
assert ut1 == ut2.convert_to(ut1)
# Mul only supports structural equality:
lp1 = Quantity("lp1")
lp1.set_dimension(length**-1)
lp1.set_scale_factor(S(2))
assert u * lp1 != 20
assert u**0 == 1
assert u**1 == u
# TODO: Pow only support structural equality:
u2 = Quantity("u2")
u3 = Quantity("u3")
u2.set_dimension(length**2)
u3.set_dimension(length**-1)
u2.set_scale_factor(S(100))
u3.set_scale_factor(S(1)/10)
assert u ** 2 != u2
assert u ** -1 != u3
assert u ** 2 == u2.convert_to(u)
assert u ** -1 == u3.convert_to(u)
def test_units():
assert convert_to((5*m/s * day) / km, 1) == 432
assert convert_to(foot / meter, meter) == Rational(3048, 10000)
# amu is a pure mass so mass/mass gives a number, not an amount (mol)
# TODO: need better simplification routine:
assert str(convert_to(grams/amu, grams).n(2)) == '6.0e+23'
# Light from the sun needs about 8.3 minutes to reach earth
t = (1*au / speed_of_light) / minute
# TODO: need a better way to simplify expressions containing units:
t = convert_to(convert_to(t, meter / minute), meter)
assert t == S(49865956897)/5995849160
# TODO: fix this, it should give `m` without `Abs`
assert sqrt(m**2) == Abs(m)
assert (sqrt(m))**2 == m
t = Symbol('t')
assert integrate(t*m/s, (t, 1*s, 5*s)) == 12*m*s
assert (t * m/s).integrate((t, 1*s, 5*s)) == 12*m*s
def test_issue_quart():
assert convert_to(4 * quart / inch ** 3, meter) == 231
assert convert_to(4 * quart / inch ** 3, millimeter) == 231
def test_issue_5565():
assert (m < s).is_Relational
def test_find_unit():
assert find_unit('coulomb') == ['coulomb', 'coulombs', 'coulomb_constant']
assert find_unit(coulomb) == ['C', 'coulomb', 'coulombs', 'planck_charge']
assert find_unit(charge) == ['C', 'coulomb', 'coulombs', 'planck_charge']
assert find_unit(inch) == [
'm', 'au', 'cm', 'dm', 'ft', 'km', 'ly', 'mi', 'mm', 'nm', 'pm', 'um',
'yd', 'nmi', 'feet', 'foot', 'inch', 'mile', 'yard', 'meter', 'miles',
'yards', 'inches', 'meters', 'micron', 'microns', 'decimeter',
'kilometer', 'lightyear', 'nanometer', 'picometer', 'centimeter',
'decimeters', 'kilometers', 'lightyears', 'micrometer', 'millimeter',
'nanometers', 'picometers', 'centimeters', 'micrometers',
'millimeters', 'nautical_mile', 'planck_length', 'nautical_miles', 'astronomical_unit',
'astronomical_units']
assert find_unit(inch**-1) == ['D', 'dioptre', 'optical_power']
assert find_unit(length**-1) == ['D', 'dioptre', 'optical_power']
assert find_unit(inch ** 3) == [
'l', 'cl', 'dl', 'ml', 'liter', 'quart', 'liters', 'quarts',
'deciliter', 'centiliter', 'deciliters', 'milliliter',
'centiliters', 'milliliters', 'planck_volume']
assert find_unit('voltage') == ['V', 'v', 'volt', 'volts', 'planck_voltage']
def test_Quantity_derivative():
x = symbols("x")
assert diff(x*meter, x) == meter
assert diff(x**3*meter**2, x) == 3*x**2*meter**2
assert diff(meter, meter) == 1
assert diff(meter**2, meter) == 2*meter
def test_quantity_postprocessing():
q1 = Quantity('q1')
q2 = Quantity('q2')
q1.set_dimension(length*pressure**2*temperature/time)
q2.set_dimension(energy*pressure*temperature/(length**2*time))
assert q1 + q2
q = q1 + q2
Dq = Dimension(Quantity.get_dimensional_expr(q))
assert dimsys_default.get_dimensional_dependencies(Dq) == {
'length': -1,
'mass': 2,
'temperature': 1,
'time': -5,
}
def test_factor_and_dimension():
assert (3000, Dimension(1)) == Quantity._collect_factor_and_dimension(3000)
assert (1001, length) == Quantity._collect_factor_and_dimension(meter + km)
assert (2, length/time) == Quantity._collect_factor_and_dimension(
meter/second + 36*km/(10*hour))
x, y = symbols('x y')
assert (x + y/100, length) == Quantity._collect_factor_and_dimension(
x*m + y*centimeter)
cH = Quantity('cH')
cH.set_dimension(amount_of_substance/volume)
pH = -log(cH)
assert (1, volume/amount_of_substance) == Quantity._collect_factor_and_dimension(
exp(pH))
v_w1 = Quantity('v_w1')
v_w2 = Quantity('v_w2')
v_w1.set_dimension(length/time)
v_w2.set_dimension(length/time)
v_w1.set_scale_factor(S(3)/2*meter/second)
v_w2.set_scale_factor(2*meter/second)
expr = Abs(v_w1/2 - v_w2)
assert (S(5)/4, length/time) == \
Quantity._collect_factor_and_dimension(expr)
expr = S(5)/2*second/meter*v_w1 - 3000
assert (-(2996 + S(1)/4), Dimension(1)) == \
Quantity._collect_factor_and_dimension(expr)
expr = v_w1**(v_w2/v_w1)
assert ((S(3)/2)**(S(4)/3), (length/time)**(S(4)/3)) == \
Quantity._collect_factor_and_dimension(expr)
@XFAIL
def test_factor_and_dimension_with_Abs():
with warns_deprecated_sympy():
v_w1 = Quantity('v_w1', length/time, S(3)/2*meter/second)
v_w1.set_dimension(length/time)
v_w1.set_scale_factor(S(3)/2*meter/second)
expr = v_w1 - Abs(v_w1)
assert (0, length/time) == Quantity._collect_factor_and_dimension(expr)
def test_dimensional_expr_of_derivative():
l = Quantity('l')
t = Quantity('t')
t1 = Quantity('t1')
l.set_dimension(length)
t.set_dimension(time)
t1.set_dimension(time)
l.set_scale_factor(36*km)
t.set_scale_factor(hour)
t1.set_scale_factor(second)
x = Symbol('x')
y = Symbol('y')
f = Function('f')
dfdx = f(x, y).diff(x, y)
dl_dt = dfdx.subs({f(x, y): l, x: t, y: t1})
assert Quantity.get_dimensional_expr(dl_dt) ==\
Quantity.get_dimensional_expr(l / t / t1) ==\
Symbol("length")/Symbol("time")**2
assert Quantity._collect_factor_and_dimension(dl_dt) ==\
Quantity._collect_factor_and_dimension(l / t / t1) ==\
(10, length/time**2)
def test_get_dimensional_expr_with_function():
v_w1 = Quantity('v_w1')
v_w2 = Quantity('v_w2')
v_w1.set_dimension(length/time)
v_w2.set_dimension(length/time)
v_w1.set_scale_factor(meter/second)
v_w2.set_scale_factor(meter/second)
assert Quantity.get_dimensional_expr(sin(v_w1)) == \
sin(Quantity.get_dimensional_expr(v_w1))
assert Quantity.get_dimensional_expr(sin(v_w1/v_w2)) == 1
def test_binary_information():
assert convert_to(kibibyte, byte) == 1024*byte
assert convert_to(mebibyte, byte) == 1024**2*byte
assert convert_to(gibibyte, byte) == 1024**3*byte
assert convert_to(tebibyte, byte) == 1024**4*byte
assert convert_to(pebibyte, byte) == 1024**5*byte
assert convert_to(exbibyte, byte) == 1024**6*byte
assert kibibyte.convert_to(bit) == 8*1024*bit
assert byte.convert_to(bit) == 8*bit
a = 10*kibibyte*hour
assert convert_to(a, byte) == 10240*byte*hour
assert convert_to(a, minute) == 600*kibibyte*minute
assert convert_to(a, [byte, minute]) == 614400*byte*minute
def test_eval_subs():
energy, mass, force = symbols('energy mass force')
expr1 = energy/mass
units = {energy: kilogram*meter**2/second**2, mass: kilogram}
assert expr1.subs(units) == meter**2/second**2
expr2 = force/mass
units = {force:gravitational_constant*kilogram**2/meter**2, mass:kilogram}
assert expr2.subs(units) == gravitational_constant*kilogram/meter**2
def test_issue_14932():
assert (log(inch) - log(2)).simplify() == log(inch/2)
assert (log(inch) - log(foot)).simplify() == -log(12)
p = symbols('p', positive=True)
assert (log(inch) - log(p)).simplify() == log(inch/p)
def test_issue_14547():
# the root issue is that an argument with dimensions should
# not raise an error when the the `arg - 1` calculation is
# performed in the assumptions system
from sympy.physics.units import foot, inch
from sympy import Eq
assert log(foot).is_zero is None
assert log(foot).is_positive is None
assert log(foot).is_nonnegative is None
assert log(foot).is_negative is None
assert log(foot).is_algebraic is None
assert log(foot).is_rational is None
# doesn't raise error
assert Eq(log(foot), log(inch)) is not None # might be False or unevaluated
x = Symbol('x')
e = foot + x
assert e.is_Add and set(e.args) == {foot, x}
e = foot + 1
assert e.is_Add and set(e.args) == {foot, 1}
|
40bc41b764b838b920786a64511a416c16215ade1ef061d1823fcf0f2a484cf3
|
from sympy.utilities.pytest import warns_deprecated_sympy
from sympy import Rational, S
from sympy.physics.units.definitions import c, kg, m, s
from sympy.physics.units.dimensions import (
Dimension, DimensionSystem, action, current, length, mass, time, velocity)
from sympy.physics.units.quantities import Quantity
from sympy.physics.units.unitsystem import UnitSystem
from sympy.utilities.pytest import raises
def test_definition():
# want to test if the system can have several units of the same dimension
dm = Quantity("dm")
dm.set_dimension(length)
dm.set_scale_factor(Rational(1, 10))
base = (m, s)
base_dim = (m.dimension, s.dimension)
ms = UnitSystem(base, (c, dm), "MS", "MS system")
assert set(ms._base_units) == set(base)
assert set(ms._units) == set((m, s, c, dm))
#assert ms._units == DimensionSystem._sort_dims(base + (velocity,))
assert ms.name == "MS"
assert ms.descr == "MS system"
assert ms._system.base_dims == base_dim
assert ms._system.derived_dims == (velocity,)
def test_error_definition():
raises(ValueError, lambda: UnitSystem((m, s, c)))
def test_str_repr():
assert str(UnitSystem((m, s), name="MS")) == "MS"
assert str(UnitSystem((m, s))) == "UnitSystem((meter, second))"
assert repr(UnitSystem((m, s))) == "<UnitSystem: (%s, %s)>" % (m, s)
def test_print_unit_base():
A = Quantity("A")
A.set_dimension(current)
A.set_scale_factor(S.One)
Js = Quantity("Js")
Js.set_dimension(action)
Js.set_scale_factor(S.One)
mksa = UnitSystem((m, kg, s, A), (Js,))
with warns_deprecated_sympy():
assert mksa.print_unit_base(Js) == m**2*kg*s**-1
def test_extend():
ms = UnitSystem((m, s), (c,))
Js = Quantity("Js")
Js.set_dimension(action)
Js.set_scale_factor(1)
mks = ms.extend((kg,), (Js,))
res = UnitSystem((m, s, kg), (c, Js))
assert set(mks._base_units) == set(res._base_units)
assert set(mks._units) == set(res._units)
def test_dim():
dimsys = UnitSystem((m, kg, s), (c,))
assert dimsys.dim == 3
def test_is_consistent():
assert UnitSystem((m, s)).is_consistent is True
|
be0f7b5e808be584cd712ca4d0330034779577fd42912d35feaebea2cdd1c321
|
from sympy.utilities.pytest import warns_deprecated_sympy
from sympy import S, Symbol, sqrt
from sympy.physics.units.dimensions import Dimension, length, time, dimsys_default
from sympy.physics.units import foot
from sympy.utilities.pytest import raises
def test_Dimension_definition():
with warns_deprecated_sympy():
assert length.get_dimensional_dependencies() == {"length": 1}
assert dimsys_default.get_dimensional_dependencies(length) == {"length": 1}
assert length.name == Symbol("length")
assert length.symbol == Symbol("L")
halflength = sqrt(length)
with warns_deprecated_sympy():
assert halflength.get_dimensional_dependencies() == {"length": S.Half}
assert dimsys_default.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_Dimension_error_regisration():
with warns_deprecated_sympy():
# tuple with more or less than two entries
raises(IndexError, lambda: length._register_as_base_dim())
with warns_deprecated_sympy():
one = Dimension(1)
raises(TypeError, lambda: one._register_as_base_dim())
def test_str():
assert str(Dimension("length")) == "Dimension(length)"
assert str(Dimension("length", "L")) == "Dimension(length, L)"
def test_Dimension_properties():
assert dimsys_default.is_dimensionless(length) is False
assert dimsys_default.is_dimensionless(length/length) is True
assert dimsys_default.is_dimensionless(Dimension("undefined")) is True
assert length.has_integer_powers(dimsys_default) is True
assert (length**(-1)).has_integer_powers(dimsys_default) is True
assert (length**1.5).has_integer_powers(dimsys_default) is False
def test_Dimension_add_sub():
assert length + length == length
assert length - length == length
assert -length == length
assert length + foot == foot + length == length
assert length - foot == foot - length == length
assert length + time == length - time != 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}
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_default.get_dimensional_dependencies(length * length) == {"length": 2}
assert dimsys_default.get_dimensional_dependencies(length ** 2) == {"length": 2}
assert dimsys_default.get_dimensional_dependencies(length * time) == { "length": 1, "time": 1}
assert dimsys_default.get_dimensional_dependencies(velo) == { "length": 1, "time": -1}
assert dimsys_default.get_dimensional_dependencies(velo ** 2) == {"length": 2, "time": -2}
assert dimsys_default.get_dimensional_dependencies(length / length) == {}
assert dimsys_default.get_dimensional_dependencies(velo / length * time) == {}
assert dimsys_default.get_dimensional_dependencies(length ** -1) == {"length": -1}
assert dimsys_default.get_dimensional_dependencies(velo ** -1.5) == {"length": -1.5, "time": 1.5}
length_a = length**"a"
assert dimsys_default.get_dimensional_dependencies(length_a) == {"length": Symbol("a")}
assert length != 1
assert length / length != 1
length_0 = length ** 0
assert dimsys_default.get_dimensional_dependencies(length_0) == {}
|
4e67bbc2cfae3d0dc8f28cdff69b6e3eea0b0f9c48bf153c095c52bdaa1d824c
|
from sympy import symbols, log, Mul, Symbol, S
from sympy.physics.units import Quantity, Dimension, length
from sympy.physics.units.prefixes import PREFIXES, Prefix, prefix_unit, kilo, \
kibi
x = Symbol('x')
def test_prefix_operations():
m = PREFIXES['m']
k = PREFIXES['k']
M = PREFIXES['M']
dodeca = Prefix('dodeca', 'dd', 1, base=12)
assert m * k == 1
assert k * k == M
assert 1 / m == k
assert k / m == M
assert dodeca * dodeca == 144
assert 1 / dodeca == S(1) / 12
assert k / dodeca == S(1000) / 12
assert dodeca / dodeca == 1
m = Quantity("fake_meter")
m.set_dimension(S.One)
m.set_scale_factor(S.One)
assert dodeca * m == 12 * m
assert dodeca / m == 12 / m
expr1 = kilo * 3
assert isinstance(expr1, Mul)
assert (expr1).args == (3, kilo)
expr2 = kilo * x
assert isinstance(expr2, Mul)
assert (expr2).args == (x, kilo)
expr3 = kilo / 3
assert isinstance(expr3, Mul)
assert (expr3).args == (S(1)/3, kilo)
expr4 = kilo / x
assert isinstance(expr4, Mul)
assert (expr4).args == (1/x, kilo)
def test_prefix_unit():
m = Quantity("fake_meter", abbrev="m")
m.set_dimension(length)
m.set_scale_factor(1)
pref = {"m": PREFIXES["m"], "c": PREFIXES["c"], "d": PREFIXES["d"]}
q1 = Quantity("millifake_meter", abbrev="mm")
q2 = Quantity("centifake_meter", abbrev="cm")
q3 = Quantity("decifake_meter", abbrev="dm")
q1.set_dimension(length)
q1.set_dimension(length)
q1.set_dimension(length)
q1.set_scale_factor(PREFIXES["m"])
q1.set_scale_factor(PREFIXES["c"])
q1.set_scale_factor(PREFIXES["d"])
res = [q1, q2, q3]
prefs = prefix_unit(m, pref)
assert set(prefs) == set(res)
assert set(map(lambda x: x.abbrev, prefs)) == set(symbols("mm,cm,dm"))
def test_bases():
assert kilo.base == 10
assert kibi.base == 2
def test_repr():
assert eval(repr(kilo)) == kilo
assert eval(repr(kibi)) == kibi
|
54a2e40cda86f0edd46468673cbf979299a7e373f53706d696e2f3d68bc995c2
|
from sympy.utilities.pytest import warns_deprecated_sympy
from sympy import (Add, Mul, Pow, Tuple, pi, sin, sqrt, sstr, sympify,
symbols)
from sympy.physics.units import (
G, centimeter, coulomb, day, degree, gram, hbar, hour, inch, joule, kelvin,
kilogram, kilometer, length, meter, mile, minute, newton, planck,
planck_length, planck_mass, planck_temperature, planck_time, radians,
second, speed_of_light, steradian, time, km)
from sympy.physics.units.dimensions import dimsys_default
from sympy.physics.units.util import convert_to, dim_simplify, check_dimensions
from sympy.utilities.pytest import raises
def NS(e, n=15, **options):
return sstr(sympify(e).evalf(n, **options), full_prec=True)
L = length
T = time
def test_dim_simplify_add():
with warns_deprecated_sympy():
assert dim_simplify(Add(L, L)) == L
with warns_deprecated_sympy():
assert dim_simplify(L + L) == L
def test_dim_simplify_mul():
with warns_deprecated_sympy():
assert dim_simplify(Mul(L, T)) == L*T
with warns_deprecated_sympy():
assert dim_simplify(L*T) == L*T
def test_dim_simplify_pow():
with warns_deprecated_sympy():
assert dim_simplify(Pow(L, 2)) == L**2
with warns_deprecated_sympy():
assert dim_simplify(L**2) == L**2
def test_dim_simplify_rec():
with warns_deprecated_sympy():
assert dim_simplify(Mul(Add(L, L), T)) == L*T
with warns_deprecated_sympy():
assert dim_simplify((L + L) * T) == L*T
def test_dim_simplify_dimless():
# TODO: this should be somehow simplified on its own,
# without the need of calling `dim_simplify`:
with warns_deprecated_sympy():
assert dim_simplify(sin(L*L**-1)**2*L).get_dimensional_dependencies()\
== dimsys_default.get_dimensional_dependencies(L)
with warns_deprecated_sympy():
assert dim_simplify(sin(L * L**(-1))**2 * L).get_dimensional_dependencies()\
== dimsys_default.get_dimensional_dependencies(L)
def test_convert_to_quantities():
assert convert_to(3, meter) == 3
assert convert_to(mile, kilometer) == 1.609344*kilometer
assert convert_to(meter/second, speed_of_light) == speed_of_light/299792458
assert convert_to(299792458*meter/second, speed_of_light) == speed_of_light
assert convert_to(2*299792458*meter/second, speed_of_light) == 2*speed_of_light
assert convert_to(speed_of_light, meter/second) == 299792458*meter/second
assert convert_to(2*speed_of_light, meter/second) == 599584916*meter/second
assert convert_to(day, second) == 86400*second
assert convert_to(2*hour, minute) == 120*minute
assert convert_to(mile, meter) == 1609.344*meter
assert convert_to(mile/hour, kilometer/hour) == 25146*kilometer/(15625*hour)
assert convert_to(3*newton, meter/second) == 3*newton
assert convert_to(3*newton, kilogram*meter/second**2) == 3*meter*kilogram/second**2
assert convert_to(kilometer + mile, meter) == 2609.344*meter
assert convert_to(2*kilometer + 3*mile, meter) == 6828.032*meter
assert convert_to(inch**2, meter**2) == 16129*meter**2/25000000
assert convert_to(3*inch**2, meter) == 48387*meter**2/25000000
assert convert_to(2*kilometer/hour + 3*mile/hour, meter/second) == 53344*meter/(28125*second)
assert convert_to(2*kilometer/hour + 3*mile/hour, centimeter/second) == 213376*centimeter/(1125*second)
assert convert_to(kilometer * (mile + kilometer), meter) == 2609344 * meter ** 2
assert convert_to(steradian, coulomb) == steradian
assert convert_to(radians, degree) == 180*degree/pi
assert convert_to(radians, [meter, degree]) == 180*degree/pi
assert convert_to(pi*radians, degree) == 180*degree
assert convert_to(pi, degree) == 180*degree
def test_convert_to_tuples_of_quantities():
assert convert_to(speed_of_light, [meter, second]) == 299792458 * meter / second
assert convert_to(speed_of_light, (meter, second)) == 299792458 * meter / second
assert convert_to(speed_of_light, Tuple(meter, second)) == 299792458 * meter / second
assert convert_to(joule, [meter, kilogram, second]) == kilogram*meter**2/second**2
assert convert_to(joule, [centimeter, gram, second]) == 10000000*centimeter**2*gram/second**2
assert convert_to(299792458*meter/second, [speed_of_light]) == speed_of_light
assert convert_to(speed_of_light / 2, [meter, second, kilogram]) == meter/second*299792458 / 2
# This doesn't make physically sense, but let's keep it as a conversion test:
assert convert_to(2 * speed_of_light, [meter, second, kilogram]) == 2 * 299792458 * meter / second
assert convert_to(G, [G, speed_of_light, planck]) == 1.0*G
assert NS(convert_to(meter, [G, speed_of_light, hbar]), n=7) == '6.187242e+34*gravitational_constant**0.5000000*hbar**0.5000000*speed_of_light**(-1.500000)'
assert NS(convert_to(planck_mass, kilogram), n=7) == '2.176471e-8*kilogram'
assert NS(convert_to(planck_length, meter), n=7) == '1.616229e-35*meter'
assert NS(convert_to(planck_time, second), n=6) == '5.39116e-44*second'
assert NS(convert_to(planck_temperature, kelvin), n=7) == '1.416809e+32*kelvin'
assert NS(convert_to(convert_to(meter, [G, speed_of_light, planck]), meter), n=10) == '1.000000000*meter'
def test_eval_simplify():
from sympy.physics.units import cm, mm, km, m, K, Quantity, kilo, foot
from sympy.simplify.simplify import simplify
from sympy.core.symbol import symbols
from sympy.utilities.pytest import raises
from sympy.core.function import Lambda
x, y = symbols('x y')
assert ((cm/mm).simplify()) == 10
assert ((km/m).simplify()) == 1000
assert ((km/cm).simplify()) == 100000
assert ((10*x*K*km**2/m/cm).simplify()) == 1000000000*x*kelvin
assert ((cm/km/m).simplify()) == 1/(10000000*centimeter)
assert (3*kilo*meter).simplify() == 3000*meter
assert (4*kilo*meter/(2*kilometer)).simplify() == 2
assert (4*kilometer**2/(kilo*meter)**2).simplify() == 4
def test_quantity_simplify():
from sympy.physics.units.util import quantity_simplify
from sympy.physics.units import kilo, foot
from sympy.core.symbol import symbols
x, y = symbols('x y')
assert quantity_simplify(x*(8*kilo*newton*meter + y)) == x*(8000*meter*newton + y)
assert quantity_simplify(foot*inch*(foot + inch)) == foot**2*(foot + foot/12)/12
assert quantity_simplify(foot*inch*(foot*foot + inch*(foot + inch))) == foot**2*(foot**2 + foot/12*(foot + foot/12))/12
assert quantity_simplify(2**(foot/inch*kilo/1000)*inch) == 4096*foot/12
assert quantity_simplify(foot**2*inch + inch**2*foot) == 13*foot**3/144
def test_check_dimensions():
x = symbols('x')
assert check_dimensions(inch + x) == inch + x
assert check_dimensions(length + x) == length + x
# after subs we get 2*length; check will clear the constant
assert check_dimensions((length + x).subs(x, length)) == length
raises(ValueError, lambda: check_dimensions(inch + 1))
raises(ValueError, lambda: check_dimensions(length + 1))
raises(ValueError, lambda: check_dimensions(length + time))
raises(ValueError, lambda: check_dimensions(meter + second))
raises(ValueError, lambda: check_dimensions(2 * meter + second))
raises(ValueError, lambda: check_dimensions(2 * meter + 3 * second))
raises(ValueError, lambda: check_dimensions(1 / second + 1 / meter))
raises(ValueError, lambda: check_dimensions(2 * meter*(mile + centimeter) + km))
|
d97b8462faad89ceb924707a9ca1a9babe11daef71cb6b211519d7b88df222ee
|
from sympy.utilities.pytest import warns_deprecated_sympy
from sympy import Matrix, eye, symbols
from sympy.physics.units.dimensions import (
Dimension, DimensionSystem, action, charge, current, length, mass, time,
velocity)
from sympy.utilities.pytest import raises
def test_call():
mksa = DimensionSystem((length, time, mass, current), (action,))
with warns_deprecated_sympy():
assert mksa(action) == mksa.print_dim_base(action)
def test_extend():
ms = DimensionSystem((length, time), (velocity,))
mks = ms.extend((mass,), (action,))
res = DimensionSystem((length, time, mass), (velocity, action))
assert mks.base_dims == res.base_dims
assert mks.derived_dims == res.derived_dims
def test_sort_dims():
with warns_deprecated_sympy():
assert (DimensionSystem.sort_dims((length, velocity, time))
== (length, time, velocity))
def test_list_dims():
dimsys = DimensionSystem((length, time, mass))
assert dimsys.list_can_dims == ("length", "mass", "time")
def test_dim_can_vector():
dimsys = DimensionSystem((length, mass, time), (velocity, action))
assert dimsys.dim_can_vector(length) == Matrix([1, 0, 0])
assert dimsys.dim_can_vector(velocity) == Matrix([1, 0, -1])
dimsys = DimensionSystem((length, velocity, action), (mass, time))
assert dimsys.dim_can_vector(length) == Matrix([1, 0, 0])
assert dimsys.dim_can_vector(velocity) == Matrix([1, 0, -1])
def test_dim_vector():
dimsys = DimensionSystem(
(length, mass, time),
(velocity, action),
{velocity: {length: 1, time: -1},
action: {mass: 1, length: 2, time: -1}})
assert dimsys.dim_vector(length) == Matrix([1, 0, 0])
assert dimsys.dim_vector(velocity) == Matrix([1, 0, -1])
dimsys = DimensionSystem((length, velocity, action), (mass, time))
assert dimsys.dim_vector(length) == Matrix([0, 1, 0])
assert dimsys.dim_vector(velocity) == Matrix([0, 0, 1])
assert dimsys.dim_vector(time) == Matrix([0, 1, -1])
def test_inv_can_transf_matrix():
dimsys = DimensionSystem((length, mass, time))
assert dimsys.inv_can_transf_matrix == eye(3)
dimsys = DimensionSystem((length, velocity, action))
assert dimsys.inv_can_transf_matrix == Matrix([[1, 2, 1], [0, 1, 0], [-1, -1, 0]])
def test_can_transf_matrix():
dimsys = DimensionSystem((length, mass, time))
assert dimsys.can_transf_matrix == eye(3)
dimsys = DimensionSystem((length, velocity, action))
assert dimsys.can_transf_matrix == Matrix([[0, 1, 0], [1, -1, 1], [0, -1, -1]])
def test_is_consistent():
assert DimensionSystem((length, time)).is_consistent is True
#assert DimensionSystem((length, time, velocity)).is_consistent is False
def test_print_dim_base():
mksa = DimensionSystem(
(length, time, mass, current),
(action,),
{action: {mass: 1, length: 2, time: -1}})
L, M, T = symbols("L M T")
assert mksa.print_dim_base(action) == L**2*M/T
def test_dim():
dimsys = DimensionSystem(
(length, mass, time),
(velocity, action),
{velocity: {length: 1, time: -1},
action: {mass: 1, length: 2, time: -1}}
)
assert dimsys.dim == 3
|
df3af28f64896c41ea25b461a5d1561de1ea16ba8ca9dd4a7a99f8e3940a236f
|
from sympy import S, Integral, sin, cos, pi, sqrt, symbols
from sympy.physics.vector import Dyadic, Point, ReferenceFrame, Vector
from sympy.physics.vector.functions import (cross, dot, express,
time_derivative,
kinematic_equations, outer,
partial_velocity,
get_motion_params, dynamicsymbols)
from sympy.utilities.pytest import raises
Vector.simp = True
q1, q2, q3, q4, q5 = symbols('q1 q2 q3 q4 q5')
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])
def test_dot():
assert dot(A.x, A.x) == 1
assert dot(A.x, A.y) == 0
assert dot(A.x, A.z) == 0
assert dot(A.y, A.x) == 0
assert dot(A.y, A.y) == 1
assert dot(A.y, A.z) == 0
assert dot(A.z, A.x) == 0
assert dot(A.z, A.y) == 0
assert dot(A.z, A.z) == 1
def test_dot_different_frames():
assert dot(N.x, A.x) == cos(q1)
assert dot(N.x, A.y) == -sin(q1)
assert dot(N.x, A.z) == 0
assert dot(N.y, A.x) == sin(q1)
assert dot(N.y, A.y) == cos(q1)
assert dot(N.y, A.z) == 0
assert dot(N.z, A.x) == 0
assert dot(N.z, A.y) == 0
assert dot(N.z, A.z) == 1
assert dot(N.x, A.x + A.y) == sqrt(2)*cos(q1 + pi/4) == dot(A.x + A.y, N.x)
assert dot(A.x, C.x) == cos(q3)
assert dot(A.x, C.y) == 0
assert dot(A.x, C.z) == sin(q3)
assert dot(A.y, C.x) == sin(q2)*sin(q3)
assert dot(A.y, C.y) == cos(q2)
assert dot(A.y, C.z) == -sin(q2)*cos(q3)
assert dot(A.z, C.x) == -cos(q2)*sin(q3)
assert dot(A.z, C.y) == sin(q2)
assert dot(A.z, C.z) == cos(q2)*cos(q3)
def test_cross():
assert cross(A.x, A.x) == 0
assert cross(A.x, A.y) == A.z
assert cross(A.x, A.z) == -A.y
assert cross(A.y, A.x) == -A.z
assert cross(A.y, A.y) == 0
assert cross(A.y, A.z) == A.x
assert cross(A.z, A.x) == A.y
assert cross(A.z, A.y) == -A.x
assert cross(A.z, A.z) == 0
def test_cross_different_frames():
assert cross(N.x, A.x) == sin(q1)*A.z
assert cross(N.x, A.y) == cos(q1)*A.z
assert cross(N.x, A.z) == -sin(q1)*A.x - cos(q1)*A.y
assert cross(N.y, A.x) == -cos(q1)*A.z
assert cross(N.y, A.y) == sin(q1)*A.z
assert cross(N.y, A.z) == cos(q1)*A.x - sin(q1)*A.y
assert cross(N.z, A.x) == A.y
assert cross(N.z, A.y) == -A.x
assert cross(N.z, A.z) == 0
assert cross(N.x, A.x) == sin(q1)*A.z
assert cross(N.x, A.y) == cos(q1)*A.z
assert cross(N.x, A.x + A.y) == sin(q1)*A.z + cos(q1)*A.z
assert cross(A.x + A.y, N.x) == -sin(q1)*A.z - cos(q1)*A.z
assert cross(A.x, C.x) == sin(q3)*C.y
assert cross(A.x, C.y) == -sin(q3)*C.x + cos(q3)*C.z
assert cross(A.x, C.z) == -cos(q3)*C.y
assert cross(C.x, A.x) == -sin(q3)*C.y
assert cross(C.y, A.x) == sin(q3)*C.x - cos(q3)*C.z
assert cross(C.z, A.x) == cos(q3)*C.y
def test_operator_match():
"""Test that the output of dot, cross, outer functions match
operator behavior.
"""
A = ReferenceFrame('A')
v = A.x + A.y
d = v | v
zerov = Vector(0)
zerod = Dyadic(0)
# dot products
assert d & d == dot(d, d)
assert d & zerod == dot(d, zerod)
assert zerod & d == dot(zerod, d)
assert d & v == dot(d, v)
assert v & d == dot(v, d)
assert d & zerov == dot(d, zerov)
assert zerov & d == dot(zerov, d)
raises(TypeError, lambda: dot(d, S(0)))
raises(TypeError, lambda: dot(S(0), d))
raises(TypeError, lambda: dot(d, 0))
raises(TypeError, lambda: dot(0, d))
assert v & v == dot(v, v)
assert v & zerov == dot(v, zerov)
assert zerov & v == dot(zerov, v)
raises(TypeError, lambda: dot(v, S(0)))
raises(TypeError, lambda: dot(S(0), v))
raises(TypeError, lambda: dot(v, 0))
raises(TypeError, lambda: dot(0, v))
# cross products
raises(TypeError, lambda: cross(d, d))
raises(TypeError, lambda: cross(d, zerod))
raises(TypeError, lambda: cross(zerod, d))
assert d ^ v == cross(d, v)
assert v ^ d == cross(v, d)
assert d ^ zerov == cross(d, zerov)
assert zerov ^ d == cross(zerov, d)
assert zerov ^ d == cross(zerov, d)
raises(TypeError, lambda: cross(d, S(0)))
raises(TypeError, lambda: cross(S(0), d))
raises(TypeError, lambda: cross(d, 0))
raises(TypeError, lambda: cross(0, d))
assert v ^ v == cross(v, v)
assert v ^ zerov == cross(v, zerov)
assert zerov ^ v == cross(zerov, v)
raises(TypeError, lambda: cross(v, S(0)))
raises(TypeError, lambda: cross(S(0), v))
raises(TypeError, lambda: cross(v, 0))
raises(TypeError, lambda: cross(0, v))
# outer products
raises(TypeError, lambda: outer(d, d))
raises(TypeError, lambda: outer(d, zerod))
raises(TypeError, lambda: outer(zerod, d))
raises(TypeError, lambda: outer(d, v))
raises(TypeError, lambda: outer(v, d))
raises(TypeError, lambda: outer(d, zerov))
raises(TypeError, lambda: outer(zerov, d))
raises(TypeError, lambda: outer(zerov, d))
raises(TypeError, lambda: outer(d, S(0)))
raises(TypeError, lambda: outer(S(0), d))
raises(TypeError, lambda: outer(d, 0))
raises(TypeError, lambda: outer(0, d))
assert v | v == outer(v, v)
assert v | zerov == outer(v, zerov)
assert zerov | v == outer(zerov, v)
raises(TypeError, lambda: outer(v, S(0)))
raises(TypeError, lambda: outer(S(0), v))
raises(TypeError, lambda: outer(v, 0))
raises(TypeError, lambda: outer(0, v))
def test_express():
assert express(Vector(0), N) == Vector(0)
assert express(S(0), N) == S(0)
assert express(A.x, C) == cos(q3)*C.x + sin(q3)*C.z
assert express(A.y, C) == sin(q2)*sin(q3)*C.x + cos(q2)*C.y - \
sin(q2)*cos(q3)*C.z
assert express(A.z, C) == -sin(q3)*cos(q2)*C.x + sin(q2)*C.y + \
cos(q2)*cos(q3)*C.z
assert express(A.x, N) == cos(q1)*N.x + sin(q1)*N.y
assert express(A.y, N) == -sin(q1)*N.x + cos(q1)*N.y
assert express(A.z, N) == N.z
assert express(A.x, A) == A.x
assert express(A.y, A) == A.y
assert express(A.z, A) == A.z
assert express(A.x, B) == B.x
assert express(A.y, B) == cos(q2)*B.y - sin(q2)*B.z
assert express(A.z, B) == sin(q2)*B.y + cos(q2)*B.z
assert express(A.x, C) == cos(q3)*C.x + sin(q3)*C.z
assert express(A.y, C) == sin(q2)*sin(q3)*C.x + cos(q2)*C.y - \
sin(q2)*cos(q3)*C.z
assert express(A.z, C) == -sin(q3)*cos(q2)*C.x + sin(q2)*C.y + \
cos(q2)*cos(q3)*C.z
# Check to make sure UnitVectors get converted properly
assert express(N.x, N) == N.x
assert express(N.y, N) == N.y
assert express(N.z, N) == N.z
assert express(N.x, A) == (cos(q1)*A.x - sin(q1)*A.y)
assert express(N.y, A) == (sin(q1)*A.x + cos(q1)*A.y)
assert express(N.z, A) == A.z
assert express(N.x, B) == (cos(q1)*B.x - sin(q1)*cos(q2)*B.y +
sin(q1)*sin(q2)*B.z)
assert express(N.y, B) == (sin(q1)*B.x + cos(q1)*cos(q2)*B.y -
sin(q2)*cos(q1)*B.z)
assert express(N.z, B) == (sin(q2)*B.y + cos(q2)*B.z)
assert express(N.x, C) == (
(cos(q1)*cos(q3) - sin(q1)*sin(q2)*sin(q3))*C.x -
sin(q1)*cos(q2)*C.y +
(sin(q3)*cos(q1) + sin(q1)*sin(q2)*cos(q3))*C.z)
assert express(N.y, C) == (
(sin(q1)*cos(q3) + sin(q2)*sin(q3)*cos(q1))*C.x +
cos(q1)*cos(q2)*C.y +
(sin(q1)*sin(q3) - sin(q2)*cos(q1)*cos(q3))*C.z)
assert express(N.z, C) == (-sin(q3)*cos(q2)*C.x + sin(q2)*C.y +
cos(q2)*cos(q3)*C.z)
assert express(A.x, N) == (cos(q1)*N.x + sin(q1)*N.y)
assert express(A.y, N) == (-sin(q1)*N.x + cos(q1)*N.y)
assert express(A.z, N) == N.z
assert express(A.x, A) == A.x
assert express(A.y, A) == A.y
assert express(A.z, A) == A.z
assert express(A.x, B) == B.x
assert express(A.y, B) == (cos(q2)*B.y - sin(q2)*B.z)
assert express(A.z, B) == (sin(q2)*B.y + cos(q2)*B.z)
assert express(A.x, C) == (cos(q3)*C.x + sin(q3)*C.z)
assert express(A.y, C) == (sin(q2)*sin(q3)*C.x + cos(q2)*C.y -
sin(q2)*cos(q3)*C.z)
assert express(A.z, C) == (-sin(q3)*cos(q2)*C.x + sin(q2)*C.y +
cos(q2)*cos(q3)*C.z)
assert express(B.x, N) == (cos(q1)*N.x + sin(q1)*N.y)
assert express(B.y, N) == (-sin(q1)*cos(q2)*N.x +
cos(q1)*cos(q2)*N.y + sin(q2)*N.z)
assert express(B.z, N) == (sin(q1)*sin(q2)*N.x -
sin(q2)*cos(q1)*N.y + cos(q2)*N.z)
assert express(B.x, A) == A.x
assert express(B.y, A) == (cos(q2)*A.y + sin(q2)*A.z)
assert express(B.z, A) == (-sin(q2)*A.y + cos(q2)*A.z)
assert express(B.x, B) == B.x
assert express(B.y, B) == B.y
assert express(B.z, B) == B.z
assert express(B.x, C) == (cos(q3)*C.x + sin(q3)*C.z)
assert express(B.y, C) == C.y
assert express(B.z, C) == (-sin(q3)*C.x + cos(q3)*C.z)
assert express(C.x, N) == (
(cos(q1)*cos(q3) - sin(q1)*sin(q2)*sin(q3))*N.x +
(sin(q1)*cos(q3) + sin(q2)*sin(q3)*cos(q1))*N.y -
sin(q3)*cos(q2)*N.z)
assert express(C.y, N) == (
-sin(q1)*cos(q2)*N.x + cos(q1)*cos(q2)*N.y + sin(q2)*N.z)
assert express(C.z, N) == (
(sin(q3)*cos(q1) + sin(q1)*sin(q2)*cos(q3))*N.x +
(sin(q1)*sin(q3) - sin(q2)*cos(q1)*cos(q3))*N.y +
cos(q2)*cos(q3)*N.z)
assert express(C.x, A) == (cos(q3)*A.x + sin(q2)*sin(q3)*A.y -
sin(q3)*cos(q2)*A.z)
assert express(C.y, A) == (cos(q2)*A.y + sin(q2)*A.z)
assert express(C.z, A) == (sin(q3)*A.x - sin(q2)*cos(q3)*A.y +
cos(q2)*cos(q3)*A.z)
assert express(C.x, B) == (cos(q3)*B.x - sin(q3)*B.z)
assert express(C.y, B) == B.y
assert express(C.z, B) == (sin(q3)*B.x + cos(q3)*B.z)
assert express(C.x, C) == C.x
assert express(C.y, C) == C.y
assert express(C.z, C) == C.z == (C.z)
# Check to make sure Vectors get converted back to UnitVectors
assert N.x == express((cos(q1)*A.x - sin(q1)*A.y), N)
assert N.y == express((sin(q1)*A.x + cos(q1)*A.y), N)
assert N.x == express((cos(q1)*B.x - sin(q1)*cos(q2)*B.y +
sin(q1)*sin(q2)*B.z), N)
assert N.y == express((sin(q1)*B.x + cos(q1)*cos(q2)*B.y -
sin(q2)*cos(q1)*B.z), N)
assert N.z == express((sin(q2)*B.y + cos(q2)*B.z), N)
"""
These don't really test our code, they instead test the auto simplification
(or lack thereof) of SymPy.
assert N.x == express((
(cos(q1)*cos(q3)-sin(q1)*sin(q2)*sin(q3))*C.x -
sin(q1)*cos(q2)*C.y +
(sin(q3)*cos(q1)+sin(q1)*sin(q2)*cos(q3))*C.z), N)
assert N.y == express((
(sin(q1)*cos(q3) + sin(q2)*sin(q3)*cos(q1))*C.x +
cos(q1)*cos(q2)*C.y +
(sin(q1)*sin(q3) - sin(q2)*cos(q1)*cos(q3))*C.z), N)
assert N.z == express((-sin(q3)*cos(q2)*C.x + sin(q2)*C.y +
cos(q2)*cos(q3)*C.z), N)
"""
assert A.x == express((cos(q1)*N.x + sin(q1)*N.y), A)
assert A.y == express((-sin(q1)*N.x + cos(q1)*N.y), A)
assert A.y == express((cos(q2)*B.y - sin(q2)*B.z), A)
assert A.z == express((sin(q2)*B.y + cos(q2)*B.z), A)
assert A.x == express((cos(q3)*C.x + sin(q3)*C.z), A)
# Tripsimp messes up here too.
#print express((sin(q2)*sin(q3)*C.x + cos(q2)*C.y -
# sin(q2)*cos(q3)*C.z), A)
assert A.y == express((sin(q2)*sin(q3)*C.x + cos(q2)*C.y -
sin(q2)*cos(q3)*C.z), A)
assert A.z == express((-sin(q3)*cos(q2)*C.x + sin(q2)*C.y +
cos(q2)*cos(q3)*C.z), A)
assert B.x == express((cos(q1)*N.x + sin(q1)*N.y), B)
assert B.y == express((-sin(q1)*cos(q2)*N.x +
cos(q1)*cos(q2)*N.y + sin(q2)*N.z), B)
assert B.z == express((sin(q1)*sin(q2)*N.x -
sin(q2)*cos(q1)*N.y + cos(q2)*N.z), B)
assert B.y == express((cos(q2)*A.y + sin(q2)*A.z), B)
assert B.z == express((-sin(q2)*A.y + cos(q2)*A.z), B)
assert B.x == express((cos(q3)*C.x + sin(q3)*C.z), B)
assert B.z == express((-sin(q3)*C.x + cos(q3)*C.z), B)
"""
assert C.x == express((
(cos(q1)*cos(q3)-sin(q1)*sin(q2)*sin(q3))*N.x +
(sin(q1)*cos(q3)+sin(q2)*sin(q3)*cos(q1))*N.y -
sin(q3)*cos(q2)*N.z), C)
assert C.y == express((
-sin(q1)*cos(q2)*N.x + cos(q1)*cos(q2)*N.y + sin(q2)*N.z), C)
assert C.z == express((
(sin(q3)*cos(q1)+sin(q1)*sin(q2)*cos(q3))*N.x +
(sin(q1)*sin(q3)-sin(q2)*cos(q1)*cos(q3))*N.y +
cos(q2)*cos(q3)*N.z), C)
"""
assert C.x == express((cos(q3)*A.x + sin(q2)*sin(q3)*A.y -
sin(q3)*cos(q2)*A.z), C)
assert C.y == express((cos(q2)*A.y + sin(q2)*A.z), C)
assert C.z == express((sin(q3)*A.x - sin(q2)*cos(q3)*A.y +
cos(q2)*cos(q3)*A.z), C)
assert C.x == express((cos(q3)*B.x - sin(q3)*B.z), C)
assert C.z == express((sin(q3)*B.x + cos(q3)*B.z), C)
def test_time_derivative():
#The use of time_derivative for calculations pertaining to scalar
#fields has been tested in test_coordinate_vars in test_essential.py
A = ReferenceFrame('A')
q = dynamicsymbols('q')
qd = dynamicsymbols('q', 1)
B = A.orientnew('B', 'Axis', [q, A.z])
d = A.x | A.x
assert time_derivative(d, B) == (-qd) * (A.y | A.x) + \
(-qd) * (A.x | A.y)
d1 = A.x | B.y
assert time_derivative(d1, A) == - qd*(A.x|B.x)
assert time_derivative(d1, B) == - qd*(A.y|B.y)
d2 = A.x | B.x
assert time_derivative(d2, A) == qd*(A.x|B.y)
assert time_derivative(d2, B) == - qd*(A.y|B.x)
d3 = A.x | B.z
assert time_derivative(d3, A) == 0
assert time_derivative(d3, B) == - qd*(A.y|B.z)
q1, q2, q3, q4 = dynamicsymbols('q1 q2 q3 q4')
q1d, q2d, q3d, q4d = dynamicsymbols('q1 q2 q3 q4', 1)
q1dd, q2dd, q3dd, q4dd = dynamicsymbols('q1 q2 q3 q4', 2)
C = B.orientnew('C', 'Axis', [q4, B.x])
v1 = q1 * A.z
v2 = q2*A.x + q3*B.y
v3 = q1*A.x + q2*A.y + q3*A.z
assert time_derivative(B.x, C) == 0
assert time_derivative(B.y, C) == - q4d*B.z
assert time_derivative(B.z, C) == q4d*B.y
assert time_derivative(v1, B) == q1d*A.z
assert time_derivative(v1, C) == - q1*sin(q)*q4d*A.x + \
q1*cos(q)*q4d*A.y + q1d*A.z
assert time_derivative(v2, A) == q2d*A.x - q3*qd*B.x + q3d*B.y
assert time_derivative(v2, C) == q2d*A.x - q2*qd*A.y + \
q2*sin(q)*q4d*A.z + q3d*B.y - q3*q4d*B.z
assert time_derivative(v3, B) == (q2*qd + q1d)*A.x + \
(-q1*qd + q2d)*A.y + q3d*A.z
assert time_derivative(d, C) == - qd*(A.y|A.x) + \
sin(q)*q4d*(A.z|A.x) - qd*(A.x|A.y) + sin(q)*q4d*(A.x|A.z)
raises(ValueError, lambda: time_derivative(B.x, C, order=0.5))
raises(ValueError, lambda: time_derivative(B.x, C, order=-1))
def test_get_motion_methods():
#Initialization
t = dynamicsymbols._t
s1, s2, s3 = symbols('s1 s2 s3')
S1, S2, S3 = symbols('S1 S2 S3')
S4, S5, S6 = symbols('S4 S5 S6')
t1, t2 = symbols('t1 t2')
a, b, c = dynamicsymbols('a b c')
ad, bd, cd = dynamicsymbols('a b c', 1)
a2d, b2d, c2d = dynamicsymbols('a b c', 2)
v0 = S1*N.x + S2*N.y + S3*N.z
v01 = S4*N.x + S5*N.y + S6*N.z
v1 = s1*N.x + s2*N.y + s3*N.z
v2 = a*N.x + b*N.y + c*N.z
v2d = ad*N.x + bd*N.y + cd*N.z
v2dd = a2d*N.x + b2d*N.y + c2d*N.z
#Test position parameter
assert get_motion_params(frame = N) == (0, 0, 0)
assert get_motion_params(N, position=v1) == (0, 0, v1)
assert get_motion_params(N, position=v2) == (v2dd, v2d, v2)
#Test velocity parameter
assert get_motion_params(N, velocity=v1) == (0, v1, v1 * t)
assert get_motion_params(N, velocity=v1, position=v0, timevalue1=t1) == \
(0, v1, v0 + v1*(t - t1))
answer = get_motion_params(N, velocity=v1, position=v2, timevalue1=t1)
answer_expected = (0, v1, v1*t - v1*t1 + v2.subs(t, t1))
assert answer == answer_expected
answer = get_motion_params(N, velocity=v2, position=v0, timevalue1=t1)
integral_vector = Integral(a, (t, t1, t))*N.x + Integral(b, (t, t1, t))*N.y \
+ Integral(c, (t, t1, t))*N.z
answer_expected = (v2d, v2, v0 + integral_vector)
assert answer == answer_expected
#Test acceleration parameter
assert get_motion_params(N, acceleration=v1) == \
(v1, v1 * t, v1 * t**2/2)
assert get_motion_params(N, acceleration=v1, velocity=v0,
position=v2, timevalue1=t1, timevalue2=t2) == \
(v1, (v0 + v1*t - v1*t2),
-v0*t1 + v1*t**2/2 + v1*t2*t1 - \
v1*t1**2/2 + t*(v0 - v1*t2) + \
v2.subs(t, t1))
assert get_motion_params(N, acceleration=v1, velocity=v0,
position=v01, timevalue1=t1, timevalue2=t2) == \
(v1, v0 + v1*t - v1*t2,
-v0*t1 + v01 + v1*t**2/2 + \
v1*t2*t1 - v1*t1**2/2 + \
t*(v0 - v1*t2))
answer = get_motion_params(N, acceleration=a*N.x, velocity=S1*N.x,
position=S2*N.x, timevalue1=t1, timevalue2=t2)
i1 = Integral(a, (t, t2, t))
answer_expected = (a*N.x, (S1 + i1)*N.x, \
(S2 + Integral(S1 + i1, (t, t1, t)))*N.x)
assert answer == answer_expected
def test_kin_eqs():
q0, q1, q2, q3 = dynamicsymbols('q0 q1 q2 q3')
q0d, q1d, q2d, q3d = dynamicsymbols('q0 q1 q2 q3', 1)
u1, u2, u3 = dynamicsymbols('u1 u2 u3')
ke = kinematic_equations([u1,u2,u3], [q1,q2,q3], 'body', 313)
assert ke == kinematic_equations([u1,u2,u3], [q1,q2,q3], 'body', '313')
kds = kinematic_equations([u1, u2, u3], [q0, q1, q2, q3], 'quaternion')
assert kds == [-0.5 * q0 * u1 - 0.5 * q2 * u3 + 0.5 * q3 * u2 + q1d,
-0.5 * q0 * u2 + 0.5 * q1 * u3 - 0.5 * q3 * u1 + q2d,
-0.5 * q0 * u3 - 0.5 * q1 * u2 + 0.5 * q2 * u1 + q3d,
0.5 * q1 * u1 + 0.5 * q2 * u2 + 0.5 * q3 * u3 + q0d]
raises(ValueError, lambda: kinematic_equations([u1, u2, u3], [q0, q1, q2], 'quaternion'))
raises(ValueError, lambda: kinematic_equations([u1, u2, u3], [q0, q1, q2, q3], 'quaternion', '123'))
raises(ValueError, lambda: kinematic_equations([u1, u2, u3], [q0, q1, q2, q3], 'foo'))
raises(TypeError, lambda: kinematic_equations(u1, [q0, q1, q2, q3], 'quaternion'))
raises(TypeError, lambda: kinematic_equations([u1], [q0, q1, q2, q3], 'quaternion'))
raises(TypeError, lambda: kinematic_equations([u1, u2, u3], q0, 'quaternion'))
raises(ValueError, lambda: kinematic_equations([u1, u2, u3], [q0, q1, q2, q3], 'body'))
raises(ValueError, lambda: kinematic_equations([u1, u2, u3], [q0, q1, q2, q3], 'space'))
raises(ValueError, lambda: kinematic_equations([u1, u2, u3], [q0, q1, q2], 'body', '222'))
assert kinematic_equations([0, 0, 0], [q0, q1, q2], 'space') == [S.Zero, S.Zero, S.Zero]
def test_partial_velocity():
q1, q2, q3, u1, u2, u3 = dynamicsymbols('q1 q2 q3 u1 u2 u3')
u4, u5 = dynamicsymbols('u4, u5')
r = symbols('r')
N = ReferenceFrame('N')
Y = N.orientnew('Y', 'Axis', [q1, N.z])
L = Y.orientnew('L', 'Axis', [q2, Y.x])
R = L.orientnew('R', 'Axis', [q3, L.y])
R.set_ang_vel(N, u1 * L.x + u2 * L.y + u3 * L.z)
C = Point('C')
C.set_vel(N, u4 * L.x + u5 * (Y.z ^ L.x))
Dmc = C.locatenew('Dmc', r * L.z)
Dmc.v2pt_theory(C, N, R)
vel_list = [Dmc.vel(N), C.vel(N), R.ang_vel_in(N)]
u_list = [u1, u2, u3, u4, u5]
assert (partial_velocity(vel_list, u_list, N) ==
[[- r*L.y, r*L.x, 0, L.x, cos(q2)*L.y - sin(q2)*L.z],
[0, 0, 0, L.x, cos(q2)*L.y - sin(q2)*L.z],
[L.x, L.y, L.z, 0, 0]])
# Make sure that partial velocities can be computed regardless if the
# orientation between frames is defined or not.
A = ReferenceFrame('A')
B = ReferenceFrame('B')
v = u4 * A.x + u5 * B.y
assert partial_velocity((v, ), (u4, u5), A) == [[A.x, B.y]]
raises(TypeError, lambda: partial_velocity(Dmc.vel(N), u_list, N))
raises(TypeError, lambda: partial_velocity(vel_list, u1, N))
|
24ceec29da18173a677f5c93886bcd4e2b82fa36938f7f41da1cd606d9b7af5b
|
from sympy import symbols, pi, sin, cos, ImmutableMatrix as Matrix
from sympy.physics.vector import ReferenceFrame, Vector, dynamicsymbols, dot
from sympy.abc import x, y, z
from sympy.utilities.pytest import raises
Vector.simp = True
A = ReferenceFrame('A')
def test_Vector():
assert A.x != A.y
assert A.y != A.z
assert A.z != A.x
assert A.x + 0 == A.x
v1 = x*A.x + y*A.y + z*A.z
v2 = x**2*A.x + y**2*A.y + z**2*A.z
v3 = v1 + v2
v4 = v1 - v2
assert isinstance(v1, Vector)
assert dot(v1, A.x) == x
assert dot(v1, A.y) == y
assert dot(v1, A.z) == z
assert isinstance(v2, Vector)
assert dot(v2, A.x) == x**2
assert dot(v2, A.y) == y**2
assert dot(v2, A.z) == z**2
assert isinstance(v3, Vector)
# We probably shouldn't be using simplify in dot...
assert dot(v3, A.x) == x**2 + x
assert dot(v3, A.y) == y**2 + y
assert dot(v3, A.z) == z**2 + z
assert isinstance(v4, Vector)
# We probably shouldn't be using simplify in dot...
assert dot(v4, A.x) == x - x**2
assert dot(v4, A.y) == y - y**2
assert dot(v4, A.z) == z - z**2
assert v1.to_matrix(A) == Matrix([[x], [y], [z]])
q = symbols('q')
B = A.orientnew('B', 'Axis', (q, A.x))
assert v1.to_matrix(B) == Matrix([[x],
[ y * cos(q) + z * sin(q)],
[-y * sin(q) + z * cos(q)]])
#Test the separate method
B = ReferenceFrame('B')
v5 = x*A.x + y*A.y + z*B.z
assert Vector(0).separate() == {}
assert v1.separate() == {A: v1}
assert v5.separate() == {A: x*A.x + y*A.y, B: z*B.z}
#Test the free_symbols property
v6 = x*A.x + y*A.y + z*A.z
assert v6.free_symbols(A) == {x,y,z}
raises(TypeError, lambda: v3.applyfunc(v1))
def test_Vector_diffs():
q1, q2, q3, q4 = dynamicsymbols('q1 q2 q3 q4')
q1d, q2d, q3d, q4d = dynamicsymbols('q1 q2 q3 q4', 1)
q1dd, q2dd, q3dd, q4dd = dynamicsymbols('q1 q2 q3 q4', 2)
N = ReferenceFrame('N')
A = N.orientnew('A', 'Axis', [q3, N.z])
B = A.orientnew('B', 'Axis', [q2, A.x])
v1 = q2 * A.x + q3 * N.y
v2 = q3 * B.x + v1
v3 = v1.dt(B)
v4 = v2.dt(B)
v5 = q1*A.x + q2*A.y + q3*A.z
assert v1.dt(N) == q2d * A.x + q2 * q3d * A.y + q3d * N.y
assert v1.dt(A) == q2d * A.x + q3 * q3d * N.x + q3d * N.y
assert v1.dt(B) == (q2d * A.x + q3 * q3d * N.x + q3d *\
N.y - q3 * cos(q3) * q2d * N.z)
assert v2.dt(N) == (q2d * A.x + (q2 + q3) * q3d * A.y + q3d * B.x + q3d *
N.y)
assert v2.dt(A) == q2d * A.x + q3d * B.x + q3 * q3d * N.x + q3d * N.y
assert v2.dt(B) == (q2d * A.x + q3d * B.x + q3 * q3d * N.x + q3d * N.y -
q3 * cos(q3) * q2d * N.z)
assert v3.dt(N) == (q2dd * A.x + q2d * q3d * A.y + (q3d**2 + q3 * q3dd) *
N.x + q3dd * N.y + (q3 * sin(q3) * q2d * q3d -
cos(q3) * q2d * q3d - q3 * cos(q3) * q2dd) * N.z)
assert v3.dt(A) == (q2dd * A.x + (2 * q3d**2 + q3 * q3dd) * N.x + (q3dd -
q3 * q3d**2) * N.y + (q3 * sin(q3) * q2d * q3d -
cos(q3) * q2d * q3d - q3 * cos(q3) * q2dd) * N.z)
assert v3.dt(B) == (q2dd * A.x - q3 * cos(q3) * q2d**2 * A.y + (2 *
q3d**2 + q3 * q3dd) * N.x + (q3dd - q3 * q3d**2) *
N.y + (2 * q3 * sin(q3) * q2d * q3d - 2 * cos(q3) *
q2d * q3d - q3 * cos(q3) * q2dd) * N.z)
assert v4.dt(N) == (q2dd * A.x + q3d * (q2d + q3d) * A.y + q3dd * B.x +
(q3d**2 + q3 * q3dd) * N.x + q3dd * N.y + (q3 *
sin(q3) * q2d * q3d - cos(q3) * q2d * q3d - q3 *
cos(q3) * q2dd) * N.z)
assert v4.dt(A) == (q2dd * A.x + q3dd * B.x + (2 * q3d**2 + q3 * q3dd) *
N.x + (q3dd - q3 * q3d**2) * N.y + (q3 * sin(q3) *
q2d * q3d - cos(q3) * q2d * q3d - q3 * cos(q3) *
q2dd) * N.z)
assert v4.dt(B) == (q2dd * A.x - q3 * cos(q3) * q2d**2 * A.y + q3dd * B.x +
(2 * q3d**2 + q3 * q3dd) * N.x + (q3dd - q3 * q3d**2) *
N.y + (2 * q3 * sin(q3) * q2d * q3d - 2 * cos(q3) *
q2d * q3d - q3 * cos(q3) * q2dd) * N.z)
assert v5.dt(B) == q1d*A.x + (q3*q2d + q2d)*A.y + (-q2*q2d + q3d)*A.z
assert v5.dt(A) == q1d*A.x + q2d*A.y + q3d*A.z
assert v5.dt(N) == (-q2*q3d + q1d)*A.x + (q1*q3d + q2d)*A.y + q3d*A.z
assert v3.diff(q1d, N) == 0
assert v3.diff(q2d, N) == A.x - q3 * cos(q3) * N.z
assert v3.diff(q3d, N) == q3 * N.x + N.y
assert v3.diff(q1d, A) == 0
assert v3.diff(q2d, A) == A.x - q3 * cos(q3) * N.z
assert v3.diff(q3d, A) == q3 * N.x + N.y
assert v3.diff(q1d, B) == 0
assert v3.diff(q2d, B) == A.x - q3 * cos(q3) * N.z
assert v3.diff(q3d, B) == q3 * N.x + N.y
assert v4.diff(q1d, N) == 0
assert v4.diff(q2d, N) == A.x - q3 * cos(q3) * N.z
assert v4.diff(q3d, N) == B.x + q3 * N.x + N.y
assert v4.diff(q1d, A) == 0
assert v4.diff(q2d, A) == A.x - q3 * cos(q3) * N.z
assert v4.diff(q3d, A) == B.x + q3 * N.x + N.y
assert v4.diff(q1d, B) == 0
assert v4.diff(q2d, B) == A.x - q3 * cos(q3) * N.z
assert v4.diff(q3d, B) == B.x + q3 * N.x + N.y
def test_vector_var_in_dcm():
N = ReferenceFrame('N')
A = ReferenceFrame('A')
B = ReferenceFrame('B')
u1, u2, u3, u4 = dynamicsymbols('u1 u2 u3 u4')
v = u1 * u2 * A.x + u3 * N.y + u4**2 * N.z
assert v.diff(u1, N, var_in_dcm=False) == u2 * A.x
assert v.diff(u1, A, var_in_dcm=False) == u2 * A.x
assert v.diff(u3, N, var_in_dcm=False) == N.y
assert v.diff(u3, A, var_in_dcm=False) == N.y
assert v.diff(u3, B, var_in_dcm=False) == N.y
assert v.diff(u4, N, var_in_dcm=False) == 2 * u4 * N.z
raises(ValueError, lambda: v.diff(u1, N))
def test_vector_simplify():
x, y, z, k, n, m, w, f, s, A = symbols('x, y, z, k, n, m, w, f, s, A')
N = ReferenceFrame('N')
test1 = (1 / x + 1 / y) * N.x
assert (test1 & N.x) != (x + y) / (x * y)
test1 = test1.simplify()
assert (test1 & N.x) == (x + y) / (x * y)
test2 = (A**2 * s**4 / (4 * pi * k * m**3)) * N.x
test2 = test2.simplify()
assert (test2 & N.x) == (A**2 * s**4 / (4 * pi * k * m**3))
test3 = ((4 + 4 * x - 2 * (2 + 2 * x)) / (2 + 2 * x)) * N.x
test3 = test3.simplify()
assert (test3 & N.x) == 0
test4 = ((-4 * x * y**2 - 2 * y**3 - 2 * x**2 * y) / (x + y)**2) * N.x
test4 = test4.simplify()
assert (test4 & N.x) == -2 * y
|
ef0312ffba4c3f36f2f493576f23d9de9c078e274d25ca2a84f6b8a5a7882e50
|
from sympy import symbols, sin, cos, pi, zeros, eye, 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.utilities.pytest import raises
Vector.simp = True
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)]])
# This test has been added to test the _w_diff_dcm() function
# for which a test was previously not included.
# Also note that the _w_diff_dcm() function was changed as part of
# PR #14758 as it was observed to be giving incorrect results
# when compared with Autolev results.
def test_w_diff_dcm():
a = ReferenceFrame('a')
b = ReferenceFrame('b')
c11, c12, c13, c21, c22, c23, c31, c32, c33 = dynamicsymbols('c11 c12 c13 c21 c22 c23 c31 c32 c33')
c11d, c12d, c13d, c21d, c22d, c23d, c31d, c32d, c33d = dynamicsymbols('c11 c12 c13 c21 c22 c23 c31 c32 c33', 1)
b.orient(a, 'DCM', Matrix([c11,c12,c13,c21,c22,c23,c31,c32,c33]).reshape(3, 3))
b1a=(b.x).express(a)
b2a=(b.y).express(a)
b3a=(b.z).express(a)
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)))
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_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')
A = I.orientnew('A', 'space', u, 'XYZ')
def test_issue_11503():
A = ReferenceFrame("A")
B = 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(TypeError, 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))
|
ad3ba7f14b1c52d7d5aeeda67d7f3ef75018a15ad7e8babeb37b6c5ad8b8814e
|
from sympy import sin, cos, symbols, pi, ImmutableMatrix as Matrix
from sympy.physics.vector import ReferenceFrame, Vector, dynamicsymbols
from sympy.physics.vector.dyadic import _check_dyadic
from sympy.utilities.pytest import raises
Vector.simp = True
A = ReferenceFrame('A')
def test_dyadic():
d1 = A.x | A.x
d2 = A.y | A.y
d3 = A.x | A.y
assert d1 * 0 == 0
assert d1 != 0
assert d1 * 2 == 2 * A.x | A.x
assert d1 / 2. == 0.5 * d1
assert d1 & (0 * d1) == 0
assert d1 & d2 == 0
assert d1 & A.x == A.x
assert d1 ^ A.x == 0
assert d1 ^ A.y == A.x | A.z
assert d1 ^ A.z == - A.x | A.y
assert d2 ^ A.x == - A.y | A.z
assert A.x ^ d1 == 0
assert A.y ^ d1 == - A.z | A.x
assert A.z ^ d1 == A.y | A.x
assert A.x & d1 == A.x
assert A.y & d1 == 0
assert A.y & d2 == A.y
assert d1 & d3 == A.x | A.y
assert d3 & d1 == 0
assert d1.dt(A) == 0
q = dynamicsymbols('q')
qd = dynamicsymbols('q', 1)
B = A.orientnew('B', 'Axis', [q, A.z])
assert d1.express(B) == d1.express(B, B)
assert d1.express(B) == ((cos(q)**2) * (B.x | B.x) + (-sin(q) * cos(q)) *
(B.x | B.y) + (-sin(q) * cos(q)) * (B.y | B.x) + (sin(q)**2) *
(B.y | B.y))
assert d1.express(B, A) == (cos(q)) * (B.x | A.x) + (-sin(q)) * (B.y | A.x)
assert d1.express(A, B) == (cos(q)) * (A.x | B.x) + (-sin(q)) * (A.x | B.y)
assert d1.dt(B) == (-qd) * (A.y | A.x) + (-qd) * (A.x | A.y)
assert d1.to_matrix(A) == Matrix([[1, 0, 0], [0, 0, 0], [0, 0, 0]])
assert d1.to_matrix(A, B) == Matrix([[cos(q), -sin(q), 0],
[0, 0, 0],
[0, 0, 0]])
assert d3.to_matrix(A) == Matrix([[0, 1, 0], [0, 0, 0], [0, 0, 0]])
a, b, c, d, e, f = symbols('a, b, c, d, e, f')
v1 = a * A.x + b * A.y + c * A.z
v2 = d * A.x + e * A.y + f * A.z
d4 = v1.outer(v2)
assert d4.to_matrix(A) == Matrix([[a * d, a * e, a * f],
[b * d, b * e, b * f],
[c * d, c * e, c * f]])
d5 = v1.outer(v1)
C = A.orientnew('C', 'Axis', [q, A.x])
for expected, actual in zip(C.dcm(A) * d5.to_matrix(A) * C.dcm(A).T,
d5.to_matrix(C)):
assert (expected - actual).simplify() == 0
raises(TypeError, lambda: d1.applyfunc(0))
def test_dyadic_simplify():
x, y, z, k, n, m, w, f, s, A = symbols('x, y, z, k, n, m, w, f, s, A')
N = ReferenceFrame('N')
dy = N.x | N.x
test1 = (1 / x + 1 / y) * dy
assert (N.x & test1 & N.x) != (x + y) / (x * y)
test1 = test1.simplify()
assert (N.x & test1 & N.x) == (x + y) / (x * y)
test2 = (A**2 * s**4 / (4 * pi * k * m**3)) * dy
test2 = test2.simplify()
assert (N.x & test2 & N.x) == (A**2 * s**4 / (4 * pi * k * m**3))
test3 = ((4 + 4 * x - 2 * (2 + 2 * x)) / (2 + 2 * x)) * dy
test3 = test3.simplify()
assert (N.x & test3 & N.x) == 0
test4 = ((-4 * x * y**2 - 2 * y**3 - 2 * x**2 * y) / (x + y)**2) * dy
test4 = test4.simplify()
assert (N.x & test4 & N.x) == -2 * y
def test_dyadic_subs():
N = ReferenceFrame('N')
s = symbols('s')
a = s*(N.x | N.x)
assert a.subs({s: 2}) == 2*(N.x | N.x)
def test_check_dyadic():
raises(TypeError, lambda: _check_dyadic(0))
|
03581e4daf70d98490b72fe1963df67e20a7b6170703fd2b1d884214321b2f07
|
# -*- coding: utf-8 -*-
from sympy import symbols, sin, asin, cos, sqrt, Function
from sympy.core.compatibility import u_decode as u
from sympy.physics.vector import ReferenceFrame, dynamicsymbols, Dyadic
from sympy.physics.vector.printing import (VectorLatexPrinter, vpprint,
vsprint, vsstrrepr)
a, b, c = symbols('a, b, c')
alpha, omega, beta = dynamicsymbols('alpha, omega, beta')
A = ReferenceFrame('A')
N = ReferenceFrame('N')
v = a ** 2 * N.x + b * N.y + c * sin(alpha) * N.z
w = alpha * N.x + sin(omega) * N.y + alpha * beta * N.z
ww = alpha * N.x + asin(omega) * N.y - alpha.diff() * beta * N.z
o = a/b * N.x + (c+b)/a * N.y + c**2/b * N.z
y = a ** 2 * (N.x | N.y) + b * (N.y | N.y) + c * sin(alpha) * (N.z | N.y)
x = alpha * (N.x | N.x) + sin(omega) * (N.y | N.z) + alpha * beta * (N.z | N.x)
xx = N.x | (-N.y - N.z)
xx2 = N.x | (N.y + N.z)
def ascii_vpretty(expr):
return vpprint(expr, use_unicode=False, wrap_line=False)
def unicode_vpretty(expr):
return vpprint(expr, use_unicode=True, wrap_line=False)
def test_latex_printer():
r = Function('r')('t')
assert VectorLatexPrinter().doprint(r ** 2) == "r^{2}"
r2 = Function('r^2')('t')
assert VectorLatexPrinter().doprint(r2.diff()) == r'\dot{r^{2}}'
ra = Function('r__a')('t')
assert VectorLatexPrinter().doprint(ra.diff().diff()) == r'\ddot{r^{a}}'
def test_vector_pretty_print():
# TODO : The unit vectors should print with subscripts but they just
# print as `n_x` instead of making `x` a subscript with unicode.
# TODO : The pretty print division does not print correctly here:
# w = alpha * N.x + sin(omega) * N.y + alpha / beta * N.z
expected = """\
2
a n_x + b n_y + c*sin(alpha) n_z\
"""
uexpected = u("""\
2
a n_x + b n_y + c⋅sin(α) n_z\
""")
assert ascii_vpretty(v) == expected
assert unicode_vpretty(v) == uexpected
expected = u('alpha n_x + sin(omega) n_y + alpha*beta n_z')
uexpected = u('α n_x + sin(ω) n_y + α⋅β n_z')
assert ascii_vpretty(w) == expected
assert unicode_vpretty(w) == uexpected
expected = """\
2
a b + c c
- n_x + ----- n_y + -- n_z
b a b\
"""
uexpected = u("""\
2
a b + c c
─ n_x + ───── n_y + ── n_z
b a b\
""")
assert ascii_vpretty(o) == expected
assert unicode_vpretty(o) == uexpected
def test_vector_latex():
a, b, c, d, omega = symbols('a, b, c, d, omega')
v = (a ** 2 + b / c) * A.x + sqrt(d) * A.y + cos(omega) * A.z
assert v._latex() == (r'(a^{2} + \frac{b}{c})\mathbf{\hat{a}_x} + '
r'\sqrt{d}\mathbf{\hat{a}_y} + '
r'\operatorname{cos}\left(\omega\right)'
r'\mathbf{\hat{a}_z}')
theta, omega, alpha, q = dynamicsymbols('theta, omega, alpha, q')
v = theta * A.x + omega * omega * A.y + (q * alpha) * A.z
assert v._latex() == (r'\theta\mathbf{\hat{a}_x} + '
r'\omega^{2}\mathbf{\hat{a}_y} + '
r'\alpha q\mathbf{\hat{a}_z}')
phi1, phi2, phi3 = dynamicsymbols('phi1, phi2, phi3')
theta1, theta2, theta3 = symbols('theta1, theta2, theta3')
v = (sin(theta1) * A.x +
cos(phi1) * cos(phi2) * A.y +
cos(theta1 + phi3) * A.z)
assert v._latex() == (r'\operatorname{sin}\left(\theta_{1}\right)'
r'\mathbf{\hat{a}_x} + \operatorname{cos}'
r'\left(\phi_{1}\right) \operatorname{cos}'
r'\left(\phi_{2}\right)\mathbf{\hat{a}_y} + '
r'\operatorname{cos}\left(\theta_{1} + '
r'\phi_{3}\right)\mathbf{\hat{a}_z}')
N = ReferenceFrame('N')
a, b, c, d, omega = symbols('a, b, c, d, omega')
v = (a ** 2 + b / c) * N.x + sqrt(d) * N.y + cos(omega) * N.z
expected = (r'(a^{2} + \frac{b}{c})\mathbf{\hat{n}_x} + '
r'\sqrt{d}\mathbf{\hat{n}_y} + '
r'\operatorname{cos}\left(\omega\right)'
r'\mathbf{\hat{n}_z}')
assert v._latex() == expected
lp = VectorLatexPrinter()
assert lp.doprint(v) == expected
# Try custom unit vectors.
N = ReferenceFrame('N', latexs=(r'\hat{i}', r'\hat{j}', r'\hat{k}'))
v = (a ** 2 + b / c) * N.x + sqrt(d) * N.y + cos(omega) * N.z
expected = (r'(a^{2} + \frac{b}{c})\hat{i} + '
r'\sqrt{d}\hat{j} + '
r'\operatorname{cos}\left(\omega\right)\hat{k}')
assert v._latex() == expected
expected = r'\alpha\mathbf{\hat{n}_x} + \operatorname{asin}\left(\omega' \
r'\right)\mathbf{\hat{n}_y} - \beta \dot{\alpha}\mathbf{\hat{n}_z}'
assert ww._latex() == expected
assert lp.doprint(ww) == expected
expected = r'- \mathbf{\hat{n}_x}\otimes \mathbf{\hat{n}_y} - ' \
r'\mathbf{\hat{n}_x}\otimes \mathbf{\hat{n}_z}'
assert xx._latex() == expected
assert lp.doprint(xx) == expected
expected = r'\mathbf{\hat{n}_x}\otimes \mathbf{\hat{n}_y} + ' \
r'\mathbf{\hat{n}_x}\otimes \mathbf{\hat{n}_z}'
assert xx2._latex() == expected
assert lp.doprint(xx2) == expected
def test_vector_latex_with_functions():
N = ReferenceFrame('N')
omega, alpha = dynamicsymbols('omega, alpha')
v = omega.diff() * N.x
assert v._latex() == r'\dot{\omega}\mathbf{\hat{n}_x}'
v = omega.diff() ** alpha * N.x
assert v._latex() == (r'\dot{\omega}^{\alpha}'
r'\mathbf{\hat{n}_x}')
def test_dyadic_pretty_print():
expected = """\
2
a n_x|n_y + b n_y|n_y + c*sin(alpha) n_z|n_y\
"""
uexpected = u("""\
2
a n_x⊗n_y + b n_y⊗n_y + c⋅sin(α) n_z⊗n_y\
""")
assert ascii_vpretty(y) == expected
assert unicode_vpretty(y) == uexpected
expected = u('alpha n_x|n_x + sin(omega) n_y|n_z + alpha*beta n_z|n_x')
uexpected = u('α n_x⊗n_x + sin(ω) n_y⊗n_z + α⋅β n_z⊗n_x')
assert ascii_vpretty(x) == expected
assert unicode_vpretty(x) == uexpected
assert ascii_vpretty(Dyadic([])) == '0'
assert unicode_vpretty(Dyadic([])) == '0'
assert ascii_vpretty(xx) == '- n_x|n_y - n_x|n_z'
assert unicode_vpretty(xx) == u('- n_x⊗n_y - n_x⊗n_z')
assert ascii_vpretty(xx2) == 'n_x|n_y + n_x|n_z'
assert unicode_vpretty(xx2) == u('n_x⊗n_y + n_x⊗n_z')
def test_dyadic_latex():
expected = (r'a^{2}\mathbf{\hat{n}_x}\otimes \mathbf{\hat{n}_y} + '
r'b\mathbf{\hat{n}_y}\otimes \mathbf{\hat{n}_y} + '
r'c \operatorname{sin}\left(\alpha\right)'
r'\mathbf{\hat{n}_z}\otimes \mathbf{\hat{n}_y}')
assert y._latex() == expected
expected = (r'\alpha\mathbf{\hat{n}_x}\otimes \mathbf{\hat{n}_x} + '
r'\operatorname{sin}\left(\omega\right)\mathbf{\hat{n}_y}'
r'\otimes \mathbf{\hat{n}_z} + '
r'\alpha \beta\mathbf{\hat{n}_z}\otimes \mathbf{\hat{n}_x}')
assert x._latex() == expected
assert Dyadic([])._latex() == '0'
def test_dyadic_str():
assert str(Dyadic([])) == '0'
assert str(y) == 'a**2*(N.x|N.y) + b*(N.y|N.y) + c*sin(alpha)*(N.z|N.y)'
assert str(x) == 'alpha*(N.x|N.x) + sin(omega)*(N.y|N.z) + alpha*beta*(N.z|N.x)'
assert str(ww) == "alpha*N.x + asin(omega)*N.y - beta*alpha'*N.z"
assert str(xx) == '- (N.x|N.y) - (N.x|N.z)'
assert str(xx2) == '(N.x|N.y) + (N.x|N.z)'
def test_vlatex(): # vlatex is broken #12078
from sympy.physics.vector import vlatex
x = symbols('x')
J = symbols('J')
f = Function('f')
g = Function('g')
h = Function('h')
expected = r'J \left(\frac{d}{d x} g{\left(x \right)} - \frac{d}{d x} h{\left(x \right)}\right)'
expr = J*f(x).diff(x).subs(f(x), g(x)-h(x))
assert vlatex(expr) == expected
def test_issue_13354():
"""
Test for proper pretty printing of physics vectors with ADD
instances in arguments.
Test is exactly the one suggested in the original bug report by
@moorepants.
"""
a, b, c = symbols('a, b, c')
A = ReferenceFrame('A')
v = a * A.x + b * A.y + c * A.z
w = b * A.x + c * A.y + a * A.z
z = w + v
expected = """(a + b) a_x + (b + c) a_y + (a + c) a_z"""
assert ascii_vpretty(z) == expected
def test_vector_derivative_printing():
# First order
v = omega.diff() * N.x
assert unicode_vpretty(v) == u('ω̇ n_x')
assert ascii_vpretty(v) == u("omega'(t) n_x")
# Second order
v = omega.diff().diff() * N.x
assert v._latex() == r'\ddot{\omega}\mathbf{\hat{n}_x}'
assert unicode_vpretty(v) == u('ω̈ n_x')
assert ascii_vpretty(v) == u("omega''(t) n_x")
# Third order
v = omega.diff().diff().diff() * N.x
assert v._latex() == r'\dddot{\omega}\mathbf{\hat{n}_x}'
assert unicode_vpretty(v) == u('ω⃛ n_x')
assert ascii_vpretty(v) == u("omega'''(t) n_x")
# Fourth order
v = omega.diff().diff().diff().diff() * N.x
assert v._latex() == r'\ddddot{\omega}\mathbf{\hat{n}_x}'
assert unicode_vpretty(v) == u('ω⃜ n_x')
assert ascii_vpretty(v) == u("omega''''(t) n_x")
# Fifth order
v = omega.diff().diff().diff().diff().diff() * N.x
assert v._latex() == r'\frac{d^{5}}{d t^{5}} \omega{\left(t \right)}\mathbf{\hat{n}_x}'
assert unicode_vpretty(v) == u(' 5\n d\n───(ω) n_x\n 5\ndt')
assert ascii_vpretty(v) == ' 5\n d\n---(omega) n_x\n 5\ndt'
def test_vector_str_printing():
assert vsprint(w) == 'alpha*N.x + sin(omega)*N.y + alpha*beta*N.z'
assert vsprint(omega.diff() * N.x) == "omega'*N.x"
assert vsstrrepr(w) == 'alpha*N.x + sin(omega)*N.y + alpha*beta*N.z'
|
585329b3dcec82f153e1d03fdfe6a23837c97884382c633c27342e26a684e304
|
from sympy import Symbol, symbols, S, simplify, Interval
from sympy.physics.continuum_mechanics.beam import Beam
from sympy.functions import SingularityFunction, Piecewise, meijerg, Abs, log
from sympy.utilities.pytest import raises, slow
from sympy.physics.units import meter, newton, kilo, giga, milli
from sympy.physics.continuum_mechanics.beam import Beam3D
x = Symbol('x')
y = Symbol('y')
R1, R2 = symbols('R1, R2')
def test_Beam():
E = Symbol('E')
E_1 = Symbol('E_1')
I = Symbol('I')
I_1 = Symbol('I_1')
b = Beam(1, E, I)
assert b.length == 1
assert b.elastic_modulus == E
assert b.second_moment == I
assert b.variable == x
# Test the length setter
b.length = 4
assert b.length == 4
# Test the E setter
b.elastic_modulus = E_1
assert b.elastic_modulus == E_1
# Test the I setter
b.second_moment = I_1
assert b.second_moment is I_1
# Test the variable setter
b.variable = y
assert b.variable is y
# Test for all boundary conditions.
b.bc_deflection = [(0, 2)]
b.bc_slope = [(0, 1)]
assert b.boundary_conditions == {'deflection': [(0, 2)], 'slope': [(0, 1)]}
# Test for slope boundary condition method
b.bc_slope.extend([(4, 3), (5, 0)])
s_bcs = b.bc_slope
assert s_bcs == [(0, 1), (4, 3), (5, 0)]
# Test for deflection boundary condition method
b.bc_deflection.extend([(4, 3), (5, 0)])
d_bcs = b.bc_deflection
assert d_bcs == [(0, 2), (4, 3), (5, 0)]
# Test for updated boundary conditions
bcs_new = b.boundary_conditions
assert bcs_new == {
'deflection': [(0, 2), (4, 3), (5, 0)],
'slope': [(0, 1), (4, 3), (5, 0)]}
b1 = Beam(30, E, I)
b1.apply_load(-8, 0, -1)
b1.apply_load(R1, 10, -1)
b1.apply_load(R2, 30, -1)
b1.apply_load(120, 30, -2)
b1.bc_deflection = [(10, 0), (30, 0)]
b1.solve_for_reaction_loads(R1, R2)
# Test for finding reaction forces
p = b1.reaction_loads
q = {R1: 6, R2: 2}
assert p == q
# Test for load distribution function.
p = b1.load
q = -8*SingularityFunction(x, 0, -1) + 6*SingularityFunction(x, 10, -1) + 120*SingularityFunction(x, 30, -2) + 2*SingularityFunction(x, 30, -1)
assert p == q
# Test for shear force distribution function
p = b1.shear_force()
q = -8*SingularityFunction(x, 0, 0) + 6*SingularityFunction(x, 10, 0) + 120*SingularityFunction(x, 30, -1) + 2*SingularityFunction(x, 30, 0)
assert p == q
# Test for bending moment distribution function
p = b1.bending_moment()
q = -8*SingularityFunction(x, 0, 1) + 6*SingularityFunction(x, 10, 1) + 120*SingularityFunction(x, 30, 0) + 2*SingularityFunction(x, 30, 1)
assert p == q
# Test for slope distribution function
p = b1.slope()
q = -4*SingularityFunction(x, 0, 2) + 3*SingularityFunction(x, 10, 2) + 120*SingularityFunction(x, 30, 1) + SingularityFunction(x, 30, 2) + S(4000)/3
assert p == q/(E*I)
# Test for deflection distribution function
p = b1.deflection()
q = 4000*x/3 - 4*SingularityFunction(x, 0, 3)/3 + SingularityFunction(x, 10, 3) + 60*SingularityFunction(x, 30, 2) + SingularityFunction(x, 30, 3)/3 - 12000
assert p == q/(E*I)
# Test using symbols
l = Symbol('l')
w0 = Symbol('w0')
w2 = Symbol('w2')
a1 = Symbol('a1')
c = Symbol('c')
c1 = Symbol('c1')
d = Symbol('d')
e = Symbol('e')
f = Symbol('f')
b2 = Beam(l, E, I)
b2.apply_load(w0, a1, 1)
b2.apply_load(w2, c1, -1)
b2.bc_deflection = [(c, d)]
b2.bc_slope = [(e, f)]
# Test for load distribution function.
p = b2.load
q = w0*SingularityFunction(x, a1, 1) + w2*SingularityFunction(x, c1, -1)
assert p == q
# Test for shear force distribution function
p = b2.shear_force()
q = w0*SingularityFunction(x, a1, 2)/2 + w2*SingularityFunction(x, c1, 0)
assert p == q
# Test for bending moment distribution function
p = b2.bending_moment()
q = w0*SingularityFunction(x, a1, 3)/6 + w2*SingularityFunction(x, c1, 1)
assert p == q
# Test for slope distribution function
p = b2.slope()
q = (w0*SingularityFunction(x, a1, 4)/24 + w2*SingularityFunction(x, c1, 2)/2)/(E*I) + (E*I*f - w0*SingularityFunction(e, a1, 4)/24 - w2*SingularityFunction(e, c1, 2)/2)/(E*I)
assert p == q
# Test for deflection distribution function
p = b2.deflection()
q = x*(E*I*f - w0*SingularityFunction(e, a1, 4)/24 - w2*SingularityFunction(e, c1, 2)/2)/(E*I) + (w0*SingularityFunction(x, a1, 5)/120 + w2*SingularityFunction(x, c1, 3)/6)/(E*I) + (E*I*(-c*f + d) + c*w0*SingularityFunction(e, a1, 4)/24 + c*w2*SingularityFunction(e, c1, 2)/2 - w0*SingularityFunction(c, a1, 5)/120 - w2*SingularityFunction(c, c1, 3)/6)/(E*I)
assert p == q
b3 = Beam(9, E, I)
b3.apply_load(value=-2, start=2, order=2, end=3)
b3.bc_slope.append((0, 2))
C3 = symbols('C3')
C4 = symbols('C4')
p = b3.load
q = -2*SingularityFunction(x, 2, 2) + 2*SingularityFunction(x, 3, 0) + 4*SingularityFunction(x, 3, 1) + 2*SingularityFunction(x, 3, 2)
assert p == q
p = b3.slope()
q = 2 + (-SingularityFunction(x, 2, 5)/30 + SingularityFunction(x, 3, 3)/3 + SingularityFunction(x, 3, 4)/6 + SingularityFunction(x, 3, 5)/30)/(E*I)
assert p == q
p = b3.deflection()
q = 2*x + (-SingularityFunction(x, 2, 6)/180 + SingularityFunction(x, 3, 4)/12 + SingularityFunction(x, 3, 5)/30 + SingularityFunction(x, 3, 6)/180)/(E*I)
assert p == q + C4
b4 = Beam(4, E, I)
b4.apply_load(-3, 0, 0, end=3)
p = b4.load
q = -3*SingularityFunction(x, 0, 0) + 3*SingularityFunction(x, 3, 0)
assert p == q
p = b4.slope()
q = -3*SingularityFunction(x, 0, 3)/6 + 3*SingularityFunction(x, 3, 3)/6
assert p == q/(E*I) + C3
p = b4.deflection()
q = -3*SingularityFunction(x, 0, 4)/24 + 3*SingularityFunction(x, 3, 4)/24
assert p == q/(E*I) + C3*x + C4
# can't use end with point loads
raises(ValueError, lambda: b4.apply_load(-3, 0, -1, end=3))
with raises(TypeError):
b4.variable = 1
def test_insufficient_bconditions():
# Test cases when required number of boundary conditions
# are not provided to solve the integration constants.
L = symbols('L', positive=True)
E, I, P, a3, a4 = symbols('E I P a3 a4')
b = Beam(L, E, I, base_char='a')
b.apply_load(R2, L, -1)
b.apply_load(R1, 0, -1)
b.apply_load(-P, L/2, -1)
b.solve_for_reaction_loads(R1, R2)
p = b.slope()
q = P*SingularityFunction(x, 0, 2)/4 - P*SingularityFunction(x, L/2, 2)/2 + P*SingularityFunction(x, L, 2)/4
assert p == q/(E*I) + a3
p = b.deflection()
q = P*SingularityFunction(x, 0, 3)/12 - P*SingularityFunction(x, L/2, 3)/6 + P*SingularityFunction(x, L, 3)/12
assert p == q/(E*I) + a3*x + a4
b.bc_deflection = [(0, 0)]
p = b.deflection()
q = a3*x + P*SingularityFunction(x, 0, 3)/12 - P*SingularityFunction(x, L/2, 3)/6 + P*SingularityFunction(x, L, 3)/12
assert p == q/(E*I)
b.bc_deflection = [(0, 0), (L, 0)]
p = b.deflection()
q = -L**2*P*x/16 + P*SingularityFunction(x, 0, 3)/12 - P*SingularityFunction(x, L/2, 3)/6 + P*SingularityFunction(x, L, 3)/12
assert p == q/(E*I)
def test_statically_indeterminate():
E = Symbol('E')
I = Symbol('I')
M1, M2 = symbols('M1, M2')
F = Symbol('F')
l = Symbol('l', positive=True)
b5 = Beam(l, E, I)
b5.bc_deflection = [(0, 0),(l, 0)]
b5.bc_slope = [(0, 0),(l, 0)]
b5.apply_load(R1, 0, -1)
b5.apply_load(M1, 0, -2)
b5.apply_load(R2, l, -1)
b5.apply_load(M2, l, -2)
b5.apply_load(-F, l/2, -1)
b5.solve_for_reaction_loads(R1, R2, M1, M2)
p = b5.reaction_loads
q = {R1: F/2, R2: F/2, M1: -F*l/8, M2: F*l/8}
assert p == q
def test_beam_units():
E = Symbol('E')
I = Symbol('I')
R1, R2 = symbols('R1, R2')
b = Beam(8*meter, 200*giga*newton/meter**2, 400*1000000*(milli*meter)**4)
b.apply_load(5*kilo*newton, 2*meter, -1)
b.apply_load(R1, 0*meter, -1)
b.apply_load(R2, 8*meter, -1)
b.apply_load(10*kilo*newton/meter, 4*meter, 0, end=8*meter)
b.bc_deflection = [(0*meter, 0*meter), (8*meter, 0*meter)]
b.solve_for_reaction_loads(R1, R2)
assert b.reaction_loads == {R1: -13750*newton, R2: -31250*newton}
b = Beam(3*meter, E*newton/meter**2, I*meter**4)
b.apply_load(8*kilo*newton, 1*meter, -1)
b.apply_load(R1, 0*meter, -1)
b.apply_load(R2, 3*meter, -1)
b.apply_load(12*kilo*newton*meter, 2*meter, -2)
b.bc_deflection = [(0*meter, 0*meter), (3*meter, 0*meter)]
b.solve_for_reaction_loads(R1, R2)
assert b.reaction_loads == {R1: -28000*newton/3, R2: 4000*newton/3}
assert b.deflection().subs(x, 1*meter) == 62000*meter/(9*E*I)
def test_variable_moment():
E = Symbol('E')
I = Symbol('I')
b = Beam(4, E, 2*(4 - x))
b.apply_load(20, 4, -1)
R, M = symbols('R, M')
b.apply_load(R, 0, -1)
b.apply_load(M, 0, -2)
b.bc_deflection = [(0, 0)]
b.bc_slope = [(0, 0)]
b.solve_for_reaction_loads(R, M)
assert b.slope().expand() == ((10*x*SingularityFunction(x, 0, 0)
- 10*(x - 4)*SingularityFunction(x, 4, 0))/E).expand()
assert b.deflection().expand() == ((5*x**2*SingularityFunction(x, 0, 0)
- 10*Piecewise((0, Abs(x)/4 < 1), (16*meijerg(((3, 1), ()), ((), (2, 0)), x/4), True))
+ 40*SingularityFunction(x, 4, 1))/E).expand()
b = Beam(4, E - x, I)
b.apply_load(20, 4, -1)
R, M = symbols('R, M')
b.apply_load(R, 0, -1)
b.apply_load(M, 0, -2)
b.bc_deflection = [(0, 0)]
b.bc_slope = [(0, 0)]
b.solve_for_reaction_loads(R, M)
assert b.slope().expand() == ((-80*(-log(-E) + log(-E + x))*SingularityFunction(x, 0, 0)
+ 80*(-log(-E + 4) + log(-E + x))*SingularityFunction(x, 4, 0) + 20*(-E*log(-E)
+ E*log(-E + x) + x)*SingularityFunction(x, 0, 0) - 20*(-E*log(-E + 4) + E*log(-E + x)
+ x - 4)*SingularityFunction(x, 4, 0))/I).expand()
def test_composite_beam():
E = Symbol('E')
I = Symbol('I')
b1 = Beam(2, E, 1.5*I)
b2 = Beam(2, E, I)
b = b1.join(b2, "fixed")
b.apply_load(-20, 0, -1)
b.apply_load(80, 0, -2)
b.apply_load(20, 4, -1)
b.bc_slope = [(0, 0)]
b.bc_deflection = [(0, 0)]
assert b.length == 4
assert b.second_moment == Piecewise((1.5*I, x <= 2), (I, x <= 4))
assert b.slope().subs(x, 4) == 120.0/(E*I)
assert b.slope().subs(x, 2) == 80.0/(E*I)
assert int(b.deflection().subs(x, 4).args[0]) == 302 # Coefficient of 1/(E*I)
l = symbols('l', positive=True)
R1, M1, R2, R3, P = symbols('R1 M1 R2 R3 P')
b1 = Beam(2*l, E, I)
b2 = Beam(2*l, E, I)
b = b1.join(b2,"hinge")
b.apply_load(M1, 0, -2)
b.apply_load(R1, 0, -1)
b.apply_load(R2, l, -1)
b.apply_load(R3, 4*l, -1)
b.apply_load(P, 3*l, -1)
b.bc_slope = [(0, 0)]
b.bc_deflection = [(0, 0), (l, 0), (4*l, 0)]
b.solve_for_reaction_loads(M1, R1, R2, R3)
assert b.reaction_loads == {R3: -P/2, R2: -5*P/4, M1: -P*l/4, R1: 3*P/4}
assert b.slope().subs(x, 3*l) == -7*P*l**2/(48*E*I)
assert b.deflection().subs(x, 2*l) == 7*P*l**3/(24*E*I)
assert b.deflection().subs(x, 3*l) == 5*P*l**3/(16*E*I)
# When beams having same second moment are joined.
b1 = Beam(2, 500, 10)
b2 = Beam(2, 500, 10)
b = b1.join(b2, "fixed")
b.apply_load(M1, 0, -2)
b.apply_load(R1, 0, -1)
b.apply_load(R2, 1, -1)
b.apply_load(R3, 4, -1)
b.apply_load(10, 3, -1)
b.bc_slope = [(0, 0)]
b.bc_deflection = [(0, 0), (1, 0), (4, 0)]
b.solve_for_reaction_loads(M1, R1, R2, R3)
assert b.slope() == -2*SingularityFunction(x, 0, 1)/5625 + SingularityFunction(x, 0, 2)/1875\
- 133*SingularityFunction(x, 1, 2)/135000 + SingularityFunction(x, 3, 2)/1000\
- 37*SingularityFunction(x, 4, 2)/67500
assert b.deflection() == -SingularityFunction(x, 0, 2)/5625 + SingularityFunction(x, 0, 3)/5625\
- 133*SingularityFunction(x, 1, 3)/405000 + SingularityFunction(x, 3, 3)/3000\
- 37*SingularityFunction(x, 4, 3)/202500
def test_point_cflexure():
E = Symbol('E')
I = Symbol('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)
assert b.point_cflexure() == [S(10)/3]
def test_remove_load():
E = Symbol('E')
I = Symbol('I')
b = Beam(4, E, I)
try:
b.remove_load(2, 1, -1)
# As no load is applied on beam, ValueError should be returned.
except ValueError:
assert True
else:
assert False
b.apply_load(-3, 0, -2)
b.apply_load(4, 2, -1)
b.apply_load(-2, 2, 2, end = 3)
b.remove_load(-2, 2, 2, end = 3)
assert b.load == -3*SingularityFunction(x, 0, -2) + 4*SingularityFunction(x, 2, -1)
assert b.applied_loads == [(-3, 0, -2, None), (4, 2, -1, None)]
try:
b.remove_load(1, 2, -1)
# As load of this magnitude was never applied at
# this position, method should return a ValueError.
except ValueError:
assert True
else:
assert False
b.remove_load(-3, 0, -2)
b.remove_load(4, 2, -1)
assert b.load == 0
assert b.applied_loads == []
def test_apply_support():
E = Symbol('E')
I = Symbol('I')
b = Beam(4, E, I)
b.apply_support(0, "cantilever")
b.apply_load(20, 4, -1)
M_0, R_0 = symbols('M_0, R_0')
b.solve_for_reaction_loads(R_0, M_0)
assert b.slope() == (80*SingularityFunction(x, 0, 1) - 10*SingularityFunction(x, 0, 2)
+ 10*SingularityFunction(x, 4, 2))/(E*I)
assert b.deflection() == (40*SingularityFunction(x, 0, 2) - 10*SingularityFunction(x, 0, 3)/3
+ 10*SingularityFunction(x, 4, 3)/3)/(E*I)
b = Beam(30, E, I)
b.apply_support(10, "pin")
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)
assert b.slope() == (-4*SingularityFunction(x, 0, 2) + 3*SingularityFunction(x, 10, 2)
+ 120*SingularityFunction(x, 30, 1) + SingularityFunction(x, 30, 2) + S(4000)/3)/(E*I)
assert 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)
P = Symbol('P', positive=True)
L = Symbol('L', positive=True)
b = Beam(L, E, I)
b.apply_support(0, type='fixed')
b.apply_support(L, type='fixed')
b.apply_load(-P, L/2, -1)
R_0, R_L, M_0, M_L = symbols('R_0, R_L, M_0, M_L')
b.solve_for_reaction_loads(R_0, R_L, M_0, M_L)
assert b.reaction_loads == {R_0: P/2, R_L: P/2, M_0: -L*P/8, M_L: L*P/8}
def test_max_shear_force():
E = Symbol('E')
I = Symbol('I')
b = Beam(3, E, I)
R, M = symbols('R, M')
b.apply_load(R, 0, -1)
b.apply_load(M, 0, -2)
b.apply_load(2, 3, -1)
b.apply_load(4, 2, -1)
b.apply_load(2, 2, 0, end=3)
b.solve_for_reaction_loads(R, M)
assert b.max_shear_force() == (Interval(0, 2), 8)
l = symbols('l', positive=True)
P = Symbol('P')
b = Beam(l, E, I)
R1, R2 = symbols('R1, R2')
b.apply_load(R1, 0, -1)
b.apply_load(R2, l, -1)
b.apply_load(P, 0, 0, end=l)
b.solve_for_reaction_loads(R1, R2)
assert b.max_shear_force() == (0, l*Abs(P)/2)
def test_max_bmoment():
E = Symbol('E')
I = Symbol('I')
l, P = symbols('l, P', positive=True)
b = Beam(l, E, I)
R1, R2 = symbols('R1, R2')
b.apply_load(R1, 0, -1)
b.apply_load(R2, l, -1)
b.apply_load(P, l/2, -1)
b.solve_for_reaction_loads(R1, R2)
b.reaction_loads
assert b.max_bmoment() == (l/2, P*l/4)
b = Beam(l, E, I)
R1, R2 = symbols('R1, R2')
b.apply_load(R1, 0, -1)
b.apply_load(R2, l, -1)
b.apply_load(P, 0, 0, end=l)
b.solve_for_reaction_loads(R1, R2)
assert b.max_bmoment() == (l/2, P*l**2/8)
def test_max_deflection():
E, I, l, F = symbols('E, I, l, F', positive=True)
b = Beam(l, E, I)
b.bc_deflection = [(0, 0),(l, 0)]
b.bc_slope = [(0, 0),(l, 0)]
b.apply_load(F/2, 0, -1)
b.apply_load(-F*l/8, 0, -2)
b.apply_load(F/2, l, -1)
b.apply_load(F*l/8, l, -2)
b.apply_load(-F, l/2, -1)
assert b.max_deflection() == (l/2, F*l**3/(192*E*I))
def test_Beam3D():
l, E, G, I, A = symbols('l, E, G, I, A')
R1, R2, R3, R4 = symbols('R1, R2, R3, R4')
b = Beam3D(l, E, G, I, A)
m, q = symbols('m, q')
b.apply_load(q, 0, 0, dir="y")
b.apply_moment_load(m, 0, 0, dir="z")
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()
assert b.shear_force() == [0, -q*x, 0]
assert b.bending_moment() == [0, 0, -m*x + q*x**2/2]
expected_deflection = (-l**2*q*x**2/(12*E*I) + l**2*x**2*(A*G*l*(l*q - 2*m)
+ 12*E*I*q)/(8*E*I*(A*G*l**2 + 12*E*I)) + l*m*x**2/(4*E*I)
- l*x**3*(A*G*l*(l*q - 2*m) + 12*E*I*q)/(12*E*I*(A*G*l**2 + 12*E*I))
- m*x**3/(6*E*I) + q*x**4/(24*E*I)
+ l*x*(A*G*l*(l*q - 2*m) + 12*E*I*q)/(2*A*G*(A*G*l**2 + 12*E*I))
- q*x**2/(2*A*G)
)
dx, dy, dz = b.deflection()
assert dx == dz == 0
assert simplify(dy - expected_deflection) == 0 # == doesn't work
b2 = Beam3D(30, E, G, I, A, x)
b2.apply_load(50, start=0, order=0, dir="y")
b2.bc_deflection = [(0, [0, 0, 0]), (30, [0, 0, 0])]
b2.apply_load(R1, start=0, order=-1, dir="y")
b2.apply_load(R2, start=30, order=-1, dir="y")
b2.solve_for_reaction_loads(R1, R2)
assert b2.reaction_loads == {R1: -750, R2: -750}
b2.solve_slope_deflection()
assert b2.slope() == [0, 0, 25*x**3/(3*E*I) - 375*x**2/(E*I) + 3750*x/(E*I)]
expected_deflection = (25*x**4/(12*E*I) - 125*x**3/(E*I) + 1875*x**2/(E*I)
- 25*x**2/(A*G) + 750*x/(A*G))
dx, dy, dz = b2.deflection()
assert dx == dz == 0
assert simplify(dy - expected_deflection) == 0 # == doesn't work
# Test for solve_for_reaction_loads
b3 = Beam3D(30, E, G, I, A, x)
b3.apply_load(8, start=0, order=0, dir="y")
b3.apply_load(9*x, start=0, order=0, dir="z")
b3.apply_load(R1, start=0, order=-1, dir="y")
b3.apply_load(R2, start=30, order=-1, dir="y")
b3.apply_load(R3, start=0, order=-1, dir="z")
b3.apply_load(R4, start=30, order=-1, dir="z")
b3.solve_for_reaction_loads(R1, R2, R3, R4)
assert b3.reaction_loads == {R1: -120, R2: -120, R3: -1350, R4: -2700}
def test_parabolic_loads():
E, I, L = symbols('E, I, L', positive=True, real=True)
R, M, P = symbols('R, M, P', real=True)
# cantilever beam fixed at x=0 and parabolic distributed loading across
# length of beam
beam = Beam(L, E, I)
beam.bc_deflection.append((0, 0))
beam.bc_slope.append((0, 0))
beam.apply_load(R, 0, -1)
beam.apply_load(M, 0, -2)
# parabolic load
beam.apply_load(1, 0, 2)
beam.solve_for_reaction_loads(R, M)
assert beam.reaction_loads[R] == -L**3 / 3
# cantilever beam fixed at x=0 and parabolic distributed loading across
# first half of beam
beam = Beam(2 * L, E, I)
beam.bc_deflection.append((0, 0))
beam.bc_slope.append((0, 0))
beam.apply_load(R, 0, -1)
beam.apply_load(M, 0, -2)
# parabolic load from x=0 to x=L
beam.apply_load(1, 0, 2, end=L)
beam.solve_for_reaction_loads(R, M)
# result should be the same as the prior example
assert beam.reaction_loads[R] == -L**3 / 3
# check constant load
beam = Beam(2 * L, E, I)
beam.apply_load(P, 0, 0, end=L)
loading = beam.load.xreplace({L: 10, E: 20, I: 30, P: 40})
assert loading.xreplace({x: 5}) == 40
assert loading.xreplace({x: 15}) == 0
# check ramp load
beam = Beam(2 * L, E, I)
beam.apply_load(P, 0, 1, end=L)
assert beam.load == (P*SingularityFunction(x, 0, 1) -
P*SingularityFunction(x, L, 1) -
P*L*SingularityFunction(x, L, 0))
# check higher order load: x**8 load from x=0 to x=L
beam = Beam(2 * L, E, I)
beam.apply_load(P, 0, 8, end=L)
loading = beam.load.xreplace({L: 10, E: 20, I: 30, P: 40})
assert loading.xreplace({x: 5}) == 40 * 5**8
assert loading.xreplace({x: 15}) == 0
|
5b917de2a74851c7174905210f3cd49075a1087db51b349a483f0b696ec59155
|
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.utilities.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
k = Symbol('k') # Wavenumber
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*cos(phi1) + A2*cos(phi2), A1*sin(phi1) + A2*sin(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) - sqrt(A1**2 + 2*A1*A2*cos(phi1 - phi2)
+ A2**2)*sin(pi*f*n*x*s/(149896229*m) - 2*pi*f*t + atan2(A1*cos(phi1)
+ A2*cos(phi2), A1*sin(phi1) + A2*sin(phi2)) + pi/2)) == 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*cos(phi1)
+ A2*cos(phi2), A1*sin(phi1) + A2*sin(phi2)))
assert w3.rewrite(exp) == sqrt(A1**2 + 2*A1*A2*cos(phi1 - phi2)
+ A2**2)*exp(I*(pi*f*n*x*s/(149896229*m) - 2*pi*f*t
+ atan2(A1*cos(phi1) + A2*cos(phi2), A1*sin(phi1) + A2*sin(phi2))))
w4 = TWave(A1, None, 0, 1/f)
assert w4.frequency == f
raises(ValueError, lambda:TWave(A1))
raises(ValueError, lambda:TWave(A1, f, phi1, t))
|
9e1da11301d896fd642d946fa473d3a68856ffb2452f24f48c5af7b7feedd6b3
|
from sympy import sqrt, simplify
from sympy.physics.optics import Medium
from sympy.abc import epsilon, mu, n
from sympy.physics.units import speed_of_light, u0, e0, m, kg, s, A
from sympy.utilities.pytest import raises
c = speed_of_light.convert_to(m/s)
e0 = e0.convert_to(A**2*s**4/(kg*m**3))
u0 = u0.convert_to(m*kg/(A**2*s**2))
def test_medium():
m1 = Medium('m1')
assert m1.intrinsic_impedance == sqrt(u0/e0)
assert m1.speed == 1/sqrt(e0*u0)
assert m1.refractive_index == c*sqrt(e0*u0)
assert m1.permittivity == e0
assert m1.permeability == u0
m2 = Medium('m2', epsilon, mu)
assert m2.intrinsic_impedance == sqrt(mu/epsilon)
assert m2.speed == 1/sqrt(epsilon*mu)
assert m2.refractive_index == c*sqrt(epsilon*mu)
assert m2.permittivity == epsilon
assert m2.permeability == mu
# Increasing electric permittivity and magnetic permeability
# by small amount from its value in vacuum.
m3 = Medium('m3', 9.0*10**(-12)*s**4*A**2/(m**3*kg), 1.45*10**(-6)*kg*m/(A**2*s**2))
assert m3.refractive_index > m1.refractive_index
assert m3 > m1
assert m3 != m1
# Decreasing electric permittivity and magnetic permeability
# by small amount from its value in vacuum.
m4 = Medium('m4', 7.0*10**(-12)*s**4*A**2/(m**3*kg), 1.15*10**(-6)*kg*m/(A**2*s**2))
assert m4.refractive_index < m1.refractive_index
assert m4 < m1
m5 = Medium('m5', permittivity=710*10**(-12)*s**4*A**2/(m**3*kg), n=1.33)
assert abs(m5.intrinsic_impedance - 6.24845417765552*kg*m**2/(A**2*s**3)) \
< 1e-12*kg*m**2/(A**2*s**3)
assert abs(m5.speed - 225407863.157895*m/s) < 1e-6*m/s
assert abs(m5.refractive_index - 1.33000000000000) < 1e-12
assert abs(m5.permittivity - 7.1e-10*A**2*s**4/(kg*m**3)) \
< 1e-20*A**2*s**4/(kg*m**3)
assert abs(m5.permeability - 2.77206575232851e-8*kg*m/(A**2*s**2)) \
< 1e-20*kg*m/(A**2*s**2)
m6 = Medium('m6', None, mu, n)
assert m6.permittivity == n**2/(c**2*mu)
assert Medium('m7') == Medium('m8', e0, u0) # test for equality
raises(ValueError, lambda:Medium('m9', e0, u0, 2))
|
c8a8be67353a35b4fcf86db522c427d45f5fed86d7e3eb28ebf9aacc3a767aea
|
from sympy.physics.optics.utils import (refraction_angle, fresnel_coefficients,
deviation, brewster_angle, critical_angle, lens_makers_formula,
mirror_formula, lens_formula, hyperfocal_distance,
transverse_magnification)
from sympy.physics.optics.medium import Medium
from sympy.physics.units import e0
from sympy import symbols, sqrt, Matrix, oo
from sympy.geometry.point import Point3D
from sympy.geometry.line import Ray3D
from sympy.geometry.plane import Plane
from sympy.core import S
from sympy.utilities.pytest import raises
def test_refraction_angle():
n1, n2 = symbols('n1, n2')
m1 = Medium('m1')
m2 = Medium('m2')
r1 = Ray3D(Point3D(-1, -1, 1), Point3D(0, 0, 0))
i = Matrix([1, 1, 1])
n = Matrix([0, 0, 1])
normal_ray = Ray3D(Point3D(0, 0, 0), Point3D(0, 0, 1))
P = Plane(Point3D(0, 0, 0), normal_vector=[0, 0, 1])
assert refraction_angle(r1, 1, 1, n) == Matrix([
[ 1],
[ 1],
[-1]])
assert refraction_angle([1, 1, 1], 1, 1, n) == Matrix([
[ 1],
[ 1],
[-1]])
assert refraction_angle((1, 1, 1), 1, 1, n) == Matrix([
[ 1],
[ 1],
[-1]])
assert refraction_angle(i, 1, 1, [0, 0, 1]) == Matrix([
[ 1],
[ 1],
[-1]])
assert refraction_angle(i, 1, 1, (0, 0, 1)) == Matrix([
[ 1],
[ 1],
[-1]])
assert refraction_angle(i, 1, 1, normal_ray) == Matrix([
[ 1],
[ 1],
[-1]])
assert refraction_angle(i, 1, 1, plane=P) == Matrix([
[ 1],
[ 1],
[-1]])
assert refraction_angle(r1, 1, 1, plane=P) == \
Ray3D(Point3D(0, 0, 0), Point3D(1, 1, -1))
assert refraction_angle(r1, m1, 1.33, plane=P) == \
Ray3D(Point3D(0, 0, 0), Point3D(S(100)/133, S(100)/133, -789378201649271*sqrt(3)/1000000000000000))
assert refraction_angle(r1, 1, m2, plane=P) == \
Ray3D(Point3D(0, 0, 0), Point3D(1, 1, -1))
assert refraction_angle(r1, n1, n2, plane=P) == \
Ray3D(Point3D(0, 0, 0), Point3D(n1/n2, n1/n2, -sqrt(3)*sqrt(-2*n1**2/(3*n2**2) + 1)))
assert refraction_angle(r1, 1.33, 1, plane=P) == 0 # TIR
assert refraction_angle(r1, 1, 1, normal_ray) == \
Ray3D(Point3D(0, 0, 0), direction_ratio=[1, 1, -1])
raises(ValueError, lambda: refraction_angle(r1, m1, m2, normal_ray, P))
raises(TypeError, lambda: refraction_angle(m1, m1, m2)) # can add other values for arg[0]
raises(TypeError, lambda: refraction_angle(r1, m1, m2, None, i))
raises(TypeError, lambda: refraction_angle(r1, m1, m2, m2))
def test_fresnel_coefficients():
assert list(round(i, 5) for i in fresnel_coefficients(0.5, 1, 1.33)) == \
[0.11163, -0.17138, 0.83581, 0.82862]
assert list(round(i, 5) for i in fresnel_coefficients(0.5, 1.33, 1)) == \
[-0.07726, 0.20482, 1.22724, 1.20482]
m1 = Medium('m1')
m2 = Medium('m2', n=2)
assert list(round(i, 5) for i in fresnel_coefficients(0.3, m1, m2)) == \
[0.31784, -0.34865, 0.65892, 0.65135]
assert list(list(round(j, 5) for j in i.as_real_imag()) for i in \
fresnel_coefficients(0.6, m2, m1)) == \
[[-0.23563, -0.97184], [0.81648, -0.57738]]
def test_deviation():
n1, n2 = symbols('n1, n2')
r1 = Ray3D(Point3D(-1, -1, 1), Point3D(0, 0, 0))
n = Matrix([0, 0, 1])
i = Matrix([-1, -1, -1])
normal_ray = Ray3D(Point3D(0, 0, 0), Point3D(0, 0, 1))
P = Plane(Point3D(0, 0, 0), normal_vector=[0, 0, 1])
assert deviation(r1, 1, 1, normal=n) == 0
assert deviation(r1, 1, 1, plane=P) == 0
assert deviation(r1, 1, 1.1, plane=P).evalf(3) + 0.119 < 1e-3
assert deviation(i, 1, 1.1, normal=normal_ray).evalf(3) + 0.119 < 1e-3
assert deviation(r1, 1.33, 1, plane=P) is None # TIR
assert deviation(r1, 1, 1, normal=[0, 0, 1]) == 0
assert deviation([-1, -1, -1], 1, 1, normal=[0, 0, 1]) == 0
def test_brewster_angle():
m1 = Medium('m1', n=1)
m2 = Medium('m2', n=1.33)
assert round(brewster_angle(m1, m2), 2) == 0.93
m1 = Medium('m1', permittivity=e0, n=1)
m2 = Medium('m2', permittivity=e0, n=1.33)
assert round(brewster_angle(m1, m2), 2) == 0.93
assert round(brewster_angle(1, 1.33), 2) == 0.93
def test_critical_angle():
m1 = Medium('m1', n=1)
m2 = Medium('m2', n=1.33)
assert round(critical_angle(m2, m1), 2) == 0.85
def test_lens_makers_formula():
n1, n2 = symbols('n1, n2')
m1 = Medium('m1', permittivity=e0, n=1)
m2 = Medium('m2', permittivity=e0, n=1.33)
assert lens_makers_formula(n1, n2, 10, -10) == 5*n2/(n1 - n2)
assert round(lens_makers_formula(m1, m2, 10, -10), 2) == -20.15
assert round(lens_makers_formula(1.33, 1, 10, -10), 2) == 15.15
def test_mirror_formula():
u, v, f = symbols('u, v, f')
assert mirror_formula(focal_length=f, u=u) == f*u/(-f + u)
assert mirror_formula(focal_length=f, v=v) == f*v/(-f + v)
assert mirror_formula(u=u, v=v) == u*v/(u + v)
assert mirror_formula(u=oo, v=v) == v
assert mirror_formula(u=oo, v=oo) == oo
assert mirror_formula(focal_length=oo, u=u) == -u
assert mirror_formula(u=u, v=oo) == u
assert mirror_formula(focal_length=oo, v=oo) == oo
assert mirror_formula(focal_length=f, v=oo) == f
assert mirror_formula(focal_length=oo, v=v) == -v
assert mirror_formula(focal_length=oo, u=oo) == oo
assert mirror_formula(focal_length=f, u=oo) == f
assert mirror_formula(focal_length=oo, u=u) == -u
raises(ValueError, lambda: mirror_formula(focal_length=f, u=u, v=v))
def test_lens_formula():
u, v, f = symbols('u, v, f')
assert lens_formula(focal_length=f, u=u) == f*u/(f + u)
assert lens_formula(focal_length=f, v=v) == f*v/(f - v)
assert lens_formula(u=u, v=v) == u*v/(u - v)
assert lens_formula(u=oo, v=v) == v
assert lens_formula(u=oo, v=oo) == oo
assert lens_formula(focal_length=oo, u=u) == u
assert lens_formula(u=u, v=oo) == -u
assert lens_formula(focal_length=oo, v=oo) == -oo
assert lens_formula(focal_length=oo, v=v) == v
assert lens_formula(focal_length=f, v=oo) == -f
assert lens_formula(focal_length=oo, u=oo) == oo
assert lens_formula(focal_length=oo, u=u) == u
assert lens_formula(focal_length=f, u=oo) == f
raises(ValueError, lambda: lens_formula(focal_length=f, u=u, v=v))
def test_hyperfocal_distance():
f, N, c = symbols('f, N, c')
assert hyperfocal_distance(f=f, N=N, c=c) == f**2/(N*c)
assert round(hyperfocal_distance(f=0.5, N=8, c=0.0033), 2) == 9.47
def test_transverse_magnification():
si, so = symbols('si, so')
assert transverse_magnification(si, so) == -si/so
assert transverse_magnification(30, 15) == -2
|
6d0c866682b25edfb1e2e73e546244e083d2cdb391673d975b81cb49566292ed
|
from sympy.assumptions.ask import Q
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)
from sympy.utilities.pytest import raises, XFAIL, slow
from sympy.utilities import cartes
from itertools import combinations
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(l, le)
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_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(1) > 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(1) > 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(Not(w), z), And(y, z)))
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(Not(w), z), And(y, z)))
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(Not(w), z), And(y, z)))
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(Not(w), z), And(y, z)))
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
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 true.is_Boolean
assert (A & B).is_Boolean
assert (A | B).is_Boolean
assert (~A).is_Boolean
assert (A ^ B).is_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))
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_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_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
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', 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 == set((y, z))
for f in (Eq, Ne):
assert f(x, 1).binary_symbols == set()
assert f(x, True).binary_symbols == set([x])
assert f(x, False).binary_symbols == set([x])
assert S.true.binary_symbols == set()
assert S.false.binary_symbols == set()
assert x.binary_symbols == set([x])
assert And(x, Eq(y, False), Eq(z, 1)).binary_symbols == set([x, y])
assert Q.prime(x).binary_symbols == set()
assert Q.is_true(x < 1).binary_symbols == set()
assert Q.is_true(x).binary_symbols == set([x])
assert Q.is_true(Eq(x, True)).binary_symbols == set([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)
|
837e2c54e408db58fdad91582196969103710945fa36e41830ff75834b821818
|
import collections
import random
from sympy.assumptions import Q
from sympy.core.add import Add
from sympy.core.compatibility import range
from sympy.core.function import (Function, diff)
from sympy.core.numbers import (E, Float, I, Integer, oo, pi)
from sympy.core.relational import (Eq, Lt)
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 (Max, Min, sqrt)
from sympy.functions.elementary.piecewise import Piecewise
from sympy.functions.elementary.trigonometric import (cos, sin, tan)
from sympy.logic.boolalg import (And, Or)
from sympy.matrices.common import (ShapeError, MatrixError, NonSquareMatrixError,
_MinimalMatrix, MatrixShaping, MatrixProperties, MatrixOperations, MatrixArithmetic,
MatrixSpecial)
from sympy.matrices.matrices import (MatrixDeterminant,
MatrixReductions, MatrixSubspaces, MatrixEigen, MatrixCalculus)
from sympy.matrices import (Matrix, diag, eye,
matrix_multiply_elementwise, ones, zeros)
from sympy.polys.polytools import Poly
from sympy.simplify.simplify import simplify
from sympy.simplify.trigsimp import trigsimp
from sympy.utilities.exceptions import SymPyDeprecationWarning
from sympy.utilities.iterables import flatten
from sympy.utilities.pytest import (raises, XFAIL, slow, skip,
warns_deprecated_sympy)
from sympy.abc import a, b, c, d, x, y, z
# classes to test the basic matrix classes
class ShapingOnlyMatrix(_MinimalMatrix, 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, 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, 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, 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 DeterminantOnlyMatrix(_MinimalMatrix, MatrixDeterminant):
pass
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)
class ReductionsOnlyMatrix(_MinimalMatrix, MatrixReductions):
pass
def eye_Reductions(n):
return ReductionsOnlyMatrix(n, n, lambda i, j: int(i == j))
def zeros_Reductions(n):
return ReductionsOnlyMatrix(n, n, lambda i, j: 0)
class SpecialOnlyMatrix(_MinimalMatrix, MatrixSpecial):
pass
class SubspaceOnlyMatrix(_MinimalMatrix, MatrixSubspaces):
pass
class EigenOnlyMatrix(_MinimalMatrix, MatrixEigen):
pass
class CalculusOnlyMatrix(_MinimalMatrix, 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_tolist():
lst = [[S.One, S.Half, x*y, S.Zero], [x, y, z, x**2], [y, -S.One, z*x, 3]]
flat_lst = [S.One, S.Half, x*y, S.Zero, x, y, z, x**2, y, -S.One, z*x, 3]
m = ShapingOnlyMatrix(3, 4, flat_lst)
assert m.tolist() == lst
def test_row_col_del():
e = ShapingOnlyMatrix(3, 3, [1, 2, 3, 4, 5, 6, 7, 8, 9])
raises(ValueError, lambda: e.row_del(5))
raises(ValueError, lambda: e.row_del(-5))
raises(ValueError, lambda: e.col_del(5))
raises(ValueError, lambda: e.col_del(-5))
assert e.row_del(2) == e.row_del(-1) == Matrix([[1, 2, 3], [4, 5, 6]])
assert e.col_del(2) == e.col_del(-1) == Matrix([[1, 2], [4, 5], [7, 8]])
assert e.row_del(1) == e.row_del(-2) == Matrix([[1, 2, 3], [7, 8, 9]])
assert e.col_del(1) == e.col_del(-2) == Matrix([[1, 3], [4, 6], [7, 9]])
def test_get_diag_blocks1():
a = Matrix([[1, 2], [2, 3]])
b = Matrix([[3, x], [y, 3]])
c = Matrix([[3, x, 3], [y, 3, z], [x, y, z]])
assert a.get_diag_blocks() == [a]
assert b.get_diag_blocks() == [b]
assert c.get_diag_blocks() == [c]
def test_get_diag_blocks2():
a = Matrix([[1, 2], [2, 3]])
b = Matrix([[3, x], [y, 3]])
c = Matrix([[3, x, 3], [y, 3, z], [x, y, z]])
A, B, C, D = diag(a, b, b), diag(a, b, c), diag(a, c, b), diag(c, c, b)
A = ShapingOnlyMatrix(A.rows, A.cols, A)
B = ShapingOnlyMatrix(B.rows, B.cols, B)
C = ShapingOnlyMatrix(C.rows, C.cols, C)
D = ShapingOnlyMatrix(D.rows, D.cols, D)
assert A.get_diag_blocks() == [a, b, b]
assert B.get_diag_blocks() == [a, b, c]
assert C.get_diag_blocks() == [a, c, b]
assert D.get_diag_blocks() == [c, c, b]
def test_shape():
m = ShapingOnlyMatrix(1, 2, [0, 0])
m.shape == (1, 2)
def test_reshape():
m0 = eye_Shaping(3)
assert m0.reshape(1, 9) == Matrix(1, 9, (1, 0, 0, 0, 1, 0, 0, 0, 1))
m1 = ShapingOnlyMatrix(3, 4, lambda i, j: i + j)
assert m1.reshape(
4, 3) == Matrix(((0, 1, 2), (3, 1, 2), (3, 4, 2), (3, 4, 5)))
assert m1.reshape(2, 6) == Matrix(((0, 1, 2, 3, 1, 2), (3, 4, 2, 3, 4, 5)))
def test_row_col():
m = ShapingOnlyMatrix(3, 3, [1, 2, 3, 4, 5, 6, 7, 8, 9])
assert m.row(0) == Matrix(1, 3, [1, 2, 3])
assert m.col(0) == Matrix(3, 1, [1, 4, 7])
def test_row_join():
assert eye_Shaping(3).row_join(Matrix([7, 7, 7])) == \
Matrix([[1, 0, 0, 7],
[0, 1, 0, 7],
[0, 0, 1, 7]])
def test_col_join():
assert eye_Shaping(3).col_join(Matrix([[7, 7, 7]])) == \
Matrix([[1, 0, 0],
[0, 1, 0],
[0, 0, 1],
[7, 7, 7]])
def test_row_insert():
r4 = Matrix([[4, 4, 4]])
for i in range(-4, 5):
l = [1, 0, 0]
l.insert(i, 4)
assert flatten(eye_Shaping(3).row_insert(i, r4).col(0).tolist()) == l
def test_col_insert():
c4 = Matrix([4, 4, 4])
for i in range(-4, 5):
l = [0, 0, 0]
l.insert(i, 4)
assert flatten(zeros_Shaping(3).col_insert(i, c4).row(0).tolist()) == l
# issue 13643
assert eye_Shaping(6).col_insert(3, Matrix([[2, 2], [2, 2], [2, 2], [2, 2], [2, 2], [2, 2]])) == \
Matrix([[1, 0, 0, 2, 2, 0, 0, 0],
[0, 1, 0, 2, 2, 0, 0, 0],
[0, 0, 1, 2, 2, 0, 0, 0],
[0, 0, 0, 2, 2, 1, 0, 0],
[0, 0, 0, 2, 2, 0, 1, 0],
[0, 0, 0, 2, 2, 0, 0, 1]])
def test_extract():
m = ShapingOnlyMatrix(4, 3, lambda i, j: i*3 + j)
assert m.extract([0, 1, 3], [0, 1]) == Matrix(3, 2, [0, 1, 3, 4, 9, 10])
assert m.extract([0, 3], [0, 0, 2]) == Matrix(2, 3, [0, 0, 2, 9, 9, 11])
assert m.extract(range(4), range(3)) == m
raises(IndexError, lambda: m.extract([4], [0]))
raises(IndexError, lambda: m.extract([0], [3]))
def test_hstack():
m = ShapingOnlyMatrix(4, 3, lambda i, j: i*3 + j)
m2 = ShapingOnlyMatrix(3, 4, lambda i, j: i*3 + j)
assert m == m.hstack(m)
assert m.hstack(m, m, m) == ShapingOnlyMatrix.hstack(m, m, m) == Matrix([
[0, 1, 2, 0, 1, 2, 0, 1, 2],
[3, 4, 5, 3, 4, 5, 3, 4, 5],
[6, 7, 8, 6, 7, 8, 6, 7, 8],
[9, 10, 11, 9, 10, 11, 9, 10, 11]])
raises(ShapeError, lambda: m.hstack(m, m2))
assert Matrix.hstack() == Matrix()
# test regression #12938
M1 = Matrix.zeros(0, 0)
M2 = Matrix.zeros(0, 1)
M3 = Matrix.zeros(0, 2)
M4 = Matrix.zeros(0, 3)
m = ShapingOnlyMatrix.hstack(M1, M2, M3, M4)
assert m.rows == 0 and m.cols == 6
def test_vstack():
m = ShapingOnlyMatrix(4, 3, lambda i, j: i*3 + j)
m2 = ShapingOnlyMatrix(3, 4, lambda i, j: i*3 + j)
assert m == m.vstack(m)
assert m.vstack(m, m, m) == ShapingOnlyMatrix.vstack(m, m, m) == Matrix([
[0, 1, 2],
[3, 4, 5],
[6, 7, 8],
[9, 10, 11],
[0, 1, 2],
[3, 4, 5],
[6, 7, 8],
[9, 10, 11],
[0, 1, 2],
[3, 4, 5],
[6, 7, 8],
[9, 10, 11]])
raises(ShapeError, lambda: m.vstack(m, m2))
assert Matrix.vstack() == Matrix()
# PropertiesOnlyMatrix tests
def test_atoms():
m = PropertiesOnlyMatrix(2, 2, [1, 2, x, 1 - 1/x])
assert m.atoms() == {S(1),S(2),S(-1), 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
assert PropertiesOnlyMatrix([[0, 0], [0, 0]]).is_zero
assert PropertiesOnlyMatrix(zeros(3, 4)).is_zero
assert not PropertiesOnlyMatrix(eye(3)).is_zero
assert PropertiesOnlyMatrix([[x, 0], [0, 0]]).is_zero == None
assert PropertiesOnlyMatrix([[x, 1], [0, 0]]).is_zero == False
a = Symbol('a', nonzero=True)
assert PropertiesOnlyMatrix([[a, 0], [0, 0]]).is_zero == False
def test_values():
assert set(PropertiesOnlyMatrix(2,2,[0,1,2,3]).values()) == set([1,2,3])
x = Symbol('x', real=True)
assert set(PropertiesOnlyMatrix(2,2,[x,0,0,1]).values()) == set([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_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]]))
b = a.permute_rows([[0, 2], [0, 1]])
assert a.permute([[0, 2], [0, 1]]) == b == Matrix([
[5, 6, 7, 8],
[9, 10, 11, 12],
[1, 2, 3, 4]])
b = a.permute_cols([[0, 2], [0, 1]])
assert a.permute([[0, 2], [0, 1]], orientation='cols') == b ==\
Matrix([
[ 2, 3, 1, 4],
[ 6, 7, 5, 8],
[10, 11, 9, 12]])
b = a.permute_cols([[0, 2], [0, 1]], direction='backward')
assert a.permute([[0, 2], [0, 1]], orientation='cols', direction='backward') == b ==\
Matrix([
[ 3, 1, 2, 4],
[ 7, 5, 6, 8],
[11, 9, 10, 12]])
assert a.permute([1, 2, 0, 3]) == Matrix([
[5, 6, 7, 8],
[9, 10, 11, 12],
[1, 2, 3, 4]])
from sympy.combinatorics import Permutation
assert a.permute(Permutation([1, 2, 0, 3])) == Matrix([
[5, 6, 7, 8],
[9, 10, 11, 12],
[1, 2, 3, 4]])
# ArithmeticOnlyMatrix tests
def test_abs():
m = ArithmeticOnlyMatrix([[1, -2], [x, y]])
assert abs(m) == ArithmeticOnlyMatrix([[1, 2], [Abs(x), Abs(y)]])
def test_add():
m = ArithmeticOnlyMatrix([[1, 2, 3], [x, y, x], [2*y, -50, z*x]])
assert m + m == ArithmeticOnlyMatrix([[2, 4, 6], [2*x, 2*y, 2*x], [4*y, -100, 2*z*x]])
n = ArithmeticOnlyMatrix(1, 2, [1, 2])
raises(ShapeError, lambda: m + n)
def test_multiplication():
a = ArithmeticOnlyMatrix((
(1, 2),
(3, 1),
(0, 6),
))
b = ArithmeticOnlyMatrix((
(1, 2),
(3, 0),
))
raises(ShapeError, lambda: b*a)
raises(TypeError, lambda: a*{})
c = a*b
assert c[0, 0] == 7
assert c[0, 1] == 2
assert c[1, 0] == 6
assert c[1, 1] == 6
assert c[2, 0] == 18
assert c[2, 1] == 0
try:
eval('c = a @ b')
except SyntaxError:
pass
else:
assert c[0, 0] == 7
assert c[0, 1] == 2
assert c[1, 0] == 6
assert c[1, 1] == 6
assert c[2, 0] == 18
assert c[2, 1] == 0
h = a.multiply_elementwise(c)
assert h == matrix_multiply_elementwise(a, c)
assert h[0, 0] == 7
assert h[0, 1] == 4
assert h[1, 0] == 18
assert h[1, 1] == 6
assert h[2, 0] == 0
assert h[2, 1] == 0
raises(ShapeError, lambda: a.multiply_elementwise(b))
c = b * Symbol("x")
assert isinstance(c, ArithmeticOnlyMatrix)
assert c[0, 0] == x
assert c[0, 1] == 2*x
assert c[1, 0] == 3*x
assert c[1, 1] == 0
c2 = x * b
assert c == c2
c = 5 * b
assert isinstance(c, ArithmeticOnlyMatrix)
assert c[0, 0] == 5
assert c[0, 1] == 2*5
assert c[1, 0] == 3*5
assert c[1, 1] == 0
try:
eval('c = 5 @ b')
except SyntaxError:
pass
else:
assert isinstance(c, ArithmeticOnlyMatrix)
assert c[0, 0] == 5
assert c[0, 1] == 2*5
assert c[1, 0] == 3*5
assert c[1, 1] == 0
def test_matmul():
a = Matrix([[1, 2], [3, 4]])
assert a.__matmul__(2) == NotImplemented
assert a.__rmatmul__(2) == NotImplemented
#This is done this way because @ is only supported in Python 3.5+
#To check 2@a case
try:
eval('2 @ a')
except SyntaxError:
pass
except TypeError: #TypeError is raised in case of NotImplemented is returned
pass
#Check a@2 case
try:
eval('a @ 2')
except SyntaxError:
pass
except TypeError: #TypeError is raised in case of NotImplemented is returned
pass
def test_power():
raises(NonSquareMatrixError, lambda: Matrix((1, 2))**2)
A = ArithmeticOnlyMatrix([[2, 3], [4, 5]])
assert (A**5)[:] == (6140, 8097, 10796, 14237)
A = ArithmeticOnlyMatrix([[2, 1, 3], [4, 2, 4], [6, 12, 1]])
assert (A**3)[:] == (290, 262, 251, 448, 440, 368, 702, 954, 433)
assert A**0 == eye(3)
assert A**1 == A
assert (ArithmeticOnlyMatrix([[2]]) ** 100)[0, 0] == 2**100
assert ArithmeticOnlyMatrix([[1, 2], [3, 4]])**Integer(2) == ArithmeticOnlyMatrix([[7, 10], [15, 22]])
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(1)/2, S(2)/2])
# DeterminantOnlyMatrix tests
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])
# 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='bareiss') == 4*x - 24
assert e.det(method='berkowitz') == 4*x - 24
raises(ValueError, lambda: e.det(iszerofunc="test"))
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_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')
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())
# ReductionsOnlyMatrix tests
def test_row_op():
e = eye_Reductions(3)
raises(ValueError, lambda: e.elementary_row_op("abc"))
raises(ValueError, lambda: e.elementary_row_op())
raises(ValueError, lambda: e.elementary_row_op('n->kn', row=5, k=5))
raises(ValueError, lambda: e.elementary_row_op('n->kn', row=-5, k=5))
raises(ValueError, lambda: e.elementary_row_op('n<->m', row1=1, row2=5))
raises(ValueError, lambda: e.elementary_row_op('n<->m', row1=5, row2=1))
raises(ValueError, lambda: e.elementary_row_op('n<->m', row1=-5, row2=1))
raises(ValueError, lambda: e.elementary_row_op('n<->m', row1=1, row2=-5))
raises(ValueError, lambda: e.elementary_row_op('n->n+km', row1=1, row2=5, k=5))
raises(ValueError, lambda: e.elementary_row_op('n->n+km', row1=5, row2=1, k=5))
raises(ValueError, lambda: e.elementary_row_op('n->n+km', row1=-5, row2=1, k=5))
raises(ValueError, lambda: e.elementary_row_op('n->n+km', row1=1, row2=-5, k=5))
raises(ValueError, lambda: e.elementary_row_op('n->n+km', row1=1, row2=1, k=5))
# test various ways to set arguments
assert e.elementary_row_op("n->kn", 0, 5) == Matrix([[5, 0, 0], [0, 1, 0], [0, 0, 1]])
assert e.elementary_row_op("n->kn", 1, 5) == Matrix([[1, 0, 0], [0, 5, 0], [0, 0, 1]])
assert e.elementary_row_op("n->kn", row=1, k=5) == Matrix([[1, 0, 0], [0, 5, 0], [0, 0, 1]])
assert e.elementary_row_op("n->kn", row1=1, k=5) == Matrix([[1, 0, 0], [0, 5, 0], [0, 0, 1]])
assert e.elementary_row_op("n<->m", 0, 1) == Matrix([[0, 1, 0], [1, 0, 0], [0, 0, 1]])
assert e.elementary_row_op("n<->m", row1=0, row2=1) == Matrix([[0, 1, 0], [1, 0, 0], [0, 0, 1]])
assert e.elementary_row_op("n<->m", row=0, row2=1) == Matrix([[0, 1, 0], [1, 0, 0], [0, 0, 1]])
assert e.elementary_row_op("n->n+km", 0, 5, 1) == Matrix([[1, 5, 0], [0, 1, 0], [0, 0, 1]])
assert e.elementary_row_op("n->n+km", row=0, k=5, row2=1) == Matrix([[1, 5, 0], [0, 1, 0], [0, 0, 1]])
assert e.elementary_row_op("n->n+km", row1=0, k=5, row2=1) == Matrix([[1, 5, 0], [0, 1, 0], [0, 0, 1]])
# make sure the matrix doesn't change size
a = ReductionsOnlyMatrix(2, 3, [0]*6)
assert a.elementary_row_op("n->kn", 1, 5) == Matrix(2, 3, [0]*6)
assert a.elementary_row_op("n<->m", 0, 1) == Matrix(2, 3, [0]*6)
assert a.elementary_row_op("n->n+km", 0, 5, 1) == Matrix(2, 3, [0]*6)
def test_col_op():
e = eye_Reductions(3)
raises(ValueError, lambda: e.elementary_col_op("abc"))
raises(ValueError, lambda: e.elementary_col_op())
raises(ValueError, lambda: e.elementary_col_op('n->kn', col=5, k=5))
raises(ValueError, lambda: e.elementary_col_op('n->kn', col=-5, k=5))
raises(ValueError, lambda: e.elementary_col_op('n<->m', col1=1, col2=5))
raises(ValueError, lambda: e.elementary_col_op('n<->m', col1=5, col2=1))
raises(ValueError, lambda: e.elementary_col_op('n<->m', col1=-5, col2=1))
raises(ValueError, lambda: e.elementary_col_op('n<->m', col1=1, col2=-5))
raises(ValueError, lambda: e.elementary_col_op('n->n+km', col1=1, col2=5, k=5))
raises(ValueError, lambda: e.elementary_col_op('n->n+km', col1=5, col2=1, k=5))
raises(ValueError, lambda: e.elementary_col_op('n->n+km', col1=-5, col2=1, k=5))
raises(ValueError, lambda: e.elementary_col_op('n->n+km', col1=1, col2=-5, k=5))
raises(ValueError, lambda: e.elementary_col_op('n->n+km', col1=1, col2=1, k=5))
# test various ways to set arguments
assert e.elementary_col_op("n->kn", 0, 5) == Matrix([[5, 0, 0], [0, 1, 0], [0, 0, 1]])
assert e.elementary_col_op("n->kn", 1, 5) == Matrix([[1, 0, 0], [0, 5, 0], [0, 0, 1]])
assert e.elementary_col_op("n->kn", col=1, k=5) == Matrix([[1, 0, 0], [0, 5, 0], [0, 0, 1]])
assert e.elementary_col_op("n->kn", col1=1, k=5) == Matrix([[1, 0, 0], [0, 5, 0], [0, 0, 1]])
assert e.elementary_col_op("n<->m", 0, 1) == Matrix([[0, 1, 0], [1, 0, 0], [0, 0, 1]])
assert e.elementary_col_op("n<->m", col1=0, col2=1) == Matrix([[0, 1, 0], [1, 0, 0], [0, 0, 1]])
assert e.elementary_col_op("n<->m", col=0, col2=1) == Matrix([[0, 1, 0], [1, 0, 0], [0, 0, 1]])
assert e.elementary_col_op("n->n+km", 0, 5, 1) == Matrix([[1, 0, 0], [5, 1, 0], [0, 0, 1]])
assert e.elementary_col_op("n->n+km", col=0, k=5, col2=1) == Matrix([[1, 0, 0], [5, 1, 0], [0, 0, 1]])
assert e.elementary_col_op("n->n+km", col1=0, k=5, col2=1) == Matrix([[1, 0, 0], [5, 1, 0], [0, 0, 1]])
# make sure the matrix doesn't change size
a = ReductionsOnlyMatrix(2, 3, [0]*6)
assert a.elementary_col_op("n->kn", 1, 5) == Matrix(2, 3, [0]*6)
assert a.elementary_col_op("n<->m", 0, 1) == Matrix(2, 3, [0]*6)
assert a.elementary_col_op("n->n+km", 0, 5, 1) == Matrix(2, 3, [0]*6)
def test_is_echelon():
zro = zeros_Reductions(3)
ident = eye_Reductions(3)
assert zro.is_echelon
assert ident.is_echelon
a = ReductionsOnlyMatrix(0, 0, [])
assert a.is_echelon
a = ReductionsOnlyMatrix(2, 3, [3, 2, 1, 0, 0, 6])
assert a.is_echelon
a = ReductionsOnlyMatrix(2, 3, [0, 0, 6, 3, 2, 1])
assert not a.is_echelon
x = Symbol('x')
a = ReductionsOnlyMatrix(3, 1, [x, 0, 0])
assert a.is_echelon
a = ReductionsOnlyMatrix(3, 1, [x, x, 0])
assert not a.is_echelon
a = ReductionsOnlyMatrix(3, 3, [0, 0, 0, 1, 2, 3, 0, 0, 0])
assert not a.is_echelon
def test_echelon_form():
# echelon form is not unique, but the result
# must be row-equivalent to the original matrix
# and it must be in echelon form.
a = zeros_Reductions(3)
e = eye_Reductions(3)
# we can assume the zero matrix and the identity matrix shouldn't change
assert a.echelon_form() == a
assert e.echelon_form() == e
a = ReductionsOnlyMatrix(0, 0, [])
assert a.echelon_form() == a
a = ReductionsOnlyMatrix(1, 1, [5])
assert a.echelon_form() == a
# now we get to the real tests
def verify_row_null_space(mat, rows, nulls):
for v in nulls:
assert all(t.is_zero for t in a_echelon*v)
for v in rows:
if not all(t.is_zero for t in v):
assert not all(t.is_zero for t in a_echelon*v.transpose())
a = ReductionsOnlyMatrix(3, 3, [1, 2, 3, 4, 5, 6, 7, 8, 9])
nulls = [Matrix([
[ 1],
[-2],
[ 1]])]
rows = [a[i,:] for i in range(a.rows)]
a_echelon = a.echelon_form()
assert a_echelon.is_echelon
verify_row_null_space(a, rows, nulls)
a = ReductionsOnlyMatrix(3, 3, [1, 2, 3, 4, 5, 6, 7, 8, 8])
nulls = []
rows = [a[i,:] for i in range(a.rows)]
a_echelon = a.echelon_form()
assert a_echelon.is_echelon
verify_row_null_space(a, rows, nulls)
a = ReductionsOnlyMatrix(3, 3, [2, 1, 3, 0, 0, 0, 2, 1, 3])
nulls = [Matrix([
[-S(1)/2],
[ 1],
[ 0]]),
Matrix([
[-S(3)/2],
[ 0],
[ 1]])]
rows = [a[i,:] for i in range(a.rows)]
a_echelon = a.echelon_form()
assert a_echelon.is_echelon
verify_row_null_space(a, rows, nulls)
# this one requires a row swap
a = ReductionsOnlyMatrix(3, 3, [2, 1, 3, 0, 0, 0, 1, 1, 3])
nulls = [Matrix([
[ 0],
[ -3],
[ 1]])]
rows = [a[i,:] for i in range(a.rows)]
a_echelon = a.echelon_form()
assert a_echelon.is_echelon
verify_row_null_space(a, rows, nulls)
a = ReductionsOnlyMatrix(3, 3, [0, 3, 3, 0, 2, 2, 0, 1, 1])
nulls = [Matrix([
[1],
[0],
[0]]),
Matrix([
[ 0],
[-1],
[ 1]])]
rows = [a[i,:] for i in range(a.rows)]
a_echelon = a.echelon_form()
assert a_echelon.is_echelon
verify_row_null_space(a, rows, nulls)
a = ReductionsOnlyMatrix(2, 3, [2, 2, 3, 3, 3, 0])
nulls = [Matrix([
[-1],
[1],
[0]])]
rows = [a[i,:] for i in range(a.rows)]
a_echelon = a.echelon_form()
assert a_echelon.is_echelon
verify_row_null_space(a, rows, nulls)
def test_rref():
e = ReductionsOnlyMatrix(0, 0, [])
assert e.rref(pivots=False) == e
e = ReductionsOnlyMatrix(1, 1, [1])
a = ReductionsOnlyMatrix(1, 1, [5])
assert e.rref(pivots=False) == a.rref(pivots=False) == e
a = ReductionsOnlyMatrix(3, 1, [1, 2, 3])
assert a.rref(pivots=False) == Matrix([[1], [0], [0]])
a = ReductionsOnlyMatrix(1, 3, [1, 2, 3])
assert a.rref(pivots=False) == Matrix([[1, 2, 3]])
a = ReductionsOnlyMatrix(3, 3, [1, 2, 3, 4, 5, 6, 7, 8, 9])
assert a.rref(pivots=False) == Matrix([
[1, 0, -1],
[0, 1, 2],
[0, 0, 0]])
a = ReductionsOnlyMatrix(3, 3, [1, 2, 3, 1, 2, 3, 1, 2, 3])
b = ReductionsOnlyMatrix(3, 3, [1, 2, 3, 0, 0, 0, 0, 0, 0])
c = ReductionsOnlyMatrix(3, 3, [0, 0, 0, 1, 2, 3, 0, 0, 0])
d = ReductionsOnlyMatrix(3, 3, [0, 0, 0, 0, 0, 0, 1, 2, 3])
assert a.rref(pivots=False) == \
b.rref(pivots=False) == \
c.rref(pivots=False) == \
d.rref(pivots=False) == b
e = eye_Reductions(3)
z = zeros_Reductions(3)
assert e.rref(pivots=False) == e
assert z.rref(pivots=False) == z
a = ReductionsOnlyMatrix([
[ 0, 0, 1, 2, 2, -5, 3],
[-1, 5, 2, 2, 1, -7, 5],
[ 0, 0, -2, -3, -3, 8, -5],
[-1, 5, 0, -1, -2, 1, 0]])
mat, pivot_offsets = a.rref()
assert mat == Matrix([
[1, -5, 0, 0, 1, 1, -1],
[0, 0, 1, 0, 0, -1, 1],
[0, 0, 0, 1, 1, -2, 1],
[0, 0, 0, 0, 0, 0, 0]])
assert pivot_offsets == (0, 2, 3)
a = ReductionsOnlyMatrix([[S(1)/19, S(1)/5, 2, 3],
[ 4, 5, 6, 7],
[ 8, 9, 10, 11],
[ 12, 13, 14, 15]])
assert a.rref(pivots=False) == Matrix([
[1, 0, 0, -S(76)/157],
[0, 1, 0, -S(5)/157],
[0, 0, 1, S(238)/157],
[0, 0, 0, 0]])
x = Symbol('x')
a = ReductionsOnlyMatrix(2, 3, [x, 1, 1, sqrt(x), x, 1])
for i, j in zip(a.rref(pivots=False),
[1, 0, sqrt(x)*(-x + 1)/(-x**(S(5)/2) + x),
0, 1, 1/(sqrt(x) + x + 1)]):
assert simplify(i - j).is_zero
# 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_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
raises(SymPyDeprecationWarning,
lambda: SpecialOnlyMatrix.jordan_block(cols=3, eigenvalue=2))
raises(SymPyDeprecationWarning,
lambda: SpecialOnlyMatrix.jordan_block(rows=3, eigenvalue=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)
# SubspaceOnlyMatrix tests
def test_columnspace():
m = SubspaceOnlyMatrix([[ 1, 2, 0, 2, 5],
[-2, -5, 1, -1, -8],
[ 0, -3, 3, 4, 1],
[ 3, 6, 0, -7, 2]])
basis = m.columnspace()
assert basis[0] == Matrix([1, -2, 0, 3])
assert basis[1] == Matrix([2, -5, -3, 6])
assert basis[2] == Matrix([2, -1, 4, -7])
assert len(basis) == 3
assert Matrix.hstack(m, *basis).columnspace() == basis
def test_rowspace():
m = SubspaceOnlyMatrix([[ 1, 2, 0, 2, 5],
[-2, -5, 1, -1, -8],
[ 0, -3, 3, 4, 1],
[ 3, 6, 0, -7, 2]])
basis = m.rowspace()
assert basis[0] == Matrix([[1, 2, 0, 2, 5]])
assert basis[1] == Matrix([[0, -1, 1, 3, 2]])
assert basis[2] == Matrix([[0, 0, 0, 5, 5]])
assert len(basis) == 3
def test_nullspace():
m = SubspaceOnlyMatrix([[ 1, 2, 0, 2, 5],
[-2, -5, 1, -1, -8],
[ 0, -3, 3, 4, 1],
[ 3, 6, 0, -7, 2]])
basis = m.nullspace()
assert basis[0] == Matrix([-2, 1, 1, 0, 0])
assert basis[1] == Matrix([-1, -1, 0, -1, 1])
# make sure the null space is really gets zeroed
assert all(e.is_zero for e in m*basis[0])
assert all(e.is_zero for e in m*basis[1])
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([[-S(12)/5], [S(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([[S(5)/26], [S(25)/26]])]
# EigenOnlyMatrix tests
def test_eigenvals():
M = EigenOnlyMatrix([[0, 1, 1],
[1, 0, 0],
[1, 1, 1]])
assert M.eigenvals() == {2*S.One: 1, -S.One: 1, S.Zero: 1}
# if we cannot factor the char poly, we raise an error
m = Matrix([
[3, 0, 0, 0, -3],
[0, -3, -3, 0, 3],
[0, 3, 0, 3, 0],
[0, 0, 3, 0, 3],
[3, 0, 0, 3, 0]])
raises(MatrixError, lambda: m.eigenvals())
def test_eigenvects():
M = EigenOnlyMatrix([[0, 1, 1],
[1, 0, 0],
[1, 1, 1]])
vecs = M.eigenvects()
for val, mult, vec_list in vecs:
assert len(vec_list) == 1
assert M*vec_list[0] == val*vec_list[0]
def test_left_eigenvects():
M = EigenOnlyMatrix([[0, 1, 1],
[1, 0, 0],
[1, 1, 1]])
vecs = M.left_eigenvects()
for val, mult, vec_list in vecs:
assert len(vec_list) == 1
assert vec_list[0]*M == val*vec_list[0]
def test_diagonalize():
m = EigenOnlyMatrix(2, 2, [0, -1, 1, 0])
raises(MatrixError, lambda: m.diagonalize(reals_only=True))
P, D = m.diagonalize()
assert D.is_diagonal()
assert D == Matrix([
[-I, 0],
[ 0, I]])
# make sure we use floats out if floats are passed in
m = EigenOnlyMatrix(2, 2, [0, .5, .5, 0])
P, D = m.diagonalize()
assert all(isinstance(e, Float) for e in D.values())
assert all(isinstance(e, Float) for e in P.values())
_, D2 = m.diagonalize(reals_only=True)
assert D == D2
def test_is_diagonalizable():
a, b, c = symbols('a b c')
m = EigenOnlyMatrix(2, 2, [a, c, c, b])
assert m.is_symmetric()
assert m.is_diagonalizable()
assert not EigenOnlyMatrix(2, 2, [1, 1, 0, 1]).is_diagonalizable()
m = EigenOnlyMatrix(2, 2, [0, -1, 1, 0])
assert m.is_diagonalizable()
assert not m.is_diagonalizable(reals_only=True)
def test_jordan_form():
m = Matrix(3, 2, [-3, 1, -3, 20, 3, 10])
raises(NonSquareMatrixError, lambda: m.jordan_form())
# the next two tests test the cases where the old
# algorithm failed due to the fact that the block structure can
# *NOT* be determined from algebraic and geometric multiplicity alone
# This can be seen most easily when one lets compute the J.c.f. of a matrix that
# is in J.c.f already.
m = EigenOnlyMatrix(4, 4, [2, 1, 0, 0,
0, 2, 1, 0,
0, 0, 2, 0,
0, 0, 0, 2
])
P, J = m.jordan_form()
assert m == J
m = EigenOnlyMatrix(4, 4, [2, 1, 0, 0,
0, 2, 0, 0,
0, 0, 2, 1,
0, 0, 0, 2
])
P, J = m.jordan_form()
assert m == J
A = Matrix([[ 2, 4, 1, 0],
[-4, 2, 0, 1],
[ 0, 0, 2, 4],
[ 0, 0, -4, 2]])
P, J = A.jordan_form()
assert simplify(P*J*P.inv()) == A
assert EigenOnlyMatrix(1,1,[1]).jordan_form() == (Matrix([1]), Matrix([1]))
assert EigenOnlyMatrix(1,1,[1]).jordan_form(calc_transform=False) == Matrix([1])
# make sure if we cannot factor the characteristic polynomial, we raise an error
m = Matrix([[3, 0, 0, 0, -3], [0, -3, -3, 0, 3], [0, 3, 0, 3, 0], [0, 0, 3, 0, 3], [3, 0, 0, 3, 0]])
raises(MatrixError, lambda: m.jordan_form())
# make sure that if the input has floats, the output does too
m = Matrix([
[ 0.6875, 0.125 + 0.1875*sqrt(3)],
[0.125 + 0.1875*sqrt(3), 0.3125]])
P, J = m.jordan_form()
assert all(isinstance(x, Float) or x == 0 for x in P)
assert all(isinstance(x, Float) or x == 0 for x in J)
def test_singular_values():
x = Symbol('x', real=True)
A = EigenOnlyMatrix([[0, 1*I], [2, 0]])
# if singular values can be sorted, they should be in decreasing order
assert A.singular_values() == [2, 1]
A = eye(3)
A[1, 1] = x
A[2, 2] = 5
vals = A.singular_values()
# since Abs(x) cannot be sorted, test set equality
assert set(vals) == set([5, 1, Abs(x)])
A = EigenOnlyMatrix([[sin(x), cos(x)], [-cos(x), sin(x)]])
vals = [sv.trigsimp() for sv in A.singular_values()]
assert vals == [S(1), S(1)]
A = EigenOnlyMatrix([
[2, 4],
[1, 3],
[0, 0],
[0, 0]
])
assert A.singular_values() == \
[sqrt(sqrt(221) + 15), sqrt(15 - sqrt(221))]
assert A.T.singular_values() == \
[sqrt(sqrt(221) + 15), sqrt(15 - sqrt(221)), 0, 0]
# 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, [S(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___eq__():
assert (EigenOnlyMatrix(
[[0, 1, 1],
[1, 0, 0],
[1, 1, 1]]) == {}) is False
|
be18afa57001c303c1edeeb2db5092724320414f6f1e018bb6e08d18b2010443
|
from sympy import Abs, S, Symbol, I, Rational, PurePoly, Float
from sympy.matrices import (Matrix, SparseMatrix, eye, ones, zeros,
ShapeError)
from sympy.utilities.pytest import raises
def test_sparse_matrix():
def sparse_eye(n):
return SparseMatrix.eye(n)
def sparse_zeros(n):
return SparseMatrix.zeros(n)
# creation args
raises(TypeError, lambda: SparseMatrix(1, 2))
a = SparseMatrix((
(1, 0),
(0, 1)
))
assert SparseMatrix(a) == a
from sympy.matrices import MutableSparseMatrix, MutableDenseMatrix
a = MutableSparseMatrix([])
b = MutableDenseMatrix([1, 2])
assert a.row_join(b) == b
assert a.col_join(b) == b
assert type(a.row_join(b)) == type(a)
assert type(a.col_join(b)) == type(a)
# make sure 0 x n matrices get stacked correctly
sparse_matrices = [SparseMatrix.zeros(0, n) for n in range(4)]
assert SparseMatrix.hstack(*sparse_matrices) == Matrix(0, 6, [])
sparse_matrices = [SparseMatrix.zeros(n, 0) for n in range(4)]
assert SparseMatrix.vstack(*sparse_matrices) == Matrix(6, 0, [])
# test element assignment
a = SparseMatrix((
(1, 0),
(0, 1)
))
a[3] = 4
assert a[1, 1] == 4
a[3] = 1
a[0, 0] = 2
assert a == SparseMatrix((
(2, 0),
(0, 1)
))
a[1, 0] = 5
assert a == SparseMatrix((
(2, 0),
(5, 1)
))
a[1, 1] = 0
assert a == SparseMatrix((
(2, 0),
(5, 0)
))
assert a._smat == {(0, 0): 2, (1, 0): 5}
# test_multiplication
a = SparseMatrix((
(1, 2),
(3, 1),
(0, 6),
))
b = SparseMatrix((
(1, 2),
(3, 0),
))
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
x = Symbol("x")
c = b * Symbol("x")
assert isinstance(c, SparseMatrix)
assert c[0, 0] == x
assert c[0, 1] == 2*x
assert c[1, 0] == 3*x
assert c[1, 1] == 0
c = 5 * b
assert isinstance(c, SparseMatrix)
assert c[0, 0] == 5
assert c[0, 1] == 2*5
assert c[1, 0] == 3*5
assert c[1, 1] == 0
#test_power
A = SparseMatrix([[2, 3], [4, 5]])
assert (A**5)[:] == [6140, 8097, 10796, 14237]
A = SparseMatrix([[2, 1, 3], [4, 2, 4], [6, 12, 1]])
assert (A**3)[:] == [290, 262, 251, 448, 440, 368, 702, 954, 433]
# test_creation
x = Symbol("x")
a = SparseMatrix([[x, 0], [0, 0]])
m = a
assert m.cols == m.rows
assert m.cols == 2
assert m[:] == [x, 0, 0, 0]
b = SparseMatrix(2, 2, [x, 0, 0, 0])
m = b
assert m.cols == m.rows
assert m.cols == 2
assert m[:] == [x, 0, 0, 0]
assert a == b
S = sparse_eye(3)
S.row_del(1)
assert S == SparseMatrix([
[1, 0, 0],
[0, 0, 1]])
S = sparse_eye(3)
S.col_del(1)
assert S == SparseMatrix([
[1, 0],
[0, 0],
[0, 1]])
S = SparseMatrix.eye(3)
S[2, 1] = 2
S.col_swap(1, 0)
assert S == SparseMatrix([
[0, 1, 0],
[1, 0, 0],
[2, 0, 1]])
a = SparseMatrix(1, 2, [1, 2])
b = a.copy()
c = a.copy()
assert a[0] == 1
a.row_del(0)
assert a == SparseMatrix(0, 2, [])
b.col_del(1)
assert b == SparseMatrix(1, 1, [1])
assert SparseMatrix([[1, 2, 3], [1, 2], [1]]) == Matrix([
[1, 2, 3],
[1, 2, 0],
[1, 0, 0]])
assert SparseMatrix(4, 4, {(1, 1): sparse_eye(2)}) == Matrix([
[0, 0, 0, 0],
[0, 1, 0, 0],
[0, 0, 1, 0],
[0, 0, 0, 0]])
raises(ValueError, lambda: SparseMatrix(1, 1, {(1, 1): 1}))
assert SparseMatrix(1, 2, [1, 2]).tolist() == [[1, 2]]
assert SparseMatrix(2, 2, [1, [2, 3]]).tolist() == [[1, 0], [2, 3]]
raises(ValueError, lambda: SparseMatrix(2, 2, [1]))
raises(ValueError, lambda: SparseMatrix(1, 1, [[1, 2]]))
assert SparseMatrix([.1]).has(Float)
# autosizing
assert SparseMatrix(None, {(0, 1): 0}).shape == (0, 0)
assert SparseMatrix(None, {(0, 1): 1}).shape == (1, 2)
assert SparseMatrix(None, None, {(0, 1): 1}).shape == (1, 2)
raises(ValueError, lambda: SparseMatrix(None, 1, [[1, 2]]))
raises(ValueError, lambda: SparseMatrix(1, None, [[1, 2]]))
raises(ValueError, lambda: SparseMatrix(3, 3, {(0, 0): ones(2), (1, 1): 2}))
# test_determinant
x, y = Symbol('x'), Symbol('y')
assert SparseMatrix(1, 1, [0]).det() == 0
assert SparseMatrix([[1]]).det() == 1
assert SparseMatrix(((-3, 2), (8, -5))).det() == -1
assert SparseMatrix(((x, 1), (y, 2*y))).det() == 2*x*y - y
assert SparseMatrix(( (1, 1, 1),
(1, 2, 3),
(1, 3, 6) )).det() == 1
assert SparseMatrix(( ( 3, -2, 0, 5),
(-2, 1, -2, 2),
( 0, -2, 5, 0),
( 5, 0, 3, 4) )).det() == -289
assert SparseMatrix(( ( 1, 2, 3, 4),
( 5, 6, 7, 8),
( 9, 10, 11, 12),
(13, 14, 15, 16) )).det() == 0
assert SparseMatrix(( (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) )).det() == 275
assert SparseMatrix(( (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) )).det() == -55
assert SparseMatrix(( (-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) )).det() == 11664
assert SparseMatrix(( ( 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) )).det() == 123
# test_slicing
m0 = sparse_eye(4)
assert m0[:3, :3] == sparse_eye(3)
assert m0[2:4, 0:2] == sparse_zeros(2)
m1 = SparseMatrix(3, 3, lambda i, j: i + j)
assert m1[0, :] == SparseMatrix(1, 3, (0, 1, 2))
assert m1[1:3, 1] == SparseMatrix(2, 1, (2, 3))
m2 = SparseMatrix(
[[0, 1, 2, 3], [4, 5, 6, 7], [8, 9, 10, 11], [12, 13, 14, 15]])
assert m2[:, -1] == SparseMatrix(4, 1, [3, 7, 11, 15])
assert m2[-2:, :] == SparseMatrix([[8, 9, 10, 11], [12, 13, 14, 15]])
assert SparseMatrix([[1, 2], [3, 4]])[[1], [1]] == Matrix([[4]])
# test_submatrix_assignment
m = sparse_zeros(4)
m[2:4, 2:4] = sparse_eye(2)
assert m == SparseMatrix([(0, 0, 0, 0),
(0, 0, 0, 0),
(0, 0, 1, 0),
(0, 0, 0, 1)])
assert len(m._smat) == 2
m[:2, :2] = sparse_eye(2)
assert m == sparse_eye(4)
m[:, 0] = SparseMatrix(4, 1, (1, 2, 3, 4))
assert m == SparseMatrix([(1, 0, 0, 0),
(2, 1, 0, 0),
(3, 0, 1, 0),
(4, 0, 0, 1)])
m[:, :] = sparse_zeros(4)
assert m == sparse_zeros(4)
m[:, :] = ((1, 2, 3, 4), (5, 6, 7, 8), (9, 10, 11, 12), (13, 14, 15, 16))
assert m == SparseMatrix((( 1, 2, 3, 4),
( 5, 6, 7, 8),
( 9, 10, 11, 12),
(13, 14, 15, 16)))
m[:2, 0] = [0, 0]
assert m == SparseMatrix((( 0, 2, 3, 4),
( 0, 6, 7, 8),
( 9, 10, 11, 12),
(13, 14, 15, 16)))
# test_reshape
m0 = sparse_eye(3)
assert m0.reshape(1, 9) == SparseMatrix(1, 9, (1, 0, 0, 0, 1, 0, 0, 0, 1))
m1 = SparseMatrix(3, 4, lambda i, j: i + j)
assert m1.reshape(4, 3) == \
SparseMatrix([(0, 1, 2), (3, 1, 2), (3, 4, 2), (3, 4, 5)])
assert m1.reshape(2, 6) == \
SparseMatrix([(0, 1, 2, 3, 1, 2), (3, 4, 2, 3, 4, 5)])
# test_applyfunc
m0 = sparse_eye(3)
assert m0.applyfunc(lambda x: 2*x) == sparse_eye(3)*2
assert m0.applyfunc(lambda x: 0 ) == sparse_zeros(3)
# test__eval_Abs
assert abs(SparseMatrix(((x, 1), (y, 2*y)))) == SparseMatrix(((Abs(x), 1), (Abs(y), 2*Abs(y))))
# test_LUdecomp
testmat = SparseMatrix([[ 0, 2, 5, 3],
[ 3, 3, 7, 4],
[ 8, 4, 0, 2],
[-2, 6, 3, 4]])
L, U, p = testmat.LUdecomposition()
assert L.is_lower
assert U.is_upper
assert (L*U).permute_rows(p, 'backward') - testmat == sparse_zeros(4)
testmat = SparseMatrix([[ 6, -2, 7, 4],
[ 0, 3, 6, 7],
[ 1, -2, 7, 4],
[-9, 2, 6, 3]])
L, U, p = testmat.LUdecomposition()
assert L.is_lower
assert U.is_upper
assert (L*U).permute_rows(p, 'backward') - testmat == sparse_zeros(4)
x, y, z = Symbol('x'), Symbol('y'), Symbol('z')
M = Matrix(((1, x, 1), (2, y, 0), (y, 0, z)))
L, U, p = M.LUdecomposition()
assert L.is_lower
assert U.is_upper
assert (L*U).permute_rows(p, 'backward') - M == sparse_zeros(3)
# test_LUsolve
A = SparseMatrix([[2, 3, 5],
[3, 6, 2],
[8, 3, 6]])
x = SparseMatrix(3, 1, [3, 7, 5])
b = A*x
soln = A.LUsolve(b)
assert soln == x
A = SparseMatrix([[0, -1, 2],
[5, 10, 7],
[8, 3, 4]])
x = SparseMatrix(3, 1, [-1, 2, 5])
b = A*x
soln = A.LUsolve(b)
assert soln == x
# test_inverse
A = sparse_eye(4)
assert A.inv() == sparse_eye(4)
assert A.inv(method="CH") == sparse_eye(4)
assert A.inv(method="LDL") == sparse_eye(4)
A = SparseMatrix([[2, 3, 5],
[3, 6, 2],
[7, 2, 6]])
Ainv = SparseMatrix(Matrix(A).inv())
assert A*Ainv == sparse_eye(3)
assert A.inv(method="CH") == Ainv
assert A.inv(method="LDL") == Ainv
A = SparseMatrix([[2, 3, 5],
[3, 6, 2],
[5, 2, 6]])
Ainv = SparseMatrix(Matrix(A).inv())
assert A*Ainv == sparse_eye(3)
assert A.inv(method="CH") == Ainv
assert A.inv(method="LDL") == Ainv
# test_cross
v1 = Matrix(1, 3, [1, 2, 3])
v2 = Matrix(1, 3, [3, 4, 5])
assert v1.cross(v2) == Matrix(1, 3, [-2, 4, -2])
assert v1.norm(2)**2 == 14
# conjugate
a = SparseMatrix(((1, 2 + I), (3, 4)))
assert a.C == SparseMatrix([
[1, 2 - I],
[3, 4]
])
# mul
assert a*Matrix(2, 2, [1, 0, 0, 1]) == a
assert a + Matrix(2, 2, [1, 1, 1, 1]) == SparseMatrix([
[2, 3 + I],
[4, 5]
])
# col join
assert a.col_join(sparse_eye(2)) == SparseMatrix([
[1, 2 + I],
[3, 4],
[1, 0],
[0, 1]
])
# symmetric
assert not a.is_symmetric(simplify=False)
# test_cofactor
assert sparse_eye(3) == sparse_eye(3).cofactor_matrix()
test = SparseMatrix([[1, 3, 2], [2, 6, 3], [2, 3, 6]])
assert test.cofactor_matrix() == \
SparseMatrix([[27, -6, -6], [-12, 2, 3], [-3, 1, 0]])
test = SparseMatrix([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
assert test.cofactor_matrix() == \
SparseMatrix([[-3, 6, -3], [6, -12, 6], [-3, 6, -3]])
# test_jacobian
x = Symbol('x')
y = Symbol('y')
L = SparseMatrix(1, 2, [x**2*y, 2*y**2 + x*y])
syms = [x, y]
assert L.jacobian(syms) == Matrix([[2*x*y, x**2], [y, 4*y + x]])
L = SparseMatrix(1, 2, [x, x**2*y**3])
assert L.jacobian(syms) == SparseMatrix([[1, 0], [2*x*y**3, x**2*3*y**2]])
# test_QR
A = Matrix([[1, 2], [2, 3]])
Q, S = A.QRdecomposition()
R = Rational
assert Q == Matrix([
[ 5**R(-1, 2), (R(2)/5)*(R(1)/5)**R(-1, 2)],
[2*5**R(-1, 2), (-R(1)/5)*(R(1)/5)**R(-1, 2)]])
assert S == Matrix([
[5**R(1, 2), 8*5**R(-1, 2)],
[ 0, (R(1)/5)**R(1, 2)]])
assert Q*S == A
assert Q.T * Q == sparse_eye(2)
R = Rational
# test nullspace
# first test reduced row-ech form
M = SparseMatrix([[5, 7, 2, 1],
[1, 6, 2, -1]])
out, tmp = M.rref()
assert out == Matrix([[1, 0, -R(2)/23, R(13)/23],
[0, 1, R(8)/23, R(-6)/23]])
M = SparseMatrix([[ 1, 3, 0, 2, 6, 3, 1],
[-2, -6, 0, -2, -8, 3, 1],
[ 3, 9, 0, 0, 6, 6, 2],
[-1, -3, 0, 1, 0, 9, 3]])
out, tmp = M.rref()
assert out == Matrix([[1, 3, 0, 0, 2, 0, 0],
[0, 0, 0, 1, 2, 0, 0],
[0, 0, 0, 0, 0, 1, R(1)/3],
[0, 0, 0, 0, 0, 0, 0]])
# now check the vectors
basis = M.nullspace()
assert basis[0] == Matrix([-3, 1, 0, 0, 0, 0, 0])
assert basis[1] == Matrix([0, 0, 1, 0, 0, 0, 0])
assert basis[2] == Matrix([-2, 0, 0, -2, 1, 0, 0])
assert basis[3] == Matrix([0, 0, 0, 0, 0, R(-1)/3, 1])
# test eigen
x = Symbol('x')
y = Symbol('y')
sparse_eye3 = sparse_eye(3)
assert sparse_eye3.charpoly(x) == PurePoly(((x - 1)**3))
assert sparse_eye3.charpoly(y) == PurePoly(((y - 1)**3))
# test values
M = Matrix([( 0, 1, -1),
( 1, 1, 0),
(-1, 0, 1)])
vals = M.eigenvals()
assert sorted(vals.keys()) == [-1, 1, 2]
R = Rational
M = Matrix([[1, 0, 0],
[0, 1, 0],
[0, 0, 1]])
assert M.eigenvects() == [(1, 3, [
Matrix([1, 0, 0]),
Matrix([0, 1, 0]),
Matrix([0, 0, 1])])]
M = Matrix([[5, 0, 2],
[3, 2, 0],
[0, 0, 1]])
assert M.eigenvects() == [(1, 1, [Matrix([R(-1)/2, R(3)/2, 1])]),
(2, 1, [Matrix([0, 1, 0])]),
(5, 1, [Matrix([1, 1, 0])])]
assert M.zeros(3, 5) == SparseMatrix(3, 5, {})
A = SparseMatrix(10, 10, {(0, 0): 18, (0, 9): 12, (1, 4): 18, (2, 7): 16, (3, 9): 12, (4, 2): 19, (5, 7): 16, (6, 2): 12, (9, 7): 18})
assert A.row_list() == [(0, 0, 18), (0, 9, 12), (1, 4, 18), (2, 7, 16), (3, 9, 12), (4, 2, 19), (5, 7, 16), (6, 2, 12), (9, 7, 18)]
assert A.col_list() == [(0, 0, 18), (4, 2, 19), (6, 2, 12), (1, 4, 18), (2, 7, 16), (5, 7, 16), (9, 7, 18), (0, 9, 12), (3, 9, 12)]
assert SparseMatrix.eye(2).nnz() == 2
def test_transpose():
assert SparseMatrix(((1, 2), (3, 4))).transpose() == \
SparseMatrix(((1, 3), (2, 4)))
def test_trace():
assert SparseMatrix(((1, 2), (3, 4))).trace() == 5
assert SparseMatrix(((0, 0), (0, 4))).trace() == 4
def test_CL_RL():
assert SparseMatrix(((1, 2), (3, 4))).row_list() == \
[(0, 0, 1), (0, 1, 2), (1, 0, 3), (1, 1, 4)]
assert SparseMatrix(((1, 2), (3, 4))).col_list() == \
[(0, 0, 1), (1, 0, 3), (0, 1, 2), (1, 1, 4)]
def test_add():
assert SparseMatrix(((1, 0), (0, 1))) + SparseMatrix(((0, 1), (1, 0))) == \
SparseMatrix(((1, 1), (1, 1)))
a = SparseMatrix(100, 100, lambda i, j: int(j != 0 and i % j == 0))
b = SparseMatrix(100, 100, lambda i, j: int(i != 0 and j % i == 0))
assert (len(a._smat) + len(b._smat) - len((a + b)._smat) > 0)
def test_errors():
raises(ValueError, lambda: SparseMatrix(1.4, 2, lambda i, j: 0))
raises(TypeError, lambda: SparseMatrix([1, 2, 3], [1, 2]))
raises(ValueError, lambda: SparseMatrix([[1, 2], [3, 4]])[(1, 2, 3)])
raises(IndexError, lambda: SparseMatrix([[1, 2], [3, 4]])[5])
raises(ValueError, lambda: SparseMatrix([[1, 2], [3, 4]])[1, 2, 3])
raises(TypeError,
lambda: SparseMatrix([[1, 2], [3, 4]]).copyin_list([0, 1], set([])))
raises(
IndexError, lambda: SparseMatrix([[1, 2], [3, 4]])[1, 2])
raises(TypeError, lambda: SparseMatrix([1, 2, 3]).cross(1))
raises(IndexError, lambda: SparseMatrix(1, 2, [1, 2])[3])
raises(ShapeError,
lambda: SparseMatrix(1, 2, [1, 2]) + SparseMatrix(2, 1, [2, 1]))
def test_len():
assert not SparseMatrix()
assert SparseMatrix() == SparseMatrix([])
assert SparseMatrix() == SparseMatrix([[]])
def test_sparse_zeros_sparse_eye():
assert SparseMatrix.eye(3) == eye(3, cls=SparseMatrix)
assert len(SparseMatrix.eye(3)._smat) == 3
assert SparseMatrix.zeros(3) == zeros(3, cls=SparseMatrix)
assert len(SparseMatrix.zeros(3)._smat) == 0
def test_copyin():
s = SparseMatrix(3, 3, {})
s[1, 0] = 1
assert s[:, 0] == SparseMatrix(Matrix([0, 1, 0]))
assert s[3] == 1
assert s[3: 4] == [1]
s[1, 1] = 42
assert s[1, 1] == 42
assert s[1, 1:] == SparseMatrix([[42, 0]])
s[1, 1:] = Matrix([[5, 6]])
assert s[1, :] == SparseMatrix([[1, 5, 6]])
s[1, 1:] = [[42, 43]]
assert s[1, :] == SparseMatrix([[1, 42, 43]])
s[0, 0] = 17
assert s[:, :1] == SparseMatrix([17, 1, 0])
s[0, 0] = [1, 1, 1]
assert s[:, 0] == SparseMatrix([1, 1, 1])
s[0, 0] = Matrix([1, 1, 1])
assert s[:, 0] == SparseMatrix([1, 1, 1])
s[0, 0] = SparseMatrix([1, 1, 1])
assert s[:, 0] == SparseMatrix([1, 1, 1])
def test_sparse_solve():
from sympy.matrices import SparseMatrix
A = SparseMatrix(((25, 15, -5), (15, 18, 0), (-5, 0, 11)))
assert A.cholesky() == Matrix([
[ 5, 0, 0],
[ 3, 3, 0],
[-1, 1, 3]])
assert A.cholesky() * A.cholesky().T == Matrix([
[25, 15, -5],
[15, 18, 0],
[-5, 0, 11]])
A = SparseMatrix(((25, 15, -5), (15, 18, 0), (-5, 0, 11)))
L, D = A.LDLdecomposition()
assert 15*L == Matrix([
[15, 0, 0],
[ 9, 15, 0],
[-3, 5, 15]])
assert D == Matrix([
[25, 0, 0],
[ 0, 9, 0],
[ 0, 0, 9]])
assert L * D * L.T == A
A = SparseMatrix(((3, 0, 2), (0, 0, 1), (1, 2, 0)))
assert A.inv() * A == SparseMatrix(eye(3))
A = SparseMatrix([
[ 2, -1, 0],
[-1, 2, -1],
[ 0, 0, 2]])
ans = SparseMatrix([
[S(2)/3, S(1)/3, S(1)/6],
[S(1)/3, S(2)/3, S(1)/3],
[ 0, 0, S(1)/2]])
assert A.inv(method='CH') == ans
assert A.inv(method='LDL') == ans
assert A * ans == SparseMatrix(eye(3))
s = A.solve(A[:, 0], 'LDL')
assert A*s == A[:, 0]
s = A.solve(A[:, 0], 'CH')
assert A*s == A[:, 0]
A = A.col_join(A)
s = A.solve_least_squares(A[:, 0], 'CH')
assert A*s == A[:, 0]
s = A.solve_least_squares(A[:, 0], 'LDL')
assert A*s == A[:, 0]
def test_hermitian():
x = Symbol('x')
a = SparseMatrix([[0, I], [-I, 0]])
assert a.is_hermitian
a = SparseMatrix([[1, I], [-I, 1]])
assert a.is_hermitian
a[0, 0] = 2*I
assert a.is_hermitian is False
a[0, 0] = x
assert a.is_hermitian is None
a[0, 1] = a[1, 0]*I
assert a.is_hermitian is False
|
747641a426d8d8ff8f39afa21e4896ee1679011b4e5bdcab50c2aca571037b25
|
import random
from sympy import (
Abs, Add, E, Float, I, Integer, Max, Min, N, Poly, Pow, PurePoly, Rational,
S, Symbol, cos, exp, expand_mul, oo, pi, signsimp, simplify, sin, sqrt, symbols,
sympify, trigsimp, tan, sstr, diff, Function)
from sympy.matrices.matrices import (ShapeError, MatrixError,
NonSquareMatrixError, DeferredVector, _find_reasonable_pivot_naive,
_simplify)
from sympy.matrices import (
GramSchmidt, ImmutableMatrix, ImmutableSparseMatrix, Matrix,
SparseMatrix, casoratian, diag, eye, hessian,
matrix_multiply_elementwise, ones, randMatrix, rot_axis1, rot_axis2,
rot_axis3, wronskian, zeros, MutableDenseMatrix, ImmutableDenseMatrix, MatrixSymbol)
from sympy.core.compatibility import long, iterable, range, Hashable
from sympy.core import Tuple, Wild
from sympy.utilities.iterables import flatten, capture
from sympy.utilities.pytest import raises, XFAIL, slow, skip, warns_deprecated_sympy
from sympy.solvers import solve
from sympy.assumptions import Q
from sympy.tensor.array import Array
from sympy.matrices.expressions import MatPow
from sympy.abc import a, b, c, d, x, y, z, t
# don't re-order this list
classes = (Matrix, SparseMatrix, ImmutableMatrix, ImmutableSparseMatrix)
def test_args():
for c, cls in enumerate(classes):
m = cls.zeros(3, 2)
# all should give back the same type of arguments, e.g. ints for shape
assert m.shape == (3, 2) and all(type(i) is int for i in m.shape)
assert m.rows == 3 and type(m.rows) is int
assert m.cols == 2 and type(m.cols) is int
if not c % 2:
assert type(m._mat) in (list, tuple, Tuple)
else:
assert type(m._smat) is dict
def test_division():
v = Matrix(1, 2, [x, y])
assert v.__div__(z) == Matrix(1, 2, [x/z, y/z])
assert v.__truediv__(z) == Matrix(1, 2, [x/z, y/z])
assert v/z == Matrix(1, 2, [x/z, y/z])
def test_sum():
m = Matrix([[1, 2, 3], [x, y, x], [2*y, -50, z*x]])
assert m + m == Matrix([[2, 4, 6], [2*x, 2*y, 2*x], [4*y, -100, 2*z*x]])
n = Matrix(1, 2, [1, 2])
raises(ShapeError, lambda: m + n)
def test_abs():
m = Matrix(1, 2, [-3, x])
n = Matrix(1, 2, [3, Abs(x)])
assert abs(m) == n
def test_addition():
a = Matrix((
(1, 2),
(3, 1),
))
b = Matrix((
(1, 2),
(3, 0),
))
assert a + b == a.add(b) == Matrix([[2, 4], [6, 1]])
def test_fancy_index_matrix():
for M in (Matrix, SparseMatrix):
a = M(3, 3, range(9))
assert a == a[:, :]
assert a[1, :] == Matrix(1, 3, [3, 4, 5])
assert a[:, 1] == Matrix([1, 4, 7])
assert a[[0, 1], :] == Matrix([[0, 1, 2], [3, 4, 5]])
assert a[[0, 1], 2] == a[[0, 1], [2]]
assert a[2, [0, 1]] == a[[2], [0, 1]]
assert a[:, [0, 1]] == Matrix([[0, 1], [3, 4], [6, 7]])
assert a[0, 0] == 0
assert a[0:2, :] == Matrix([[0, 1, 2], [3, 4, 5]])
assert a[:, 0:2] == Matrix([[0, 1], [3, 4], [6, 7]])
assert a[::2, 1] == a[[0, 2], 1]
assert a[1, ::2] == a[1, [0, 2]]
a = M(3, 3, range(9))
assert a[[0, 2, 1, 2, 1], :] == Matrix([
[0, 1, 2],
[6, 7, 8],
[3, 4, 5],
[6, 7, 8],
[3, 4, 5]])
assert a[:, [0,2,1,2,1]] == Matrix([
[0, 2, 1, 2, 1],
[3, 5, 4, 5, 4],
[6, 8, 7, 8, 7]])
a = SparseMatrix.zeros(3)
a[1, 2] = 2
a[0, 1] = 3
a[2, 0] = 4
assert a.extract([1, 1], [2]) == Matrix([
[2],
[2]])
assert a.extract([1, 0], [2, 2, 2]) == Matrix([
[2, 2, 2],
[0, 0, 0]])
assert a.extract([1, 0, 1, 2], [2, 0, 1, 0]) == Matrix([
[2, 0, 0, 0],
[0, 0, 3, 0],
[2, 0, 0, 0],
[0, 4, 0, 4]])
def test_multiplication():
a = Matrix((
(1, 2),
(3, 1),
(0, 6),
))
b = Matrix((
(1, 2),
(3, 0),
))
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 = matrix_multiply_elementwise(a, c)
assert h == a.multiply_elementwise(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: matrix_multiply_elementwise(a, b))
c = b * Symbol("x")
assert isinstance(c, Matrix)
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, Matrix)
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, Matrix)
assert c[0, 0] == 5
assert c[0, 1] == 2*5
assert c[1, 0] == 3*5
assert c[1, 1] == 0
def test_power():
raises(NonSquareMatrixError, lambda: Matrix((1, 2))**2)
R = Rational
A = Matrix([[2, 3], [4, 5]])
assert (A**-3)[:] == [R(-269)/8, R(153)/8, R(51)/2, R(-29)/2]
assert (A**5)[:] == [6140, 8097, 10796, 14237]
A = Matrix([[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 (Matrix([[2]]) ** 100)[0, 0] == 2**100
assert eye(2)**10000000 == eye(2)
assert Matrix([[1, 2], [3, 4]])**Integer(2) == Matrix([[7, 10], [15, 22]])
A = Matrix([[33, 24], [48, 57]])
assert (A**(S(1)/2))[:] == [5, 2, 4, 7]
A = Matrix([[0, 4], [-1, 5]])
assert (A**(S(1)/2))**2 == A
assert Matrix([[1, 0], [1, 1]])**(S(1)/2) == Matrix([[1, 0], [S.Half, 1]])
assert Matrix([[1, 0], [1, 1]])**0.5 == Matrix([[1.0, 0], [0.5, 1.0]])
from sympy.abc import a, b, n
assert Matrix([[1, a], [0, 1]])**n == Matrix([[1, a*n], [0, 1]])
assert Matrix([[b, a], [0, b]])**n == Matrix([[b**n, a*b**(n-1)*n], [0, b**n]])
assert Matrix([[a, 1, 0], [0, a, 1], [0, 0, a]])**n == Matrix([
[a**n, a**(n-1)*n, a**(n-2)*(n-1)*n/2],
[0, a**n, a**(n-1)*n],
[0, 0, a**n]])
assert Matrix([[a, 1, 0], [0, a, 0], [0, 0, b]])**n == Matrix([
[a**n, a**(n-1)*n, 0],
[0, a**n, 0],
[0, 0, b**n]])
A = Matrix([[1, 0], [1, 7]])
assert A._matrix_pow_by_jordan_blocks(3) == A._eval_pow_by_recursion(3)
A = Matrix([[2]])
assert A**10 == Matrix([[2**10]]) == A._matrix_pow_by_jordan_blocks(10) == \
A._eval_pow_by_recursion(10)
# testing a matrix that cannot be jordan blocked issue 11766
m = Matrix([[3, 0, 0, 0, -3], [0, -3, -3, 0, 3], [0, 3, 0, 3, 0], [0, 0, 3, 0, 3], [3, 0, 0, 3, 0]])
raises(MatrixError, lambda: m._matrix_pow_by_jordan_blocks(10))
# test issue 11964
raises(ValueError, lambda: Matrix([[1, 1], [3, 3]])._matrix_pow_by_jordan_blocks(-10))
A = Matrix([[0, 1, 0], [0, 0, 1], [0, 0, 0]]) # Nilpotent jordan block size 3
assert A**10.0 == Matrix([[0, 0, 0], [0, 0, 0], [0, 0, 0]])
raises(ValueError, lambda: A**2.1)
raises(ValueError, lambda: A**(S(3)/2))
A = Matrix([[8, 1], [3, 2]])
assert A**10.0 == Matrix([[1760744107, 272388050], [817164150, 126415807]])
A = Matrix([[0, 0, 1], [0, 0, 1], [0, 0, 1]]) # Nilpotent jordan block size 1
assert A**10.2 == Matrix([[0, 0, 1], [0, 0, 1], [0, 0, 1]])
A = Matrix([[0, 1, 0], [0, 0, 1], [0, 0, 1]]) # Nilpotent jordan block size 2
assert A**10.0 == Matrix([[0, 0, 1], [0, 0, 1], [0, 0, 1]])
n = Symbol('n', integer=True)
assert isinstance(A**n, MatPow)
n = Symbol('n', integer=True, nonnegative=True)
raises(ValueError, lambda: A**n)
assert A**(n + 2) == Matrix([[0, 0, 1], [0, 0, 1], [0, 0, 1]])
raises(ValueError, lambda: A**(S(3)/2))
A = Matrix([[0, 0, 1], [3, 0, 1], [4, 3, 1]])
assert A**5.0 == Matrix([[168, 72, 89], [291, 144, 161], [572, 267, 329]])
assert A**5.0 == A**5
A = Matrix([[0, 1, 0],[-1, 0, 0],[0, 0, 0]])
n = Symbol("n")
An = A**n
assert An.subs(n, 2).doit() == A**2
raises(ValueError, lambda: An.subs(n, -2).doit())
assert An * An == A**(2*n)
def test_creation():
raises(ValueError, lambda: Matrix(5, 5, range(20)))
raises(ValueError, lambda: Matrix(5, -1, []))
raises(IndexError, lambda: Matrix((1, 2))[2])
with raises(IndexError):
Matrix((1, 2))[1:2] = 5
with raises(IndexError):
Matrix((1, 2))[3] = 5
assert Matrix() == Matrix([]) == Matrix([[]]) == Matrix(0, 0, [])
a = Matrix([[x, 0], [0, 0]])
m = a
assert m.cols == m.rows
assert m.cols == 2
assert m[:] == [x, 0, 0, 0]
b = Matrix(2, 2, [x, 0, 0, 0])
m = b
assert m.cols == m.rows
assert m.cols == 2
assert m[:] == [x, 0, 0, 0]
assert a == b
assert Matrix(b) == b
c = Matrix((
Matrix((
(1, 2, 3),
(4, 5, 6)
)),
(7, 8, 9)
))
assert c.cols == 3
assert c.rows == 3
assert c[:] == [1, 2, 3, 4, 5, 6, 7, 8, 9]
assert Matrix(eye(2)) == eye(2)
assert ImmutableMatrix(ImmutableMatrix(eye(2))) == ImmutableMatrix(eye(2))
assert ImmutableMatrix(c) == c.as_immutable()
assert Matrix(ImmutableMatrix(c)) == ImmutableMatrix(c).as_mutable()
assert c is not Matrix(c)
def test_tolist():
lst = [[S.One, S.Half, x*y, S.Zero], [x, y, z, x**2], [y, -S.One, z*x, 3]]
m = Matrix(lst)
assert m.tolist() == lst
def test_as_mutable():
assert zeros(0, 3).as_mutable() == zeros(0, 3)
assert zeros(0, 3).as_immutable() == ImmutableMatrix(zeros(0, 3))
assert zeros(3, 0).as_immutable() == ImmutableMatrix(zeros(3, 0))
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="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="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="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="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="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="bareiss") == 275
assert M.det(method="berkowitz") == 275
assert M.det(method="lu") == 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="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="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="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="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_slicing():
m0 = eye(4)
assert m0[:3, :3] == eye(3)
assert m0[2:4, 0:2] == zeros(2)
m1 = Matrix(3, 3, lambda i, j: i + j)
assert m1[0, :] == Matrix(1, 3, (0, 1, 2))
assert m1[1:3, 1] == Matrix(2, 1, (2, 3))
m2 = Matrix([[0, 1, 2, 3], [4, 5, 6, 7], [8, 9, 10, 11], [12, 13, 14, 15]])
assert m2[:, -1] == Matrix(4, 1, [3, 7, 11, 15])
assert m2[-2:, :] == Matrix([[8, 9, 10, 11], [12, 13, 14, 15]])
def test_submatrix_assignment():
m = zeros(4)
m[2:4, 2:4] = eye(2)
assert m == Matrix(((0, 0, 0, 0),
(0, 0, 0, 0),
(0, 0, 1, 0),
(0, 0, 0, 1)))
m[:2, :2] = eye(2)
assert m == eye(4)
m[:, 0] = Matrix(4, 1, (1, 2, 3, 4))
assert m == Matrix(((1, 0, 0, 0),
(2, 1, 0, 0),
(3, 0, 1, 0),
(4, 0, 0, 1)))
m[:, :] = zeros(4)
assert m == zeros(4)
m[:, :] = [(1, 2, 3, 4), (5, 6, 7, 8), (9, 10, 11, 12), (13, 14, 15, 16)]
assert m == Matrix(((1, 2, 3, 4),
(5, 6, 7, 8),
(9, 10, 11, 12),
(13, 14, 15, 16)))
m[:2, 0] = [0, 0]
assert m == Matrix(((0, 2, 3, 4),
(0, 6, 7, 8),
(9, 10, 11, 12),
(13, 14, 15, 16)))
def test_extract():
m = Matrix(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_reshape():
m0 = eye(3)
assert m0.reshape(1, 9) == Matrix(1, 9, (1, 0, 0, 0, 1, 0, 0, 0, 1))
m1 = Matrix(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_applyfunc():
m0 = eye(3)
assert m0.applyfunc(lambda x: 2*x) == eye(3)*2
assert m0.applyfunc(lambda x: 0) == zeros(3)
def test_expand():
m0 = Matrix([[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 Matrix([exp(I*a)]).expand(complex=True) == \
Matrix([cos(a) + I*sin(a)])
assert Matrix([[0, 1, 2], [0, 0, -1], [0, 0, 0]]).exp() == Matrix([
[1, 1, Rational(3, 2)],
[0, 1, -1],
[0, 0, 1]]
)
def test_refine():
m0 = Matrix([[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_random():
M = randMatrix(3, 3)
M = randMatrix(3, 3, seed=3)
assert M == randMatrix(3, 3, seed=3)
M = randMatrix(3, 4, 0, 150)
M = randMatrix(3, seed=4, symmetric=True)
assert M == randMatrix(3, seed=4, symmetric=True)
S = M.copy()
S.simplify()
assert S == M # doesn't fail when elements are Numbers, not int
rng = random.Random(4)
assert M == randMatrix(3, symmetric=True, prng=rng)
# Ensure symmetry
for size in (10, 11): # Test odd and even
for percent in (100, 70, 30):
M = randMatrix(size, symmetric=True, percent=percent, prng=rng)
assert M == M.T
M = randMatrix(10, min=1, percent=70)
zero_count = 0
for i in range(M.shape[0]):
for j in range(M.shape[1]):
if M[i, j] == 0:
zero_count += 1
assert zero_count == 30
def test_LUdecomp():
testmat = Matrix([[0, 2, 5, 3],
[3, 3, 7, 4],
[8, 4, 0, 2],
[-2, 6, 3, 4]])
L, U, p = testmat.LUdecomposition()
assert L.is_lower
assert U.is_upper
assert (L*U).permute_rows(p, 'backward') - testmat == zeros(4)
testmat = Matrix([[6, -2, 7, 4],
[0, 3, 6, 7],
[1, -2, 7, 4],
[-9, 2, 6, 3]])
L, U, p = testmat.LUdecomposition()
assert L.is_lower
assert U.is_upper
assert (L*U).permute_rows(p, 'backward') - testmat == zeros(4)
# non-square
testmat = Matrix([[1, 2, 3],
[4, 5, 6],
[7, 8, 9],
[10, 11, 12]])
L, U, p = testmat.LUdecomposition(rankcheck=False)
assert L.is_lower
assert U.is_upper
assert (L*U).permute_rows(p, 'backward') - testmat == zeros(4, 3)
# square and singular
testmat = Matrix([[1, 2, 3],
[2, 4, 6],
[4, 5, 6]])
L, U, p = testmat.LUdecomposition(rankcheck=False)
assert L.is_lower
assert U.is_upper
assert (L*U).permute_rows(p, 'backward') - testmat == zeros(3)
M = Matrix(((1, x, 1), (2, y, 0), (y, 0, z)))
L, U, p = M.LUdecomposition()
assert L.is_lower
assert U.is_upper
assert (L*U).permute_rows(p, 'backward') - M == zeros(3)
mL = Matrix((
(1, 0, 0),
(2, 3, 0),
))
assert mL.is_lower is True
assert mL.is_upper is False
mU = Matrix((
(1, 2, 3),
(0, 4, 5),
))
assert mU.is_lower is False
assert mU.is_upper is True
# test FF LUdecomp
M = Matrix([[1, 3, 3],
[3, 2, 6],
[3, 2, 2]])
P, L, Dee, U = M.LUdecompositionFF()
assert P*M == L*Dee.inv()*U
M = Matrix([[1, 2, 3, 4],
[3, -1, 2, 3],
[3, 1, 3, -2],
[6, -1, 0, 2]])
P, L, Dee, U = M.LUdecompositionFF()
assert P*M == L*Dee.inv()*U
M = Matrix([[0, 0, 1],
[2, 3, 0],
[3, 1, 4]])
P, L, Dee, U = M.LUdecompositionFF()
assert P*M == L*Dee.inv()*U
# issue 15794
M = Matrix(
[[1, 2, 3],
[4, 5, 6],
[7, 8, 9]]
)
raises(ValueError, lambda : M.LUdecomposition_Simple(rankcheck=True))
def test_LUsolve():
A = Matrix([[2, 3, 5],
[3, 6, 2],
[8, 3, 6]])
x = Matrix(3, 1, [3, 7, 5])
b = A*x
soln = A.LUsolve(b)
assert soln == x
A = Matrix([[0, -1, 2],
[5, 10, 7],
[8, 3, 4]])
x = Matrix(3, 1, [-1, 2, 5])
b = A*x
soln = A.LUsolve(b)
assert soln == x
A = Matrix([[2, 1], [1, 0], [1, 0]]) # issue 14548
b = Matrix([3, 1, 1])
assert A.LUsolve(b) == Matrix([1, 1])
b = Matrix([3, 1, 2]) # inconsistent
raises(ValueError, lambda: A.LUsolve(b))
A = Matrix([[0, -1, 2],
[5, 10, 7],
[8, 3, 4],
[2, 3, 5],
[3, 6, 2],
[8, 3, 6]])
x = Matrix([2, 1, -4])
b = A*x
soln = A.LUsolve(b)
assert soln == x
A = Matrix([[0, -1, 2], [5, 10, 7]]) # underdetermined
x = Matrix([-1, 2, 0])
b = A*x
raises(NotImplementedError, lambda: A.LUsolve(b))
A = Matrix(4, 4, lambda i, j: 1/(i+j+1) if i != 3 else 0)
b = Matrix.zeros(4, 1)
raises(NotImplementedError, lambda: A.LUsolve(b))
def test_QRsolve():
A = Matrix([[2, 3, 5],
[3, 6, 2],
[8, 3, 6]])
x = Matrix(3, 1, [3, 7, 5])
b = A*x
soln = A.QRsolve(b)
assert soln == x
x = Matrix([[1, 2], [3, 4], [5, 6]])
b = A*x
soln = A.QRsolve(b)
assert soln == x
A = Matrix([[0, -1, 2],
[5, 10, 7],
[8, 3, 4]])
x = Matrix(3, 1, [-1, 2, 5])
b = A*x
soln = A.QRsolve(b)
assert soln == x
x = Matrix([[7, 8], [9, 10], [11, 12]])
b = A*x
soln = A.QRsolve(b)
assert soln == x
def test_inverse():
A = eye(4)
assert A.inv() == eye(4)
assert A.inv(method="LU") == eye(4)
assert A.inv(method="ADJ") == eye(4)
A = Matrix([[2, 3, 5],
[3, 6, 2],
[8, 3, 6]])
Ainv = A.inv()
assert A*Ainv == eye(3)
assert A.inv(method="LU") == Ainv
assert A.inv(method="ADJ") == Ainv
# test that immutability is not a problem
cls = ImmutableMatrix
m = cls([[48, 49, 31],
[ 9, 71, 94],
[59, 28, 65]])
assert all(type(m.inv(s)) is cls for s in 'GE ADJ LU'.split())
cls = ImmutableSparseMatrix
m = cls([[48, 49, 31],
[ 9, 71, 94],
[59, 28, 65]])
assert all(type(m.inv(s)) is cls for s in 'CH LDL'.split())
def test_matrix_inverse_mod():
A = Matrix(2, 1, [1, 0])
raises(NonSquareMatrixError, lambda: A.inv_mod(2))
A = Matrix(2, 2, [1, 0, 0, 0])
raises(ValueError, lambda: A.inv_mod(2))
A = Matrix(2, 2, [1, 2, 3, 4])
Ai = Matrix(2, 2, [1, 1, 0, 1])
assert A.inv_mod(3) == Ai
A = Matrix(2, 2, [1, 0, 0, 1])
assert A.inv_mod(2) == A
A = Matrix(3, 3, [1, 2, 3, 4, 5, 6, 7, 8, 9])
raises(ValueError, lambda: A.inv_mod(5))
A = Matrix(3, 3, [5, 1, 3, 2, 6, 0, 2, 1, 1])
Ai = Matrix(3, 3, [6, 8, 0, 1, 5, 6, 5, 6, 4])
assert A.inv_mod(9) == Ai
A = Matrix(3, 3, [1, 6, -3, 4, 1, -5, 3, -5, 5])
Ai = Matrix(3, 3, [4, 3, 3, 1, 2, 5, 1, 5, 1])
assert A.inv_mod(6) == Ai
A = Matrix(3, 3, [1, 6, 1, 4, 1, 5, 3, 2, 5])
Ai = Matrix(3, 3, [6, 0, 3, 6, 6, 4, 1, 6, 1])
assert A.inv_mod(7) == Ai
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_jacobian_hessian():
L = Matrix(1, 2, [x**2*y, 2*y**2 + x*y])
syms = [x, y]
assert L.jacobian(syms) == Matrix([[2*x*y, x**2], [y, 4*y + x]])
L = Matrix(1, 2, [x, x**2*y**3])
assert L.jacobian(syms) == Matrix([[1, 0], [2*x*y**3, x**2*3*y**2]])
f = x**2*y
syms = [x, y]
assert hessian(f, syms) == Matrix([[2*y, 2*x], [2*x, 0]])
f = x**2*y**3
assert hessian(f, syms) == \
Matrix([[2*y**3, 6*x*y**2], [6*x*y**2, 6*x**2*y]])
f = z + x*y**2
g = x**2 + 2*y**3
ans = Matrix([[0, 2*y],
[2*y, 2*x]])
assert ans == hessian(f, Matrix([x, y]))
assert ans == hessian(f, Matrix([x, y]).T)
assert hessian(f, (y, x), [g]) == Matrix([
[ 0, 6*y**2, 2*x],
[6*y**2, 2*x, 2*y],
[ 2*x, 2*y, 0]])
def test_QR():
A = Matrix([[1, 2], [2, 3]])
Q, S = A.QRdecomposition()
R = Rational
assert Q == Matrix([
[ 5**R(-1, 2), (R(2)/5)*(R(1)/5)**R(-1, 2)],
[2*5**R(-1, 2), (-R(1)/5)*(R(1)/5)**R(-1, 2)]])
assert S == Matrix([[5**R(1, 2), 8*5**R(-1, 2)], [0, (R(1)/5)**R(1, 2)]])
assert Q*S == A
assert Q.T * Q == eye(2)
A = Matrix([[1, 1, 1], [1, 1, 3], [2, 3, 4]])
Q, R = A.QRdecomposition()
assert Q.T * Q == eye(Q.cols)
assert R.is_upper
assert A == Q*R
def test_QR_non_square():
# Narrow (cols < rows) matrices
A = Matrix([[9, 0, 26], [12, 0, -7], [0, 4, 4], [0, -3, -3]])
Q, R = A.QRdecomposition()
assert Q.T * Q == eye(Q.cols)
assert R.is_upper
assert A == Q*R
A = Matrix([[1, -1, 4], [1, 4, -2], [1, 4, 2], [1, -1, 0]])
Q, R = A.QRdecomposition()
assert Q.T * Q == eye(Q.cols)
assert R.is_upper
assert A == Q*R
A = Matrix(2, 1, [1, 2])
Q, R = A.QRdecomposition()
assert Q.T * Q == eye(Q.cols)
assert R.is_upper
assert A == Q*R
# Wide (cols > rows) matrices
A = Matrix([[1, 2, 3], [4, 5, 6]])
Q, R = A.QRdecomposition()
assert Q.T * Q == eye(Q.cols)
assert R.is_upper
assert A == Q*R
A = Matrix([[1, 2, 3, 4], [1, 4, 9, 16], [1, 8, 27, 64]])
Q, R = A.QRdecomposition()
assert Q.T * Q == eye(Q.cols)
assert R.is_upper
assert A == Q*R
A = Matrix(1, 2, [1, 2])
Q, R = A.QRdecomposition()
assert Q.T * Q == eye(Q.cols)
assert R.is_upper
assert A == Q*R
def test_QR_trivial():
# Rank deficient matrices
A = Matrix([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
Q, R = A.QRdecomposition()
assert Q.T * Q == eye(Q.cols)
assert R.is_upper
assert A == Q*R
A = Matrix([[1, 1, 1], [2, 2, 2], [3, 3, 3], [4, 4, 4]])
Q, R = A.QRdecomposition()
assert Q.T * Q == eye(Q.cols)
assert R.is_upper
assert A == Q*R
A = Matrix([[1, 1, 1], [2, 2, 2], [3, 3, 3], [4, 4, 4]]).T
Q, R = A.QRdecomposition()
assert Q.T * Q == eye(Q.cols)
assert R.is_upper
assert A == Q*R
# Zero rank matrices
A = Matrix([[0, 0, 0]])
Q, R = A.QRdecomposition()
assert Q.T * Q == eye(Q.cols)
assert R.is_upper
assert A == Q*R
A = Matrix([[0, 0, 0]]).T
Q, R = A.QRdecomposition()
assert Q.T * Q == eye(Q.cols)
assert R.is_upper
assert A == Q*R
A = Matrix([[0, 0, 0], [0, 0, 0]])
Q, R = A.QRdecomposition()
assert Q.T * Q == eye(Q.cols)
assert R.is_upper
assert A == Q*R
A = Matrix([[0, 0, 0], [0, 0, 0]]).T
Q, R = A.QRdecomposition()
assert Q.T * Q == eye(Q.cols)
assert R.is_upper
assert A == Q*R
# Rank deficient matrices with zero norm from beginning columns
A = Matrix([[0, 0, 0], [1, 2, 3]]).T
Q, R = A.QRdecomposition()
assert Q.T * Q == eye(Q.cols)
assert R.is_upper
assert A == Q*R
A = Matrix([[0, 0, 0, 0], [1, 2, 3, 4], [0, 0, 0, 0]]).T
Q, R = A.QRdecomposition()
assert Q.T * Q == eye(Q.cols)
assert R.is_upper
assert A == Q*R
A = Matrix([[0, 0, 0, 0], [1, 2, 3, 4], [0, 0, 0, 0], [2, 4, 6, 8]]).T
Q, R = A.QRdecomposition()
assert Q.T * Q == eye(Q.cols)
assert R.is_upper
assert A == Q*R
A = Matrix([[0, 0, 0], [0, 0, 0], [0, 0, 0], [1, 2, 3]]).T
Q, R = A.QRdecomposition()
assert Q.T * Q == eye(Q.cols)
assert R.is_upper
assert A == Q*R
def test_nullspace():
# first test reduced row-ech form
R = Rational
M = Matrix([[5, 7, 2, 1],
[1, 6, 2, -1]])
out, tmp = M.rref()
assert out == Matrix([[1, 0, -R(2)/23, R(13)/23],
[0, 1, R(8)/23, R(-6)/23]])
M = Matrix([[-5, -1, 4, -3, -1],
[ 1, -1, -1, 1, 0],
[-1, 0, 0, 0, 0],
[ 4, 1, -4, 3, 1],
[-2, 0, 2, -2, -1]])
assert M*M.nullspace()[0] == Matrix(5, 1, [0]*5)
M = Matrix([[ 1, 3, 0, 2, 6, 3, 1],
[-2, -6, 0, -2, -8, 3, 1],
[ 3, 9, 0, 0, 6, 6, 2],
[-1, -3, 0, 1, 0, 9, 3]])
out, tmp = M.rref()
assert out == Matrix([[1, 3, 0, 0, 2, 0, 0],
[0, 0, 0, 1, 2, 0, 0],
[0, 0, 0, 0, 0, 1, R(1)/3],
[0, 0, 0, 0, 0, 0, 0]])
# now check the vectors
basis = M.nullspace()
assert basis[0] == Matrix([-3, 1, 0, 0, 0, 0, 0])
assert basis[1] == Matrix([0, 0, 1, 0, 0, 0, 0])
assert basis[2] == Matrix([-2, 0, 0, -2, 1, 0, 0])
assert basis[3] == Matrix([0, 0, 0, 0, 0, R(-1)/3, 1])
# issue 4797; just see that we can do it when rows > cols
M = Matrix([[1, 2], [2, 4], [3, 6]])
assert M.nullspace()
def test_columnspace():
M = Matrix([[ 1, 2, 0, 2, 5],
[-2, -5, 1, -1, -8],
[ 0, -3, 3, 4, 1],
[ 3, 6, 0, -7, 2]])
# now check the vectors
basis = M.columnspace()
assert basis[0] == Matrix([1, -2, 0, 3])
assert basis[1] == Matrix([2, -5, -3, 6])
assert basis[2] == Matrix([2, -1, 4, -7])
#check by columnspace definition
a, b, c, d, e = symbols('a b c d e')
X = Matrix([a, b, c, d, e])
for i in range(len(basis)):
eq=M*X-basis[i]
assert len(solve(eq, X)) != 0
#check if rank-nullity theorem holds
assert M.rank() == len(basis)
assert len(M.nullspace()) + len(M.columnspace()) == M.cols
def test_wronskian():
assert wronskian([cos(x), sin(x)], x) == cos(x)**2 + sin(x)**2
assert wronskian([exp(x), exp(2*x)], x) == exp(3*x)
assert wronskian([exp(x), x], x) == exp(x) - x*exp(x)
assert wronskian([1, x, x**2], x) == 2
w1 = -6*exp(x)*sin(x)*x + 6*cos(x)*exp(x)*x**2 - 6*exp(x)*cos(x)*x - \
exp(x)*cos(x)*x**3 + exp(x)*sin(x)*x**3
assert wronskian([exp(x), cos(x), x**3], x).expand() == w1
assert wronskian([exp(x), cos(x), x**3], x, method='berkowitz').expand() \
== w1
w2 = -x**3*cos(x)**2 - x**3*sin(x)**2 - 6*x*cos(x)**2 - 6*x*sin(x)**2
assert wronskian([sin(x), cos(x), x**3], x).expand() == w2
assert wronskian([sin(x), cos(x), x**3], x, method='berkowitz').expand() \
== w2
assert wronskian([], x) == 1
def test_eigen():
R = Rational
assert eye(3).charpoly(x) == Poly((x - 1)**3, x)
assert eye(3).charpoly(y) == Poly((y - 1)**3, y)
M = Matrix([[1, 0, 0],
[0, 1, 0],
[0, 0, 1]])
assert M.eigenvals(multiple=False) == {S.One: 3}
assert M.eigenvals(multiple=True) == [1, 1, 1]
assert M.eigenvects() == (
[(1, 3, [Matrix([1, 0, 0]),
Matrix([0, 1, 0]),
Matrix([0, 0, 1])])])
assert M.left_eigenvects() == (
[(1, 3, [Matrix([[1, 0, 0]]),
Matrix([[0, 1, 0]]),
Matrix([[0, 0, 1]])])])
M = Matrix([[0, 1, 1],
[1, 0, 0],
[1, 1, 1]])
assert M.eigenvals() == {2*S.One: 1, -S.One: 1, S.Zero: 1}
assert M.eigenvects() == (
[
(-1, 1, [Matrix([-1, 1, 0])]),
( 0, 1, [Matrix([0, -1, 1])]),
( 2, 1, [Matrix([R(2, 3), R(1, 3), 1])])
])
assert M.left_eigenvects() == (
[
(-1, 1, [Matrix([[-2, 1, 1]])]),
(0, 1, [Matrix([[-1, -1, 1]])]),
(2, 1, [Matrix([[1, 1, 1]])])
])
a = Symbol('a')
M = Matrix([[a, 0],
[0, 1]])
assert M.eigenvals() == {a: 1, S.One: 1}
M = Matrix([[1, -1],
[1, 3]])
assert M.eigenvects() == ([(2, 2, [Matrix(2, 1, [-1, 1])])])
assert M.left_eigenvects() == ([(2, 2, [Matrix([[1, 1]])])])
M = Matrix([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
a = R(15, 2)
b = 3*33**R(1, 2)
c = R(13, 2)
d = (R(33, 8) + 3*b/8)
e = (R(33, 8) - 3*b/8)
def NS(e, n):
return str(N(e, n))
r = [
(a - b/2, 1, [Matrix([(12 + 24/(c - b/2))/((c - b/2)*e) + 3/(c - b/2),
(6 + 12/(c - b/2))/e, 1])]),
( 0, 1, [Matrix([1, -2, 1])]),
(a + b/2, 1, [Matrix([(12 + 24/(c + b/2))/((c + b/2)*d) + 3/(c + b/2),
(6 + 12/(c + b/2))/d, 1])]),
]
r1 = [(NS(r[i][0], 2), NS(r[i][1], 2),
[NS(j, 2) for j in r[i][2][0]]) for i in range(len(r))]
r = M.eigenvects()
r2 = [(NS(r[i][0], 2), NS(r[i][1], 2),
[NS(j, 2) for j in r[i][2][0]]) for i in range(len(r))]
assert sorted(r1) == sorted(r2)
eps = Symbol('eps', real=True)
M = Matrix([[abs(eps), I*eps ],
[-I*eps, abs(eps) ]])
assert M.eigenvects() == (
[
( 0, 1, [Matrix([[-I*eps/abs(eps)], [1]])]),
( 2*abs(eps), 1, [ Matrix([[I*eps/abs(eps)], [1]]) ] ),
])
assert M.left_eigenvects() == (
[
(0, 1, [Matrix([[I*eps/Abs(eps), 1]])]),
(2*Abs(eps), 1, [Matrix([[-I*eps/Abs(eps), 1]])])
])
M = Matrix(3, 3, [1, 2, 0, 0, 3, 0, 2, -4, 2])
M._eigenvects = M.eigenvects(simplify=False)
assert max(i.q for i in M._eigenvects[0][2][0]) > 1
M._eigenvects = M.eigenvects(simplify=True)
assert max(i.q for i in M._eigenvects[0][2][0]) == 1
M = Matrix([[S(1)/4, 1], [1, 1]])
assert M.eigenvects(simplify=True) == [
(S(5)/8 - sqrt(73)/8, 1, [Matrix([[-sqrt(73)/8 - S(3)/8], [1]])]),
(S(5)/8 + sqrt(73)/8, 1, [Matrix([[-S(3)/8 + sqrt(73)/8], [1]])])]
assert M.eigenvects(simplify=False) ==[
(S(5)/8 - sqrt(73)/8, 1, [Matrix([[-1/(-S(3)/8 + sqrt(73)/8)],
[ 1]])]),
(S(5)/8 + sqrt(73)/8, 1, [Matrix([[-1/(-sqrt(73)/8 - S(3)/8)],
[ 1]])])]
m = Matrix([[1, .6, .6], [.6, .9, .9], [.9, .6, .6]])
evals = { S(5)/4 - sqrt(385)/20: 1, sqrt(385)/20 + S(5)/4: 1, S.Zero: 1}
assert m.eigenvals() == evals
nevals = list(sorted(m.eigenvals(rational=False).keys()))
sevals = list(sorted(evals.keys()))
assert all(abs(nevals[i] - sevals[i]) < 1e-9 for i in range(len(nevals)))
# issue 10719
assert Matrix([]).eigenvals() == {}
assert Matrix([]).eigenvects() == []
# issue 15119
raises(NonSquareMatrixError, lambda : Matrix([[1, 2], [0, 4], [0, 0]]).eigenvals())
raises(NonSquareMatrixError, lambda : Matrix([[1, 0], [3, 4], [5, 6]]).eigenvals())
raises(NonSquareMatrixError, lambda : Matrix([[1, 2, 3], [0, 5, 6]]).eigenvals())
raises(NonSquareMatrixError, lambda : Matrix([[1, 0, 0], [4, 5, 0]]).eigenvals())
raises(NonSquareMatrixError, lambda : Matrix([[1, 2, 3], [0, 5, 6]]).eigenvals(error_when_incomplete = False))
raises(NonSquareMatrixError, lambda : Matrix([[1, 0, 0], [4, 5, 0]]).eigenvals(error_when_incomplete = False))
# issue 15125
from sympy.core.function import count_ops
q = Symbol("q", positive = True)
m = Matrix([[-2, exp(-q), 1], [exp(q), -2, 1], [1, 1, -2]])
assert count_ops(m.eigenvals(simplify=False)) > count_ops(m.eigenvals(simplify=True))
assert count_ops(m.eigenvals(simplify=lambda x: x)) > count_ops(m.eigenvals(simplify=True))
assert isinstance(m.eigenvals(simplify=True, multiple=False), dict)
assert isinstance(m.eigenvals(simplify=True, multiple=True), list)
assert isinstance(m.eigenvals(simplify=lambda x: x, multiple=False), dict)
assert isinstance(m.eigenvals(simplify=lambda x: x, multiple=True), list)
def test_subs():
assert Matrix([[1, x], [x, 4]]).subs(x, 5) == Matrix([[1, 5], [5, 4]])
assert Matrix([[x, 2], [x + y, 4]]).subs([[x, -1], [y, -2]]) == \
Matrix([[-1, 2], [-3, 4]])
assert Matrix([[x, 2], [x + y, 4]]).subs([(x, -1), (y, -2)]) == \
Matrix([[-1, 2], [-3, 4]])
assert Matrix([[x, 2], [x + y, 4]]).subs({x: -1, y: -2}) == \
Matrix([[-1, 2], [-3, 4]])
assert Matrix([x*y]).subs({x: y - 1, y: x - 1}, simultaneous=True) == \
Matrix([(x - 1)*(y - 1)])
for cls in classes:
assert Matrix([[2, 0], [0, 2]]) == cls.eye(2).subs(1, 2)
def test_xreplace():
assert Matrix([[1, x], [x, 4]]).xreplace({x: 5}) == \
Matrix([[1, 5], [5, 4]])
assert Matrix([[x, 2], [x + y, 4]]).xreplace({x: -1, y: -2}) == \
Matrix([[-1, 2], [-3, 4]])
for cls in classes:
assert Matrix([[2, 0], [0, 2]]) == cls.eye(2).xreplace({1: 2})
def test_simplify():
n = Symbol('n')
f = Function('f')
M = Matrix([[ 1/x + 1/y, (x + x*y) / x ],
[ (f(x) + y*f(x))/f(x), 2 * (1/n - cos(n * pi)/n) / pi ]])
M.simplify()
assert M == Matrix([[ (x + y)/(x * y), 1 + y ],
[ 1 + y, 2*((1 - 1*cos(pi*n))/(pi*n)) ]])
eq = (1 + x)**2
M = Matrix([[eq]])
M.simplify()
assert M == Matrix([[eq]])
M.simplify(ratio=oo) == M
assert M == Matrix([[eq.simplify(ratio=oo)]])
def test_transpose():
M = Matrix([[1, 2, 3, 4, 5, 6, 7, 8, 9, 0],
[1, 2, 3, 4, 5, 6, 7, 8, 9, 0]])
assert M.T == Matrix( [ [1, 1],
[2, 2],
[3, 3],
[4, 4],
[5, 5],
[6, 6],
[7, 7],
[8, 8],
[9, 9],
[0, 0] ])
assert M.T.T == M
assert M.T == M.transpose()
def test_conjugate():
M = Matrix([[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_conj_dirac():
raises(AttributeError, lambda: eye(3).D)
M = Matrix([[1, I, I, I],
[0, 1, I, I],
[0, 0, 1, I],
[0, 0, 0, 1]])
assert M.D == Matrix([[ 1, 0, 0, 0],
[-I, 1, 0, 0],
[-I, -I, -1, 0],
[-I, -I, I, -1]])
def test_trace():
M = Matrix([[1, 0, 0],
[0, 5, 0],
[0, 0, 8]])
assert M.trace() == 14
def test_shape():
M = Matrix([[x, 0, 0],
[0, y, 0]])
assert M.shape == (2, 3)
def test_col_row_op():
M = Matrix([[x, 0, 0],
[0, y, 0]])
M.row_op(1, lambda r, j: r + j + 1)
assert M == Matrix([[x, 0, 0],
[1, y + 2, 3]])
M.col_op(0, lambda c, j: c + y**j)
assert M == Matrix([[x + 1, 0, 0],
[1 + y, y + 2, 3]])
# neither row nor slice give copies that allow the original matrix to
# be changed
assert M.row(0) == Matrix([[x + 1, 0, 0]])
r1 = M.row(0)
r1[0] = 42
assert M[0, 0] == x + 1
r1 = M[0, :-1] # also testing negative slice
r1[0] = 42
assert M[0, 0] == x + 1
c1 = M.col(0)
assert c1 == Matrix([x + 1, 1 + y])
c1[0] = 0
assert M[0, 0] == x + 1
c1 = M[:, 0]
c1[0] = 42
assert M[0, 0] == x + 1
def test_zip_row_op():
for cls in classes[:2]: # XXX: immutable matrices don't support row ops
M = cls.eye(3)
M.zip_row_op(1, 0, lambda v, u: v + 2*u)
assert M == cls([[1, 0, 0],
[2, 1, 0],
[0, 0, 1]])
M = cls.eye(3)*2
M[0, 1] = -1
M.zip_row_op(1, 0, lambda v, u: v + 2*u); M
assert M == cls([[2, -1, 0],
[4, 0, 0],
[0, 0, 2]])
def test_issue_3950():
m = Matrix([1, 2, 3])
a = Matrix([1, 2, 3])
b = Matrix([2, 2, 3])
assert not (m in [])
assert not (m in [1])
assert m != 1
assert m == a
assert m != b
def test_issue_3981():
class Index1(object):
def __index__(self):
return 1
class Index2(object):
def __index__(self):
return 2
index1 = Index1()
index2 = Index2()
m = Matrix([1, 2, 3])
assert m[index2] == 3
m[index2] = 5
assert m[2] == 5
m = Matrix([[1, 2, 3], [4, 5, 6]])
assert m[index1, index2] == 6
assert m[1, index2] == 6
assert m[index1, 2] == 6
m[index1, index2] = 4
assert m[1, 2] == 4
m[1, index2] = 6
assert m[1, 2] == 6
m[index1, 2] = 8
assert m[1, 2] == 8
def test_evalf():
a = Matrix([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_is_symbolic():
a = Matrix([[x, x], [x, x]])
assert a.is_symbolic() is True
a = Matrix([[1, 2, 3, 4], [5, 6, 7, 8]])
assert a.is_symbolic() is False
a = Matrix([[1, 2, 3, 4], [5, 6, x, 8]])
assert a.is_symbolic() is True
a = Matrix([[1, x, 3]])
assert a.is_symbolic() is True
a = Matrix([[1, 2, 3]])
assert a.is_symbolic() is False
a = Matrix([[1], [x], [3]])
assert a.is_symbolic() is True
a = Matrix([[1], [2], [3]])
assert a.is_symbolic() is False
def test_is_upper():
a = Matrix([[1, 2, 3]])
assert a.is_upper is True
a = Matrix([[1], [2], [3]])
assert a.is_upper is False
a = zeros(4, 2)
assert a.is_upper is True
def test_is_lower():
a = Matrix([[1, 2, 3]])
assert a.is_lower is False
a = Matrix([[1], [2], [3]])
assert a.is_lower is True
def test_is_nilpotent():
a = Matrix(4, 4, [0, 2, 1, 6, 0, 0, 1, 2, 0, 0, 0, 3, 0, 0, 0, 0])
assert a.is_nilpotent()
a = Matrix([[1, 0], [0, 1]])
assert not a.is_nilpotent()
a = Matrix([])
assert a.is_nilpotent()
def test_zeros_ones_fill():
n, m = 3, 5
a = zeros(n, m)
a.fill( 5 )
b = 5 * ones(n, m)
assert a == b
assert a.rows == b.rows == 3
assert a.cols == b.cols == 5
assert a.shape == b.shape == (3, 5)
assert zeros(2) == zeros(2, 2)
assert ones(2) == ones(2, 2)
assert zeros(2, 3) == Matrix(2, 3, [0]*6)
assert ones(2, 3) == Matrix(2, 3, [1]*6)
def test_empty_zeros():
a = zeros(0)
assert a == Matrix()
a = zeros(0, 2)
assert a.rows == 0
assert a.cols == 2
a = zeros(2, 0)
assert a.rows == 2
assert a.cols == 0
def test_issue_3749():
a = Matrix([[x**2, x*y], [x*sin(y), x*cos(y)]])
assert a.diff(x) == Matrix([[2*x, y], [sin(y), cos(y)]])
assert Matrix([
[x, -x, x**2],
[exp(x), 1/x - exp(-x), x + 1/x]]).limit(x, oo) == \
Matrix([[oo, -oo, oo], [oo, 0, oo]])
assert Matrix([
[(exp(x) - 1)/x, 2*x + y*x, x**x ],
[1/x, abs(x), abs(sin(x + 1))]]).limit(x, 0) == \
Matrix([[1, 0, 1], [oo, 0, sin(1)]])
assert a.integrate(x) == Matrix([
[Rational(1, 3)*x**3, y*x**2/2],
[x**2*sin(y)/2, x**2*cos(y)/2]])
def test_inv_iszerofunc():
A = eye(4)
A.col_swap(0, 1)
for method in "GE", "LU":
assert A.inv(method=method, iszerofunc=lambda x: x == 0) == \
A.inv(method="ADJ")
def test_jacobian_metrics():
rho, phi = symbols("rho,phi")
X = Matrix([rho*cos(phi), rho*sin(phi)])
Y = Matrix([rho, phi])
J = X.jacobian(Y)
assert J == X.jacobian(Y.T)
assert J == (X.T).jacobian(Y)
assert J == (X.T).jacobian(Y.T)
g = J.T*eye(J.shape[0])*J
g = g.applyfunc(trigsimp)
assert g == Matrix([[1, 0], [0, rho**2]])
def test_jacobian2():
rho, phi = symbols("rho,phi")
X = Matrix([rho*cos(phi), rho*sin(phi), rho**2])
Y = Matrix([rho, phi])
J = Matrix([
[cos(phi), -rho*sin(phi)],
[sin(phi), rho*cos(phi)],
[ 2*rho, 0],
])
assert X.jacobian(Y) == J
def test_issue_4564():
X = Matrix([exp(x + y + z), exp(x + y + z), exp(x + y + z)])
Y = Matrix([x, y, z])
for i in range(1, 3):
for j in range(1, 3):
X_slice = X[:i, :]
Y_slice = Y[:j, :]
J = X_slice.jacobian(Y_slice)
assert J.rows == i
assert J.cols == j
for k in range(j):
assert J[:, k] == X_slice
def test_nonvectorJacobian():
X = Matrix([[exp(x + y + z), exp(x + y + z)],
[exp(x + y + z), exp(x + y + z)]])
raises(TypeError, lambda: X.jacobian(Matrix([x, y, z])))
X = X[0, :]
Y = Matrix([[x, y], [x, z]])
raises(TypeError, lambda: X.jacobian(Y))
raises(TypeError, lambda: X.jacobian(Matrix([ [x, y], [x, z] ])))
def test_vec():
m = Matrix([[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_vech():
m = Matrix([[1, 2], [2, 3]])
m_vech = m.vech()
assert m_vech.cols == 1
for i in range(3):
assert m_vech[i] == i + 1
m_vech = m.vech(diagonal=False)
assert m_vech[0] == 2
m = Matrix([[1, x*(x + y)], [y*x + x**2, 1]])
m_vech = m.vech(diagonal=False)
assert m_vech[0] == x*(x + y)
m = Matrix([[1, x*(x + y)], [y*x, 1]])
m_vech = m.vech(diagonal=False, check_symmetry=False)
assert m_vech[0] == y*x
def test_vech_errors():
m = Matrix([[1, 3]])
raises(ShapeError, lambda: m.vech())
m = Matrix([[1, 3], [2, 4]])
raises(ValueError, lambda: m.vech())
raises(ShapeError, lambda: Matrix([ [1, 3] ]).vech())
raises(ValueError, lambda: Matrix([ [1, 3], [2, 4] ]).vech())
def test_diag():
# mostly tested in testcommonmatrix.py
assert diag([1, 2, 3]) == Matrix([1, 2, 3])
m = [1, 2, [3]]
raises(ValueError, lambda: diag(m))
assert diag(m, strict=False) == Matrix([1, 2, 3])
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]])
assert diag(a, b, b).get_diag_blocks() == [a, b, b]
assert diag(a, b, c).get_diag_blocks() == [a, b, c]
assert diag(a, c, b).get_diag_blocks() == [a, c, b]
assert diag(c, c, b).get_diag_blocks() == [c, c, b]
def test_inv_block():
a = Matrix([[1, 2], [2, 3]])
b = Matrix([[3, x], [y, 3]])
c = Matrix([[3, x, 3], [y, 3, z], [x, y, z]])
A = diag(a, b, b)
assert A.inv(try_block_diag=True) == diag(a.inv(), b.inv(), b.inv())
A = diag(a, b, c)
assert A.inv(try_block_diag=True) == diag(a.inv(), b.inv(), c.inv())
A = diag(a, c, b)
assert A.inv(try_block_diag=True) == diag(a.inv(), c.inv(), b.inv())
A = diag(a, a, b, a, c, a)
assert A.inv(try_block_diag=True) == diag(
a.inv(), a.inv(), b.inv(), a.inv(), c.inv(), a.inv())
assert A.inv(try_block_diag=True, method="ADJ") == diag(
a.inv(method="ADJ"), a.inv(method="ADJ"), b.inv(method="ADJ"),
a.inv(method="ADJ"), c.inv(method="ADJ"), a.inv(method="ADJ"))
def test_creation_args():
"""
Check that matrix dimensions can be specified using any reasonable type
(see issue 4614).
"""
raises(ValueError, lambda: zeros(3, -1))
raises(TypeError, lambda: zeros(1, 2, 3, 4))
assert zeros(long(3)) == zeros(3)
assert zeros(Integer(3)) == zeros(3)
raises(ValueError, lambda: zeros(3.))
assert eye(long(3)) == eye(3)
assert eye(Integer(3)) == eye(3)
raises(ValueError, lambda: eye(3.))
assert ones(long(3), Integer(4)) == ones(3, 4)
raises(TypeError, lambda: Matrix(5))
raises(TypeError, lambda: Matrix(1, 2))
raises(ValueError, lambda: Matrix([1, [2]]))
def test_diagonal_symmetrical():
m = Matrix(2, 2, [0, 1, 1, 0])
assert not m.is_diagonal()
assert m.is_symmetric()
assert m.is_symmetric(simplify=False)
m = Matrix(2, 2, [1, 0, 0, 1])
assert m.is_diagonal()
m = diag(1, 2, 3)
assert m.is_diagonal()
assert m.is_symmetric()
m = Matrix(3, 3, [1, 0, 0, 0, 2, 0, 0, 0, 3])
assert m == diag(1, 2, 3)
m = Matrix(2, 3, zeros(2, 3))
assert not m.is_symmetric()
assert m.is_diagonal()
m = Matrix(((5, 0), (0, 6), (0, 0)))
assert m.is_diagonal()
m = Matrix(((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_diagonalization():
m = Matrix(3, 2, [-3, 1, -3, 20, 3, 10])
assert not m.is_diagonalizable()
assert not m.is_symmetric()
raises(NonSquareMatrixError, lambda: m.diagonalize())
# diagonalizable
m = diag(1, 2, 3)
(P, D) = m.diagonalize()
assert P == eye(3)
assert D == m
m = Matrix(2, 2, [0, 1, 1, 0])
assert m.is_symmetric()
assert m.is_diagonalizable()
(P, D) = m.diagonalize()
assert P.inv() * m * P == D
m = Matrix(2, 2, [1, 0, 0, 3])
assert m.is_symmetric()
assert m.is_diagonalizable()
(P, D) = m.diagonalize()
assert P.inv() * m * P == D
assert P == eye(2)
assert D == m
m = Matrix(2, 2, [1, 1, 0, 0])
assert m.is_diagonalizable()
(P, D) = m.diagonalize()
assert P.inv() * m * P == D
m = Matrix(3, 3, [1, 2, 0, 0, 3, 0, 2, -4, 2])
assert m.is_diagonalizable()
(P, D) = m.diagonalize()
assert P.inv() * m * P == D
for i in P:
assert i.as_numer_denom()[1] == 1
m = Matrix(2, 2, [1, 0, 0, 0])
assert m.is_diagonal()
assert m.is_diagonalizable()
(P, D) = m.diagonalize()
assert P.inv() * m * P == D
assert P == Matrix([[0, 1], [1, 0]])
# diagonalizable, complex only
m = Matrix(2, 2, [0, 1, -1, 0])
assert not m.is_diagonalizable(True)
raises(MatrixError, lambda: m.diagonalize(True))
assert m.is_diagonalizable()
(P, D) = m.diagonalize()
assert P.inv() * m * P == D
# not diagonalizable
m = Matrix(2, 2, [0, 1, 0, 0])
assert not m.is_diagonalizable()
raises(MatrixError, lambda: m.diagonalize())
m = Matrix(3, 3, [-3, 1, -3, 20, 3, 10, 2, -2, 4])
assert not m.is_diagonalizable()
raises(MatrixError, lambda: m.diagonalize())
# symbolic
a, b, c, d = symbols('a b c d')
m = Matrix(2, 2, [a, c, c, b])
assert m.is_symmetric()
assert m.is_diagonalizable()
def test_issue_15887():
# Mutable matrix should not use cache
a = MutableDenseMatrix([[0, 1], [1, 0]])
assert a.is_diagonalizable() is True
a[1, 0] = 0
assert a.is_diagonalizable() is False
a = MutableDenseMatrix([[0, 1], [1, 0]])
a.diagonalize()
a[1, 0] = 0
raises(MatrixError, lambda: a.diagonalize())
# Test deprecated cache and kwargs
with warns_deprecated_sympy():
a._cache_eigenvects
with warns_deprecated_sympy():
a._cache_is_diagonalizable
with warns_deprecated_sympy():
a.is_diagonalizable(clear_cache=True)
with warns_deprecated_sympy():
a.is_diagonalizable(clear_subproducts=True)
@XFAIL
def test_eigen_vects():
m = Matrix(2, 2, [1, 0, 0, I])
raises(NotImplementedError, lambda: m.is_diagonalizable(True))
# !!! bug because of eigenvects() or roots(x**2 + (-1 - I)*x + I, x)
# see issue 5292
assert not m.is_diagonalizable(True)
raises(MatrixError, lambda: m.diagonalize(True))
(P, D) = m.diagonalize(True)
def test_jordan_form():
m = Matrix(3, 2, [-3, 1, -3, 20, 3, 10])
raises(NonSquareMatrixError, lambda: m.jordan_form())
# diagonalizable
m = Matrix(3, 3, [7, -12, 6, 10, -19, 10, 12, -24, 13])
Jmust = Matrix(3, 3, [-1, 0, 0, 0, 1, 0, 0, 0, 1])
P, J = m.jordan_form()
assert Jmust == J
assert Jmust == m.diagonalize()[1]
# m = Matrix(3, 3, [0, 6, 3, 1, 3, 1, -2, 2, 1])
# m.jordan_form() # very long
# m.jordan_form() #
# diagonalizable, complex only
# Jordan cells
# complexity: one of eigenvalues is zero
m = Matrix(3, 3, [0, 1, 0, -4, 4, 0, -2, 1, 2])
# The blocks are ordered according to the value of their eigenvalues,
# in order to make the matrix compatible with .diagonalize()
Jmust = Matrix(3, 3, [2, 1, 0, 0, 2, 0, 0, 0, 2])
P, J = m.jordan_form()
assert Jmust == J
# complexity: all of eigenvalues are equal
m = Matrix(3, 3, [2, 6, -15, 1, 1, -5, 1, 2, -6])
# Jmust = Matrix(3, 3, [-1, 0, 0, 0, -1, 1, 0, 0, -1])
# same here see 1456ff
Jmust = Matrix(3, 3, [-1, 1, 0, 0, -1, 0, 0, 0, -1])
P, J = m.jordan_form()
assert Jmust == J
# complexity: two of eigenvalues are zero
m = Matrix(3, 3, [4, -5, 2, 5, -7, 3, 6, -9, 4])
Jmust = Matrix(3, 3, [0, 1, 0, 0, 0, 0, 0, 0, 1])
P, J = m.jordan_form()
assert Jmust == J
m = Matrix(4, 4, [6, 5, -2, -3, -3, -1, 3, 3, 2, 1, -2, -3, -1, 1, 5, 5])
Jmust = Matrix(4, 4, [2, 1, 0, 0,
0, 2, 0, 0,
0, 0, 2, 1,
0, 0, 0, 2]
)
P, J = m.jordan_form()
assert Jmust == J
m = Matrix(4, 4, [6, 2, -8, -6, -3, 2, 9, 6, 2, -2, -8, -6, -1, 0, 3, 4])
# Jmust = Matrix(4, 4, [2, 0, 0, 0, 0, 2, 1, 0, 0, 0, 2, 0, 0, 0, 0, -2])
# same here see 1456ff
Jmust = Matrix(4, 4, [-2, 0, 0, 0,
0, 2, 1, 0,
0, 0, 2, 0,
0, 0, 0, 2])
P, J = m.jordan_form()
assert Jmust == J
m = Matrix(4, 4, [5, 4, 2, 1, 0, 1, -1, -1, -1, -1, 3, 0, 1, 1, -1, 2])
assert not m.is_diagonalizable()
Jmust = Matrix(4, 4, [1, 0, 0, 0, 0, 2, 0, 0, 0, 0, 4, 1, 0, 0, 0, 4])
P, J = m.jordan_form()
assert Jmust == J
# checking for maximum precision to remain unchanged
m = Matrix([[Float('1.0', precision=110), Float('2.0', precision=110)],
[Float('3.14159265358979323846264338327', precision=110), Float('4.0', precision=110)]])
P, J = m.jordan_form()
for term in J._mat:
if isinstance(term, Float):
assert term._prec == 110
def test_jordan_form_complex_issue_9274():
A = Matrix([[ 2, 4, 1, 0],
[-4, 2, 0, 1],
[ 0, 0, 2, 4],
[ 0, 0, -4, 2]])
p = 2 - 4*I;
q = 2 + 4*I;
Jmust1 = Matrix([[p, 1, 0, 0],
[0, p, 0, 0],
[0, 0, q, 1],
[0, 0, 0, q]])
Jmust2 = Matrix([[q, 1, 0, 0],
[0, q, 0, 0],
[0, 0, p, 1],
[0, 0, 0, p]])
P, J = A.jordan_form()
assert J == Jmust1 or J == Jmust2
assert simplify(P*J*P.inv()) == A
def test_issue_10220():
# two non-orthogonal Jordan blocks with eigenvalue 1
M = Matrix([[1, 0, 0, 1],
[0, 1, 1, 0],
[0, 0, 1, 1],
[0, 0, 0, 1]])
P, J = M.jordan_form()
assert P == Matrix([[0, 1, 0, 1],
[1, 0, 0, 0],
[0, 1, 0, 0],
[0, 0, 1, 0]])
assert J == Matrix([
[1, 1, 0, 0],
[0, 1, 1, 0],
[0, 0, 1, 0],
[0, 0, 0, 1]])
def test_jordan_form_issue_15858():
A = Matrix([
[1, 1, 1, 0],
[-2, -1, 0, -1],
[0, 0, -1, -1],
[0, 0, 2, 1]])
(P, J) = A.jordan_form()
assert simplify(P) == Matrix([
[-I, -I/2, I, I/2],
[-1 + I, 0, -1 - I, 0],
[0, I*(-1 + I)/2, 0, I*(1 + I)/2],
[0, 1, 0, 1]])
assert J == Matrix([
[-I, 1, 0, 0],
[0, -I, 0, 0],
[0, 0, I, 1],
[0, 0, 0, I]])
def test_Matrix_berkowitz_charpoly():
UA, K_i, K_w = symbols('UA K_i K_w')
A = Matrix([[-K_i - UA + K_i**2/(K_i + K_w), K_i*K_w/(K_i + K_w)],
[ K_i*K_w/(K_i + K_w), -K_w + K_w**2/(K_i + K_w)]])
charpoly = A.charpoly(x)
assert charpoly == \
Poly(x**2 + (K_i*UA + K_w*UA + 2*K_i*K_w)/(K_i + K_w)*x +
K_i*K_w*UA/(K_i + K_w), x, domain='ZZ(K_i,K_w,UA)')
assert type(charpoly) is PurePoly
A = Matrix([[1, 3], [2, 0]])
assert A.charpoly() == A.charpoly(x) == PurePoly(x**2 - x - 6)
A = Matrix([[1, 2], [x, 0]])
p = A.charpoly(x)
assert p.gen != x
assert p.as_expr().subs(p.gen, x) == x**2 - 3*x
def test_exp():
m = Matrix([[3, 4], [0, -2]])
m_exp = Matrix([[exp(3), -4*exp(-2)/5 + 4*exp(3)/5], [0, exp(-2)]])
assert m.exp() == m_exp
assert exp(m) == m_exp
m = Matrix([[1, 0], [0, 1]])
assert m.exp() == Matrix([[E, 0], [0, E]])
assert exp(m) == Matrix([[E, 0], [0, E]])
m = Matrix([[1, -1], [1, 1]])
assert m.exp() == Matrix([[E*cos(1), -E*sin(1)], [E*sin(1), E*cos(1)]])
def test_has():
A = Matrix(((x, y), (2, 3)))
assert A.has(x)
assert not A.has(z)
assert A.has(Symbol)
A = A.subs(x, 2)
assert not A.has(x)
def test_LUdecomposition_Simple_iszerofunc():
# Test if callable passed to matrices.LUdecomposition_Simple() as iszerofunc keyword argument is used inside
# matrices.LUdecomposition_Simple()
magic_string = "I got passed in!"
def goofyiszero(value):
raise ValueError(magic_string)
try:
lu, p = Matrix([[1, 0], [0, 1]]).LUdecomposition_Simple(iszerofunc=goofyiszero)
except ValueError as err:
assert magic_string == err.args[0]
return
assert False
def test_LUdecomposition_iszerofunc():
# Test if callable passed to matrices.LUdecomposition() as iszerofunc keyword argument is used inside
# matrices.LUdecomposition_Simple()
magic_string = "I got passed in!"
def goofyiszero(value):
raise ValueError(magic_string)
try:
l, u, p = Matrix([[1, 0], [0, 1]]).LUdecomposition(iszerofunc=goofyiszero)
except ValueError as err:
assert magic_string == err.args[0]
return
assert False
def test_find_reasonable_pivot_naive_finds_guaranteed_nonzero1():
# Test if matrices._find_reasonable_pivot_naive()
# finds a guaranteed non-zero pivot when the
# some of the candidate pivots are symbolic expressions.
# Keyword argument: simpfunc=None indicates that no simplifications
# should be performed during the search.
x = Symbol('x')
column = Matrix(3, 1, [x, cos(x)**2 + sin(x)**2, Rational(1, 2)])
pivot_offset, pivot_val, pivot_assumed_nonzero, simplified =\
_find_reasonable_pivot_naive(column)
assert pivot_val == Rational(1, 2)
def test_find_reasonable_pivot_naive_finds_guaranteed_nonzero2():
# Test if matrices._find_reasonable_pivot_naive()
# finds a guaranteed non-zero pivot when the
# some of the candidate pivots are symbolic expressions.
# Keyword argument: simpfunc=_simplify indicates that the search
# should attempt to simplify candidate pivots.
x = Symbol('x')
column = Matrix(3, 1,
[x,
cos(x)**2+sin(x)**2+x**2,
cos(x)**2+sin(x)**2])
pivot_offset, pivot_val, pivot_assumed_nonzero, simplified =\
_find_reasonable_pivot_naive(column, simpfunc=_simplify)
assert pivot_val == 1
def test_find_reasonable_pivot_naive_simplifies():
# Test if matrices._find_reasonable_pivot_naive()
# simplifies candidate pivots, and reports
# their offsets correctly.
x = Symbol('x')
column = Matrix(3, 1,
[x,
cos(x)**2+sin(x)**2+x,
cos(x)**2+sin(x)**2])
pivot_offset, pivot_val, pivot_assumed_nonzero, simplified =\
_find_reasonable_pivot_naive(column, simpfunc=_simplify)
assert len(simplified) == 2
assert simplified[0][0] == 1
assert simplified[0][1] == 1+x
assert simplified[1][0] == 2
assert simplified[1][1] == 1
def test_errors():
raises(ValueError, lambda: Matrix([[1, 2], [1]]))
raises(IndexError, lambda: Matrix([[1, 2]])[1.2, 5])
raises(IndexError, lambda: Matrix([[1, 2]])[1, 5.2])
raises(ValueError, lambda: randMatrix(3, c=4, symmetric=True))
raises(ValueError, lambda: Matrix([1, 2]).reshape(4, 6))
raises(ShapeError,
lambda: Matrix([[1, 2], [3, 4]]).copyin_matrix([1, 0], Matrix([1, 2])))
raises(TypeError, lambda: Matrix([[1, 2], [3, 4]]).copyin_list([0,
1], set([])))
raises(NonSquareMatrixError, lambda: Matrix([[1, 2, 3], [2, 3, 0]]).inv())
raises(ShapeError,
lambda: Matrix(1, 2, [1, 2]).row_join(Matrix([[1, 2], [3, 4]])))
raises(
ShapeError, lambda: Matrix([1, 2]).col_join(Matrix([[1, 2], [3, 4]])))
raises(ShapeError, lambda: Matrix([1]).row_insert(1, Matrix([[1,
2], [3, 4]])))
raises(ShapeError, lambda: Matrix([1]).col_insert(1, Matrix([[1,
2], [3, 4]])))
raises(NonSquareMatrixError, lambda: Matrix([1, 2]).trace())
raises(TypeError, lambda: Matrix([1]).applyfunc(1))
raises(ShapeError, lambda: Matrix([1]).LUsolve(Matrix([[1, 2], [3, 4]])))
raises(ValueError, lambda: Matrix([[1, 2], [3, 4]]).minor(4, 5))
raises(ValueError, lambda: Matrix([[1, 2], [3, 4]]).minor_submatrix(4, 5))
raises(TypeError, lambda: Matrix([1, 2, 3]).cross(1))
raises(TypeError, lambda: Matrix([1, 2, 3]).dot(1))
raises(ShapeError, lambda: Matrix([1, 2, 3]).dot(Matrix([1, 2])))
raises(ShapeError, lambda: Matrix([1, 2]).dot([]))
raises(TypeError, lambda: Matrix([1, 2]).dot('a'))
with warns_deprecated_sympy():
Matrix([[1, 2], [3, 4]]).dot(Matrix([[4, 3], [1, 2]]))
raises(ShapeError, lambda: Matrix([1, 2]).dot([1, 2, 3]))
raises(NonSquareMatrixError, lambda: Matrix([1, 2, 3]).exp())
raises(ShapeError, lambda: Matrix([[1, 2], [3, 4]]).normalized())
raises(ValueError, lambda: Matrix([1, 2]).inv(method='not a method'))
raises(NonSquareMatrixError, lambda: Matrix([1, 2]).inverse_GE())
raises(ValueError, lambda: Matrix([[1, 2], [1, 2]]).inverse_GE())
raises(NonSquareMatrixError, lambda: Matrix([1, 2]).inverse_ADJ())
raises(ValueError, lambda: Matrix([[1, 2], [1, 2]]).inverse_ADJ())
raises(NonSquareMatrixError, lambda: Matrix([1, 2]).inverse_LU())
raises(NonSquareMatrixError, lambda: Matrix([1, 2]).is_nilpotent())
raises(NonSquareMatrixError, lambda: Matrix([1, 2]).det())
raises(ValueError,
lambda: Matrix([[1, 2], [3, 4]]).det(method='Not a real method'))
raises(ValueError,
lambda: Matrix([[1, 2, 3, 4], [5, 6, 7, 8],
[9, 10, 11, 12], [13, 14, 15, 16]]).det(iszerofunc="Not function"))
raises(ValueError,
lambda: Matrix([[1, 2, 3, 4], [5, 6, 7, 8],
[9, 10, 11, 12], [13, 14, 15, 16]]).det(iszerofunc=False))
raises(ValueError,
lambda: hessian(Matrix([[1, 2], [3, 4]]), Matrix([[1, 2], [2, 1]])))
raises(ValueError, lambda: hessian(Matrix([[1, 2], [3, 4]]), []))
raises(ValueError, lambda: hessian(Symbol('x')**2, 'a'))
raises(IndexError, lambda: eye(3)[5, 2])
raises(IndexError, lambda: eye(3)[2, 5])
M = Matrix(((1, 2, 3, 4), (5, 6, 7, 8), (9, 10, 11, 12), (13, 14, 15, 16)))
raises(ValueError, lambda: M.det('method=LU_decomposition()'))
V = Matrix([[10, 10, 10]])
M = Matrix([[1, 2, 3], [2, 3, 4], [3, 4, 5]])
raises(ValueError, lambda: M.row_insert(4.7, V))
M = Matrix([[1, 2, 3], [2, 3, 4], [3, 4, 5]])
raises(ValueError, lambda: M.col_insert(-4.2, V))
def test_len():
assert len(Matrix()) == 0
assert len(Matrix([[1, 2]])) == len(Matrix([[1], [2]])) == 2
assert len(Matrix(0, 2, lambda i, j: 0)) == \
len(Matrix(2, 0, lambda i, j: 0)) == 0
assert len(Matrix([[0, 1, 2], [3, 4, 5]])) == 6
assert Matrix([1]) == Matrix([[1]])
assert not Matrix()
assert Matrix() == Matrix([])
def test_integrate():
A = Matrix(((1, 4, x), (y, 2, 4), (10, 5, x**2)))
assert A.integrate(x) == \
Matrix(((x, 4*x, x**2/2), (x*y, 2*x, 4*x), (10*x, 5*x, x**3/3)))
assert A.integrate(y) == \
Matrix(((y, 4*y, x*y), (y**2/2, 2*y, 4*y), (10*y, 5*y, y*x**2)))
def test_limit():
A = Matrix(((1, 4, sin(x)/x), (y, 2, 4), (10, 5, x**2 + 1)))
assert A.limit(x, 0) == Matrix(((1, 4, 1), (y, 2, 4), (10, 5, 1)))
def test_diff():
A = MutableDenseMatrix(((1, 4, x), (y, 2, 4), (10, 5, x**2 + 1)))
assert isinstance(A.diff(x), type(A))
assert A.diff(x) == MutableDenseMatrix(((0, 0, 1), (0, 0, 0), (0, 0, 2*x)))
assert A.diff(y) == MutableDenseMatrix(((0, 0, 0), (1, 0, 0), (0, 0, 0)))
assert diff(A, x) == MutableDenseMatrix(((0, 0, 1), (0, 0, 0), (0, 0, 2*x)))
assert diff(A, y) == MutableDenseMatrix(((0, 0, 0), (1, 0, 0), (0, 0, 0)))
A_imm = A.as_immutable()
assert isinstance(A_imm.diff(x), type(A_imm))
assert A_imm.diff(x) == ImmutableDenseMatrix(((0, 0, 1), (0, 0, 0), (0, 0, 2*x)))
assert A_imm.diff(y) == ImmutableDenseMatrix(((0, 0, 0), (1, 0, 0), (0, 0, 0)))
assert diff(A_imm, x) == ImmutableDenseMatrix(((0, 0, 1), (0, 0, 0), (0, 0, 2*x)))
assert diff(A_imm, y) == ImmutableDenseMatrix(((0, 0, 0), (1, 0, 0), (0, 0, 0)))
def test_diff_by_matrix():
# Derive matrix by matrix:
A = MutableDenseMatrix([[x, y], [z, t]])
assert A.diff(A) == Array([[[[1, 0], [0, 0]], [[0, 1], [0, 0]]], [[[0, 0], [1, 0]], [[0, 0], [0, 1]]]])
assert diff(A, A) == Array([[[[1, 0], [0, 0]], [[0, 1], [0, 0]]], [[[0, 0], [1, 0]], [[0, 0], [0, 1]]]])
A_imm = A.as_immutable()
assert A_imm.diff(A_imm) == Array([[[[1, 0], [0, 0]], [[0, 1], [0, 0]]], [[[0, 0], [1, 0]], [[0, 0], [0, 1]]]])
assert diff(A_imm, A_imm) == Array([[[[1, 0], [0, 0]], [[0, 1], [0, 0]]], [[[0, 0], [1, 0]], [[0, 0], [0, 1]]]])
# Derive a constant matrix:
assert A.diff(a) == MutableDenseMatrix([[0, 0], [0, 0]])
B = ImmutableDenseMatrix([a, b])
assert A.diff(B) == Array.zeros(2, 1, 2, 2)
assert A.diff(A) == Array([[[[1, 0], [0, 0]], [[0, 1], [0, 0]]], [[[0, 0], [1, 0]], [[0, 0], [0, 1]]]])
# Test diff with tuples:
dB = B.diff([[a, b]])
assert dB.shape == (2, 2, 1)
assert dB == Array([[[1], [0]], [[0], [1]]])
f = Function("f")
fxyz = f(x, y, z)
assert fxyz.diff([[x, y, z]]) == Array([fxyz.diff(x), fxyz.diff(y), fxyz.diff(z)])
assert fxyz.diff(([x, y, z], 2)) == Array([
[fxyz.diff(x, 2), fxyz.diff(x, y), fxyz.diff(x, z)],
[fxyz.diff(x, y), fxyz.diff(y, 2), fxyz.diff(y, z)],
[fxyz.diff(x, z), fxyz.diff(z, y), fxyz.diff(z, 2)],
])
expr = sin(x)*exp(y)
assert expr.diff([[x, y]]) == Array([cos(x)*exp(y), sin(x)*exp(y)])
assert expr.diff(y, ((x, y),)) == Array([cos(x)*exp(y), sin(x)*exp(y)])
assert expr.diff(x, ((x, y),)) == Array([-sin(x)*exp(y), cos(x)*exp(y)])
assert expr.diff(((y, x),), [[x, y]]) == Array([[cos(x)*exp(y), -sin(x)*exp(y)], [sin(x)*exp(y), cos(x)*exp(y)]])
# Test different notations:
fxyz.diff(x).diff(y).diff(x) == fxyz.diff(((x, y, z),), 3)[0, 1, 0]
fxyz.diff(z).diff(y).diff(x) == fxyz.diff(((x, y, z),), 3)[2, 1, 0]
fxyz.diff([[x, y, z]], ((z, y, x),)) == Array([[fxyz.diff(i).diff(j) for i in (x, y, z)] for j in (z, y, x)])
# Test scalar derived by matrix remains matrix:
res = x.diff(Matrix([[x, y]]))
assert isinstance(res, ImmutableDenseMatrix)
assert res == Matrix([[1, 0]])
res = (x**3).diff(Matrix([[x, y]]))
assert isinstance(res, ImmutableDenseMatrix)
assert res == Matrix([[3*x**2, 0]])
def test_getattr():
A = Matrix(((1, 4, x), (y, 2, 4), (10, 5, x**2 + 1)))
raises(AttributeError, lambda: A.nonexistantattribute)
assert getattr(A, 'diff')(x) == Matrix(((0, 0, 1), (0, 0, 0), (0, 0, 2*x)))
def test_hessenberg():
A = Matrix([[3, 4, 1], [2, 4, 5], [0, 1, 2]])
assert A.is_upper_hessenberg
A = A.T
assert A.is_lower_hessenberg
A[0, -1] = 1
assert A.is_lower_hessenberg is False
A = Matrix([[3, 4, 1], [2, 4, 5], [3, 1, 2]])
assert not A.is_upper_hessenberg
A = zeros(5, 2)
assert A.is_upper_hessenberg
def test_cholesky():
raises(NonSquareMatrixError, lambda: Matrix((1, 2)).cholesky())
raises(ValueError, lambda: Matrix(((1, 2), (3, 4))).cholesky())
raises(ValueError, lambda: Matrix(((5 + I, 0), (0, 1))).cholesky())
raises(ValueError, lambda: Matrix(((1, 5), (5, 1))).cholesky())
raises(ValueError, lambda: Matrix(((1, 2), (3, 4))).cholesky(hermitian=False))
assert Matrix(((5 + I, 0), (0, 1))).cholesky(hermitian=False) == Matrix([
[sqrt(5 + I), 0], [0, 1]])
A = Matrix(((1, 5), (5, 1)))
L = A.cholesky(hermitian=False)
assert L == Matrix([[1, 0], [5, 2*sqrt(6)*I]])
assert L*L.T == A
A = Matrix(((25, 15, -5), (15, 18, 0), (-5, 0, 11)))
L = A.cholesky()
assert L * L.T == A
assert L.is_lower
assert L == Matrix([[5, 0, 0], [3, 3, 0], [-1, 1, 3]])
A = Matrix(((4, -2*I, 2 + 2*I), (2*I, 2, -1 + I), (2 - 2*I, -1 - I, 11)))
assert A.cholesky() == Matrix(((2, 0, 0), (I, 1, 0), (1 - I, 0, 3)))
def test_LDLdecomposition():
raises(NonSquareMatrixError, lambda: Matrix((1, 2)).LDLdecomposition())
raises(ValueError, lambda: Matrix(((1, 2), (3, 4))).LDLdecomposition())
raises(ValueError, lambda: Matrix(((5 + I, 0), (0, 1))).LDLdecomposition())
raises(ValueError, lambda: Matrix(((1, 5), (5, 1))).LDLdecomposition())
raises(ValueError, lambda: Matrix(((1, 2), (3, 4))).LDLdecomposition(hermitian=False))
A = Matrix(((1, 5), (5, 1)))
L, D = A.LDLdecomposition(hermitian=False)
assert L * D * L.T == A
A = Matrix(((25, 15, -5), (15, 18, 0), (-5, 0, 11)))
L, D = A.LDLdecomposition()
assert L * D * L.T == A
assert L.is_lower
assert L == Matrix([[1, 0, 0], [ S(3)/5, 1, 0], [S(-1)/5, S(1)/3, 1]])
assert D.is_diagonal()
assert D == Matrix([[25, 0, 0], [0, 9, 0], [0, 0, 9]])
A = Matrix(((4, -2*I, 2 + 2*I), (2*I, 2, -1 + I), (2 - 2*I, -1 - I, 11)))
L, D = A.LDLdecomposition()
assert expand_mul(L * D * L.H) == A
assert L == Matrix(((1, 0, 0), (I/2, 1, 0), (S(1)/2 - I/2, 0, 1)))
assert D == Matrix(((4, 0, 0), (0, 1, 0), (0, 0, 9)))
def test_cholesky_solve():
A = Matrix([[2, 3, 5],
[3, 6, 2],
[8, 3, 6]])
x = Matrix(3, 1, [3, 7, 5])
b = A*x
soln = A.cholesky_solve(b)
assert soln == x
A = Matrix([[0, -1, 2],
[5, 10, 7],
[8, 3, 4]])
x = Matrix(3, 1, [-1, 2, 5])
b = A*x
soln = A.cholesky_solve(b)
assert soln == x
A = Matrix(((1, 5), (5, 1)))
x = Matrix((4, -3))
b = A*x
soln = A.cholesky_solve(b)
assert soln == x
A = Matrix(((9, 3*I), (-3*I, 5)))
x = Matrix((-2, 1))
b = A*x
soln = A.cholesky_solve(b)
assert expand_mul(soln) == x
A = Matrix(((9*I, 3), (-3 + I, 5)))
x = Matrix((2 + 3*I, -1))
b = A*x
soln = A.cholesky_solve(b)
assert expand_mul(soln) == x
a00, a01, a11, b0, b1 = symbols('a00, a01, a11, b0, b1')
A = Matrix(((a00, a01), (a01, a11)))
b = Matrix((b0, b1))
x = A.cholesky_solve(b)
assert simplify(A*x) == b
def test_LDLsolve():
A = Matrix([[2, 3, 5],
[3, 6, 2],
[8, 3, 6]])
x = Matrix(3, 1, [3, 7, 5])
b = A*x
soln = A.LDLsolve(b)
assert soln == x
A = Matrix([[0, -1, 2],
[5, 10, 7],
[8, 3, 4]])
x = Matrix(3, 1, [-1, 2, 5])
b = A*x
soln = A.LDLsolve(b)
assert soln == x
A = Matrix(((9, 3*I), (-3*I, 5)))
x = Matrix((-2, 1))
b = A*x
soln = A.LDLsolve(b)
assert expand_mul(soln) == x
A = Matrix(((9*I, 3), (-3 + I, 5)))
x = Matrix((2 + 3*I, -1))
b = A*x
soln = A.LDLsolve(b)
assert expand_mul(soln) == x
A = Matrix(((9, 3), (3, 9)))
x = Matrix((1, 1))
b = A * x
soln = A.LDLsolve(b)
assert expand_mul(soln) == x
A = Matrix([[-5, -3, -4], [-3, -7, 7]])
x = Matrix([[8], [7], [-2]])
b = A * x
raises(NotImplementedError, lambda: A.LDLsolve(b))
def test_lower_triangular_solve():
raises(NonSquareMatrixError,
lambda: Matrix([1, 0]).lower_triangular_solve(Matrix([0, 1])))
raises(ShapeError,
lambda: Matrix([[1, 0], [0, 1]]).lower_triangular_solve(Matrix([1])))
raises(ValueError,
lambda: Matrix([[2, 1], [1, 2]]).lower_triangular_solve(
Matrix([[1, 0], [0, 1]])))
A = Matrix([[1, 0], [0, 1]])
B = Matrix([[x, y], [y, x]])
C = Matrix([[4, 8], [2, 9]])
assert A.lower_triangular_solve(B) == B
assert A.lower_triangular_solve(C) == C
def test_upper_triangular_solve():
raises(NonSquareMatrixError,
lambda: Matrix([1, 0]).upper_triangular_solve(Matrix([0, 1])))
raises(TypeError,
lambda: Matrix([[1, 0], [0, 1]]).upper_triangular_solve(Matrix([1])))
raises(TypeError,
lambda: Matrix([[2, 1], [1, 2]]).upper_triangular_solve(
Matrix([[1, 0], [0, 1]])))
A = Matrix([[1, 0], [0, 1]])
B = Matrix([[x, y], [y, x]])
C = Matrix([[2, 4], [3, 8]])
assert A.upper_triangular_solve(B) == B
assert A.upper_triangular_solve(C) == C
def test_diagonal_solve():
raises(TypeError, lambda: Matrix([1, 1]).diagonal_solve(Matrix([1])))
A = Matrix([[1, 0], [0, 1]])*2
B = Matrix([[x, y], [y, x]])
assert A.diagonal_solve(B) == B/2
A = Matrix([[1, 0], [1, 2]])
raises(TypeError, lambda: A.diagonal_solve(B))
def test_matrix_norm():
# Vector Tests
# Test columns and symbols
x = Symbol('x', real=True)
v = Matrix([cos(x), sin(x)])
assert trigsimp(v.norm(2)) == 1
assert v.norm(10) == Pow(cos(x)**10 + sin(x)**10, S(1)/10)
# Test Rows
A = Matrix([[5, Rational(3, 2)]])
assert A.norm() == Pow(25 + Rational(9, 4), S(1)/2)
assert A.norm(oo) == max(A._mat)
assert A.norm(-oo) == min(A._mat)
# Matrix Tests
# Intuitive test
A = Matrix([[1, 1], [1, 1]])
assert A.norm(2) == 2
assert A.norm(-2) == 0
assert A.norm('frobenius') == 2
assert eye(10).norm(2) == eye(10).norm(-2) == 1
assert A.norm(oo) == 2
# Test with Symbols and more complex entries
A = Matrix([[3, y, y], [x, S(1)/2, -pi]])
assert (A.norm('fro')
== sqrt(S(37)/4 + 2*abs(y)**2 + pi**2 + x**2))
# Check non-square
A = Matrix([[1, 2, -3], [4, 5, Rational(13, 2)]])
assert A.norm(2) == sqrt(S(389)/8 + sqrt(78665)/8)
assert A.norm(-2) == S(0)
assert A.norm('frobenius') == sqrt(389)/2
# Test properties of matrix norms
# https://en.wikipedia.org/wiki/Matrix_norm#Definition
# Two matrices
A = Matrix([[1, 2], [3, 4]])
B = Matrix([[5, 5], [-2, 2]])
C = Matrix([[0, -I], [I, 0]])
D = Matrix([[1, 0], [0, -1]])
L = [A, B, C, D]
alpha = Symbol('alpha', real=True)
for order in ['fro', 2, -2]:
# Zero Check
assert zeros(3).norm(order) == S(0)
# Check Triangle Inequality for all Pairs of Matrices
for X in L:
for Y in L:
dif = (X.norm(order) + Y.norm(order) -
(X + Y).norm(order))
assert (dif >= 0)
# Scalar multiplication linearity
for M in [A, B, C, D]:
dif = simplify((alpha*M).norm(order) -
abs(alpha) * M.norm(order))
assert dif == 0
# Test Properties of Vector Norms
# https://en.wikipedia.org/wiki/Vector_norm
# Two column vectors
a = Matrix([1, 1 - 1*I, -3])
b = Matrix([S(1)/2, 1*I, 1])
c = Matrix([-1, -1, -1])
d = Matrix([3, 2, I])
e = Matrix([Integer(1e2), Rational(1, 1e2), 1])
L = [a, b, c, d, e]
alpha = Symbol('alpha', real=True)
for order in [1, 2, -1, -2, S.Infinity, S.NegativeInfinity, pi]:
# Zero Check
if order > 0:
assert Matrix([0, 0, 0]).norm(order) == S(0)
# Triangle inequality on all pairs
if order >= 1: # Triangle InEq holds only for these norms
for X in L:
for Y in L:
dif = (X.norm(order) + Y.norm(order) -
(X + Y).norm(order))
assert simplify(dif >= 0) is S.true
# Linear to scalar multiplication
if order in [1, 2, -1, -2, S.Infinity, S.NegativeInfinity]:
for X in L:
dif = simplify((alpha*X).norm(order) -
(abs(alpha) * X.norm(order)))
assert dif == 0
# ord=1
M = Matrix(3, 3, [1, 3, 0, -2, -1, 0, 3, 9, 6])
assert M.norm(1) == 13
def test_condition_number():
x = Symbol('x', real=True)
A = eye(3)
A[0, 0] = 10
A[2, 2] = S(1)/10
assert A.condition_number() == 100
A[1, 1] = x
assert A.condition_number() == Max(10, Abs(x)) / Min(S(1)/10, Abs(x))
M = Matrix([[cos(x), sin(x)], [-sin(x), cos(x)]])
Mc = M.condition_number()
assert all(Float(1.).epsilon_eq(Mc.subs(x, val).evalf()) for val in
[Rational(1, 5), Rational(1, 2), Rational(1, 10), pi/2, pi, 7*pi/4 ])
#issue 10782
assert Matrix([]).condition_number() == 0
def test_equality():
A = Matrix(((1, 2, 3), (4, 5, 6), (7, 8, 9)))
B = Matrix(((9, 8, 7), (6, 5, 4), (3, 2, 1)))
assert A == A[:, :]
assert not A != A[:, :]
assert not A == B
assert A != B
assert A != 10
assert not A == 10
# A SparseMatrix can be equal to a Matrix
C = SparseMatrix(((1, 0, 0), (0, 1, 0), (0, 0, 1)))
D = Matrix(((1, 0, 0), (0, 1, 0), (0, 0, 1)))
assert C == D
assert not C != D
def test_col_join():
assert eye(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(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(3).col_insert(i, c4).row(0).tolist()) == l
def test_normalized():
assert Matrix([3, 4]).normalized() == \
Matrix([Rational(3, 5), Rational(4, 5)])
# Zero vector trivial cases
assert Matrix([0, 0, 0]).normalized() == Matrix([0, 0, 0])
# Machine precision error truncation trivial cases
m = Matrix([0,0,1.e-100])
assert m.normalized(
iszerofunc=lambda x: x.evalf(n=10, chop=True).is_zero
) == Matrix([0, 0, 0])
def test_print_nonzero():
assert capture(lambda: eye(3).print_nonzero()) == \
'[X ]\n[ X ]\n[ X]\n'
assert capture(lambda: eye(3).print_nonzero('.')) == \
'[. ]\n[ . ]\n[ .]\n'
def test_zeros_eye():
assert Matrix.eye(3) == eye(3)
assert Matrix.zeros(3) == zeros(3)
assert ones(3, 4) == Matrix(3, 4, [1]*12)
i = Matrix([[1, 0], [0, 1]])
z = Matrix([[0, 0], [0, 0]])
for cls in classes:
m = cls.eye(2)
assert i == m # but m == i will fail if m is immutable
assert i == eye(2, cls=cls)
assert type(m) == cls
m = cls.zeros(2)
assert z == m
assert z == zeros(2, cls=cls)
assert type(m) == cls
def test_is_zero():
assert Matrix().is_zero
assert Matrix([[0, 0], [0, 0]]).is_zero
assert zeros(3, 4).is_zero
assert not eye(3).is_zero
assert Matrix([[x, 0], [0, 0]]).is_zero == None
assert SparseMatrix([[x, 0], [0, 0]]).is_zero == None
assert ImmutableMatrix([[x, 0], [0, 0]]).is_zero == None
assert ImmutableSparseMatrix([[x, 0], [0, 0]]).is_zero == None
assert Matrix([[x, 1], [0, 0]]).is_zero == False
a = Symbol('a', nonzero=True)
assert Matrix([[a, 0], [0, 0]]).is_zero == False
def test_rotation_matrices():
# This tests the rotation matrices by rotating about an axis and back.
theta = pi/3
r3_plus = rot_axis3(theta)
r3_minus = rot_axis3(-theta)
r2_plus = rot_axis2(theta)
r2_minus = rot_axis2(-theta)
r1_plus = rot_axis1(theta)
r1_minus = rot_axis1(-theta)
assert r3_minus*r3_plus*eye(3) == eye(3)
assert r2_minus*r2_plus*eye(3) == eye(3)
assert r1_minus*r1_plus*eye(3) == eye(3)
# Check the correctness of the trace of the rotation matrix
assert r1_plus.trace() == 1 + 2*cos(theta)
assert r2_plus.trace() == 1 + 2*cos(theta)
assert r3_plus.trace() == 1 + 2*cos(theta)
# Check that a rotation with zero angle doesn't change anything.
assert rot_axis1(0) == eye(3)
assert rot_axis2(0) == eye(3)
assert rot_axis3(0) == eye(3)
def test_DeferredVector():
assert str(DeferredVector("vector")[4]) == "vector[4]"
assert sympify(DeferredVector("d")) == DeferredVector("d")
raises(IndexError, lambda: DeferredVector("d")[-1])
assert str(DeferredVector("d")) == "d"
assert repr(DeferredVector("test")) == "DeferredVector('test')"
def test_DeferredVector_not_iterable():
assert not iterable(DeferredVector('X'))
def test_DeferredVector_Matrix():
raises(TypeError, lambda: Matrix(DeferredVector("V")))
def test_GramSchmidt():
R = Rational
m1 = Matrix(1, 2, [1, 2])
m2 = Matrix(1, 2, [2, 3])
assert GramSchmidt([m1, m2]) == \
[Matrix(1, 2, [1, 2]), Matrix(1, 2, [R(2)/5, R(-1)/5])]
assert GramSchmidt([m1.T, m2.T]) == \
[Matrix(2, 1, [1, 2]), Matrix(2, 1, [R(2)/5, R(-1)/5])]
# from wikipedia
assert GramSchmidt([Matrix([3, 1]), Matrix([2, 2])], True) == [
Matrix([3*sqrt(10)/10, sqrt(10)/10]),
Matrix([-sqrt(10)/10, 3*sqrt(10)/10])]
def test_casoratian():
assert casoratian([1, 2, 3, 4], 1) == 0
assert casoratian([1, 2, 3, 4], 1, zero=False) == 0
def test_zero_dimension_multiply():
assert (Matrix()*zeros(0, 3)).shape == (0, 3)
assert zeros(3, 0)*zeros(0, 3) == zeros(3, 3)
assert zeros(0, 3)*zeros(3, 0) == Matrix()
def test_slice_issue_2884():
m = Matrix(2, 2, range(4))
assert m[1, :] == Matrix([[2, 3]])
assert m[-1, :] == Matrix([[2, 3]])
assert m[:, 1] == Matrix([[1, 3]]).T
assert m[:, -1] == Matrix([[1, 3]]).T
raises(IndexError, lambda: m[2, :])
raises(IndexError, lambda: m[2, 2])
def test_slice_issue_3401():
assert zeros(0, 3)[:, -1].shape == (0, 1)
assert zeros(3, 0)[0, :] == Matrix(1, 0, [])
def test_copyin():
s = zeros(3, 3)
s[3] = 1
assert s[:, 0] == Matrix([0, 1, 0])
assert s[3] == 1
assert s[3: 4] == [1]
s[1, 1] = 42
assert s[1, 1] == 42
assert s[1, 1:] == Matrix([[42, 0]])
s[1, 1:] = Matrix([[5, 6]])
assert s[1, :] == Matrix([[1, 5, 6]])
s[1, 1:] = [[42, 43]]
assert s[1, :] == Matrix([[1, 42, 43]])
s[0, 0] = 17
assert s[:, :1] == Matrix([17, 1, 0])
s[0, 0] = [1, 1, 1]
assert s[:, 0] == Matrix([1, 1, 1])
s[0, 0] = Matrix([1, 1, 1])
assert s[:, 0] == Matrix([1, 1, 1])
s[0, 0] = SparseMatrix([1, 1, 1])
assert s[:, 0] == Matrix([1, 1, 1])
def test_invertible_check():
# sometimes a singular matrix will have a pivot vector shorter than
# the number of rows in a matrix...
assert Matrix([[1, 2], [1, 2]]).rref() == (Matrix([[1, 2], [0, 0]]), (0,))
raises(ValueError, lambda: Matrix([[1, 2], [1, 2]]).inv())
m = Matrix([
[-1, -1, 0],
[ x, 1, 1],
[ 1, x, -1],
])
assert len(m.rref()[1]) != m.rows
# in addition, unless simplify=True in the call to rref, the identity
# matrix will be returned even though m is not invertible
assert m.rref()[0] != eye(3)
assert m.rref(simplify=signsimp)[0] != eye(3)
raises(ValueError, lambda: m.inv(method="ADJ"))
raises(ValueError, lambda: m.inv(method="GE"))
raises(ValueError, lambda: m.inv(method="LU"))
@XFAIL
def test_issue_3959():
x, y = symbols('x, y')
e = x*y
assert e.subs(x, Matrix([3, 5, 3])) == Matrix([3, 5, 3])*y
def test_issue_5964():
assert str(Matrix([[1, 2], [3, 4]])) == 'Matrix([[1, 2], [3, 4]])'
def test_issue_7604():
x, y = symbols(u"x y")
assert sstr(Matrix([[x, 2*y], [y**2, x + 3]])) == \
'Matrix([\n[ x, 2*y],\n[y**2, x + 3]])'
def test_is_Identity():
assert eye(3).is_Identity
assert eye(3).as_immutable().is_Identity
assert not zeros(3).is_Identity
assert not ones(3).is_Identity
# issue 6242
assert not Matrix([[1, 0, 0]]).is_Identity
# issue 8854
assert SparseMatrix(3,3, {(0,0):1, (1,1):1, (2,2):1}).is_Identity
assert not SparseMatrix(2,3, range(6)).is_Identity
assert not SparseMatrix(3,3, {(0,0):1, (1,1):1}).is_Identity
assert not SparseMatrix(3,3, {(0,0):1, (1,1):1, (2,2):1, (0,1):2, (0,2):3}).is_Identity
def test_dot():
assert ones(1, 3).dot(ones(3, 1)) == 3
assert ones(1, 3).dot([1, 1, 1]) == 3
assert Matrix([1, 2, 3]).dot(Matrix([1, 2, 3])) == 14
assert Matrix([1, 2, 3*I]).dot(Matrix([I, 2, 3*I])) == -5 + I
assert Matrix([1, 2, 3*I]).dot(Matrix([I, 2, 3*I]), hermitian=False) == -5 + I
assert Matrix([1, 2, 3*I]).dot(Matrix([I, 2, 3*I]), hermitian=True) == 13 + I
assert Matrix([1, 2, 3*I]).dot(Matrix([I, 2, 3*I]), hermitian=True, conjugate_convention="physics") == 13 - I
assert Matrix([1, 2, 3*I]).dot(Matrix([4, 5*I, 6]), hermitian=True, conjugate_convention="right") == 4 + 8*I
assert Matrix([1, 2, 3*I]).dot(Matrix([4, 5*I, 6]), hermitian=True, conjugate_convention="left") == 4 - 8*I
assert Matrix([I, 2*I]).dot(Matrix([I, 2*I]), hermitian=False, conjugate_convention="left") == -5
assert Matrix([I, 2*I]).dot(Matrix([I, 2*I]), conjugate_convention="left") == 5
raises(ValueError, lambda: Matrix([1, 2]).dot(Matrix([3, 4]), hermitian=True, conjugate_convention="test"))
def test_dual():
B_x, B_y, B_z, E_x, E_y, E_z = symbols(
'B_x B_y B_z E_x E_y E_z', real=True)
F = Matrix((
( 0, E_x, E_y, E_z),
(-E_x, 0, B_z, -B_y),
(-E_y, -B_z, 0, B_x),
(-E_z, B_y, -B_x, 0)
))
Fd = Matrix((
( 0, -B_x, -B_y, -B_z),
(B_x, 0, E_z, -E_y),
(B_y, -E_z, 0, E_x),
(B_z, E_y, -E_x, 0)
))
assert F.dual().equals(Fd)
assert eye(3).dual().equals(zeros(3))
assert F.dual().dual().equals(-F)
def test_anti_symmetric():
assert Matrix([1, 2]).is_anti_symmetric() is False
m = Matrix(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
# tweak to fail
m[2, 1] = -m[2, 1]
assert m.is_anti_symmetric() is False
# untweak
m[2, 1] = -m[2, 1]
m = m.expand()
assert m.is_anti_symmetric(simplify=False) is True
m[0, 0] = 1
assert m.is_anti_symmetric() is False
def test_normalize_sort_diogonalization():
A = Matrix(((1, 2), (2, 1)))
P, Q = A.diagonalize(normalize=True)
assert P*P.T == P.T*P == eye(P.cols)
P, Q = A.diagonalize(normalize=True, sort=True)
assert P*P.T == P.T*P == eye(P.cols)
assert P*Q*P.inv() == A
def test_issue_5321():
raises(ValueError, lambda: Matrix([[1, 2, 3], Matrix(0, 1, [])]))
def test_issue_5320():
assert Matrix.hstack(eye(2), 2*eye(2)) == Matrix([
[1, 0, 2, 0],
[0, 1, 0, 2]
])
assert Matrix.vstack(eye(2), 2*eye(2)) == Matrix([
[1, 0],
[0, 1],
[2, 0],
[0, 2]
])
cls = SparseMatrix
assert cls.hstack(cls(eye(2)), cls(2*eye(2))) == Matrix([
[1, 0, 2, 0],
[0, 1, 0, 2]
])
def test_issue_11944():
A = Matrix([[1]])
AIm = sympify(A)
assert Matrix.hstack(AIm, A) == Matrix([[1, 1]])
assert Matrix.vstack(AIm, A) == Matrix([[1], [1]])
def test_cross():
a = [1, 2, 3]
b = [3, 4, 5]
col = Matrix([-2, 4, -2])
row = col.T
def test(M, ans):
assert ans == M
assert type(M) == cls
for cls in classes:
A = cls(a)
B = cls(b)
test(A.cross(B), col)
test(A.cross(B.T), col)
test(A.T.cross(B.T), row)
test(A.T.cross(B), row)
raises(ShapeError, lambda:
Matrix(1, 2, [1, 1]).cross(Matrix(1, 2, [1, 1])))
def test_hash():
for cls in classes[-2:]:
s = {cls.eye(1), cls.eye(1)}
assert len(s) == 1 and s.pop() == cls.eye(1)
# issue 3979
for cls in classes[:2]:
assert not isinstance(cls.eye(1), Hashable)
@XFAIL
def test_issue_3979():
# when this passes, delete this and change the [1:2]
# to [:2] in the test_hash above for issue 3979
cls = classes[0]
raises(AttributeError, lambda: hash(cls.eye(1)))
def test_adjoint():
dat = [[0, I], [1, 0]]
ans = Matrix([[0, 1], [-I, 0]])
for cls in classes:
assert ans == cls(dat).adjoint()
def test_simplify_immutable():
from sympy import simplify, sin, cos
assert simplify(ImmutableMatrix([[sin(x)**2 + cos(x)**2]])) == \
ImmutableMatrix([[1]])
def test_rank():
from sympy.abc import x
m = Matrix([[1, 2], [x, 1 - 1/x]])
assert m.rank() == 2
n = Matrix(3, 3, range(1, 10))
assert n.rank() == 2
p = zeros(3)
assert p.rank() == 0
def test_issue_11434():
ax, ay, bx, by, cx, cy, dx, dy, ex, ey, t0, t1 = \
symbols('a_x a_y b_x b_y c_x c_y d_x d_y e_x e_y t_0 t_1')
M = Matrix([[ax, ay, ax*t0, ay*t0, 0],
[bx, by, bx*t0, by*t0, 0],
[cx, cy, cx*t0, cy*t0, 1],
[dx, dy, dx*t0, dy*t0, 1],
[ex, ey, 2*ex*t1 - ex*t0, 2*ey*t1 - ey*t0, 0]])
assert M.rank() == 4
def test_rank_regression_from_so():
# see:
# https://stackoverflow.com/questions/19072700/why-does-sympy-give-me-the-wrong-answer-when-i-row-reduce-a-symbolic-matrix
nu, lamb = symbols('nu, lambda')
A = Matrix([[-3*nu, 1, 0, 0],
[ 3*nu, -2*nu - 1, 2, 0],
[ 0, 2*nu, (-1*nu) - lamb - 2, 3],
[ 0, 0, nu + lamb, -3]])
expected_reduced = Matrix([[1, 0, 0, 1/(nu**2*(-lamb - nu))],
[0, 1, 0, 3/(nu*(-lamb - nu))],
[0, 0, 1, 3/(-lamb - nu)],
[0, 0, 0, 0]])
expected_pivots = (0, 1, 2)
reduced, pivots = A.rref()
assert simplify(expected_reduced - reduced) == zeros(*A.shape)
assert pivots == expected_pivots
def test_replace():
from sympy import symbols, Function, Matrix
F, G = symbols('F, G', cls=Function)
K = Matrix(2, 2, lambda i, j: G(i+j))
M = Matrix(2, 2, lambda i, j: F(i+j))
N = M.replace(F, G)
assert N == K
def test_replace_map():
from sympy import symbols, Function, Matrix
F, G = symbols('F, G', cls=Function)
K = Matrix(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 = Matrix(2, 2, lambda i, j: F(i+j))
N = M.replace(F, G, True)
assert N == K
def test_atoms():
m = Matrix([[1, 2], [x, 1 - 1/x]])
assert m.atoms() == {S(1),S(2),S(-1), x}
assert m.atoms(Symbol) == {x}
def test_pinv():
# Pseudoinverse of an invertible matrix is the inverse.
A1 = Matrix([[a, b], [c, d]])
assert simplify(A1.pinv()) == simplify(A1.inv())
# Test the four properties of the pseudoinverse for various matrices.
As = [Matrix([[13, 104], [2212, 3], [-3, 5]]),
Matrix([[1, 7, 9], [11, 17, 19]]),
Matrix([a, b])]
for A in As:
A_pinv = A.pinv()
AAp = A * A_pinv
ApA = A_pinv * A
assert simplify(AAp * A) == A
assert simplify(ApA * A_pinv) == A_pinv
assert AAp.H == AAp
assert ApA.H == ApA
def test_pinv_solve():
# Fully determined system (unique result, identical to other solvers).
A = Matrix([[1, 5], [7, 9]])
B = Matrix([12, 13])
assert A.pinv_solve(B) == A.cholesky_solve(B)
assert A.pinv_solve(B) == A.LDLsolve(B)
assert A.pinv_solve(B) == Matrix([sympify('-43/26'), sympify('71/26')])
assert A * A.pinv() * B == B
# Fully determined, with two-dimensional B matrix.
B = Matrix([[12, 13, 14], [15, 16, 17]])
assert A.pinv_solve(B) == A.cholesky_solve(B)
assert A.pinv_solve(B) == A.LDLsolve(B)
assert A.pinv_solve(B) == Matrix([[-33, -37, -41], [69, 75, 81]]) / 26
assert A * A.pinv() * B == B
# Underdetermined system (infinite results).
A = Matrix([[1, 0, 1], [0, 1, 1]])
B = Matrix([5, 7])
solution = A.pinv_solve(B)
w = {}
for s in solution.atoms(Symbol):
# Extract dummy symbols used in the solution.
w[s.name] = s
assert solution == Matrix([[w['w0_0']/3 + w['w1_0']/3 - w['w2_0']/3 + 1],
[w['w0_0']/3 + w['w1_0']/3 - w['w2_0']/3 + 3],
[-w['w0_0']/3 - w['w1_0']/3 + w['w2_0']/3 + 4]])
assert A * A.pinv() * B == B
# Overdetermined system (least squares results).
A = Matrix([[1, 0], [0, 0], [0, 1]])
B = Matrix([3, 2, 1])
assert A.pinv_solve(B) == Matrix([3, 1])
# Proof the solution is not exact.
assert A * A.pinv() * B != B
def test_pinv_rank_deficient():
# Test the four properties of the pseudoinverse for various matrices.
As = [Matrix([[1, 1, 1], [2, 2, 2]]),
Matrix([[1, 0], [0, 0]]),
Matrix([[1, 2], [2, 4], [3, 6]])]
for A in As:
A_pinv = A.pinv()
AAp = A * A_pinv
ApA = A_pinv * A
assert simplify(AAp * A) == A
assert simplify(ApA * A_pinv) == A_pinv
assert AAp.H == AAp
assert ApA.H == ApA
# Test solving with rank-deficient matrices.
A = Matrix([[1, 0], [0, 0]])
# Exact, non-unique solution.
B = Matrix([3, 0])
solution = A.pinv_solve(B)
w1 = solution.atoms(Symbol).pop()
assert w1.name == 'w1_0'
assert solution == Matrix([3, w1])
assert A * A.pinv() * B == B
# Least squares, non-unique solution.
B = Matrix([3, 1])
solution = A.pinv_solve(B)
w1 = solution.atoms(Symbol).pop()
assert w1.name == 'w1_0'
assert solution == Matrix([3, w1])
assert A * A.pinv() * B != B
@XFAIL
def test_pinv_rank_deficient_when_diagonalization_fails():
# Test the four properties of the pseudoinverse for matrices when
# diagonalization of A.H*A fails.'
As = [Matrix([
[61, 89, 55, 20, 71, 0],
[62, 96, 85, 85, 16, 0],
[69, 56, 17, 4, 54, 0],
[10, 54, 91, 41, 71, 0],
[ 7, 30, 10, 48, 90, 0],
[0,0,0,0,0,0]])]
for A in As:
A_pinv = A.pinv()
AAp = A * A_pinv
ApA = A_pinv * A
assert simplify(AAp * A) == A
assert simplify(ApA * A_pinv) == A_pinv
assert AAp.H == AAp
assert ApA.H == ApA
def test_gauss_jordan_solve():
# Square, full rank, unique solution
A = Matrix([[1, 2, 3], [4, 5, 6], [7, 8, 10]])
b = Matrix([3, 6, 9])
sol, params = A.gauss_jordan_solve(b)
assert sol == Matrix([[-1], [2], [0]])
assert params == Matrix(0, 1, [])
# Square, full rank, unique solution, B has more columns than rows
A = eye(3)
B = Matrix([[1, 2, 3, 4], [5, 6, 7, 8], [9, 10, 11, 12]])
sol, params = A.gauss_jordan_solve(B)
assert sol == B
assert params == Matrix(0, 4, [])
# Square, reduced rank, parametrized solution
A = Matrix([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
b = Matrix([3, 6, 9])
sol, params, freevar = A.gauss_jordan_solve(b, freevar=True)
w = {}
for s in sol.atoms(Symbol):
# Extract dummy symbols used in the solution.
w[s.name] = s
assert sol == Matrix([[w['tau0'] - 1], [-2*w['tau0'] + 2], [w['tau0']]])
assert params == Matrix([[w['tau0']]])
assert freevar == [2]
# Square, reduced rank, parametrized solution, B has two columns
A = Matrix([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
B = Matrix([[3, 4], [6, 8], [9, 12]])
sol, params, freevar = A.gauss_jordan_solve(B, freevar=True)
w = {}
for s in sol.atoms(Symbol):
# Extract dummy symbols used in the solution.
w[s.name] = s
assert sol == Matrix([[w['tau0'] - 1, w['tau1'] - S(4)/3],
[-2*w['tau0'] + 2, -2*w['tau1'] + S(8)/3],
[w['tau0'], w['tau1']],])
assert params == Matrix([[w['tau0'], w['tau1']]])
assert freevar == [2]
# Square, reduced rank, parametrized solution
A = Matrix([[1, 2, 3], [2, 4, 6], [3, 6, 9]])
b = Matrix([0, 0, 0])
sol, params = A.gauss_jordan_solve(b)
w = {}
for s in sol.atoms(Symbol):
w[s.name] = s
assert sol == Matrix([[-2*w['tau0'] - 3*w['tau1']],
[w['tau0']], [w['tau1']]])
assert params == Matrix([[w['tau0']], [w['tau1']]])
# Square, reduced rank, parametrized solution
A = Matrix([[0, 0, 0], [0, 0, 0], [0, 0, 0]])
b = Matrix([0, 0, 0])
sol, params = A.gauss_jordan_solve(b)
w = {}
for s in sol.atoms(Symbol):
w[s.name] = s
assert sol == Matrix([[w['tau0']], [w['tau1']], [w['tau2']]])
assert params == Matrix([[w['tau0']], [w['tau1']], [w['tau2']]])
# Square, reduced rank, no solution
A = Matrix([[1, 2, 3], [2, 4, 6], [3, 6, 9]])
b = Matrix([0, 0, 1])
raises(ValueError, lambda: A.gauss_jordan_solve(b))
# Rectangular, tall, full rank, unique solution
A = Matrix([[1, 5, 3], [2, 1, 6], [1, 7, 9], [1, 4, 3]])
b = Matrix([0, 0, 1, 0])
sol, params = A.gauss_jordan_solve(b)
assert sol == Matrix([[-S(1)/2], [0], [S(1)/6]])
assert params == Matrix(0, 1, [])
# Rectangular, tall, full rank, unique solution, B has less columns than rows
A = Matrix([[1, 5, 3], [2, 1, 6], [1, 7, 9], [1, 4, 3]])
B = Matrix([[0,0], [0, 0], [1, 2], [0, 0]])
sol, params = A.gauss_jordan_solve(B)
assert sol == Matrix([[-S(1)/2, -S(2)/2], [0, 0], [S(1)/6, S(2)/6]])
assert params == Matrix(0, 2, [])
# Rectangular, tall, full rank, no solution
A = Matrix([[1, 5, 3], [2, 1, 6], [1, 7, 9], [1, 4, 3]])
b = Matrix([0, 0, 0, 1])
raises(ValueError, lambda: A.gauss_jordan_solve(b))
# Rectangular, tall, full rank, no solution, B has two columns (2nd has no solution)
A = Matrix([[1, 5, 3], [2, 1, 6], [1, 7, 9], [1, 4, 3]])
B = Matrix([[0,0], [0, 0], [1, 0], [0, 1]])
raises(ValueError, lambda: A.gauss_jordan_solve(B))
# Rectangular, tall, full rank, no solution, B has two columns (1st has no solution)
A = Matrix([[1, 5, 3], [2, 1, 6], [1, 7, 9], [1, 4, 3]])
B = Matrix([[0,0], [0, 0], [0, 1], [1, 0]])
raises(ValueError, lambda: A.gauss_jordan_solve(B))
# Rectangular, tall, reduced rank, parametrized solution
A = Matrix([[1, 5, 3], [2, 10, 6], [3, 15, 9], [1, 4, 3]])
b = Matrix([0, 0, 0, 1])
sol, params = A.gauss_jordan_solve(b)
w = {}
for s in sol.atoms(Symbol):
w[s.name] = s
assert sol == Matrix([[-3*w['tau0'] + 5], [-1], [w['tau0']]])
assert params == Matrix([[w['tau0']]])
# Rectangular, tall, reduced rank, no solution
A = Matrix([[1, 5, 3], [2, 10, 6], [3, 15, 9], [1, 4, 3]])
b = Matrix([0, 0, 1, 1])
raises(ValueError, lambda: A.gauss_jordan_solve(b))
# Rectangular, wide, full rank, parametrized solution
A = Matrix([[1, 2, 3, 4], [5, 6, 7, 8], [9, 10, 1, 12]])
b = Matrix([1, 1, 1])
sol, params = A.gauss_jordan_solve(b)
w = {}
for s in sol.atoms(Symbol):
w[s.name] = s
assert sol == Matrix([[2*w['tau0'] - 1], [-3*w['tau0'] + 1], [0],
[w['tau0']]])
assert params == Matrix([[w['tau0']]])
# Rectangular, wide, reduced rank, parametrized solution
A = Matrix([[1, 2, 3, 4], [5, 6, 7, 8], [2, 4, 6, 8]])
b = Matrix([0, 1, 0])
sol, params = A.gauss_jordan_solve(b)
w = {}
for s in sol.atoms(Symbol):
w[s.name] = s
assert sol == Matrix([[w['tau0'] + 2*w['tau1'] + 1/S(2)],
[-2*w['tau0'] - 3*w['tau1'] - 1/S(4)],
[w['tau0']], [w['tau1']]])
assert params == Matrix([[w['tau0']], [w['tau1']]])
# watch out for clashing symbols
x0, x1, x2, _x0 = symbols('_tau0 _tau1 _tau2 tau1')
M = Matrix([[0, 1, 0, 0, 0, 0], [0, 0, 0, 1, 0, _x0]])
A = M[:, :-1]
b = M[:, -1:]
sol, params = A.gauss_jordan_solve(b)
assert params == Matrix(3, 1, [x0, x1, x2])
assert sol == Matrix(5, 1, [x1, 0, x0, _x0, x2])
# Rectangular, wide, reduced rank, no solution
A = Matrix([[1, 2, 3, 4], [5, 6, 7, 8], [2, 4, 6, 8]])
b = Matrix([1, 1, 1])
raises(ValueError, lambda: A.gauss_jordan_solve(b))
def test_solve():
A = Matrix([[1,2], [2,4]])
b = Matrix([[3], [4]])
raises(ValueError, lambda: A.solve(b)) #no solution
b = Matrix([[ 4], [8]])
raises(ValueError, lambda: A.solve(b)) #infinite solution
def test_issue_7201():
assert ones(0, 1) + ones(0, 1) == Matrix(0, 1, [])
assert ones(1, 0) + ones(1, 0) == Matrix(1, 0, [])
def test_free_symbols():
for M in ImmutableMatrix, ImmutableSparseMatrix, Matrix, SparseMatrix:
assert M([[x], [0]]).free_symbols == {x}
def test_from_ndarray():
"""See issue 7465."""
try:
from numpy import array
except ImportError:
skip('NumPy must be available to test creating matrices from ndarrays')
assert Matrix(array([1, 2, 3])) == Matrix([1, 2, 3])
assert Matrix(array([[1, 2, 3]])) == Matrix([[1, 2, 3]])
assert Matrix(array([[1, 2, 3], [4, 5, 6]])) == \
Matrix([[1, 2, 3], [4, 5, 6]])
assert Matrix(array([x, y, z])) == Matrix([x, y, z])
raises(NotImplementedError, lambda: Matrix(array([[
[1, 2], [3, 4]], [[5, 6], [7, 8]]])))
def test_hermitian():
a = Matrix([[1, I], [-I, 1]])
assert a.is_hermitian
a[0, 0] = 2*I
assert a.is_hermitian is False
a[0, 0] = x
assert a.is_hermitian is None
a[0, 1] = a[1, 0]*I
assert a.is_hermitian is False
def test_doit():
a = Matrix([[Add(x,x, evaluate=False)]])
assert a[0] != 2*x
assert a.doit() == Matrix([[2*x]])
def test_issue_9457_9467_9876():
# for row_del(index)
M = Matrix([[1, 2, 3], [2, 3, 4], [3, 4, 5]])
M.row_del(1)
assert M == Matrix([[1, 2, 3], [3, 4, 5]])
N = Matrix([[1, 2, 3], [2, 3, 4], [3, 4, 5]])
N.row_del(-2)
assert N == Matrix([[1, 2, 3], [3, 4, 5]])
O = Matrix([[1, 2, 3], [5, 6, 7], [9, 10, 11]])
O.row_del(-1)
assert O == Matrix([[1, 2, 3], [5, 6, 7]])
P = Matrix([[1, 2, 3], [2, 3, 4], [3, 4, 5]])
raises(IndexError, lambda: P.row_del(10))
Q = Matrix([[1, 2, 3], [2, 3, 4], [3, 4, 5]])
raises(IndexError, lambda: Q.row_del(-10))
# for col_del(index)
M = Matrix([[1, 2, 3], [2, 3, 4], [3, 4, 5]])
M.col_del(1)
assert M == Matrix([[1, 3], [2, 4], [3, 5]])
N = Matrix([[1, 2, 3], [2, 3, 4], [3, 4, 5]])
N.col_del(-2)
assert N == Matrix([[1, 3], [2, 4], [3, 5]])
P = Matrix([[1, 2, 3], [2, 3, 4], [3, 4, 5]])
raises(IndexError, lambda: P.col_del(10))
Q = Matrix([[1, 2, 3], [2, 3, 4], [3, 4, 5]])
raises(IndexError, lambda: Q.col_del(-10))
def test_issue_9422():
x, y = symbols('x y', commutative=False)
a, b = symbols('a b')
M = eye(2)
M1 = Matrix(2, 2, [x, y, y, z])
assert y*x*M != x*y*M
assert b*a*M == a*b*M
assert x*M1 != M1*x
assert a*M1 == M1*a
assert y*x*M == Matrix([[y*x, 0], [0, y*x]])
def test_issue_10770():
M = Matrix([])
a = ['col_insert', 'row_join'], Matrix([9, 6, 3])
b = ['row_insert', 'col_join'], a[1].T
c = ['row_insert', 'col_insert'], Matrix([[1, 2], [3, 4]])
for ops, m in (a, b, c):
for op in ops:
f = getattr(M, op)
new = f(m) if 'join' in op else f(42, m)
assert new == m and id(new) != id(m)
def test_issue_10658():
A = Matrix([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
assert A.extract([0, 1, 2], [True, True, False]) == \
Matrix([[1, 2], [4, 5], [7, 8]])
assert A.extract([0, 1, 2], [True, False, False]) == Matrix([[1], [4], [7]])
assert A.extract([True, False, False], [0, 1, 2]) == Matrix([[1, 2, 3]])
assert A.extract([True, False, True], [0, 1, 2]) == \
Matrix([[1, 2, 3], [7, 8, 9]])
assert A.extract([0, 1, 2], [False, False, False]) == Matrix(3, 0, [])
assert A.extract([False, False, False], [0, 1, 2]) == Matrix(0, 3, [])
assert A.extract([True, False, True], [False, True, False]) == \
Matrix([[2], [8]])
def test_opportunistic_simplification():
# this test relates to issue #10718, #9480, #11434
# issue #9480
m = Matrix([[-5 + 5*sqrt(2), -5], [-5*sqrt(2)/2 + 5, -5*sqrt(2)/2]])
assert m.rank() == 1
# issue #10781
m = Matrix([[3+3*sqrt(3)*I, -9],[4,-3+3*sqrt(3)*I]])
assert simplify(m.rref()[0] - Matrix([[1, -9/(3 + 3*sqrt(3)*I)], [0, 0]])) == zeros(2, 2)
# issue #11434
ax,ay,bx,by,cx,cy,dx,dy,ex,ey,t0,t1 = symbols('a_x a_y b_x b_y c_x c_y d_x d_y e_x e_y t_0 t_1')
m = Matrix([[ax,ay,ax*t0,ay*t0,0],[bx,by,bx*t0,by*t0,0],[cx,cy,cx*t0,cy*t0,1],[dx,dy,dx*t0,dy*t0,1],[ex,ey,2*ex*t1-ex*t0,2*ey*t1-ey*t0,0]])
assert m.rank() == 4
def test_partial_pivoting():
# example from https://en.wikipedia.org/wiki/Pivot_element
# partial pivoting with back subsitution gives a perfect result
# naive pivoting give an error ~1e-13, so anything better than
# 1e-15 is good
mm=Matrix([[0.003 ,59.14, 59.17],[ 5.291, -6.13,46.78]])
assert (mm.rref()[0] - Matrix([[1.0, 0, 10.0], [ 0, 1.0, 1.0]])).norm() < 1e-15
# issue #11549
m_mixed = Matrix([[6e-17, 1.0, 4],[ -1.0, 0, 8],[ 0, 0, 1]])
m_float = Matrix([[6e-17, 1.0, 4.],[ -1.0, 0., 8.],[ 0., 0., 1.]])
m_inv = Matrix([[ 0, -1.0, 8.0],[1.0, 6.0e-17, -4.0],[ 0, 0, 1]])
# this example is numerically unstable and involves a matrix with a norm >= 8,
# this comparing the difference of the results with 1e-15 is numerically sound.
assert (m_mixed.inv() - m_inv).norm() < 1e-15
assert (m_float.inv() - m_inv).norm() < 1e-15
def test_iszero_substitution():
""" When doing numerical computations, all elements that pass
the iszerofunc test should be set to numerically zero if they
aren't already. """
# Matrix from issue #9060
m = Matrix([[0.9, -0.1, -0.2, 0],[-0.8, 0.9, -0.4, 0],[-0.1, -0.8, 0.6, 0]])
m_rref = m.rref(iszerofunc=lambda x: abs(x)<6e-15)[0]
m_correct = Matrix([[1.0, 0, -0.301369863013699, 0],[ 0, 1.0, -0.712328767123288, 0],[ 0, 0, 0, 0]])
m_diff = m_rref - m_correct
assert m_diff.norm() < 1e-15
# if a zero-substitution wasn't made, this entry will be -1.11022302462516e-16
assert m_rref[2,2] == 0
def test_rank_decomposition():
a = Matrix(0, 0, [])
c, f = a.rank_decomposition()
assert f.is_echelon
assert c.cols == f.rows == a.rank()
assert c * f == a
a = Matrix(1, 1, [5])
c, f = a.rank_decomposition()
assert f.is_echelon
assert c.cols == f.rows == a.rank()
assert c * f == a
a = Matrix(3, 3, [1, 2, 3, 1, 2, 3, 1, 2, 3])
c, f = a.rank_decomposition()
assert f.is_echelon
assert c.cols == f.rows == a.rank()
assert c * f == a
a = Matrix([
[0, 0, 1, 2, 2, -5, 3],
[-1, 5, 2, 2, 1, -7, 5],
[0, 0, -2, -3, -3, 8, -5],
[-1, 5, 0, -1, -2, 1, 0]])
c, f = a.rank_decomposition()
assert f.is_echelon
assert c.cols == f.rows == a.rank()
assert c * f == a
@slow
def test_issue_11238():
from sympy import Point
xx = 8*tan(13*pi/45)/(tan(13*pi/45) + sqrt(3))
yy = (-8*sqrt(3)*tan(13*pi/45)**2 + 24*tan(13*pi/45))/(-3 + tan(13*pi/45)**2)
p1 = Point(0, 0)
p2 = Point(1, -sqrt(3))
p0 = Point(xx,yy)
m1 = Matrix([p1 - simplify(p0), p2 - simplify(p0)])
m2 = Matrix([p1 - p0, p2 - p0])
m3 = Matrix([simplify(p1 - p0), simplify(p2 - p0)])
assert m1.rank(simplify=True) == 1
assert m2.rank(simplify=True) == 1
assert m3.rank(simplify=True) == 1
def test_as_real_imag():
m1 = Matrix(2,2,[1,2,3,4])
m2 = m1*S.ImaginaryUnit
m3 = m1 + m2
for kls in classes:
a,b = kls(m3).as_real_imag()
assert list(a) == list(m1)
assert list(b) == list(m1)
def test_deprecated():
# Maintain tests for deprecated functions. We must capture
# the deprecation warnings. When the deprecated functionality is
# removed, the corresponding tests should be removed.
m = Matrix(3, 3, [0, 1, 0, -4, 4, 0, -2, 1, 2])
P, Jcells = m.jordan_cells()
assert Jcells[1] == Matrix(1, 1, [2])
assert Jcells[0] == Matrix(2, 2, [2, 1, 0, 2])
with warns_deprecated_sympy():
assert Matrix([[1,2],[3,4]]).dot(Matrix([[1,3],[4,5]])) == [10, 19, 14, 28]
def test_issue_14489():
from sympy import Mod
A = Matrix([-1, 1, 2])
B = Matrix([10, 20, -15])
assert Mod(A, 3) == Matrix([2, 1, 2])
assert Mod(B, 4) == Matrix([2, 0, 1])
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_issue_14943():
# Test that __array__ accepts the optional dtype argument
try:
from numpy import array
except ImportError:
skip('NumPy must be available to test creating matrices from ndarrays')
M = Matrix([[1,2], [3,4]])
assert array(M, dtype=float).dtype.name == 'float64'
def test_issue_8240():
# Eigenvalues of large triangular matrices
n = 200
diagonal_variables = [Symbol('x%s' % i) for i in range(n)]
M = [[0 for i in range(n)] for j in range(n)]
for i in range(n):
M[i][i] = diagonal_variables[i]
M = Matrix(M)
eigenvals = M.eigenvals()
assert len(eigenvals) == n
for i in range(n):
assert eigenvals[diagonal_variables[i]] == 1
eigenvals = M.eigenvals(multiple=True)
assert set(eigenvals) == set(diagonal_variables)
# with multiplicity
M = Matrix([[x, 0, 0], [1, y, 0], [2, 3, x]])
eigenvals = M.eigenvals()
assert eigenvals == {x: 2, y: 1}
eigenvals = M.eigenvals(multiple=True)
assert len(eigenvals) == 3
assert eigenvals.count(x) == 2
assert eigenvals.count(y) == 1
def test_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(( (-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 test_case_6913():
m = MatrixSymbol('m', 1, 1)
a = Symbol("a")
a = m[0, 0]>0
assert str(a) == 'm[0, 0] > 0'
def test_issue_15872():
A = Matrix([[1, 1, 1, 0], [-2, -1, 0, -1], [0, 0, -1, -1], [0, 0, 2, 1]])
B = A - Matrix.eye(4) * I
assert B.rank() == 3
assert (B**2).rank() == 2
assert (B**3).rank() == 2
def test_issue_11948():
A = MatrixSymbol('A', 3, 3)
a = Wild('a')
assert A.match(a) == {a: A}
|
8d9a31c8ed70c177cbec3555e72112eb65be1f6aa65ac8b210d36297fac7e600
|
from __future__ import print_function, division
from sympy import Basic
from sympy.functions import adjoint, conjugate
from sympy.matrices.expressions.matexpr import MatrixExpr
class Transpose(MatrixExpr):
"""
The transpose of a matrix expression.
This is a symbolic object that simply stores its argument without
evaluating it. To actually compute the transpose, use the ``transpose()``
function, or the ``.T`` attribute of matrices.
Examples
========
>>> from sympy.matrices import MatrixSymbol, Transpose
>>> from sympy.functions import transpose
>>> A = MatrixSymbol('A', 3, 5)
>>> B = MatrixSymbol('B', 5, 3)
>>> Transpose(A)
A.T
>>> A.T == transpose(A) == Transpose(A)
True
>>> Transpose(A*B)
(A*B).T
>>> transpose(A*B)
B.T*A.T
"""
is_Transpose = True
def doit(self, **hints):
arg = self.arg
if hints.get('deep', True) and isinstance(arg, Basic):
arg = arg.doit(**hints)
_eval_transpose = getattr(arg, '_eval_transpose', None)
if _eval_transpose is not None:
result = _eval_transpose()
return result if result is not None else Transpose(arg)
else:
return Transpose(arg)
@property
def arg(self):
return self.args[0]
@property
def shape(self):
return self.arg.shape[::-1]
def _entry(self, i, j, expand=False):
return self.arg._entry(j, i, expand=expand)
def _eval_adjoint(self):
return conjugate(self.arg)
def _eval_conjugate(self):
return adjoint(self.arg)
def _eval_transpose(self):
return self.arg
def _eval_trace(self):
from .trace import Trace
return Trace(self.arg) # Trace(X.T) => Trace(X)
def _eval_determinant(self):
from sympy.matrices.expressions.determinant import det
return det(self.arg)
def _eval_derivative_matrix_lines(self, x):
lines = self.args[0]._eval_derivative_matrix_lines(x)
return [i.transpose() for i in lines]
def transpose(expr):
"""Matrix transpose"""
return Transpose(expr).doit(deep=False)
from sympy.assumptions.ask import ask, Q
from sympy.assumptions.refine import handlers_dict
def refine_Transpose(expr, assumptions):
"""
>>> from sympy import MatrixSymbol, Q, assuming, refine
>>> X = MatrixSymbol('X', 2, 2)
>>> X.T
X.T
>>> with assuming(Q.symmetric(X)):
... print(refine(X.T))
X
"""
if ask(Q.symmetric(expr), assumptions):
return expr.arg
return expr
handlers_dict['Transpose'] = refine_Transpose
|
e2f5c4b6baca0e0ba16dac363d1f1df59549d00fdff0d8637c72b36cd6e927e0
|
from __future__ import print_function, division
from sympy import Number
from sympy.core import Mul, Basic, sympify, Add
from sympy.core.compatibility import range
from sympy.functions import adjoint
from sympy.matrices.expressions.transpose import transpose
from sympy.strategies import (rm_id, unpack, typed, flatten, exhaust,
do_one, new)
from sympy.matrices.expressions.matexpr import (MatrixExpr, ShapeError,
Identity, ZeroMatrix, GenericIdentity)
from sympy.matrices.expressions.matpow import MatPow
from sympy.matrices.matrices import MatrixBase
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
def __new__(cls, *args, **kwargs):
check = kwargs.get('check', True)
if not args:
return GenericIdentity()
# This must be removed aggressively in the constructor to avoid
# TypeErrors from GenericIdentity().shape
args = filter(lambda i: GenericIdentity() != i, args)
args = list(map(sympify, args))
obj = Basic.__new__(cls, *args)
factor, matrices = obj.as_coeff_matrices()
if check:
validate(*matrices)
if not matrices:
# Should it be
#
# return Basic.__neq__(cls, factor, GenericIdentity()) ?
return factor
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):
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
for i in range(1, len(matrices)):
indices[i] = Dummy("i_%i" % i)
for i, arg in enumerate(matrices[:-1]):
ind_ranges[i] = arg.shape[1] - 1
matrices = [arg[indices[i], indices[i+1]] 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):
return MatMul(*[transpose(arg) for arg in self.args[::-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:
from sympy.matrices.expressions.inverse import Inverse
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
mats = [arg for arg in self.args if arg.is_Matrix]
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:]
right_mat = MatMul.fromiter(right_args)
right_rev = MatMul.fromiter([Transpose(i).doit() for i in reversed(right_args)])
left_mat = MatMul.fromiter(left_args)
left_rev = MatMul.fromiter([Transpose(i).doit() for i in reversed(left_args)])
d = self.args[ind]._eval_derivative_matrix_lines(x)
for i in d:
if i.transposed:
i.append_first(right_mat)
i.append_second(left_rev)
else:
i.append_first(left_rev)
i.append_second(right_mat)
lines.append(i)
return lines
def validate(*matrices):
""" Checks for valid shapes for args of MatMul """
for i in range(len(matrices)-1):
A, B = matrices[i:i+2]
if A.cols != B.rows:
raise ShapeError("Matrices %s and %s are not aligned"%(A, B))
# Rules
def newmul(*args):
if args[0] == 1:
args = args[1:]
return new(MatMul, *args)
def any_zeros(mul):
if any([arg.is_zero or (arg.is_Matrix and arg.is_ZeroMatrix)
for arg in mul.args]):
matrices = [arg for arg in mul.args if arg.is_Matrix]
return ZeroMatrix(matrices[0].rows, matrices[-1].cols)
return mul
def merge_explicit(matmul):
""" Merge explicit MatrixBase arguments
>>> from sympy import MatrixSymbol, eye, 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 xxinv(mul):
""" Y * X * X.I -> Y """
from sympy.matrices.expressions.inverse import Inverse
factor, matrices = mul.as_coeff_matrices()
for i, (X, Y) in enumerate(zip(matrices[:-1], matrices[1:])):
try:
if X.is_square and Y.is_square:
_X, x_exp = X, 1
_Y, y_exp = Y, 1
if isinstance(X, MatPow) and not isinstance(X, Inverse):
_X, x_exp = X.args
if isinstance(Y, MatPow) and not isinstance(Y, Inverse):
_Y, y_exp = Y.args
if _X == _Y.inverse():
if x_exp - y_exp > 0:
I = _X**(x_exp-y_exp)
else:
I = _Y**(y_exp-x_exp)
return newmul(factor, *(matrices[:i] + [I] + matrices[i+2:]))
except ValueError: # Y might not be invertible
pass
return mul
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
from sympy.matrices.expressions import MatPow
factor, mmul = mul.as_coeff_mmul()
args = []
base = None
exp = 0
for arg in mmul.args:
if isinstance(arg, MatPow):
current_base = arg.args[0]
current_exp = arg.args[1]
else:
current_base = arg
current_exp = 1
if current_base == base:
exp += current_exp
else:
if not base is None:
if exp == 1:
args.append(base)
else:
args.append(base**exp)
exp = current_exp
base = current_base
if exp == 1:
args.append(base)
else:
args.append(base**exp)
return newmul(factor, *args)
rules = (any_zeros, remove_ids, xxinv, unpack, rm_id(lambda x: x == 1),
merge_explicit, factor_in_front, flatten, combine_powers)
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
|
67ac964406f0721549ac2b00b763e79a1ae6346508a345c374dfc09395b5fd82
|
from __future__ import print_function, division
from .matexpr import MatrixExpr, ShapeError, Identity, ZeroMatrix
from sympy.core import S
from sympy.core.compatibility import range
from sympy.core.sympify import _sympify
from sympy.matrices import MatrixBase
class MatPow(MatrixExpr):
def __new__(cls, base, exp):
base = _sympify(base)
if not base.is_Matrix:
raise TypeError("Function parameter should be a matrix")
exp = _sympify(exp)
return super(MatPow, cls).__new__(cls, base, exp)
@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 not A.base.is_square:
raise ShapeError("Power of non-square matrix %s" % A.base)
elif 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._entry(i, j)
def doit(self, **kwargs):
from sympy.matrices.expressions import Inverse
deep = kwargs.get('deep', True)
if deep:
args = [arg.doit(**kwargs) for arg in self.args]
else:
args = self.args
base, exp = args
# combine all powers, e.g. (A**2)**3 = A**6
while isinstance(base, MatPow):
exp = exp*base.args[1]
base = base.args[0]
if exp.is_zero and base.is_square:
if isinstance(base, MatrixBase):
return base.func(Identity(base.shape[0]))
return Identity(base.shape[0])
elif isinstance(base, ZeroMatrix) and exp.is_negative:
raise ValueError("Matrix determinant is 0, not invertible.")
elif isinstance(base, (Identity, ZeroMatrix)):
return base
elif isinstance(base, MatrixBase) and exp.is_number:
if exp is S.One:
return base
return base**exp
# Note: just evaluate cases we know, return unevaluated on others.
# E.g., MatrixSymbol('x', n, m) to power 0 is not an error.
elif exp is S(-1) and base.is_square:
return Inverse(base).doit(**kwargs)
elif exp is S.One:
return base
return MatPow(base, exp)
def _eval_transpose(self):
base, exp = self.args
return MatPow(base.T, exp)
def _eval_derivative_matrix_lines(self, x):
from .matmul import MatMul
from .inverse import Inverse
exp = self.exp
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)
|
f90634722d612eafb6a7e1de321e2242929abfd82db263c8ea2ff86bae2a500c
|
from __future__ import print_function, division
from functools import wraps, reduce
import collections
from sympy.core import S, Symbol, Tuple, Integer, Basic, Expr, Eq, Mul, Add
from sympy.core.decorators import call_highest_priority
from sympy.core.compatibility import range, SYMPY_INTS, default_sort_key
from sympy.core.sympify import SympifyError, sympify
from sympy.functions import conjugate, adjoint
from sympy.functions.special.tensor_functions import KroneckerDelta
from sympy.matrices import ShapeError
from sympy.simplify import simplify
from sympy.utilities.misc import filldedent
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, strict=True)
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
is_MatrixExpr = True
is_Identity = None
is_Inverse = False
is_Transpose = False
is_ZeroMatrix = False
is_MatAdd = False
is_MatMul = False
is_commutative = False
is_number = False
is_symbol = False
def __new__(cls, *args, **kwargs):
args = map(sympify, args)
return Basic.__new__(cls, *args, **kwargs)
# The following is adapted from the core Expr object
def __neg__(self):
return MatMul(S.NegativeOne, self).doit()
def __abs__(self):
raise NotImplementedError
@_sympifyit('other', NotImplemented)
@call_highest_priority('__radd__')
def __add__(self, other):
return MatAdd(self, other, check=True).doit()
@_sympifyit('other', NotImplemented)
@call_highest_priority('__add__')
def __radd__(self, other):
return MatAdd(other, self, check=True).doit()
@_sympifyit('other', NotImplemented)
@call_highest_priority('__rsub__')
def __sub__(self, other):
return MatAdd(self, -other, check=True).doit()
@_sympifyit('other', NotImplemented)
@call_highest_priority('__sub__')
def __rsub__(self, other):
return MatAdd(other, -self, check=True).doit()
@_sympifyit('other', NotImplemented)
@call_highest_priority('__rmul__')
def __mul__(self, other):
return MatMul(self, other).doit()
@_sympifyit('other', NotImplemented)
@call_highest_priority('__rmul__')
def __matmul__(self, other):
return MatMul(self, other).doit()
@_sympifyit('other', NotImplemented)
@call_highest_priority('__mul__')
def __rmul__(self, other):
return MatMul(other, self).doit()
@_sympifyit('other', NotImplemented)
@call_highest_priority('__mul__')
def __rmatmul__(self, other):
return MatMul(other, self).doit()
@_sympifyit('other', NotImplemented)
@call_highest_priority('__rpow__')
def __pow__(self, other):
if not self.is_square:
raise ShapeError("Power of non-square matrix %s" % self)
elif self.is_Identity:
return self
elif other is S.Zero:
return Identity(self.rows)
elif other is S.One:
return self
return MatPow(self, other).doit(deep=False)
@_sympifyit('other', NotImplemented)
@call_highest_priority('__pow__')
def __rpow__(self, other):
raise NotImplementedError("Matrix Power not defined")
@_sympifyit('other', NotImplemented)
@call_highest_priority('__rdiv__')
def __div__(self, other):
return self * other**S.NegativeOne
@_sympifyit('other', NotImplemented)
@call_highest_priority('__div__')
def __rdiv__(self, other):
raise NotImplementedError()
#return MatMul(other, Pow(self, S.NegativeOne))
__truediv__ = __div__
__rtruediv__ = __rdiv__
@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):
from sympy import I
real = (S(1)/2) * (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):
return MatPow(self, exp)
def _eval_simplify(self, **kwargs):
if self.is_Atom:
return self
else:
return self.__class__(*[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(self, x):
return _matrix_derivative(self, x)
def _eval_derivative_n_times(self, x, n):
return Basic._eval_derivative_n_times(self, x, n)
def _visit_eval_derivative_scalar(self, x):
# `x` is a scalar:
if x.has(self):
return _matrix_derivative(x, self)
else:
return ZeroMatrix(*self.shape)
def _visit_eval_derivative_array(self, x):
if x.has(self):
return _matrix_derivative(x, self)
else:
from sympy import Derivative
return Derivative(x, self)
def _accept_eval_derivative(self, s):
return s._visit_eval_derivative_array(self)
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)
T = property(transpose, None, None, 'Matrix transposition.')
def inverse(self):
return self._eval_inverse()
inv = 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
"""
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, Symbol
>>> 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
ret_indices = list(j for i in lines for j in i)
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: {0} vs. (0, {1})".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: {0} vs. (0, {1})".format(
(r1, r2), matrix_symbol.shape[1]))
if (i1 == i2) and (i1 in index_ranges):
return [(trace(matrix_symbol), None)]
return [(MatrixElement(matrix_symbol, i1, i2), (i1, i2))]
elif isinstance(expr, Sum):
return recurse_expr(
expr.args[0],
index_ranges={i[0]: i[1:] for i in expr.args[1:]}
)
else:
return [(expr, None)]
retvals = recurse_expr(expr)
factors, indices = zip(*retvals)
retexpr = Mul.fromiter(factors)
if len(indices) == 0 or list(set(indices)) == [None]:
return retexpr
if first_index is None:
for i in indices:
if i is not None:
ind0 = i
break
return remove_matelement(retexpr, *ind0)
else:
return remove_matelement(retexpr, first_index, last_index)
def applyfunc(self, func):
from .applyfunc import ElementwiseApplyFunction
return ElementwiseApplyFunction(func, self)
def _eval_Eq(self, other):
if not isinstance(other, MatrixExpr):
return False
if self.shape != other.shape:
return False
if (self - other).is_ZeroMatrix:
return True
return Eq(self, other, evaluate=False)
def get_postprocessor(cls):
def _postprocessor(expr):
# To avoid circular imports, we can't have MatMul/MatAdd on the top level
mat_class = {Mul: MatMul, Add: MatAdd}[cls]
nonmatrices = []
matrices = []
for term in expr.args:
if isinstance(term, MatrixExpr):
matrices.append(term)
else:
nonmatrices.append(term)
if not matrices:
return cls._from_args(nonmatrices)
if nonmatrices:
if cls == Mul:
for i in range(len(matrices)):
if not matrices[i].is_MatrixExpr:
# If one of the matrices explicit, absorb the scalar into it
# (doit will combine all explicit matrices into one, so it
# doesn't matter which)
matrices[i] = matrices[i].__mul__(cls._from_args(nonmatrices))
nonmatrices = []
break
else:
# Maintain the ability to create Add(scalar, matrix) without
# raising an exception. That way different algorithms can
# replace matrix expressions with non-commutative symbols to
# manipulate them like non-commutative scalars.
return cls._from_args(nonmatrices + [mat_class(*matrices).doit(deep=False)])
return mat_class(cls._from_args(nonmatrices), *matrices).doit(deep=False)
return _postprocessor
Basic._constructor_postprocessor_mapping[MatrixExpr] = {
"Mul": [get_postprocessor(Mul)],
"Add": [get_postprocessor(Add)],
}
def _matrix_derivative(expr, x):
from sympy import Derivative
lines = expr._eval_derivative_matrix_lines(x)
ranks = [i.rank() for i in lines]
assert len(set(ranks)) == 1
rank = ranks[0]
if rank <= 2:
return Add.fromiter([i.matrix_form() for i in lines])
return Derivative(expr, x)
class MatrixElement(Expr):
parent = property(lambda self: self.args[0])
i = property(lambda self: self.args[1])
j = property(lambda self: self.args[2])
_diff_wrt = True
is_symbol = True
is_commutative = True
def __new__(cls, name, n, m):
n, m = map(sympify, (n, m))
from sympy import MatrixBase
if isinstance(name, (MatrixBase,)):
if n.is_Integer and m.is_Integer:
return name[n, m]
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]
if M == v.args[0]:
return KroneckerDelta(self.args[1], v.args[1])*KroneckerDelta(self.args[2], v.args[2])
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)
obj = Basic.__new__(cls, name, n, m)
return obj
def _hashable_content(self):
return (self.name, self.shape)
@property
def shape(self):
return self.args[1:3]
@property
def name(self):
return self.args[0]
def _eval_subs(self, old, new):
# only do substitutions in shape
shape = Tuple(*self.shape)._subs(old, new)
return MatrixSymbol(self.name, *shape)
def __call__(self, *args):
raise TypeError("%s object is not callable" % self.__class__)
def _entry(self, i, j, **kwargs):
return MatrixElement(self, i, j)
@property
def free_symbols(self):
return set((self,))
def doit(self, **hints):
if hints.get('deep', True):
return type(self)(self.name, self.args[1].doit(**hints),
self.args[2].doit(**hints))
else:
return self
def _eval_simplify(self, **kwargs):
return self
def _eval_derivative_matrix_lines(self, x):
if self != x:
return [_LeftRightArgs(
ZeroMatrix(x.shape[0], self.shape[0]),
ZeroMatrix(x.shape[1], self.shape[1]),
transposed=False,
)]
else:
first = Identity(self.shape[0])
second = Identity(self.shape[1])
return [_LeftRightArgs(
first=first,
second=second,
transposed=False,
)]
class Identity(MatrixExpr):
"""The Matrix Identity I - multiplicative identity
Examples
========
>>> from sympy.matrices import Identity, MatrixSymbol
>>> A = MatrixSymbol('A', 3, 5)
>>> I = Identity(3)
>>> I*A
A
"""
is_Identity = True
def __new__(cls, n):
return super(Identity, cls).__new__(cls, sympify(n))
@property
def rows(self):
return self.args[0]
@property
def cols(self):
return self.args[0]
@property
def shape(self):
return (self.args[0], self.args[0])
@property
def is_square(self):
return True
def _eval_transpose(self):
return self
def _eval_trace(self):
return self.rows
def _eval_inverse(self):
return self
def conjugate(self):
return self
def _entry(self, i, j, **kwargs):
eq = Eq(i, j)
if eq is S.true:
return S.One
elif eq is S.false:
return S.Zero
return KroneckerDelta(i, j)
def _eval_determinant(self):
return S.One
class GenericIdentity(Identity):
"""
An identity matrix without a specified shape
This exists primarily so MatMul() with no arguments can return something
meaningful.
"""
def __new__(cls):
# super(Identity, cls) instead of super(GenericIdentity, cls) because
# Identity.__new__ doesn't have the same signature
return super(Identity, cls).__new__(cls)
@property
def rows(self):
raise TypeError("GenericIdentity does not have a specified shape")
@property
def cols(self):
raise TypeError("GenericIdentity does not have a specified shape")
@property
def shape(self):
raise TypeError("GenericIdentity does not have a specified shape")
# Avoid Matrix.__eq__ which might call .shape
def __eq__(self, other):
return isinstance(other, GenericIdentity)
def __ne__(self, other):
return not (self == other)
def __hash__(self):
return super(GenericIdentity, self).__hash__()
class ZeroMatrix(MatrixExpr):
"""The Matrix Zero 0 - additive identity
Examples
========
>>> from sympy import MatrixSymbol, ZeroMatrix
>>> A = MatrixSymbol('A', 3, 5)
>>> Z = ZeroMatrix(3, 5)
>>> A + Z
A
>>> Z*A.T
0
"""
is_ZeroMatrix = True
def __new__(cls, m, n):
return super(ZeroMatrix, cls).__new__(cls, m, n)
@property
def shape(self):
return (self.args[0], self.args[1])
@_sympifyit('other', NotImplemented)
@call_highest_priority('__rpow__')
def __pow__(self, other):
if other != 1 and not self.is_square:
raise ShapeError("Power of non-square matrix %s" % self)
if other == 0:
return Identity(self.rows)
if other < 1:
raise ValueError("Matrix det == 0; not invertible.")
return self
def _eval_transpose(self):
return ZeroMatrix(self.cols, self.rows)
def _eval_trace(self):
return S.Zero
def _eval_determinant(self):
return S.Zero
def conjugate(self):
return self
def _entry(self, i, j, **kwargs):
return S.Zero
def __nonzero__(self):
return False
__bool__ = __nonzero__
class GenericZeroMatrix(ZeroMatrix):
"""
A zero matrix without a specified shape
This exists primarily so MatAdd() with no arguments can return something
meaningful.
"""
def __new__(cls):
# super(ZeroMatrix, cls) instead of super(GenericZeroMatrix, cls)
# because ZeroMatrix.__new__ doesn't have the same signature
return super(ZeroMatrix, cls).__new__(cls)
@property
def rows(self):
raise TypeError("GenericZeroMatrix does not have a specified shape")
@property
def cols(self):
raise TypeError("GenericZeroMatrix does not have a specified shape")
@property
def shape(self):
raise TypeError("GenericZeroMatrix does not have a specified shape")
# Avoid Matrix.__eq__ which might call .shape
def __eq__(self, other):
return isinstance(other, GenericZeroMatrix)
def __ne__(self, other):
return not (self == other)
def __hash__(self):
return super(GenericZeroMatrix, self).__hash__()
def matrix_symbols(expr):
return [sym for sym in expr.free_symbols if sym.is_Matrix]
class _LeftRightArgs(object):
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, first, second, higher=S.One, transposed=False):
self.first = first
self.second = second
self.higher = higher
self.transposed = transposed
def __repr__(self):
return "_LeftRightArgs(first=%s[%s], second=%s[%s], higher=%s, transposed=%s)" % (
self.first, self.first.shape if isinstance(self.first, MatrixExpr) else None,
self.second, self.second.shape if isinstance(self.second, MatrixExpr) else None,
self.higher,
self.transposed,
)
def transpose(self):
self.transposed = not self.transposed
return self
def matrix_form(self):
if self.first != 1 and self.higher != 1:
raise ValueError("higher dimensional array cannot be represented")
# Remove one-dimensional identity matrices:
# (this is needed by `a.diff(a)` where `a` is a vector)
if self.first == Identity(1):
return self.second.T
if self.second == Identity(1):
return self.first
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 append_first(self, other):
self.first *= other
def append_second(self, other):
self.second *= other
def __hash__(self):
return hash((self.first, self.second, self.transposed))
def __eq__(self, other):
if not isinstance(other, _LeftRightArgs):
return False
return (self.first == other.first) and (self.second == other.second) and (self.transposed == other.transposed)
from .matmul import MatMul
from .matadd import MatAdd
from .matpow import MatPow
from .transpose import Transpose
from .inverse import Inverse
|
9816d41736ba07bcca219c951df29de0b0bbd40a14227d69650829167eb287e1
|
"""Implementation of the Kronecker product"""
from __future__ import division, print_function
from sympy.core import Add, Mul, Pow, prod, sympify
from sympy.core.compatibility import range
from sympy.functions import adjoint
from sympy.matrices.expressions.matexpr import MatrixExpr, ShapeError, Identity
from sympy.matrices.expressions.transpose import transpose
from sympy.matrices.matrices import MatrixBase
from sympy.strategies import (
canon, condition, distribute, do_one, exhaust, flatten, typed, unpack)
from sympy.strategies.traverse import bottom_up
from sympy.utilities import sift
from .matadd import MatAdd
from .matmul import MatMul
from .matpow import MatPow
def kronecker_product(*matrices):
"""
The Kronecker product of two or more arguments.
This computes the explicit Kronecker product for subclasses of
``MatrixBase`` i.e. explicit matrices. Otherwise, a symbolic
``KroneckerProduct`` object is returned.
Examples
========
For ``MatrixSymbol`` arguments a ``KroneckerProduct`` object is returned.
Elements of this matrix can be obtained by indexing, or for MatrixSymbols
with known dimension the explicit matrix can be obtained with
``.as_explicit()``
>>> from sympy.matrices import kronecker_product, MatrixSymbol
>>> A = MatrixSymbol('A', 2, 2)
>>> B = MatrixSymbol('B', 2, 2)
>>> kronecker_product(A)
A
>>> kronecker_product(A, B)
KroneckerProduct(A, B)
>>> kronecker_product(A, B)[0, 1]
A[0, 0]*B[0, 1]
>>> kronecker_product(A, B).as_explicit()
Matrix([
[A[0, 0]*B[0, 0], A[0, 0]*B[0, 1], A[0, 1]*B[0, 0], A[0, 1]*B[0, 1]],
[A[0, 0]*B[1, 0], A[0, 0]*B[1, 1], A[0, 1]*B[1, 0], A[0, 1]*B[1, 1]],
[A[1, 0]*B[0, 0], A[1, 0]*B[0, 1], A[1, 1]*B[0, 0], A[1, 1]*B[0, 1]],
[A[1, 0]*B[1, 0], A[1, 0]*B[1, 1], A[1, 1]*B[1, 0], A[1, 1]*B[1, 1]]])
For explicit matrices the Kronecker product is returned as a Matrix
>>> from sympy.matrices import Matrix, kronecker_product
>>> sigma_x = Matrix([
... [0, 1],
... [1, 0]])
...
>>> Isigma_y = Matrix([
... [0, 1],
... [-1, 0]])
...
>>> kronecker_product(sigma_x, Isigma_y)
Matrix([
[ 0, 0, 0, 1],
[ 0, 0, -1, 0],
[ 0, 1, 0, 0],
[-1, 0, 0, 0]])
See Also
========
KroneckerProduct
"""
if not matrices:
raise TypeError("Empty Kronecker product is undefined")
validate(*matrices)
if len(matrices) == 1:
return matrices[0]
else:
return KroneckerProduct(*matrices).doit()
class KroneckerProduct(MatrixExpr):
"""
The Kronecker product of two or more arguments.
The Kronecker product is a non-commutative product of matrices.
Given two matrices of dimension (m, n) and (s, t) it produces a matrix
of dimension (m s, n t).
This is a symbolic object that simply stores its argument without
evaluating it. To actually compute the product, use the function
``kronecker_product()`` or call the the ``.doit()`` or ``.as_explicit()``
methods.
>>> from sympy.matrices import KroneckerProduct, MatrixSymbol
>>> A = MatrixSymbol('A', 5, 5)
>>> B = MatrixSymbol('B', 5, 5)
>>> isinstance(KroneckerProduct(A, B), KroneckerProduct)
True
"""
is_KroneckerProduct = True
def __new__(cls, *args, **kwargs):
args = list(map(sympify, args))
if all(a.is_Identity for a in args):
ret = Identity(prod(a.rows for a in args))
if all(isinstance(a, MatrixBase) for a in args):
return ret.as_explicit()
else:
return ret
check = kwargs.get('check', True)
if check:
validate(*args)
return super(KroneckerProduct, cls).__new__(cls, *args)
@property
def shape(self):
rows, cols = self.args[0].shape
for mat in self.args[1:]:
rows *= mat.rows
cols *= mat.cols
return (rows, cols)
def _entry(self, i, j):
result = 1
for mat in reversed(self.args):
i, m = divmod(i, mat.rows)
j, n = divmod(j, mat.cols)
result *= mat[m, n]
return result
def _eval_adjoint(self):
return KroneckerProduct(*list(map(adjoint, self.args))).doit()
def _eval_conjugate(self):
return KroneckerProduct(*[a.conjugate() for a in self.args]).doit()
def _eval_transpose(self):
return KroneckerProduct(*list(map(transpose, self.args))).doit()
def _eval_trace(self):
from .trace import trace
return prod(trace(a) for a in self.args)
def _eval_determinant(self):
from .determinant import det, Determinant
if not all(a.is_square for a in self.args):
return Determinant(self)
m = self.rows
return prod(det(a)**(m/a.rows) for a in self.args)
def _eval_inverse(self):
try:
return KroneckerProduct(*[a.inverse() for a in self.args])
except ShapeError:
from sympy.matrices.expressions.inverse import Inverse
return Inverse(self)
def structurally_equal(self, other):
'''Determine whether two matrices have the same Kronecker product structure
Examples
========
>>> from sympy import KroneckerProduct, MatrixSymbol, symbols
>>> m, n = symbols(r'm, n', integer=True)
>>> A = MatrixSymbol('A', m, m)
>>> B = MatrixSymbol('B', n, n)
>>> C = MatrixSymbol('C', m, m)
>>> D = MatrixSymbol('D', n, n)
>>> KroneckerProduct(A, B).structurally_equal(KroneckerProduct(C, D))
True
>>> KroneckerProduct(A, B).structurally_equal(KroneckerProduct(D, C))
False
>>> KroneckerProduct(A, B).structurally_equal(C)
False
'''
# Inspired by BlockMatrix
return (isinstance(other, KroneckerProduct)
and self.shape == other.shape
and len(self.args) == len(other.args)
and all(a.shape == b.shape for (a, b) in zip(self.args, other.args)))
def has_matching_shape(self, other):
'''Determine whether two matrices have the appropriate structure to bring matrix
multiplication inside the KroneckerProdut
Examples
========
>>> from sympy import KroneckerProduct, MatrixSymbol, symbols
>>> m, n = symbols(r'm, n', integer=True)
>>> A = MatrixSymbol('A', m, n)
>>> B = MatrixSymbol('B', n, m)
>>> KroneckerProduct(A, B).has_matching_shape(KroneckerProduct(B, A))
True
>>> KroneckerProduct(A, B).has_matching_shape(KroneckerProduct(A, B))
False
>>> KroneckerProduct(A, B).has_matching_shape(A)
False
'''
return (isinstance(other, KroneckerProduct)
and self.cols == other.rows
and len(self.args) == len(other.args)
and all(a.cols == b.rows for (a, b) in zip(self.args, other.args)))
def _eval_expand_kroneckerproduct(self, **hints):
return flatten(canon(typed({KroneckerProduct: distribute(KroneckerProduct, MatAdd)}))(self))
def _kronecker_add(self, other):
if self.structurally_equal(other):
return self.__class__(*[a + b for (a, b) in zip(self.args, other.args)])
else:
return self + other
def _kronecker_mul(self, other):
if self.has_matching_shape(other):
return self.__class__(*[a*b for (a, b) in zip(self.args, other.args)])
else:
return self * other
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 canonicalize(KroneckerProduct(*args))
def validate(*args):
if not all(arg.is_Matrix for arg in args):
raise TypeError("Mix of Matrix and Scalar symbols")
# rules
def extract_commutative(kron):
c_part = []
nc_part = []
for arg in kron.args:
c, nc = arg.args_cnc()
c_part.extend(c)
nc_part.append(Mul._from_args(nc))
c_part = Mul(*c_part)
if c_part != 1:
return c_part*KroneckerProduct(*nc_part)
return kron
def matrix_kronecker_product(*matrices):
"""Compute the Kronecker product of a sequence of SymPy Matrices.
This is the standard Kronecker product of matrices [1].
Parameters
==========
matrices : tuple of MatrixBase instances
The matrices to take the Kronecker product of.
Returns
=======
matrix : MatrixBase
The Kronecker product matrix.
Examples
========
>>> from sympy import Matrix
>>> from sympy.matrices.expressions.kronecker import (
... matrix_kronecker_product)
>>> m1 = Matrix([[1,2],[3,4]])
>>> m2 = Matrix([[1,0],[0,1]])
>>> matrix_kronecker_product(m1, m2)
Matrix([
[1, 0, 2, 0],
[0, 1, 0, 2],
[3, 0, 4, 0],
[0, 3, 0, 4]])
>>> matrix_kronecker_product(m2, m1)
Matrix([
[1, 2, 0, 0],
[3, 4, 0, 0],
[0, 0, 1, 2],
[0, 0, 3, 4]])
References
==========
[1] https://en.wikipedia.org/wiki/Kronecker_product
"""
# Make sure we have a sequence of Matrices
if not all(isinstance(m, MatrixBase) for m in matrices):
raise TypeError(
'Sequence of Matrices expected, got: %s' % repr(matrices)
)
# Pull out the first element in the product.
matrix_expansion = matrices[-1]
# Do the kronecker product working from right to left.
for mat in reversed(matrices[:-1]):
rows = mat.rows
cols = mat.cols
# Go through each row appending kronecker product to.
# running matrix_expansion.
for i in range(rows):
start = matrix_expansion*mat[i*cols]
# Go through each column joining each item
for j in range(cols - 1):
start = start.row_join(
matrix_expansion*mat[i*cols + j + 1]
)
# If this is the first element, make it the start of the
# new row.
if i == 0:
next = start
else:
next = next.col_join(start)
matrix_expansion = next
MatrixClass = max(matrices, key=lambda M: M._class_priority).__class__
if isinstance(matrix_expansion, MatrixClass):
return matrix_expansion
else:
return MatrixClass(matrix_expansion)
def explicit_kronecker_product(kron):
# Make sure we have a sequence of Matrices
if not all(isinstance(m, MatrixBase) for m in kron.args):
return kron
return matrix_kronecker_product(*kron.args)
rules = (unpack,
explicit_kronecker_product,
flatten,
extract_commutative)
canonicalize = exhaust(condition(lambda x: isinstance(x, KroneckerProduct),
do_one(*rules)))
def _kronecker_dims_key(expr):
if isinstance(expr, KroneckerProduct):
return tuple(a.shape for a in expr.args)
else:
return (0,)
def kronecker_mat_add(expr):
from functools import reduce
args = sift(expr.args, _kronecker_dims_key)
nonkrons = args.pop((0,), None)
if not args:
return expr
krons = [reduce(lambda x, y: x._kronecker_add(y), group)
for group in args.values()]
if not nonkrons:
return MatAdd(*krons)
else:
return MatAdd(*krons) + nonkrons
def kronecker_mat_mul(expr):
# modified from block matrix code
factor, matrices = expr.as_coeff_matrices()
i = 0
while i < len(matrices) - 1:
A, B = matrices[i:i+2]
if isinstance(A, KroneckerProduct) and isinstance(B, KroneckerProduct):
matrices[i] = A._kronecker_mul(B)
matrices.pop(i+1)
else:
i += 1
return factor*MatMul(*matrices)
def kronecker_mat_pow(expr):
if isinstance(expr.base, KroneckerProduct):
return KroneckerProduct(*[MatPow(a, expr.exp) for a in expr.base.args])
else:
return expr
def combine_kronecker(expr):
"""Combine KronekeckerProduct with expression.
If possible write operations on KroneckerProducts of compatible shapes
as a single KroneckerProduct.
Examples
========
>>> from sympy.matrices.expressions import MatrixSymbol, KroneckerProduct, combine_kronecker
>>> from sympy import symbols
>>> m, n = symbols(r'm, n', integer=True)
>>> A = MatrixSymbol('A', m, n)
>>> B = MatrixSymbol('B', n, m)
>>> combine_kronecker(KroneckerProduct(A, B)*KroneckerProduct(B, A))
KroneckerProduct(A*B, B*A)
>>> combine_kronecker(KroneckerProduct(A, B)+KroneckerProduct(B.T, A.T))
KroneckerProduct(A + B.T, B + A.T)
>>> combine_kronecker(KroneckerProduct(A, B)**m)
KroneckerProduct(A**m, B**m)
"""
def haskron(expr):
return isinstance(expr, MatrixExpr) and expr.has(KroneckerProduct)
rule = exhaust(
bottom_up(exhaust(condition(haskron, typed(
{MatAdd: kronecker_mat_add,
MatMul: kronecker_mat_mul,
MatPow: kronecker_mat_pow})))))
result = rule(expr)
doit = getattr(result, 'doit', None)
if doit is not None:
return doit()
else:
return result
|
0efc5417f52580e935fd5f9ca695051d2d3a2f6b36051915dde18655ea41056e
|
from __future__ import print_function, division
from sympy import Basic, Expr, sympify, S
from sympy.matrices.matrices import MatrixBase
from .matexpr import ShapeError
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)
"""
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 ShapeError("Trace of a non-square matrix")
return Basic.__new__(cls, mat)
def _eval_transpose(self):
return self
def _eval_derivative(self, v):
from sympy.matrices.expressions.matexpr import _matrix_derivative
return _matrix_derivative(self, v)
def _eval_derivative_matrix_lines(self, x):
r = self.args[0]._eval_derivative_matrix_lines(x)
for lr in r:
if lr.higher == 1:
lr.higher *= lr.first * lr.second.T
else:
# This is not a matrix line:
lr.higher *= Trace(lr.first * lr.second.T)
lr.first = S.One
lr.second = S.One
return r
@property
def arg(self):
return self.args[0]
def doit(self, **kwargs):
if kwargs.get('deep', True):
arg = self.arg.doit(**kwargs)
try:
return arg._eval_trace()
except (AttributeError, NotImplementedError):
return Trace(arg)
else:
# _eval_trace would go too deep here
if isinstance(self.arg, MatrixBase):
return trace(self.arg)
else:
return Trace(self.arg)
def _eval_rewrite_as_Sum(self, expr, **kwargs):
from sympy import Sum, Dummy
i = Dummy('i')
return Sum(self.arg[i, i], (i, 0, self.arg.rows-1)).doit()
def trace(expr):
"""Trace of a Matrix. Sum of the diagonal elements.
Examples
========
>>> from sympy import trace, Symbol, MatrixSymbol, pprint, 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()
|
a0a57a1f50baa7745d94110480df55f392b26cd5dfeacf93e991fb625bac5c78
|
from __future__ import print_function, division
from sympy import ask, Q
from sympy.core import Basic, Add, sympify
from sympy.core.compatibility import range
from sympy.strategies import typed, exhaust, condition, do_one, unpack
from sympy.strategies.traverse import bottom_up
from sympy.utilities import sift
from sympy.matrices.expressions.matexpr import MatrixExpr, ZeroMatrix, Identity
from sympy.matrices.expressions.matmul import MatMul
from sympy.matrices.expressions.matadd import MatAdd
from sympy.matrices.expressions.matpow import MatPow
from sympy.matrices.expressions.transpose import Transpose, transpose
from sympy.matrices.expressions.trace import Trace
from sympy.matrices.expressions.determinant import det, Determinant
from sympy.matrices.expressions.slice import MatrixSlice
from sympy.matrices.expressions.inverse import Inverse
from sympy.matrices import Matrix, ShapeError
from sympy.functions.elementary.complexes import re, im
class BlockMatrix(MatrixExpr):
"""A BlockMatrix is a Matrix composed of other smaller, submatrices
The submatrices are stored in a SymPy Matrix object but accessed as part of
a Matrix Expression
>>> from sympy import (MatrixSymbol, BlockMatrix, symbols,
... Identity, ZeroMatrix, block_collapse)
>>> n,m,l = symbols('n m l')
>>> X = MatrixSymbol('X', n, n)
>>> Y = MatrixSymbol('Y', m ,m)
>>> Z = MatrixSymbol('Z', n, m)
>>> B = BlockMatrix([[X, Z], [ZeroMatrix(m,n), Y]])
>>> print(B)
Matrix([
[X, Z],
[0, Y]])
>>> C = BlockMatrix([[Identity(n), Z]])
>>> print(C)
Matrix([[I, Z]])
>>> print(block_collapse(C*B))
Matrix([[X, Z + Z*Y]])
"""
def __new__(cls, *args):
from sympy.matrices.immutable import ImmutableDenseMatrix
args = map(sympify, args)
mat = ImmutableDenseMatrix(*args)
obj = Basic.__new__(cls, mat)
return obj
@property
def shape(self):
numrows = numcols = 0
M = self.blocks
for i in range(M.shape[0]):
numrows += M[i, 0].shape[0]
for i in range(M.shape[1]):
numcols += M[0, i].shape[1]
return (numrows, numcols)
@property
def blockshape(self):
return self.blocks.shape
@property
def blocks(self):
return self.args[0]
@property
def rowblocksizes(self):
return [self.blocks[i, 0].rows for i in range(self.blockshape[0])]
@property
def colblocksizes(self):
return [self.blocks[0, i].cols for i in range(self.blockshape[1])]
def structurally_equal(self, other):
return (isinstance(other, BlockMatrix)
and self.shape == other.shape
and self.blockshape == other.blockshape
and self.rowblocksizes == other.rowblocksizes
and self.colblocksizes == other.colblocksizes)
def _blockmul(self, other):
if (isinstance(other, BlockMatrix) and
self.colblocksizes == other.rowblocksizes):
return BlockMatrix(self.blocks*other.blocks)
return self * other
def _blockadd(self, other):
if (isinstance(other, BlockMatrix)
and self.structurally_equal(other)):
return BlockMatrix(self.blocks + other.blocks)
return self + other
def _eval_transpose(self):
# Flip all the individual matrices
matrices = [transpose(matrix) for matrix in self.blocks]
# Make a copy
M = Matrix(self.blockshape[0], self.blockshape[1], matrices)
# Transpose the block structure
M = M.transpose()
return BlockMatrix(M)
def _eval_trace(self):
if self.rowblocksizes == self.colblocksizes:
return Add(*[Trace(self.blocks[i, i])
for i in range(self.blockshape[0])])
raise NotImplementedError(
"Can't perform trace of irregular blockshape")
def _eval_determinant(self):
if self.blockshape == (2, 2):
[[A, B],
[C, D]] = self.blocks.tolist()
if ask(Q.invertible(A)):
return det(A)*det(D - C*A.I*B)
elif ask(Q.invertible(D)):
return det(D)*det(A - B*D.I*C)
return Determinant(self)
def as_real_imag(self):
real_matrices = [re(matrix) for matrix in self.blocks]
real_matrices = Matrix(self.blockshape[0], self.blockshape[1], real_matrices)
im_matrices = [im(matrix) for matrix in self.blocks]
im_matrices = Matrix(self.blockshape[0], self.blockshape[1], im_matrices)
return (real_matrices, im_matrices)
def transpose(self):
"""Return transpose of matrix.
Examples
========
>>> from sympy import MatrixSymbol, BlockMatrix, ZeroMatrix
>>> from sympy.abc import l, m, n
>>> X = MatrixSymbol('X', n, n)
>>> Y = MatrixSymbol('Y', m ,m)
>>> Z = MatrixSymbol('Z', n, m)
>>> B = BlockMatrix([[X, Z], [ZeroMatrix(m,n), Y]])
>>> B.transpose()
Matrix([
[X.T, 0],
[Z.T, Y.T]])
>>> _.transpose()
Matrix([
[X, Z],
[0, Y]])
"""
return self._eval_transpose()
def _entry(self, i, j):
# Find row entry
for row_block, numrows in enumerate(self.rowblocksizes):
if (i < numrows) != False:
break
else:
i -= numrows
for col_block, numcols in enumerate(self.colblocksizes):
if (j < numcols) != False:
break
else:
j -= numcols
return self.blocks[row_block, col_block][i, j]
@property
def is_Identity(self):
if self.blockshape[0] != self.blockshape[1]:
return False
for i in range(self.blockshape[0]):
for j in range(self.blockshape[1]):
if i==j and not self.blocks[i, j].is_Identity:
return False
if i!=j and not self.blocks[i, j].is_ZeroMatrix:
return False
return True
@property
def is_structurally_symmetric(self):
return self.rowblocksizes == self.colblocksizes
def equals(self, other):
if self == other:
return True
if (isinstance(other, BlockMatrix) and self.blocks == other.blocks):
return True
return super(BlockMatrix, self).equals(other)
class BlockDiagMatrix(BlockMatrix):
"""
A BlockDiagMatrix is a BlockMatrix with matrices only along the diagonal
>>> from sympy import MatrixSymbol, BlockDiagMatrix, symbols, Identity
>>> n,m,l = symbols('n m l')
>>> X = MatrixSymbol('X', n, n)
>>> Y = MatrixSymbol('Y', m ,m)
>>> BlockDiagMatrix(X, Y)
Matrix([
[X, 0],
[0, Y]])
"""
def __new__(cls, *mats):
return Basic.__new__(BlockDiagMatrix, *mats)
@property
def diag(self):
return self.args
@property
def blocks(self):
from sympy.matrices.immutable import ImmutableDenseMatrix
mats = self.args
data = [[mats[i] if i == j else ZeroMatrix(mats[i].rows, mats[j].cols)
for j in range(len(mats))]
for i in range(len(mats))]
return ImmutableDenseMatrix(data)
@property
def shape(self):
return (sum(block.rows for block in self.args),
sum(block.cols for block in self.args))
@property
def blockshape(self):
n = len(self.args)
return (n, n)
@property
def rowblocksizes(self):
return [block.rows for block in self.args]
@property
def colblocksizes(self):
return [block.cols for block in self.args]
def _eval_inverse(self, expand='ignored'):
return BlockDiagMatrix(*[mat.inverse() for mat in self.args])
def _blockmul(self, other):
if (isinstance(other, BlockDiagMatrix) and
self.colblocksizes == other.rowblocksizes):
return BlockDiagMatrix(*[a*b for a, b in zip(self.args, other.args)])
else:
return BlockMatrix._blockmul(self, other)
def _blockadd(self, other):
if (isinstance(other, BlockDiagMatrix) and
self.blockshape == other.blockshape and
self.rowblocksizes == other.rowblocksizes and
self.colblocksizes == other.colblocksizes):
return BlockDiagMatrix(*[a + b for a, b in zip(self.args, other.args)])
else:
return BlockMatrix._blockadd(self, other)
def block_collapse(expr):
"""Evaluates a block matrix expression
>>> from sympy import MatrixSymbol, BlockMatrix, symbols, \
Identity, Matrix, ZeroMatrix, block_collapse
>>> n,m,l = symbols('n m l')
>>> X = MatrixSymbol('X', n, n)
>>> Y = MatrixSymbol('Y', m ,m)
>>> Z = MatrixSymbol('Z', n, m)
>>> B = BlockMatrix([[X, Z], [ZeroMatrix(m, n), Y]])
>>> print(B)
Matrix([
[X, Z],
[0, Y]])
>>> C = BlockMatrix([[Identity(n), Z]])
>>> print(C)
Matrix([[I, Z]])
>>> print(block_collapse(C*B))
Matrix([[X, Z + Z*Y]])
"""
hasbm = lambda expr: isinstance(expr, MatrixExpr) and expr.has(BlockMatrix)
rule = exhaust(
bottom_up(exhaust(condition(hasbm, typed(
{MatAdd: do_one(bc_matadd, bc_block_plus_ident),
MatMul: do_one(bc_matmul, bc_dist),
MatPow: bc_matmul,
Transpose: bc_transpose,
Inverse: bc_inverse,
BlockMatrix: do_one(bc_unpack, deblock)})))))
result = rule(expr)
doit = getattr(result, 'doit', None)
if doit is not None:
return doit()
else:
return result
def bc_unpack(expr):
if expr.blockshape == (1, 1):
return expr.blocks[0, 0]
return expr
def bc_matadd(expr):
args = sift(expr.args, lambda M: isinstance(M, BlockMatrix))
blocks = args[True]
if not blocks:
return expr
nonblocks = args[False]
block = blocks[0]
for b in blocks[1:]:
block = block._blockadd(b)
if nonblocks:
return MatAdd(*nonblocks) + block
else:
return block
def bc_block_plus_ident(expr):
idents = [arg for arg in expr.args if arg.is_Identity]
if not idents:
return expr
blocks = [arg for arg in expr.args if isinstance(arg, BlockMatrix)]
if (blocks and all(b.structurally_equal(blocks[0]) for b in blocks)
and blocks[0].is_structurally_symmetric):
block_id = BlockDiagMatrix(*[Identity(k)
for k in blocks[0].rowblocksizes])
return MatAdd(block_id * len(idents), *blocks).doit()
return expr
def bc_dist(expr):
""" Turn a*[X, Y] into [a*X, a*Y] """
factor, mat = expr.as_coeff_mmul()
if factor != 1 and isinstance(unpack(mat), BlockMatrix):
B = unpack(mat).blocks
return BlockMatrix([[factor * B[i, j] for j in range(B.cols)]
for i in range(B.rows)])
return expr
def bc_matmul(expr):
if isinstance(expr, MatPow):
if expr.args[1].is_Integer:
factor, matrices = (1, [expr.args[0]]*expr.args[1])
else:
return expr
else:
factor, matrices = expr.as_coeff_matrices()
i = 0
while (i+1 < len(matrices)):
A, B = matrices[i:i+2]
if isinstance(A, BlockMatrix) and isinstance(B, BlockMatrix):
matrices[i] = A._blockmul(B)
matrices.pop(i+1)
elif isinstance(A, BlockMatrix):
matrices[i] = A._blockmul(BlockMatrix([[B]]))
matrices.pop(i+1)
elif isinstance(B, BlockMatrix):
matrices[i] = BlockMatrix([[A]])._blockmul(B)
matrices.pop(i+1)
else:
i+=1
return MatMul(factor, *matrices).doit()
def bc_transpose(expr):
return BlockMatrix(block_collapse(expr.arg).blocks.applyfunc(transpose).T)
def bc_inverse(expr):
expr2 = blockinverse_1x1(expr)
if expr != expr2:
return expr2
return blockinverse_2x2(Inverse(reblock_2x2(expr.arg)))
def blockinverse_1x1(expr):
if isinstance(expr.arg, BlockMatrix) and expr.arg.blockshape == (1, 1):
mat = Matrix([[expr.arg.blocks[0].inverse()]])
return BlockMatrix(mat)
return expr
def blockinverse_2x2(expr):
if isinstance(expr.arg, BlockMatrix) and expr.arg.blockshape == (2, 2):
# Cite: The Matrix Cookbook Section 9.1.3
[[A, B],
[C, D]] = expr.arg.blocks.tolist()
return BlockMatrix([[ (A - B*D.I*C).I, (-A).I*B*(D - C*A.I*B).I],
[-(D - C*A.I*B).I*C*A.I, (D - C*A.I*B).I]])
else:
return expr
def deblock(B):
""" Flatten a BlockMatrix of BlockMatrices """
if not isinstance(B, BlockMatrix) or not B.blocks.has(BlockMatrix):
return B
wrap = lambda x: x if isinstance(x, BlockMatrix) else BlockMatrix([[x]])
bb = B.blocks.applyfunc(wrap) # everything is a block
from sympy import Matrix
try:
MM = Matrix(0, sum(bb[0, i].blocks.shape[1] for i in range(bb.shape[1])), [])
for row in range(0, bb.shape[0]):
M = Matrix(bb[row, 0].blocks)
for col in range(1, bb.shape[1]):
M = M.row_join(bb[row, col].blocks)
MM = MM.col_join(M)
return BlockMatrix(MM)
except ShapeError:
return B
def reblock_2x2(B):
""" Reblock a BlockMatrix so that it has 2x2 blocks of block matrices """
if not isinstance(B, BlockMatrix) or not all(d > 2 for d in B.blocks.shape):
return B
BM = BlockMatrix # for brevity's sake
return BM([[ B.blocks[0, 0], BM(B.blocks[0, 1:])],
[BM(B.blocks[1:, 0]), BM(B.blocks[1:, 1:])]])
def bounds(sizes):
""" Convert sequence of numbers into pairs of low-high pairs
>>> from sympy.matrices.expressions.blockmatrix import bounds
>>> bounds((1, 10, 50))
[(0, 1), (1, 11), (11, 61)]
"""
low = 0
rv = []
for size in sizes:
rv.append((low, low + size))
low += size
return rv
def blockcut(expr, rowsizes, colsizes):
""" Cut a matrix expression into Blocks
>>> from sympy import ImmutableMatrix, blockcut
>>> M = ImmutableMatrix(4, 4, range(16))
>>> B = blockcut(M, (1, 3), (1, 3))
>>> type(B).__name__
'BlockMatrix'
>>> ImmutableMatrix(B.blocks[0, 1])
Matrix([[1, 2, 3]])
"""
rowbounds = bounds(rowsizes)
colbounds = bounds(colsizes)
return BlockMatrix([[MatrixSlice(expr, rowbound, colbound)
for colbound in colbounds]
for rowbound in rowbounds])
|
7cfcebb06226a3fcd07be80e4dfffee54d498b10114f8f68823fb56950ab0c9e
|
from __future__ import print_function, division
from sympy.core.compatibility import reduce
from operator import add
from sympy.core import Add, Basic, sympify
from sympy.functions import adjoint
from sympy.matrices.matrices import MatrixBase
from sympy.matrices.expressions.transpose import transpose
from sympy.strategies import (rm_id, unpack, flatten, sort, condition,
exhaust, do_one, glom)
from sympy.matrices.expressions.matexpr import (MatrixExpr, ShapeError,
ZeroMatrix, GenericZeroMatrix)
from sympy.utilities import default_sort_key, sift
from sympy.core.operations import AssocOp
class MatAdd(MatrixExpr, Add):
"""A Sum of Matrix Expressions
MatAdd inherits from and operates like SymPy Add
Examples
========
>>> from sympy import MatAdd, MatrixSymbol
>>> A = MatrixSymbol('A', 5, 5)
>>> B = MatrixSymbol('B', 5, 5)
>>> C = MatrixSymbol('C', 5, 5)
>>> MatAdd(A, B, C)
A + B + C
"""
is_MatAdd = True
def __new__(cls, *args, **kwargs):
if not args:
return GenericZeroMatrix()
# This must be removed aggressively in the constructor to avoid
# TypeErrors from GenericZeroMatrix().shape
args = filter(lambda i: GenericZeroMatrix() != i, args)
args = list(map(sympify, args))
check = kwargs.get('check', False)
obj = Basic.__new__(cls, *args)
if check:
if all(not isinstance(i, MatrixExpr) for i in args):
return Add.fromiter(args)
validate(*args)
return obj
@property
def shape(self):
return self.args[0].shape
def _entry(self, i, j, expand=None):
return Add(*[arg._entry(i, j) for arg in self.args])
def _eval_transpose(self):
return MatAdd(*[transpose(arg) for arg in self.args]).doit()
def _eval_adjoint(self):
return MatAdd(*[adjoint(arg) for arg in self.args]).doit()
def _eval_trace(self):
from .trace import trace
return Add(*[trace(arg) for arg in self.args]).doit()
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 canonicalize(MatAdd(*args))
def _eval_derivative_matrix_lines(self, x):
add_lines = [arg._eval_derivative_matrix_lines(x) for arg in self.args]
return [j for i in add_lines for j in i]
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))
factor_of = lambda arg: arg.as_coeff_mmul()[0]
matrix_of = lambda arg: unpack(arg.as_coeff_mmul()[1])
def combine(cnt, mat):
if cnt == 1:
return mat
else:
return cnt * mat
def merge_explicit(matadd):
""" Merge explicit MatrixBase arguments
Examples
========
>>> from sympy import MatrixSymbol, eye, Matrix, MatAdd, pprint
>>> from sympy.matrices.expressions.matadd import merge_explicit
>>> A = MatrixSymbol('A', 2, 2)
>>> B = eye(2)
>>> C = Matrix([[1, 2], [3, 4]])
>>> X = MatAdd(A, B, C)
>>> pprint(X)
[1 0] [1 2]
A + [ ] + [ ]
[0 1] [3 4]
>>> pprint(merge_explicit(X))
[2 2]
A + [ ]
[3 5]
"""
groups = sift(matadd.args, lambda arg: isinstance(arg, MatrixBase))
if len(groups[True]) > 1:
return MatAdd(*(groups[False] + [reduce(add, groups[True])]))
else:
return matadd
rules = (rm_id(lambda x: x == 0 or isinstance(x, ZeroMatrix)),
unpack,
flatten,
glom(matrix_of, factor_of, combine),
merge_explicit,
sort(default_sort_key))
canonicalize = exhaust(condition(lambda x: isinstance(x, MatAdd),
do_one(*rules)))
|
e10f314c0198be0b44ba329fd4665e8f984707302ef6929e81fc3a33ca76299a
|
from __future__ import print_function, division
from sympy.matrices.expressions.matexpr import MatrixExpr
from sympy import Tuple, Basic
from sympy.functions.elementary.integers import floor
def normalize(i, parentsize):
if isinstance(i, slice):
i = (i.start, i.stop, i.step)
if not isinstance(i, (tuple, list, Tuple)):
if (i < 0) == True:
i += parentsize
i = (i, i+1, 1)
i = list(i)
if len(i) == 2:
i.append(1)
start, stop, step = i
start = start or 0
if stop is None:
stop = parentsize
if (start < 0) == True:
start += parentsize
if (stop < 0) == True:
stop += parentsize
step = step or 1
if ((stop - start) * step < 1) == True:
raise IndexError()
return (start, stop, step)
class MatrixSlice(MatrixExpr):
""" A MatrixSlice of a Matrix Expression
Examples
========
>>> from sympy import MatrixSlice, ImmutableMatrix
>>> M = ImmutableMatrix(4, 4, range(16))
>>> M
Matrix([
[ 0, 1, 2, 3],
[ 4, 5, 6, 7],
[ 8, 9, 10, 11],
[12, 13, 14, 15]])
>>> B = MatrixSlice(M, (0, 2), (2, 4))
>>> ImmutableMatrix(B)
Matrix([
[2, 3],
[6, 7]])
"""
parent = property(lambda self: self.args[0])
rowslice = property(lambda self: self.args[1])
colslice = property(lambda self: self.args[2])
def __new__(cls, parent, rowslice, colslice):
rowslice = normalize(rowslice, parent.shape[0])
colslice = normalize(colslice, parent.shape[1])
if not (len(rowslice) == len(colslice) == 3):
raise IndexError()
if ((0 > rowslice[0]) == True or
(parent.shape[0] < rowslice[1]) == True or
(0 > colslice[0]) == True or
(parent.shape[1] < colslice[1]) == True):
raise IndexError()
if isinstance(parent, MatrixSlice):
return mat_slice_of_slice(parent, rowslice, colslice)
return Basic.__new__(cls, parent, Tuple(*rowslice), Tuple(*colslice))
@property
def shape(self):
rows = self.rowslice[1] - self.rowslice[0]
rows = rows if self.rowslice[2] == 1 else floor(rows/self.rowslice[2])
cols = self.colslice[1] - self.colslice[0]
cols = cols if self.colslice[2] == 1 else floor(cols/self.colslice[2])
return rows, cols
def _entry(self, i, j):
return self.parent._entry(i*self.rowslice[2] + self.rowslice[0],
j*self.colslice[2] + self.colslice[0])
@property
def on_diag(self):
return self.rowslice == self.colslice
def slice_of_slice(s, t):
start1, stop1, step1 = s
start2, stop2, step2 = t
start = start1 + start2*step1
step = step1 * step2
stop = start1 + step1*stop2
if stop > stop1:
raise IndexError()
return start, stop, step
def mat_slice_of_slice(parent, rowslice, colslice):
""" Collapse nested matrix slices
>>> from sympy import MatrixSymbol
>>> X = MatrixSymbol('X', 10, 10)
>>> X[:, 1:5][5:8, :]
X[5:8, 1:5]
>>> X[1:9:2, 2:6][1:3, 2]
X[3:7:2, 4]
"""
row = slice_of_slice(parent.rowslice, rowslice)
col = slice_of_slice(parent.colslice, colslice)
return MatrixSlice(parent.parent, row, col)
|
c15432a27101bf70b3c422104a6ec2bc556a08ded2717137240c836561726075
|
from sympy.core import I, symbols, Basic, Mul
from sympy.functions import adjoint, transpose
from sympy.matrices import (Identity, Inverse, Matrix, MatrixSymbol, ZeroMatrix,
eye, ImmutableMatrix)
from sympy.matrices.expressions import Adjoint, Transpose, det, MatPow
from sympy.matrices.expressions.matmul import (factor_in_front, remove_ids,
MatMul, xxinv, any_zeros, unpack, only_squares)
from sympy.strategies import null_safe
from sympy import refine, Q, Symbol
from sympy.utilities.pytest import XFAIL
n, m, l, k = symbols('n m l k', 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)
def test_adjoint():
assert adjoint(A*B) == Adjoint(B)*Adjoint(A)
assert adjoint(2*A*B) == 2*Adjoint(B)*Adjoint(A)
assert adjoint(2*I*C) == -2*I*Adjoint(C)
M = Matrix(2, 2, [1, 2 + I, 3, 4])
MA = Matrix(2, 2, [1, 3, 2 - I, 4])
assert adjoint(M) == MA
assert adjoint(2*M) == 2*MA
assert adjoint(MatMul(2, M)) == MatMul(2, MA).doit()
def test_transpose():
assert transpose(A*B) == Transpose(B)*Transpose(A)
assert transpose(2*A*B) == 2*Transpose(B)*Transpose(A)
assert transpose(2*I*C) == 2*I*Transpose(C)
M = Matrix(2, 2, [1, 2 + I, 3, 4])
MT = Matrix(2, 2, [1, 3, 2 + I, 4])
assert transpose(M) == MT
assert transpose(2*M) == 2*MT
assert transpose(MatMul(2, M)) == MatMul(2, MT).doit()
def test_factor_in_front():
assert factor_in_front(MatMul(A, 2, B, evaluate=False)) ==\
MatMul(2, A, B, evaluate=False)
def test_remove_ids():
assert remove_ids(MatMul(A, Identity(m), B, evaluate=False)) == \
MatMul(A, B, evaluate=False)
assert null_safe(remove_ids)(MatMul(Identity(n), evaluate=False)) == \
MatMul(Identity(n), evaluate=False)
def test_xxinv():
assert xxinv(MatMul(D, Inverse(D), D, evaluate=False)) == \
MatMul(Identity(n), D, evaluate=False)
def test_any_zeros():
assert any_zeros(MatMul(A, ZeroMatrix(m, k), evaluate=False)) == \
ZeroMatrix(n, k)
def test_unpack():
assert unpack(MatMul(A, evaluate=False)) == A
x = MatMul(A, B)
assert unpack(x) == x
def test_only_squares():
assert only_squares(C) == [C]
assert only_squares(C, D) == [C, D]
assert only_squares(C, A, A.T, D) == [C, A*A.T, D]
def test_determinant():
assert det(2*C) == 2**n*det(C)
assert det(2*C*D) == 2**n*det(C)*det(D)
assert det(3*C*A*A.T*D) == 3**n*det(C)*det(A*A.T)*det(D)
def test_doit():
assert MatMul(C, 2, D).args == (C, 2, D)
assert MatMul(C, 2, D).doit().args == (2, C, D)
assert MatMul(C, Transpose(D*C)).args == (C, Transpose(D*C))
assert MatMul(C, Transpose(D*C)).doit(deep=True).args == (C, C.T, D.T)
def test_doit_drills_down():
X = ImmutableMatrix([[1, 2], [3, 4]])
Y = ImmutableMatrix([[2, 3], [4, 5]])
assert MatMul(X, MatPow(Y, 2)).doit() == X*Y**2
assert MatMul(C, Transpose(D*C)).doit().args == (C, C.T, D.T)
def test_doit_deep_false_still_canonical():
assert (MatMul(C, Transpose(D*C), 2).doit(deep=False).args ==
(2, C, Transpose(D*C)))
def test_matmul_scalar_Matrix_doit():
# Issue 9053
X = Matrix([[1, 2], [3, 4]])
assert MatMul(2, X).doit() == 2*X
def test_matmul_sympify():
assert isinstance(MatMul(eye(1), eye(1)).args[0], Basic)
def test_collapse_MatrixBase():
A = Matrix([[1, 1], [1, 1]])
B = Matrix([[1, 2], [3, 4]])
assert MatMul(A, B).doit() == ImmutableMatrix([[4, 6], [4, 6]])
def test_refine():
assert refine(C*C.T*D, Q.orthogonal(C)).doit() == D
kC = k*C
assert refine(kC*C.T, Q.orthogonal(C)).doit() == k*Identity(n)
assert refine(kC* kC.T, Q.orthogonal(C)).doit() == (k**2)*Identity(n)
def test_matmul_no_matrices():
assert MatMul(1) == 1
assert MatMul(n, m) == n*m
assert not isinstance(MatMul(n, m), MatMul)
def test_matmul_args_cnc():
assert MatMul(n, A, A.T).args_cnc() == [[n], [A, A.T]]
assert MatMul(A, A.T).args_cnc() == [[], [A, A.T]]
@XFAIL
def test_matmul_args_cnc_symbols():
# Not currently supported
a, b = symbols('a b', commutative=False)
assert MatMul(n, a, b, A, A.T).args_cnc() == [[n], [a, b, A, A.T]]
assert MatMul(n, a, A, b, A.T).args_cnc() == [[n], [a, A, b, A.T]]
def test_issue_12950():
M = Matrix([[Symbol("x")]]) * MatrixSymbol("A", 1, 1)
assert MatrixSymbol("A", 1, 1).as_explicit()[0]*Symbol('x') == M.as_explicit()[0]
def test_construction_with_Mul():
assert Mul(C, D) == MatMul(C, D)
assert Mul(D, C) == MatMul(D, C)
|
2fc6ddb6a3daa7c1577eb0c94bdb4a259eca2d76994a6caad311ba308a5d30ed
|
"""
Some examples have been taken from:
http://www.math.uwaterloo.ca/~hwolkowi//matrixcookbook.pdf
"""
from sympy import MatrixSymbol, Inverse, symbols, Determinant, Trace, Derivative
from sympy import MatAdd, Identity, MatMul, ZeroMatrix
k = symbols("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)
a = MatrixSymbol("a", k, 1)
b = MatrixSymbol("b", k, 1)
c = MatrixSymbol("c", k, 1)
d = MatrixSymbol("d", k, 1)
def test_matrix_derivative_non_matrix_result():
# This is a 4-dimensional array:
assert A.diff(A) == Derivative(A, A)
assert A.T.diff(A) == Derivative(A.T, A)
assert (2*A).diff(A) == Derivative(2*A, A)
assert MatAdd(A, A).diff(A) == Derivative(MatAdd(A, A), A)
assert (A + B).diff(A) == Derivative(A + B, A) # TODO: `B` can be removed.
def test_matrix_derivative_trivial_cases():
# Cookbook example 33:
# TODO: find a way to represent a four-dimensional zero-array:
assert X.diff(A) == Derivative(X, A)
def test_matrix_derivative_with_inverse():
# Cookbook example 61:
expr = a.T*Inverse(X)*b
assert expr.diff(X) == -Inverse(X).T*a*b.T*Inverse(X).T
# Cookbook example 62:
expr = Determinant(Inverse(X))
# Not implemented yet:
# assert expr.diff(X) == -Determinant(X.inv())*(X.inv()).T
# Cookbook example 63:
expr = Trace(A*Inverse(X)*B)
assert expr.diff(X) == -(X**(-1)*B*A*X**(-1)).T
# Cookbook example 64:
expr = Trace(Inverse(X + A))
assert expr.diff(X) == -(Inverse(X + A)).T**2
def test_matrix_derivative_vectors_and_scalars():
assert x.diff(x) == Identity(k)
assert x.T.diff(x) == Identity(k)
# Cookbook example 69:
expr = x.T*a
assert expr.diff(x) == a
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
def test_matrix_derivatives_of_traces():
expr = Trace(A)*A
assert expr.diff(A) == Derivative(Trace(A)*A, A)
## First order:
# Cookbook example 99:
expr = Trace(X)
assert expr.diff(X) == Identity(k)
# Cookbook example 100:
expr = Trace(X*A)
assert expr.diff(X) == A.T
# Cookbook example 101:
expr = Trace(A*X*B)
assert expr.diff(X) == A.T*B.T
# Cookbook example 102:
expr = Trace(A*X.T*B)
assert expr.diff(X) == B*A
# Cookbook example 103:
expr = Trace(X.T*A)
assert expr.diff(X) == A
# Cookbook example 104:
expr = Trace(A*X.T)
assert expr.diff(X) == A
# Cookbook example 105:
# TODO: TensorProduct is not supported
#expr = Trace(TensorProduct(A, X))
#assert expr.diff(X) == Trace(A)*Identity(k)
## Second order:
# Cookbook example 106:
expr = Trace(X**2)
assert expr.diff(X) == 2*X.T
# Cookbook example 107:
expr = Trace(X**2*B)
assert expr.diff(X) == (X*B + B*X).T
expr = Trace(MatMul(X, X, B))
assert expr.diff(X) == (X*B + B*X).T
# Cookbook example 108:
expr = Trace(X.T*B*X)
assert expr.diff(X) == B*X + B.T*X
# Cookbook example 109:
expr = Trace(B*X*X.T)
assert expr.diff(X) == B*X + B.T*X
# Cookbook example 110:
expr = Trace(X*X.T*B)
assert expr.diff(X) == B*X + B.T*X
# Cookbook example 111:
expr = Trace(X*B*X.T)
assert expr.diff(X) == X*B.T + X*B
# Cookbook example 112:
expr = Trace(B*X.T*X)
assert expr.diff(X) == X*B.T + X*B
# Cookbook example 113:
expr = Trace(X.T*X*B)
assert expr.diff(X) == X*B.T + X*B
# Cookbook example 114:
expr = Trace(A*X*B*X)
assert expr.diff(X) == A.T*X.T*B.T + B.T*X.T*A.T
# Cookbook example 115:
expr = Trace(X.T*X)
assert expr.diff(X) == 2*X
expr = Trace(X*X.T)
assert expr.diff(X) == 2*X
# Cookbook example 116:
expr = Trace(B.T*X.T*C*X*B)
assert expr.diff(X) == C.T*X*B*B.T + C*X*B*B.T
# Cookbook example 117:
expr = Trace(X.T*B*X*C)
assert expr.diff(X) == B*X*C + B.T*X*C.T
# Cookbook example 118:
expr = Trace(A*X*B*X.T*C)
assert expr.diff(X) == A.T*C.T*X*B.T + C*A*X*B
# Cookbook example 119:
expr = Trace((A*X*B + C)*(A*X*B + C).T)
assert expr.diff(X) == 2*A.T*(A*X*B + C)*B.T
# Cookbook example 120:
# TODO: no support for TensorProduct.
# expr = Trace(TensorProduct(X, X))
# expr = Trace(X)*Trace(X)
# expr.diff(X) == 2*Trace(X)*Identity(k)
# Higher Order
# Cookbook example 121:
expr = Trace(X**k)
#assert expr.diff(X) == k*(X**(k-1)).T
# Cookbook example 122:
expr = Trace(A*X**k)
#assert expr.diff(X) == # Needs indices
# Cookbook example 123:
expr = Trace(B.T*X.T*C*X*X.T*C*X*B)
assert expr.diff(X) == C*X*X.T*C*X*B*B.T + C.T*X*B*B.T*X.T*C.T*X + C*X*B*B.T*X.T*C*X + C.T*X*X.T*C.T*X*B*B.T
# Other
# Cookbook example 124:
expr = Trace(A*X**(-1)*B)
assert expr.diff(X) == -Inverse(X).T*A.T*B.T*Inverse(X).T
# Cookbook example 125:
expr = Trace(Inverse(X.T*C*X)*A)
# Warning: result in the cookbook is equivalent if B and C are symmetric:
assert expr.diff(X) == - X.inv().T*A.T*X.inv()*C.inv().T*X.inv().T - X.inv().T*A*X.inv()*C.inv()*X.inv().T
# Cookbook example 126:
expr = Trace((X.T*C*X).inv()*(X.T*B*X))
assert expr.diff(X) == -2*C*X*(X.T*C*X).inv()*X.T*B*X*(X.T*C*X).inv() + 2*B*X*(X.T*C*X).inv()
# Cookbook example 127:
expr = Trace((A + X.T*C*X).inv()*(X.T*B*X))
# Warning: result in the cookbook is equivalent if B and C are symmetric:
assert expr.diff(X) == B*X*Inverse(A + X.T*C*X) - C*X*Inverse(A + X.T*C*X)*X.T*B*X*Inverse(A + X.T*C*X) - C.T*X*Inverse(A.T + (C*X).T*X)*X.T*B.T*X*Inverse(A.T + (C*X).T*X) + B.T*X*Inverse(A.T + (C*X).T*X)
def test_derivatives_of_complicated_matrix_expr():
expr = a.T*(A*X*(X.T*B + X*A) + B.T*X.T*(a*b.T*(X*D*X.T + X*(X.T*B + A*X)*D*B - X.T*C.T*A)*B + B*(X*D.T + B*A*X*A.T - 3*X*D))*B + 42*X*B*X.T*A.T*(X + X.T))*b
result = (B*(B*A*X*A.T - 3*X*D + X*D.T) + a*b.T*(X*(A*X + X.T*B)*D*B + X*D*X.T - X.T*C.T*A)*B)*B*b*a.T*B.T + B**2*b*a.T*B.T*X.T*a*b.T*X*D + 42*A*X*B.T*X.T*a*b.T + B*D*B**3*b*a.T*B.T*X.T*a*b.T*X + B*b*a.T*A*X + 42*a*b.T*(X + X.T)*A*X*B.T + b*a.T*X*B*a*b.T*B.T**2*X*D.T + b*a.T*X*B*a*b.T*B.T**3*D.T*(B.T*X + X.T*A.T) + 42*b*a.T*X*B*X.T*A.T + 42*A.T*(X + X.T)*b*a.T*X*B + A.T*B.T**2*X*B*a*b.T*B.T*A + A.T*a*b.T*(A.T*X.T + B.T*X) + A.T*X.T*b*a.T*X*B*a*b.T*B.T**3*D.T + B.T*X*B*a*b.T*B.T*D - 3*B.T*X*B*a*b.T*B.T*D.T - C.T*A*B**2*b*a.T*B.T*X.T*a*b.T + X.T*A.T*a*b.T*A.T
assert expr.diff(X) == result
def test_mixed_deriv_mixed_expressions():
expr = 3*Trace(A)
assert expr.diff(A) == 3*Identity(k)
expr = k
deriv = expr.diff(A)
assert isinstance(deriv, ZeroMatrix)
assert deriv == ZeroMatrix(k, k)
expr = Trace(A)**2
assert expr.diff(A) == (2*Trace(A))*Identity(k)
expr = Trace(A)*A
# TODO: this is not yet supported:
assert expr.diff(A) == Derivative(expr, A)
expr = Trace(Trace(A)*A)
assert expr.diff(A) == (2*Trace(A))*Identity(k)
expr = Trace(Trace(Trace(A)*A)*A)
assert expr.diff(A) == (3*Trace(A)**2)*Identity(k)
|
679b05fbcd0e1f20d8e9a3c6c295a0962429ce1d81ff1c5f34aa10726e13d4c7
|
from sympy import (KroneckerDelta, diff, Piecewise, Sum, Dummy, factor,
expand, zeros, gcd_terms, Eq)
from sympy.core import S, symbols, Add, Mul
from sympy.core.compatibility import long
from sympy.functions import transpose, sin, cos, sqrt, cbrt
from sympy.simplify import simplify
from sympy.matrices import (Identity, ImmutableMatrix, Inverse, MatAdd, MatMul,
MatPow, Matrix, MatrixExpr, MatrixSymbol, ShapeError, ZeroMatrix,
SparseMatrix, Transpose, Adjoint)
from sympy.matrices.expressions.matexpr import (MatrixElement,
GenericZeroMatrix, GenericIdentity)
from sympy.utilities.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_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
def test_ZeroMatrix():
A = MatrixSymbol('A', n, m)
Z = ZeroMatrix(n, m)
assert A + Z == A
assert A*Z.T == ZeroMatrix(n, n)
assert Z*A.T == ZeroMatrix(n, n)
assert A - A == ZeroMatrix(*A.shape)
assert not Z
assert transpose(Z) == ZeroMatrix(m, n)
assert Z.conjugate() == Z
assert ZeroMatrix(n, n)**0 == Identity(n)
with raises(ShapeError):
Z**0
with raises(ShapeError):
Z**2
def test_ZeroMatrix_doit():
Znn = ZeroMatrix(Add(n, n, evaluate=False), n)
assert isinstance(Znn.rows, Add)
assert Znn.doit() == ZeroMatrix(2*n, n)
assert isinstance(Znn.doit().rows, Mul)
def test_Identity():
A = MatrixSymbol('A', n, m)
i, j = symbols('i j')
In = Identity(n)
Im = Identity(m)
assert A*Im == A
assert In*A == A
assert transpose(In) == In
assert In.inverse() == In
assert In.conjugate() == In
assert In[i, j] != 0
assert Sum(In[i, j], (i, 0, n-1), (j, 0, n-1)).subs(n,3).doit() == 3
assert Sum(Sum(In[i, j], (i, 0, n-1)), (j, 0, n-1)).subs(n,3).doit() == 3
def test_Identity_doit():
Inn = Identity(Add(n, n, evaluate=False))
assert isinstance(Inn.rows, Add)
assert Inn.doit() == Identity(2*n)
assert isinstance(Inn.doit().rows, Mul)
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(0)
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**(S(1)/3) == cbrt(A)
raises(ShapeError, 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 == set((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, ImmutableMatrix([[1]]))
def test_matmul_simplify():
A = MatrixSymbol('A', 1, 1)
assert simplify(MatMul(A, ImmutableMatrix([[sin(x)**2 + cos(x)**2]]))) == \
MatMul(A, ImmutableMatrix([[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[long(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)*KroneckerDelta(0, p)
assert str(dexpr) == "Sum(KroneckerDelta(_i_1, p)*D[k, _i_1], (_i_1, 0, n - 1))"
assert str(dexpr.doit()) == 'Piecewise((D[k, p], (p >= 0) & (p <= n - 1)), (0, True))'
# TODO: bug with .dummy_eq( ), the previous 2 lines should be replaced by:
return # stop eval
_i_1 = Dummy("_i_1")
assert dexpr.dummy_eq(Sum(KroneckerDelta(_i_1, p)*D[k, _i_1], (_i_1, 0, n - 1)))
assert dexpr.doit().dummy_eq(Piecewise((D[k, p], (p >= 0) & (p <= n - 1)), (0, True)))
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_generic_zero_matrix():
z = GenericZeroMatrix()
A = MatrixSymbol("A", n, n)
assert z == z
assert z != A
assert A != z
assert z.is_ZeroMatrix
raises(TypeError, lambda: z.shape)
raises(TypeError, lambda: z.rows)
raises(TypeError, lambda: z.cols)
assert MatAdd() == z
assert MatAdd(z, A) == MatAdd(A)
# Make sure it is hashable
hash(z)
def test_generic_identity():
I = GenericIdentity()
A = MatrixSymbol("A", n, n)
assert I == I
assert I != A
assert A != I
assert I.is_Identity
assert I**-1 == I
raises(TypeError, lambda: I.shape)
raises(TypeError, lambda: I.rows)
raises(TypeError, lambda: I.cols)
assert MatMul() == I
assert MatMul(I, A) == MatMul(A)
# Make sure it is hashable
hash(I)
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) == MatMul
assert a.args == (2, (C*D + 2*D*C))
|
36a6b6893030173b794585eede958a2b57bec1d2613beb0e33bf4645e305e760
|
from sympy.core import Lambda, S, symbols
from sympy.concrete import Sum
from sympy.functions import adjoint, conjugate, transpose
from sympy.matrices import eye, Matrix, ShapeError, ImmutableMatrix
from sympy.matrices.expressions import (
Adjoint, Identity, FunctionMatrix, MatrixExpr, MatrixSymbol, Trace,
ZeroMatrix, trace, MatPow, MatAdd, MatMul
)
from sympy.utilities.pytest import raises, XFAIL
n = symbols('n', integer=True)
A = MatrixSymbol('A', n, n)
B = MatrixSymbol('B', n, n)
C = MatrixSymbol('C', 3, 4)
def test_Trace():
assert isinstance(Trace(A), Trace)
assert not isinstance(Trace(A), MatrixExpr)
raises(ShapeError, lambda: Trace(C))
assert trace(eye(3)) == 3
assert trace(Matrix(3, 3, [1, 2, 3, 4, 5, 6, 7, 8, 9])) == 15
assert adjoint(Trace(A)) == trace(Adjoint(A))
assert conjugate(Trace(A)) == trace(Adjoint(A))
assert transpose(Trace(A)) == Trace(A)
A / Trace(A) # Make sure this is possible
# Some easy simplifications
assert trace(Identity(5)) == 5
assert trace(ZeroMatrix(5, 5)) == 0
assert trace(2*A*B) == 2*Trace(A*B)
assert trace(A.T) == trace(A)
i, j = symbols('i j')
F = FunctionMatrix(3, 3, Lambda((i, j), i + j))
assert trace(F) == (0 + 0) + (1 + 1) + (2 + 2)
raises(TypeError, lambda: Trace(S.One))
assert Trace(A).arg is A
assert str(trace(A)) == str(Trace(A).doit())
assert Trace(A).is_commutative is True
def test_Trace_A_plus_B():
assert trace(A + B) == Trace(A) + Trace(B)
assert Trace(A + B).arg == MatAdd(A, B)
assert Trace(A + B).doit() == Trace(A) + Trace(B)
def test_Trace_MatAdd_doit():
# See issue #9028
X = ImmutableMatrix([[1, 2, 3]]*3)
Y = MatrixSymbol('Y', 3, 3)
q = MatAdd(X, 2*X, Y, -3*Y)
assert Trace(q).arg == q
assert Trace(q).doit() == 18 - 2*Trace(Y)
def test_Trace_MatPow_doit():
X = Matrix([[1, 2], [3, 4]])
assert Trace(X).doit() == 5
q = MatPow(X, 2)
assert Trace(q).arg == q
assert Trace(q).doit() == 29
def test_Trace_MutableMatrix_plus():
# See issue #9043
X = Matrix([[1, 2], [3, 4]])
assert Trace(X) + Trace(X) == 2*Trace(X)
def test_Trace_doit_deep_False():
X = Matrix([[1, 2], [3, 4]])
q = MatPow(X, 2)
assert Trace(q).doit(deep=False).arg == q
q = MatAdd(X, 2*X)
assert Trace(q).doit(deep=False).arg == q
q = MatMul(X, 2*X)
assert Trace(q).doit(deep=False).arg == q
def test_trace_constant_factor():
# Issue 9052: gave 2*Trace(MatMul(A)) instead of 2*Trace(A)
assert trace(2*A) == 2*Trace(A)
X = ImmutableMatrix([[1, 2], [3, 4]])
assert trace(MatMul(2, X)) == 10
def test_rewrite():
assert isinstance(trace(A).rewrite(Sum), Sum)
|
e4da49515a5c9d87e83aae51ce2d7701c276d84d165d350fed757bbb70dd776b
|
from sympy import (Interval, Intersection, Set, EmptySet,
FiniteSet, Union, ComplexRegion, ProductSet)
from sympy.sets.fancysets import Integers, Naturals, Reals
from sympy.sets.sets import UniversalSet
from sympy import S, sympify
from sympy.multipledispatch import dispatch
@dispatch(Integers, Set)
def union_sets(a, b):
intersect = Intersection(a, b)
if intersect == a:
return b
elif intersect == b:
return a
@dispatch(ComplexRegion, Set)
def union_sets(a, b):
if b.is_subset(S.Reals):
# treat a subset of reals as a complex region
b = ComplexRegion.from_real(b)
if b.is_ComplexRegion:
# a in rectangular form
if (not a.polar) and (not b.polar):
return ComplexRegion(Union(a.sets, b.sets))
# a in polar form
elif a.polar and b.polar:
return ComplexRegion(Union(a.sets, b.sets), polar=True)
return None
@dispatch(EmptySet, Set)
def union_sets(a, b):
return b
@dispatch(UniversalSet, Set)
def union_sets(a, b):
return a
@dispatch(ProductSet, ProductSet)
def union_sets(a, b):
if b.is_subset(a):
return a
if len(b.args) != len(a.args):
return None
if a.args[0] == b.args[0]:
return a.args[0] * Union(ProductSet(a.args[1:]),
ProductSet(b.args[1:]))
if a.args[-1] == b.args[-1]:
return Union(ProductSet(a.args[:-1]),
ProductSet(b.args[:-1])) * a.args[-1]
return None
@dispatch(ProductSet, Set)
def union_sets(a, b):
if b.is_subset(a):
return a
return None
@dispatch(Interval, Interval)
def union_sets(a, b):
if a._is_comparable(b):
from sympy.functions.elementary.miscellaneous import Min, Max
# Non-overlapping intervals
end = Min(a.end, b.end)
start = Max(a.start, b.start)
if (end < start or
(end == start and (end not in a and end not in b))):
return None
else:
start = Min(a.start, b.start)
end = Max(a.end, b.end)
left_open = ((a.start != start or a.left_open) and
(b.start != start or b.left_open))
right_open = ((a.end != end or a.right_open) and
(b.end != end or b.right_open))
return Interval(start, end, left_open, right_open)
@dispatch(Interval, UniversalSet)
def union_sets(a, b):
return S.UniversalSet
@dispatch(Interval, Set)
def union_sets(a, b):
# If I have open end points and these endpoints are contained in b
# But only in case, when endpoints are finite. Because
# interval does not contain oo or -oo.
open_left_in_b_and_finite = (a.left_open and
sympify(b.contains(a.start)) is S.true and
a.start.is_finite)
open_right_in_b_and_finite = (a.right_open and
sympify(b.contains(a.end)) is S.true and
a.end.is_finite)
if open_left_in_b_and_finite or open_right_in_b_and_finite:
# Fill in my end points and return
open_left = a.left_open and a.start not in b
open_right = a.right_open and a.end not in b
new_a = Interval(a.start, a.end, open_left, open_right)
return set((new_a, b))
return None
@dispatch(FiniteSet, FiniteSet)
def union_sets(a, b):
return FiniteSet(*(a._elements | b._elements))
@dispatch(FiniteSet, Set)
def union_sets(a, b):
# If `b` set contains one of my elements, remove it from `a`
if any(b.contains(x) == True for x in a):
return set((
FiniteSet(*[x for x in a if not b.contains(x)]), b))
return None
@dispatch(Set, Set)
def union_sets(a, b):
return None
|
f8ad01936591118abc1592c835d60e8f731899807786ba97593964c55ae4943d
|
from sympy.multipledispatch import dispatch, Dispatcher
from sympy.core import Basic, Expr, Function, Add, Mul, Pow, Dummy, Integer
from sympy import Min, Max, Set, sympify, symbols, exp, log, S, Wild
from sympy.sets import (imageset, Interval, FiniteSet, Union, ImageSet,
ProductSet, EmptySet, Intersection, Range)
from sympy.core.function import Lambda, _coeff_isneg
from sympy.sets.fancysets import Integers
from sympy.core.function import FunctionClass
from sympy.logic.boolalg import And, Or, Not, true, false
_x, _y = symbols("x y")
FunctionUnion = (FunctionClass, Lambda)
@dispatch(FunctionClass, Set)
def _set_function(f, x):
return None
@dispatch(FunctionUnion, FiniteSet)
def _set_function(f, x):
return FiniteSet(*map(f, x))
@dispatch(Lambda, Interval)
def _set_function(f, x):
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 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_EmptySet:
break
return result
if not x.start.is_comparable or not x.end.is_comparable:
return
try:
sing = [i for i in singularities(expr, var)
if i.is_real and i in x]
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:
solns = list(solveset(diff(expr, var), var))
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)
def _set_function(f, x):
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)
def _set_function(f, x):
return Union(*(imageset(f, arg) for arg in x.args))
@dispatch(FunctionUnion, Intersection)
def _set_function(f, x):
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, EmptySet)
def _set_function(f, x):
return x
@dispatch(FunctionUnion, Set)
def _set_function(f, x):
return ImageSet(Lambda(_x, f(_x)), x)
@dispatch(FunctionUnion, Range)
def _set_function(f, self):
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)
def _set_function(f, self):
expr = f.expr
if not isinstance(expr, Expr):
return
if len(f.variables) > 1:
return
n = f.variables[0]
# 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]:
# canonical shift
expr = match[a]*n + match[b] % match[a]
if expr != f.expr:
return ImageSet(Lambda(n, expr), S.Integers)
|
c12f4b3dc0a6fe889063e7524c2b88737e2e2030828fdbce6e420b884ea7aa77
|
from sympy.sets import (ConditionSet, Intersection, FiniteSet,
EmptySet, Union)
from sympy import (Symbol, Eq, S, Abs, sin, pi, Interval,
And, Mod, oo, Function)
from sympy.utilities.pytest import raises
w = Symbol('w')
x = Symbol('x')
y = Symbol('y')
z = Symbol('z')
L = Symbol('lambda')
f = Function('f')
def test_CondSet():
sin_sols_principal = ConditionSet(x, Eq(sin(x), 0),
Interval(0, 2*pi, False, True))
assert pi in sin_sols_principal
assert pi/2 not in sin_sols_principal
assert 3*pi not in sin_sols_principal
assert 5 in ConditionSet(x, x**2 > 4, S.Reals)
assert 1 not in ConditionSet(x, x**2 > 4, S.Reals)
# in this case, 0 is not part of the base set so
# it can't be in any subset selected by the condition
assert 0 not in ConditionSet(x, y > 5, Interval(1, 7))
# since 'in' requires a true/false, the following raises
# an error because the given value provides no information
# for the condition to evaluate (since the condition does
# not depend on the dummy symbol): the result is `y > 5`.
# In this case, ConditionSet is just acting like
# Piecewise((Interval(1, 7), y > 5), (S.EmptySet, True)).
raises(TypeError, lambda: 6 in ConditionSet(x, y > 5, Interval(1, 7)))
assert isinstance(ConditionSet(x, x < 1, {x, y}).base_set, FiniteSet)
raises(TypeError, lambda: ConditionSet(x, x + 1, {x, y}))
raises(TypeError, lambda: ConditionSet(x, x, 1))
I = S.Integers
C = ConditionSet
assert C(x, x < 1, C(x, x < 2, I)
) == C(x, (x < 1) & (x < 2), I)
assert C(y, y < 1, C(x, y < 2, I)
) == C(x, (x < 1) & (y < 2), I)
assert C(y, y < 1, C(x, x < 2, I)
) == C(y, (y < 1) & (y < 2), I)
assert C(y, y < 1, C(x, y < x, I)
) == C(x, (x < 1) & (y < x), I)
assert C(y, x < 1, C(x, y < x, I)
) == C(L, (x < 1) & (y < L), I)
c = C(y, x < 1, C(x, L < y, I))
assert c == C(c.sym, (L < y) & (x < 1), I)
assert c.sym not in (x, y, L)
c = C(y, x < 1, C(x, y < x, FiniteSet(L)))
assert c == C(L, And(x < 1, y < L), FiniteSet(L))
def test_CondSet_intersect():
input_conditionset = ConditionSet(x, x**2 > 4, Interval(1, 4, False, False))
other_domain = Interval(0, 3, False, False)
output_conditionset = ConditionSet(x, x**2 > 4, Interval(1, 3, False, False))
assert Intersection(input_conditionset, other_domain) == output_conditionset
def test_issue_9849():
assert ConditionSet(x, Eq(x, x), S.Naturals) == S.Naturals
assert ConditionSet(x, Eq(Abs(sin(x)), -1), S.Naturals) == S.EmptySet
def test_simplified_FiniteSet_in_CondSet():
assert ConditionSet(x, And(x < 1, x > -3), FiniteSet(0, 1, 2)) == FiniteSet(0)
assert ConditionSet(x, x < 0, FiniteSet(0, 1, 2)) == EmptySet()
assert ConditionSet(x, And(x < -3), EmptySet()) == EmptySet()
y = Symbol('y')
assert (ConditionSet(x, And(x > 0), FiniteSet(-1, 0, 1, y)) ==
Union(FiniteSet(1), ConditionSet(x, And(x > 0), FiniteSet(y))))
assert (ConditionSet(x, Eq(Mod(x, 3), 1), FiniteSet(1, 4, 2, y)) ==
Union(FiniteSet(1, 4), ConditionSet(x, Eq(Mod(x, 3), 1), FiniteSet(y))))
def test_free_symbols():
assert ConditionSet(x, Eq(y, 0), FiniteSet(z)
).free_symbols == {y, z}
assert ConditionSet(x, Eq(x, 0), FiniteSet(z)
).free_symbols == {z}
assert ConditionSet(x, Eq(x, 0), FiniteSet(x, z)
).free_symbols == {x, z}
def test_subs_CondSet():
s = FiniteSet(z, y)
c = ConditionSet(x, x < 2, s)
# you can only replace sym with a symbol that is not in
# the free symbols
assert c.subs(x, 1) == c
assert c.subs(x, y) == ConditionSet(y, y < 2, s)
# double subs needed to change dummy if the base set
# also contains the dummy
orig = ConditionSet(y, y < 2, s)
base = orig.subs(y, w)
and_dummy = base.subs(y, w)
assert base == ConditionSet(y, y < 2, {w, z})
assert and_dummy == ConditionSet(w, w < 2, {w, z})
assert c.subs(x, w) == ConditionSet(w, w < 2, s)
assert ConditionSet(x, x < y, s
).subs(y, w) == ConditionSet(x, x < w, s.subs(y, w))
# if the user uses assumptions that cause the condition
# to evaluate, that can't be helped from SymPy's end
n = Symbol('n', negative=True)
assert ConditionSet(n, 0 < n, S.Integers) is S.EmptySet
p = Symbol('p', positive=True)
assert ConditionSet(n, n < y, S.Integers
).subs(n, x) == ConditionSet(x, x < y, S.Integers)
nc = Symbol('nc', commutative=False)
raises(ValueError, lambda: ConditionSet(
x, x < p, S.Integers).subs(x, nc))
raises(ValueError, lambda: ConditionSet(
x, x < p, S.Integers).subs(x, n))
raises(ValueError, lambda: ConditionSet(
x + 1, x < 1, S.Integers))
raises(ValueError, lambda: ConditionSet(
x + 1, x < 1, s))
assert ConditionSet(
n, n < x, Interval(0, oo)).subs(x, p) == Interval(0, oo)
assert ConditionSet(
n, n < x, Interval(-oo, 0)).subs(x, p) == S.EmptySet
assert ConditionSet(f(x), f(x) < 1, {w, z}
).subs(f(x), y) == ConditionSet(y, y < 1, {w, z})
def test_subs_CondSet_tebr():
# to eventually be removed
c = ConditionSet((x, y), {x + 1, x + y}, S.Reals)
assert c.subs(x, z) == c
def test_dummy_eq():
C = ConditionSet
I = S.Integers
c = C(x, x < 1, I)
assert c.dummy_eq(C(y, y < 1, I))
assert c.dummy_eq(1) == False
assert c.dummy_eq(C(x, x < 1, S.Reals)) == False
raises(ValueError, lambda: c.dummy_eq(C(x, x < 1, S.Reals), z))
# to eventually be removed
c1 = ConditionSet((x, y), {x + 1, x + y}, S.Reals)
c2 = ConditionSet((x, y), {x + 1, x + y}, S.Reals)
c3 = ConditionSet((x, y), {x + 1, x + y}, S.Complexes)
assert c1.dummy_eq(c2)
assert c1.dummy_eq(c3) is False
assert c.dummy_eq(c1) is False
assert c1.dummy_eq(c) is False
def test_contains():
assert 6 in ConditionSet(x, x > 5, Interval(1, 7))
assert (8 in ConditionSet(x, y > 5, Interval(1, 7))) is False
# `in` should give True or False; in this case there is not
# enough information for that result
raises(TypeError,
lambda: 6 in ConditionSet(x, y > 5, Interval(1, 7)))
assert ConditionSet(x, y > 5, Interval(1, 7)
).contains(6) == (y > 5)
assert ConditionSet(x, y > 5, Interval(1, 7)
).contains(8) is S.false
assert ConditionSet(x, y > 5, Interval(1, 7)
).contains(w) == And(w >= 1, w <= 7, y > 5)
|
f0a764d573179f0624dff3004d6385cf803664539da11b13ccd0c30c594e1fb0
|
from sympy.sets.setexpr import SetExpr
from sympy.sets import Interval, FiniteSet, Intersection, ImageSet, Union
from sympy import (Expr, Set, exp, log, cos, Symbol, Min, Max, S, oo,
symbols, Lambda, Dummy)
I = Interval(0, 2)
a, x = symbols("a, x")
_d = Dummy("d")
def test_setexpr():
se = SetExpr(Interval(0, 1))
assert isinstance(se.set, Set)
assert isinstance(se, Expr)
def test_scalar_funcs():
assert SetExpr(Interval(0, 1)).set == Interval(0, 1)
a, b = Symbol('a', real=True), Symbol('b', real=True)
a, b = 1, 2
# TODO: add support for more functions in the future:
for f in [exp, log]:
input_se = f(SetExpr(Interval(a, b)))
output = input_se.set
expected = Interval(Min(f(a), f(b)), Max(f(a), f(b)))
assert output == expected
def test_Add_Mul():
assert (SetExpr(Interval(0, 1)) + 1).set == Interval(1, 2)
assert (SetExpr(Interval(0, 1)) * 2).set == Interval(0, 2)
def test_Pow():
assert (SetExpr(Interval(0, 2))**2).set == Interval(0, 4)
def test_compound():
assert (exp(SetExpr(Interval(0, 1)) * 2 + 1)).set == \
Interval(exp(1), exp(3))
def test_Interval_Interval():
assert (SetExpr(Interval(1, 2)) + SetExpr(Interval(10, 20))).set == \
Interval(11, 22)
assert (SetExpr(Interval(1, 2)) * SetExpr(Interval(10, 20))).set == \
Interval(10, 40)
def test_FiniteSet_FiniteSet():
assert (SetExpr(FiniteSet(1, 2, 3)) + SetExpr(FiniteSet(1, 2))).set ==\
FiniteSet(2, 3, 4, 5)
assert (SetExpr(FiniteSet(1, 2, 3)) * SetExpr(FiniteSet(1, 2))).set ==\
FiniteSet(1, 2, 3, 4, 6)
def test_Interval_FiniteSet():
assert (SetExpr(FiniteSet(1, 2)) + SetExpr(Interval(0, 10))).set == \
Interval(1, 12)
def test_Many_Sets():
assert (SetExpr(Interval(0, 1)) +
SetExpr(Interval(2, 3)) +
SetExpr(FiniteSet(10, 11, 12))).set == Interval(12, 16)
def test_same_setexprs_are_not_identical():
a = SetExpr(FiniteSet(0, 1))
b = SetExpr(FiniteSet(0, 1))
assert (a + b).set == FiniteSet(0, 1, 2)
# Cannont detect the set being the same:
# assert (a + a).set == FiniteSet(0, 2)
def test_Interval_arithmetic():
i12cc = SetExpr(Interval(1, 2))
i12lo = SetExpr(Interval.Lopen(1, 2))
i12ro = SetExpr(Interval.Ropen(1, 2))
i12o = SetExpr(Interval.open(1, 2))
n23cc = SetExpr(Interval(-2, 3))
n23lo = SetExpr(Interval.Lopen(-2, 3))
n23ro = SetExpr(Interval.Ropen(-2, 3))
n23o = SetExpr(Interval.open(-2, 3))
n3n2cc = SetExpr(Interval(-3, -2))
assert i12cc + i12cc == SetExpr(Interval(2, 4))
assert i12cc - i12cc == SetExpr(Interval(-1, 1))
assert i12cc * i12cc == SetExpr(Interval(1, 4))
assert i12cc / i12cc == SetExpr(Interval(S.Half, 2))
assert i12cc ** 2 == SetExpr(Interval(1, 4))
assert i12cc ** 3 == SetExpr(Interval(1, 8))
assert i12lo + i12ro == SetExpr(Interval.open(2, 4))
assert i12lo - i12ro == SetExpr(Interval.Lopen(-1, 1))
assert i12lo * i12ro == SetExpr(Interval.open(1, 4))
assert i12lo / i12ro == SetExpr(Interval.Lopen(S.Half, 2))
assert i12lo + i12lo == SetExpr(Interval.Lopen(2, 4))
assert i12lo - i12lo == SetExpr(Interval.open(-1, 1))
assert i12lo * i12lo == SetExpr(Interval.Lopen(1, 4))
assert i12lo / i12lo == SetExpr(Interval.open(S.Half, 2))
assert i12lo + i12cc == SetExpr(Interval.Lopen(2, 4))
assert i12lo - i12cc == SetExpr(Interval.Lopen(-1, 1))
assert i12lo * i12cc == SetExpr(Interval.Lopen(1, 4))
assert i12lo / i12cc == SetExpr(Interval.Lopen(S.Half, 2))
assert i12lo + i12o == SetExpr(Interval.open(2, 4))
assert i12lo - i12o == SetExpr(Interval.open(-1, 1))
assert i12lo * i12o == SetExpr(Interval.open(1, 4))
assert i12lo / i12o == SetExpr(Interval.open(S.Half, 2))
assert i12lo ** 2 == SetExpr(Interval.Lopen(1, 4))
assert i12lo ** 3 == SetExpr(Interval.Lopen(1, 8))
assert i12ro + i12ro == SetExpr(Interval.Ropen(2, 4))
assert i12ro - i12ro == SetExpr(Interval.open(-1, 1))
assert i12ro * i12ro == SetExpr(Interval.Ropen(1, 4))
assert i12ro / i12ro == SetExpr(Interval.open(S.Half, 2))
assert i12ro + i12cc == SetExpr(Interval.Ropen(2, 4))
assert i12ro - i12cc == SetExpr(Interval.Ropen(-1, 1))
assert i12ro * i12cc == SetExpr(Interval.Ropen(1, 4))
assert i12ro / i12cc == SetExpr(Interval.Ropen(S.Half, 2))
assert i12ro + i12o == SetExpr(Interval.open(2, 4))
assert i12ro - i12o == SetExpr(Interval.open(-1, 1))
assert i12ro * i12o == SetExpr(Interval.open(1, 4))
assert i12ro / i12o == SetExpr(Interval.open(S.Half, 2))
assert i12ro ** 2 == SetExpr(Interval.Ropen(1, 4))
assert i12ro ** 3 == SetExpr(Interval.Ropen(1, 8))
assert i12o + i12lo == SetExpr(Interval.open(2, 4))
assert i12o - i12lo == SetExpr(Interval.open(-1, 1))
assert i12o * i12lo == SetExpr(Interval.open(1, 4))
assert i12o / i12lo == SetExpr(Interval.open(S.Half, 2))
assert i12o + i12ro == SetExpr(Interval.open(2, 4))
assert i12o - i12ro == SetExpr(Interval.open(-1, 1))
assert i12o * i12ro == SetExpr(Interval.open(1, 4))
assert i12o / i12ro == SetExpr(Interval.open(S.Half, 2))
assert i12o + i12cc == SetExpr(Interval.open(2, 4))
assert i12o - i12cc == SetExpr(Interval.open(-1, 1))
assert i12o * i12cc == SetExpr(Interval.open(1, 4))
assert i12o / i12cc == SetExpr(Interval.open(S.Half, 2))
assert i12o ** 2 == SetExpr(Interval.open(1, 4))
assert i12o ** 3 == SetExpr(Interval.open(1, 8))
assert n23cc + n23cc == SetExpr(Interval(-4, 6))
assert n23cc - n23cc == SetExpr(Interval(-5, 5))
assert n23cc * n23cc == SetExpr(Interval(-6, 9))
assert n23cc / n23cc == SetExpr(Interval.open(-oo, oo))
assert n23cc + n23ro == SetExpr(Interval.Ropen(-4, 6))
assert n23cc - n23ro == SetExpr(Interval.Lopen(-5, 5))
assert n23cc * n23ro == SetExpr(Interval.Ropen(-6, 9))
assert n23cc / n23ro == SetExpr(Interval.Lopen(-oo, oo))
assert n23cc + n23lo == SetExpr(Interval.Lopen(-4, 6))
assert n23cc - n23lo == SetExpr(Interval.Ropen(-5, 5))
assert n23cc * n23lo == SetExpr(Interval(-6, 9))
assert n23cc / n23lo == SetExpr(Interval.open(-oo, oo))
assert n23cc + n23o == SetExpr(Interval.open(-4, 6))
assert n23cc - n23o == SetExpr(Interval.open(-5, 5))
assert n23cc * n23o == SetExpr(Interval.open(-6, 9))
assert n23cc / n23o == SetExpr(Interval.open(-oo, oo))
assert n23cc ** 2 == SetExpr(Interval(0, 9))
assert n23cc ** 3 == SetExpr(Interval(-8, 27))
n32cc = SetExpr(Interval(-3, 2))
n32lo = SetExpr(Interval.Lopen(-3, 2))
n32ro = SetExpr(Interval.Ropen(-3, 2))
assert n32cc * n32lo == SetExpr(Interval.Ropen(-6, 9))
assert n32cc * n32cc == SetExpr(Interval(-6, 9))
assert n32lo * n32cc == SetExpr(Interval.Ropen(-6, 9))
assert n32cc * n32ro == SetExpr(Interval(-6, 9))
assert n32lo * n32ro == SetExpr(Interval.Ropen(-6, 9))
assert n32cc / n32lo == SetExpr(Interval.Ropen(-oo, oo))
assert i12cc / n32lo == SetExpr(Interval.Ropen(-oo, oo))
assert n3n2cc ** 2 == SetExpr(Interval(4, 9))
assert n3n2cc ** 3 == SetExpr(Interval(-27, -8))
assert n23cc + i12cc == SetExpr(Interval(-1, 5))
assert n23cc - i12cc == SetExpr(Interval(-4, 2))
assert n23cc * i12cc == SetExpr(Interval(-4, 6))
assert n23cc / i12cc == SetExpr(Interval(-2, 3))
def test_SetExpr_Intersection():
x, y, z, w = symbols("x y z w")
set1 = Interval(x, y)
set2 = Interval(w, z)
inter = Intersection(set1, set2)
se = SetExpr(inter)
assert exp(se).set == Intersection(
ImageSet(Lambda(x, exp(x)), set1),
ImageSet(Lambda(x, exp(x)), set2))
assert cos(se).set == ImageSet(Lambda(x, cos(x)), inter)
def test_SetExpr_Interval_div():
# TODO: some expressions cannot be calculated due to bugs (currently
# commented):
assert SetExpr(Interval(-3, -2))/SetExpr(Interval(-2, 1)) == SetExpr(Interval(-oo, oo))
assert SetExpr(Interval(2, 3))/SetExpr(Interval(-2, 2)) == SetExpr(Interval(-oo, oo))
assert SetExpr(Interval(-3, -2))/SetExpr(Interval(0, 4)) == SetExpr(Interval(-oo, -S(1)/2))
assert SetExpr(Interval(2, 4))/SetExpr(Interval(-3, 0)) == SetExpr(Interval(-oo, -S(2)/3))
assert SetExpr(Interval(2, 4))/SetExpr(Interval(0, 3)) == SetExpr(Interval(S(2)/3, oo))
#assert SetExpr(Interval(0, 1))/SetExpr(Interval(0, 1)) == SetExpr(Interval(0, oo))
#assert SetExpr(Interval(-1, 0))/SetExpr(Interval(0, 1)) == SetExpr(Interval(-oo, 0))
assert SetExpr(Interval(-1, 2))/SetExpr(Interval(-2, 2)) == SetExpr(Interval(-oo, oo))
assert 1/SetExpr(Interval(-1, 2)) == SetExpr(Union(Interval(-oo, -1), Interval(S.Half, oo)))
assert 1/SetExpr(Interval(0, 2)) == SetExpr(Interval(S(1)/2, oo))
assert (-1)/SetExpr(Interval(0, 2)) == SetExpr(Interval(-oo, -S(1)/2))
#assert 1/SetExpr(Interval(-oo, 0)) == SetExpr(Interval.open(-oo, 0))
assert 1/SetExpr(Interval(-1, 0)) == SetExpr(Interval(-oo, -1))
#assert (-2)/SetExpr(Interval(-oo, 0)) == SetExpr(Interval(0, oo))
#assert 1/SetExpr(Interval(-oo, -1)) == SetExpr(Interval(-1, 0))
#assert SetExpr(Interval(1, 2))/a == Mul(SetExpr(Interval(1, 2)), 1/a, evaluate=False)
#assert SetExpr(Interval(1, 2))/0 == SetExpr(Interval(1, 2))*zoo
#assert SetExpr(Interval(1, oo))/oo == SetExpr(Interval(0, oo))
#assert SetExpr(Interval(1, oo))/(-oo) == SetExpr(Interval(-oo, 0))
#assert SetExpr(Interval(-oo, -1))/oo == SetExpr(Interval(-oo, 0))
#assert SetExpr(Interval(-oo, -1))/(-oo) == SetExpr(Interval(0, oo))
#assert SetExpr(Interval(-oo, oo))/oo == SetExpr(Interval(-oo, oo))
#assert SetExpr(Interval(-oo, oo))/(-oo) == SetExpr(Interval(-oo, oo))
#assert SetExpr(Interval(-1, oo))/oo == SetExpr(Interval(0, oo))
#assert SetExpr(Interval(-1, oo))/(-oo) == SetExpr(Interval(-oo, 0))
#assert SetExpr(Interval(-oo, 1))/oo == SetExpr(Interval(-oo, 0))
#assert SetExpr(Interval(-oo, 1))/(-oo) == SetExpr(Interval(0, oo))
def test_SetExpr_Interval_pow():
assert SetExpr(Interval(0, 2))**2 == SetExpr(Interval(0, 4))
assert SetExpr(Interval(-1, 1))**2 == SetExpr(Interval(0, 1))
assert SetExpr(Interval(1, 2))**2 == SetExpr(Interval(1, 4))
assert SetExpr(Interval(-1, 2))**3 == SetExpr(Interval(-1, 8))
assert SetExpr(Interval(-1, 1))**0 == SetExpr(FiniteSet(1))
#assert SetExpr(Interval(1, 2))**(S(5)/2) == SetExpr(Interval(1, 4*sqrt(2)))
#assert SetExpr(Interval(-1, 2))**(S.One/3) == SetExpr(Interval(-1, 2**(S.One/3)))
#assert SetExpr(Interval(0, 2))**(S.One/2) == SetExpr(Interval(0, sqrt(2)))
#assert SetExpr(Interval(-4, 2))**(S(2)/3) == SetExpr(Interval(0, 2*2**(S.One/3)))
#assert SetExpr(Interval(-1, 5))**(S.One/2) == SetExpr(Interval(0, sqrt(5)))
#assert SetExpr(Interval(-oo, 2))**(S.One/2) == SetExpr(Interval(0, sqrt(2)))
#assert SetExpr(Interval(-2, 3))**(S(-1)/4) == SetExpr(Interval(0, oo))
assert SetExpr(Interval(1, 5))**(-2) == SetExpr(Interval(S.One/25, 1))
assert SetExpr(Interval(-1, 3))**(-2) == SetExpr(Interval(0, oo))
assert SetExpr(Interval(0, 2))**(-2) == SetExpr(Interval(S.One/4, oo))
assert SetExpr(Interval(-1, 2))**(-3) == SetExpr(Union(Interval(-oo, -1), Interval(S(1)/8, oo)))
assert SetExpr(Interval(-3, -2))**(-3) == SetExpr(Interval(S(-1)/8, -S.One/27))
assert SetExpr(Interval(-3, -2))**(-2) == SetExpr(Interval(S.One/9, S.One/4))
#assert SetExpr(Interval(0, oo))**(S.One/2) == SetExpr(Interval(0, oo))
#assert SetExpr(Interval(-oo, -1))**(S.One/3) == SetExpr(Interval(-oo, -1))
#assert SetExpr(Interval(-2, 3))**(-S.One/3) == SetExpr(Interval(-oo, oo))
assert SetExpr(Interval(-oo, 0))**(-2) == SetExpr(Interval.open(0, oo))
assert SetExpr(Interval(-2, 0))**(-2) == SetExpr(Interval(S.One/4, oo))
assert SetExpr(Interval(S.One/3, S.One/2))**oo == SetExpr(FiniteSet(0))
assert SetExpr(Interval(0, S.One/2))**oo == SetExpr(FiniteSet(0))
assert SetExpr(Interval(S.One/2, 1))**oo == SetExpr(Interval(0, oo))
assert SetExpr(Interval(0, 1))**oo == SetExpr(Interval(0, oo))
assert SetExpr(Interval(2, 3))**oo == SetExpr(FiniteSet(oo))
assert SetExpr(Interval(1, 2))**oo == SetExpr(Interval(0, oo))
assert SetExpr(Interval(S.One/2, 3))**oo == SetExpr(Interval(0, oo))
assert SetExpr(Interval(-S.One/3, -S.One/4))**oo == SetExpr(FiniteSet(0))
assert SetExpr(Interval(-1, -S.One/2))**oo == SetExpr(Interval(-oo, oo))
assert SetExpr(Interval(-3, -2))**oo == SetExpr(FiniteSet(-oo, oo))
assert SetExpr(Interval(-2, -1))**oo == SetExpr(Interval(-oo, oo))
assert SetExpr(Interval(-2, -S.One/2))**oo == SetExpr(Interval(-oo, oo))
assert SetExpr(Interval(-S.One/2, S.One/2))**oo == SetExpr(FiniteSet(0))
assert SetExpr(Interval(-S.One/2, 1))**oo == SetExpr(Interval(0, oo))
assert SetExpr(Interval(-S(2)/3, 2))**oo == SetExpr(Interval(0, oo))
assert SetExpr(Interval(-1, 1))**oo == SetExpr(Interval(-oo, oo))
assert SetExpr(Interval(-1, S.One/2))**oo == SetExpr(Interval(-oo, oo))
assert SetExpr(Interval(-1, 2))**oo == SetExpr(Interval(-oo, oo))
assert SetExpr(Interval(-2, S.One/2))**oo == SetExpr(Interval(-oo, oo))
assert (SetExpr(Interval(1, 2))**x).dummy_eq(SetExpr(ImageSet(Lambda(_d, _d**x), Interval(1, 2))))
assert SetExpr(Interval(2, 3))**(-oo) == SetExpr(FiniteSet(0))
assert SetExpr(Interval(0, 2))**(-oo) == SetExpr(Interval(0, oo))
assert (SetExpr(Interval(-1, 2))**(-oo)).dummy_eq(SetExpr(ImageSet(Lambda(_d, _d**(-oo)), Interval(-1, 2))))
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