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ai-content-maker/.venv/Lib/site-packages/sympy/solvers/tests/test_solveset.py

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2024-05-03 04:18:51 +03:00
from math import isclose
from sympy.core.add import Add
from sympy.core.containers import Tuple
from sympy.core.function import (Function, Lambda, nfloat, diff)
from sympy.core.mod import Mod
from sympy.core.numbers import (E, I, Rational, oo, pi, Integer)
from sympy.core.relational import (Eq, Gt, Ne, Ge)
from sympy.core.singleton import S
from sympy.core.sorting import ordered
from sympy.core.symbol import (Dummy, Symbol, symbols)
from sympy.core.sympify import sympify
from sympy.functions.elementary.complexes import (Abs, arg, im, re, sign, conjugate)
from sympy.functions.elementary.exponential import (LambertW, exp, log)
from sympy.functions.elementary.hyperbolic import (HyperbolicFunction,
sinh, tanh, cosh, sech, coth)
from sympy.functions.elementary.miscellaneous import sqrt, Min, Max
from sympy.functions.elementary.piecewise import Piecewise
from sympy.functions.elementary.trigonometric import (
TrigonometricFunction, acos, acot, acsc, asec, asin, atan, atan2,
cos, cot, csc, sec, sin, tan)
from sympy.functions.special.error_functions import (erf, erfc,
erfcinv, erfinv)
from sympy.logic.boolalg import And
from sympy.matrices.dense import MutableDenseMatrix as Matrix
from sympy.matrices.immutable import ImmutableDenseMatrix
from sympy.polys.polytools import Poly
from sympy.polys.rootoftools import CRootOf
from sympy.sets.contains import Contains
from sympy.sets.conditionset import ConditionSet
from sympy.sets.fancysets import ImageSet, Range
from sympy.sets.sets import (Complement, FiniteSet,
Intersection, Interval, Union, imageset, ProductSet)
from sympy.simplify import simplify
from sympy.tensor.indexed import Indexed
from sympy.utilities.iterables import numbered_symbols
from sympy.testing.pytest import (XFAIL, raises, skip, slow, SKIP, _both_exp_pow)
from sympy.core.random import verify_numerically as tn
from sympy.physics.units import cm
from sympy.solvers import solve
from sympy.solvers.solveset import (
solveset_real, domain_check, solveset_complex, linear_eq_to_matrix,
linsolve, _is_function_class_equation, invert_real, invert_complex,
solveset, solve_decomposition, substitution, nonlinsolve, solvify,
_is_finite_with_finite_vars, _transolve, _is_exponential,
_solve_exponential, _is_logarithmic, _is_lambert,
_solve_logarithm, _term_factors, _is_modular, NonlinearError)
from sympy.abc import (a, b, c, d, e, f, g, h, i, j, k, l, m, n, q, r,
t, w, x, y, z)
def dumeq(i, j):
if type(i) in (list, tuple):
return all(dumeq(i, j) for i, j in zip(i, j))
return i == j or i.dummy_eq(j)
def assert_close_ss(sol1, sol2):
"""Test solutions with floats from solveset are close"""
sol1 = sympify(sol1)
sol2 = sympify(sol2)
assert isinstance(sol1, FiniteSet)
assert isinstance(sol2, FiniteSet)
assert len(sol1) == len(sol2)
assert all(isclose(v1, v2) for v1, v2 in zip(sol1, sol2))
def assert_close_nl(sol1, sol2):
"""Test solutions with floats from nonlinsolve are close"""
sol1 = sympify(sol1)
sol2 = sympify(sol2)
assert isinstance(sol1, FiniteSet)
assert isinstance(sol2, FiniteSet)
assert len(sol1) == len(sol2)
for s1, s2 in zip(sol1, sol2):
assert len(s1) == len(s2)
assert all(isclose(v1, v2) for v1, v2 in zip(s1, s2))
@_both_exp_pow
def test_invert_real():
x = Symbol('x', real=True)
def ireal(x, s=S.Reals):
return Intersection(s, x)
assert invert_real(exp(x), z, x) == (x, ireal(FiniteSet(log(z))))
y = Symbol('y', positive=True)
n = Symbol('n', real=True)
assert invert_real(x + 3, y, x) == (x, FiniteSet(y - 3))
assert invert_real(x*3, y, x) == (x, FiniteSet(y / 3))
assert invert_real(exp(x), y, x) == (x, FiniteSet(log(y)))
assert invert_real(exp(3*x), y, x) == (x, FiniteSet(log(y) / 3))
assert invert_real(exp(x + 3), y, x) == (x, FiniteSet(log(y) - 3))
assert invert_real(exp(x) + 3, y, x) == (x, ireal(FiniteSet(log(y - 3))))
assert invert_real(exp(x)*3, y, x) == (x, FiniteSet(log(y / 3)))
assert invert_real(log(x), y, x) == (x, FiniteSet(exp(y)))
assert invert_real(log(3*x), y, x) == (x, FiniteSet(exp(y) / 3))
assert invert_real(log(x + 3), y, x) == (x, FiniteSet(exp(y) - 3))
assert invert_real(Abs(x), y, x) == (x, FiniteSet(y, -y))
assert invert_real(2**x, y, x) == (x, FiniteSet(log(y)/log(2)))
assert invert_real(2**exp(x), y, x) == (x, ireal(FiniteSet(log(log(y)/log(2)))))
assert invert_real(x**2, y, x) == (x, FiniteSet(sqrt(y), -sqrt(y)))
assert invert_real(x**S.Half, y, x) == (x, FiniteSet(y**2))
raises(ValueError, lambda: invert_real(x, x, x))
# issue 21236
assert invert_real(x**pi, y, x) == (x, FiniteSet(y**(1/pi)))
assert invert_real(x**pi, -E, x) == (x, S.EmptySet)
assert invert_real(x**Rational(3/2), 1000, x) == (x, FiniteSet(100))
assert invert_real(x**1.0, 1, x) == (x**1.0, FiniteSet(1))
raises(ValueError, lambda: invert_real(S.One, y, x))
assert invert_real(x**31 + x, y, x) == (x**31 + x, FiniteSet(y))
lhs = x**31 + x
base_values = FiniteSet(y - 1, -y - 1)
assert invert_real(Abs(x**31 + x + 1), y, x) == (lhs, base_values)
assert dumeq(invert_real(sin(x), y, x),
(x, imageset(Lambda(n, n*pi + (-1)**n*asin(y)), S.Integers)))
assert dumeq(invert_real(sin(exp(x)), y, x),
(x, imageset(Lambda(n, log((-1)**n*asin(y) + n*pi)), S.Integers)))
assert dumeq(invert_real(csc(x), y, x),
(x, imageset(Lambda(n, n*pi + (-1)**n*acsc(y)), S.Integers)))
assert dumeq(invert_real(csc(exp(x)), y, x),
(x, imageset(Lambda(n, log((-1)**n*acsc(y) + n*pi)), S.Integers)))
assert dumeq(invert_real(cos(x), y, x),
(x, Union(imageset(Lambda(n, 2*n*pi + acos(y)), S.Integers), \
imageset(Lambda(n, 2*n*pi - acos(y)), S.Integers))))
assert dumeq(invert_real(cos(exp(x)), y, x),
(x, Union(imageset(Lambda(n, log(2*n*pi + acos(y))), S.Integers), \
imageset(Lambda(n, log(2*n*pi - acos(y))), S.Integers))))
assert dumeq(invert_real(sec(x), y, x),
(x, Union(imageset(Lambda(n, 2*n*pi + asec(y)), S.Integers), \
imageset(Lambda(n, 2*n*pi - asec(y)), S.Integers))))
assert dumeq(invert_real(sec(exp(x)), y, x),
(x, Union(imageset(Lambda(n, log(2*n*pi + asec(y))), S.Integers), \
imageset(Lambda(n, log(2*n*pi - asec(y))), S.Integers))))
assert dumeq(invert_real(tan(x), y, x),
(x, imageset(Lambda(n, n*pi + atan(y)), S.Integers)))
assert dumeq(invert_real(tan(exp(x)), y, x),
(x, imageset(Lambda(n, log(n*pi + atan(y))), S.Integers)))
assert dumeq(invert_real(cot(x), y, x),
(x, imageset(Lambda(n, n*pi + acot(y)), S.Integers)))
assert dumeq(invert_real(cot(exp(x)), y, x),
(x, imageset(Lambda(n, log(n*pi + acot(y))), S.Integers)))
assert dumeq(invert_real(tan(tan(x)), y, x),
(tan(x), imageset(Lambda(n, n*pi + atan(y)), S.Integers)))
x = Symbol('x', positive=True)
assert invert_real(x**pi, y, x) == (x, FiniteSet(y**(1/pi)))
def test_invert_complex():
assert invert_complex(x + 3, y, x) == (x, FiniteSet(y - 3))
assert invert_complex(x*3, y, x) == (x, FiniteSet(y / 3))
assert invert_complex((x - 1)**3, 0, x) == (x, FiniteSet(1))
assert dumeq(invert_complex(exp(x), y, x),
(x, imageset(Lambda(n, I*(2*pi*n + arg(y)) + log(Abs(y))), S.Integers)))
assert invert_complex(log(x), y, x) == (x, FiniteSet(exp(y)))
raises(ValueError, lambda: invert_real(1, y, x))
raises(ValueError, lambda: invert_complex(x, x, x))
raises(ValueError, lambda: invert_complex(x, x, 1))
# https://github.com/skirpichev/omg/issues/16
assert invert_complex(sinh(x), 0, x) != (x, FiniteSet(0))
def test_domain_check():
assert domain_check(1/(1 + (1/(x+1))**2), x, -1) is False
assert domain_check(x**2, x, 0) is True
assert domain_check(x, x, oo) is False
assert domain_check(0, x, oo) is False
def test_issue_11536():
assert solveset(0**x - 100, x, S.Reals) == S.EmptySet
assert solveset(0**x - 1, x, S.Reals) == FiniteSet(0)
def test_issue_17479():
f = (x**2 + y**2)**2 + (x**2 + z**2)**2 - 2*(2*x**2 + y**2 + z**2)
fx = f.diff(x)
fy = f.diff(y)
fz = f.diff(z)
sol = nonlinsolve([fx, fy, fz], [x, y, z])
assert len(sol) >= 4 and len(sol) <= 20
# nonlinsolve has been giving a varying number of solutions
# (originally 18, then 20, now 19) due to various internal changes.
# Unfortunately not all the solutions are actually valid and some are
# redundant. Since the original issue was that an exception was raised,
# this first test only checks that nonlinsolve returns a "plausible"
# solution set. The next test checks the result for correctness.
@XFAIL
def test_issue_18449():
x, y, z = symbols("x, y, z")
f = (x**2 + y**2)**2 + (x**2 + z**2)**2 - 2*(2*x**2 + y**2 + z**2)
fx = diff(f, x)
fy = diff(f, y)
fz = diff(f, z)
sol = nonlinsolve([fx, fy, fz], [x, y, z])
for (xs, ys, zs) in sol:
d = {x: xs, y: ys, z: zs}
assert tuple(_.subs(d).simplify() for _ in (fx, fy, fz)) == (0, 0, 0)
# After simplification and removal of duplicate elements, there should
# only be 4 parametric solutions left:
# simplifiedsolutions = FiniteSet((sqrt(1 - z**2), z, z),
# (-sqrt(1 - z**2), z, z),
# (sqrt(1 - z**2), -z, z),
# (-sqrt(1 - z**2), -z, z))
# TODO: Is the above solution set definitely complete?
def test_issue_21047():
f = (2 - x)**2 + (sqrt(x - 1) - 1)**6
assert solveset(f, x, S.Reals) == FiniteSet(2)
f = (sqrt(x)-1)**2 + (sqrt(x)+1)**2 -2*x**2 + sqrt(2)
assert solveset(f, x, S.Reals) == FiniteSet(
S.Half - sqrt(2*sqrt(2) + 5)/2, S.Half + sqrt(2*sqrt(2) + 5)/2)
def test_is_function_class_equation():
assert _is_function_class_equation(TrigonometricFunction,
tan(x), x) is True
assert _is_function_class_equation(TrigonometricFunction,
tan(x) - 1, x) is True
assert _is_function_class_equation(TrigonometricFunction,
tan(x) + sin(x), x) is True
assert _is_function_class_equation(TrigonometricFunction,
tan(x) + sin(x) - a, x) is True
assert _is_function_class_equation(TrigonometricFunction,
sin(x)*tan(x) + sin(x), x) is True
assert _is_function_class_equation(TrigonometricFunction,
sin(x)*tan(x + a) + sin(x), x) is True
assert _is_function_class_equation(TrigonometricFunction,
sin(x)*tan(x*a) + sin(x), x) is True
assert _is_function_class_equation(TrigonometricFunction,
a*tan(x) - 1, x) is True
assert _is_function_class_equation(TrigonometricFunction,
tan(x)**2 + sin(x) - 1, x) is True
assert _is_function_class_equation(TrigonometricFunction,
tan(x) + x, x) is False
assert _is_function_class_equation(TrigonometricFunction,
tan(x**2), x) is False
assert _is_function_class_equation(TrigonometricFunction,
tan(x**2) + sin(x), x) is False
assert _is_function_class_equation(TrigonometricFunction,
tan(x)**sin(x), x) is False
assert _is_function_class_equation(TrigonometricFunction,
tan(sin(x)) + sin(x), x) is False
assert _is_function_class_equation(HyperbolicFunction,
tanh(x), x) is True
assert _is_function_class_equation(HyperbolicFunction,
tanh(x) - 1, x) is True
assert _is_function_class_equation(HyperbolicFunction,
tanh(x) + sinh(x), x) is True
assert _is_function_class_equation(HyperbolicFunction,
tanh(x) + sinh(x) - a, x) is True
assert _is_function_class_equation(HyperbolicFunction,
sinh(x)*tanh(x) + sinh(x), x) is True
assert _is_function_class_equation(HyperbolicFunction,
sinh(x)*tanh(x + a) + sinh(x), x) is True
assert _is_function_class_equation(HyperbolicFunction,
sinh(x)*tanh(x*a) + sinh(x), x) is True
assert _is_function_class_equation(HyperbolicFunction,
a*tanh(x) - 1, x) is True
assert _is_function_class_equation(HyperbolicFunction,
tanh(x)**2 + sinh(x) - 1, x) is True
assert _is_function_class_equation(HyperbolicFunction,
tanh(x) + x, x) is False
assert _is_function_class_equation(HyperbolicFunction,
tanh(x**2), x) is False
assert _is_function_class_equation(HyperbolicFunction,
tanh(x**2) + sinh(x), x) is False
assert _is_function_class_equation(HyperbolicFunction,
tanh(x)**sinh(x), x) is False
assert _is_function_class_equation(HyperbolicFunction,
tanh(sinh(x)) + sinh(x), x) is False
def test_garbage_input():
raises(ValueError, lambda: solveset_real([y], y))
x = Symbol('x', real=True)
assert solveset_real(x, 1) == S.EmptySet
assert solveset_real(x - 1, 1) == FiniteSet(x)
assert solveset_real(x, pi) == S.EmptySet
assert solveset_real(x, x**2) == S.EmptySet
raises(ValueError, lambda: solveset_complex([x], x))
assert solveset_complex(x, pi) == S.EmptySet
raises(ValueError, lambda: solveset((x, y), x))
raises(ValueError, lambda: solveset(x + 1, S.Reals))
raises(ValueError, lambda: solveset(x + 1, x, 2))
def test_solve_mul():
assert solveset_real((a*x + b)*(exp(x) - 3), x) == \
Union({log(3)}, Intersection({-b/a}, S.Reals))
anz = Symbol('anz', nonzero=True)
bb = Symbol('bb', real=True)
assert solveset_real((anz*x + bb)*(exp(x) - 3), x) == \
FiniteSet(-bb/anz, log(3))
assert solveset_real((2*x + 8)*(8 + exp(x)), x) == FiniteSet(S(-4))
assert solveset_real(x/log(x), x) is S.EmptySet
def test_solve_invert():
assert solveset_real(exp(x) - 3, x) == FiniteSet(log(3))
assert solveset_real(log(x) - 3, x) == FiniteSet(exp(3))
assert solveset_real(3**(x + 2), x) == FiniteSet()
assert solveset_real(3**(2 - x), x) == FiniteSet()
assert solveset_real(y - b*exp(a/x), x) == Intersection(
S.Reals, FiniteSet(a/log(y/b)))
# issue 4504
assert solveset_real(2**x - 10, x) == FiniteSet(1 + log(5)/log(2))
def test_errorinverses():
assert solveset_real(erf(x) - S.Half, x) == \
FiniteSet(erfinv(S.Half))
assert solveset_real(erfinv(x) - 2, x) == \
FiniteSet(erf(2))
assert solveset_real(erfc(x) - S.One, x) == \
FiniteSet(erfcinv(S.One))
assert solveset_real(erfcinv(x) - 2, x) == FiniteSet(erfc(2))
def test_solve_polynomial():
x = Symbol('x', real=True)
y = Symbol('y', real=True)
assert solveset_real(3*x - 2, x) == FiniteSet(Rational(2, 3))
assert solveset_real(x**2 - 1, x) == FiniteSet(-S.One, S.One)
assert solveset_real(x - y**3, x) == FiniteSet(y ** 3)
assert solveset_real(x**3 - 15*x - 4, x) == FiniteSet(
-2 + 3 ** S.Half,
S(4),
-2 - 3 ** S.Half)
assert solveset_real(sqrt(x) - 1, x) == FiniteSet(1)
assert solveset_real(sqrt(x) - 2, x) == FiniteSet(4)
assert solveset_real(x**Rational(1, 4) - 2, x) == FiniteSet(16)
assert solveset_real(x**Rational(1, 3) - 3, x) == FiniteSet(27)
assert len(solveset_real(x**5 + x**3 + 1, x)) == 1
assert len(solveset_real(-2*x**3 + 4*x**2 - 2*x + 6, x)) > 0
assert solveset_real(x**6 + x**4 + I, x) is S.EmptySet
def test_return_root_of():
f = x**5 - 15*x**3 - 5*x**2 + 10*x + 20
s = list(solveset_complex(f, x))
for root in s:
assert root.func == CRootOf
# if one uses solve to get the roots of a polynomial that has a CRootOf
# solution, make sure that the use of nfloat during the solve process
# doesn't fail. Note: if you want numerical solutions to a polynomial
# it is *much* faster to use nroots to get them than to solve the
# equation only to get CRootOf solutions which are then numerically
# evaluated. So for eq = x**5 + 3*x + 7 do Poly(eq).nroots() rather
# than [i.n() for i in solve(eq)] to get the numerical roots of eq.
assert nfloat(list(solveset_complex(x**5 + 3*x**3 + 7, x))[0],
exponent=False) == CRootOf(x**5 + 3*x**3 + 7, 0).n()
sol = list(solveset_complex(x**6 - 2*x + 2, x))
assert all(isinstance(i, CRootOf) for i in sol) and len(sol) == 6
f = x**5 - 15*x**3 - 5*x**2 + 10*x + 20
s = list(solveset_complex(f, x))
for root in s:
assert root.func == CRootOf
s = x**5 + 4*x**3 + 3*x**2 + Rational(7, 4)
assert solveset_complex(s, x) == \
FiniteSet(*Poly(s*4, domain='ZZ').all_roots())
# Refer issue #7876
eq = x*(x - 1)**2*(x + 1)*(x**6 - x + 1)
assert solveset_complex(eq, x) == \
FiniteSet(-1, 0, 1, CRootOf(x**6 - x + 1, 0),
CRootOf(x**6 - x + 1, 1),
CRootOf(x**6 - x + 1, 2),
CRootOf(x**6 - x + 1, 3),
CRootOf(x**6 - x + 1, 4),
CRootOf(x**6 - x + 1, 5))
def test_solveset_sqrt_1():
assert solveset_real(sqrt(5*x + 6) - 2 - x, x) == \
FiniteSet(-S.One, S(2))
assert solveset_real(sqrt(x - 1) - x + 7, x) == FiniteSet(10)
assert solveset_real(sqrt(x - 2) - 5, x) == FiniteSet(27)
assert solveset_real(sqrt(x) - 2 - 5, x) == FiniteSet(49)
assert solveset_real(sqrt(x**3), x) == FiniteSet(0)
assert solveset_real(sqrt(x - 1), x) == FiniteSet(1)
assert solveset_real(sqrt((x-3)/x), x) == FiniteSet(3)
assert solveset_real(sqrt((x-3)/x)-Rational(1, 2), x) == \
FiniteSet(4)
def test_solveset_sqrt_2():
x = Symbol('x', real=True)
y = Symbol('y', real=True)
# http://tutorial.math.lamar.edu/Classes/Alg/SolveRadicalEqns.aspx#Solve_Rad_Ex2_a
assert solveset_real(sqrt(2*x - 1) - sqrt(x - 4) - 2, x) == \
FiniteSet(S(5), S(13))
assert solveset_real(sqrt(x + 7) + 2 - sqrt(3 - x), x) == \
FiniteSet(-6)
# http://www.purplemath.com/modules/solverad.htm
assert solveset_real(sqrt(17*x - sqrt(x**2 - 5)) - 7, x) == \
FiniteSet(3)
eq = x + 1 - (x**4 + 4*x**3 - x)**Rational(1, 4)
assert solveset_real(eq, x) == FiniteSet(Rational(-1, 2), Rational(-1, 3))
eq = sqrt(2*x + 9) - sqrt(x + 1) - sqrt(x + 4)
assert solveset_real(eq, x) == FiniteSet(0)
eq = sqrt(x + 4) + sqrt(2*x - 1) - 3*sqrt(x - 1)
assert solveset_real(eq, x) == FiniteSet(5)
eq = sqrt(x)*sqrt(x - 7) - 12
assert solveset_real(eq, x) == FiniteSet(16)
eq = sqrt(x - 3) + sqrt(x) - 3
assert solveset_real(eq, x) == FiniteSet(4)
eq = sqrt(2*x**2 - 7) - (3 - x)
assert solveset_real(eq, x) == FiniteSet(-S(8), S(2))
# others
eq = sqrt(9*x**2 + 4) - (3*x + 2)
assert solveset_real(eq, x) == FiniteSet(0)
assert solveset_real(sqrt(x - 3) - sqrt(x) - 3, x) == FiniteSet()
eq = (2*x - 5)**Rational(1, 3) - 3
assert solveset_real(eq, x) == FiniteSet(16)
assert solveset_real(sqrt(x) + sqrt(sqrt(x)) - 4, x) == \
FiniteSet((Rational(-1, 2) + sqrt(17)/2)**4)
eq = sqrt(x) - sqrt(x - 1) + sqrt(sqrt(x))
assert solveset_real(eq, x) == FiniteSet()
eq = (x - 4)**2 + (sqrt(x) - 2)**4
assert solveset_real(eq, x) == FiniteSet(-4, 4)
eq = (sqrt(x) + sqrt(x + 1) + sqrt(1 - x) - 6*sqrt(5)/5)
ans = solveset_real(eq, x)
ra = S('''-1484/375 - 4*(-S(1)/2 + sqrt(3)*I/2)*(-12459439/52734375 +
114*sqrt(12657)/78125)**(S(1)/3) - 172564/(140625*(-S(1)/2 +
sqrt(3)*I/2)*(-12459439/52734375 + 114*sqrt(12657)/78125)**(S(1)/3))''')
rb = Rational(4, 5)
assert all(abs(eq.subs(x, i).n()) < 1e-10 for i in (ra, rb)) and \
len(ans) == 2 and \
{i.n(chop=True) for i in ans} == \
{i.n(chop=True) for i in (ra, rb)}
assert solveset_real(sqrt(x) + x**Rational(1, 3) +
x**Rational(1, 4), x) == FiniteSet(0)
assert solveset_real(x/sqrt(x**2 + 1), x) == FiniteSet(0)
eq = (x - y**3)/((y**2)*sqrt(1 - y**2))
assert solveset_real(eq, x) == FiniteSet(y**3)
# issue 4497
assert solveset_real(1/(5 + x)**Rational(1, 5) - 9, x) == \
FiniteSet(Rational(-295244, 59049))
@XFAIL
def test_solve_sqrt_fail():
# this only works if we check real_root(eq.subs(x, Rational(1, 3)))
# but checksol doesn't work like that
eq = (x**3 - 3*x**2)**Rational(1, 3) + 1 - x
assert solveset_real(eq, x) == FiniteSet(Rational(1, 3))
@slow
def test_solve_sqrt_3():
R = Symbol('R')
eq = sqrt(2)*R*sqrt(1/(R + 1)) + (R + 1)*(sqrt(2)*sqrt(1/(R + 1)) - 1)
sol = solveset_complex(eq, R)
fset = [Rational(5, 3) + 4*sqrt(10)*cos(atan(3*sqrt(111)/251)/3)/3,
-sqrt(10)*cos(atan(3*sqrt(111)/251)/3)/3 +
40*re(1/((Rational(-1, 2) - sqrt(3)*I/2)*(Rational(251, 27) + sqrt(111)*I/9)**Rational(1, 3)))/9 +
sqrt(30)*sin(atan(3*sqrt(111)/251)/3)/3 + Rational(5, 3) +
I*(-sqrt(30)*cos(atan(3*sqrt(111)/251)/3)/3 -
sqrt(10)*sin(atan(3*sqrt(111)/251)/3)/3 +
40*im(1/((Rational(-1, 2) - sqrt(3)*I/2)*(Rational(251, 27) + sqrt(111)*I/9)**Rational(1, 3)))/9)]
cset = [40*re(1/((Rational(-1, 2) + sqrt(3)*I/2)*(Rational(251, 27) + sqrt(111)*I/9)**Rational(1, 3)))/9 -
sqrt(10)*cos(atan(3*sqrt(111)/251)/3)/3 - sqrt(30)*sin(atan(3*sqrt(111)/251)/3)/3 +
Rational(5, 3) +
I*(40*im(1/((Rational(-1, 2) + sqrt(3)*I/2)*(Rational(251, 27) + sqrt(111)*I/9)**Rational(1, 3)))/9 -
sqrt(10)*sin(atan(3*sqrt(111)/251)/3)/3 +
sqrt(30)*cos(atan(3*sqrt(111)/251)/3)/3)]
fs = FiniteSet(*fset)
cs = ConditionSet(R, Eq(eq, 0), FiniteSet(*cset))
assert sol == (fs - {-1}) | (cs - {-1})
# the number of real roots will depend on the value of m: for m=1 there are 4
# and for m=-1 there are none.
eq = -sqrt((m - q)**2 + (-m/(2*q) + S.Half)**2) + sqrt((-m**2/2 - sqrt(
4*m**4 - 4*m**2 + 8*m + 1)/4 - Rational(1, 4))**2 + (m**2/2 - m - sqrt(
4*m**4 - 4*m**2 + 8*m + 1)/4 - Rational(1, 4))**2)
unsolved_object = ConditionSet(q, Eq(sqrt((m - q)**2 + (-m/(2*q) + S.Half)**2) -
sqrt((-m**2/2 - sqrt(4*m**4 - 4*m**2 + 8*m + 1)/4 - Rational(1, 4))**2 + (m**2/2 - m -
sqrt(4*m**4 - 4*m**2 + 8*m + 1)/4 - Rational(1, 4))**2), 0), S.Reals)
assert solveset_real(eq, q) == unsolved_object
def test_solve_polynomial_symbolic_param():
assert solveset_complex((x**2 - 1)**2 - a, x) == \
FiniteSet(sqrt(1 + sqrt(a)), -sqrt(1 + sqrt(a)),
sqrt(1 - sqrt(a)), -sqrt(1 - sqrt(a)))
# issue 4507
assert solveset_complex(y - b/(1 + a*x), x) == \
FiniteSet((b/y - 1)/a) - FiniteSet(-1/a)
# issue 4508
assert solveset_complex(y - b*x/(a + x), x) == \
FiniteSet(-a*y/(y - b)) - FiniteSet(-a)
def test_solve_rational():
assert solveset_real(1/x + 1, x) == FiniteSet(-S.One)
assert solveset_real(1/exp(x) - 1, x) == FiniteSet(0)
assert solveset_real(x*(1 - 5/x), x) == FiniteSet(5)
assert solveset_real(2*x/(x + 2) - 1, x) == FiniteSet(2)
assert solveset_real((x**2/(7 - x)).diff(x), x) == \
FiniteSet(S.Zero, S(14))
def test_solveset_real_gen_is_pow():
assert solveset_real(sqrt(1) + 1, x) is S.EmptySet
def test_no_sol():
assert solveset(1 - oo*x) is S.EmptySet
assert solveset(oo*x, x) is S.EmptySet
assert solveset(oo*x - oo, x) is S.EmptySet
assert solveset_real(4, x) is S.EmptySet
assert solveset_real(exp(x), x) is S.EmptySet
assert solveset_real(x**2 + 1, x) is S.EmptySet
assert solveset_real(-3*a/sqrt(x), x) is S.EmptySet
assert solveset_real(1/x, x) is S.EmptySet
assert solveset_real(-(1 + x)/(2 + x)**2 + 1/(2 + x), x
) is S.EmptySet
def test_sol_zero_real():
assert solveset_real(0, x) == S.Reals
assert solveset(0, x, Interval(1, 2)) == Interval(1, 2)
assert solveset_real(-x**2 - 2*x + (x + 1)**2 - 1, x) == S.Reals
def test_no_sol_rational_extragenous():
assert solveset_real((x/(x + 1) + 3)**(-2), x) is S.EmptySet
assert solveset_real((x - 1)/(1 + 1/(x - 1)), x) is S.EmptySet
def test_solve_polynomial_cv_1a():
"""
Test for solving on equations that can be converted to
a polynomial equation using the change of variable y -> x**Rational(p, q)
"""
assert solveset_real(sqrt(x) - 1, x) == FiniteSet(1)
assert solveset_real(sqrt(x) - 2, x) == FiniteSet(4)
assert solveset_real(x**Rational(1, 4) - 2, x) == FiniteSet(16)
assert solveset_real(x**Rational(1, 3) - 3, x) == FiniteSet(27)
assert solveset_real(x*(x**(S.One / 3) - 3), x) == \
FiniteSet(S.Zero, S(27))
def test_solveset_real_rational():
"""Test solveset_real for rational functions"""
x = Symbol('x', real=True)
y = Symbol('y', real=True)
assert solveset_real((x - y**3) / ((y**2)*sqrt(1 - y**2)), x) \
== FiniteSet(y**3)
# issue 4486
assert solveset_real(2*x/(x + 2) - 1, x) == FiniteSet(2)
def test_solveset_real_log():
assert solveset_real(log((x-1)*(x+1)), x) == \
FiniteSet(sqrt(2), -sqrt(2))
def test_poly_gens():
assert solveset_real(4**(2*(x**2) + 2*x) - 8, x) == \
FiniteSet(Rational(-3, 2), S.Half)
def test_solve_abs():
n = Dummy('n')
raises(ValueError, lambda: solveset(Abs(x) - 1, x))
assert solveset(Abs(x) - n, x, S.Reals).dummy_eq(
ConditionSet(x, Contains(n, Interval(0, oo)), {-n, n}))
assert solveset_real(Abs(x) - 2, x) == FiniteSet(-2, 2)
assert solveset_real(Abs(x) + 2, x) is S.EmptySet
assert solveset_real(Abs(x + 3) - 2*Abs(x - 3), x) == \
FiniteSet(1, 9)
assert solveset_real(2*Abs(x) - Abs(x - 1), x) == \
FiniteSet(-1, Rational(1, 3))
sol = ConditionSet(
x,
And(
Contains(b, Interval(0, oo)),
Contains(a + b, Interval(0, oo)),
Contains(a - b, Interval(0, oo))),
FiniteSet(-a - b - 3, -a + b - 3, a - b - 3, a + b - 3))
eq = Abs(Abs(x + 3) - a) - b
assert invert_real(eq, 0, x)[1] == sol
reps = {a: 3, b: 1}
eqab = eq.subs(reps)
for si in sol.subs(reps):
assert not eqab.subs(x, si)
assert dumeq(solveset(Eq(sin(Abs(x)), 1), x, domain=S.Reals), Union(
Intersection(Interval(0, oo),
ImageSet(Lambda(n, (-1)**n*pi/2 + n*pi), S.Integers)),
Intersection(Interval(-oo, 0),
ImageSet(Lambda(n, n*pi - (-1)**(-n)*pi/2), S.Integers))))
def test_issue_9824():
assert dumeq(solveset(sin(x)**2 - 2*sin(x) + 1, x), ImageSet(Lambda(n, 2*n*pi + pi/2), S.Integers))
assert dumeq(solveset(cos(x)**2 - 2*cos(x) + 1, x), ImageSet(Lambda(n, 2*n*pi), S.Integers))
def test_issue_9565():
assert solveset_real(Abs((x - 1)/(x - 5)) <= Rational(1, 3), x) == Interval(-1, 2)
def test_issue_10069():
eq = abs(1/(x - 1)) - 1 > 0
assert solveset_real(eq, x) == Union(
Interval.open(0, 1), Interval.open(1, 2))
def test_real_imag_splitting():
a, b = symbols('a b', real=True)
assert solveset_real(sqrt(a**2 - b**2) - 3, a) == \
FiniteSet(-sqrt(b**2 + 9), sqrt(b**2 + 9))
assert solveset_real(sqrt(a**2 + b**2) - 3, a) != \
S.EmptySet
def test_units():
assert solveset_real(1/x - 1/(2*cm), x) == FiniteSet(2*cm)
def test_solve_only_exp_1():
y = Symbol('y', positive=True)
assert solveset_real(exp(x) - y, x) == FiniteSet(log(y))
assert solveset_real(exp(x) + exp(-x) - 4, x) == \
FiniteSet(log(-sqrt(3) + 2), log(sqrt(3) + 2))
assert solveset_real(exp(x) + exp(-x) - y, x) != S.EmptySet
def test_atan2():
# The .inverse() method on atan2 works only if x.is_real is True and the
# second argument is a real constant
assert solveset_real(atan2(x, 2) - pi/3, x) == FiniteSet(2*sqrt(3))
def test_piecewise_solveset():
eq = Piecewise((x - 2, Gt(x, 2)), (2 - x, True)) - 3
assert set(solveset_real(eq, x)) == set(FiniteSet(-1, 5))
absxm3 = Piecewise(
(x - 3, 0 <= x - 3),
(3 - x, 0 > x - 3))
y = Symbol('y', positive=True)
assert solveset_real(absxm3 - y, x) == FiniteSet(-y + 3, y + 3)
f = Piecewise(((x - 2)**2, x >= 0), (0, True))
assert solveset(f, x, domain=S.Reals) == Union(FiniteSet(2), Interval(-oo, 0, True, True))
assert solveset(
Piecewise((x + 1, x > 0), (I, True)) - I, x, S.Reals
) == Interval(-oo, 0)
assert solveset(Piecewise((x - 1, Ne(x, I)), (x, True)), x) == FiniteSet(1)
# issue 19718
g = Piecewise((1, x > 10), (0, True))
assert solveset(g > 0, x, S.Reals) == Interval.open(10, oo)
from sympy.logic.boolalg import BooleanTrue
f = BooleanTrue()
assert solveset(f, x, domain=Interval(-3, 10)) == Interval(-3, 10)
# issue 20552
f = Piecewise((0, Eq(x, 0)), (x**2/Abs(x), True))
g = Piecewise((0, Eq(x, pi)), ((x - pi)/sin(x), True))
assert solveset(f, x, domain=S.Reals) == FiniteSet(0)
assert solveset(g) == FiniteSet(pi)
def test_solveset_complex_polynomial():
assert solveset_complex(a*x**2 + b*x + c, x) == \
FiniteSet(-b/(2*a) - sqrt(-4*a*c + b**2)/(2*a),
-b/(2*a) + sqrt(-4*a*c + b**2)/(2*a))
assert solveset_complex(x - y**3, y) == FiniteSet(
(-x**Rational(1, 3))/2 + I*sqrt(3)*x**Rational(1, 3)/2,
x**Rational(1, 3),
(-x**Rational(1, 3))/2 - I*sqrt(3)*x**Rational(1, 3)/2)
assert solveset_complex(x + 1/x - 1, x) == \
FiniteSet(S.Half + I*sqrt(3)/2, S.Half - I*sqrt(3)/2)
def test_sol_zero_complex():
assert solveset_complex(0, x) is S.Complexes
def test_solveset_complex_rational():
assert solveset_complex((x - 1)*(x - I)/(x - 3), x) == \
FiniteSet(1, I)
assert solveset_complex((x - y**3)/((y**2)*sqrt(1 - y**2)), x) == \
FiniteSet(y**3)
assert solveset_complex(-x**2 - I, x) == \
FiniteSet(-sqrt(2)/2 + sqrt(2)*I/2, sqrt(2)/2 - sqrt(2)*I/2)
def test_solve_quintics():
skip("This test is too slow")
f = x**5 - 110*x**3 - 55*x**2 + 2310*x + 979
s = solveset_complex(f, x)
for root in s:
res = f.subs(x, root.n()).n()
assert tn(res, 0)
f = x**5 + 15*x + 12
s = solveset_complex(f, x)
for root in s:
res = f.subs(x, root.n()).n()
assert tn(res, 0)
def test_solveset_complex_exp():
assert dumeq(solveset_complex(exp(x) - 1, x),
imageset(Lambda(n, I*2*n*pi), S.Integers))
assert dumeq(solveset_complex(exp(x) - I, x),
imageset(Lambda(n, I*(2*n*pi + pi/2)), S.Integers))
assert solveset_complex(1/exp(x), x) == S.EmptySet
assert dumeq(solveset_complex(sinh(x).rewrite(exp), x),
imageset(Lambda(n, n*pi*I), S.Integers))
def test_solveset_real_exp():
assert solveset(Eq((-2)**x, 4), x, S.Reals) == FiniteSet(2)
assert solveset(Eq(-2**x, 4), x, S.Reals) == S.EmptySet
assert solveset(Eq((-3)**x, 27), x, S.Reals) == S.EmptySet
assert solveset(Eq((-5)**(x+1), 625), x, S.Reals) == FiniteSet(3)
assert solveset(Eq(2**(x-3), -16), x, S.Reals) == S.EmptySet
assert solveset(Eq((-3)**(x - 3), -3**39), x, S.Reals) == FiniteSet(42)
assert solveset(Eq(2**x, y), x, S.Reals) == Intersection(S.Reals, FiniteSet(log(y)/log(2)))
assert invert_real((-2)**(2*x) - 16, 0, x) == (x, FiniteSet(2))
def test_solve_complex_log():
assert solveset_complex(log(x), x) == FiniteSet(1)
assert solveset_complex(1 - log(a + 4*x**2), x) == \
FiniteSet(-sqrt(-a + E)/2, sqrt(-a + E)/2)
def test_solve_complex_sqrt():
assert solveset_complex(sqrt(5*x + 6) - 2 - x, x) == \
FiniteSet(-S.One, S(2))
assert solveset_complex(sqrt(5*x + 6) - (2 + 2*I) - x, x) == \
FiniteSet(-S(2), 3 - 4*I)
assert solveset_complex(4*x*(1 - a * sqrt(x)), x) == \
FiniteSet(S.Zero, 1 / a ** 2)
def test_solveset_complex_tan():
s = solveset_complex(tan(x).rewrite(exp), x)
assert dumeq(s, imageset(Lambda(n, pi*n), S.Integers) - \
imageset(Lambda(n, pi*n + pi/2), S.Integers))
@_both_exp_pow
def test_solve_trig():
assert dumeq(solveset_real(sin(x), x),
Union(imageset(Lambda(n, 2*pi*n), S.Integers),
imageset(Lambda(n, 2*pi*n + pi), S.Integers)))
assert dumeq(solveset_real(sin(x) - 1, x),
imageset(Lambda(n, 2*pi*n + pi/2), S.Integers))
assert dumeq(solveset_real(cos(x), x),
Union(imageset(Lambda(n, 2*pi*n + pi/2), S.Integers),
imageset(Lambda(n, 2*pi*n + pi*Rational(3, 2)), S.Integers)))
assert dumeq(solveset_real(sin(x) + cos(x), x),
Union(imageset(Lambda(n, 2*n*pi + pi*Rational(3, 4)), S.Integers),
imageset(Lambda(n, 2*n*pi + pi*Rational(7, 4)), S.Integers)))
assert solveset_real(sin(x)**2 + cos(x)**2, x) == S.EmptySet
assert dumeq(solveset_complex(cos(x) - S.Half, x),
Union(imageset(Lambda(n, 2*n*pi + pi*Rational(5, 3)), S.Integers),
imageset(Lambda(n, 2*n*pi + pi/3), S.Integers)))
assert dumeq(solveset(sin(y + a) - sin(y), a, domain=S.Reals),
Union(ImageSet(Lambda(n, 2*n*pi), S.Integers),
Intersection(ImageSet(Lambda(n, -I*(I*(
2*n*pi + arg(-exp(-2*I*y))) +
2*im(y))), S.Integers), S.Reals)))
assert dumeq(solveset_real(sin(2*x)*cos(x) + cos(2*x)*sin(x)-1, x),
ImageSet(Lambda(n, n*pi*Rational(2, 3) + pi/6), S.Integers))
assert dumeq(solveset_real(2*tan(x)*sin(x) + 1, x), Union(
ImageSet(Lambda(n, 2*n*pi + atan(sqrt(2)*sqrt(-1 + sqrt(17))/
(1 - sqrt(17))) + pi), S.Integers),
ImageSet(Lambda(n, 2*n*pi - atan(sqrt(2)*sqrt(-1 + sqrt(17))/
(1 - sqrt(17))) + pi), S.Integers)))
assert dumeq(solveset_real(cos(2*x)*cos(4*x) - 1, x),
ImageSet(Lambda(n, n*pi), S.Integers))
assert dumeq(solveset(sin(x/10) + Rational(3, 4)), Union(
ImageSet(Lambda(n, 20*n*pi + 10*atan(3*sqrt(7)/7) + 10*pi), S.Integers),
ImageSet(Lambda(n, 20*n*pi - 10*atan(3*sqrt(7)/7) + 20*pi), S.Integers)))
assert dumeq(solveset(cos(x/15) + cos(x/5)), Union(
ImageSet(Lambda(n, 30*n*pi + 15*pi/2), S.Integers),
ImageSet(Lambda(n, 30*n*pi + 45*pi/2), S.Integers),
ImageSet(Lambda(n, 30*n*pi + 75*pi/4), S.Integers),
ImageSet(Lambda(n, 30*n*pi + 45*pi/4), S.Integers),
ImageSet(Lambda(n, 30*n*pi + 105*pi/4), S.Integers),
ImageSet(Lambda(n, 30*n*pi + 15*pi/4), S.Integers)))
assert dumeq(solveset(sec(sqrt(2)*x/3) + 5), Union(
ImageSet(Lambda(n, 3*sqrt(2)*(2*n*pi - pi + atan(2*sqrt(6)))/2), S.Integers),
ImageSet(Lambda(n, 3*sqrt(2)*(2*n*pi - atan(2*sqrt(6)) + pi)/2), S.Integers)))
assert dumeq(simplify(solveset(tan(pi*x) - cot(pi/2*x))), Union(
ImageSet(Lambda(n, 4*n + 1), S.Integers),
ImageSet(Lambda(n, 4*n + 3), S.Integers),
ImageSet(Lambda(n, 4*n + Rational(7, 3)), S.Integers),
ImageSet(Lambda(n, 4*n + Rational(5, 3)), S.Integers),
ImageSet(Lambda(n, 4*n + Rational(11, 3)), S.Integers),
ImageSet(Lambda(n, 4*n + Rational(1, 3)), S.Integers)))
assert dumeq(solveset(cos(9*x)), Union(
ImageSet(Lambda(n, 2*n*pi/9 + pi/18), S.Integers),
ImageSet(Lambda(n, 2*n*pi/9 + pi/6), S.Integers)))
assert dumeq(solveset(sin(8*x) + cot(12*x), x, S.Reals), Union(
ImageSet(Lambda(n, n*pi/2 + pi/8), S.Integers),
ImageSet(Lambda(n, n*pi/2 + 3*pi/8), S.Integers),
ImageSet(Lambda(n, n*pi/2 + 5*pi/16), S.Integers),
ImageSet(Lambda(n, n*pi/2 + 3*pi/16), S.Integers),
ImageSet(Lambda(n, n*pi/2 + 7*pi/16), S.Integers),
ImageSet(Lambda(n, n*pi/2 + pi/16), S.Integers)))
# This is the only remaining solveset test that actually ends up being solved
# by _solve_trig2(). All others are handled by the improved _solve_trig1.
assert dumeq(solveset_real(2*cos(x)*cos(2*x) - 1, x),
Union(ImageSet(Lambda(n, 2*n*pi + 2*atan(sqrt(-2*2**Rational(1, 3)*(67 +
9*sqrt(57))**Rational(2, 3) + 8*2**Rational(2, 3) + 11*(67 +
9*sqrt(57))**Rational(1, 3))/(3*(67 + 9*sqrt(57))**Rational(1, 6)))), S.Integers),
ImageSet(Lambda(n, 2*n*pi - 2*atan(sqrt(-2*2**Rational(1, 3)*(67 +
9*sqrt(57))**Rational(2, 3) + 8*2**Rational(2, 3) + 11*(67 +
9*sqrt(57))**Rational(1, 3))/(3*(67 + 9*sqrt(57))**Rational(1, 6))) +
2*pi), S.Integers)))
# issue #16870
assert dumeq(simplify(solveset(sin(x/180*pi) - S.Half, x, S.Reals)), Union(
ImageSet(Lambda(n, 360*n + 150), S.Integers),
ImageSet(Lambda(n, 360*n + 30), S.Integers)))
def test_solve_hyperbolic():
# actual solver: _solve_trig1
n = Dummy('n')
assert solveset(sinh(x) + cosh(x), x) == S.EmptySet
assert solveset(sinh(x) + cos(x), x) == ConditionSet(x,
Eq(cos(x) + sinh(x), 0), S.Complexes)
assert solveset_real(sinh(x) + sech(x), x) == FiniteSet(
log(sqrt(sqrt(5) - 2)))
assert solveset_real(3*cosh(2*x) - 5, x) == FiniteSet(
-log(3)/2, log(3)/2)
assert solveset_real(sinh(x - 3) - 2, x) == FiniteSet(
log((2 + sqrt(5))*exp(3)))
assert solveset_real(cosh(2*x) + 2*sinh(x) - 5, x) == FiniteSet(
log(-2 + sqrt(5)), log(1 + sqrt(2)))
assert solveset_real((coth(x) + sinh(2*x))/cosh(x) - 3, x) == FiniteSet(
log(S.Half + sqrt(5)/2), log(1 + sqrt(2)))
assert solveset_real(cosh(x)*sinh(x) - 2, x) == FiniteSet(
log(4 + sqrt(17))/2)
assert solveset_real(sinh(x) + tanh(x) - 1, x) == FiniteSet(
log(sqrt(2)/2 + sqrt(-S(1)/2 + sqrt(2))))
assert dumeq(solveset_complex(sinh(x) - I/2, x), Union(
ImageSet(Lambda(n, I*(2*n*pi + 5*pi/6)), S.Integers),
ImageSet(Lambda(n, I*(2*n*pi + pi/6)), S.Integers)))
assert dumeq(solveset_complex(sinh(x) + sech(x), x), Union(
ImageSet(Lambda(n, 2*n*I*pi + log(sqrt(-2 + sqrt(5)))), S.Integers),
ImageSet(Lambda(n, I*(2*n*pi + pi/2) + log(sqrt(2 + sqrt(5)))), S.Integers),
ImageSet(Lambda(n, I*(2*n*pi + pi) + log(sqrt(-2 + sqrt(5)))), S.Integers),
ImageSet(Lambda(n, I*(2*n*pi - pi/2) + log(sqrt(2 + sqrt(5)))), S.Integers)))
assert dumeq(solveset(sinh(x/10) + Rational(3, 4)), Union(
ImageSet(Lambda(n, 10*I*(2*n*pi + pi) + 10*log(2)), S.Integers),
ImageSet(Lambda(n, 20*n*I*pi - 10*log(2)), S.Integers)))
assert dumeq(solveset(cosh(x/15) + cosh(x/5)), Union(
ImageSet(Lambda(n, 15*I*(2*n*pi + pi/2)), S.Integers),
ImageSet(Lambda(n, 15*I*(2*n*pi - pi/2)), S.Integers),
ImageSet(Lambda(n, 15*I*(2*n*pi - 3*pi/4)), S.Integers),
ImageSet(Lambda(n, 15*I*(2*n*pi + 3*pi/4)), S.Integers),
ImageSet(Lambda(n, 15*I*(2*n*pi - pi/4)), S.Integers),
ImageSet(Lambda(n, 15*I*(2*n*pi + pi/4)), S.Integers)))
assert dumeq(solveset(sech(sqrt(2)*x/3) + 5), Union(
ImageSet(Lambda(n, 3*sqrt(2)*I*(2*n*pi - pi + atan(2*sqrt(6)))/2), S.Integers),
ImageSet(Lambda(n, 3*sqrt(2)*I*(2*n*pi - atan(2*sqrt(6)) + pi)/2), S.Integers)))
assert dumeq(solveset(tanh(pi*x) - coth(pi/2*x)), Union(
ImageSet(Lambda(n, 2*I*(2*n*pi + pi/2)/pi), S.Integers),
ImageSet(Lambda(n, 2*I*(2*n*pi - pi/2)/pi), S.Integers)))
assert dumeq(solveset(cosh(9*x)), Union(
ImageSet(Lambda(n, I*(2*n*pi + pi/2)/9), S.Integers),
ImageSet(Lambda(n, I*(2*n*pi - pi/2)/9), S.Integers)))
# issues #9606 / #9531:
assert solveset(sinh(x), x, S.Reals) == FiniteSet(0)
assert dumeq(solveset(sinh(x), x, S.Complexes), Union(
ImageSet(Lambda(n, I*(2*n*pi + pi)), S.Integers),
ImageSet(Lambda(n, 2*n*I*pi), S.Integers)))
# issues #11218 / #18427
assert dumeq(solveset(sin(pi*x), x, S.Reals), Union(
ImageSet(Lambda(n, (2*n*pi + pi)/pi), S.Integers),
ImageSet(Lambda(n, 2*n), S.Integers)))
assert dumeq(solveset(sin(pi*x), x), Union(
ImageSet(Lambda(n, (2*n*pi + pi)/pi), S.Integers),
ImageSet(Lambda(n, 2*n), S.Integers)))
# issue #17543
assert dumeq(simplify(solveset(I*cot(8*x - 8*E), x)), Union(
ImageSet(Lambda(n, n*pi/4 - 13*pi/16 + E), S.Integers),
ImageSet(Lambda(n, n*pi/4 - 11*pi/16 + E), S.Integers)))
# issues #18490 / #19489
assert solveset(cosh(x) + cosh(3*x) - cosh(5*x), x, S.Reals
).dummy_eq(ConditionSet(x,
Eq(cosh(x) + cosh(3*x) - cosh(5*x), 0), S.Reals))
assert solveset(sinh(8*x) + coth(12*x)).dummy_eq(
ConditionSet(x, Eq(sinh(8*x) + coth(12*x), 0), S.Complexes))
def test_solve_trig_hyp_symbolic():
# actual solver: _solve_trig1
assert dumeq(solveset(sin(a*x), x), ConditionSet(x, Ne(a, 0), Union(
ImageSet(Lambda(n, (2*n*pi + pi)/a), S.Integers),
ImageSet(Lambda(n, 2*n*pi/a), S.Integers))))
assert dumeq(solveset(cosh(x/a), x), ConditionSet(x, Ne(a, 0), Union(
ImageSet(Lambda(n, I*a*(2*n*pi + pi/2)), S.Integers),
ImageSet(Lambda(n, I*a*(2*n*pi - pi/2)), S.Integers))))
assert dumeq(solveset(sin(2*sqrt(3)/3*a**2/(b*pi)*x)
+ cos(4*sqrt(3)/3*a**2/(b*pi)*x), x),
ConditionSet(x, Ne(b, 0) & Ne(a**2, 0), Union(
ImageSet(Lambda(n, sqrt(3)*pi*b*(2*n*pi + pi/2)/(2*a**2)), S.Integers),
ImageSet(Lambda(n, sqrt(3)*pi*b*(2*n*pi - 5*pi/6)/(2*a**2)), S.Integers),
ImageSet(Lambda(n, sqrt(3)*pi*b*(2*n*pi - pi/6)/(2*a**2)), S.Integers))))
assert dumeq(simplify(solveset(cot((1 + I)*x) - cot((3 + 3*I)*x), x)), Union(
ImageSet(Lambda(n, pi*(1 - I)*(4*n + 1)/4), S.Integers),
ImageSet(Lambda(n, pi*(1 - I)*(4*n - 1)/4), S.Integers)))
assert dumeq(solveset(cosh((a**2 + 1)*x) - 3, x),
ConditionSet(x, Ne(a**2 + 1, 0), Union(
ImageSet(Lambda(n, (2*n*I*pi + log(3 - 2*sqrt(2)))/(a**2 + 1)), S.Integers),
ImageSet(Lambda(n, (2*n*I*pi + log(2*sqrt(2) + 3))/(a**2 + 1)), S.Integers))))
ar = Symbol('ar', real=True)
assert solveset(cosh((ar**2 + 1)*x) - 2, x, S.Reals) == FiniteSet(
log(sqrt(3) + 2)/(ar**2 + 1), log(2 - sqrt(3))/(ar**2 + 1))
def test_issue_9616():
assert dumeq(solveset(sinh(x) + tanh(x) - 1, x), Union(
ImageSet(Lambda(n, 2*n*I*pi + log(sqrt(2)/2 + sqrt(-S.Half + sqrt(2)))), S.Integers),
ImageSet(Lambda(n, I*(2*n*pi - atan(sqrt(2)*sqrt(S.Half + sqrt(2))) + pi)
+ log(sqrt(1 + sqrt(2)))), S.Integers),
ImageSet(Lambda(n, I*(2*n*pi + pi) + log(-sqrt(2)/2 + sqrt(-S.Half + sqrt(2)))), S.Integers),
ImageSet(Lambda(n, I*(2*n*pi - pi + atan(sqrt(2)*sqrt(S.Half + sqrt(2))))
+ log(sqrt(1 + sqrt(2)))), S.Integers)))
f1 = (sinh(x)).rewrite(exp)
f2 = (tanh(x)).rewrite(exp)
assert dumeq(solveset(f1 + f2 - 1, x), Union(
Complement(ImageSet(
Lambda(n, I*(2*n*pi + pi) + log(-sqrt(2)/2 + sqrt(-S.Half + sqrt(2)))), S.Integers),
ImageSet(Lambda(n, I*(2*n*pi + pi)/2), S.Integers)),
Complement(ImageSet(Lambda(n, I*(2*n*pi - pi + atan(sqrt(2)*sqrt(S.Half + sqrt(2))))
+ log(sqrt(1 + sqrt(2)))), S.Integers),
ImageSet(Lambda(n, I*(2*n*pi + pi)/2), S.Integers)),
Complement(ImageSet(Lambda(n, I*(2*n*pi - atan(sqrt(2)*sqrt(S.Half + sqrt(2))) + pi)
+ log(sqrt(1 + sqrt(2)))), S.Integers),
ImageSet(Lambda(n, I*(2*n*pi + pi)/2), S.Integers)),
Complement(
ImageSet(Lambda(n, 2*n*I*pi + log(sqrt(2)/2 + sqrt(-S.Half + sqrt(2)))), S.Integers),
ImageSet(Lambda(n, I*(2*n*pi + pi)/2), S.Integers))))
def test_solve_invalid_sol():
assert 0 not in solveset_real(sin(x)/x, x)
assert 0 not in solveset_complex((exp(x) - 1)/x, x)
@XFAIL
def test_solve_trig_simplified():
n = Dummy('n')
assert dumeq(solveset_real(sin(x), x),
imageset(Lambda(n, n*pi), S.Integers))
assert dumeq(solveset_real(cos(x), x),
imageset(Lambda(n, n*pi + pi/2), S.Integers))
assert dumeq(solveset_real(cos(x) + sin(x), x),
imageset(Lambda(n, n*pi - pi/4), S.Integers))
@XFAIL
def test_solve_lambert():
assert solveset_real(x*exp(x) - 1, x) == FiniteSet(LambertW(1))
assert solveset_real(exp(x) + x, x) == FiniteSet(-LambertW(1))
assert solveset_real(x + 2**x, x) == \
FiniteSet(-LambertW(log(2))/log(2))
# issue 4739
ans = solveset_real(3*x + 5 + 2**(-5*x + 3), x)
assert ans == FiniteSet(Rational(-5, 3) +
LambertW(-10240*2**Rational(1, 3)*log(2)/3)/(5*log(2)))
eq = 2*(3*x + 4)**5 - 6*7**(3*x + 9)
result = solveset_real(eq, x)
ans = FiniteSet((log(2401) +
5*LambertW(-log(7**(7*3**Rational(1, 5)/5))))/(3*log(7))/-1)
assert result == ans
assert solveset_real(eq.expand(), x) == result
assert solveset_real(5*x - 1 + 3*exp(2 - 7*x), x) == \
FiniteSet(Rational(1, 5) + LambertW(-21*exp(Rational(3, 5))/5)/7)
assert solveset_real(2*x + 5 + log(3*x - 2), x) == \
FiniteSet(Rational(2, 3) + LambertW(2*exp(Rational(-19, 3))/3)/2)
assert solveset_real(3*x + log(4*x), x) == \
FiniteSet(LambertW(Rational(3, 4))/3)
assert solveset_real(x**x - 2) == FiniteSet(exp(LambertW(log(2))))
a = Symbol('a')
assert solveset_real(-a*x + 2*x*log(x), x) == FiniteSet(exp(a/2))
a = Symbol('a', real=True)
assert solveset_real(a/x + exp(x/2), x) == \
FiniteSet(2*LambertW(-a/2))
assert solveset_real((a/x + exp(x/2)).diff(x), x) == \
FiniteSet(4*LambertW(sqrt(2)*sqrt(a)/4))
# coverage test
assert solveset_real(tanh(x + 3)*tanh(x - 3) - 1, x) is S.EmptySet
assert solveset_real((x**2 - 2*x + 1).subs(x, log(x) + 3*x), x) == \
FiniteSet(LambertW(3*S.Exp1)/3)
assert solveset_real((x**2 - 2*x + 1).subs(x, (log(x) + 3*x)**2 - 1), x) == \
FiniteSet(LambertW(3*exp(-sqrt(2)))/3, LambertW(3*exp(sqrt(2)))/3)
assert solveset_real((x**2 - 2*x - 2).subs(x, log(x) + 3*x), x) == \
FiniteSet(LambertW(3*exp(1 + sqrt(3)))/3, LambertW(3*exp(-sqrt(3) + 1))/3)
assert solveset_real(x*log(x) + 3*x + 1, x) == \
FiniteSet(exp(-3 + LambertW(-exp(3))))
eq = (x*exp(x) - 3).subs(x, x*exp(x))
assert solveset_real(eq, x) == \
FiniteSet(LambertW(3*exp(-LambertW(3))))
assert solveset_real(3*log(a**(3*x + 5)) + a**(3*x + 5), x) == \
FiniteSet(-((log(a**5) + LambertW(Rational(1, 3)))/(3*log(a))))
p = symbols('p', positive=True)
assert solveset_real(3*log(p**(3*x + 5)) + p**(3*x + 5), x) == \
FiniteSet(
log((-3**Rational(1, 3) - 3**Rational(5, 6)*I)*LambertW(Rational(1, 3))**Rational(1, 3)/(2*p**Rational(5, 3)))/log(p),
log((-3**Rational(1, 3) + 3**Rational(5, 6)*I)*LambertW(Rational(1, 3))**Rational(1, 3)/(2*p**Rational(5, 3)))/log(p),
log((3*LambertW(Rational(1, 3))/p**5)**(1/(3*log(p)))),) # checked numerically
# check collection
b = Symbol('b')
eq = 3*log(a**(3*x + 5)) + b*log(a**(3*x + 5)) + a**(3*x + 5)
assert solveset_real(eq, x) == FiniteSet(
-((log(a**5) + LambertW(1/(b + 3)))/(3*log(a))))
# issue 4271
assert solveset_real((a/x + exp(x/2)).diff(x, 2), x) == FiniteSet(
6*LambertW((-1)**Rational(1, 3)*a**Rational(1, 3)/3))
assert solveset_real(x**3 - 3**x, x) == \
FiniteSet(-3/log(3)*LambertW(-log(3)/3))
assert solveset_real(3**cos(x) - cos(x)**3) == FiniteSet(
acos(-3*LambertW(-log(3)/3)/log(3)))
assert solveset_real(x**2 - 2**x, x) == \
solveset_real(-x**2 + 2**x, x)
assert solveset_real(3*log(x) - x*log(3)) == FiniteSet(
-3*LambertW(-log(3)/3)/log(3),
-3*LambertW(-log(3)/3, -1)/log(3))
assert solveset_real(LambertW(2*x) - y) == FiniteSet(
y*exp(y)/2)
@XFAIL
def test_other_lambert():
a = Rational(6, 5)
assert solveset_real(x**a - a**x, x) == FiniteSet(
a, -a*LambertW(-log(a)/a)/log(a))
@_both_exp_pow
def test_solveset():
f = Function('f')
raises(ValueError, lambda: solveset(x + y))
assert solveset(x, 1) == S.EmptySet
assert solveset(f(1)**2 + y + 1, f(1)
) == FiniteSet(-sqrt(-y - 1), sqrt(-y - 1))
assert solveset(f(1)**2 - 1, f(1), S.Reals) == FiniteSet(-1, 1)
assert solveset(f(1)**2 + 1, f(1)) == FiniteSet(-I, I)
assert solveset(x - 1, 1) == FiniteSet(x)
assert solveset(sin(x) - cos(x), sin(x)) == FiniteSet(cos(x))
assert solveset(0, domain=S.Reals) == S.Reals
assert solveset(1) == S.EmptySet
assert solveset(True, domain=S.Reals) == S.Reals # issue 10197
assert solveset(False, domain=S.Reals) == S.EmptySet
assert solveset(exp(x) - 1, domain=S.Reals) == FiniteSet(0)
assert solveset(exp(x) - 1, x, S.Reals) == FiniteSet(0)
assert solveset(Eq(exp(x), 1), x, S.Reals) == FiniteSet(0)
assert solveset(exp(x) - 1, exp(x), S.Reals) == FiniteSet(1)
A = Indexed('A', x)
assert solveset(A - 1, A, S.Reals) == FiniteSet(1)
assert solveset(x - 1 >= 0, x, S.Reals) == Interval(1, oo)
assert solveset(exp(x) - 1 >= 0, x, S.Reals) == Interval(0, oo)
assert dumeq(solveset(exp(x) - 1, x), imageset(Lambda(n, 2*I*pi*n), S.Integers))
assert dumeq(solveset(Eq(exp(x), 1), x), imageset(Lambda(n, 2*I*pi*n),
S.Integers))
# issue 13825
assert solveset(x**2 + f(0) + 1, x) == {-sqrt(-f(0) - 1), sqrt(-f(0) - 1)}
# issue 19977
assert solveset(atan(log(x)) > 0, x, domain=Interval.open(0, oo)) == Interval.open(1, oo)
@_both_exp_pow
def test_multi_exp():
k1, k2, k3 = symbols('k1, k2, k3')
assert dumeq(solveset(exp(exp(x)) - 5, x),\
imageset(Lambda(((k1, n),), I*(2*k1*pi + arg(2*n*I*pi + log(5))) + log(Abs(2*n*I*pi + log(5)))),\
ProductSet(S.Integers, S.Integers)))
assert dumeq(solveset((d*exp(exp(a*x + b)) + c), x),\
imageset(Lambda(x, (-b + x)/a), ImageSet(Lambda(((k1, n),), \
I*(2*k1*pi + arg(I*(2*n*pi + arg(-c/d)) + log(Abs(c/d)))) + log(Abs(I*(2*n*pi + arg(-c/d)) + log(Abs(c/d))))), \
ProductSet(S.Integers, S.Integers))))
assert dumeq(solveset((d*exp(exp(exp(a*x + b))) + c), x),\
imageset(Lambda(x, (-b + x)/a), ImageSet(Lambda(((k2, k1, n),), \
I*(2*k2*pi + arg(I*(2*k1*pi + arg(I*(2*n*pi + arg(-c/d)) + log(Abs(c/d)))) + \
log(Abs(I*(2*n*pi + arg(-c/d)) + log(Abs(c/d)))))) + log(Abs(I*(2*k1*pi + arg(I*(2*n*pi + arg(-c/d)) + \
log(Abs(c/d)))) + log(Abs(I*(2*n*pi + arg(-c/d)) + log(Abs(c/d))))))), \
ProductSet(S.Integers, S.Integers, S.Integers))))
assert dumeq(solveset((d*exp(exp(exp(exp(a*x + b)))) + c), x),\
ImageSet(Lambda(x, (-b + x)/a), ImageSet(Lambda(((k3, k2, k1, n),), \
I*(2*k3*pi + arg(I*(2*k2*pi + arg(I*(2*k1*pi + arg(I*(2*n*pi + arg(-c/d)) + log(Abs(c/d)))) + \
log(Abs(I*(2*n*pi + arg(-c/d)) + log(Abs(c/d)))))) + log(Abs(I*(2*k1*pi + arg(I*(2*n*pi + arg(-c/d)) + \
log(Abs(c/d)))) + log(Abs(I*(2*n*pi + arg(-c/d)) + log(Abs(c/d)))))))) + log(Abs(I*(2*k2*pi + \
arg(I*(2*k1*pi + arg(I*(2*n*pi + arg(-c/d)) + log(Abs(c/d)))) + log(Abs(I*(2*n*pi + arg(-c/d)) + log(Abs(c/d)))))) + \
log(Abs(I*(2*k1*pi + arg(I*(2*n*pi + arg(-c/d)) + log(Abs(c/d)))) + log(Abs(I*(2*n*pi + arg(-c/d)) + log(Abs(c/d))))))))), \
ProductSet(S.Integers, S.Integers, S.Integers, S.Integers))))
def test__solveset_multi():
from sympy.solvers.solveset import _solveset_multi
from sympy.sets import Reals
# Basic univariate case:
assert _solveset_multi([x**2-1], [x], [S.Reals]) == FiniteSet((1,), (-1,))
# Linear systems of two equations
assert _solveset_multi([x+y, x+1], [x, y], [Reals, Reals]) == FiniteSet((-1, 1))
assert _solveset_multi([x+y, x+1], [y, x], [Reals, Reals]) == FiniteSet((1, -1))
assert _solveset_multi([x+y, x-y-1], [x, y], [Reals, Reals]) == FiniteSet((S(1)/2, -S(1)/2))
assert _solveset_multi([x-1, y-2], [x, y], [Reals, Reals]) == FiniteSet((1, 2))
# assert dumeq(_solveset_multi([x+y], [x, y], [Reals, Reals]), ImageSet(Lambda(x, (x, -x)), Reals))
assert dumeq(_solveset_multi([x+y], [x, y], [Reals, Reals]), Union(
ImageSet(Lambda(((x,),), (x, -x)), ProductSet(Reals)),
ImageSet(Lambda(((y,),), (-y, y)), ProductSet(Reals))))
assert _solveset_multi([x+y, x+y+1], [x, y], [Reals, Reals]) == S.EmptySet
assert _solveset_multi([x+y, x-y, x-1], [x, y], [Reals, Reals]) == S.EmptySet
assert _solveset_multi([x+y, x-y, x-1], [y, x], [Reals, Reals]) == S.EmptySet
# Systems of three equations:
assert _solveset_multi([x+y+z-1, x+y-z-2, x-y-z-3], [x, y, z], [Reals,
Reals, Reals]) == FiniteSet((2, -S.Half, -S.Half))
# Nonlinear systems:
from sympy.abc import theta
assert _solveset_multi([x**2+y**2-2, x+y], [x, y], [Reals, Reals]) == FiniteSet((-1, 1), (1, -1))
assert _solveset_multi([x**2-1, y], [x, y], [Reals, Reals]) == FiniteSet((1, 0), (-1, 0))
#assert _solveset_multi([x**2-y**2], [x, y], [Reals, Reals]) == Union(
# ImageSet(Lambda(x, (x, -x)), Reals), ImageSet(Lambda(x, (x, x)), Reals))
assert dumeq(_solveset_multi([x**2-y**2], [x, y], [Reals, Reals]), Union(
ImageSet(Lambda(((x,),), (x, -Abs(x))), ProductSet(Reals)),
ImageSet(Lambda(((x,),), (x, Abs(x))), ProductSet(Reals)),
ImageSet(Lambda(((y,),), (-Abs(y), y)), ProductSet(Reals)),
ImageSet(Lambda(((y,),), (Abs(y), y)), ProductSet(Reals))))
assert _solveset_multi([r*cos(theta)-1, r*sin(theta)], [theta, r],
[Interval(0, pi), Interval(-1, 1)]) == FiniteSet((0, 1), (pi, -1))
assert _solveset_multi([r*cos(theta)-1, r*sin(theta)], [r, theta],
[Interval(0, 1), Interval(0, pi)]) == FiniteSet((1, 0))
#assert _solveset_multi([r*cos(theta)-r, r*sin(theta)], [r, theta],
# [Interval(0, 1), Interval(0, pi)]) == ?
assert dumeq(_solveset_multi([r*cos(theta)-r, r*sin(theta)], [r, theta],
[Interval(0, 1), Interval(0, pi)]), Union(
ImageSet(Lambda(((r,),), (r, 0)), ImageSet(Lambda(r, (r,)), Interval(0, 1))),
ImageSet(Lambda(((theta,),), (0, theta)), ImageSet(Lambda(theta, (theta,)), Interval(0, pi)))))
def test_conditionset():
assert solveset(Eq(sin(x)**2 + cos(x)**2, 1), x, domain=S.Reals
) is S.Reals
assert solveset(Eq(x**2 + x*sin(x), 1), x, domain=S.Reals
).dummy_eq(ConditionSet(x, Eq(x**2 + x*sin(x) - 1, 0), S.Reals))
assert dumeq(solveset(Eq(-I*(exp(I*x) - exp(-I*x))/2, 1), x
), imageset(Lambda(n, 2*n*pi + pi/2), S.Integers))
assert solveset(x + sin(x) > 1, x, domain=S.Reals
).dummy_eq(ConditionSet(x, x + sin(x) > 1, S.Reals))
assert solveset(Eq(sin(Abs(x)), x), x, domain=S.Reals
).dummy_eq(ConditionSet(x, Eq(-x + sin(Abs(x)), 0), S.Reals))
assert solveset(y**x-z, x, S.Reals
).dummy_eq(ConditionSet(x, Eq(y**x - z, 0), S.Reals))
@XFAIL
def test_conditionset_equality():
''' Checking equality of different representations of ConditionSet'''
assert solveset(Eq(tan(x), y), x) == ConditionSet(x, Eq(tan(x), y), S.Complexes)
def test_solveset_domain():
assert solveset(x**2 - x - 6, x, Interval(0, oo)) == FiniteSet(3)
assert solveset(x**2 - 1, x, Interval(0, oo)) == FiniteSet(1)
assert solveset(x**4 - 16, x, Interval(0, 10)) == FiniteSet(2)
def test_improve_coverage():
solution = solveset(exp(x) + sin(x), x, S.Reals)
unsolved_object = ConditionSet(x, Eq(exp(x) + sin(x), 0), S.Reals)
assert solution.dummy_eq(unsolved_object)
def test_issue_9522():
expr1 = Eq(1/(x**2 - 4) + x, 1/(x**2 - 4) + 2)
expr2 = Eq(1/x + x, 1/x)
assert solveset(expr1, x, S.Reals) is S.EmptySet
assert solveset(expr2, x, S.Reals) is S.EmptySet
def test_solvify():
assert solvify(x**2 + 10, x, S.Reals) == []
assert solvify(x**3 + 1, x, S.Complexes) == [-1, S.Half - sqrt(3)*I/2,
S.Half + sqrt(3)*I/2]
assert solvify(log(x), x, S.Reals) == [1]
assert solvify(cos(x), x, S.Reals) == [pi/2, pi*Rational(3, 2)]
assert solvify(sin(x) + 1, x, S.Reals) == [pi*Rational(3, 2)]
raises(NotImplementedError, lambda: solvify(sin(exp(x)), x, S.Complexes))
def test_solvify_piecewise():
p1 = Piecewise((0, x < -1), (x**2, x <= 1), (log(x), True))
p2 = Piecewise((0, x < -10), (x**2 + 5*x - 6, x >= -9))
p3 = Piecewise((0, Eq(x, 0)), (x**2/Abs(x), True))
p4 = Piecewise((0, Eq(x, pi)), ((x - pi)/sin(x), True))
# issue 21079
assert solvify(p1, x, S.Reals) == [0]
assert solvify(p2, x, S.Reals) == [-6, 1]
assert solvify(p3, x, S.Reals) == [0]
assert solvify(p4, x, S.Reals) == [pi]
def test_abs_invert_solvify():
x = Symbol('x',positive=True)
assert solvify(sin(Abs(x)), x, S.Reals) == [0, pi]
x = Symbol('x')
assert solvify(sin(Abs(x)), x, S.Reals) is None
def test_linear_eq_to_matrix():
assert linear_eq_to_matrix(0, x) == (Matrix([[0]]), Matrix([[0]]))
assert linear_eq_to_matrix(1, x) == (Matrix([[0]]), Matrix([[-1]]))
# integer coefficients
eqns1 = [2*x + y - 2*z - 3, x - y - z, x + y + 3*z - 12]
eqns2 = [Eq(3*x + 2*y - z, 1), Eq(2*x - 2*y + 4*z, -2), -2*x + y - 2*z]
A, B = linear_eq_to_matrix(eqns1, x, y, z)
assert A == Matrix([[2, 1, -2], [1, -1, -1], [1, 1, 3]])
assert B == Matrix([[3], [0], [12]])
A, B = linear_eq_to_matrix(eqns2, x, y, z)
assert A == Matrix([[3, 2, -1], [2, -2, 4], [-2, 1, -2]])
assert B == Matrix([[1], [-2], [0]])
# Pure symbolic coefficients
eqns3 = [a*b*x + b*y + c*z - d, e*x + d*x + f*y + g*z - h, i*x + j*y + k*z - l]
A, B = linear_eq_to_matrix(eqns3, x, y, z)
assert A == Matrix([[a*b, b, c], [d + e, f, g], [i, j, k]])
assert B == Matrix([[d], [h], [l]])
# raise Errors if
# 1) no symbols are given
raises(ValueError, lambda: linear_eq_to_matrix(eqns3))
# 2) there are duplicates
raises(ValueError, lambda: linear_eq_to_matrix(eqns3, [x, x, y]))
# 3) a nonlinear term is detected in the original expression
raises(NonlinearError, lambda: linear_eq_to_matrix(Eq(1/x + x, 1/x), [x]))
raises(NonlinearError, lambda: linear_eq_to_matrix([x**2], [x]))
raises(NonlinearError, lambda: linear_eq_to_matrix([x*y], [x, y]))
# 4) Eq being used to represent equations autoevaluates
# (use unevaluated Eq instead)
raises(ValueError, lambda: linear_eq_to_matrix(Eq(x, x), x))
raises(ValueError, lambda: linear_eq_to_matrix(Eq(x, x + 1), x))
# if non-symbols are passed, the user is responsible for interpreting
assert linear_eq_to_matrix([x], [1/x]) == (Matrix([[0]]), Matrix([[-x]]))
# issue 15195
assert linear_eq_to_matrix(x + y*(z*(3*x + 2) + 3), x) == (
Matrix([[3*y*z + 1]]), Matrix([[-y*(2*z + 3)]]))
assert linear_eq_to_matrix(Matrix(
[[a*x + b*y - 7], [5*x + 6*y - c]]), x, y) == (
Matrix([[a, b], [5, 6]]), Matrix([[7], [c]]))
# issue 15312
assert linear_eq_to_matrix(Eq(x + 2, 1), x) == (
Matrix([[1]]), Matrix([[-1]]))
def test_issue_16577():
assert linear_eq_to_matrix(Eq(a*(2*x + 3*y) + 4*y, 5), x, y) == (
Matrix([[2*a, 3*a + 4]]), Matrix([[5]]))
def test_issue_10085():
assert invert_real(exp(x),0,x) == (x, S.EmptySet)
def test_linsolve():
x1, x2, x3, x4 = symbols('x1, x2, x3, x4')
# Test for different input forms
M = Matrix([[1, 2, 1, 1, 7], [1, 2, 2, -1, 12], [2, 4, 0, 6, 4]])
system1 = A, B = M[:, :-1], M[:, -1]
Eqns = [x1 + 2*x2 + x3 + x4 - 7, x1 + 2*x2 + 2*x3 - x4 - 12,
2*x1 + 4*x2 + 6*x4 - 4]
sol = FiniteSet((-2*x2 - 3*x4 + 2, x2, 2*x4 + 5, x4))
assert linsolve(Eqns, (x1, x2, x3, x4)) == sol
assert linsolve(Eqns, *(x1, x2, x3, x4)) == sol
assert linsolve(system1, (x1, x2, x3, x4)) == sol
assert linsolve(system1, *(x1, x2, x3, x4)) == sol
# issue 9667 - symbols can be Dummy symbols
x1, x2, x3, x4 = symbols('x:4', cls=Dummy)
assert linsolve(system1, x1, x2, x3, x4) == FiniteSet(
(-2*x2 - 3*x4 + 2, x2, 2*x4 + 5, x4))
# raise ValueError for garbage value
raises(ValueError, lambda: linsolve(Eqns))
raises(ValueError, lambda: linsolve(x1))
raises(ValueError, lambda: linsolve(x1, x2))
raises(ValueError, lambda: linsolve((A,), x1, x2))
raises(ValueError, lambda: linsolve(A, B, x1, x2))
raises(ValueError, lambda: linsolve([x1], x1, x1))
raises(ValueError, lambda: linsolve([x1], (i for i in (x1, x1))))
#raise ValueError if equations are non-linear in given variables
raises(NonlinearError, lambda: linsolve([x + y - 1, x ** 2 + y - 3], [x, y]))
raises(NonlinearError, lambda: linsolve([cos(x) + y, x + y], [x, y]))
assert linsolve([x + z - 1, x ** 2 + y - 3], [z, y]) == {(-x + 1, -x**2 + 3)}
# Fully symbolic test
A = Matrix([[a, b], [c, d]])
B = Matrix([[e], [g]])
system2 = (A, B)
sol = FiniteSet(((-b*g + d*e)/(a*d - b*c), (a*g - c*e)/(a*d - b*c)))
assert linsolve(system2, [x, y]) == sol
# No solution
A = Matrix([[1, 2, 3], [2, 4, 6], [3, 6, 9]])
B = Matrix([0, 0, 1])
assert linsolve((A, B), (x, y, z)) is S.EmptySet
# Issue #10056
A, B, J1, J2 = symbols('A B J1 J2')
Augmatrix = Matrix([
[2*I*J1, 2*I*J2, -2/J1],
[-2*I*J2, -2*I*J1, 2/J2],
[0, 2, 2*I/(J1*J2)],
[2, 0, 0],
])
assert linsolve(Augmatrix, A, B) == FiniteSet((0, I/(J1*J2)))
# Issue #10121 - Assignment of free variables
Augmatrix = Matrix([[0, 1, 0, 0, 0, 0], [0, 0, 0, 1, 0, 0]])
assert linsolve(Augmatrix, a, b, c, d, e) == FiniteSet((a, 0, c, 0, e))
#raises(IndexError, lambda: linsolve(Augmatrix, a, b, c))
x0, x1, x2, _x0 = symbols('tau0 tau1 tau2 _tau0')
assert linsolve(Matrix([[0, 1, 0, 0, 0, 0], [0, 0, 0, 1, 0, _x0]])
) == FiniteSet((x0, 0, x1, _x0, x2))
x0, x1, x2, _x0 = symbols('tau00 tau01 tau02 tau0')
assert linsolve(Matrix([[0, 1, 0, 0, 0, 0], [0, 0, 0, 1, 0, _x0]])
) == FiniteSet((x0, 0, x1, _x0, x2))
x0, x1, x2, _x0 = symbols('tau00 tau01 tau02 tau1')
assert linsolve(Matrix([[0, 1, 0, 0, 0, 0], [0, 0, 0, 1, 0, _x0]])
) == FiniteSet((x0, 0, x1, _x0, x2))
# symbols can be given as generators
x0, x2, x4 = symbols('x0, x2, x4')
assert linsolve(Augmatrix, numbered_symbols('x')
) == FiniteSet((x0, 0, x2, 0, x4))
Augmatrix[-1, -1] = x0
# use Dummy to avoid clash; the names may clash but the symbols
# will not
Augmatrix[-1, -1] = symbols('_x0')
assert len(linsolve(
Augmatrix, numbered_symbols('x', cls=Dummy)).free_symbols) == 4
# Issue #12604
f = Function('f')
assert linsolve([f(x) - 5], f(x)) == FiniteSet((5,))
# Issue #14860
from sympy.physics.units import meter, newton, kilo
kN = kilo*newton
Eqns = [8*kN + x + y, 28*kN*meter + 3*x*meter]
assert linsolve(Eqns, x, y) == {
(kilo*newton*Rational(-28, 3), kN*Rational(4, 3))}
# linsolve does not allow expansion (real or implemented)
# to remove singularities, but it will cancel linear terms
assert linsolve([Eq(x, x + y)], [x, y]) == {(x, 0)}
assert linsolve([Eq(x + x*y, 1 + y)], [x]) == {(1,)}
assert linsolve([Eq(1 + y, x + x*y)], [x]) == {(1,)}
raises(NonlinearError, lambda:
linsolve([Eq(x**2, x**2 + y)], [x, y]))
# corner cases
#
# XXX: The case below should give the same as for [0]
# assert linsolve([], [x]) == {(x,)}
assert linsolve([], [x]) is S.EmptySet
assert linsolve([0], [x]) == {(x,)}
assert linsolve([x], [x, y]) == {(0, y)}
assert linsolve([x, 0], [x, y]) == {(0, y)}
def test_linsolve_large_sparse():
#
# This is mainly a performance test
#
def _mk_eqs_sol(n):
xs = symbols('x:{}'.format(n))
ys = symbols('y:{}'.format(n))
syms = xs + ys
eqs = []
sol = (-S.Half,) * n + (S.Half,) * n
for xi, yi in zip(xs, ys):
eqs.extend([xi + yi, xi - yi + 1])
return eqs, syms, FiniteSet(sol)
n = 500
eqs, syms, sol = _mk_eqs_sol(n)
assert linsolve(eqs, syms) == sol
def test_linsolve_immutable():
A = ImmutableDenseMatrix([[1, 1, 2], [0, 1, 2], [0, 0, 1]])
B = ImmutableDenseMatrix([2, 1, -1])
assert linsolve([A, B], (x, y, z)) == FiniteSet((1, 3, -1))
A = ImmutableDenseMatrix([[1, 1, 7], [1, -1, 3]])
assert linsolve(A) == FiniteSet((5, 2))
def test_solve_decomposition():
n = Dummy('n')
f1 = exp(3*x) - 6*exp(2*x) + 11*exp(x) - 6
f2 = sin(x)**2 - 2*sin(x) + 1
f3 = sin(x)**2 - sin(x)
f4 = sin(x + 1)
f5 = exp(x + 2) - 1
f6 = 1/log(x)
f7 = 1/x
s1 = ImageSet(Lambda(n, 2*n*pi), S.Integers)
s2 = ImageSet(Lambda(n, 2*n*pi + pi), S.Integers)
s3 = ImageSet(Lambda(n, 2*n*pi + pi/2), S.Integers)
s4 = ImageSet(Lambda(n, 2*n*pi - 1), S.Integers)
s5 = ImageSet(Lambda(n, 2*n*pi - 1 + pi), S.Integers)
assert solve_decomposition(f1, x, S.Reals) == FiniteSet(0, log(2), log(3))
assert dumeq(solve_decomposition(f2, x, S.Reals), s3)
assert dumeq(solve_decomposition(f3, x, S.Reals), Union(s1, s2, s3))
assert dumeq(solve_decomposition(f4, x, S.Reals), Union(s4, s5))
assert solve_decomposition(f5, x, S.Reals) == FiniteSet(-2)
assert solve_decomposition(f6, x, S.Reals) == S.EmptySet
assert solve_decomposition(f7, x, S.Reals) == S.EmptySet
assert solve_decomposition(x, x, Interval(1, 2)) == S.EmptySet
# nonlinsolve testcases
def test_nonlinsolve_basic():
assert nonlinsolve([],[]) == S.EmptySet
assert nonlinsolve([],[x, y]) == S.EmptySet
system = [x, y - x - 5]
assert nonlinsolve([x],[x, y]) == FiniteSet((0, y))
assert nonlinsolve(system, [y]) == S.EmptySet
soln = (ImageSet(Lambda(n, 2*n*pi + pi/2), S.Integers),)
assert dumeq(nonlinsolve([sin(x) - 1], [x]), FiniteSet(tuple(soln)))
soln = ((ImageSet(Lambda(n, 2*n*pi + pi), S.Integers), FiniteSet(1)),
(ImageSet(Lambda(n, 2*n*pi), S.Integers), FiniteSet(1,)))
assert dumeq(nonlinsolve([sin(x), y - 1], [x, y]), FiniteSet(*soln))
assert nonlinsolve([x**2 - 1], [x]) == FiniteSet((-1,), (1,))
soln = FiniteSet((y, y))
assert nonlinsolve([x - y, 0], x, y) == soln
assert nonlinsolve([0, x - y], x, y) == soln
assert nonlinsolve([x - y, x - y], x, y) == soln
assert nonlinsolve([x, 0], x, y) == FiniteSet((0, y))
f = Function('f')
assert nonlinsolve([f(x), 0], f(x), y) == FiniteSet((0, y))
assert nonlinsolve([f(x), 0], f(x), f(y)) == FiniteSet((0, f(y)))
A = Indexed('A', x)
assert nonlinsolve([A, 0], A, y) == FiniteSet((0, y))
assert nonlinsolve([x**2 -1], [sin(x)]) == FiniteSet((S.EmptySet,))
assert nonlinsolve([x**2 -1], sin(x)) == FiniteSet((S.EmptySet,))
assert nonlinsolve([x**2 -1], 1) == FiniteSet((x**2,))
assert nonlinsolve([x**2 -1], x + y) == FiniteSet((S.EmptySet,))
assert nonlinsolve([Eq(1, x + y), Eq(1, -x + y - 1), Eq(1, -x + y - 1)], x, y) == FiniteSet(
(-S.Half, 3*S.Half))
def test_nonlinsolve_abs():
soln = FiniteSet((y, y), (-y, y))
assert nonlinsolve([Abs(x) - y], x, y) == soln
def test_raise_exception_nonlinsolve():
raises(IndexError, lambda: nonlinsolve([x**2 -1], []))
raises(ValueError, lambda: nonlinsolve([x**2 -1]))
def test_trig_system():
# TODO: add more simple testcases when solveset returns
# simplified soln for Trig eq
assert nonlinsolve([sin(x) - 1, cos(x) -1 ], x) == S.EmptySet
soln1 = (ImageSet(Lambda(n, 2*n*pi + pi/2), S.Integers),)
soln = FiniteSet(soln1)
assert dumeq(nonlinsolve([sin(x) - 1, cos(x)], x), soln)
@XFAIL
def test_trig_system_fail():
# fails because solveset trig solver is not much smart.
sys = [x + y - pi/2, sin(x) + sin(y) - 1]
# solveset returns conditionset for sin(x) + sin(y) - 1
soln_1 = (ImageSet(Lambda(n, n*pi + pi/2), S.Integers),
ImageSet(Lambda(n, n*pi), S.Integers))
soln_1 = FiniteSet(soln_1)
soln_2 = (ImageSet(Lambda(n, n*pi), S.Integers),
ImageSet(Lambda(n, n*pi+ pi/2), S.Integers))
soln_2 = FiniteSet(soln_2)
soln = soln_1 + soln_2
assert dumeq(nonlinsolve(sys, [x, y]), soln)
# Add more cases from here
# http://www.vitutor.com/geometry/trigonometry/equations_systems.html#uno
sys = [sin(x) + sin(y) - (sqrt(3)+1)/2, sin(x) - sin(y) - (sqrt(3) - 1)/2]
soln_x = Union(ImageSet(Lambda(n, 2*n*pi + pi/3), S.Integers),
ImageSet(Lambda(n, 2*n*pi + pi*Rational(2, 3)), S.Integers))
soln_y = Union(ImageSet(Lambda(n, 2*n*pi + pi/6), S.Integers),
ImageSet(Lambda(n, 2*n*pi + pi*Rational(5, 6)), S.Integers))
assert dumeq(nonlinsolve(sys, [x, y]), FiniteSet((soln_x, soln_y)))
def test_nonlinsolve_positive_dimensional():
x, y, a, b, c, d = symbols('x, y, a, b, c, d', extended_real=True)
assert nonlinsolve([x*y, x*y - x], [x, y]) == FiniteSet((0, y))
system = [a**2 + a*c, a - b]
assert nonlinsolve(system, [a, b]) == FiniteSet((0, 0), (-c, -c))
# here (a= 0, b = 0) is independent soln so both is printed.
# if symbols = [a, b, c] then only {a : -c ,b : -c}
eq1 = a + b + c + d
eq2 = a*b + b*c + c*d + d*a
eq3 = a*b*c + b*c*d + c*d*a + d*a*b
eq4 = a*b*c*d - 1
system = [eq1, eq2, eq3, eq4]
sol1 = (-1/d, -d, 1/d, FiniteSet(d) - FiniteSet(0))
sol2 = (1/d, -d, -1/d, FiniteSet(d) - FiniteSet(0))
soln = FiniteSet(sol1, sol2)
assert nonlinsolve(system, [a, b, c, d]) == soln
assert nonlinsolve([x**4 - 3*x**2 + y*x, x*z**2, y*z - 1], [x, y, z]) == \
{(0, 1/z, z)}
def test_nonlinsolve_polysys():
x, y, z = symbols('x, y, z', real=True)
assert nonlinsolve([x**2 + y - 2, x**2 + y], [x, y]) == S.EmptySet
s = (-y + 2, y)
assert nonlinsolve([(x + y)**2 - 4, x + y - 2], [x, y]) == FiniteSet(s)
system = [x**2 - y**2]
soln_real = FiniteSet((-y, y), (y, y))
soln_complex = FiniteSet((-Abs(y), y), (Abs(y), y))
soln =soln_real + soln_complex
assert nonlinsolve(system, [x, y]) == soln
system = [x**2 - y**2]
soln_real= FiniteSet((y, -y), (y, y))
soln_complex = FiniteSet((y, -Abs(y)), (y, Abs(y)))
soln = soln_real + soln_complex
assert nonlinsolve(system, [y, x]) == soln
system = [x**2 + y - 3, x - y - 4]
assert nonlinsolve(system, (x, y)) != nonlinsolve(system, (y, x))
assert nonlinsolve([-x**2 - y**2 + z, -2*x, -2*y, S.One], [x, y, z]) == S.EmptySet
assert nonlinsolve([x + y + z, S.One, S.One, S.One], [x, y, z]) == S.EmptySet
system = [-x**2*z**2 + x*y*z + y**4, -2*x*z**2 + y*z, x*z + 4*y**3, -2*x**2*z + x*y]
assert nonlinsolve(system, [x, y, z]) == FiniteSet((0, 0, z), (x, 0, 0))
def test_nonlinsolve_using_substitution():
x, y, z, n = symbols('x, y, z, n', real = True)
system = [(x + y)*n - y**2 + 2]
s_x = (n*y - y**2 + 2)/n
soln = (-s_x, y)
assert nonlinsolve(system, [x, y]) == FiniteSet(soln)
system = [z**2*x**2 - z**2*y**2/exp(x)]
soln_real_1 = (y, x, 0)
soln_real_2 = (-exp(x/2)*Abs(x), x, z)
soln_real_3 = (exp(x/2)*Abs(x), x, z)
soln_complex_1 = (-x*exp(x/2), x, z)
soln_complex_2 = (x*exp(x/2), x, z)
syms = [y, x, z]
soln = FiniteSet(soln_real_1, soln_complex_1, soln_complex_2,\
soln_real_2, soln_real_3)
assert nonlinsolve(system,syms) == soln
def test_nonlinsolve_complex():
n = Dummy('n')
assert dumeq(nonlinsolve([exp(x) - sin(y), 1/y - 3], [x, y]), {
(ImageSet(Lambda(n, 2*n*I*pi + log(sin(Rational(1, 3)))), S.Integers), Rational(1, 3))})
system = [exp(x) - sin(y), 1/exp(y) - 3]
assert dumeq(nonlinsolve(system, [x, y]), {
(ImageSet(Lambda(n, I*(2*n*pi + pi)
+ log(sin(log(3)))), S.Integers), -log(3)),
(ImageSet(Lambda(n, I*(2*n*pi + arg(sin(2*n*I*pi - log(3))))
+ log(Abs(sin(2*n*I*pi - log(3))))), S.Integers),
ImageSet(Lambda(n, 2*n*I*pi - log(3)), S.Integers))})
system = [exp(x) - sin(y), y**2 - 4]
assert dumeq(nonlinsolve(system, [x, y]), {
(ImageSet(Lambda(n, I*(2*n*pi + pi) + log(sin(2))), S.Integers), -2),
(ImageSet(Lambda(n, 2*n*I*pi + log(sin(2))), S.Integers), 2)})
system = [exp(x) - 2, y ** 2 - 2]
assert dumeq(nonlinsolve(system, [x, y]), {
(log(2), -sqrt(2)), (log(2), sqrt(2)),
(ImageSet(Lambda(n, 2*n*I*pi + log(2)), S.Integers), FiniteSet(-sqrt(2))),
(ImageSet(Lambda(n, 2 * n * I * pi + log(2)), S.Integers), FiniteSet(sqrt(2)))})
def test_nonlinsolve_radical():
assert nonlinsolve([sqrt(y) - x - z, y - 1], [x, y, z]) == {(1 - z, 1, z)}
def test_nonlinsolve_inexact():
sol = [(-1.625, -1.375), (1.625, 1.375)]
res = nonlinsolve([(x + y)**2 - 9, x**2 - y**2 - 0.75], [x, y])
assert all(abs(res.args[i][j]-sol[i][j]) < 1e-9
for i in range(2) for j in range(2))
assert nonlinsolve([(x + y)**2 - 9, (x + y)**2 - 0.75], [x, y]) == S.EmptySet
assert nonlinsolve([y**2 + (x - 0.5)**2 - 0.0625, 2*x - 1.0, 2*y], [x, y]) == \
S.EmptySet
res = nonlinsolve([x**2 + y - 0.5, (x + y)**2, log(z)], [x, y, z])
sol = [(-0.366025403784439, 0.366025403784439, 1),
(-0.366025403784439, 0.366025403784439, 1),
(1.36602540378444, -1.36602540378444, 1)]
assert all(abs(res.args[i][j]-sol[i][j]) < 1e-9
for i in range(3) for j in range(3))
res = nonlinsolve([y - x**2, x**5 - x + 1.0], [x, y])
sol = [(-1.16730397826142, 1.36259857766493),
(-0.181232444469876 - 1.08395410131771*I,
-1.14211129483496 + 0.392895302949911*I),
(-0.181232444469876 + 1.08395410131771*I,
-1.14211129483496 - 0.392895302949911*I),
(0.764884433600585 - 0.352471546031726*I,
0.460812006002492 - 0.539199997693599*I),
(0.764884433600585 + 0.352471546031726*I,
0.460812006002492 + 0.539199997693599*I)]
assert all(abs(res.args[i][j] - sol[i][j]) < 1e-9
for i in range(5) for j in range(2))
@XFAIL
def test_solve_nonlinear_trans():
# After the transcendental equation solver these will work
x, y = symbols('x, y', real=True)
soln1 = FiniteSet((2*LambertW(y/2), y))
soln2 = FiniteSet((-x*sqrt(exp(x)), y), (x*sqrt(exp(x)), y))
soln3 = FiniteSet((x*exp(x/2), x))
soln4 = FiniteSet(2*LambertW(y/2), y)
assert nonlinsolve([x**2 - y**2/exp(x)], [x, y]) == soln1
assert nonlinsolve([x**2 - y**2/exp(x)], [y, x]) == soln2
assert nonlinsolve([x**2 - y**2/exp(x)], [y, x]) == soln3
assert nonlinsolve([x**2 - y**2/exp(x)], [x, y]) == soln4
def test_issue_14642():
x = Symbol('x')
n1 = 0.5*x**3+x**2+0.5+I #add I in the Polynomials
solution = solveset(n1, x)
assert abs(solution.args[0] - (-2.28267560928153 - 0.312325580497716*I)) <= 1e-9
assert abs(solution.args[1] - (-0.297354141679308 + 1.01904778618762*I)) <= 1e-9
assert abs(solution.args[2] - (0.580029750960839 - 0.706722205689907*I)) <= 1e-9
# Symbolic
n1 = S.Half*x**3+x**2+S.Half+I
res = FiniteSet(-((3*sqrt(3)*31985**(S(1)/4)*sin(atan(S(172)/49)/2)/2 +
S(43)/2)**2 + (27 + 3*sqrt(3)*31985**(S(1)/4)*cos(atan(S(172)/49)
/2)/2)**2)**(S(1)/6)*cos(atan((27 + 3*sqrt(3)*31985**(S(1)/4)*
cos(atan(S(172)/49)/2)/2)/(3*sqrt(3)*31985**(S(1)/4)*sin(atan(
S(172)/49)/2)/2 + S(43)/2))/3)/3 - S(2)/3 - 4*cos(atan((27 +
3*sqrt(3)*31985**(S(1)/4)*cos(atan(S(172)/49)/2)/2)/(3*sqrt(3)*
31985**(S(1)/4)*sin(atan(S(172)/49)/2)/2 + S(43)/2))/3)/(3*((3*
sqrt(3)*31985**(S(1)/4)*sin(atan(S(172)/49)/2)/2 + S(43)/2)**2 +
(27 + 3*sqrt(3)*31985**(S(1)/4)*cos(atan(S(172)/49)/2)/2)**2)**(S(1)/
6)) + I*(-((3*sqrt(3)*31985**(S(1)/4)*sin(atan(S(172)/49)/2)/2 +
S(43)/2)**2 + (27 + 3*sqrt(3)*31985**(S(1)/4)*cos(atan(S(172)/49)/
2)/2)**2)**(S(1)/6)*sin(atan((27 + 3*sqrt(3)*31985**(S(1)/4)*cos(
atan(S(172)/49)/2)/2)/(3*sqrt(3)*31985**(S(1)/4)*sin(atan(S(172)/49)
/2)/2 + S(43)/2))/3)/3 + 4*sin(atan((27 + 3*sqrt(3)*31985**(S(1)/4)*
cos(atan(S(172)/49)/2)/2)/(3*sqrt(3)*31985**(S(1)/4)*sin(atan(S(172)
/49)/2)/2 + S(43)/2))/3)/(3*((3*sqrt(3)*31985**(S(1)/4)*sin(atan(
S(172)/49)/2)/2 + S(43)/2)**2 + (27 + 3*sqrt(3)*31985**(S(1)/4)*
cos(atan(S(172)/49)/2)/2)**2)**(S(1)/6))), -S(2)/3 - sqrt(3)*((3*
sqrt(3)*31985**(S(1)/4)*sin(atan(S(172)/49)/2)/2 + S(43)/2)**2 +
(27 + 3*sqrt(3)*31985**(S(1)/4)*cos(atan(S(172)/49)/2)/2)**2)**(S(1)
/6)*sin(atan((27 + 3*sqrt(3)*31985**(S(1)/4)*cos(atan(S(172)/49)/2)
/2)/(3*sqrt(3)*31985**(S(1)/4)*sin(atan(S(172)/49)/2)/2 + S(43)/2))
/3)/6 - 4*re(1/((-S(1)/2 - sqrt(3)*I/2)*(S(43)/2 + 27*I + sqrt(-256 +
(43 + 54*I)**2)/2)**(S(1)/3)))/3 + ((3*sqrt(3)*31985**(S(1)/4)*sin(
atan(S(172)/49)/2)/2 + S(43)/2)**2 + (27 + 3*sqrt(3)*31985**(S(1)/4)*
cos(atan(S(172)/49)/2)/2)**2)**(S(1)/6)*cos(atan((27 + 3*sqrt(3)*
31985**(S(1)/4)*cos(atan(S(172)/49)/2)/2)/(3*sqrt(3)*31985**(S(1)/4)*
sin(atan(S(172)/49)/2)/2 + S(43)/2))/3)/6 + I*(-4*im(1/((-S(1)/2 -
sqrt(3)*I/2)*(S(43)/2 + 27*I + sqrt(-256 + (43 + 54*I)**2)/2)**(S(1)/
3)))/3 + ((3*sqrt(3)*31985**(S(1)/4)*sin(atan(S(172)/49)/2)/2 +
S(43)/2)**2 + (27 + 3*sqrt(3)*31985**(S(1)/4)*cos(atan(S(172)/49)/2)
/2)**2)**(S(1)/6)*sin(atan((27 + 3*sqrt(3)*31985**(S(1)/4)*cos(atan(
S(172)/49)/2)/2)/(3*sqrt(3)*31985**(S(1)/4)*sin(atan(S(172)/49)/2)/2 +
S(43)/2))/3)/6 + sqrt(3)*((3*sqrt(3)*31985**(S(1)/4)*sin(atan(S(172)/
49)/2)/2 + S(43)/2)**2 + (27 + 3*sqrt(3)*31985**(S(1)/4)*cos(atan(
S(172)/49)/2)/2)**2)**(S(1)/6)*cos(atan((27 + 3*sqrt(3)*31985**(S(1)/
4)*cos(atan(S(172)/49)/2)/2)/(3*sqrt(3)*31985**(S(1)/4)*sin(atan(
S(172)/49)/2)/2 + S(43)/2))/3)/6), -S(2)/3 - 4*re(1/((-S(1)/2 +
sqrt(3)*I/2)*(S(43)/2 + 27*I + sqrt(-256 + (43 + 54*I)**2)/2)**(S(1)
/3)))/3 + sqrt(3)*((3*sqrt(3)*31985**(S(1)/4)*sin(atan(S(172)/49)/2)/2 +
S(43)/2)**2 + (27 + 3*sqrt(3)*31985**(S(1)/4)*cos(atan(S(172)/49)/2)
/2)**2)**(S(1)/6)*sin(atan((27 + 3*sqrt(3)*31985**(S(1)/4)*cos(atan(
S(172)/49)/2)/2)/(3*sqrt(3)*31985**(S(1)/4)*sin(atan(S(172)/49)/2)/2 +
S(43)/2))/3)/6 + ((3*sqrt(3)*31985**(S(1)/4)*sin(atan(S(172)/49)/2)/2 +
S(43)/2)**2 + (27 + 3*sqrt(3)*31985**(S(1)/4)*cos(atan(S(172)/49)/2)
/2)**2)**(S(1)/6)*cos(atan((27 + 3*sqrt(3)*31985**(S(1)/4)*cos(atan(
S(172)/49)/2)/2)/(3*sqrt(3)*31985**(S(1)/4)*sin(atan(S(172)/49)/2)/2 +
S(43)/2))/3)/6 + I*(-sqrt(3)*((3*sqrt(3)*31985**(S(1)/4)*sin(atan(
S(172)/49)/2)/2 + S(43)/2)**2 + (27 + 3*sqrt(3)*31985**(S(1)/4)*cos(
atan(S(172)/49)/2)/2)**2)**(S(1)/6)*cos(atan((27 + 3*sqrt(3)*31985**(
S(1)/4)*cos(atan(S(172)/49)/2)/2)/(3*sqrt(3)*31985**(S(1)/4)*sin(
atan(S(172)/49)/2)/2 + S(43)/2))/3)/6 + ((3*sqrt(3)*31985**(S(1)/4)*
sin(atan(S(172)/49)/2)/2 + S(43)/2)**2 + (27 + 3*sqrt(3)*31985**(S(1)/4)*
cos(atan(S(172)/49)/2)/2)**2)**(S(1)/6)*sin(atan((27 + 3*sqrt(3)*31985**(
S(1)/4)*cos(atan(S(172)/49)/2)/2)/(3*sqrt(3)*31985**(S(1)/4)*sin(
atan(S(172)/49)/2)/2 + S(43)/2))/3)/6 - 4*im(1/((-S(1)/2 + sqrt(3)*I/2)*
(S(43)/2 + 27*I + sqrt(-256 + (43 + 54*I)**2)/2)**(S(1)/3)))/3))
assert solveset(n1, x) == res
def test_issue_13961():
V = (ax, bx, cx, gx, jx, lx, mx, nx, q) = symbols('ax bx cx gx jx lx mx nx q')
S = (ax*q - lx*q - mx, ax - gx*q - lx, bx*q**2 + cx*q - jx*q - nx, q*(-ax*q + lx*q + mx), q*(-ax + gx*q + lx))
sol = FiniteSet((lx + mx/q, (-cx*q + jx*q + nx)/q**2, cx, mx/q**2, jx, lx, mx, nx, Complement({q}, {0})),
(lx + mx/q, (cx*q - jx*q - nx)/q**2*-1, cx, mx/q**2, jx, lx, mx, nx, Complement({q}, {0})))
assert nonlinsolve(S, *V) == sol
# The two solutions are in fact identical, so even better if only one is returned
def test_issue_14541():
solutions = solveset(sqrt(-x**2 - 2.0), x)
assert abs(solutions.args[0]+1.4142135623731*I) <= 1e-9
assert abs(solutions.args[1]-1.4142135623731*I) <= 1e-9
def test_issue_13396():
expr = -2*y*exp(-x**2 - y**2)*Abs(x)
sol = FiniteSet(0)
assert solveset(expr, y, domain=S.Reals) == sol
# Related type of equation also solved here
assert solveset(atan(x**2 - y**2)-pi/2, y, S.Reals) is S.EmptySet
def test_issue_12032():
sol = FiniteSet(-sqrt(-2/(3*(Rational(1, 16) + sqrt(849)/144)**(Rational(1, 3))) +
2*(Rational(1, 16) + sqrt(849)/144)**(Rational(1, 3)))/2 +
sqrt(Abs(-2*(Rational(1, 16) + sqrt(849)/144)**(Rational(1, 3)) +
2/(3*(Rational(1, 16) + sqrt(849)/144)**(Rational(1, 3))) +
2/sqrt(-2/(3*(Rational(1, 16) + sqrt(849)/144)**(Rational(1, 3))) +
2*(Rational(1, 16) + sqrt(849)/144)**(Rational(1, 3)))))/2,
-sqrt(Abs(-2*(Rational(1, 16) + sqrt(849)/144)**(Rational(1, 3)) +
2/(3*(Rational(1, 16) + sqrt(849)/144)**(Rational(1, 3))) +
2/sqrt(-2/(3*(Rational(1, 16) + sqrt(849)/144)**(Rational(1, 3))) +
2*(Rational(1, 16) + sqrt(849)/144)**(Rational(1, 3)))))/2 -
sqrt(-2/(3*(Rational(1, 16) + sqrt(849)/144)**(Rational(1, 3))) +
2*(Rational(1, 16) + sqrt(849)/144)**(Rational(1, 3)))/2,
sqrt(-2/(3*(Rational(1, 16) + sqrt(849)/144)**(Rational(1, 3))) +
2*(Rational(1, 16) + sqrt(849)/144)**(Rational(1, 3)))/2 -
I*sqrt(Abs(-2/sqrt(-2/(3*(Rational(1, 16) + sqrt(849)/144)**(Rational(1, 3))) +
2*(Rational(1, 16) + sqrt(849)/144)**(Rational(1, 3))) -
2*(Rational(1, 16) + sqrt(849)/144)**(Rational(1, 3)) +
2/(3*(Rational(1, 16) + sqrt(849)/144)**(Rational(1, 3)))))/2,
sqrt(-2/(3*(Rational(1, 16) + sqrt(849)/144)**(Rational(1, 3))) +
2*(Rational(1, 16) + sqrt(849)/144)**(Rational(1, 3)))/2 +
I*sqrt(Abs(-2/sqrt(-2/(3*(Rational(1, 16) + sqrt(849)/144)**(Rational(1, 3))) +
2*(Rational(1, 16) + sqrt(849)/144)**(Rational(1, 3))) -
2*(Rational(1, 16) + sqrt(849)/144)**(Rational(1, 3)) +
2/(3*(Rational(1, 16) + sqrt(849)/144)**(Rational(1,3)))))/2)
assert solveset(x**4 + x - 1, x) == sol
def test_issue_10876():
assert solveset(1/sqrt(x), x) == S.EmptySet
def test_issue_19050():
# test_issue_19050 --> TypeError removed
assert dumeq(nonlinsolve([x + y, sin(y)], [x, y]),
FiniteSet((ImageSet(Lambda(n, -2*n*pi), S.Integers), ImageSet(Lambda(n, 2*n*pi), S.Integers)),\
(ImageSet(Lambda(n, -2*n*pi - pi), S.Integers), ImageSet(Lambda(n, 2*n*pi + pi), S.Integers))))
assert dumeq(nonlinsolve([x + y, sin(y) + cos(y)], [x, y]),
FiniteSet((ImageSet(Lambda(n, -2*n*pi - 3*pi/4), S.Integers), ImageSet(Lambda(n, 2*n*pi + 3*pi/4), S.Integers)), \
(ImageSet(Lambda(n, -2*n*pi - 7*pi/4), S.Integers), ImageSet(Lambda(n, 2*n*pi + 7*pi/4), S.Integers))))
def test_issue_16618():
# AttributeError is removed !
eqn = [sin(x)*sin(y), cos(x)*cos(y) - 1]
ans = FiniteSet((x, 2*n*pi), (2*n*pi, y), (x, 2*n*pi + pi), (2*n*pi + pi, y))
sol = nonlinsolve(eqn, [x, y])
for i0, j0 in zip(ordered(sol), ordered(ans)):
assert len(i0) == len(j0) == 2
assert all(a.dummy_eq(b) for a, b in zip(i0, j0))
assert len(sol) == len(ans)
def test_issue_17566():
assert nonlinsolve([32*(2**x)/2**(-y) - 4**y, 27*(3**x) - S(1)/3**y], x, y) ==\
FiniteSet((-log(81)/log(3), 1))
def test_issue_16643():
n = Dummy('n')
assert solveset(x**2*sin(x), x).dummy_eq(Union(ImageSet(Lambda(n, 2*n*pi + pi), S.Integers),
ImageSet(Lambda(n, 2*n*pi), S.Integers)))
def test_issue_19587():
n,m = symbols('n m')
assert nonlinsolve([32*2**m*2**n - 4**n, 27*3**m - 3**(-n)], m, n) ==\
FiniteSet((-log(81)/log(3), 1))
def test_issue_5132_1():
system = [sqrt(x**2 + y**2) - sqrt(10), x + y - 4]
assert nonlinsolve(system, [x, y]) == FiniteSet((1, 3), (3, 1))
n = Dummy('n')
eqs = [exp(x)**2 - sin(y) + z**2, 1/exp(y) - 3]
s_real_y = -log(3)
s_real_z = sqrt(-exp(2*x) - sin(log(3)))
soln_real = FiniteSet((s_real_y, s_real_z), (s_real_y, -s_real_z))
lam = Lambda(n, 2*n*I*pi + -log(3))
s_complex_y = ImageSet(lam, S.Integers)
lam = Lambda(n, sqrt(-exp(2*x) + sin(2*n*I*pi + -log(3))))
s_complex_z_1 = ImageSet(lam, S.Integers)
lam = Lambda(n, -sqrt(-exp(2*x) + sin(2*n*I*pi + -log(3))))
s_complex_z_2 = ImageSet(lam, S.Integers)
soln_complex = FiniteSet(
(s_complex_y, s_complex_z_1),
(s_complex_y, s_complex_z_2)
)
soln = soln_real + soln_complex
assert dumeq(nonlinsolve(eqs, [y, z]), soln)
def test_issue_5132_2():
x, y = symbols('x, y', real=True)
eqs = [exp(x)**2 - sin(y) + z**2]
n = Dummy('n')
soln_real = (log(-z**2 + sin(y))/2, z)
lam = Lambda( n, I*(2*n*pi + arg(-z**2 + sin(y)))/2 + log(Abs(z**2 - sin(y)))/2)
img = ImageSet(lam, S.Integers)
# not sure about the complex soln. But it looks correct.
soln_complex = (img, z)
soln = FiniteSet(soln_real, soln_complex)
assert dumeq(nonlinsolve(eqs, [x, z]), soln)
system = [r - x**2 - y**2, tan(t) - y/x]
s_x = sqrt(r/(tan(t)**2 + 1))
s_y = sqrt(r/(tan(t)**2 + 1))*tan(t)
soln = FiniteSet((s_x, s_y), (-s_x, -s_y))
assert nonlinsolve(system, [x, y]) == soln
def test_issue_6752():
a, b = symbols('a, b', real=True)
assert nonlinsolve([a**2 + a, a - b], [a, b]) == {(-1, -1), (0, 0)}
@SKIP("slow")
def test_issue_5114_solveset():
# slow testcase
from sympy.abc import o, p
# there is no 'a' in the equation set but this is how the
# problem was originally posed
syms = [a, b, c, f, h, k, n]
eqs = [b + r/d - c/d,
c*(1/d + 1/e + 1/g) - f/g - r/d,
f*(1/g + 1/i + 1/j) - c/g - h/i,
h*(1/i + 1/l + 1/m) - f/i - k/m,
k*(1/m + 1/o + 1/p) - h/m - n/p,
n*(1/p + 1/q) - k/p]
assert len(nonlinsolve(eqs, syms)) == 1
@SKIP("Hangs")
def _test_issue_5335():
# Not able to check zero dimensional system.
# is_zero_dimensional Hangs
lam, a0, conc = symbols('lam a0 conc')
eqs = [lam + 2*y - a0*(1 - x/2)*x - 0.005*x/2*x,
a0*(1 - x/2)*x - 1*y - 0.743436700916726*y,
x + y - conc]
sym = [x, y, a0]
# there are 4 solutions but only two are valid
assert len(nonlinsolve(eqs, sym)) == 2
# float
eqs = [lam + 2*y - a0*(1 - x/2)*x - 0.005*x/2*x,
a0*(1 - x/2)*x - 1*y - 0.743436700916726*y,
x + y - conc]
sym = [x, y, a0]
assert len(nonlinsolve(eqs, sym)) == 2
def test_issue_2777():
# the equations represent two circles
x, y = symbols('x y', real=True)
e1, e2 = sqrt(x**2 + y**2) - 10, sqrt(y**2 + (-x + 10)**2) - 3
a, b = Rational(191, 20), 3*sqrt(391)/20
ans = {(a, -b), (a, b)}
assert nonlinsolve((e1, e2), (x, y)) == ans
assert nonlinsolve((e1, e2/(x - a)), (x, y)) == S.EmptySet
# make the 2nd circle's radius be -3
e2 += 6
assert nonlinsolve((e1, e2), (x, y)) == S.EmptySet
def test_issue_8828():
x1 = 0
y1 = -620
r1 = 920
x2 = 126
y2 = 276
x3 = 51
y3 = 205
r3 = 104
v = [x, y, z]
f1 = (x - x1)**2 + (y - y1)**2 - (r1 - z)**2
f2 = (x2 - x)**2 + (y2 - y)**2 - z**2
f3 = (x - x3)**2 + (y - y3)**2 - (r3 - z)**2
F = [f1, f2, f3]
g1 = sqrt((x - x1)**2 + (y - y1)**2) + z - r1
g2 = f2
g3 = sqrt((x - x3)**2 + (y - y3)**2) + z - r3
G = [g1, g2, g3]
# both soln same
A = nonlinsolve(F, v)
B = nonlinsolve(G, v)
assert A == B
def test_nonlinsolve_conditionset():
# when solveset failed to solve all the eq
# return conditionset
f = Function('f')
f1 = f(x) - pi/2
f2 = f(y) - pi*Rational(3, 2)
intermediate_system = Eq(2*f(x) - pi, 0) & Eq(2*f(y) - 3*pi, 0)
syms = Tuple(x, y)
soln = ConditionSet(
syms,
intermediate_system,
S.Complexes**2)
assert nonlinsolve([f1, f2], [x, y]) == soln
def test_substitution_basic():
assert substitution([], [x, y]) == S.EmptySet
assert substitution([], []) == S.EmptySet
system = [2*x**2 + 3*y**2 - 30, 3*x**2 - 2*y**2 - 19]
soln = FiniteSet((-3, -2), (-3, 2), (3, -2), (3, 2))
assert substitution(system, [x, y]) == soln
soln = FiniteSet((-1, 1))
assert substitution([x + y], [x], [{y: 1}], [y], set(), [x, y]) == soln
assert substitution(
[x + y], [x], [{y: 1}], [y],
{x + 1}, [y, x]) == S.EmptySet
def test_substitution_incorrect():
# the solutions in the following two tests are incorrect. The
# correct result is EmptySet in both cases.
assert substitution([h - 1, k - 1, f - 2, f - 4, -2 * k],
[h, k, f]) == {(1, 1, f)}
assert substitution([x + y + z, S.One, S.One, S.One], [x, y, z]) == \
{(-y - z, y, z)}
# the correct result in the test below is {(-I, I, I, -I),
# (I, -I, -I, I)}
assert substitution([a - d, b + d, c + d, d**2 + 1], [a, b, c, d]) == \
{(d, -d, -d, d)}
# the result in the test below is incomplete. The complete result
# is {(0, b), (log(2), 2)}
assert substitution([a*(a - log(b)), a*(b - 2)], [a, b]) == \
{(0, b)}
# The system in the test below is zero-dimensional, so the result
# should have no free symbols
assert substitution([-k*y + 6*x - 4*y, -81*k + 49*y**2 - 270,
-3*k*z + k + z**3, k**2 - 2*k + 4],
[x, y, z, k]).free_symbols == {z}
def test_substitution_redundant():
# the third and fourth solutions are redundant in the test below
assert substitution([x**2 - y**2, z - 1], [x, z]) == \
{(-y, 1), (y, 1), (-sqrt(y**2), 1), (sqrt(y**2), 1)}
# the system below has three solutions. Two of the solutions
# returned by substitution are redundant.
res = substitution([x - y, y**3 - 3*y**2 + 1], [x, y])
assert len(res) == 5
def test_issue_5132_substitution():
x, y, z, r, t = symbols('x, y, z, r, t', real=True)
system = [r - x**2 - y**2, tan(t) - y/x]
s_x_1 = Complement(FiniteSet(-sqrt(r/(tan(t)**2 + 1))), FiniteSet(0))
s_x_2 = Complement(FiniteSet(sqrt(r/(tan(t)**2 + 1))), FiniteSet(0))
s_y = sqrt(r/(tan(t)**2 + 1))*tan(t)
soln = FiniteSet((s_x_2, s_y)) + FiniteSet((s_x_1, -s_y))
assert substitution(system, [x, y]) == soln
n = Dummy('n')
eqs = [exp(x)**2 - sin(y) + z**2, 1/exp(y) - 3]
s_real_y = -log(3)
s_real_z = sqrt(-exp(2*x) - sin(log(3)))
soln_real = FiniteSet((s_real_y, s_real_z), (s_real_y, -s_real_z))
lam = Lambda(n, 2*n*I*pi + -log(3))
s_complex_y = ImageSet(lam, S.Integers)
lam = Lambda(n, sqrt(-exp(2*x) + sin(2*n*I*pi + -log(3))))
s_complex_z_1 = ImageSet(lam, S.Integers)
lam = Lambda(n, -sqrt(-exp(2*x) + sin(2*n*I*pi + -log(3))))
s_complex_z_2 = ImageSet(lam, S.Integers)
soln_complex = FiniteSet(
(s_complex_y, s_complex_z_1),
(s_complex_y, s_complex_z_2))
soln = soln_real + soln_complex
assert dumeq(substitution(eqs, [y, z]), soln)
def test_raises_substitution():
raises(ValueError, lambda: substitution([x**2 -1], []))
raises(TypeError, lambda: substitution([x**2 -1]))
raises(ValueError, lambda: substitution([x**2 -1], [sin(x)]))
raises(TypeError, lambda: substitution([x**2 -1], x))
raises(TypeError, lambda: substitution([x**2 -1], 1))
def test_issue_21022():
from sympy.core.sympify import sympify
eqs = [
'k-16',
'p-8',
'y*y+z*z-x*x',
'd - x + p',
'd*d+k*k-y*y',
'z*z-p*p-k*k',
'abc-efg',
]
efg = Symbol('efg')
eqs = [sympify(x) for x in eqs]
syb = list(ordered(set.union(*[x.free_symbols for x in eqs])))
res = nonlinsolve(eqs, syb)
ans = FiniteSet(
(efg, 32, efg, 16, 8, 40, -16*sqrt(5), -8*sqrt(5)),
(efg, 32, efg, 16, 8, 40, -16*sqrt(5), 8*sqrt(5)),
(efg, 32, efg, 16, 8, 40, 16*sqrt(5), -8*sqrt(5)),
(efg, 32, efg, 16, 8, 40, 16*sqrt(5), 8*sqrt(5)),
)
assert len(res) == len(ans) == 4
assert res == ans
for result in res.args:
assert len(result) == 8
def test_issue_17940():
n = Dummy('n')
k1 = Dummy('k1')
sol = ImageSet(Lambda(((k1, n),), I*(2*k1*pi + arg(2*n*I*pi + log(5)))
+ log(Abs(2*n*I*pi + log(5)))),
ProductSet(S.Integers, S.Integers))
assert solveset(exp(exp(x)) - 5, x).dummy_eq(sol)
def test_issue_17906():
assert solveset(7**(x**2 - 80) - 49**x, x) == FiniteSet(-8, 10)
def test_issue_17933():
eq1 = x*sin(45) - y*cos(q)
eq2 = x*cos(45) - y*sin(q)
eq3 = 9*x*sin(45)/10 + y*cos(q)
eq4 = 9*x*cos(45)/10 + y*sin(z) - z
assert nonlinsolve([eq1, eq2, eq3, eq4], x, y, z, q) ==\
FiniteSet((0, 0, 0, q))
def test_issue_14565():
# removed redundancy
assert dumeq(nonlinsolve([k + m, k + m*exp(-2*pi*k)], [k, m]) ,
FiniteSet((-n*I, ImageSet(Lambda(n, n*I), S.Integers))))
# end of tests for nonlinsolve
def test_issue_9556():
b = Symbol('b', positive=True)
assert solveset(Abs(x) + 1, x, S.Reals) is S.EmptySet
assert solveset(Abs(x) + b, x, S.Reals) is S.EmptySet
assert solveset(Eq(b, -1), b, S.Reals) is S.EmptySet
def test_issue_9611():
assert solveset(Eq(x - x + a, a), x, S.Reals) == S.Reals
assert solveset(Eq(y - y + a, a), y) == S.Complexes
def test_issue_9557():
assert solveset(x**2 + a, x, S.Reals) == Intersection(S.Reals,
FiniteSet(-sqrt(-a), sqrt(-a)))
def test_issue_9778():
x = Symbol('x', real=True)
y = Symbol('y', real=True)
assert solveset(x**3 + 1, x, S.Reals) == FiniteSet(-1)
assert solveset(x**Rational(3, 5) + 1, x, S.Reals) == S.EmptySet
assert solveset(x**3 + y, x, S.Reals) == \
FiniteSet(-Abs(y)**Rational(1, 3)*sign(y))
def test_issue_10214():
assert solveset(x**Rational(3, 2) + 4, x, S.Reals) == S.EmptySet
assert solveset(x**(Rational(-3, 2)) + 4, x, S.Reals) == S.EmptySet
ans = FiniteSet(-2**Rational(2, 3))
assert solveset(x**(S(3)) + 4, x, S.Reals) == ans
assert (x**(S(3)) + 4).subs(x,list(ans)[0]) == 0 # substituting ans and verifying the result.
assert (x**(S(3)) + 4).subs(x,-(-2)**Rational(2, 3)) == 0
def test_issue_9849():
assert solveset(Abs(sin(x)) + 1, x, S.Reals) == S.EmptySet
def test_issue_9953():
assert linsolve([ ], x) == S.EmptySet
def test_issue_9913():
assert solveset(2*x + 1/(x - 10)**2, x, S.Reals) == \
FiniteSet(-(3*sqrt(24081)/4 + Rational(4027, 4))**Rational(1, 3)/3 - 100/
(3*(3*sqrt(24081)/4 + Rational(4027, 4))**Rational(1, 3)) + Rational(20, 3))
def test_issue_10397():
assert solveset(sqrt(x), x, S.Complexes) == FiniteSet(0)
def test_issue_14987():
raises(ValueError, lambda: linear_eq_to_matrix(
[x**2], x))
raises(ValueError, lambda: linear_eq_to_matrix(
[x*(-3/x + 1) + 2*y - a], [x, y]))
raises(ValueError, lambda: linear_eq_to_matrix(
[(x**2 - 3*x)/(x - 3) - 3], x))
raises(ValueError, lambda: linear_eq_to_matrix(
[(x + 1)**3 - x**3 - 3*x**2 + 7], x))
raises(ValueError, lambda: linear_eq_to_matrix(
[x*(1/x + 1) + y], [x, y]))
raises(ValueError, lambda: linear_eq_to_matrix(
[(x + 1)*y], [x, y]))
raises(ValueError, lambda: linear_eq_to_matrix(
[Eq(1/x, 1/x + y)], [x, y]))
raises(ValueError, lambda: linear_eq_to_matrix(
[Eq(y/x, y/x + y)], [x, y]))
raises(ValueError, lambda: linear_eq_to_matrix(
[Eq(x*(x + 1), x**2 + y)], [x, y]))
def test_simplification():
eq = x + (a - b)/(-2*a + 2*b)
assert solveset(eq, x) == FiniteSet(S.Half)
assert solveset(eq, x, S.Reals) == Intersection({-((a - b)/(-2*a + 2*b))}, S.Reals)
# So that ap - bn is not zero:
ap = Symbol('ap', positive=True)
bn = Symbol('bn', negative=True)
eq = x + (ap - bn)/(-2*ap + 2*bn)
assert solveset(eq, x) == FiniteSet(S.Half)
assert solveset(eq, x, S.Reals) == FiniteSet(S.Half)
def test_integer_domain_relational():
eq1 = 2*x + 3 > 0
eq2 = x**2 + 3*x - 2 >= 0
eq3 = x + 1/x > -2 + 1/x
eq4 = x + sqrt(x**2 - 5) > 0
eq = x + 1/x > -2 + 1/x
eq5 = eq.subs(x,log(x))
eq6 = log(x)/x <= 0
eq7 = log(x)/x < 0
eq8 = x/(x-3) < 3
eq9 = x/(x**2-3) < 3
assert solveset(eq1, x, S.Integers) == Range(-1, oo, 1)
assert solveset(eq2, x, S.Integers) == Union(Range(-oo, -3, 1), Range(1, oo, 1))
assert solveset(eq3, x, S.Integers) == Union(Range(-1, 0, 1), Range(1, oo, 1))
assert solveset(eq4, x, S.Integers) == Range(3, oo, 1)
assert solveset(eq5, x, S.Integers) == Range(2, oo, 1)
assert solveset(eq6, x, S.Integers) == Range(1, 2, 1)
assert solveset(eq7, x, S.Integers) == S.EmptySet
assert solveset(eq8, x, domain=Range(0,5)) == Range(0, 3, 1)
assert solveset(eq9, x, domain=Range(0,5)) == Union(Range(0, 2, 1), Range(2, 5, 1))
# test_issue_19794
assert solveset(x + 2 < 0, x, S.Integers) == Range(-oo, -2, 1)
def test_issue_10555():
f = Function('f')
g = Function('g')
assert solveset(f(x) - pi/2, x, S.Reals).dummy_eq(
ConditionSet(x, Eq(f(x) - pi/2, 0), S.Reals))
assert solveset(f(g(x)) - pi/2, g(x), S.Reals).dummy_eq(
ConditionSet(g(x), Eq(f(g(x)) - pi/2, 0), S.Reals))
def test_issue_8715():
eq = x + 1/x > -2 + 1/x
assert solveset(eq, x, S.Reals) == \
(Interval.open(-2, oo) - FiniteSet(0))
assert solveset(eq.subs(x,log(x)), x, S.Reals) == \
Interval.open(exp(-2), oo) - FiniteSet(1)
def test_issue_11174():
eq = z**2 + exp(2*x) - sin(y)
soln = Intersection(S.Reals, FiniteSet(log(-z**2 + sin(y))/2))
assert solveset(eq, x, S.Reals) == soln
eq = sqrt(r)*Abs(tan(t))/sqrt(tan(t)**2 + 1) + x*tan(t)
s = -sqrt(r)*Abs(tan(t))/(sqrt(tan(t)**2 + 1)*tan(t))
soln = Intersection(S.Reals, FiniteSet(s))
assert solveset(eq, x, S.Reals) == soln
def test_issue_11534():
# eq1 and eq2 should not have the same solutions because squaring both
# sides of the radical equation introduces a spurious solution branch.
# The equations have a symbolic parameter y and it is easy to see that for
# y != 0 the solution s1 will not be valid for eq1.
x = Symbol('x', real=True)
y = Symbol('y', real=True)
eq1 = -y + x/sqrt(-x**2 + 1)
eq2 = -y**2 + x**2/(-x**2 + 1)
# We get a ConditionSet here because s1 works in eq1 if y is equal to zero
# although not for any other value of y. That case is redundant though
# because if y=0 then s1=s2 so the solution for eq1 could just be returned
# as s2 - {-1, 1}. In fact we have
# |y/sqrt(y**2 + 1)| < 1
# So the complements are not needed either. The ideal output here would be
# sol1 = s2
# sol2 = s1 | s2.
s1, s2 = FiniteSet(-y/sqrt(y**2 + 1)), FiniteSet(y/sqrt(y**2 + 1))
cset = ConditionSet(x, Eq(eq1, 0), s1)
sol1 = (s2 - {-1, 1}) | (cset - {-1, 1})
sol2 = (s1 | s2) - {-1, 1}
assert solveset(eq1, x, S.Reals) == sol1
assert solveset(eq2, x, S.Reals) == sol2
def test_issue_10477():
assert solveset((x**2 + 4*x - 3)/x < 2, x, S.Reals) == \
Union(Interval.open(-oo, -3), Interval.open(0, 1))
def test_issue_10671():
assert solveset(sin(y), y, Interval(0, pi)) == FiniteSet(0, pi)
i = Interval(1, 10)
assert solveset((1/x).diff(x) < 0, x, i) == i
def test_issue_11064():
eq = x + sqrt(x**2 - 5)
assert solveset(eq > 0, x, S.Reals) == \
Interval(sqrt(5), oo)
assert solveset(eq < 0, x, S.Reals) == \
Interval(-oo, -sqrt(5))
assert solveset(eq > sqrt(5), x, S.Reals) == \
Interval.Lopen(sqrt(5), oo)
def test_issue_12478():
eq = sqrt(x - 2) + 2
soln = solveset_real(eq, x)
assert soln is S.EmptySet
assert solveset(eq < 0, x, S.Reals) is S.EmptySet
assert solveset(eq > 0, x, S.Reals) == Interval(2, oo)
def test_issue_12429():
eq = solveset(log(x)/x <= 0, x, S.Reals)
sol = Interval.Lopen(0, 1)
assert eq == sol
def test_issue_19506():
eq = arg(x + I)
C = Dummy('C')
assert solveset(eq).dummy_eq(Intersection(ConditionSet(C, Eq(im(C) + 1, 0), S.Complexes),
ConditionSet(C, re(C) > 0, S.Complexes)))
def test_solveset_arg():
assert solveset(arg(x), x, S.Reals) == Interval.open(0, oo)
assert solveset(arg(4*x -3), x, S.Reals) == Interval.open(Rational(3, 4), oo)
def test__is_finite_with_finite_vars():
f = _is_finite_with_finite_vars
# issue 12482
assert all(f(1/x) is None for x in (
Dummy(), Dummy(real=True), Dummy(complex=True)))
assert f(1/Dummy(real=False)) is True # b/c it's finite but not 0
def test_issue_13550():
assert solveset(x**2 - 2*x - 15, symbol = x, domain = Interval(-oo, 0)) == FiniteSet(-3)
def test_issue_13849():
assert nonlinsolve((t*(sqrt(5) + sqrt(2)) - sqrt(2), t), t) is S.EmptySet
def test_issue_14223():
assert solveset((Abs(x + Min(x, 2)) - 2).rewrite(Piecewise), x,
S.Reals) == FiniteSet(-1, 1)
assert solveset((Abs(x + Min(x, 2)) - 2).rewrite(Piecewise), x,
Interval(0, 2)) == FiniteSet(1)
assert solveset(x, x, FiniteSet(1, 2)) is S.EmptySet
def test_issue_10158():
dom = S.Reals
assert solveset(x*Max(x, 15) - 10, x, dom) == FiniteSet(Rational(2, 3))
assert solveset(x*Min(x, 15) - 10, x, dom) == FiniteSet(-sqrt(10), sqrt(10))
assert solveset(Max(Abs(x - 3) - 1, x + 2) - 3, x, dom) == FiniteSet(-1, 1)
assert solveset(Abs(x - 1) - Abs(y), x, dom) == FiniteSet(-Abs(y) + 1, Abs(y) + 1)
assert solveset(Abs(x + 4*Abs(x + 1)), x, dom) == FiniteSet(Rational(-4, 3), Rational(-4, 5))
assert solveset(2*Abs(x + Abs(x + Max(3, x))) - 2, x, S.Reals) == FiniteSet(-1, -2)
dom = S.Complexes
raises(ValueError, lambda: solveset(x*Max(x, 15) - 10, x, dom))
raises(ValueError, lambda: solveset(x*Min(x, 15) - 10, x, dom))
raises(ValueError, lambda: solveset(Max(Abs(x - 3) - 1, x + 2) - 3, x, dom))
raises(ValueError, lambda: solveset(Abs(x - 1) - Abs(y), x, dom))
raises(ValueError, lambda: solveset(Abs(x + 4*Abs(x + 1)), x, dom))
def test_issue_14300():
f = 1 - exp(-18000000*x) - y
a1 = FiniteSet(-log(-y + 1)/18000000)
assert solveset(f, x, S.Reals) == \
Intersection(S.Reals, a1)
assert dumeq(solveset(f, x),
ImageSet(Lambda(n, -I*(2*n*pi + arg(-y + 1))/18000000 -
log(Abs(y - 1))/18000000), S.Integers))
def test_issue_14454():
number = CRootOf(x**4 + x - 1, 2)
raises(ValueError, lambda: invert_real(number, 0, x))
assert invert_real(x**2, number, x) # no error
def test_issue_17882():
assert solveset(-8*x**2/(9*(x**2 - 1)**(S(4)/3)) + 4/(3*(x**2 - 1)**(S(1)/3)), x, S.Complexes) == \
FiniteSet(sqrt(3), -sqrt(3))
def test_term_factors():
assert list(_term_factors(3**x - 2)) == [-2, 3**x]
expr = 4**(x + 1) + 4**(x + 2) + 4**(x - 1) - 3**(x + 2) - 3**(x + 3)
assert set(_term_factors(expr)) == {
3**(x + 2), 4**(x + 2), 3**(x + 3), 4**(x - 1), -1, 4**(x + 1)}
#################### tests for transolve and its helpers ###############
def test_transolve():
assert _transolve(3**x, x, S.Reals) == S.EmptySet
assert _transolve(3**x - 9**(x + 5), x, S.Reals) == FiniteSet(-10)
def test_issue_21276():
eq = (2*x*(y - z) - y*erf(y - z) - y + z*erf(y - z) + z)**2
assert solveset(eq.expand(), y) == FiniteSet(z, z + erfinv(2*x - 1))
# exponential tests
def test_exponential_real():
from sympy.abc import y
e1 = 3**(2*x) - 2**(x + 3)
e2 = 4**(5 - 9*x) - 8**(2 - x)
e3 = 2**x + 4**x
e4 = exp(log(5)*x) - 2**x
e5 = exp(x/y)*exp(-z/y) - 2
e6 = 5**(x/2) - 2**(x/3)
e7 = 4**(x + 1) + 4**(x + 2) + 4**(x - 1) - 3**(x + 2) - 3**(x + 3)
e8 = -9*exp(-2*x + 5) + 4*exp(3*x + 1)
e9 = 2**x + 4**x + 8**x - 84
e10 = 29*2**(x + 1)*615**(x) - 123*2726**(x)
assert solveset(e1, x, S.Reals) == FiniteSet(
-3*log(2)/(-2*log(3) + log(2)))
assert solveset(e2, x, S.Reals) == FiniteSet(Rational(4, 15))
assert solveset(e3, x, S.Reals) == S.EmptySet
assert solveset(e4, x, S.Reals) == FiniteSet(0)
assert solveset(e5, x, S.Reals) == Intersection(
S.Reals, FiniteSet(y*log(2*exp(z/y))))
assert solveset(e6, x, S.Reals) == FiniteSet(0)
assert solveset(e7, x, S.Reals) == FiniteSet(2)
assert solveset(e8, x, S.Reals) == FiniteSet(-2*log(2)/5 + 2*log(3)/5 + Rational(4, 5))
assert solveset(e9, x, S.Reals) == FiniteSet(2)
assert solveset(e10,x, S.Reals) == FiniteSet((-log(29) - log(2) + log(123))/(-log(2726) + log(2) + log(615)))
assert solveset_real(-9*exp(-2*x + 5) + 2**(x + 1), x) == FiniteSet(
-((-5 - 2*log(3) + log(2))/(log(2) + 2)))
assert solveset_real(4**(x/2) - 2**(x/3), x) == FiniteSet(0)
b = sqrt(6)*sqrt(log(2))/sqrt(log(5))
assert solveset_real(5**(x/2) - 2**(3/x), x) == FiniteSet(-b, b)
# coverage test
C1, C2 = symbols('C1 C2')
f = Function('f')
assert solveset_real(C1 + C2/x**2 - exp(-f(x)), f(x)) == Intersection(
S.Reals, FiniteSet(-log(C1 + C2/x**2)))
y = symbols('y', positive=True)
assert solveset_real(x**2 - y**2/exp(x), y) == Intersection(
S.Reals, FiniteSet(-sqrt(x**2*exp(x)), sqrt(x**2*exp(x))))
p = Symbol('p', positive=True)
assert solveset_real((1/p + 1)**(p + 1), p).dummy_eq(
ConditionSet(x, Eq((1 + 1/x)**(x + 1), 0), S.Reals))
assert solveset(2**x - 4**x + 12, x, S.Reals) == {2}
assert solveset(2**x - 2**(2*x) + 12, x, S.Reals) == {2}
@XFAIL
def test_exponential_complex():
n = Dummy('n')
assert dumeq(solveset_complex(2**x + 4**x, x),imageset(
Lambda(n, I*(2*n*pi + pi)/log(2)), S.Integers))
assert solveset_complex(x**z*y**z - 2, z) == FiniteSet(
log(2)/(log(x) + log(y)))
assert dumeq(solveset_complex(4**(x/2) - 2**(x/3), x), imageset(
Lambda(n, 3*n*I*pi/log(2)), S.Integers))
assert dumeq(solveset(2**x + 32, x), imageset(
Lambda(n, (I*(2*n*pi + pi) + 5*log(2))/log(2)), S.Integers))
eq = (2**exp(y**2/x) + 2)/(x**2 + 15)
a = sqrt(x)*sqrt(-log(log(2)) + log(log(2) + 2*n*I*pi))
assert solveset_complex(eq, y) == FiniteSet(-a, a)
union1 = imageset(Lambda(n, I*(2*n*pi - pi*Rational(2, 3))/log(2)), S.Integers)
union2 = imageset(Lambda(n, I*(2*n*pi + pi*Rational(2, 3))/log(2)), S.Integers)
assert dumeq(solveset(2**x + 4**x + 8**x, x), Union(union1, union2))
eq = 4**(x + 1) + 4**(x + 2) + 4**(x - 1) - 3**(x + 2) - 3**(x + 3)
res = solveset(eq, x)
num = 2*n*I*pi - 4*log(2) + 2*log(3)
den = -2*log(2) + log(3)
ans = imageset(Lambda(n, num/den), S.Integers)
assert dumeq(res, ans)
def test_expo_conditionset():
f1 = (exp(x) + 1)**x - 2
f2 = (x + 2)**y*x - 3
f3 = 2**x - exp(x) - 3
f4 = log(x) - exp(x)
f5 = 2**x + 3**x - 5**x
assert solveset(f1, x, S.Reals).dummy_eq(ConditionSet(
x, Eq((exp(x) + 1)**x - 2, 0), S.Reals))
assert solveset(f2, x, S.Reals).dummy_eq(ConditionSet(
x, Eq(x*(x + 2)**y - 3, 0), S.Reals))
assert solveset(f3, x, S.Reals).dummy_eq(ConditionSet(
x, Eq(2**x - exp(x) - 3, 0), S.Reals))
assert solveset(f4, x, S.Reals).dummy_eq(ConditionSet(
x, Eq(-exp(x) + log(x), 0), S.Reals))
assert solveset(f5, x, S.Reals).dummy_eq(ConditionSet(
x, Eq(2**x + 3**x - 5**x, 0), S.Reals))
def test_exponential_symbols():
x, y, z = symbols('x y z', positive=True)
xr, zr = symbols('xr, zr', real=True)
assert solveset(z**x - y, x, S.Reals) == Intersection(
S.Reals, FiniteSet(log(y)/log(z)))
f1 = 2*x**w - 4*y**w
f2 = (x/y)**w - 2
sol1 = Intersection({log(2)/(log(x) - log(y))}, S.Reals)
sol2 = Intersection({log(2)/log(x/y)}, S.Reals)
assert solveset(f1, w, S.Reals) == sol1, solveset(f1, w, S.Reals)
assert solveset(f2, w, S.Reals) == sol2, solveset(f2, w, S.Reals)
assert solveset(x**x, x, Interval.Lopen(0,oo)).dummy_eq(
ConditionSet(w, Eq(w**w, 0), Interval.open(0, oo)))
assert solveset(x**y - 1, y, S.Reals) == FiniteSet(0)
assert solveset(exp(x/y)*exp(-z/y) - 2, y, S.Reals) == \
Complement(ConditionSet(y, Eq(im(x)/y, 0) & Eq(im(z)/y, 0), \
Complement(Intersection(FiniteSet((x - z)/log(2)), S.Reals), FiniteSet(0))), FiniteSet(0))
assert solveset(exp(xr/y)*exp(-zr/y) - 2, y, S.Reals) == \
Complement(FiniteSet((xr - zr)/log(2)), FiniteSet(0))
assert solveset(a**x - b**x, x).dummy_eq(ConditionSet(
w, Ne(a, 0) & Ne(b, 0), FiniteSet(0)))
def test_ignore_assumptions():
# make sure assumptions are ignored
xpos = symbols('x', positive=True)
x = symbols('x')
assert solveset_complex(xpos**2 - 4, xpos
) == solveset_complex(x**2 - 4, x)
@XFAIL
def test_issue_10864():
assert solveset(x**(y*z) - x, x, S.Reals) == FiniteSet(1)
@XFAIL
def test_solve_only_exp_2():
assert solveset_real(sqrt(exp(x)) + sqrt(exp(-x)) - 4, x) == \
FiniteSet(2*log(-sqrt(3) + 2), 2*log(sqrt(3) + 2))
def test_is_exponential():
assert _is_exponential(y, x) is False
assert _is_exponential(3**x - 2, x) is True
assert _is_exponential(5**x - 7**(2 - x), x) is True
assert _is_exponential(sin(2**x) - 4*x, x) is False
assert _is_exponential(x**y - z, y) is True
assert _is_exponential(x**y - z, x) is False
assert _is_exponential(2**x + 4**x - 1, x) is True
assert _is_exponential(x**(y*z) - x, x) is False
assert _is_exponential(x**(2*x) - 3**x, x) is False
assert _is_exponential(x**y - y*z, y) is False
assert _is_exponential(x**y - x*z, y) is True
def test_solve_exponential():
assert _solve_exponential(3**(2*x) - 2**(x + 3), 0, x, S.Reals) == \
FiniteSet(-3*log(2)/(-2*log(3) + log(2)))
assert _solve_exponential(2**y + 4**y, 1, y, S.Reals) == \
FiniteSet(log(Rational(-1, 2) + sqrt(5)/2)/log(2))
assert _solve_exponential(2**y + 4**y, 0, y, S.Reals) == \
S.EmptySet
assert _solve_exponential(2**x + 3**x - 5**x, 0, x, S.Reals) == \
ConditionSet(x, Eq(2**x + 3**x - 5**x, 0), S.Reals)
# end of exponential tests
# logarithmic tests
def test_logarithmic():
assert solveset_real(log(x - 3) + log(x + 3), x) == FiniteSet(
-sqrt(10), sqrt(10))
assert solveset_real(log(x + 1) - log(2*x - 1), x) == FiniteSet(2)
assert solveset_real(log(x + 3) + log(1 + 3/x) - 3, x) == FiniteSet(
-3 + sqrt(-12 + exp(3))*exp(Rational(3, 2))/2 + exp(3)/2,
-sqrt(-12 + exp(3))*exp(Rational(3, 2))/2 - 3 + exp(3)/2)
eq = z - log(x) + log(y/(x*(-1 + y**2/x**2)))
assert solveset_real(eq, x) == \
Intersection(S.Reals, FiniteSet(-sqrt(y**2 - y*exp(z)),
sqrt(y**2 - y*exp(z)))) - \
Intersection(S.Reals, FiniteSet(-sqrt(y**2), sqrt(y**2)))
assert solveset_real(
log(3*x) - log(-x + 1) - log(4*x + 1), x) == FiniteSet(Rational(-1, 2), S.Half)
assert solveset(log(x**y) - y*log(x), x, S.Reals) == S.Reals
@XFAIL
def test_uselogcombine_2():
eq = log(exp(2*x) + 1) + log(-tanh(x) + 1) - log(2)
assert solveset_real(eq, x) is S.EmptySet
eq = log(8*x) - log(sqrt(x) + 1) - 2
assert solveset_real(eq, x) is S.EmptySet
def test_is_logarithmic():
assert _is_logarithmic(y, x) is False
assert _is_logarithmic(log(x), x) is True
assert _is_logarithmic(log(x) - 3, x) is True
assert _is_logarithmic(log(x)*log(y), x) is True
assert _is_logarithmic(log(x)**2, x) is False
assert _is_logarithmic(log(x - 3) + log(x + 3), x) is True
assert _is_logarithmic(log(x**y) - y*log(x), x) is True
assert _is_logarithmic(sin(log(x)), x) is False
assert _is_logarithmic(x + y, x) is False
assert _is_logarithmic(log(3*x) - log(1 - x) + 4, x) is True
assert _is_logarithmic(log(x) + log(y) + x, x) is False
assert _is_logarithmic(log(log(x - 3)) + log(x - 3), x) is True
assert _is_logarithmic(log(log(3) + x) + log(x), x) is True
assert _is_logarithmic(log(x)*(y + 3) + log(x), y) is False
def test_solve_logarithm():
y = Symbol('y')
assert _solve_logarithm(log(x**y) - y*log(x), 0, x, S.Reals) == S.Reals
y = Symbol('y', positive=True)
assert _solve_logarithm(log(x)*log(y), 0, x, S.Reals) == FiniteSet(1)
# end of logarithmic tests
# lambert tests
def test_is_lambert():
a, b, c = symbols('a,b,c')
assert _is_lambert(x**2, x) is False
assert _is_lambert(a**x**2+b*x+c, x) is True
assert _is_lambert(E**2, x) is False
assert _is_lambert(x*E**2, x) is False
assert _is_lambert(3*log(x) - x*log(3), x) is True
assert _is_lambert(log(log(x - 3)) + log(x-3), x) is True
assert _is_lambert(5*x - 1 + 3*exp(2 - 7*x), x) is True
assert _is_lambert((a/x + exp(x/2)).diff(x, 2), x) is True
assert _is_lambert((x**2 - 2*x + 1).subs(x, (log(x) + 3*x)**2 - 1), x) is True
assert _is_lambert(x*sinh(x) - 1, x) is True
assert _is_lambert(x*cos(x) - 5, x) is True
assert _is_lambert(tanh(x) - 5*x, x) is True
assert _is_lambert(cosh(x) - sinh(x), x) is False
# end of lambert tests
def test_linear_coeffs():
from sympy.solvers.solveset import linear_coeffs
assert linear_coeffs(0, x) == [0, 0]
assert all(i is S.Zero for i in linear_coeffs(0, x))
assert linear_coeffs(x + 2*y + 3, x, y) == [1, 2, 3]
assert linear_coeffs(x + 2*y + 3, y, x) == [2, 1, 3]
assert linear_coeffs(x + 2*x**2 + 3, x, x**2) == [1, 2, 3]
raises(ValueError, lambda:
linear_coeffs(x + 2*x**2 + x**3, x, x**2))
raises(ValueError, lambda:
linear_coeffs(1/x*(x - 1) + 1/x, x))
raises(ValueError, lambda:
linear_coeffs(x, x, x))
assert linear_coeffs(a*(x + y), x, y) == [a, a, 0]
assert linear_coeffs(1.0, x, y) == [0, 0, 1.0]
# don't include coefficients of 0
assert linear_coeffs(Eq(x, x + y), x, y, dict=True) == {y: -1}
assert linear_coeffs(0, x, y, dict=True) == {}
def test_is_modular():
assert _is_modular(y, x) is False
assert _is_modular(Mod(x, 3) - 1, x) is True
assert _is_modular(Mod(x**3 - 3*x**2 - x + 1, 3) - 1, x) is True
assert _is_modular(Mod(exp(x + y), 3) - 2, x) is True
assert _is_modular(Mod(exp(x + y), 3) - log(x), x) is True
assert _is_modular(Mod(x, 3) - 1, y) is False
assert _is_modular(Mod(x, 3)**2 - 5, x) is False
assert _is_modular(Mod(x, 3)**2 - y, x) is False
assert _is_modular(exp(Mod(x, 3)) - 1, x) is False
assert _is_modular(Mod(3, y) - 1, y) is False
def test_invert_modular():
n = Dummy('n', integer=True)
from sympy.solvers.solveset import _invert_modular as invert_modular
# non invertible cases
assert invert_modular(Mod(sin(x), 7), S(5), n, x) == (Mod(sin(x), 7), 5)
assert invert_modular(Mod(exp(x), 7), S(5), n, x) == (Mod(exp(x), 7), 5)
assert invert_modular(Mod(log(x), 7), S(5), n, x) == (Mod(log(x), 7), 5)
# a is symbol
assert dumeq(invert_modular(Mod(x, 7), S(5), n, x),
(x, ImageSet(Lambda(n, 7*n + 5), S.Integers)))
# a.is_Add
assert dumeq(invert_modular(Mod(x + 8, 7), S(5), n, x),
(x, ImageSet(Lambda(n, 7*n + 4), S.Integers)))
assert invert_modular(Mod(x**2 + x, 7), S(5), n, x) == \
(Mod(x**2 + x, 7), 5)
# a.is_Mul
assert dumeq(invert_modular(Mod(3*x, 7), S(5), n, x),
(x, ImageSet(Lambda(n, 7*n + 4), S.Integers)))
assert invert_modular(Mod((x + 1)*(x + 2), 7), S(5), n, x) == \
(Mod((x + 1)*(x + 2), 7), 5)
# a.is_Pow
assert invert_modular(Mod(x**4, 7), S(5), n, x) == \
(x, S.EmptySet)
assert dumeq(invert_modular(Mod(3**x, 4), S(3), n, x),
(x, ImageSet(Lambda(n, 2*n + 1), S.Naturals0)))
assert dumeq(invert_modular(Mod(2**(x**2 + x + 1), 7), S(2), n, x),
(x**2 + x + 1, ImageSet(Lambda(n, 3*n + 1), S.Naturals0)))
assert invert_modular(Mod(sin(x)**4, 7), S(5), n, x) == (x, S.EmptySet)
def test_solve_modular():
n = Dummy('n', integer=True)
# if rhs has symbol (need to be implemented in future).
assert solveset(Mod(x, 4) - x, x, S.Integers
).dummy_eq(
ConditionSet(x, Eq(-x + Mod(x, 4), 0),
S.Integers))
# when _invert_modular fails to invert
assert solveset(3 - Mod(sin(x), 7), x, S.Integers
).dummy_eq(
ConditionSet(x, Eq(Mod(sin(x), 7) - 3, 0), S.Integers))
assert solveset(3 - Mod(log(x), 7), x, S.Integers
).dummy_eq(
ConditionSet(x, Eq(Mod(log(x), 7) - 3, 0), S.Integers))
assert solveset(3 - Mod(exp(x), 7), x, S.Integers
).dummy_eq(ConditionSet(x, Eq(Mod(exp(x), 7) - 3, 0),
S.Integers))
# EmptySet solution definitely
assert solveset(7 - Mod(x, 5), x, S.Integers) is S.EmptySet
assert solveset(5 - Mod(x, 5), x, S.Integers) is S.EmptySet
# Negative m
assert dumeq(solveset(2 + Mod(x, -3), x, S.Integers),
ImageSet(Lambda(n, -3*n - 2), S.Integers))
assert solveset(4 + Mod(x, -3), x, S.Integers) is S.EmptySet
# linear expression in Mod
assert dumeq(solveset(3 - Mod(x, 5), x, S.Integers),
ImageSet(Lambda(n, 5*n + 3), S.Integers))
assert dumeq(solveset(3 - Mod(5*x - 8, 7), x, S.Integers),
ImageSet(Lambda(n, 7*n + 5), S.Integers))
assert dumeq(solveset(3 - Mod(5*x, 7), x, S.Integers),
ImageSet(Lambda(n, 7*n + 2), S.Integers))
# higher degree expression in Mod
assert dumeq(solveset(Mod(x**2, 160) - 9, x, S.Integers),
Union(ImageSet(Lambda(n, 160*n + 3), S.Integers),
ImageSet(Lambda(n, 160*n + 13), S.Integers),
ImageSet(Lambda(n, 160*n + 67), S.Integers),
ImageSet(Lambda(n, 160*n + 77), S.Integers),
ImageSet(Lambda(n, 160*n + 83), S.Integers),
ImageSet(Lambda(n, 160*n + 93), S.Integers),
ImageSet(Lambda(n, 160*n + 147), S.Integers),
ImageSet(Lambda(n, 160*n + 157), S.Integers)))
assert solveset(3 - Mod(x**4, 7), x, S.Integers) is S.EmptySet
assert dumeq(solveset(Mod(x**4, 17) - 13, x, S.Integers),
Union(ImageSet(Lambda(n, 17*n + 3), S.Integers),
ImageSet(Lambda(n, 17*n + 5), S.Integers),
ImageSet(Lambda(n, 17*n + 12), S.Integers),
ImageSet(Lambda(n, 17*n + 14), S.Integers)))
# a.is_Pow tests
assert dumeq(solveset(Mod(7**x, 41) - 15, x, S.Integers),
ImageSet(Lambda(n, 40*n + 3), S.Naturals0))
assert dumeq(solveset(Mod(12**x, 21) - 18, x, S.Integers),
ImageSet(Lambda(n, 6*n + 2), S.Naturals0))
assert dumeq(solveset(Mod(3**x, 4) - 3, x, S.Integers),
ImageSet(Lambda(n, 2*n + 1), S.Naturals0))
assert dumeq(solveset(Mod(2**x, 7) - 2 , x, S.Integers),
ImageSet(Lambda(n, 3*n + 1), S.Naturals0))
assert dumeq(solveset(Mod(3**(3**x), 4) - 3, x, S.Integers),
Intersection(ImageSet(Lambda(n, Intersection({log(2*n + 1)/log(3)},
S.Integers)), S.Naturals0), S.Integers))
# Implemented for m without primitive root
assert solveset(Mod(x**3, 7) - 2, x, S.Integers) is S.EmptySet
assert dumeq(solveset(Mod(x**3, 8) - 1, x, S.Integers),
ImageSet(Lambda(n, 8*n + 1), S.Integers))
assert dumeq(solveset(Mod(x**4, 9) - 4, x, S.Integers),
Union(ImageSet(Lambda(n, 9*n + 4), S.Integers),
ImageSet(Lambda(n, 9*n + 5), S.Integers)))
# domain intersection
assert dumeq(solveset(3 - Mod(5*x - 8, 7), x, S.Naturals0),
Intersection(ImageSet(Lambda(n, 7*n + 5), S.Integers), S.Naturals0))
# Complex args
assert solveset(Mod(x, 3) - I, x, S.Integers) == \
S.EmptySet
assert solveset(Mod(I*x, 3) - 2, x, S.Integers
).dummy_eq(
ConditionSet(x, Eq(Mod(I*x, 3) - 2, 0), S.Integers))
assert solveset(Mod(I + x, 3) - 2, x, S.Integers
).dummy_eq(
ConditionSet(x, Eq(Mod(x + I, 3) - 2, 0), S.Integers))
# issue 17373 (https://github.com/sympy/sympy/issues/17373)
assert dumeq(solveset(Mod(x**4, 14) - 11, x, S.Integers),
Union(ImageSet(Lambda(n, 14*n + 3), S.Integers),
ImageSet(Lambda(n, 14*n + 11), S.Integers)))
assert dumeq(solveset(Mod(x**31, 74) - 43, x, S.Integers),
ImageSet(Lambda(n, 74*n + 31), S.Integers))
# issue 13178
n = symbols('n', integer=True)
a = 742938285
b = 1898888478
m = 2**31 - 1
c = 20170816
assert dumeq(solveset(c - Mod(a**n*b, m), n, S.Integers),
ImageSet(Lambda(n, 2147483646*n + 100), S.Naturals0))
assert dumeq(solveset(c - Mod(a**n*b, m), n, S.Naturals0),
Intersection(ImageSet(Lambda(n, 2147483646*n + 100), S.Naturals0),
S.Naturals0))
assert dumeq(solveset(c - Mod(a**(2*n)*b, m), n, S.Integers),
Intersection(ImageSet(Lambda(n, 1073741823*n + 50), S.Naturals0),
S.Integers))
assert solveset(c - Mod(a**(2*n + 7)*b, m), n, S.Integers) is S.EmptySet
assert dumeq(solveset(c - Mod(a**(n - 4)*b, m), n, S.Integers),
Intersection(ImageSet(Lambda(n, 2147483646*n + 104), S.Naturals0),
S.Integers))
# end of modular tests
def test_issue_17276():
assert nonlinsolve([Eq(x, 5**(S(1)/5)), Eq(x*y, 25*sqrt(5))], x, y) == \
FiniteSet((5**(S(1)/5), 25*5**(S(3)/10)))
def test_issue_10426():
x = Dummy('x')
a = Symbol('a')
n = Dummy('n')
assert (solveset(sin(x + a) - sin(x), a)).dummy_eq(Dummy('x')) == (Union(
ImageSet(Lambda(n, 2*n*pi), S.Integers),
Intersection(S.Complexes, ImageSet(Lambda(n, -I*(I*(2*n*pi + arg(-exp(-2*I*x))) + 2*im(x))),
S.Integers)))).dummy_eq(Dummy('x,n'))
def test_solveset_conjugate():
"""Test solveset for simple conjugate functions"""
assert solveset(conjugate(x) -3 + I) == FiniteSet(3 + I)
def test_issue_18208():
variables = symbols('x0:16') + symbols('y0:12')
x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15,\
y0, y1, y2, y3, y4, y5, y6, y7, y8, y9, y10, y11 = variables
eqs = [x0 + x1 + x2 + x3 - 51,
x0 + x1 + x4 + x5 - 46,
x2 + x3 + x6 + x7 - 39,
x0 + x3 + x4 + x7 - 50,
x1 + x2 + x5 + x6 - 35,
x4 + x5 + x6 + x7 - 34,
x4 + x5 + x8 + x9 - 46,
x10 + x11 + x6 + x7 - 23,
x11 + x4 + x7 + x8 - 25,
x10 + x5 + x6 + x9 - 44,
x10 + x11 + x8 + x9 - 35,
x12 + x13 + x8 + x9 - 35,
x10 + x11 + x14 + x15 - 29,
x11 + x12 + x15 + x8 - 35,
x10 + x13 + x14 + x9 - 29,
x12 + x13 + x14 + x15 - 29,
y0 + y1 + y2 + y3 - 55,
y0 + y1 + y4 + y5 - 53,
y2 + y3 + y6 + y7 - 56,
y0 + y3 + y4 + y7 - 57,
y1 + y2 + y5 + y6 - 52,
y4 + y5 + y6 + y7 - 54,
y4 + y5 + y8 + y9 - 48,
y10 + y11 + y6 + y7 - 60,
y11 + y4 + y7 + y8 - 51,
y10 + y5 + y6 + y9 - 57,
y10 + y11 + y8 + y9 - 54,
x10 - 2,
x11 - 5,
x12 - 1,
x13 - 6,
x14 - 1,
x15 - 21,
y0 - 12,
y1 - 20]
expected = [38 - x3, x3 - 10, 23 - x3, x3, 12 - x7, x7 + 6, 16 - x7, x7,
8, 20, 2, 5, 1, 6, 1, 21, 12, 20, -y11 + y9 + 2, y11 - y9 + 21,
-y11 - y7 + y9 + 24, y11 + y7 - y9 - 3, 33 - y7, y7, 27 - y9, y9,
27 - y11, y11]
A, b = linear_eq_to_matrix(eqs, variables)
# solve
solve_expected = {v:eq for v, eq in zip(variables, expected) if v != eq}
assert solve(eqs, variables) == solve_expected
# linsolve
linsolve_expected = FiniteSet(Tuple(*expected))
assert linsolve(eqs, variables) == linsolve_expected
assert linsolve((A, b), variables) == linsolve_expected
# gauss_jordan_solve
gj_solve, new_vars = A.gauss_jordan_solve(b)
gj_solve = list(gj_solve)
gj_expected = linsolve_expected.subs(zip([x3, x7, y7, y9, y11], new_vars))
assert FiniteSet(Tuple(*gj_solve)) == gj_expected
# nonlinsolve
# The solution set of nonlinsolve is currently equivalent to linsolve and is
# also correct. However, we would prefer to use the same symbols as parameters
# for the solution to the underdetermined system in all cases if possible.
# We want a solution that is not just equivalent but also given in the same form.
# This test may be changed should nonlinsolve be modified in this way.
nonlinsolve_expected = FiniteSet((38 - x3, x3 - 10, 23 - x3, x3, 12 - x7, x7 + 6,
16 - x7, x7, 8, 20, 2, 5, 1, 6, 1, 21, 12, 20,
-y5 + y7 - 1, y5 - y7 + 24, 21 - y5, y5, 33 - y7,
y7, 27 - y9, y9, -y5 + y7 - y9 + 24, y5 - y7 + y9 + 3))
assert nonlinsolve(eqs, variables) == nonlinsolve_expected
def test_substitution_with_infeasible_solution():
a00, a01, a10, a11, l0, l1, l2, l3, m0, m1, m2, m3, m4, m5, m6, m7, c00, c01, c10, c11, p00, p01, p10, p11 = symbols(
'a00, a01, a10, a11, l0, l1, l2, l3, m0, m1, m2, m3, m4, m5, m6, m7, c00, c01, c10, c11, p00, p01, p10, p11'
)
solvefor = [p00, p01, p10, p11, c00, c01, c10, c11, m0, m1, m3, l0, l1, l2, l3]
system = [
-l0 * c00 - l1 * c01 + m0 + c00 + c01,
-l0 * c10 - l1 * c11 + m1,
-l2 * c00 - l3 * c01 + c00 + c01,
-l2 * c10 - l3 * c11 + m3,
-l0 * p00 - l2 * p10 + p00 + p10,
-l1 * p00 - l3 * p10 + p00 + p10,
-l0 * p01 - l2 * p11,
-l1 * p01 - l3 * p11,
-a00 + c00 * p00 + c10 * p01,
-a01 + c01 * p00 + c11 * p01,
-a10 + c00 * p10 + c10 * p11,
-a11 + c01 * p10 + c11 * p11,
-m0 * p00,
-m1 * p01,
-m2 * p10,
-m3 * p11,
-m4 * c00,
-m5 * c01,
-m6 * c10,
-m7 * c11,
m2,
m4,
m5,
m6,
m7
]
sol = FiniteSet(
(0, Complement(FiniteSet(p01), FiniteSet(0)), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, l2, l3),
(p00, Complement(FiniteSet(p01), FiniteSet(0)), 0, p11, 0, 0, 0, 0, 0, 0, 0, 1, 1, -p01/p11, -p01/p11),
(0, Complement(FiniteSet(p01), FiniteSet(0)), 0, p11, 0, 0, 0, 0, 0, 0, 0, 1, -l3*p11/p01, -p01/p11, l3),
(0, Complement(FiniteSet(p01), FiniteSet(0)), 0, p11, 0, 0, 0, 0, 0, 0, 0, -l2*p11/p01, -l3*p11/p01, l2, l3),
)
assert sol != nonlinsolve(system, solvefor)
def test_issue_20097():
assert solveset(1/sqrt(x)) is S.EmptySet
def test_issue_15350():
assert solveset(diff(sqrt(1/x+x))) == FiniteSet(-1, 1)
def test_issue_18359():
c1 = Piecewise((0, x < 0), (Min(1, x)/2 - Min(2, x)/2 + Min(3, x)/2, True))
c2 = Piecewise((Piecewise((0, x < 0), (Min(1, x)/2 - Min(2, x)/2 + Min(3, x)/2, True)), x >= 0), (0, True))
correct_result = Interval(1, 2)
result1 = solveset(c1 - Rational(1, 2), x, Interval(0, 3))
result2 = solveset(c2 - Rational(1, 2), x, Interval(0, 3))
assert result1 == correct_result
assert result2 == correct_result
def test_issue_17604():
lhs = -2**(3*x/11)*exp(x/11) + pi**(x/11)
assert _is_exponential(lhs, x)
assert _solve_exponential(lhs, 0, x, S.Complexes) == FiniteSet(0)
def test_issue_17580():
assert solveset(1/(1 - x**3)**2, x, S.Reals) is S.EmptySet
def test_issue_17566_actual():
sys = [2**x + 2**y - 3, 4**x + 9**y - 5]
# Not clear this is the correct result, but at least no recursion error
assert nonlinsolve(sys, x, y) == FiniteSet((log(3 - 2**y)/log(2), y))
def test_issue_17565():
eq = Ge(2*(x - 2)**2/(3*(x + 1)**(Integer(1)/3)) + 2*(x - 2)*(x + 1)**(Integer(2)/3), 0)
res = Union(Interval.Lopen(-1, -Rational(1, 4)), Interval(2, oo))
assert solveset(eq, x, S.Reals) == res
def test_issue_15024():
function = (x + 5)/sqrt(-x**2 - 10*x)
assert solveset(function, x, S.Reals) == FiniteSet(Integer(-5))
def test_issue_16877():
assert dumeq(nonlinsolve([x - 1, sin(y)], x, y),
FiniteSet((FiniteSet(1), ImageSet(Lambda(n, 2*n*pi), S.Integers)),
(FiniteSet(1), ImageSet(Lambda(n, 2*n*pi + pi), S.Integers))))
# Even better if (FiniteSet(1), ImageSet(Lambda(n, n*pi), S.Integers)) is obtained
def test_issue_16876():
assert dumeq(nonlinsolve([sin(x), 2*x - 4*y], x, y),
FiniteSet((ImageSet(Lambda(n, 2*n*pi), S.Integers),
ImageSet(Lambda(n, n*pi), S.Integers)),
(ImageSet(Lambda(n, 2*n*pi + pi), S.Integers),
ImageSet(Lambda(n, n*pi + pi/2), S.Integers))))
# Even better if (ImageSet(Lambda(n, n*pi), S.Integers),
# ImageSet(Lambda(n, n*pi/2), S.Integers)) is obtained
def test_issue_21236():
x, z = symbols("x z")
y = symbols('y', rational=True)
assert solveset(x**y - z, x, S.Reals) == ConditionSet(x, Eq(x**y - z, 0), S.Reals)
e1, e2 = symbols('e1 e2', even=True)
y = e1/e2 # don't know if num or den will be odd and the other even
assert solveset(x**y - z, x, S.Reals) == ConditionSet(x, Eq(x**y - z, 0), S.Reals)
def test_issue_21908():
assert nonlinsolve([(x**2 + 2*x - y**2)*exp(x), -2*y*exp(x)], x, y
) == {(-2, 0), (0, 0)}
def test_issue_19144():
# test case 1
expr1 = [x + y - 1, y**2 + 1]
eq1 = [Eq(i, 0) for i in expr1]
soln1 = {(1 - I, I), (1 + I, -I)}
soln_expr1 = nonlinsolve(expr1, [x, y])
soln_eq1 = nonlinsolve(eq1, [x, y])
assert soln_eq1 == soln_expr1 == soln1
# test case 2 - with denoms
expr2 = [x/y - 1, y**2 + 1]
eq2 = [Eq(i, 0) for i in expr2]
soln2 = {(-I, -I), (I, I)}
soln_expr2 = nonlinsolve(expr2, [x, y])
soln_eq2 = nonlinsolve(eq2, [x, y])
assert soln_eq2 == soln_expr2 == soln2
# denominators that cancel in expression
assert nonlinsolve([Eq(x + 1/x, 1/x)], [x]) == FiniteSet((S.EmptySet,))
def test_issue_22413():
res = nonlinsolve((4*y*(2*x + 2*exp(y) + 1)*exp(2*x),
4*x*exp(2*x) + 4*y*exp(2*x + y) + 4*exp(2*x + y) + 1),
x, y)
# First solution is not correct, but the issue was an exception
sols = FiniteSet((x, S.Zero), (-exp(y) - S.Half, y))
assert res == sols
def test_issue_23318():
eqs_eq = [
Eq(53.5780461486929, x * log(y / (5.0 - y) + 1) / y),
Eq(x, 0.0015 * z),
Eq(0.0015, 7845.32 * y / z),
]
eqs_expr = [eq.rewrite(Add) for eq in eqs_eq]
sol = {(266.97755814852, 0.0340301680681629, 177985.03876568)}
assert_close_nl(nonlinsolve(eqs_eq, [x, y, z]), sol)
assert_close_nl(nonlinsolve(eqs_expr, [x, y, z]), sol)
logterm = log(1.91196789933362e-7*z/(5.0 - 1.91196789933362e-7*z) + 1)
eq = -0.0015*z*logterm + 1.02439504345316e-5*z
assert_close_ss(solveset(eq, z), {0, 177985.038765679})
def test_issue_19814():
assert nonlinsolve([ 2**m - 2**(2*n), 4*2**m - 2**(4*n)], m, n
) == FiniteSet((log(2**(2*n))/log(2), S.Complexes))
def test_issue_22058():
sol = solveset(-sqrt(t)*x**2 + 2*x + sqrt(t), x, S.Reals)
# doesn't fail (and following numerical check)
assert sol.xreplace({t: 1}) == {1 - sqrt(2), 1 + sqrt(2)}, sol.xreplace({t: 1})
def test_issue_11184():
assert solveset(20*sqrt(y**2 + (sqrt(-(y - 10)*(y + 10)) + 10)**2) - 60, y, S.Reals) is S.EmptySet
def test_issue_21890():
e = S(2)/3
assert nonlinsolve([4*x**3*y**4 - 2*y, 4*x**4*y**3 - 2*x], x, y) == {
(2**e/(2*y), y), ((-2**e/4 - 2**e*sqrt(3)*I/4)/y, y),
((-2**e/4 + 2**e*sqrt(3)*I/4)/y, y)}
assert nonlinsolve([(1 - 4*x**2)*exp(-2*x**2 - 2*y**2),
-4*x*y*exp(-2*x**2)*exp(-2*y**2)], x, y) == {(-S(1)/2, 0), (S(1)/2, 0)}
rx, ry = symbols('x y', real=True)
sol = nonlinsolve([4*rx**3*ry**4 - 2*ry, 4*rx**4*ry**3 - 2*rx], rx, ry)
ans = {(2**(S(2)/3)/(2*ry), ry),
((-2**(S(2)/3)/4 - 2**(S(2)/3)*sqrt(3)*I/4)/ry, ry),
((-2**(S(2)/3)/4 + 2**(S(2)/3)*sqrt(3)*I/4)/ry, ry)}
assert sol == ans
def test_issue_22628():
assert nonlinsolve([h - 1, k - 1, f - 2, f - 4, -2*k], h, k, f) == S.EmptySet
assert nonlinsolve([x**3 - 1, x + y, x**2 - 4], [x, y]) == S.EmptySet