489 lines
20 KiB
Python
489 lines
20 KiB
Python
|
"""Tests for tools for solving inequalities and systems of inequalities. """
|
||
|
|
|
||
|
|
from sympy.concrete.summations import Sum
|
||
|
|
from sympy.core.function import Function
|
||
|
|
from sympy.core.numbers import (I, Rational, oo, pi)
|
||
|
|
from sympy.core.relational import (Eq, Ge, Gt, Le, Lt, Ne)
|
||
|
|
from sympy.core.singleton import S
|
||
|
|
from sympy.core.symbol import (Dummy, Symbol)
|
||
|
|
from sympy.functions.elementary.complexes import Abs
|
||
|
|
from sympy.functions.elementary.exponential import (exp, log)
|
||
|
|
from sympy.functions.elementary.miscellaneous import (root, sqrt)
|
||
|
|
from sympy.functions.elementary.piecewise import Piecewise
|
||
|
|
from sympy.functions.elementary.trigonometric import (cos, sin, tan)
|
||
|
|
from sympy.integrals.integrals import Integral
|
||
|
|
from sympy.logic.boolalg import (And, Or)
|
||
|
|
from sympy.polys.polytools import (Poly, PurePoly)
|
||
|
|
from sympy.sets.sets import (FiniteSet, Interval, Union)
|
||
|
|
from sympy.solvers.inequalities import (reduce_inequalities,
|
||
|
|
solve_poly_inequality as psolve,
|
||
|
|
reduce_rational_inequalities,
|
||
|
|
solve_univariate_inequality as isolve,
|
||
|
|
reduce_abs_inequality,
|
||
|
|
_solve_inequality)
|
||
|
|
from sympy.polys.rootoftools import rootof
|
||
|
|
from sympy.solvers.solvers import solve
|
||
|
|
from sympy.solvers.solveset import solveset
|
||
|
|
from sympy.abc import x, y
|
||
|
|
|
||
|
|
from sympy.core.mod import Mod
|
||
|
|
|
||
|
|
from sympy.testing.pytest import raises, XFAIL
|
||
|
|
|
||
|
|
|
||
|
|
inf = oo.evalf()
|
||
|
|
|
||
|
|
|
||
|
|
def test_solve_poly_inequality():
|
||
|
|
assert psolve(Poly(0, x), '==') == [S.Reals]
|
||
|
|
assert psolve(Poly(1, x), '==') == [S.EmptySet]
|
||
|
|
assert psolve(PurePoly(x + 1, x), ">") == [Interval(-1, oo, True, False)]
|
||
|
|
|
||
|
|
|
||
|
|
def test_reduce_poly_inequalities_real_interval():
|
||
|
|
assert reduce_rational_inequalities(
|
||
|
|
[[Eq(x**2, 0)]], x, relational=False) == FiniteSet(0)
|
||
|
|
assert reduce_rational_inequalities(
|
||
|
|
[[Le(x**2, 0)]], x, relational=False) == FiniteSet(0)
|
||
|
|
assert reduce_rational_inequalities(
|
||
|
|
[[Lt(x**2, 0)]], x, relational=False) == S.EmptySet
|
||
|
|
assert reduce_rational_inequalities(
|
||
|
|
[[Ge(x**2, 0)]], x, relational=False) == \
|
||
|
|
S.Reals if x.is_real else Interval(-oo, oo)
|
||
|
|
assert reduce_rational_inequalities(
|
||
|
|
[[Gt(x**2, 0)]], x, relational=False) == \
|
||
|
|
FiniteSet(0).complement(S.Reals)
|
||
|
|
assert reduce_rational_inequalities(
|
||
|
|
[[Ne(x**2, 0)]], x, relational=False) == \
|
||
|
|
FiniteSet(0).complement(S.Reals)
|
||
|
|
|
||
|
|
assert reduce_rational_inequalities(
|
||
|
|
[[Eq(x**2, 1)]], x, relational=False) == FiniteSet(-1, 1)
|
||
|
|
assert reduce_rational_inequalities(
|
||
|
|
[[Le(x**2, 1)]], x, relational=False) == Interval(-1, 1)
|
||
|
|
assert reduce_rational_inequalities(
|
||
|
|
[[Lt(x**2, 1)]], x, relational=False) == Interval(-1, 1, True, True)
|
||
|
|
assert reduce_rational_inequalities(
|
||
|
|
[[Ge(x**2, 1)]], x, relational=False) == \
|
||
|
|
Union(Interval(-oo, -1), Interval(1, oo))
|
||
|
|
assert reduce_rational_inequalities(
|
||
|
|
[[Gt(x**2, 1)]], x, relational=False) == \
|
||
|
|
Interval(-1, 1).complement(S.Reals)
|
||
|
|
assert reduce_rational_inequalities(
|
||
|
|
[[Ne(x**2, 1)]], x, relational=False) == \
|
||
|
|
FiniteSet(-1, 1).complement(S.Reals)
|
||
|
|
assert reduce_rational_inequalities([[Eq(
|
||
|
|
x**2, 1.0)]], x, relational=False) == FiniteSet(-1.0, 1.0).evalf()
|
||
|
|
assert reduce_rational_inequalities(
|
||
|
|
[[Le(x**2, 1.0)]], x, relational=False) == Interval(-1.0, 1.0)
|
||
|
|
assert reduce_rational_inequalities([[Lt(
|
||
|
|
x**2, 1.0)]], x, relational=False) == Interval(-1.0, 1.0, True, True)
|
||
|
|
assert reduce_rational_inequalities(
|
||
|
|
[[Ge(x**2, 1.0)]], x, relational=False) == \
|
||
|
|
Union(Interval(-inf, -1.0), Interval(1.0, inf))
|
||
|
|
assert reduce_rational_inequalities(
|
||
|
|
[[Gt(x**2, 1.0)]], x, relational=False) == \
|
||
|
|
Union(Interval(-inf, -1.0, right_open=True),
|
||
|
|
Interval(1.0, inf, left_open=True))
|
||
|
|
assert reduce_rational_inequalities([[Ne(
|
||
|
|
x**2, 1.0)]], x, relational=False) == \
|
||
|
|
FiniteSet(-1.0, 1.0).complement(S.Reals)
|
||
|
|
|
||
|
|
s = sqrt(2)
|
||
|
|
|
||
|
|
assert reduce_rational_inequalities([[Lt(
|
||
|
|
x**2 - 1, 0), Gt(x**2 - 1, 0)]], x, relational=False) == S.EmptySet
|
||
|
|
assert reduce_rational_inequalities([[Le(x**2 - 1, 0), Ge(
|
||
|
|
x**2 - 1, 0)]], x, relational=False) == FiniteSet(-1, 1)
|
||
|
|
assert reduce_rational_inequalities(
|
||
|
|
[[Le(x**2 - 2, 0), Ge(x**2 - 1, 0)]], x, relational=False
|
||
|
|
) == Union(Interval(-s, -1, False, False), Interval(1, s, False, False))
|
||
|
|
assert reduce_rational_inequalities(
|
||
|
|
[[Le(x**2 - 2, 0), Gt(x**2 - 1, 0)]], x, relational=False
|
||
|
|
) == Union(Interval(-s, -1, False, True), Interval(1, s, True, False))
|
||
|
|
assert reduce_rational_inequalities(
|
||
|
|
[[Lt(x**2 - 2, 0), Ge(x**2 - 1, 0)]], x, relational=False
|
||
|
|
) == Union(Interval(-s, -1, True, False), Interval(1, s, False, True))
|
||
|
|
assert reduce_rational_inequalities(
|
||
|
|
[[Lt(x**2 - 2, 0), Gt(x**2 - 1, 0)]], x, relational=False
|
||
|
|
) == Union(Interval(-s, -1, True, True), Interval(1, s, True, True))
|
||
|
|
assert reduce_rational_inequalities(
|
||
|
|
[[Lt(x**2 - 2, 0), Ne(x**2 - 1, 0)]], x, relational=False
|
||
|
|
) == Union(Interval(-s, -1, True, True), Interval(-1, 1, True, True),
|
||
|
|
Interval(1, s, True, True))
|
||
|
|
|
||
|
|
assert reduce_rational_inequalities([[Lt(x**2, -1.)]], x) is S.false
|
||
|
|
|
||
|
|
|
||
|
|
def test_reduce_poly_inequalities_complex_relational():
|
||
|
|
assert reduce_rational_inequalities(
|
||
|
|
[[Eq(x**2, 0)]], x, relational=True) == Eq(x, 0)
|
||
|
|
assert reduce_rational_inequalities(
|
||
|
|
[[Le(x**2, 0)]], x, relational=True) == Eq(x, 0)
|
||
|
|
assert reduce_rational_inequalities(
|
||
|
|
[[Lt(x**2, 0)]], x, relational=True) == False
|
||
|
|
assert reduce_rational_inequalities(
|
||
|
|
[[Ge(x**2, 0)]], x, relational=True) == And(Lt(-oo, x), Lt(x, oo))
|
||
|
|
assert reduce_rational_inequalities(
|
||
|
|
[[Gt(x**2, 0)]], x, relational=True) == \
|
||
|
|
And(Gt(x, -oo), Lt(x, oo), Ne(x, 0))
|
||
|
|
assert reduce_rational_inequalities(
|
||
|
|
[[Ne(x**2, 0)]], x, relational=True) == \
|
||
|
|
And(Gt(x, -oo), Lt(x, oo), Ne(x, 0))
|
||
|
|
|
||
|
|
for one in (S.One, S(1.0)):
|
||
|
|
inf = one*oo
|
||
|
|
assert reduce_rational_inequalities(
|
||
|
|
[[Eq(x**2, one)]], x, relational=True) == \
|
||
|
|
Or(Eq(x, -one), Eq(x, one))
|
||
|
|
assert reduce_rational_inequalities(
|
||
|
|
[[Le(x**2, one)]], x, relational=True) == \
|
||
|
|
And(And(Le(-one, x), Le(x, one)))
|
||
|
|
assert reduce_rational_inequalities(
|
||
|
|
[[Lt(x**2, one)]], x, relational=True) == \
|
||
|
|
And(And(Lt(-one, x), Lt(x, one)))
|
||
|
|
assert reduce_rational_inequalities(
|
||
|
|
[[Ge(x**2, one)]], x, relational=True) == \
|
||
|
|
And(Or(And(Le(one, x), Lt(x, inf)), And(Le(x, -one), Lt(-inf, x))))
|
||
|
|
assert reduce_rational_inequalities(
|
||
|
|
[[Gt(x**2, one)]], x, relational=True) == \
|
||
|
|
And(Or(And(Lt(-inf, x), Lt(x, -one)), And(Lt(one, x), Lt(x, inf))))
|
||
|
|
assert reduce_rational_inequalities(
|
||
|
|
[[Ne(x**2, one)]], x, relational=True) == \
|
||
|
|
Or(And(Lt(-inf, x), Lt(x, -one)),
|
||
|
|
And(Lt(-one, x), Lt(x, one)),
|
||
|
|
And(Lt(one, x), Lt(x, inf)))
|
||
|
|
|
||
|
|
|
||
|
|
def test_reduce_rational_inequalities_real_relational():
|
||
|
|
assert reduce_rational_inequalities([], x) == False
|
||
|
|
assert reduce_rational_inequalities(
|
||
|
|
[[(x**2 + 3*x + 2)/(x**2 - 16) >= 0]], x, relational=False) == \
|
||
|
|
Union(Interval.open(-oo, -4), Interval(-2, -1), Interval.open(4, oo))
|
||
|
|
|
||
|
|
assert reduce_rational_inequalities(
|
||
|
|
[[((-2*x - 10)*(3 - x))/((x**2 + 5)*(x - 2)**2) < 0]], x,
|
||
|
|
relational=False) == \
|
||
|
|
Union(Interval.open(-5, 2), Interval.open(2, 3))
|
||
|
|
|
||
|
|
assert reduce_rational_inequalities([[(x + 1)/(x - 5) <= 0]], x,
|
||
|
|
relational=False) == \
|
||
|
|
Interval.Ropen(-1, 5)
|
||
|
|
|
||
|
|
assert reduce_rational_inequalities([[(x**2 + 4*x + 3)/(x - 1) > 0]], x,
|
||
|
|
relational=False) == \
|
||
|
|
Union(Interval.open(-3, -1), Interval.open(1, oo))
|
||
|
|
|
||
|
|
assert reduce_rational_inequalities([[(x**2 - 16)/(x - 1)**2 < 0]], x,
|
||
|
|
relational=False) == \
|
||
|
|
Union(Interval.open(-4, 1), Interval.open(1, 4))
|
||
|
|
|
||
|
|
assert reduce_rational_inequalities([[(3*x + 1)/(x + 4) >= 1]], x,
|
||
|
|
relational=False) == \
|
||
|
|
Union(Interval.open(-oo, -4), Interval.Ropen(Rational(3, 2), oo))
|
||
|
|
|
||
|
|
assert reduce_rational_inequalities([[(x - 8)/x <= 3 - x]], x,
|
||
|
|
relational=False) == \
|
||
|
|
Union(Interval.Lopen(-oo, -2), Interval.Lopen(0, 4))
|
||
|
|
|
||
|
|
# issue sympy/sympy#10237
|
||
|
|
assert reduce_rational_inequalities(
|
||
|
|
[[x < oo, x >= 0, -oo < x]], x, relational=False) == Interval(0, oo)
|
||
|
|
|
||
|
|
|
||
|
|
def test_reduce_abs_inequalities():
|
||
|
|
e = abs(x - 5) < 3
|
||
|
|
ans = And(Lt(2, x), Lt(x, 8))
|
||
|
|
assert reduce_inequalities(e) == ans
|
||
|
|
assert reduce_inequalities(e, x) == ans
|
||
|
|
assert reduce_inequalities(abs(x - 5)) == Eq(x, 5)
|
||
|
|
assert reduce_inequalities(
|
||
|
|
abs(2*x + 3) >= 8) == Or(And(Le(Rational(5, 2), x), Lt(x, oo)),
|
||
|
|
And(Le(x, Rational(-11, 2)), Lt(-oo, x)))
|
||
|
|
assert reduce_inequalities(abs(x - 4) + abs(
|
||
|
|
3*x - 5) < 7) == And(Lt(S.Half, x), Lt(x, 4))
|
||
|
|
assert reduce_inequalities(abs(x - 4) + abs(3*abs(x) - 5) < 7) == \
|
||
|
|
Or(And(S(-2) < x, x < -1), And(S.Half < x, x < 4))
|
||
|
|
|
||
|
|
nr = Symbol('nr', extended_real=False)
|
||
|
|
raises(TypeError, lambda: reduce_inequalities(abs(nr - 5) < 3))
|
||
|
|
assert reduce_inequalities(x < 3, symbols=[x, nr]) == And(-oo < x, x < 3)
|
||
|
|
|
||
|
|
|
||
|
|
def test_reduce_inequalities_general():
|
||
|
|
assert reduce_inequalities(Ge(sqrt(2)*x, 1)) == And(sqrt(2)/2 <= x, x < oo)
|
||
|
|
assert reduce_inequalities(x + 1 > 0) == And(S.NegativeOne < x, x < oo)
|
||
|
|
|
||
|
|
|
||
|
|
def test_reduce_inequalities_boolean():
|
||
|
|
assert reduce_inequalities(
|
||
|
|
[Eq(x**2, 0), True]) == Eq(x, 0)
|
||
|
|
assert reduce_inequalities([Eq(x**2, 0), False]) == False
|
||
|
|
assert reduce_inequalities(x**2 >= 0) is S.true # issue 10196
|
||
|
|
|
||
|
|
|
||
|
|
def test_reduce_inequalities_multivariate():
|
||
|
|
assert reduce_inequalities([Ge(x**2, 1), Ge(y**2, 1)]) == And(
|
||
|
|
Or(And(Le(S.One, x), Lt(x, oo)), And(Le(x, -1), Lt(-oo, x))),
|
||
|
|
Or(And(Le(S.One, y), Lt(y, oo)), And(Le(y, -1), Lt(-oo, y))))
|
||
|
|
|
||
|
|
|
||
|
|
def test_reduce_inequalities_errors():
|
||
|
|
raises(NotImplementedError, lambda: reduce_inequalities(Ge(sin(x) + x, 1)))
|
||
|
|
raises(NotImplementedError, lambda: reduce_inequalities(Ge(x**2*y + y, 1)))
|
||
|
|
|
||
|
|
|
||
|
|
def test__solve_inequalities():
|
||
|
|
assert reduce_inequalities(x + y < 1, symbols=[x]) == (x < 1 - y)
|
||
|
|
assert reduce_inequalities(x + y >= 1, symbols=[x]) == (x < oo) & (x >= -y + 1)
|
||
|
|
assert reduce_inequalities(Eq(0, x - y), symbols=[x]) == Eq(x, y)
|
||
|
|
assert reduce_inequalities(Ne(0, x - y), symbols=[x]) == Ne(x, y)
|
||
|
|
|
||
|
|
|
||
|
|
def test_issue_6343():
|
||
|
|
eq = -3*x**2/2 - x*Rational(45, 4) + Rational(33, 2) > 0
|
||
|
|
assert reduce_inequalities(eq) == \
|
||
|
|
And(x < Rational(-15, 4) + sqrt(401)/4, -sqrt(401)/4 - Rational(15, 4) < x)
|
||
|
|
|
||
|
|
|
||
|
|
def test_issue_8235():
|
||
|
|
assert reduce_inequalities(x**2 - 1 < 0) == \
|
||
|
|
And(S.NegativeOne < x, x < 1)
|
||
|
|
assert reduce_inequalities(x**2 - 1 <= 0) == \
|
||
|
|
And(S.NegativeOne <= x, x <= 1)
|
||
|
|
assert reduce_inequalities(x**2 - 1 > 0) == \
|
||
|
|
Or(And(-oo < x, x < -1), And(x < oo, S.One < x))
|
||
|
|
assert reduce_inequalities(x**2 - 1 >= 0) == \
|
||
|
|
Or(And(-oo < x, x <= -1), And(S.One <= x, x < oo))
|
||
|
|
|
||
|
|
eq = x**8 + x - 9 # we want CRootOf solns here
|
||
|
|
sol = solve(eq >= 0)
|
||
|
|
tru = Or(And(rootof(eq, 1) <= x, x < oo), And(-oo < x, x <= rootof(eq, 0)))
|
||
|
|
assert sol == tru
|
||
|
|
|
||
|
|
# recast vanilla as real
|
||
|
|
assert solve(sqrt((-x + 1)**2) < 1) == And(S.Zero < x, x < 2)
|
||
|
|
|
||
|
|
|
||
|
|
def test_issue_5526():
|
||
|
|
assert reduce_inequalities(0 <=
|
||
|
|
x + Integral(y**2, (y, 1, 3)) - 1, [x]) == \
|
||
|
|
(x >= -Integral(y**2, (y, 1, 3)) + 1)
|
||
|
|
f = Function('f')
|
||
|
|
e = Sum(f(x), (x, 1, 3))
|
||
|
|
assert reduce_inequalities(0 <= x + e + y**2, [x]) == \
|
||
|
|
(x >= -y**2 - Sum(f(x), (x, 1, 3)))
|
||
|
|
|
||
|
|
|
||
|
|
def test_solve_univariate_inequality():
|
||
|
|
assert isolve(x**2 >= 4, x, relational=False) == Union(Interval(-oo, -2),
|
||
|
|
Interval(2, oo))
|
||
|
|
assert isolve(x**2 >= 4, x) == Or(And(Le(2, x), Lt(x, oo)), And(Le(x, -2),
|
||
|
|
Lt(-oo, x)))
|
||
|
|
assert isolve((x - 1)*(x - 2)*(x - 3) >= 0, x, relational=False) == \
|
||
|
|
Union(Interval(1, 2), Interval(3, oo))
|
||
|
|
assert isolve((x - 1)*(x - 2)*(x - 3) >= 0, x) == \
|
||
|
|
Or(And(Le(1, x), Le(x, 2)), And(Le(3, x), Lt(x, oo)))
|
||
|
|
assert isolve((x - 1)*(x - 2)*(x - 4) < 0, x, domain = FiniteSet(0, 3)) == \
|
||
|
|
Or(Eq(x, 0), Eq(x, 3))
|
||
|
|
# issue 2785:
|
||
|
|
assert isolve(x**3 - 2*x - 1 > 0, x, relational=False) == \
|
||
|
|
Union(Interval(-1, -sqrt(5)/2 + S.Half, True, True),
|
||
|
|
Interval(S.Half + sqrt(5)/2, oo, True, True))
|
||
|
|
# issue 2794:
|
||
|
|
assert isolve(x**3 - x**2 + x - 1 > 0, x, relational=False) == \
|
||
|
|
Interval(1, oo, True)
|
||
|
|
#issue 13105
|
||
|
|
assert isolve((x + I)*(x + 2*I) < 0, x) == Eq(x, 0)
|
||
|
|
assert isolve(((x - 1)*(x - 2) + I)*((x - 1)*(x - 2) + 2*I) < 0, x) == Or(Eq(x, 1), Eq(x, 2))
|
||
|
|
assert isolve((((x - 1)*(x - 2) + I)*((x - 1)*(x - 2) + 2*I))/(x - 2) > 0, x) == Eq(x, 1)
|
||
|
|
raises (ValueError, lambda: isolve((x**2 - 3*x*I + 2)/x < 0, x))
|
||
|
|
|
||
|
|
# numerical testing in valid() is needed
|
||
|
|
assert isolve(x**7 - x - 2 > 0, x) == \
|
||
|
|
And(rootof(x**7 - x - 2, 0) < x, x < oo)
|
||
|
|
|
||
|
|
# handle numerator and denominator; although these would be handled as
|
||
|
|
# rational inequalities, these test confirm that the right thing is done
|
||
|
|
# when the domain is EX (e.g. when 2 is replaced with sqrt(2))
|
||
|
|
assert isolve(1/(x - 2) > 0, x) == And(S(2) < x, x < oo)
|
||
|
|
den = ((x - 1)*(x - 2)).expand()
|
||
|
|
assert isolve((x - 1)/den <= 0, x) == \
|
||
|
|
(x > -oo) & (x < 2) & Ne(x, 1)
|
||
|
|
|
||
|
|
n = Dummy('n')
|
||
|
|
raises(NotImplementedError, lambda: isolve(Abs(x) <= n, x, relational=False))
|
||
|
|
c1 = Dummy("c1", positive=True)
|
||
|
|
raises(NotImplementedError, lambda: isolve(n/c1 < 0, c1))
|
||
|
|
n = Dummy('n', negative=True)
|
||
|
|
assert isolve(n/c1 > -2, c1) == (-n/2 < c1)
|
||
|
|
assert isolve(n/c1 < 0, c1) == True
|
||
|
|
assert isolve(n/c1 > 0, c1) == False
|
||
|
|
|
||
|
|
zero = cos(1)**2 + sin(1)**2 - 1
|
||
|
|
raises(NotImplementedError, lambda: isolve(x**2 < zero, x))
|
||
|
|
raises(NotImplementedError, lambda: isolve(
|
||
|
|
x**2 < zero*I, x))
|
||
|
|
raises(NotImplementedError, lambda: isolve(1/(x - y) < 2, x))
|
||
|
|
raises(NotImplementedError, lambda: isolve(1/(x - y) < 0, x))
|
||
|
|
raises(TypeError, lambda: isolve(x - I < 0, x))
|
||
|
|
|
||
|
|
zero = x**2 + x - x*(x + 1)
|
||
|
|
assert isolve(zero < 0, x, relational=False) is S.EmptySet
|
||
|
|
assert isolve(zero <= 0, x, relational=False) is S.Reals
|
||
|
|
|
||
|
|
# make sure iter_solutions gets a default value
|
||
|
|
raises(NotImplementedError, lambda: isolve(
|
||
|
|
Eq(cos(x)**2 + sin(x)**2, 1), x))
|
||
|
|
|
||
|
|
|
||
|
|
def test_trig_inequalities():
|
||
|
|
# all the inequalities are solved in a periodic interval.
|
||
|
|
assert isolve(sin(x) < S.Half, x, relational=False) == \
|
||
|
|
Union(Interval(0, pi/6, False, True), Interval.open(pi*Rational(5, 6), 2*pi))
|
||
|
|
assert isolve(sin(x) > S.Half, x, relational=False) == \
|
||
|
|
Interval(pi/6, pi*Rational(5, 6), True, True)
|
||
|
|
assert isolve(cos(x) < S.Zero, x, relational=False) == \
|
||
|
|
Interval(pi/2, pi*Rational(3, 2), True, True)
|
||
|
|
assert isolve(cos(x) >= S.Zero, x, relational=False) == \
|
||
|
|
Union(Interval(0, pi/2), Interval.Ropen(pi*Rational(3, 2), 2*pi))
|
||
|
|
|
||
|
|
assert isolve(tan(x) < S.One, x, relational=False) == \
|
||
|
|
Union(Interval.Ropen(0, pi/4), Interval.open(pi/2, pi))
|
||
|
|
|
||
|
|
assert isolve(sin(x) <= S.Zero, x, relational=False) == \
|
||
|
|
Union(FiniteSet(S.Zero), Interval.Ropen(pi, 2*pi))
|
||
|
|
|
||
|
|
assert isolve(sin(x) <= S.One, x, relational=False) == S.Reals
|
||
|
|
assert isolve(cos(x) < S(-2), x, relational=False) == S.EmptySet
|
||
|
|
assert isolve(sin(x) >= S.NegativeOne, x, relational=False) == S.Reals
|
||
|
|
assert isolve(cos(x) > S.One, x, relational=False) == S.EmptySet
|
||
|
|
|
||
|
|
|
||
|
|
def test_issue_9954():
|
||
|
|
assert isolve(x**2 >= 0, x, relational=False) == S.Reals
|
||
|
|
assert isolve(x**2 >= 0, x, relational=True) == S.Reals.as_relational(x)
|
||
|
|
assert isolve(x**2 < 0, x, relational=False) == S.EmptySet
|
||
|
|
assert isolve(x**2 < 0, x, relational=True) == S.EmptySet.as_relational(x)
|
||
|
|
|
||
|
|
|
||
|
|
@XFAIL
|
||
|
|
def test_slow_general_univariate():
|
||
|
|
r = rootof(x**5 - x**2 + 1, 0)
|
||
|
|
assert solve(sqrt(x) + 1/root(x, 3) > 1) == \
|
||
|
|
Or(And(0 < x, x < r**6), And(r**6 < x, x < oo))
|
||
|
|
|
||
|
|
|
||
|
|
def test_issue_8545():
|
||
|
|
eq = 1 - x - abs(1 - x)
|
||
|
|
ans = And(Lt(1, x), Lt(x, oo))
|
||
|
|
assert reduce_abs_inequality(eq, '<', x) == ans
|
||
|
|
eq = 1 - x - sqrt((1 - x)**2)
|
||
|
|
assert reduce_inequalities(eq < 0) == ans
|
||
|
|
|
||
|
|
|
||
|
|
def test_issue_8974():
|
||
|
|
assert isolve(-oo < x, x) == And(-oo < x, x < oo)
|
||
|
|
assert isolve(oo > x, x) == And(-oo < x, x < oo)
|
||
|
|
|
||
|
|
|
||
|
|
def test_issue_10198():
|
||
|
|
assert reduce_inequalities(
|
||
|
|
-1 + 1/abs(1/x - 1) < 0) == (x > -oo) & (x < S(1)/2) & Ne(x, 0)
|
||
|
|
|
||
|
|
assert reduce_inequalities(abs(1/sqrt(x)) - 1, x) == Eq(x, 1)
|
||
|
|
assert reduce_abs_inequality(-3 + 1/abs(1 - 1/x), '<', x) == \
|
||
|
|
Or(And(-oo < x, x < 0),
|
||
|
|
And(S.Zero < x, x < Rational(3, 4)), And(Rational(3, 2) < x, x < oo))
|
||
|
|
raises(ValueError,lambda: reduce_abs_inequality(-3 + 1/abs(
|
||
|
|
1 - 1/sqrt(x)), '<', x))
|
||
|
|
|
||
|
|
|
||
|
|
def test_issue_10047():
|
||
|
|
# issue 10047: this must remain an inequality, not True, since if x
|
||
|
|
# is not real the inequality is invalid
|
||
|
|
# assert solve(sin(x) < 2) == (x <= oo)
|
||
|
|
|
||
|
|
# with PR 16956, (x <= oo) autoevaluates when x is extended_real
|
||
|
|
# which is assumed in the current implementation of inequality solvers
|
||
|
|
assert solve(sin(x) < 2) == True
|
||
|
|
assert solveset(sin(x) < 2, domain=S.Reals) == S.Reals
|
||
|
|
|
||
|
|
|
||
|
|
def test_issue_10268():
|
||
|
|
assert solve(log(x) < 1000) == And(S.Zero < x, x < exp(1000))
|
||
|
|
|
||
|
|
|
||
|
|
@XFAIL
|
||
|
|
def test_isolve_Sets():
|
||
|
|
n = Dummy('n')
|
||
|
|
assert isolve(Abs(x) <= n, x, relational=False) == \
|
||
|
|
Piecewise((S.EmptySet, n < 0), (Interval(-n, n), True))
|
||
|
|
|
||
|
|
|
||
|
|
def test_integer_domain_relational_isolve():
|
||
|
|
|
||
|
|
dom = FiniteSet(0, 3)
|
||
|
|
x = Symbol('x',zero=False)
|
||
|
|
assert isolve((x - 1)*(x - 2)*(x - 4) < 0, x, domain=dom) == Eq(x, 3)
|
||
|
|
|
||
|
|
x = Symbol('x')
|
||
|
|
assert isolve(x + 2 < 0, x, domain=S.Integers) == \
|
||
|
|
(x <= -3) & (x > -oo) & Eq(Mod(x, 1), 0)
|
||
|
|
assert isolve(2 * x + 3 > 0, x, domain=S.Integers) == \
|
||
|
|
(x >= -1) & (x < oo) & Eq(Mod(x, 1), 0)
|
||
|
|
assert isolve((x ** 2 + 3 * x - 2) < 0, x, domain=S.Integers) == \
|
||
|
|
(x >= -3) & (x <= 0) & Eq(Mod(x, 1), 0)
|
||
|
|
assert isolve((x ** 2 + 3 * x - 2) > 0, x, domain=S.Integers) == \
|
||
|
|
((x >= 1) & (x < oo) & Eq(Mod(x, 1), 0)) | (
|
||
|
|
(x <= -4) & (x > -oo) & Eq(Mod(x, 1), 0))
|
||
|
|
|
||
|
|
|
||
|
|
def test_issue_10671_12466():
|
||
|
|
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
|
||
|
|
assert solveset((log(x - 6)/x) <= 0, x, S.Reals) == \
|
||
|
|
Interval.Lopen(6, 7)
|
||
|
|
|
||
|
|
|
||
|
|
def test__solve_inequality():
|
||
|
|
for op in (Gt, Lt, Le, Ge, Eq, Ne):
|
||
|
|
assert _solve_inequality(op(x, 1), x).lhs == x
|
||
|
|
assert _solve_inequality(op(S.One, x), x).lhs == x
|
||
|
|
# don't get tricked by symbol on right: solve it
|
||
|
|
assert _solve_inequality(Eq(2*x - 1, x), x) == Eq(x, 1)
|
||
|
|
ie = Eq(S.One, y)
|
||
|
|
assert _solve_inequality(ie, x) == ie
|
||
|
|
for fx in (x**2, exp(x), sin(x) + cos(x), x*(1 + x)):
|
||
|
|
for c in (0, 1):
|
||
|
|
e = 2*fx - c > 0
|
||
|
|
assert _solve_inequality(e, x, linear=True) == (
|
||
|
|
fx > c/S(2))
|
||
|
|
assert _solve_inequality(2*x**2 + 2*x - 1 < 0, x, linear=True) == (
|
||
|
|
x*(x + 1) < S.Half)
|
||
|
|
assert _solve_inequality(Eq(x*y, 1), x) == Eq(x*y, 1)
|
||
|
|
nz = Symbol('nz', nonzero=True)
|
||
|
|
assert _solve_inequality(Eq(x*nz, 1), x) == Eq(x, 1/nz)
|
||
|
|
assert _solve_inequality(x*nz < 1, x) == (x*nz < 1)
|
||
|
|
a = Symbol('a', positive=True)
|
||
|
|
assert _solve_inequality(a/x > 1, x) == (S.Zero < x) & (x < a)
|
||
|
|
assert _solve_inequality(a/x > 1, x, linear=True) == (1/x > 1/a)
|
||
|
|
# make sure to include conditions under which solution is valid
|
||
|
|
e = Eq(1 - x, x*(1/x - 1))
|
||
|
|
assert _solve_inequality(e, x) == Ne(x, 0)
|
||
|
|
assert _solve_inequality(x < x*(1/x - 1), x) == (x < S.Half) & Ne(x, 0)
|
||
|
|
|
||
|
|
|
||
|
|
def test__pt():
|
||
|
|
from sympy.solvers.inequalities import _pt
|
||
|
|
assert _pt(-oo, oo) == 0
|
||
|
|
assert _pt(S.One, S(3)) == 2
|
||
|
|
assert _pt(S.One, oo) == _pt(oo, S.One) == 2
|
||
|
|
assert _pt(S.One, -oo) == _pt(-oo, S.One) == S.Half
|
||
|
|
assert _pt(S.NegativeOne, oo) == _pt(oo, S.NegativeOne) == Rational(-1, 2)
|
||
|
|
assert _pt(S.NegativeOne, -oo) == _pt(-oo, S.NegativeOne) == -2
|
||
|
|
assert _pt(x, oo) == _pt(oo, x) == x + 1
|
||
|
|
assert _pt(x, -oo) == _pt(-oo, x) == x - 1
|
||
|
|
raises(ValueError, lambda: _pt(Dummy('i', infinite=True), S.One))
|