Vehicle-cpp/hkpc/software/sympy/calculus/tests/test_singularities.py

from sympy.core.numbers import (I, Rational, oo)
from sympy.core.singleton import S
from sympy.core.symbol import Symbol
from sympy.functions.elementary.exponential import (exp, log)
from sympy.functions.elementary.miscellaneous import sqrt
from sympy.calculus.singularities import (
    singularities,
    is_increasing,
    is_strictly_increasing,
    is_decreasing,
    is_strictly_decreasing,
    is_monotonic
)
from sympy.sets import Interval, FiniteSet
from sympy.testing.pytest import raises
from sympy.abc import x, y


def test_singularities():
    x = Symbol('x')
    assert singularities(x**2, x) == S.EmptySet
    assert singularities(x/(x**2 + 3*x + 2), x) == FiniteSet(-2, -1)
    assert singularities(1/(x**2 + 1), x) == FiniteSet(I, -I)
    assert singularities(x/(x**3 + 1), x) == \
        FiniteSet(-1, (1 - sqrt(3) * I) / 2, (1 + sqrt(3) * I) / 2)
    assert singularities(1/(y**2 + 2*I*y + 1), y) == \
        FiniteSet(-I + sqrt(2)*I, -I - sqrt(2)*I)

    x = Symbol('x', real=True)
    assert singularities(1/(x**2 + 1), x) == S.EmptySet
    assert singularities(exp(1/x), x, S.Reals) == FiniteSet(0)
    assert singularities(exp(1/x), x, Interval(1, 2)) == S.EmptySet
    assert singularities(log((x - 2)**2), x, Interval(1, 3)) == FiniteSet(2)
    raises(NotImplementedError, lambda: singularities(x**-oo, x))


def test_is_increasing():
    """Test whether is_increasing returns correct value."""
    a = Symbol('a', negative=True)

    assert is_increasing(x**3 - 3*x**2 + 4*x, S.Reals)
    assert is_increasing(-x**2, Interval(-oo, 0))
    assert not is_increasing(-x**2, Interval(0, oo))
    assert not is_increasing(4*x**3 - 6*x**2 - 72*x + 30, Interval(-2, 3))
    assert is_increasing(x**2 + y, Interval(1, oo), x)
    assert is_increasing(-x**2*a, Interval(1, oo), x)
    assert is_increasing(1)

    assert is_increasing(4*x**3 - 6*x**2 - 72*x + 30, Interval(-2, 3)) is False


def test_is_strictly_increasing():
    """Test whether is_strictly_increasing returns correct value."""
    assert is_strictly_increasing(
        4*x**3 - 6*x**2 - 72*x + 30, Interval.Ropen(-oo, -2))
    assert is_strictly_increasing(
        4*x**3 - 6*x**2 - 72*x + 30, Interval.Lopen(3, oo))
    assert not is_strictly_increasing(
        4*x**3 - 6*x**2 - 72*x + 30, Interval.open(-2, 3))
    assert not is_strictly_increasing(-x**2, Interval(0, oo))
    assert not is_strictly_decreasing(1)

    assert is_strictly_increasing(4*x**3 - 6*x**2 - 72*x + 30, Interval.open(-2, 3)) is False


def test_is_decreasing():
    """Test whether is_decreasing returns correct value."""
    b = Symbol('b', positive=True)

    assert is_decreasing(1/(x**2 - 3*x), Interval.open(Rational(3,2), 3))
    assert is_decreasing(1/(x**2 - 3*x), Interval.open(1.5, 3))
    assert is_decreasing(1/(x**2 - 3*x), Interval.Lopen(3, oo))
    assert not is_decreasing(1/(x**2 - 3*x), Interval.Ropen(-oo, Rational(3, 2)))
    assert not is_decreasing(-x**2, Interval(-oo, 0))
    assert not is_decreasing(-x**2*b, Interval(-oo, 0), x)


def test_is_strictly_decreasing():
    """Test whether is_strictly_decreasing returns correct value."""
    assert is_strictly_decreasing(1/(x**2 - 3*x), Interval.Lopen(3, oo))
    assert not is_strictly_decreasing(
        1/(x**2 - 3*x), Interval.Ropen(-oo, Rational(3, 2)))
    assert not is_strictly_decreasing(-x**2, Interval(-oo, 0))
    assert not is_strictly_decreasing(1)
    assert is_strictly_decreasing(1/(x**2 - 3*x), Interval.open(Rational(3,2), 3))
    assert is_strictly_decreasing(1/(x**2 - 3*x), Interval.open(1.5, 3))


def test_is_monotonic():
    """Test whether is_monotonic returns correct value."""
    assert is_monotonic(1/(x**2 - 3*x), Interval.open(Rational(3,2), 3))
    assert is_monotonic(1/(x**2 - 3*x), Interval.open(1.5, 3))
    assert is_monotonic(1/(x**2 - 3*x), Interval.Lopen(3, oo))
    assert is_monotonic(x**3 - 3*x**2 + 4*x, S.Reals)
    assert not is_monotonic(-x**2, S.Reals)
    assert is_monotonic(x**2 + y + 1, Interval(1, 2), x)
    raises(NotImplementedError, lambda: is_monotonic(x**2 + y + 1))


def test_issue_23401():
    x = Symbol('x')
    expr = (x + 1)/(-1.0e-3*x**2 + 0.1*x + 0.1)
    assert is_increasing(expr, Interval(1,2), x)