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

from sympy.core.numbers import (E, I, Rational, oo, pi)
from sympy.core.singleton import S
from sympy.core.symbol import (Symbol, symbols)
from sympy.functions.elementary.complexes import (Abs, re)
from sympy.functions.elementary.exponential import (exp, log)
from sympy.functions.elementary.miscellaneous import sqrt
from sympy.functions.elementary.piecewise import Piecewise
from sympy.functions.elementary.trigonometric import (cos, cot, csc, sec, sin, tan)
from sympy.functions.special.error_functions import expint
from sympy.matrices.expressions.matexpr import MatrixSymbol
from sympy.simplify.simplify import simplify
from sympy.calculus.util import (function_range, continuous_domain, not_empty_in,
                                 periodicity, lcim, is_convex,
                                 stationary_points, minimum, maximum)
from sympy.sets.sets import (Interval, FiniteSet, Complement, Union)
from sympy.testing.pytest import raises, _both_exp_pow
from sympy.abc import x

a = Symbol('a', real=True)

def test_function_range():
    x, y, a, b = symbols('x y a b')
    assert function_range(sin(x), x, Interval(-pi/2, pi/2)
        ) == Interval(-1, 1)
    assert function_range(sin(x), x, Interval(0, pi)
        ) == Interval(0, 1)
    assert function_range(tan(x), x, Interval(0, pi)
        ) == Interval(-oo, oo)
    assert function_range(tan(x), x, Interval(pi/2, pi)
        ) == Interval(-oo, 0)
    assert function_range((x + 3)/(x - 2), x, Interval(-5, 5)
        ) == Union(Interval(-oo, Rational(2, 7)), Interval(Rational(8, 3), oo))
    assert function_range(1/(x**2), x, Interval(-1, 1)
        ) == Interval(1, oo)
    assert function_range(exp(x), x, Interval(-1, 1)
        ) == Interval(exp(-1), exp(1))
    assert function_range(log(x) - x, x, S.Reals
        ) == Interval(-oo, -1)
    assert function_range(sqrt(3*x - 1), x, Interval(0, 2)
        ) == Interval(0, sqrt(5))
    assert function_range(x*(x - 1) - (x**2 - x), x, S.Reals
        ) == FiniteSet(0)
    assert function_range(x*(x - 1) - (x**2 - x) + y, x, S.Reals
        ) == FiniteSet(y)
    assert function_range(sin(x), x, Union(Interval(-5, -3), FiniteSet(4))
        ) == Union(Interval(-sin(3), 1), FiniteSet(sin(4)))
    assert function_range(cos(x), x, Interval(-oo, -4)
        ) == Interval(-1, 1)
    assert function_range(cos(x), x, S.EmptySet) == S.EmptySet
    assert function_range(x/sqrt(x**2+1), x, S.Reals) == Interval.open(-1,1)
    raises(NotImplementedError, lambda : function_range(
        exp(x)*(sin(x) - cos(x))/2 - x, x, S.Reals))
    raises(NotImplementedError, lambda : function_range(
        sin(x) + x, x, S.Reals)) # issue 13273
    raises(NotImplementedError, lambda : function_range(
        log(x), x, S.Integers))
    raises(NotImplementedError, lambda : function_range(
        sin(x)/2, x, S.Naturals))


def test_continuous_domain():
    x = Symbol('x')
    assert continuous_domain(sin(x), x, Interval(0, 2*pi)) == Interval(0, 2*pi)
    assert continuous_domain(tan(x), x, Interval(0, 2*pi)) == \
        Union(Interval(0, pi/2, False, True), Interval(pi/2, pi*Rational(3, 2), True, True),
              Interval(pi*Rational(3, 2), 2*pi, True, False))
    assert continuous_domain((x - 1)/((x - 1)**2), x, S.Reals) == \
        Union(Interval(-oo, 1, True, True), Interval(1, oo, True, True))
    assert continuous_domain(log(x) + log(4*x - 1), x, S.Reals) == \
        Interval(Rational(1, 4), oo, True, True)
    assert continuous_domain(1/sqrt(x - 3), x, S.Reals) == Interval(3, oo, True, True)
    assert continuous_domain(1/x - 2, x, S.Reals) == \
        Union(Interval.open(-oo, 0), Interval.open(0, oo))
    assert continuous_domain(1/(x**2 - 4) + 2, x, S.Reals) == \
        Union(Interval.open(-oo, -2), Interval.open(-2, 2), Interval.open(2, oo))
    domain = continuous_domain(log(tan(x)**2 + 1), x, S.Reals)
    assert not domain.contains(3*pi/2)
    assert domain.contains(5)
    d = Symbol('d', even=True, zero=False)
    assert continuous_domain(x**(1/d), x, S.Reals) == Interval(0, oo)


def test_not_empty_in():
    assert not_empty_in(FiniteSet(x, 2*x).intersect(Interval(1, 2, True, False)), x) == \
        Interval(S.Half, 2, True, False)
    assert not_empty_in(FiniteSet(x, x**2).intersect(Interval(1, 2)), x) == \
        Union(Interval(-sqrt(2), -1), Interval(1, 2))
    assert not_empty_in(FiniteSet(x**2 + x, x).intersect(Interval(2, 4)), x) == \
        Union(Interval(-sqrt(17)/2 - S.Half, -2),
              Interval(1, Rational(-1, 2) + sqrt(17)/2), Interval(2, 4))
    assert not_empty_in(FiniteSet(x/(x - 1)).intersect(S.Reals), x) == \
        Complement(S.Reals, FiniteSet(1))
    assert not_empty_in(FiniteSet(a/(a - 1)).intersect(S.Reals), a) == \
        Complement(S.Reals, FiniteSet(1))
    assert not_empty_in(FiniteSet((x**2 - 3*x + 2)/(x - 1)).intersect(S.Reals), x) == \
        Complement(S.Reals, FiniteSet(1))
    assert not_empty_in(FiniteSet(3, 4, x/(x - 1)).intersect(Interval(2, 3)), x) == \
        Interval(-oo, oo)
    assert not_empty_in(FiniteSet(4, x/(x - 1)).intersect(Interval(2, 3)), x) == \
        Interval(S(3)/2, 2)
    assert not_empty_in(FiniteSet(x/(x**2 - 1)).intersect(S.Reals), x) == \
        Complement(S.Reals, FiniteSet(-1, 1))
    assert not_empty_in(FiniteSet(x, x**2).intersect(Union(Interval(1, 3, True, True),
                                                           Interval(4, 5))), x) == \
        Union(Interval(-sqrt(5), -2), Interval(-sqrt(3), -1, True, True),
              Interval(1, 3, True, True), Interval(4, 5))
    assert not_empty_in(FiniteSet(1).intersect(Interval(3, 4)), x) == S.EmptySet
    assert not_empty_in(FiniteSet(x**2/(x + 2)).intersect(Interval(1, oo)), x) == \
        Union(Interval(-2, -1, True, False), Interval(2, oo))
    raises(ValueError, lambda: not_empty_in(x))
    raises(ValueError, lambda: not_empty_in(Interval(0, 1), x))
    raises(NotImplementedError,
           lambda: not_empty_in(FiniteSet(x).intersect(S.Reals), x, a))


@_both_exp_pow
def test_periodicity():
    x = Symbol('x')
    y = Symbol('y')
    z = Symbol('z', real=True)

    assert periodicity(sin(2*x), x) == pi
    assert periodicity((-2)*tan(4*x), x) == pi/4
    assert periodicity(sin(x)**2, x) == 2*pi
    assert periodicity(3**tan(3*x), x) == pi/3
    assert periodicity(tan(x)*cos(x), x) == 2*pi
    assert periodicity(sin(x)**(tan(x)), x) == 2*pi
    assert periodicity(tan(x)*sec(x), x) == 2*pi
    assert periodicity(sin(2*x)*cos(2*x) - y, x) == pi/2
    assert periodicity(tan(x) + cot(x), x) == pi
    assert periodicity(sin(x) - cos(2*x), x) == 2*pi
    assert periodicity(sin(x) - 1, x) == 2*pi
    assert periodicity(sin(4*x) + sin(x)*cos(x), x) == pi
    assert periodicity(exp(sin(x)), x) == 2*pi
    assert periodicity(log(cot(2*x)) - sin(cos(2*x)), x) == pi
    assert periodicity(sin(2*x)*exp(tan(x) - csc(2*x)), x) == pi
    assert periodicity(cos(sec(x) - csc(2*x)), x) == 2*pi
    assert periodicity(tan(sin(2*x)), x) == pi
    assert periodicity(2*tan(x)**2, x) == pi
    assert periodicity(sin(x%4), x) == 4
    assert periodicity(sin(x)%4, x) == 2*pi
    assert periodicity(tan((3*x-2)%4), x) == Rational(4, 3)
    assert periodicity((sqrt(2)*(x+1)+x) % 3, x) == 3 / (sqrt(2)+1)
    assert periodicity((x**2+1) % x, x) is None
    assert periodicity(sin(re(x)), x) == 2*pi
    assert periodicity(sin(x)**2 + cos(x)**2, x) is S.Zero
    assert periodicity(tan(x), y) is S.Zero
    assert periodicity(sin(x) + I*cos(x), x) == 2*pi
    assert periodicity(x - sin(2*y), y) == pi

    assert periodicity(exp(x), x) is None
    assert periodicity(exp(I*x), x) == 2*pi
    assert periodicity(exp(I*z), z) == 2*pi
    assert periodicity(exp(z), z) is None
    assert periodicity(exp(log(sin(z) + I*cos(2*z)), evaluate=False), z) == 2*pi
    assert periodicity(exp(log(sin(2*z) + I*cos(z)), evaluate=False), z) == 2*pi
    assert periodicity(exp(sin(z)), z) == 2*pi
    assert periodicity(exp(2*I*z), z) == pi
    assert periodicity(exp(z + I*sin(z)), z) is None
    assert periodicity(exp(cos(z/2) + sin(z)), z) == 4*pi
    assert periodicity(log(x), x) is None
    assert periodicity(exp(x)**sin(x), x) is None
    assert periodicity(sin(x)**y, y) is None

    assert periodicity(Abs(sin(Abs(sin(x)))), x) == pi
    assert all(periodicity(Abs(f(x)), x) == pi for f in (
        cos, sin, sec, csc, tan, cot))
    assert periodicity(Abs(sin(tan(x))), x) == pi
    assert periodicity(Abs(sin(sin(x) + tan(x))), x) == 2*pi
    assert periodicity(sin(x) > S.Half, x) == 2*pi

    assert periodicity(x > 2, x) is None
    assert periodicity(x**3 - x**2 + 1, x) is None
    assert periodicity(Abs(x), x) is None
    assert periodicity(Abs(x**2 - 1), x) is None

    assert periodicity((x**2 + 4)%2, x) is None
    assert periodicity((E**x)%3, x) is None

    assert periodicity(sin(expint(1, x))/expint(1, x), x) is None
    # returning `None` for any Piecewise
    p = Piecewise((0, x < -1), (x**2, x <= 1), (log(x), True))
    assert periodicity(p, x) is None

    m = MatrixSymbol('m', 3, 3)
    raises(NotImplementedError, lambda: periodicity(sin(m), m))
    raises(NotImplementedError, lambda: periodicity(sin(m[0, 0]), m))
    raises(NotImplementedError, lambda: periodicity(sin(m), m[0, 0]))
    raises(NotImplementedError, lambda: periodicity(sin(m[0, 0]), m[0, 0]))


def test_periodicity_check():
    x = Symbol('x')
    y = Symbol('y')

    assert periodicity(tan(x), x, check=True) == pi
    assert periodicity(sin(x) + cos(x), x, check=True) == 2*pi
    assert periodicity(sec(x), x) == 2*pi
    assert periodicity(sin(x*y), x) == 2*pi/abs(y)
    assert periodicity(Abs(sec(sec(x))), x) == pi


def test_lcim():
    assert lcim([S.Half, S(2), S(3)]) == 6
    assert lcim([pi/2, pi/4, pi]) == pi
    assert lcim([2*pi, pi/2]) == 2*pi
    assert lcim([S.One, 2*pi]) is None
    assert lcim([S(2) + 2*E, E/3 + Rational(1, 3), S.One + E]) == S(2) + 2*E


def test_is_convex():
    assert is_convex(1/x, x, domain=Interval.open(0, oo)) == True
    assert is_convex(1/x, x, domain=Interval(-oo, 0)) == False
    assert is_convex(x**2, x, domain=Interval(0, oo)) == True
    assert is_convex(1/x**3, x, domain=Interval.Lopen(0, oo)) == True
    assert is_convex(-1/x**3, x, domain=Interval.Ropen(-oo, 0)) == True
    assert is_convex(log(x), x) == False
    raises(NotImplementedError, lambda: is_convex(log(x), x, a))


def test_stationary_points():
    x, y = symbols('x y')

    assert stationary_points(sin(x), x, Interval(-pi/2, pi/2)
        ) == {-pi/2, pi/2}
    assert  stationary_points(sin(x), x, Interval.Ropen(0, pi/4)
        ) is S.EmptySet
    assert stationary_points(tan(x), x,
        ) is S.EmptySet
    assert stationary_points(sin(x)*cos(x), x, Interval(0, pi)
        ) == {pi/4, pi*Rational(3, 4)}
    assert stationary_points(sec(x), x, Interval(0, pi)
        ) == {0, pi}
    assert stationary_points((x+3)*(x-2), x
        ) == FiniteSet(Rational(-1, 2))
    assert stationary_points((x + 3)/(x - 2), x, Interval(-5, 5)
        ) is S.EmptySet
    assert stationary_points((x**2+3)/(x-2), x
        ) == {2 - sqrt(7), 2 + sqrt(7)}
    assert stationary_points((x**2+3)/(x-2), x, Interval(0, 5)
        ) == {2 + sqrt(7)}
    assert stationary_points(x**4 + x**3 - 5*x**2, x, S.Reals
        ) == FiniteSet(-2, 0, Rational(5, 4))
    assert stationary_points(exp(x), x
        ) is S.EmptySet
    assert stationary_points(log(x) - x, x, S.Reals
        ) == {1}
    assert stationary_points(cos(x), x, Union(Interval(0, 5), Interval(-6, -3))
        ) == {0, -pi, pi}
    assert stationary_points(y, x, S.Reals
        ) == S.Reals
    assert stationary_points(y, x, S.EmptySet) == S.EmptySet


def test_maximum():
    x, y = symbols('x y')
    assert maximum(sin(x), x) is S.One
    assert maximum(sin(x), x, Interval(0, 1)) == sin(1)
    assert maximum(tan(x), x) is oo
    assert maximum(tan(x), x, Interval(-pi/4, pi/4)) is S.One
    assert maximum(sin(x)*cos(x), x, S.Reals) == S.Half
    assert simplify(maximum(sin(x)*cos(x), x, Interval(pi*Rational(3, 8), pi*Rational(5, 8)))
        ) == sqrt(2)/4
    assert maximum((x+3)*(x-2), x) is oo
    assert maximum((x+3)*(x-2), x, Interval(-5, 0)) == S(14)
    assert maximum((x+3)/(x-2), x, Interval(-5, 0)) == Rational(2, 7)
    assert simplify(maximum(-x**4-x**3+x**2+10, x)
        ) == 41*sqrt(41)/512 + Rational(5419, 512)
    assert maximum(exp(x), x, Interval(-oo, 2)) == exp(2)
    assert maximum(log(x) - x, x, S.Reals) is S.NegativeOne
    assert maximum(cos(x), x, Union(Interval(0, 5), Interval(-6, -3))
        ) is S.One
    assert maximum(cos(x)-sin(x), x, S.Reals) == sqrt(2)
    assert maximum(y, x, S.Reals) == y
    assert maximum(abs(a**3 + a), a, Interval(0, 2)) == 10
    assert maximum(abs(60*a**3 + 24*a), a, Interval(0, 2)) == 528
    assert maximum(abs(12*a*(5*a**2 + 2)), a, Interval(0, 2)) == 528
    assert maximum(x/sqrt(x**2+1), x, S.Reals) == 1

    raises(ValueError, lambda : maximum(sin(x), x, S.EmptySet))
    raises(ValueError, lambda : maximum(log(cos(x)), x, S.EmptySet))
    raises(ValueError, lambda : maximum(1/(x**2 + y**2 + 1), x, S.EmptySet))
    raises(ValueError, lambda : maximum(sin(x), sin(x)))
    raises(ValueError, lambda : maximum(sin(x), x*y, S.EmptySet))
    raises(ValueError, lambda : maximum(sin(x), S.One))


def test_minimum():
    x, y = symbols('x y')

    assert minimum(sin(x), x) is S.NegativeOne
    assert minimum(sin(x), x, Interval(1, 4)) == sin(4)
    assert minimum(tan(x), x) is -oo
    assert minimum(tan(x), x, Interval(-pi/4, pi/4)) is S.NegativeOne
    assert minimum(sin(x)*cos(x), x, S.Reals) == Rational(-1, 2)
    assert simplify(minimum(sin(x)*cos(x), x, Interval(pi*Rational(3, 8), pi*Rational(5, 8)))
        ) == -sqrt(2)/4
    assert minimum((x+3)*(x-2), x) == Rational(-25, 4)
    assert minimum((x+3)/(x-2), x, Interval(-5, 0)) == Rational(-3, 2)
    assert minimum(x**4-x**3+x**2+10, x) == S(10)
    assert minimum(exp(x), x, Interval(-2, oo)) == exp(-2)
    assert minimum(log(x) - x, x, S.Reals) is -oo
    assert minimum(cos(x), x, Union(Interval(0, 5), Interval(-6, -3))
        ) is S.NegativeOne
    assert minimum(cos(x)-sin(x), x, S.Reals) == -sqrt(2)
    assert minimum(y, x, S.Reals) == y
    assert minimum(x/sqrt(x**2+1), x, S.Reals) == -1

    raises(ValueError, lambda : minimum(sin(x), x, S.EmptySet))
    raises(ValueError, lambda : minimum(log(cos(x)), x, S.EmptySet))
    raises(ValueError, lambda : minimum(1/(x**2 + y**2 + 1), x, S.EmptySet))
    raises(ValueError, lambda : minimum(sin(x), sin(x)))
    raises(ValueError, lambda : minimum(sin(x), x*y, S.EmptySet))
    raises(ValueError, lambda : minimum(sin(x), S.One))


def test_issue_19869():
    t = symbols('t')
    assert (maximum(sqrt(3)*(t - 1)/(3*sqrt(t**2 + 1)), t)
        ) == sqrt(3)/3


def test_issue_16469():
    x = Symbol("x", real=True)
    f = abs(x)
    assert function_range(f, x, S.Reals) == Interval(0, oo, False, True)


@_both_exp_pow
def test_issue_18747():
    assert periodicity(exp(pi*I*(x/4+S.Half/2)), x) == 8