venv/lib/python3.10/site-packages/sympy/integrals/tests/test_rde.py

"""Most of these tests come from the examples in Bronstein's book."""
from sympy.core.numbers import (I, Rational, oo)
from sympy.core.symbol import symbols
from sympy.polys.polytools import Poly
from sympy.integrals.risch import (DifferentialExtension,
    NonElementaryIntegralException)
from sympy.integrals.rde import (order_at, order_at_oo, weak_normalizer,
    normal_denom, special_denom, bound_degree, spde, solve_poly_rde,
    no_cancel_equal, cancel_primitive, cancel_exp, rischDE)

from sympy.testing.pytest import raises
from sympy.abc import x, t, z, n

t0, t1, t2, k = symbols('t:3 k')


def test_order_at():
    a = Poly(t**4, t)
    b = Poly((t**2 + 1)**3*t, t)
    c = Poly((t**2 + 1)**6*t, t)
    d = Poly((t**2 + 1)**10*t**10, t)
    e = Poly((t**2 + 1)**100*t**37, t)
    p1 = Poly(t, t)
    p2 = Poly(1 + t**2, t)
    assert order_at(a, p1, t) == 4
    assert order_at(b, p1, t) == 1
    assert order_at(c, p1, t) == 1
    assert order_at(d, p1, t) == 10
    assert order_at(e, p1, t) == 37
    assert order_at(a, p2, t) == 0
    assert order_at(b, p2, t) == 3
    assert order_at(c, p2, t) == 6
    assert order_at(d, p1, t) == 10
    assert order_at(e, p2, t) == 100
    assert order_at(Poly(0, t), Poly(t, t), t) is oo
    assert order_at_oo(Poly(t**2 - 1, t), Poly(t + 1), t) == \
        order_at_oo(Poly(t - 1, t), Poly(1, t), t) == -1
    assert order_at_oo(Poly(0, t), Poly(1, t), t) is oo

def test_weak_normalizer():
    a = Poly((1 + x)*t**5 + 4*t**4 + (-1 - 3*x)*t**3 - 4*t**2 + (-2 + 2*x)*t, t)
    d = Poly(t**4 - 3*t**2 + 2, t)
    DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t, t)]})
    r = weak_normalizer(a, d, DE, z)
    assert r == (Poly(t**5 - t**4 - 4*t**3 + 4*t**2 + 4*t - 4, t, domain='ZZ[x]'),
        (Poly((1 + x)*t**2 + x*t, t, domain='ZZ[x]'),
         Poly(t + 1, t, domain='ZZ[x]')))
    assert weak_normalizer(r[1][0], r[1][1], DE) == (Poly(1, t), r[1])
    r = weak_normalizer(Poly(1 + t**2), Poly(t**2 - 1, t), DE, z)
    assert r == (Poly(t**4 - 2*t**2 + 1, t), (Poly(-3*t**2 + 1, t), Poly(t**2 - 1, t)))
    assert weak_normalizer(r[1][0], r[1][1], DE, z) == (Poly(1, t), r[1])
    DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(1 + t**2)]})
    r = weak_normalizer(Poly(1 + t**2), Poly(t, t), DE, z)
    assert r == (Poly(t, t), (Poly(0, t), Poly(1, t)))
    assert weak_normalizer(r[1][0], r[1][1], DE, z) == (Poly(1, t), r[1])


def test_normal_denom():
    DE = DifferentialExtension(extension={'D': [Poly(1, x)]})
    raises(NonElementaryIntegralException, lambda: normal_denom(Poly(1, x), Poly(1, x),
    Poly(1, x), Poly(x, x), DE))
    fa, fd = Poly(t**2 + 1, t), Poly(1, t)
    ga, gd = Poly(1, t), Poly(t**2, t)
    DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t**2 + 1, t)]})
    assert normal_denom(fa, fd, ga, gd, DE) == \
        (Poly(t, t), (Poly(t**3 - t**2 + t - 1, t), Poly(1, t)), (Poly(1, t),
        Poly(1, t)), Poly(t, t))


def test_special_denom():
    # TODO: add more tests here
    DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t, t)]})
    assert special_denom(Poly(1, t), Poly(t**2, t), Poly(1, t), Poly(t**2 - 1, t),
    Poly(t, t), DE) == \
        (Poly(1, t), Poly(t**2 - 1, t), Poly(t**2 - 1, t), Poly(t, t))
#    assert special_denom(Poly(1, t), Poly(2*x, t), Poly((1 + 2*x)*t, t), DE) == 1

    # issue 3940
    # Note, this isn't a very good test, because the denominator is just 1,
    # but at least it tests the exp cancellation case
    DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(-2*x*t0, t0),
        Poly(I*k*t1, t1)]})
    DE.decrement_level()
    assert special_denom(Poly(1, t0), Poly(I*k, t0), Poly(1, t0), Poly(t0, t0),
    Poly(1, t0), DE) == \
        (Poly(1, t0, domain='ZZ'), Poly(I*k, t0, domain='ZZ_I[k,x]'),
                Poly(t0, t0, domain='ZZ'), Poly(1, t0, domain='ZZ'))


    assert special_denom(Poly(1, t), Poly(t**2, t), Poly(1, t), Poly(t**2 - 1, t),
    Poly(t, t), DE, case='tan') == \
           (Poly(1, t, t0, domain='ZZ'), Poly(t**2, t0, t, domain='ZZ[x]'),
            Poly(t, t, t0, domain='ZZ'), Poly(1, t0, domain='ZZ'))

    raises(ValueError, lambda: special_denom(Poly(1, t), Poly(t**2, t), Poly(1, t), Poly(t**2 - 1, t),
    Poly(t, t), DE, case='unrecognized_case'))


def test_bound_degree_fail():
    # Primitive
    DE = DifferentialExtension(extension={'D': [Poly(1, x),
        Poly(t0/x**2, t0), Poly(1/x, t)]})
    assert bound_degree(Poly(t**2, t), Poly(-(1/x**2*t**2 + 1/x), t),
        Poly((2*x - 1)*t**4 + (t0 + x)/x*t**3 - (t0 + 4*x**2)/2*x*t**2 + x*t,
        t), DE) == 3


def test_bound_degree():
    # Base
    DE = DifferentialExtension(extension={'D': [Poly(1, x)]})
    assert bound_degree(Poly(1, x), Poly(-2*x, x), Poly(1, x), DE) == 0

    # Primitive (see above test_bound_degree_fail)
    # TODO: Add test for when the degree bound becomes larger after limited_integrate
    # TODO: Add test for db == da - 1 case

    # Exp
    # TODO: Add tests
    # TODO: Add test for when the degree becomes larger after parametric_log_deriv()

    # Nonlinear
    DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t**2 + 1, t)]})
    assert bound_degree(Poly(t, t), Poly((t - 1)*(t**2 + 1), t), Poly(1, t), DE) == 0


def test_spde():
    DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t**2 + 1, t)]})
    raises(NonElementaryIntegralException, lambda: spde(Poly(t, t), Poly((t - 1)*(t**2 + 1), t), Poly(1, t), 0, DE))
    DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t, t)]})
    assert spde(Poly(t**2 + x*t*2 + x**2, t), Poly(t**2/x**2 + (2/x - 1)*t, t),
        Poly(t**2/x**2 + (2/x - 1)*t, t), 0, DE) == \
        (Poly(0, t), Poly(0, t), 0, Poly(0, t), Poly(1, t, domain='ZZ(x)'))
    DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t0/x**2, t0), Poly(1/x, t)]})
    assert spde(Poly(t**2, t), Poly(-t**2/x**2 - 1/x, t),
    Poly((2*x - 1)*t**4 + (t0 + x)/x*t**3 - (t0 + 4*x**2)/(2*x)*t**2 + x*t, t), 3, DE) == \
        (Poly(0, t), Poly(0, t), 0, Poly(0, t),
        Poly(t0*t**2/2 + x**2*t**2 - x**2*t, t, domain='ZZ(x,t0)'))
    DE = DifferentialExtension(extension={'D': [Poly(1, x)]})
    assert spde(Poly(x**2 + x + 1, x), Poly(-2*x - 1, x), Poly(x**5/2 +
    3*x**4/4 + x**3 - x**2 + 1, x), 4, DE) == \
        (Poly(0, x, domain='QQ'), Poly(x/2 - Rational(1, 4), x), 2, Poly(x**2 + x + 1, x), Poly(x*Rational(5, 4), x))
    assert spde(Poly(x**2 + x + 1, x), Poly(-2*x - 1, x), Poly(x**5/2 +
    3*x**4/4 + x**3 - x**2 + 1, x), n, DE) == \
        (Poly(0, x, domain='QQ'), Poly(x/2 - Rational(1, 4), x), -2 + n, Poly(x**2 + x + 1, x), Poly(x*Rational(5, 4), x))
    DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(1, t)]})
    raises(NonElementaryIntegralException, lambda: spde(Poly((t - 1)*(t**2 + 1)**2, t), Poly((t - 1)*(t**2 + 1), t), Poly(1, t), 0, DE))
    DE = DifferentialExtension(extension={'D': [Poly(1, x)]})
    assert spde(Poly(x**2 - x, x), Poly(1, x), Poly(9*x**4 - 10*x**3 + 2*x**2, x), 4, DE) == \
        (Poly(0, x, domain='ZZ'), Poly(0, x), 0, Poly(0, x), Poly(3*x**3 - 2*x**2, x, domain='QQ'))
    assert spde(Poly(x**2 - x, x), Poly(x**2 - 5*x + 3, x), Poly(x**7 - x**6 - 2*x**4 + 3*x**3 - x**2, x), 5, DE) == \
        (Poly(1, x, domain='QQ'), Poly(x + 1, x, domain='QQ'), 1, Poly(x**4 - x**3, x), Poly(x**3 - x**2, x, domain='QQ'))

def test_solve_poly_rde_no_cancel():
    # deg(b) large
    DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(1 + t**2, t)]})
    assert solve_poly_rde(Poly(t**2 + 1, t), Poly(t**3 + (x + 1)*t**2 + t + x + 2, t),
    oo, DE) == Poly(t + x, t)
    # deg(b) small
    DE = DifferentialExtension(extension={'D': [Poly(1, x)]})
    assert solve_poly_rde(Poly(0, x), Poly(x/2 - Rational(1, 4), x), oo, DE) == \
        Poly(x**2/4 - x/4, x)
    DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t**2 + 1, t)]})
    assert solve_poly_rde(Poly(2, t), Poly(t**2 + 2*t + 3, t), 1, DE) == \
        Poly(t + 1, t, x)
    # deg(b) == deg(D) - 1
    DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t**2 + 1, t)]})
    assert no_cancel_equal(Poly(1 - t, t),
    Poly(t**3 + t**2 - 2*x*t - 2*x, t), oo, DE) == \
        (Poly(t**2, t), 1, Poly((-2 - 2*x)*t - 2*x, t))


def test_solve_poly_rde_cancel():
    # exp
    DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t, t)]})
    assert cancel_exp(Poly(2*x, t), Poly(2*x, t), 0, DE) == \
        Poly(1, t)
    assert cancel_exp(Poly(2*x, t), Poly((1 + 2*x)*t, t), 1, DE) == \
        Poly(t, t)
    # TODO: Add more exp tests, including tests that require is_deriv_in_field()

    # primitive
    DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(1/x, t)]})

    # If the DecrementLevel context manager is working correctly, this shouldn't
    # cause any problems with the further tests.
    raises(NonElementaryIntegralException, lambda: cancel_primitive(Poly(1, t), Poly(t, t), oo, DE))

    assert cancel_primitive(Poly(1, t), Poly(t + 1/x, t), 2, DE) == \
        Poly(t, t)
    assert cancel_primitive(Poly(4*x, t), Poly(4*x*t**2 + 2*t/x, t), 3, DE) == \
        Poly(t**2, t)

    # TODO: Add more primitive tests, including tests that require is_deriv_in_field()


def test_rischDE():
    # TODO: Add more tests for rischDE, including ones from the text
    DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t, t)]})
    DE.decrement_level()
    assert rischDE(Poly(-2*x, x), Poly(1, x), Poly(1 - 2*x - 2*x**2, x),
    Poly(1, x), DE) == \
        (Poly(x + 1, x), Poly(1, x))