203 lines
9.3 KiB
Python
203 lines
9.3 KiB
Python
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"""Most of these tests come from the examples in Bronstein's book."""
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from sympy.core.numbers import (I, Rational, oo)
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from sympy.core.symbol import symbols
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from sympy.polys.polytools import Poly
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from sympy.integrals.risch import (DifferentialExtension,
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NonElementaryIntegralException)
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from sympy.integrals.rde import (order_at, order_at_oo, weak_normalizer,
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normal_denom, special_denom, bound_degree, spde, solve_poly_rde,
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no_cancel_equal, cancel_primitive, cancel_exp, rischDE)
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from sympy.testing.pytest import raises
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from sympy.abc import x, t, z, n
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t0, t1, t2, k = symbols('t:3 k')
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def test_order_at():
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a = Poly(t**4, t)
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b = Poly((t**2 + 1)**3*t, t)
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c = Poly((t**2 + 1)**6*t, t)
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d = Poly((t**2 + 1)**10*t**10, t)
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e = Poly((t**2 + 1)**100*t**37, t)
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p1 = Poly(t, t)
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p2 = Poly(1 + t**2, t)
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assert order_at(a, p1, t) == 4
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assert order_at(b, p1, t) == 1
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assert order_at(c, p1, t) == 1
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assert order_at(d, p1, t) == 10
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assert order_at(e, p1, t) == 37
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assert order_at(a, p2, t) == 0
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assert order_at(b, p2, t) == 3
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assert order_at(c, p2, t) == 6
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assert order_at(d, p1, t) == 10
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assert order_at(e, p2, t) == 100
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assert order_at(Poly(0, t), Poly(t, t), t) is oo
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assert order_at_oo(Poly(t**2 - 1, t), Poly(t + 1), t) == \
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order_at_oo(Poly(t - 1, t), Poly(1, t), t) == -1
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assert order_at_oo(Poly(0, t), Poly(1, t), t) is oo
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def test_weak_normalizer():
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a = Poly((1 + x)*t**5 + 4*t**4 + (-1 - 3*x)*t**3 - 4*t**2 + (-2 + 2*x)*t, t)
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d = Poly(t**4 - 3*t**2 + 2, t)
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DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t, t)]})
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r = weak_normalizer(a, d, DE, z)
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assert r == (Poly(t**5 - t**4 - 4*t**3 + 4*t**2 + 4*t - 4, t, domain='ZZ[x]'),
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(Poly((1 + x)*t**2 + x*t, t, domain='ZZ[x]'),
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Poly(t + 1, t, domain='ZZ[x]')))
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assert weak_normalizer(r[1][0], r[1][1], DE) == (Poly(1, t), r[1])
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r = weak_normalizer(Poly(1 + t**2), Poly(t**2 - 1, t), DE, z)
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assert r == (Poly(t**4 - 2*t**2 + 1, t), (Poly(-3*t**2 + 1, t), Poly(t**2 - 1, t)))
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assert weak_normalizer(r[1][0], r[1][1], DE, z) == (Poly(1, t), r[1])
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DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(1 + t**2)]})
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r = weak_normalizer(Poly(1 + t**2), Poly(t, t), DE, z)
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assert r == (Poly(t, t), (Poly(0, t), Poly(1, t)))
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assert weak_normalizer(r[1][0], r[1][1], DE, z) == (Poly(1, t), r[1])
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def test_normal_denom():
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DE = DifferentialExtension(extension={'D': [Poly(1, x)]})
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raises(NonElementaryIntegralException, lambda: normal_denom(Poly(1, x), Poly(1, x),
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Poly(1, x), Poly(x, x), DE))
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fa, fd = Poly(t**2 + 1, t), Poly(1, t)
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ga, gd = Poly(1, t), Poly(t**2, t)
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DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t**2 + 1, t)]})
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assert normal_denom(fa, fd, ga, gd, DE) == \
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(Poly(t, t), (Poly(t**3 - t**2 + t - 1, t), Poly(1, t)), (Poly(1, t),
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Poly(1, t)), Poly(t, t))
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def test_special_denom():
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# TODO: add more tests here
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DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t, t)]})
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assert special_denom(Poly(1, t), Poly(t**2, t), Poly(1, t), Poly(t**2 - 1, t),
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Poly(t, t), DE) == \
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(Poly(1, t), Poly(t**2 - 1, t), Poly(t**2 - 1, t), Poly(t, t))
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# assert special_denom(Poly(1, t), Poly(2*x, t), Poly((1 + 2*x)*t, t), DE) == 1
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# issue 3940
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# Note, this isn't a very good test, because the denominator is just 1,
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# but at least it tests the exp cancellation case
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DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(-2*x*t0, t0),
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Poly(I*k*t1, t1)]})
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DE.decrement_level()
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assert special_denom(Poly(1, t0), Poly(I*k, t0), Poly(1, t0), Poly(t0, t0),
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Poly(1, t0), DE) == \
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(Poly(1, t0, domain='ZZ'), Poly(I*k, t0, domain='ZZ_I[k,x]'),
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Poly(t0, t0, domain='ZZ'), Poly(1, t0, domain='ZZ'))
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assert special_denom(Poly(1, t), Poly(t**2, t), Poly(1, t), Poly(t**2 - 1, t),
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Poly(t, t), DE, case='tan') == \
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(Poly(1, t, t0, domain='ZZ'), Poly(t**2, t0, t, domain='ZZ[x]'),
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Poly(t, t, t0, domain='ZZ'), Poly(1, t0, domain='ZZ'))
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raises(ValueError, lambda: special_denom(Poly(1, t), Poly(t**2, t), Poly(1, t), Poly(t**2 - 1, t),
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Poly(t, t), DE, case='unrecognized_case'))
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def test_bound_degree_fail():
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# Primitive
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DE = DifferentialExtension(extension={'D': [Poly(1, x),
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Poly(t0/x**2, t0), Poly(1/x, t)]})
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assert bound_degree(Poly(t**2, t), Poly(-(1/x**2*t**2 + 1/x), t),
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Poly((2*x - 1)*t**4 + (t0 + x)/x*t**3 - (t0 + 4*x**2)/2*x*t**2 + x*t,
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t), DE) == 3
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def test_bound_degree():
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# Base
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DE = DifferentialExtension(extension={'D': [Poly(1, x)]})
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assert bound_degree(Poly(1, x), Poly(-2*x, x), Poly(1, x), DE) == 0
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# Primitive (see above test_bound_degree_fail)
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# TODO: Add test for when the degree bound becomes larger after limited_integrate
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# TODO: Add test for db == da - 1 case
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# Exp
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# TODO: Add tests
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# TODO: Add test for when the degree becomes larger after parametric_log_deriv()
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# Nonlinear
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DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t**2 + 1, t)]})
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assert bound_degree(Poly(t, t), Poly((t - 1)*(t**2 + 1), t), Poly(1, t), DE) == 0
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def test_spde():
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DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t**2 + 1, t)]})
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raises(NonElementaryIntegralException, lambda: spde(Poly(t, t), Poly((t - 1)*(t**2 + 1), t), Poly(1, t), 0, DE))
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DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t, t)]})
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assert spde(Poly(t**2 + x*t*2 + x**2, t), Poly(t**2/x**2 + (2/x - 1)*t, t),
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Poly(t**2/x**2 + (2/x - 1)*t, t), 0, DE) == \
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(Poly(0, t), Poly(0, t), 0, Poly(0, t), Poly(1, t, domain='ZZ(x)'))
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DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t0/x**2, t0), Poly(1/x, t)]})
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assert spde(Poly(t**2, t), Poly(-t**2/x**2 - 1/x, t),
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Poly((2*x - 1)*t**4 + (t0 + x)/x*t**3 - (t0 + 4*x**2)/(2*x)*t**2 + x*t, t), 3, DE) == \
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(Poly(0, t), Poly(0, t), 0, Poly(0, t),
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Poly(t0*t**2/2 + x**2*t**2 - x**2*t, t, domain='ZZ(x,t0)'))
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DE = DifferentialExtension(extension={'D': [Poly(1, x)]})
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assert spde(Poly(x**2 + x + 1, x), Poly(-2*x - 1, x), Poly(x**5/2 +
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3*x**4/4 + x**3 - x**2 + 1, x), 4, DE) == \
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(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))
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assert spde(Poly(x**2 + x + 1, x), Poly(-2*x - 1, x), Poly(x**5/2 +
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3*x**4/4 + x**3 - x**2 + 1, x), n, DE) == \
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(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))
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DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(1, t)]})
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raises(NonElementaryIntegralException, lambda: spde(Poly((t - 1)*(t**2 + 1)**2, t), Poly((t - 1)*(t**2 + 1), t), Poly(1, t), 0, DE))
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DE = DifferentialExtension(extension={'D': [Poly(1, x)]})
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assert spde(Poly(x**2 - x, x), Poly(1, x), Poly(9*x**4 - 10*x**3 + 2*x**2, x), 4, DE) == \
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(Poly(0, x, domain='ZZ'), Poly(0, x), 0, Poly(0, x), Poly(3*x**3 - 2*x**2, x, domain='QQ'))
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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) == \
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(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'))
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def test_solve_poly_rde_no_cancel():
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# deg(b) large
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DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(1 + t**2, t)]})
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assert solve_poly_rde(Poly(t**2 + 1, t), Poly(t**3 + (x + 1)*t**2 + t + x + 2, t),
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oo, DE) == Poly(t + x, t)
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# deg(b) small
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DE = DifferentialExtension(extension={'D': [Poly(1, x)]})
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assert solve_poly_rde(Poly(0, x), Poly(x/2 - Rational(1, 4), x), oo, DE) == \
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Poly(x**2/4 - x/4, x)
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DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t**2 + 1, t)]})
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assert solve_poly_rde(Poly(2, t), Poly(t**2 + 2*t + 3, t), 1, DE) == \
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Poly(t + 1, t, x)
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# deg(b) == deg(D) - 1
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DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t**2 + 1, t)]})
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assert no_cancel_equal(Poly(1 - t, t),
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Poly(t**3 + t**2 - 2*x*t - 2*x, t), oo, DE) == \
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(Poly(t**2, t), 1, Poly((-2 - 2*x)*t - 2*x, t))
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def test_solve_poly_rde_cancel():
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# exp
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DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t, t)]})
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assert cancel_exp(Poly(2*x, t), Poly(2*x, t), 0, DE) == \
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Poly(1, t)
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assert cancel_exp(Poly(2*x, t), Poly((1 + 2*x)*t, t), 1, DE) == \
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Poly(t, t)
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# TODO: Add more exp tests, including tests that require is_deriv_in_field()
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# primitive
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DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(1/x, t)]})
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# If the DecrementLevel context manager is working correctly, this shouldn't
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# cause any problems with the further tests.
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raises(NonElementaryIntegralException, lambda: cancel_primitive(Poly(1, t), Poly(t, t), oo, DE))
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assert cancel_primitive(Poly(1, t), Poly(t + 1/x, t), 2, DE) == \
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Poly(t, t)
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assert cancel_primitive(Poly(4*x, t), Poly(4*x*t**2 + 2*t/x, t), 3, DE) == \
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Poly(t**2, t)
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# TODO: Add more primitive tests, including tests that require is_deriv_in_field()
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def test_rischDE():
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# TODO: Add more tests for rischDE, including ones from the text
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DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t, t)]})
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DE.decrement_level()
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assert rischDE(Poly(-2*x, x), Poly(1, x), Poly(1 - 2*x - 2*x**2, x),
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Poly(1, x), DE) == \
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(Poly(x + 1, x), Poly(1, x))
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