def test_param_poly_rischDE():
DE = DifferentialExtension(extension={'D': [Poly(1, x)]})
a = Poly(x**2 - x, x, field=True)
b = Poly(1, x, field=True)
q = [Poly(x, x, field=True), Poly(x**2, x, field=True)]
h, A = param_poly_rischDE(a, b, q, 3, DE)
assert A.nullspace() == [Matrix([0, 1, 1, 1])] # c1, c2, d1, d2
# Solution of a*Dp + b*p = c1*q1 + c2*q2 = q2 = x**2
# is d1*h1 + d2*h2 = h1 + h2 = x.
assert h[0] + h[1] == Poly(x, x)
# a*Dp + b*p = q1 = x has no solution.
a = Poly(x**2 - x, x, field=True)
b = Poly(x**2 - 5 * x + 3, x, field=True)
q = [
Poly(1, x, field=True),
Poly(x, x, field=True),
Poly(x**2, x, field=True)
]
h, A = param_poly_rischDE(a, b, q, 3, DE)
assert A.nullspace() == [Matrix([3, -5, 1, -5, 1, 1])]
p = -5 * h[0] + h[1] + h[2] # Poly(1, x)
assert a * derivation(p, DE) + b * p == Poly(x**2 - 5 * x + 3, x)
def test_param_poly_rischDE():
DE = DifferentialExtension(extension={'D': [Poly(1, x)]})
a = Poly(x**2 - x, x, field=True)
b = Poly(1, x, field=True)
q = [Poly(x, x, field=True), Poly(x**2, x, field=True)]
h, A = param_poly_rischDE(a, b, q, 3, DE)
assert A.nullspace() == [Matrix([0, 1, 1, 1])] # c1, c2, d1, d2
# Solution of a*Dp + b*p = c1*q1 + c2*q2 = q2 = x**2
# is d1*h1 + d2*h2 = h1 + h2 = x.
assert h[0] + h[1] == Poly(x, x)
# a*Dp + b*p = q1 = x has no solution.
a = Poly(x**2 - x, x, field=True)
b = Poly(x**2 - 5*x + 3, x, field=True)
q = [Poly(1, x, field=True), Poly(x, x, field=True),
Poly(x**2, x, field=True)]
h, A = param_poly_rischDE(a, b, q, 3, DE)
assert A.nullspace() == [Matrix([3, -5, 1, -5, 1, 1])]
p = -5*h[0] + h[1] + h[2] # Poly(1, x)
assert a*derivation(p, DE) + b*p == Poly(x**2 - 5*x + 3, x)
