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))
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))
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), Poly(x/2 - S(1)/4, x), 2, Poly(x**2 + x + 1, x), Poly(5*x/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), Poly(x/2 - S(1)/4, x), -2 + n, Poly(x**2 + x + 1, x), Poly(5*x/4, x))
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))
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))
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), Poly(x/2 - S(1)/4, x), 2, Poly(x**2 + x + 1, x), Poly(5*x/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), Poly(x/2 - S(1)/4, x), -2 + n, Poly(x**2 + x + 1, x), Poly(5*x/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), Poly(0, x), 0, Poly(0, x), Poly(3*x**3 - 2*x**2, x))
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), Poly(x + 1, x), 1, Poly(x**4 - x**3, x), Poly(x**3 - x**2, x))
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 limited_integrate(fa, fd, G, DE):
"""
Solves the limited integration problem: f = Dv + Sum(ci*wi, (i, 1, n))
"""
fa, fd = fa * Poly(1 / fd.LC(), DE.t), fd.monic()
A, B, h, N, g, V = limited_integrate_reduce(fa, fd, G, DE)
V = [g] + V
g = A.gcd(B)
A, B, V = A.quo(g), B.quo(g), [
via.cancel(vid * g, include=True) for via, vid in V
]
Q, M = prde_linear_constraints(A, B, V, DE)
M, _ = constant_system(M, zeros(M.rows, 1), DE)
l = M.nullspace()
if M == Matrix() or len(l) > 1:
# Continue with param_rischDE()
raise NotImplementedError("param_rischDE() is required to solve this "
"integral.")
elif len(l) == 0:
raise NonElementaryIntegralException
elif len(l) == 1:
# The c1 == 1. In this case, we can assume a normal Risch DE
if l[0][0].is_zero:
raise NonElementaryIntegralException
else:
l[0] *= 1 / l[0][0]
C = sum([Poly(i, DE.t) * q for (i, q) in zip(l[0], Q)])
# Custom version of rischDE() that uses the already computed
# denominator and degree bound from above.
B, C, m, alpha, beta = spde(A, B, C, N, DE)
y = solve_poly_rde(B, C, m, DE)
return ((alpha * y + beta, h), list(l[0][1:]))
else:
raise NotImplementedError
def limited_integrate(fa, fd, G, DE):
"""
Solves the limited integration problem: f = Dv + Sum(ci*wi, (i, 1, n))
"""
fa, fd = fa*Poly(1/fd.LC(), DE.t), fd.monic()
A, B, h, N, g, V = limited_integrate_reduce(fa, fd, G, DE)
V = [g] + V
g = A.gcd(B)
A, B, V = A.quo(g), B.quo(g), [via.cancel(vid*g, include=True) for
via, vid in V]
Q, M = prde_linear_constraints(A, B, V, DE)
M, _ = constant_system(M, zeros(M.rows, 1), DE)
l = M.nullspace()
if M == Matrix() or len(l) > 1:
# Continue with param_rischDE()
raise NotImplementedError("param_rischDE() is required to solve this "
"integral.")
elif len(l) == 0:
raise NonElementaryIntegralException
elif len(l) == 1:
# The c1 == 1. In this case, we can assume a normal Risch DE
if l[0][0].is_zero:
raise NonElementaryIntegralException
else:
l[0] *= 1/l[0][0]
C = sum([Poly(i, DE.t)*q for (i, q) in zip(l[0], Q)])
# Custom version of rischDE() that uses the already computed
# denominator and degree bound from above.
B, C, m, alpha, beta = spde(A, B, C, N, DE)
y = solve_poly_rde(B, C, m, DE)
return ((alpha*y + beta, h), list(l[0][1:]))
else:
raise NotImplementedError