def test_recognize_log_derivative():
a = Poly(2*x**2 + 4*x*t - 2*t - x**2*t, t)
d = Poly((2*x + t)*(t + x**2), t)
DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t, t)]})
assert recognize_log_derivative(a, d, DE, z) is True
DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(1/x, t)]})
assert recognize_log_derivative(Poly(t + 1, t), Poly(t + x, t), DE) is True
assert recognize_log_derivative(Poly(2, t), Poly(t**2 - 1, t), DE) is True
DE = DifferentialExtension(extension={'D': [Poly(1, x)]})
assert recognize_log_derivative(Poly(1, x), Poly(x**2 - 2, x), DE) is False
assert recognize_log_derivative(Poly(1, x), Poly(x**2 + x, x), DE) is True
DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t**2 + 1, t)]})
assert recognize_log_derivative(Poly(1, t), Poly(t**2 - 2, t), DE) is False
assert recognize_log_derivative(Poly(1, t), Poly(t**2 + t, t), DE) is False
def test_recognize_log_derivative():
a = Poly(2*x**2 + 4*x*t - 2*t - x**2*t, t)
d = Poly((2*x + t)*(t + x**2), t)
DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t, t)]})
assert recognize_log_derivative(a, d, DE, z) is True
DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(1/x, t)]})
assert recognize_log_derivative(Poly(t + 1, t), Poly(t + x, t), DE) is True
assert recognize_log_derivative(Poly(2, t), Poly(t**2 - 1, t), DE) is True
DE = DifferentialExtension(extension={'D': [Poly(1, x)]})
assert recognize_log_derivative(Poly(1, x), Poly(x**2 - 2, x), DE) is False
assert recognize_log_derivative(Poly(1, x), Poly(x**2 + x, x), DE) is True
DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t**2 + 1, t)]})
assert recognize_log_derivative(Poly(1, t), Poly(t**2 - 2, t), DE) is False
assert recognize_log_derivative(Poly(1, t), Poly(t**2 + t, t), DE) is False
etaa, etad = frac_in(dcoeff, DE.t)
A = parametric_log_deriv(alphaa, alphad, etaa, etad, DE)
if A is not None:
a, m, z = A
if a == 1:
n = min(n, m)
elif case == 'tan':
dcoeff = DE.d.quo(Poly(DE.t**2 + 1, DE.t))
# We are guaranteed to not have problems,
# because case != 'base'.
with DecrementLevel(DE):
betaa, alphaa, alphad = real_imag(ba, bd*a, DE.t)
betad = alphad
etaa, etad = frac_in(dcoeff, DE.t)
if recognize_log_derivative(2*betaa, betad, DE):
A = parametric_log_deriv(alphaa, alphad, etaa, etad, DE)
B = parametric_log_deriv(betaa, betad, etaa, etad, DE)
if A is not None and B is not None:
a, s, z = A
if a == 1:
n = min(n, s/2)
N = max(0, -nb)
pN = p**N
pn = p**-n # This is 1/h
A = a*pN
B = ba*pN.quo(bd) + Poly(n, DE.t)*a*derivation(p, DE).quo(p)*pN
G = [(Ga*pN*pn).cancel(Gd, include=True) for Ga, Gd in G]
h = pn