def test_residue_reduce():
a = Poly(2*t**2 - t - x**2, t)
d = Poly(t**3 - x**2*t, t)
DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(1/x, t)], 'Tfuncs': [log]})
assert residue_reduce(a, d, DE, z, invert=False) == \
([(Poly(z**2 - S(1)/4, z), Poly((1 + 3*x*z - 6*z**2 -
2*x**2 + 4*x**2*z**2)*t - x*z + x**2 + 2*x**2*z**2 - 2*z*x**3, t))], False)
assert residue_reduce(a, d, DE, z, invert=True) == \
([(Poly(z**2 - S(1)/4, z), Poly(t + 2*x*z, t))], False)
assert residue_reduce(Poly(-2/x, t), Poly(t**2 - 1, t,), DE, z, invert=False) == \
([(Poly(z**2 - 1, z), Poly(-2*z*t/x - 2/x, t))], True)
ans = residue_reduce(Poly(-2/x, t), Poly(t**2 - 1, t), DE, z, invert=True)
assert ans == ([(Poly(z**2 - 1, z), Poly(t + z, t))], True)
assert residue_reduce_to_basic(ans[0], DE, z) == -log(-1 + log(x)) + log(1 + log(x))
DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(-t**2 - t/x - (1 - nu**2/x**2), t)]})
# TODO: Skip or make faster
assert residue_reduce(Poly((-2*nu**2 - x**4)/(2*x**2)*t - (1 + x**2)/x, t),
Poly(t**2 + 1 + x**2/2, t), DE, z) == \
([(Poly(z + S(1)/2, z, domain='QQ'), Poly(t**2 + 1 + x**2/2, t, domain='EX'))], True)
DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(1 + t**2, t)]})
assert residue_reduce(Poly(-2*x*t + 1 - x**2, t),
Poly(t**2 + 2*x*t + 1 + x**2, t), DE, z) == \
([(Poly(z**2 + S(1)/4, z), Poly(t + x + 2*z, t))], True)
DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t, t)]})
assert residue_reduce(Poly(t, t), Poly(t + sqrt(2), t), DE, z) == \
([(Poly(z - 1, z), Poly(t + sqrt(2), t))], True)
def is_log_deriv_k_t_radical_in_field(fa, fd, DE, case='auto', z=None):
"""
Checks if f can be written as the logarithmic derivative of a k(t)-radical.
f in k(t) can be written as the logarithmic derivative of a k(t) radical if
there exist n in ZZ and u in k(t) with n, u != 0 such that n*f == Du/u.
Either returns (n, u) or None, which means that f cannot be written as the
logarithmic derivative of a k(t)-radical.
case is one of {'primitive', 'exp', 'tan', 'auto'} for the primitive,
hyperexponential, and hypertangent cases, respectively. If case it 'auto',
it will attempt to determine the type of the derivation automatically.
"""
fa, fd = fa.cancel(fd, include=True)
# f must be simple
n, s = splitfactor(fd, DE)
if not s.is_one:
pass
#return None
z = z or Dummy('z')
H, b = residue_reduce(fa, fd, DE, z=z)
if not b:
# I will have to verify, but I believe that the answer should be
# None in this case. This should never happen for the
# functions given when solving the parametric logarithmic
# derivative problem when integration elementary functions (see
# Bronstein's book, page 255), so most likely this indicates a bug.
return None
roots = [(i, i.real_roots()) for i, _ in H]
if not all(len(j) == i.degree() and all(k.is_Rational for k in j) for
i, j in roots):
# If f is the logarithmic derivative of a k(t)-radical, then all the
# roots of the resultant must be rational numbers.
return None
# [(a, i), ...], where i*log(a) is a term in the log-part of the integral
# of f
respolys, residues = zip(*roots) or [[], []]
# Note: this might be empty, but everything below should work find in that
# case (it should be the same as if it were [[1, 1]])
residueterms = [(H[j][1].subs(z, i), i) for j in xrange(len(H)) for
i in residues[j]]
# TODO: finish writing this and write tests
p = cancel(fa.as_expr()/fd.as_expr() - residue_reduce_derivation(H, DE, z))
p = p.as_poly(DE.t)
if p is None:
# f - Dg will be in k[t] if f is the logarithmic derivative of a k(t)-radical
return None
if p.degree(DE.t) >= max(1, DE.d.degree(DE.t)):
return None
if case == 'auto':
case = DE.case
if case == 'exp':
wa, wd = derivation(DE.t, DE).cancel(Poly(DE.t, DE.t), include=True)
with DecrementLevel(DE):
pa, pd = frac_in(p, DE.t, cancel=True)
wa, wd = frac_in((wa, wd), DE.t)
A = parametric_log_deriv(pa, pd, wa, wd, DE)
if A is None:
return None
n, e, u = A
u *= DE.t**e
# raise NotImplementedError("The hyperexponential case is "
# "not yet completely implemented for is_log_deriv_k_t_radical_in_field().")
elif case == 'primitive':
with DecrementLevel(DE):
pa, pd = frac_in(p, DE.t)
A = is_log_deriv_k_t_radical_in_field(pa, pd, DE, case='auto')
if A is None:
return None
n, u = A
elif case == 'base':
# TODO: we can use more efficient residue reduction from ratint()
if not fd.is_sqf or fa.degree() >= fd.degree():
# f is the logarithmic derivative in the base case if and only if
# f = fa/fd, fd is square-free, deg(fa) < deg(fd), and
# gcd(fa, fd) == 1. The last condition is handled by cancel() above.
return None
# Note: if residueterms = [], returns (1, 1)
# f had better be 0 in that case.
n = reduce(ilcm, [i.as_numer_denom()[1] for _, i in residueterms], S(1))
u = Mul(*[Pow(i, j*n) for i, j in residueterms])
return (n, u)
elif case == 'tan':
raise NotImplementedError("The hypertangent case is "
"not yet implemented for is_log_deriv_k_t_radical_in_field()")
elif case in ['other_linear', 'other_nonlinear']:
# XXX: If these are supported by the structure theorems, change to NotImplementedError.
raise ValueError("The %s case is not supported in this function." % case)
else:
raise ValueError("case must be one of {'primitive', 'exp', 'tan', "
"'base', 'auto'}, not %s" % case)
common_denom = reduce(ilcm, [i.as_numer_denom()[1] for i in [j for _, j in
residueterms]] + [n], S(1))
residueterms = [(i, j*common_denom) for i, j in residueterms]
m = common_denom//n
assert common_denom == n*m # Verify exact division
u = cancel(u**m*Mul(*[Pow(i, j) for i, j in residueterms]))
return (common_denom, u)
it will attempt to determine the type of the derivation automatically.
See also
========
is_log_deriv_k_t_radical, is_deriv_k
"""
fa, fd = fa.cancel(fd, include=True)
# f must be simple
n, s = splitfactor(fd, DE)
if not s.is_one:
pass
z = z or Dummy('z')
H, b = residue_reduce(fa, fd, DE, z=z)
if not b:
# I will have to verify, but I believe that the answer should be
# None in this case. This should never happen for the
# functions given when solving the parametric logarithmic
# derivative problem when integration elementary functions (see
# Bronstein's book, page 255), so most likely this indicates a bug.
return None
roots = [(i, i.real_roots()) for i, _ in H]
if not all(len(j) == i.degree() and all(k.is_Rational for k in j) for
i, j in roots):
# If f is the logarithmic derivative of a k(t)-radical, then all the
# roots of the resultant must be rational numbers.
return None