ai-content-maker/.venv/Lib/site-packages/sympy/integrals/rde.py

801 lines
27 KiB
Python

"""
Algorithms for solving the Risch differential equation.
Given a differential field K of characteristic 0 that is a simple
monomial extension of a base field k and f, g in K, the Risch
Differential Equation problem is to decide if there exist y in K such
that Dy + f*y == g and to find one if there are some. If t is a
monomial over k and the coefficients of f and g are in k(t), then y is
in k(t), and the outline of the algorithm here is given as:
1. Compute the normal part n of the denominator of y. The problem is
then reduced to finding y' in k<t>, where y == y'/n.
2. Compute the special part s of the denominator of y. The problem is
then reduced to finding y'' in k[t], where y == y''/(n*s)
3. Bound the degree of y''.
4. Reduce the equation Dy + f*y == g to a similar equation with f, g in
k[t].
5. Find the solutions in k[t] of bounded degree of the reduced equation.
See Chapter 6 of "Symbolic Integration I: Transcendental Functions" by
Manuel Bronstein. See also the docstring of risch.py.
"""
from operator import mul
from functools import reduce
from sympy.core import oo
from sympy.core.symbol import Dummy
from sympy.polys import Poly, gcd, ZZ, cancel
from sympy.functions.elementary.complexes import (im, re)
from sympy.functions.elementary.miscellaneous import sqrt
from sympy.integrals.risch import (gcdex_diophantine, frac_in, derivation,
splitfactor, NonElementaryIntegralException, DecrementLevel, recognize_log_derivative)
# TODO: Add messages to NonElementaryIntegralException errors
def order_at(a, p, t):
"""
Computes the order of a at p, with respect to t.
Explanation
===========
For a, p in k[t], the order of a at p is defined as nu_p(a) = max({n
in Z+ such that p**n|a}), where a != 0. If a == 0, nu_p(a) = +oo.
To compute the order at a rational function, a/b, use the fact that
nu_p(a/b) == nu_p(a) - nu_p(b).
"""
if a.is_zero:
return oo
if p == Poly(t, t):
return a.as_poly(t).ET()[0][0]
# Uses binary search for calculating the power. power_list collects the tuples
# (p^k,k) where each k is some power of 2. After deciding the largest k
# such that k is power of 2 and p^k|a the loop iteratively calculates
# the actual power.
power_list = []
p1 = p
r = a.rem(p1)
tracks_power = 1
while r.is_zero:
power_list.append((p1,tracks_power))
p1 = p1*p1
tracks_power *= 2
r = a.rem(p1)
n = 0
product = Poly(1, t)
while len(power_list) != 0:
final = power_list.pop()
productf = product*final[0]
r = a.rem(productf)
if r.is_zero:
n += final[1]
product = productf
return n
def order_at_oo(a, d, t):
"""
Computes the order of a/d at oo (infinity), with respect to t.
For f in k(t), the order or f at oo is defined as deg(d) - deg(a), where
f == a/d.
"""
if a.is_zero:
return oo
return d.degree(t) - a.degree(t)
def weak_normalizer(a, d, DE, z=None):
"""
Weak normalization.
Explanation
===========
Given a derivation D on k[t] and f == a/d in k(t), return q in k[t]
such that f - Dq/q is weakly normalized with respect to t.
f in k(t) is said to be "weakly normalized" with respect to t if
residue_p(f) is not a positive integer for any normal irreducible p
in k[t] such that f is in R_p (Definition 6.1.1). If f has an
elementary integral, this is equivalent to no logarithm of
integral(f) whose argument depends on t has a positive integer
coefficient, where the arguments of the logarithms not in k(t) are
in k[t].
Returns (q, f - Dq/q)
"""
z = z or Dummy('z')
dn, ds = splitfactor(d, DE)
# Compute d1, where dn == d1*d2**2*...*dn**n is a square-free
# factorization of d.
g = gcd(dn, dn.diff(DE.t))
d_sqf_part = dn.quo(g)
d1 = d_sqf_part.quo(gcd(d_sqf_part, g))
a1, b = gcdex_diophantine(d.quo(d1).as_poly(DE.t), d1.as_poly(DE.t),
a.as_poly(DE.t))
r = (a - Poly(z, DE.t)*derivation(d1, DE)).as_poly(DE.t).resultant(
d1.as_poly(DE.t))
r = Poly(r, z)
if not r.expr.has(z):
return (Poly(1, DE.t), (a, d))
N = [i for i in r.real_roots() if i in ZZ and i > 0]
q = reduce(mul, [gcd(a - Poly(n, DE.t)*derivation(d1, DE), d1) for n in N],
Poly(1, DE.t))
dq = derivation(q, DE)
sn = q*a - d*dq
sd = q*d
sn, sd = sn.cancel(sd, include=True)
return (q, (sn, sd))
def normal_denom(fa, fd, ga, gd, DE):
"""
Normal part of the denominator.
Explanation
===========
Given a derivation D on k[t] and f, g in k(t) with f weakly
normalized with respect to t, either raise NonElementaryIntegralException,
in which case the equation Dy + f*y == g has no solution in k(t), or the
quadruplet (a, b, c, h) such that a, h in k[t], b, c in k<t>, and for any
solution y in k(t) of Dy + f*y == g, q = y*h in k<t> satisfies
a*Dq + b*q == c.
This constitutes step 1 in the outline given in the rde.py docstring.
"""
dn, ds = splitfactor(fd, DE)
en, es = splitfactor(gd, DE)
p = dn.gcd(en)
h = en.gcd(en.diff(DE.t)).quo(p.gcd(p.diff(DE.t)))
a = dn*h
c = a*h
if c.div(en)[1]:
# en does not divide dn*h**2
raise NonElementaryIntegralException
ca = c*ga
ca, cd = ca.cancel(gd, include=True)
ba = a*fa - dn*derivation(h, DE)*fd
ba, bd = ba.cancel(fd, include=True)
# (dn*h, dn*h*f - dn*Dh, dn*h**2*g, h)
return (a, (ba, bd), (ca, cd), h)
def special_denom(a, ba, bd, ca, cd, DE, case='auto'):
"""
Special part of the denominator.
Explanation
===========
case is one of {'exp', 'tan', 'primitive'} for the hyperexponential,
hypertangent, and primitive cases, respectively. For the
hyperexponential (resp. hypertangent) case, given a derivation D on
k[t] and a in k[t], b, c, in k<t> with Dt/t in k (resp. Dt/(t**2 + 1) in
k, sqrt(-1) not in k), a != 0, and gcd(a, t) == 1 (resp.
gcd(a, t**2 + 1) == 1), return the quadruplet (A, B, C, 1/h) such that
A, B, C, h in k[t] and for any solution q in k<t> of a*Dq + b*q == c,
r = qh in k[t] satisfies A*Dr + B*r == C.
For ``case == 'primitive'``, k<t> == k[t], so it returns (a, b, c, 1) in
this case.
This constitutes step 2 of the outline given in the rde.py docstring.
"""
# TODO: finish writing this and write tests
if case == 'auto':
case = DE.case
if case == 'exp':
p = Poly(DE.t, DE.t)
elif case == 'tan':
p = Poly(DE.t**2 + 1, DE.t)
elif case in ('primitive', 'base'):
B = ba.to_field().quo(bd)
C = ca.to_field().quo(cd)
return (a, B, C, Poly(1, DE.t))
else:
raise ValueError("case must be one of {'exp', 'tan', 'primitive', "
"'base'}, not %s." % case)
nb = order_at(ba, p, DE.t) - order_at(bd, p, DE.t)
nc = order_at(ca, p, DE.t) - order_at(cd, p, DE.t)
n = min(0, nc - min(0, nb))
if not nb:
# Possible cancellation.
from .prde import parametric_log_deriv
if case == 'exp':
dcoeff = DE.d.quo(Poly(DE.t, DE.t))
with DecrementLevel(DE): # We are guaranteed to not have problems,
# because case != 'base'.
alphaa, alphad = frac_in(-ba.eval(0)/bd.eval(0)/a.eval(0), DE.t)
etaa, etad = frac_in(dcoeff, DE.t)
A = parametric_log_deriv(alphaa, alphad, etaa, etad, DE)
if A is not None:
Q, m, z = A
if Q == 1:
n = min(n, m)
elif case == 'tan':
dcoeff = DE.d.quo(Poly(DE.t**2+1, DE.t))
with DecrementLevel(DE): # We are guaranteed to not have problems,
# because case != 'base'.
alphaa, alphad = frac_in(im(-ba.eval(sqrt(-1))/bd.eval(sqrt(-1))/a.eval(sqrt(-1))), DE.t)
betaa, betad = frac_in(re(-ba.eval(sqrt(-1))/bd.eval(sqrt(-1))/a.eval(sqrt(-1))), DE.t)
etaa, etad = frac_in(dcoeff, DE.t)
if recognize_log_derivative(Poly(2, DE.t)*betaa, betad, DE):
A = parametric_log_deriv(alphaa*Poly(sqrt(-1), DE.t)*betad+alphad*betaa, alphad*betad, etaa, etad, DE)
if A is not None:
Q, m, z = A
if Q == 1:
n = min(n, m)
N = max(0, -nb, n - nc)
pN = p**N
pn = p**-n
A = a*pN
B = ba*pN.quo(bd) + Poly(n, DE.t)*a*derivation(p, DE).quo(p)*pN
C = (ca*pN*pn).quo(cd)
h = pn
# (a*p**N, (b + n*a*Dp/p)*p**N, c*p**(N - n), p**-n)
return (A, B, C, h)
def bound_degree(a, b, cQ, DE, case='auto', parametric=False):
"""
Bound on polynomial solutions.
Explanation
===========
Given a derivation D on k[t] and ``a``, ``b``, ``c`` in k[t] with ``a != 0``, return
n in ZZ such that deg(q) <= n for any solution q in k[t] of
a*Dq + b*q == c, when parametric=False, or deg(q) <= n for any solution
c1, ..., cm in Const(k) and q in k[t] of a*Dq + b*q == Sum(ci*gi, (i, 1, m))
when parametric=True.
For ``parametric=False``, ``cQ`` is ``c``, a ``Poly``; for ``parametric=True``, ``cQ`` is Q ==
[q1, ..., qm], a list of Polys.
This constitutes step 3 of the outline given in the rde.py docstring.
"""
# TODO: finish writing this and write tests
if case == 'auto':
case = DE.case
da = a.degree(DE.t)
db = b.degree(DE.t)
# The parametric and regular cases are identical, except for this part
if parametric:
dc = max([i.degree(DE.t) for i in cQ])
else:
dc = cQ.degree(DE.t)
alpha = cancel(-b.as_poly(DE.t).LC().as_expr()/
a.as_poly(DE.t).LC().as_expr())
if case == 'base':
n = max(0, dc - max(db, da - 1))
if db == da - 1 and alpha.is_Integer:
n = max(0, alpha, dc - db)
elif case == 'primitive':
if db > da:
n = max(0, dc - db)
else:
n = max(0, dc - da + 1)
etaa, etad = frac_in(DE.d, DE.T[DE.level - 1])
t1 = DE.t
with DecrementLevel(DE):
alphaa, alphad = frac_in(alpha, DE.t)
if db == da - 1:
from .prde import limited_integrate
# if alpha == m*Dt + Dz for z in k and m in ZZ:
try:
(za, zd), m = limited_integrate(alphaa, alphad, [(etaa, etad)],
DE)
except NonElementaryIntegralException:
pass
else:
if len(m) != 1:
raise ValueError("Length of m should be 1")
n = max(n, m[0])
elif db == da:
# if alpha == Dz/z for z in k*:
# beta = -lc(a*Dz + b*z)/(z*lc(a))
# if beta == m*Dt + Dw for w in k and m in ZZ:
# n = max(n, m)
from .prde import is_log_deriv_k_t_radical_in_field
A = is_log_deriv_k_t_radical_in_field(alphaa, alphad, DE)
if A is not None:
aa, z = A
if aa == 1:
beta = -(a*derivation(z, DE).as_poly(t1) +
b*z.as_poly(t1)).LC()/(z.as_expr()*a.LC())
betaa, betad = frac_in(beta, DE.t)
from .prde import limited_integrate
try:
(za, zd), m = limited_integrate(betaa, betad,
[(etaa, etad)], DE)
except NonElementaryIntegralException:
pass
else:
if len(m) != 1:
raise ValueError("Length of m should be 1")
n = max(n, m[0].as_expr())
elif case == 'exp':
from .prde import parametric_log_deriv
n = max(0, dc - max(db, da))
if da == db:
etaa, etad = frac_in(DE.d.quo(Poly(DE.t, DE.t)), DE.T[DE.level - 1])
with DecrementLevel(DE):
alphaa, alphad = frac_in(alpha, DE.t)
A = parametric_log_deriv(alphaa, alphad, etaa, etad, DE)
if A is not None:
# if alpha == m*Dt/t + Dz/z for z in k* and m in ZZ:
# n = max(n, m)
a, m, z = A
if a == 1:
n = max(n, m)
elif case in ('tan', 'other_nonlinear'):
delta = DE.d.degree(DE.t)
lam = DE.d.LC()
alpha = cancel(alpha/lam)
n = max(0, dc - max(da + delta - 1, db))
if db == da + delta - 1 and alpha.is_Integer:
n = max(0, alpha, dc - db)
else:
raise ValueError("case must be one of {'exp', 'tan', 'primitive', "
"'other_nonlinear', 'base'}, not %s." % case)
return n
def spde(a, b, c, n, DE):
"""
Rothstein's Special Polynomial Differential Equation algorithm.
Explanation
===========
Given a derivation D on k[t], an integer n and ``a``,``b``,``c`` in k[t] with
``a != 0``, either raise NonElementaryIntegralException, in which case the
equation a*Dq + b*q == c has no solution of degree at most ``n`` in
k[t], or return the tuple (B, C, m, alpha, beta) such that B, C,
alpha, beta in k[t], m in ZZ, and any solution q in k[t] of degree
at most n of a*Dq + b*q == c must be of the form
q == alpha*h + beta, where h in k[t], deg(h) <= m, and Dh + B*h == C.
This constitutes step 4 of the outline given in the rde.py docstring.
"""
zero = Poly(0, DE.t)
alpha = Poly(1, DE.t)
beta = Poly(0, DE.t)
while True:
if c.is_zero:
return (zero, zero, 0, zero, beta) # -1 is more to the point
if (n < 0) is True:
raise NonElementaryIntegralException
g = a.gcd(b)
if not c.rem(g).is_zero: # g does not divide c
raise NonElementaryIntegralException
a, b, c = a.quo(g), b.quo(g), c.quo(g)
if a.degree(DE.t) == 0:
b = b.to_field().quo(a)
c = c.to_field().quo(a)
return (b, c, n, alpha, beta)
r, z = gcdex_diophantine(b, a, c)
b += derivation(a, DE)
c = z - derivation(r, DE)
n -= a.degree(DE.t)
beta += alpha * r
alpha *= a
def no_cancel_b_large(b, c, n, DE):
"""
Poly Risch Differential Equation - No cancellation: deg(b) large enough.
Explanation
===========
Given a derivation D on k[t], ``n`` either an integer or +oo, and ``b``,``c``
in k[t] with ``b != 0`` and either D == d/dt or
deg(b) > max(0, deg(D) - 1), either raise NonElementaryIntegralException, in
which case the equation ``Dq + b*q == c`` has no solution of degree at
most n in k[t], or a solution q in k[t] of this equation with
``deg(q) < n``.
"""
q = Poly(0, DE.t)
while not c.is_zero:
m = c.degree(DE.t) - b.degree(DE.t)
if not 0 <= m <= n: # n < 0 or m < 0 or m > n
raise NonElementaryIntegralException
p = Poly(c.as_poly(DE.t).LC()/b.as_poly(DE.t).LC()*DE.t**m, DE.t,
expand=False)
q = q + p
n = m - 1
c = c - derivation(p, DE) - b*p
return q
def no_cancel_b_small(b, c, n, DE):
"""
Poly Risch Differential Equation - No cancellation: deg(b) small enough.
Explanation
===========
Given a derivation D on k[t], ``n`` either an integer or +oo, and ``b``,``c``
in k[t] with deg(b) < deg(D) - 1 and either D == d/dt or
deg(D) >= 2, either raise NonElementaryIntegralException, in which case the
equation Dq + b*q == c has no solution of degree at most n in k[t],
or a solution q in k[t] of this equation with deg(q) <= n, or the
tuple (h, b0, c0) such that h in k[t], b0, c0, in k, and for any
solution q in k[t] of degree at most n of Dq + bq == c, y == q - h
is a solution in k of Dy + b0*y == c0.
"""
q = Poly(0, DE.t)
while not c.is_zero:
if n == 0:
m = 0
else:
m = c.degree(DE.t) - DE.d.degree(DE.t) + 1
if not 0 <= m <= n: # n < 0 or m < 0 or m > n
raise NonElementaryIntegralException
if m > 0:
p = Poly(c.as_poly(DE.t).LC()/(m*DE.d.as_poly(DE.t).LC())*DE.t**m,
DE.t, expand=False)
else:
if b.degree(DE.t) != c.degree(DE.t):
raise NonElementaryIntegralException
if b.degree(DE.t) == 0:
return (q, b.as_poly(DE.T[DE.level - 1]),
c.as_poly(DE.T[DE.level - 1]))
p = Poly(c.as_poly(DE.t).LC()/b.as_poly(DE.t).LC(), DE.t,
expand=False)
q = q + p
n = m - 1
c = c - derivation(p, DE) - b*p
return q
# TODO: better name for this function
def no_cancel_equal(b, c, n, DE):
"""
Poly Risch Differential Equation - No cancellation: deg(b) == deg(D) - 1
Explanation
===========
Given a derivation D on k[t] with deg(D) >= 2, n either an integer
or +oo, and b, c in k[t] with deg(b) == deg(D) - 1, either raise
NonElementaryIntegralException, in which case the equation Dq + b*q == c has
no solution of degree at most n in k[t], or a solution q in k[t] of
this equation with deg(q) <= n, or the tuple (h, m, C) such that h
in k[t], m in ZZ, and C in k[t], and for any solution q in k[t] of
degree at most n of Dq + b*q == c, y == q - h is a solution in k[t]
of degree at most m of Dy + b*y == C.
"""
q = Poly(0, DE.t)
lc = cancel(-b.as_poly(DE.t).LC()/DE.d.as_poly(DE.t).LC())
if lc.is_Integer and lc.is_positive:
M = lc
else:
M = -1
while not c.is_zero:
m = max(M, c.degree(DE.t) - DE.d.degree(DE.t) + 1)
if not 0 <= m <= n: # n < 0 or m < 0 or m > n
raise NonElementaryIntegralException
u = cancel(m*DE.d.as_poly(DE.t).LC() + b.as_poly(DE.t).LC())
if u.is_zero:
return (q, m, c)
if m > 0:
p = Poly(c.as_poly(DE.t).LC()/u*DE.t**m, DE.t, expand=False)
else:
if c.degree(DE.t) != DE.d.degree(DE.t) - 1:
raise NonElementaryIntegralException
else:
p = c.as_poly(DE.t).LC()/b.as_poly(DE.t).LC()
q = q + p
n = m - 1
c = c - derivation(p, DE) - b*p
return q
def cancel_primitive(b, c, n, DE):
"""
Poly Risch Differential Equation - Cancellation: Primitive case.
Explanation
===========
Given a derivation D on k[t], n either an integer or +oo, ``b`` in k, and
``c`` in k[t] with Dt in k and ``b != 0``, either raise
NonElementaryIntegralException, in which case the equation Dq + b*q == c
has no solution of degree at most n in k[t], or a solution q in k[t] of
this equation with deg(q) <= n.
"""
# Delayed imports
from .prde import is_log_deriv_k_t_radical_in_field
with DecrementLevel(DE):
ba, bd = frac_in(b, DE.t)
A = is_log_deriv_k_t_radical_in_field(ba, bd, DE)
if A is not None:
n, z = A
if n == 1: # b == Dz/z
raise NotImplementedError("is_deriv_in_field() is required to "
" solve this problem.")
# if z*c == Dp for p in k[t] and deg(p) <= n:
# return p/z
# else:
# raise NonElementaryIntegralException
if c.is_zero:
return c # return 0
if n < c.degree(DE.t):
raise NonElementaryIntegralException
q = Poly(0, DE.t)
while not c.is_zero:
m = c.degree(DE.t)
if n < m:
raise NonElementaryIntegralException
with DecrementLevel(DE):
a2a, a2d = frac_in(c.LC(), DE.t)
sa, sd = rischDE(ba, bd, a2a, a2d, DE)
stm = Poly(sa.as_expr()/sd.as_expr()*DE.t**m, DE.t, expand=False)
q += stm
n = m - 1
c -= b*stm + derivation(stm, DE)
return q
def cancel_exp(b, c, n, DE):
"""
Poly Risch Differential Equation - Cancellation: Hyperexponential case.
Explanation
===========
Given a derivation D on k[t], n either an integer or +oo, ``b`` in k, and
``c`` in k[t] with Dt/t in k and ``b != 0``, either raise
NonElementaryIntegralException, in which case the equation Dq + b*q == c
has no solution of degree at most n in k[t], or a solution q in k[t] of
this equation with deg(q) <= n.
"""
from .prde import parametric_log_deriv
eta = DE.d.quo(Poly(DE.t, DE.t)).as_expr()
with DecrementLevel(DE):
etaa, etad = frac_in(eta, DE.t)
ba, bd = frac_in(b, DE.t)
A = parametric_log_deriv(ba, bd, etaa, etad, DE)
if A is not None:
a, m, z = A
if a == 1:
raise NotImplementedError("is_deriv_in_field() is required to "
"solve this problem.")
# if c*z*t**m == Dp for p in k<t> and q = p/(z*t**m) in k[t] and
# deg(q) <= n:
# return q
# else:
# raise NonElementaryIntegralException
if c.is_zero:
return c # return 0
if n < c.degree(DE.t):
raise NonElementaryIntegralException
q = Poly(0, DE.t)
while not c.is_zero:
m = c.degree(DE.t)
if n < m:
raise NonElementaryIntegralException
# a1 = b + m*Dt/t
a1 = b.as_expr()
with DecrementLevel(DE):
# TODO: Write a dummy function that does this idiom
a1a, a1d = frac_in(a1, DE.t)
a1a = a1a*etad + etaa*a1d*Poly(m, DE.t)
a1d = a1d*etad
a2a, a2d = frac_in(c.LC(), DE.t)
sa, sd = rischDE(a1a, a1d, a2a, a2d, DE)
stm = Poly(sa.as_expr()/sd.as_expr()*DE.t**m, DE.t, expand=False)
q += stm
n = m - 1
c -= b*stm + derivation(stm, DE) # deg(c) becomes smaller
return q
def solve_poly_rde(b, cQ, n, DE, parametric=False):
"""
Solve a Polynomial Risch Differential Equation with degree bound ``n``.
This constitutes step 4 of the outline given in the rde.py docstring.
For parametric=False, cQ is c, a Poly; for parametric=True, cQ is Q ==
[q1, ..., qm], a list of Polys.
"""
# No cancellation
if not b.is_zero and (DE.case == 'base' or
b.degree(DE.t) > max(0, DE.d.degree(DE.t) - 1)):
if parametric:
# Delayed imports
from .prde import prde_no_cancel_b_large
return prde_no_cancel_b_large(b, cQ, n, DE)
return no_cancel_b_large(b, cQ, n, DE)
elif (b.is_zero or b.degree(DE.t) < DE.d.degree(DE.t) - 1) and \
(DE.case == 'base' or DE.d.degree(DE.t) >= 2):
if parametric:
from .prde import prde_no_cancel_b_small
return prde_no_cancel_b_small(b, cQ, n, DE)
R = no_cancel_b_small(b, cQ, n, DE)
if isinstance(R, Poly):
return R
else:
# XXX: Might k be a field? (pg. 209)
h, b0, c0 = R
with DecrementLevel(DE):
b0, c0 = b0.as_poly(DE.t), c0.as_poly(DE.t)
if b0 is None: # See above comment
raise ValueError("b0 should be a non-Null value")
if c0 is None:
raise ValueError("c0 should be a non-Null value")
y = solve_poly_rde(b0, c0, n, DE).as_poly(DE.t)
return h + y
elif DE.d.degree(DE.t) >= 2 and b.degree(DE.t) == DE.d.degree(DE.t) - 1 and \
n > -b.as_poly(DE.t).LC()/DE.d.as_poly(DE.t).LC():
# TODO: Is this check necessary, and if so, what should it do if it fails?
# b comes from the first element returned from spde()
if not b.as_poly(DE.t).LC().is_number:
raise TypeError("Result should be a number")
if parametric:
raise NotImplementedError("prde_no_cancel_b_equal() is not yet "
"implemented.")
R = no_cancel_equal(b, cQ, n, DE)
if isinstance(R, Poly):
return R
else:
h, m, C = R
# XXX: Or should it be rischDE()?
y = solve_poly_rde(b, C, m, DE)
return h + y
else:
# Cancellation
if b.is_zero:
raise NotImplementedError("Remaining cases for Poly (P)RDE are "
"not yet implemented (is_deriv_in_field() required).")
else:
if DE.case == 'exp':
if parametric:
raise NotImplementedError("Parametric RDE cancellation "
"hyperexponential case is not yet implemented.")
return cancel_exp(b, cQ, n, DE)
elif DE.case == 'primitive':
if parametric:
raise NotImplementedError("Parametric RDE cancellation "
"primitive case is not yet implemented.")
return cancel_primitive(b, cQ, n, DE)
else:
raise NotImplementedError("Other Poly (P)RDE cancellation "
"cases are not yet implemented (%s)." % DE.case)
if parametric:
raise NotImplementedError("Remaining cases for Poly PRDE not yet "
"implemented.")
raise NotImplementedError("Remaining cases for Poly RDE not yet "
"implemented.")
def rischDE(fa, fd, ga, gd, DE):
"""
Solve a Risch Differential Equation: Dy + f*y == g.
Explanation
===========
See the outline in the docstring of rde.py for more information
about the procedure used. Either raise NonElementaryIntegralException, in
which case there is no solution y in the given differential field,
or return y in k(t) satisfying Dy + f*y == g, or raise
NotImplementedError, in which case, the algorithms necessary to
solve the given Risch Differential Equation have not yet been
implemented.
"""
_, (fa, fd) = weak_normalizer(fa, fd, DE)
a, (ba, bd), (ca, cd), hn = normal_denom(fa, fd, ga, gd, DE)
A, B, C, hs = special_denom(a, ba, bd, ca, cd, DE)
try:
# Until this is fully implemented, use oo. Note that this will almost
# certainly cause non-termination in spde() (unless A == 1), and
# *might* lead to non-termination in the next step for a nonelementary
# integral (I don't know for certain yet). Fortunately, spde() is
# currently written recursively, so this will just give
# RuntimeError: maximum recursion depth exceeded.
n = bound_degree(A, B, C, DE)
except NotImplementedError:
# Useful for debugging:
# import warnings
# warnings.warn("rischDE: Proceeding with n = oo; may cause "
# "non-termination.")
n = oo
B, C, m, alpha, beta = spde(A, B, C, n, DE)
if C.is_zero:
y = C
else:
y = solve_poly_rde(B, C, m, DE)
return (alpha*y + beta, hn*hs)