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1333 lines
51 KiB
1333 lines
51 KiB
5 months ago
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"""
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Algorithms for solving Parametric Risch Differential Equations.
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The methods used for solving Parametric Risch Differential Equations parallel
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those for solving Risch Differential Equations. See the outline in the
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docstring of rde.py for more information.
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The Parametric Risch Differential Equation problem is, given f, g1, ..., gm in
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K(t), to determine if there exist y in K(t) and c1, ..., cm in Const(K) such
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that Dy + f*y == Sum(ci*gi, (i, 1, m)), and to find such y and ci if they exist.
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For the algorithms here G is a list of tuples of factions of the terms on the
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right hand side of the equation (i.e., gi in k(t)), and Q is a list of terms on
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the right hand side of the equation (i.e., qi in k[t]). See the docstring of
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each function for more information.
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"""
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import itertools
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from functools import reduce
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from sympy.core import Dummy, ilcm, Add, Mul, Pow, S
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from sympy.integrals.rde import (order_at, order_at_oo, weak_normalizer,
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bound_degree)
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from sympy.integrals.risch import (gcdex_diophantine, frac_in, derivation,
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residue_reduce, splitfactor, residue_reduce_derivation, DecrementLevel,
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recognize_log_derivative)
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from sympy.polys import Poly, lcm, cancel, sqf_list
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from sympy.polys.polymatrix import PolyMatrix as Matrix
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from sympy.solvers import solve
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zeros = Matrix.zeros
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eye = Matrix.eye
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def prde_normal_denom(fa, fd, G, DE):
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"""
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Parametric Risch Differential Equation - Normal part of the denominator.
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Explanation
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===========
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Given a derivation D on k[t] and f, g1, ..., gm in k(t) with f weakly
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normalized with respect to t, return the tuple (a, b, G, h) such that
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a, h in k[t], b in k<t>, G = [g1, ..., gm] in k(t)^m, and for any solution
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c1, ..., cm in Const(k) and y in k(t) of Dy + f*y == Sum(ci*gi, (i, 1, m)),
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q == y*h in k<t> satisfies a*Dq + b*q == Sum(ci*Gi, (i, 1, m)).
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"""
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dn, ds = splitfactor(fd, DE)
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Gas, Gds = list(zip(*G))
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gd = reduce(lambda i, j: i.lcm(j), Gds, Poly(1, DE.t))
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en, es = splitfactor(gd, DE)
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p = dn.gcd(en)
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h = en.gcd(en.diff(DE.t)).quo(p.gcd(p.diff(DE.t)))
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a = dn*h
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c = a*h
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ba = a*fa - dn*derivation(h, DE)*fd
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ba, bd = ba.cancel(fd, include=True)
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G = [(c*A).cancel(D, include=True) for A, D in G]
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return (a, (ba, bd), G, h)
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def real_imag(ba, bd, gen):
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"""
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Helper function, to get the real and imaginary part of a rational function
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evaluated at sqrt(-1) without actually evaluating it at sqrt(-1).
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Explanation
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===========
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Separates the even and odd power terms by checking the degree of terms wrt
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mod 4. Returns a tuple (ba[0], ba[1], bd) where ba[0] is real part
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of the numerator ba[1] is the imaginary part and bd is the denominator
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of the rational function.
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"""
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bd = bd.as_poly(gen).as_dict()
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ba = ba.as_poly(gen).as_dict()
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denom_real = [value if key[0] % 4 == 0 else -value if key[0] % 4 == 2 else 0 for key, value in bd.items()]
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denom_imag = [value if key[0] % 4 == 1 else -value if key[0] % 4 == 3 else 0 for key, value in bd.items()]
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bd_real = sum(r for r in denom_real)
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bd_imag = sum(r for r in denom_imag)
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num_real = [value if key[0] % 4 == 0 else -value if key[0] % 4 == 2 else 0 for key, value in ba.items()]
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num_imag = [value if key[0] % 4 == 1 else -value if key[0] % 4 == 3 else 0 for key, value in ba.items()]
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ba_real = sum(r for r in num_real)
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ba_imag = sum(r for r in num_imag)
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ba = ((ba_real*bd_real + ba_imag*bd_imag).as_poly(gen), (ba_imag*bd_real - ba_real*bd_imag).as_poly(gen))
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bd = (bd_real*bd_real + bd_imag*bd_imag).as_poly(gen)
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return (ba[0], ba[1], bd)
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def prde_special_denom(a, ba, bd, G, DE, case='auto'):
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"""
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Parametric Risch Differential Equation - Special part of the denominator.
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Explanation
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===========
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Case is one of {'exp', 'tan', 'primitive'} for the hyperexponential,
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hypertangent, and primitive cases, respectively. For the hyperexponential
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(resp. hypertangent) case, given a derivation D on k[t] and a in k[t],
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b in k<t>, and g1, ..., gm in k(t) with Dt/t in k (resp. Dt/(t**2 + 1) in
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k, sqrt(-1) not in k), a != 0, and gcd(a, t) == 1 (resp.
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gcd(a, t**2 + 1) == 1), return the tuple (A, B, GG, h) such that A, B, h in
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k[t], GG = [gg1, ..., ggm] in k(t)^m, and for any solution c1, ..., cm in
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Const(k) and q in k<t> of a*Dq + b*q == Sum(ci*gi, (i, 1, m)), r == q*h in
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k[t] satisfies A*Dr + B*r == Sum(ci*ggi, (i, 1, m)).
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For case == 'primitive', k<t> == k[t], so it returns (a, b, G, 1) in this
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case.
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"""
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# TODO: Merge this with the very similar special_denom() in rde.py
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if case == 'auto':
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case = DE.case
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if case == 'exp':
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p = Poly(DE.t, DE.t)
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elif case == 'tan':
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p = Poly(DE.t**2 + 1, DE.t)
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elif case in ('primitive', 'base'):
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B = ba.quo(bd)
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return (a, B, G, Poly(1, DE.t))
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else:
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raise ValueError("case must be one of {'exp', 'tan', 'primitive', "
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"'base'}, not %s." % case)
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nb = order_at(ba, p, DE.t) - order_at(bd, p, DE.t)
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nc = min([order_at(Ga, p, DE.t) - order_at(Gd, p, DE.t) for Ga, Gd in G])
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n = min(0, nc - min(0, nb))
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if not nb:
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# Possible cancellation.
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if case == 'exp':
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dcoeff = DE.d.quo(Poly(DE.t, DE.t))
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with DecrementLevel(DE): # We are guaranteed to not have problems,
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# because case != 'base'.
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alphaa, alphad = frac_in(-ba.eval(0)/bd.eval(0)/a.eval(0), DE.t)
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etaa, etad = frac_in(dcoeff, DE.t)
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A = parametric_log_deriv(alphaa, alphad, etaa, etad, DE)
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if A is not None:
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Q, m, z = A
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if Q == 1:
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n = min(n, m)
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elif case == 'tan':
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dcoeff = DE.d.quo(Poly(DE.t**2 + 1, DE.t))
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with DecrementLevel(DE): # We are guaranteed to not have problems,
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# because case != 'base'.
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betaa, alphaa, alphad = real_imag(ba, bd*a, DE.t)
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betad = alphad
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etaa, etad = frac_in(dcoeff, DE.t)
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if recognize_log_derivative(Poly(2, DE.t)*betaa, betad, DE):
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A = parametric_log_deriv(alphaa, alphad, etaa, etad, DE)
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B = parametric_log_deriv(betaa, betad, etaa, etad, DE)
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if A is not None and B is not None:
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Q, s, z = A
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# TODO: Add test
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if Q == 1:
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n = min(n, s/2)
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N = max(0, -nb)
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pN = p**N
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pn = p**-n # This is 1/h
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A = a*pN
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B = ba*pN.quo(bd) + Poly(n, DE.t)*a*derivation(p, DE).quo(p)*pN
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G = [(Ga*pN*pn).cancel(Gd, include=True) for Ga, Gd in G]
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h = pn
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# (a*p**N, (b + n*a*Dp/p)*p**N, g1*p**(N - n), ..., gm*p**(N - n), p**-n)
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return (A, B, G, h)
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def prde_linear_constraints(a, b, G, DE):
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"""
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Parametric Risch Differential Equation - Generate linear constraints on the constants.
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Explanation
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===========
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Given a derivation D on k[t], a, b, in k[t] with gcd(a, b) == 1, and
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G = [g1, ..., gm] in k(t)^m, return Q = [q1, ..., qm] in k[t]^m and a
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matrix M with entries in k(t) such that for any solution c1, ..., cm in
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Const(k) and p in k[t] of a*Dp + b*p == Sum(ci*gi, (i, 1, m)),
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(c1, ..., cm) is a solution of Mx == 0, and p and the ci satisfy
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a*Dp + b*p == Sum(ci*qi, (i, 1, m)).
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Because M has entries in k(t), and because Matrix does not play well with
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Poly, M will be a Matrix of Basic expressions.
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"""
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m = len(G)
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Gns, Gds = list(zip(*G))
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d = reduce(lambda i, j: i.lcm(j), Gds)
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d = Poly(d, field=True)
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Q = [(ga*(d).quo(gd)).div(d) for ga, gd in G]
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if not all(ri.is_zero for _, ri in Q):
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N = max(ri.degree(DE.t) for _, ri in Q)
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M = Matrix(N + 1, m, lambda i, j: Q[j][1].nth(i), DE.t)
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else:
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M = Matrix(0, m, [], DE.t) # No constraints, return the empty matrix.
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qs, _ = list(zip(*Q))
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return (qs, M)
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def poly_linear_constraints(p, d):
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"""
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Given p = [p1, ..., pm] in k[t]^m and d in k[t], return
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q = [q1, ..., qm] in k[t]^m and a matrix M with entries in k such
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that Sum(ci*pi, (i, 1, m)), for c1, ..., cm in k, is divisible
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by d if and only if (c1, ..., cm) is a solution of Mx = 0, in
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which case the quotient is Sum(ci*qi, (i, 1, m)).
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"""
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m = len(p)
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q, r = zip(*[pi.div(d) for pi in p])
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if not all(ri.is_zero for ri in r):
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n = max(ri.degree() for ri in r)
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M = Matrix(n + 1, m, lambda i, j: r[j].nth(i), d.gens)
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else:
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M = Matrix(0, m, [], d.gens) # No constraints.
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return q, M
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def constant_system(A, u, DE):
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"""
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Generate a system for the constant solutions.
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Explanation
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===========
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Given a differential field (K, D) with constant field C = Const(K), a Matrix
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A, and a vector (Matrix) u with coefficients in K, returns the tuple
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(B, v, s), where B is a Matrix with coefficients in C and v is a vector
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(Matrix) such that either v has coefficients in C, in which case s is True
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and the solutions in C of Ax == u are exactly all the solutions of Bx == v,
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or v has a non-constant coefficient, in which case s is False Ax == u has no
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constant solution.
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This algorithm is used both in solving parametric problems and in
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determining if an element a of K is a derivative of an element of K or the
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logarithmic derivative of a K-radical using the structure theorem approach.
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Because Poly does not play well with Matrix yet, this algorithm assumes that
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all matrix entries are Basic expressions.
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"""
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if not A:
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return A, u
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Au = A.row_join(u)
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Au, _ = Au.rref()
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# Warning: This will NOT return correct results if cancel() cannot reduce
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# an identically zero expression to 0. The danger is that we might
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# incorrectly prove that an integral is nonelementary (such as
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# risch_integrate(exp((sin(x)**2 + cos(x)**2 - 1)*x**2), x).
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# But this is a limitation in computer algebra in general, and implicit
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# in the correctness of the Risch Algorithm is the computability of the
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# constant field (actually, this same correctness problem exists in any
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# algorithm that uses rref()).
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#
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# We therefore limit ourselves to constant fields that are computable
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# via the cancel() function, in order to prevent a speed bottleneck from
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# calling some more complex simplification function (rational function
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# coefficients will fall into this class). Furthermore, (I believe) this
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# problem will only crop up if the integral explicitly contains an
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# expression in the constant field that is identically zero, but cannot
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# be reduced to such by cancel(). Therefore, a careful user can avoid this
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# problem entirely by being careful with the sorts of expressions that
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# appear in his integrand in the variables other than the integration
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# variable (the structure theorems should be able to completely decide these
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# problems in the integration variable).
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A, u = Au[:, :-1], Au[:, -1]
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D = lambda x: derivation(x, DE, basic=True)
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for j, i in itertools.product(range(A.cols), range(A.rows)):
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if A[i, j].expr.has(*DE.T):
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# This assumes that const(F(t0, ..., tn) == const(K) == F
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Ri = A[i, :]
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# Rm+1; m = A.rows
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DAij = D(A[i, j])
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Rm1 = Ri.applyfunc(lambda x: D(x) / DAij)
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um1 = D(u[i]) / DAij
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Aj = A[:, j]
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A = A - Aj * Rm1
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u = u - Aj * um1
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A = A.col_join(Rm1)
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u = u.col_join(Matrix([um1], u.gens))
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return (A, u)
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def prde_spde(a, b, Q, n, DE):
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"""
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Special Polynomial Differential Equation algorithm: Parametric Version.
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Explanation
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===========
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Given a derivation D on k[t], an integer n, and a, b, q1, ..., qm in k[t]
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with deg(a) > 0 and gcd(a, b) == 1, return (A, B, Q, R, n1), with
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Qq = [q1, ..., qm] and R = [r1, ..., rm], such that for any solution
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c1, ..., cm in Const(k) and q in k[t] of degree at most n of
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a*Dq + b*q == Sum(ci*gi, (i, 1, m)), p = (q - Sum(ci*ri, (i, 1, m)))/a has
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degree at most n1 and satisfies A*Dp + B*p == Sum(ci*qi, (i, 1, m))
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"""
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R, Z = list(zip(*[gcdex_diophantine(b, a, qi) for qi in Q]))
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A = a
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B = b + derivation(a, DE)
|
||
|
|
Qq = [zi - derivation(ri, DE) for ri, zi in zip(R, Z)]
|
||
|
|
R = list(R)
|
||
|
|
n1 = n - a.degree(DE.t)
|
||
|
|
|
||
|
|
return (A, B, Qq, R, n1)
|
||
|
|
|
||
|
|
|
||
|
|
def prde_no_cancel_b_large(b, Q, n, DE):
|
||
|
|
"""
|
||
|
|
Parametric Poly Risch Differential Equation - No cancellation: deg(b) large enough.
|
||
|
|
|
||
|
|
Explanation
|
||
|
|
===========
|
||
|
|
|
||
|
|
Given a derivation D on k[t], n in ZZ, and b, q1, ..., qm in k[t] with
|
||
|
|
b != 0 and either D == d/dt or deg(b) > max(0, deg(D) - 1), returns
|
||
|
|
h1, ..., hr in k[t] and a matrix A with coefficients in Const(k) such that
|
||
|
|
if c1, ..., cm in Const(k) and q in k[t] satisfy deg(q) <= n and
|
||
|
|
Dq + b*q == Sum(ci*qi, (i, 1, m)), then q = Sum(dj*hj, (j, 1, r)), where
|
||
|
|
d1, ..., dr in Const(k) and A*Matrix([[c1, ..., cm, d1, ..., dr]]).T == 0.
|
||
|
|
"""
|
||
|
|
db = b.degree(DE.t)
|
||
|
|
m = len(Q)
|
||
|
|
H = [Poly(0, DE.t)]*m
|
||
|
|
|
||
|
|
for N, i in itertools.product(range(n, -1, -1), range(m)): # [n, ..., 0]
|
||
|
|
si = Q[i].nth(N + db)/b.LC()
|
||
|
|
sitn = Poly(si*DE.t**N, DE.t)
|
||
|
|
H[i] = H[i] + sitn
|
||
|
|
Q[i] = Q[i] - derivation(sitn, DE) - b*sitn
|
||
|
|
|
||
|
|
if all(qi.is_zero for qi in Q):
|
||
|
|
dc = -1
|
||
|
|
else:
|
||
|
|
dc = max([qi.degree(DE.t) for qi in Q])
|
||
|
|
M = Matrix(dc + 1, m, lambda i, j: Q[j].nth(i), DE.t)
|
||
|
|
A, u = constant_system(M, zeros(dc + 1, 1, DE.t), DE)
|
||
|
|
c = eye(m, DE.t)
|
||
|
|
A = A.row_join(zeros(A.rows, m, DE.t)).col_join(c.row_join(-c))
|
||
|
|
|
||
|
|
return (H, A)
|
||
|
|
|
||
|
|
|
||
|
|
def prde_no_cancel_b_small(b, Q, n, DE):
|
||
|
|
"""
|
||
|
|
Parametric Poly Risch Differential Equation - No cancellation: deg(b) small enough.
|
||
|
|
|
||
|
|
Explanation
|
||
|
|
===========
|
||
|
|
|
||
|
|
Given a derivation D on k[t], n in ZZ, and b, q1, ..., qm in k[t] with
|
||
|
|
deg(b) < deg(D) - 1 and either D == d/dt or deg(D) >= 2, returns
|
||
|
|
h1, ..., hr in k[t] and a matrix A with coefficients in Const(k) such that
|
||
|
|
if c1, ..., cm in Const(k) and q in k[t] satisfy deg(q) <= n and
|
||
|
|
Dq + b*q == Sum(ci*qi, (i, 1, m)) then q = Sum(dj*hj, (j, 1, r)) where
|
||
|
|
d1, ..., dr in Const(k) and A*Matrix([[c1, ..., cm, d1, ..., dr]]).T == 0.
|
||
|
|
"""
|
||
|
|
m = len(Q)
|
||
|
|
H = [Poly(0, DE.t)]*m
|
||
|
|
|
||
|
|
for N, i in itertools.product(range(n, 0, -1), range(m)): # [n, ..., 1]
|
||
|
|
si = Q[i].nth(N + DE.d.degree(DE.t) - 1)/(N*DE.d.LC())
|
||
|
|
sitn = Poly(si*DE.t**N, DE.t)
|
||
|
|
H[i] = H[i] + sitn
|
||
|
|
Q[i] = Q[i] - derivation(sitn, DE) - b*sitn
|
||
|
|
|
||
|
|
if b.degree(DE.t) > 0:
|
||
|
|
for i in range(m):
|
||
|
|
si = Poly(Q[i].nth(b.degree(DE.t))/b.LC(), DE.t)
|
||
|
|
H[i] = H[i] + si
|
||
|
|
Q[i] = Q[i] - derivation(si, DE) - b*si
|
||
|
|
if all(qi.is_zero for qi in Q):
|
||
|
|
dc = -1
|
||
|
|
else:
|
||
|
|
dc = max([qi.degree(DE.t) for qi in Q])
|
||
|
|
M = Matrix(dc + 1, m, lambda i, j: Q[j].nth(i), DE.t)
|
||
|
|
A, u = constant_system(M, zeros(dc + 1, 1, DE.t), DE)
|
||
|
|
c = eye(m, DE.t)
|
||
|
|
A = A.row_join(zeros(A.rows, m, DE.t)).col_join(c.row_join(-c))
|
||
|
|
return (H, A)
|
||
|
|
|
||
|
|
# else: b is in k, deg(qi) < deg(Dt)
|
||
|
|
|
||
|
|
t = DE.t
|
||
|
|
if DE.case != 'base':
|
||
|
|
with DecrementLevel(DE):
|
||
|
|
t0 = DE.t # k = k0(t0)
|
||
|
|
ba, bd = frac_in(b, t0, field=True)
|
||
|
|
Q0 = [frac_in(qi.TC(), t0, field=True) for qi in Q]
|
||
|
|
f, B = param_rischDE(ba, bd, Q0, DE)
|
||
|
|
|
||
|
|
# f = [f1, ..., fr] in k^r and B is a matrix with
|
||
|
|
# m + r columns and entries in Const(k) = Const(k0)
|
||
|
|
# such that Dy0 + b*y0 = Sum(ci*qi, (i, 1, m)) has
|
||
|
|
# a solution y0 in k with c1, ..., cm in Const(k)
|
||
|
|
# if and only y0 = Sum(dj*fj, (j, 1, r)) where
|
||
|
|
# d1, ..., dr ar in Const(k) and
|
||
|
|
# B*Matrix([c1, ..., cm, d1, ..., dr]) == 0.
|
||
|
|
|
||
|
|
# Transform fractions (fa, fd) in f into constant
|
||
|
|
# polynomials fa/fd in k[t].
|
||
|
|
# (Is there a better way?)
|
||
|
|
f = [Poly(fa.as_expr()/fd.as_expr(), t, field=True)
|
||
|
|
for fa, fd in f]
|
||
|
|
B = Matrix.from_Matrix(B.to_Matrix(), t)
|
||
|
|
else:
|
||
|
|
# Base case. Dy == 0 for all y in k and b == 0.
|
||
|
|
# Dy + b*y = Sum(ci*qi) is solvable if and only if
|
||
|
|
# Sum(ci*qi) == 0 in which case the solutions are
|
||
|
|
# y = d1*f1 for f1 = 1 and any d1 in Const(k) = k.
|
||
|
|
|
||
|
|
f = [Poly(1, t, field=True)] # r = 1
|
||
|
|
B = Matrix([[qi.TC() for qi in Q] + [S.Zero]], DE.t)
|
||
|
|
# The condition for solvability is
|
||
|
|
# B*Matrix([c1, ..., cm, d1]) == 0
|
||
|
|
# There are no constraints on d1.
|
||
|
|
|
||
|
|
# Coefficients of t^j (j > 0) in Sum(ci*qi) must be zero.
|
||
|
|
d = max([qi.degree(DE.t) for qi in Q])
|
||
|
|
if d > 0:
|
||
|
|
M = Matrix(d, m, lambda i, j: Q[j].nth(i + 1), DE.t)
|
||
|
|
A, _ = constant_system(M, zeros(d, 1, DE.t), DE)
|
||
|
|
else:
|
||
|
|
# No constraints on the hj.
|
||
|
|
A = Matrix(0, m, [], DE.t)
|
||
|
|
|
||
|
|
# Solutions of the original equation are
|
||
|
|
# y = Sum(dj*fj, (j, 1, r) + Sum(ei*hi, (i, 1, m)),
|
||
|
|
# where ei == ci (i = 1, ..., m), when
|
||
|
|
# A*Matrix([c1, ..., cm]) == 0 and
|
||
|
|
# B*Matrix([c1, ..., cm, d1, ..., dr]) == 0
|
||
|
|
|
||
|
|
# Build combined constraint matrix with m + r + m columns.
|
||
|
|
|
||
|
|
r = len(f)
|
||
|
|
I = eye(m, DE.t)
|
||
|
|
A = A.row_join(zeros(A.rows, r + m, DE.t))
|
||
|
|
B = B.row_join(zeros(B.rows, m, DE.t))
|
||
|
|
C = I.row_join(zeros(m, r, DE.t)).row_join(-I)
|
||
|
|
|
||
|
|
return f + H, A.col_join(B).col_join(C)
|
||
|
|
|
||
|
|
|
||
|
|
def prde_cancel_liouvillian(b, Q, n, DE):
|
||
|
|
"""
|
||
|
|
Pg, 237.
|
||
|
|
"""
|
||
|
|
H = []
|
||
|
|
|
||
|
|
# Why use DecrementLevel? Below line answers that:
|
||
|
|
# Assuming that we can solve such problems over 'k' (not k[t])
|
||
|
|
if DE.case == 'primitive':
|
||
|
|
with DecrementLevel(DE):
|
||
|
|
ba, bd = frac_in(b, DE.t, field=True)
|
||
|
|
|
||
|
|
for i in range(n, -1, -1):
|
||
|
|
if DE.case == 'exp': # this re-checking can be avoided
|
||
|
|
with DecrementLevel(DE):
|
||
|
|
ba, bd = frac_in(b + (i*(derivation(DE.t, DE)/DE.t)).as_poly(b.gens),
|
||
|
|
DE.t, field=True)
|
||
|
|
with DecrementLevel(DE):
|
||
|
|
Qy = [frac_in(q.nth(i), DE.t, field=True) for q in Q]
|
||
|
|
fi, Ai = param_rischDE(ba, bd, Qy, DE)
|
||
|
|
fi = [Poly(fa.as_expr()/fd.as_expr(), DE.t, field=True)
|
||
|
|
for fa, fd in fi]
|
||
|
|
Ai = Ai.set_gens(DE.t)
|
||
|
|
|
||
|
|
ri = len(fi)
|
||
|
|
|
||
|
|
if i == n:
|
||
|
|
M = Ai
|
||
|
|
else:
|
||
|
|
M = Ai.col_join(M.row_join(zeros(M.rows, ri, DE.t)))
|
||
|
|
|
||
|
|
Fi, hi = [None]*ri, [None]*ri
|
||
|
|
|
||
|
|
# from eq. on top of p.238 (unnumbered)
|
||
|
|
for j in range(ri):
|
||
|
|
hji = fi[j] * (DE.t**i).as_poly(fi[j].gens)
|
||
|
|
hi[j] = hji
|
||
|
|
# building up Sum(djn*(D(fjn*t^n) - b*fjnt^n))
|
||
|
|
Fi[j] = -(derivation(hji, DE) - b*hji)
|
||
|
|
|
||
|
|
H += hi
|
||
|
|
# in the next loop instead of Q it has
|
||
|
|
# to be Q + Fi taking its place
|
||
|
|
Q = Q + Fi
|
||
|
|
|
||
|
|
return (H, M)
|
||
|
|
|
||
|
|
|
||
|
|
def param_poly_rischDE(a, b, q, n, DE):
|
||
|
|
"""Polynomial solutions of a parametric Risch differential equation.
|
||
|
|
|
||
|
|
Explanation
|
||
|
|
===========
|
||
|
|
|
||
|
|
Given a derivation D in k[t], a, b in k[t] relatively prime, and q
|
||
|
|
= [q1, ..., qm] in k[t]^m, return h = [h1, ..., hr] in k[t]^r and
|
||
|
|
a matrix A with m + r columns and entries in Const(k) such that
|
||
|
|
a*Dp + b*p = Sum(ci*qi, (i, 1, m)) has a solution p of degree <= n
|
||
|
|
in k[t] with c1, ..., cm in Const(k) if and only if p = Sum(dj*hj,
|
||
|
|
(j, 1, r)) where d1, ..., dr are in Const(k) and (c1, ..., cm,
|
||
|
|
d1, ..., dr) is a solution of Ax == 0.
|
||
|
|
"""
|
||
|
|
m = len(q)
|
||
|
|
if n < 0:
|
||
|
|
# Only the trivial zero solution is possible.
|
||
|
|
# Find relations between the qi.
|
||
|
|
if all(qi.is_zero for qi in q):
|
||
|
|
return [], zeros(1, m, DE.t) # No constraints.
|
||
|
|
|
||
|
|
N = max([qi.degree(DE.t) for qi in q])
|
||
|
|
M = Matrix(N + 1, m, lambda i, j: q[j].nth(i), DE.t)
|
||
|
|
A, _ = constant_system(M, zeros(M.rows, 1, DE.t), DE)
|
||
|
|
|
||
|
|
return [], A
|
||
|
|
|
||
|
|
if a.is_ground:
|
||
|
|
# Normalization: a = 1.
|
||
|
|
a = a.LC()
|
||
|
|
b, q = b.quo_ground(a), [qi.quo_ground(a) for qi in q]
|
||
|
|
|
||
|
|
if not b.is_zero and (DE.case == 'base' or
|
||
|
|
b.degree() > max(0, DE.d.degree() - 1)):
|
||
|
|
return prde_no_cancel_b_large(b, q, n, DE)
|
||
|
|
|
||
|
|
elif ((b.is_zero or b.degree() < DE.d.degree() - 1)
|
||
|
|
and (DE.case == 'base' or DE.d.degree() >= 2)):
|
||
|
|
return prde_no_cancel_b_small(b, q, n, DE)
|
||
|
|
|
||
|
|
elif (DE.d.degree() >= 2 and
|
||
|
|
b.degree() == DE.d.degree() - 1 and
|
||
|
|
n > -b.as_poly().LC()/DE.d.as_poly().LC()):
|
||
|
|
raise NotImplementedError("prde_no_cancel_b_equal() is "
|
||
|
|
"not yet implemented.")
|
||
|
|
|
||
|
|
else:
|
||
|
|
# Liouvillian cases
|
||
|
|
if DE.case in ('primitive', 'exp'):
|
||
|
|
return prde_cancel_liouvillian(b, q, n, DE)
|
||
|
|
else:
|
||
|
|
raise NotImplementedError("non-linear and hypertangent "
|
||
|
|
"cases have not yet been implemented")
|
||
|
|
|
||
|
|
# else: deg(a) > 0
|
||
|
|
|
||
|
|
# Iterate SPDE as long as possible cumulating coefficient
|
||
|
|
# and terms for the recovery of original solutions.
|
||
|
|
alpha, beta = a.one, [a.zero]*m
|
||
|
|
while n >= 0: # and a, b relatively prime
|
||
|
|
a, b, q, r, n = prde_spde(a, b, q, n, DE)
|
||
|
|
beta = [betai + alpha*ri for betai, ri in zip(beta, r)]
|
||
|
|
alpha *= a
|
||
|
|
# Solutions p of a*Dp + b*p = Sum(ci*qi) correspond to
|
||
|
|
# solutions alpha*p + Sum(ci*betai) of the initial equation.
|
||
|
|
d = a.gcd(b)
|
||
|
|
if not d.is_ground:
|
||
|
|
break
|
||
|
|
|
||
|
|
# a*Dp + b*p = Sum(ci*qi) may have a polynomial solution
|
||
|
|
# only if the sum is divisible by d.
|
||
|
|
|
||
|
|
qq, M = poly_linear_constraints(q, d)
|
||
|
|
# qq = [qq1, ..., qqm] where qqi = qi.quo(d).
|
||
|
|
# M is a matrix with m columns an entries in k.
|
||
|
|
# Sum(fi*qi, (i, 1, m)), where f1, ..., fm are elements of k, is
|
||
|
|
# divisible by d if and only if M*Matrix([f1, ..., fm]) == 0,
|
||
|
|
# in which case the quotient is Sum(fi*qqi).
|
||
|
|
|
||
|
|
A, _ = constant_system(M, zeros(M.rows, 1, DE.t), DE)
|
||
|
|
# A is a matrix with m columns and entries in Const(k).
|
||
|
|
# Sum(ci*qqi) is Sum(ci*qi).quo(d), and the remainder is zero
|
||
|
|
# for c1, ..., cm in Const(k) if and only if
|
||
|
|
# A*Matrix([c1, ...,cm]) == 0.
|
||
|
|
|
||
|
|
V = A.nullspace()
|
||
|
|
# V = [v1, ..., vu] where each vj is a column matrix with
|
||
|
|
# entries aj1, ..., ajm in Const(k).
|
||
|
|
# Sum(aji*qi) is divisible by d with exact quotient Sum(aji*qqi).
|
||
|
|
# Sum(ci*qi) is divisible by d if and only if ci = Sum(dj*aji)
|
||
|
|
# (i = 1, ..., m) for some d1, ..., du in Const(k).
|
||
|
|
# In that case, solutions of
|
||
|
|
# a*Dp + b*p = Sum(ci*qi) = Sum(dj*Sum(aji*qi))
|
||
|
|
# are the same as those of
|
||
|
|
# (a/d)*Dp + (b/d)*p = Sum(dj*rj)
|
||
|
|
# where rj = Sum(aji*qqi).
|
||
|
|
|
||
|
|
if not V: # No non-trivial solution.
|
||
|
|
return [], eye(m, DE.t) # Could return A, but this has
|
||
|
|
# the minimum number of rows.
|
||
|
|
|
||
|
|
Mqq = Matrix([qq]) # A single row.
|
||
|
|
r = [(Mqq*vj)[0] for vj in V] # [r1, ..., ru]
|
||
|
|
|
||
|
|
# Solutions of (a/d)*Dp + (b/d)*p = Sum(dj*rj) correspond to
|
||
|
|
# solutions alpha*p + Sum(Sum(dj*aji)*betai) of the initial
|
||
|
|
# equation. These are equal to alpha*p + Sum(dj*fj) where
|
||
|
|
# fj = Sum(aji*betai).
|
||
|
|
Mbeta = Matrix([beta])
|
||
|
|
f = [(Mbeta*vj)[0] for vj in V] # [f1, ..., fu]
|
||
|
|
|
||
|
|
#
|
||
|
|
# Solve the reduced equation recursively.
|
||
|
|
#
|
||
|
|
g, B = param_poly_rischDE(a.quo(d), b.quo(d), r, n, DE)
|
||
|
|
|
||
|
|
# g = [g1, ..., gv] in k[t]^v and and B is a matrix with u + v
|
||
|
|
# columns and entries in Const(k) such that
|
||
|
|
# (a/d)*Dp + (b/d)*p = Sum(dj*rj) has a solution p of degree <= n
|
||
|
|
# in k[t] if and only if p = Sum(ek*gk) where e1, ..., ev are in
|
||
|
|
# Const(k) and B*Matrix([d1, ..., du, e1, ..., ev]) == 0.
|
||
|
|
# The solutions of the original equation are then
|
||
|
|
# Sum(dj*fj, (j, 1, u)) + alpha*Sum(ek*gk, (k, 1, v)).
|
||
|
|
|
||
|
|
# Collect solution components.
|
||
|
|
h = f + [alpha*gk for gk in g]
|
||
|
|
|
||
|
|
# Build combined relation matrix.
|
||
|
|
A = -eye(m, DE.t)
|
||
|
|
for vj in V:
|
||
|
|
A = A.row_join(vj)
|
||
|
|
A = A.row_join(zeros(m, len(g), DE.t))
|
||
|
|
A = A.col_join(zeros(B.rows, m, DE.t).row_join(B))
|
||
|
|
|
||
|
|
return h, A
|
||
|
|
|
||
|
|
|
||
|
|
def param_rischDE(fa, fd, G, DE):
|
||
|
|
"""
|
||
|
|
Solve a Parametric Risch Differential Equation: Dy + f*y == Sum(ci*Gi, (i, 1, m)).
|
||
|
|
|
||
|
|
Explanation
|
||
|
|
===========
|
||
|
|
|
||
|
|
Given a derivation D in k(t), f in k(t), and G
|
||
|
|
= [G1, ..., Gm] in k(t)^m, return h = [h1, ..., hr] in k(t)^r and
|
||
|
|
a matrix A with m + r columns and entries in Const(k) such that
|
||
|
|
Dy + f*y = Sum(ci*Gi, (i, 1, m)) has a solution y
|
||
|
|
in k(t) with c1, ..., cm in Const(k) if and only if y = Sum(dj*hj,
|
||
|
|
(j, 1, r)) where d1, ..., dr are in Const(k) and (c1, ..., cm,
|
||
|
|
d1, ..., dr) is a solution of Ax == 0.
|
||
|
|
|
||
|
|
Elements of k(t) are tuples (a, d) with a and d in k[t].
|
||
|
|
"""
|
||
|
|
m = len(G)
|
||
|
|
q, (fa, fd) = weak_normalizer(fa, fd, DE)
|
||
|
|
# Solutions of the weakly normalized equation Dz + f*z = q*Sum(ci*Gi)
|
||
|
|
# correspond to solutions y = z/q of the original equation.
|
||
|
|
gamma = q
|
||
|
|
G = [(q*ga).cancel(gd, include=True) for ga, gd in G]
|
||
|
|
|
||
|
|
a, (ba, bd), G, hn = prde_normal_denom(fa, fd, G, DE)
|
||
|
|
# Solutions q in k<t> of a*Dq + b*q = Sum(ci*Gi) correspond
|
||
|
|
# to solutions z = q/hn of the weakly normalized equation.
|
||
|
|
gamma *= hn
|
||
|
|
|
||
|
|
A, B, G, hs = prde_special_denom(a, ba, bd, G, DE)
|
||
|
|
# Solutions p in k[t] of A*Dp + B*p = Sum(ci*Gi) correspond
|
||
|
|
# to solutions q = p/hs of the previous equation.
|
||
|
|
gamma *= hs
|
||
|
|
|
||
|
|
g = A.gcd(B)
|
||
|
|
a, b, g = A.quo(g), B.quo(g), [gia.cancel(gid*g, include=True) for
|
||
|
|
gia, gid in G]
|
||
|
|
|
||
|
|
# a*Dp + b*p = Sum(ci*gi) may have a polynomial solution
|
||
|
|
# only if the sum is in k[t].
|
||
|
|
|
||
|
|
q, M = prde_linear_constraints(a, b, g, DE)
|
||
|
|
|
||
|
|
# q = [q1, ..., qm] where qi in k[t] is the polynomial component
|
||
|
|
# of the partial fraction expansion of gi.
|
||
|
|
# M is a matrix with m columns and entries in k.
|
||
|
|
# Sum(fi*gi, (i, 1, m)), where f1, ..., fm are elements of k,
|
||
|
|
# is a polynomial if and only if M*Matrix([f1, ..., fm]) == 0,
|
||
|
|
# in which case the sum is equal to Sum(fi*qi).
|
||
|
|
|
||
|
|
M, _ = constant_system(M, zeros(M.rows, 1, DE.t), DE)
|
||
|
|
# M is a matrix with m columns and entries in Const(k).
|
||
|
|
# Sum(ci*gi) is in k[t] for c1, ..., cm in Const(k)
|
||
|
|
# if and only if M*Matrix([c1, ..., cm]) == 0,
|
||
|
|
# in which case the sum is Sum(ci*qi).
|
||
|
|
|
||
|
|
## Reduce number of constants at this point
|
||
|
|
|
||
|
|
V = M.nullspace()
|
||
|
|
# V = [v1, ..., vu] where each vj is a column matrix with
|
||
|
|
# entries aj1, ..., ajm in Const(k).
|
||
|
|
# Sum(aji*gi) is in k[t] and equal to Sum(aji*qi) (j = 1, ..., u).
|
||
|
|
# Sum(ci*gi) is in k[t] if and only is ci = Sum(dj*aji)
|
||
|
|
# (i = 1, ..., m) for some d1, ..., du in Const(k).
|
||
|
|
# In that case,
|
||
|
|
# Sum(ci*gi) = Sum(ci*qi) = Sum(dj*Sum(aji*qi)) = Sum(dj*rj)
|
||
|
|
# where rj = Sum(aji*qi) (j = 1, ..., u) in k[t].
|
||
|
|
|
||
|
|
if not V: # No non-trivial solution
|
||
|
|
return [], eye(m, DE.t)
|
||
|
|
|
||
|
|
Mq = Matrix([q]) # A single row.
|
||
|
|
r = [(Mq*vj)[0] for vj in V] # [r1, ..., ru]
|
||
|
|
|
||
|
|
# Solutions of a*Dp + b*p = Sum(dj*rj) correspond to solutions
|
||
|
|
# y = p/gamma of the initial equation with ci = Sum(dj*aji).
|
||
|
|
|
||
|
|
try:
|
||
|
|
# We try n=5. At least for prde_spde, it will always
|
||
|
|
# terminate no matter what n is.
|
||
|
|
n = bound_degree(a, b, r, DE, parametric=True)
|
||
|
|
except NotImplementedError:
|
||
|
|
# A temporary bound is set. Eventually, it will be removed.
|
||
|
|
# the currently added test case takes large time
|
||
|
|
# even with n=5, and much longer with large n's.
|
||
|
|
n = 5
|
||
|
|
|
||
|
|
h, B = param_poly_rischDE(a, b, r, n, DE)
|
||
|
|
|
||
|
|
# h = [h1, ..., hv] in k[t]^v and and B is a matrix with u + v
|
||
|
|
# columns and entries in Const(k) such that
|
||
|
|
# a*Dp + b*p = Sum(dj*rj) has a solution p of degree <= n
|
||
|
|
# in k[t] if and only if p = Sum(ek*hk) where e1, ..., ev are in
|
||
|
|
# Const(k) and B*Matrix([d1, ..., du, e1, ..., ev]) == 0.
|
||
|
|
# The solutions of the original equation for ci = Sum(dj*aji)
|
||
|
|
# (i = 1, ..., m) are then y = Sum(ek*hk, (k, 1, v))/gamma.
|
||
|
|
|
||
|
|
## Build combined relation matrix with m + u + v columns.
|
||
|
|
|
||
|
|
A = -eye(m, DE.t)
|
||
|
|
for vj in V:
|
||
|
|
A = A.row_join(vj)
|
||
|
|
A = A.row_join(zeros(m, len(h), DE.t))
|
||
|
|
A = A.col_join(zeros(B.rows, m, DE.t).row_join(B))
|
||
|
|
|
||
|
|
## Eliminate d1, ..., du.
|
||
|
|
|
||
|
|
W = A.nullspace()
|
||
|
|
|
||
|
|
# W = [w1, ..., wt] where each wl is a column matrix with
|
||
|
|
# entries blk (k = 1, ..., m + u + v) in Const(k).
|
||
|
|
# The vectors (bl1, ..., blm) generate the space of those
|
||
|
|
# constant families (c1, ..., cm) for which a solution of
|
||
|
|
# the equation Dy + f*y == Sum(ci*Gi) exists. They generate
|
||
|
|
# the space and form a basis except possibly when Dy + f*y == 0
|
||
|
|
# is solvable in k(t}. The corresponding solutions are
|
||
|
|
# y = Sum(blk'*hk, (k, 1, v))/gamma, where k' = k + m + u.
|
||
|
|
|
||
|
|
v = len(h)
|
||
|
|
shape = (len(W), m+v)
|
||
|
|
elements = [wl[:m] + wl[-v:] for wl in W] # excise dj's.
|
||
|
|
items = [e for row in elements for e in row]
|
||
|
|
|
||
|
|
# Need to set the shape in case W is empty
|
||
|
|
M = Matrix(*shape, items, DE.t)
|
||
|
|
N = M.nullspace()
|
||
|
|
|
||
|
|
# N = [n1, ..., ns] where the ni in Const(k)^(m + v) are column
|
||
|
|
# vectors generating the space of linear relations between
|
||
|
|
# c1, ..., cm, e1, ..., ev.
|
||
|
|
|
||
|
|
C = Matrix([ni[:] for ni in N], DE.t) # rows n1, ..., ns.
|
||
|
|
|
||
|
|
return [hk.cancel(gamma, include=True) for hk in h], C
|
||
|
|
|
||
|
|
|
||
|
|
def limited_integrate_reduce(fa, fd, G, DE):
|
||
|
|
"""
|
||
|
|
Simpler version of step 1 & 2 for the limited integration problem.
|
||
|
|
|
||
|
|
Explanation
|
||
|
|
===========
|
||
|
|
|
||
|
|
Given a derivation D on k(t) and f, g1, ..., gn in k(t), return
|
||
|
|
(a, b, h, N, g, V) such that a, b, h in k[t], N is a non-negative integer,
|
||
|
|
g in k(t), V == [v1, ..., vm] in k(t)^m, and for any solution v in k(t),
|
||
|
|
c1, ..., cm in C of f == Dv + Sum(ci*wi, (i, 1, m)), p = v*h is in k<t>, and
|
||
|
|
p and the ci satisfy a*Dp + b*p == g + Sum(ci*vi, (i, 1, m)). Furthermore,
|
||
|
|
if S1irr == Sirr, then p is in k[t], and if t is nonlinear or Liouvillian
|
||
|
|
over k, then deg(p) <= N.
|
||
|
|
|
||
|
|
So that the special part is always computed, this function calls the more
|
||
|
|
general prde_special_denom() automatically if it cannot determine that
|
||
|
|
S1irr == Sirr. Furthermore, it will automatically call bound_degree() when
|
||
|
|
t is linear and non-Liouvillian, which for the transcendental case, implies
|
||
|
|
that Dt == a*t + b with for some a, b in k*.
|
||
|
|
"""
|
||
|
|
dn, ds = splitfactor(fd, DE)
|
||
|
|
E = [splitfactor(gd, DE) for _, gd in G]
|
||
|
|
En, Es = list(zip(*E))
|
||
|
|
c = reduce(lambda i, j: i.lcm(j), (dn,) + En) # lcm(dn, en1, ..., enm)
|
||
|
|
hn = c.gcd(c.diff(DE.t))
|
||
|
|
a = hn
|
||
|
|
b = -derivation(hn, DE)
|
||
|
|
N = 0
|
||
|
|
|
||
|
|
# These are the cases where we know that S1irr = Sirr, but there could be
|
||
|
|
# others, and this algorithm will need to be extended to handle them.
|
||
|
|
if DE.case in ('base', 'primitive', 'exp', 'tan'):
|
||
|
|
hs = reduce(lambda i, j: i.lcm(j), (ds,) + Es) # lcm(ds, es1, ..., esm)
|
||
|
|
a = hn*hs
|
||
|
|
b -= (hn*derivation(hs, DE)).quo(hs)
|
||
|
|
mu = min(order_at_oo(fa, fd, DE.t), min([order_at_oo(ga, gd, DE.t) for
|
||
|
|
ga, gd in G]))
|
||
|
|
# So far, all the above are also nonlinear or Liouvillian, but if this
|
||
|
|
# changes, then this will need to be updated to call bound_degree()
|
||
|
|
# as per the docstring of this function (DE.case == 'other_linear').
|
||
|
|
N = hn.degree(DE.t) + hs.degree(DE.t) + max(0, 1 - DE.d.degree(DE.t) - mu)
|
||
|
|
else:
|
||
|
|
# TODO: implement this
|
||
|
|
raise NotImplementedError
|
||
|
|
|
||
|
|
V = [(-a*hn*ga).cancel(gd, include=True) for ga, gd in G]
|
||
|
|
return (a, b, a, N, (a*hn*fa).cancel(fd, include=True), V)
|
||
|
|
|
||
|
|
|
||
|
|
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()
|
||
|
|
# interpreting limited integration problem as a
|
||
|
|
# parametric Risch DE problem
|
||
|
|
Fa = Poly(0, DE.t)
|
||
|
|
Fd = Poly(1, DE.t)
|
||
|
|
G = [(fa, fd)] + G
|
||
|
|
h, A = param_rischDE(Fa, Fd, G, DE)
|
||
|
|
V = A.nullspace()
|
||
|
|
V = [v for v in V if v[0] != 0]
|
||
|
|
if not V:
|
||
|
|
return None
|
||
|
|
else:
|
||
|
|
# we can take any vector from V, we take V[0]
|
||
|
|
c0 = V[0][0]
|
||
|
|
# v = [-1, c1, ..., cm, d1, ..., dr]
|
||
|
|
v = V[0]/(-c0)
|
||
|
|
r = len(h)
|
||
|
|
m = len(v) - r - 1
|
||
|
|
C = list(v[1: m + 1])
|
||
|
|
y = -sum([v[m + 1 + i]*h[i][0].as_expr()/h[i][1].as_expr() \
|
||
|
|
for i in range(r)])
|
||
|
|
y_num, y_den = y.as_numer_denom()
|
||
|
|
Ya, Yd = Poly(y_num, DE.t), Poly(y_den, DE.t)
|
||
|
|
Y = Ya*Poly(1/Yd.LC(), DE.t), Yd.monic()
|
||
|
|
return Y, C
|
||
|
|
|
||
|
|
|
||
|
|
def parametric_log_deriv_heu(fa, fd, wa, wd, DE, c1=None):
|
||
|
|
"""
|
||
|
|
Parametric logarithmic derivative heuristic.
|
||
|
|
|
||
|
|
Explanation
|
||
|
|
===========
|
||
|
|
|
||
|
|
Given a derivation D on k[t], f in k(t), and a hyperexponential monomial
|
||
|
|
theta over k(t), raises either NotImplementedError, in which case the
|
||
|
|
heuristic failed, or returns None, in which case it has proven that no
|
||
|
|
solution exists, or returns a solution (n, m, v) of the equation
|
||
|
|
n*f == Dv/v + m*Dtheta/theta, with v in k(t)* and n, m in ZZ with n != 0.
|
||
|
|
|
||
|
|
If this heuristic fails, the structure theorem approach will need to be
|
||
|
|
used.
|
||
|
|
|
||
|
|
The argument w == Dtheta/theta
|
||
|
|
"""
|
||
|
|
# TODO: finish writing this and write tests
|
||
|
|
c1 = c1 or Dummy('c1')
|
||
|
|
|
||
|
|
p, a = fa.div(fd)
|
||
|
|
q, b = wa.div(wd)
|
||
|
|
|
||
|
|
B = max(0, derivation(DE.t, DE).degree(DE.t) - 1)
|
||
|
|
C = max(p.degree(DE.t), q.degree(DE.t))
|
||
|
|
|
||
|
|
if q.degree(DE.t) > B:
|
||
|
|
eqs = [p.nth(i) - c1*q.nth(i) for i in range(B + 1, C + 1)]
|
||
|
|
s = solve(eqs, c1)
|
||
|
|
if not s or not s[c1].is_Rational:
|
||
|
|
# deg(q) > B, no solution for c.
|
||
|
|
return None
|
||
|
|
|
||
|
|
M, N = s[c1].as_numer_denom()
|
||
|
|
M_poly = M.as_poly(q.gens)
|
||
|
|
N_poly = N.as_poly(q.gens)
|
||
|
|
|
||
|
|
nfmwa = N_poly*fa*wd - M_poly*wa*fd
|
||
|
|
nfmwd = fd*wd
|
||
|
|
Qv = is_log_deriv_k_t_radical_in_field(nfmwa, nfmwd, DE, 'auto')
|
||
|
|
if Qv is None:
|
||
|
|
# (N*f - M*w) is not the logarithmic derivative of a k(t)-radical.
|
||
|
|
return None
|
||
|
|
|
||
|
|
Q, v = Qv
|
||
|
|
|
||
|
|
if Q.is_zero or v.is_zero:
|
||
|
|
return None
|
||
|
|
|
||
|
|
return (Q*N, Q*M, v)
|
||
|
|
|
||
|
|
if p.degree(DE.t) > B:
|
||
|
|
return None
|
||
|
|
|
||
|
|
c = lcm(fd.as_poly(DE.t).LC(), wd.as_poly(DE.t).LC())
|
||
|
|
l = fd.monic().lcm(wd.monic())*Poly(c, DE.t)
|
||
|
|
ln, ls = splitfactor(l, DE)
|
||
|
|
z = ls*ln.gcd(ln.diff(DE.t))
|
||
|
|
|
||
|
|
if not z.has(DE.t):
|
||
|
|
# TODO: We treat this as 'no solution', until the structure
|
||
|
|
# theorem version of parametric_log_deriv is implemented.
|
||
|
|
return None
|
||
|
|
|
||
|
|
u1, r1 = (fa*l.quo(fd)).div(z) # (l*f).div(z)
|
||
|
|
u2, r2 = (wa*l.quo(wd)).div(z) # (l*w).div(z)
|
||
|
|
|
||
|
|
eqs = [r1.nth(i) - c1*r2.nth(i) for i in range(z.degree(DE.t))]
|
||
|
|
s = solve(eqs, c1)
|
||
|
|
if not s or not s[c1].is_Rational:
|
||
|
|
# deg(q) <= B, no solution for c.
|
||
|
|
return None
|
||
|
|
|
||
|
|
M, N = s[c1].as_numer_denom()
|
||
|
|
|
||
|
|
nfmwa = N.as_poly(DE.t)*fa*wd - M.as_poly(DE.t)*wa*fd
|
||
|
|
nfmwd = fd*wd
|
||
|
|
Qv = is_log_deriv_k_t_radical_in_field(nfmwa, nfmwd, DE)
|
||
|
|
if Qv is None:
|
||
|
|
# (N*f - M*w) is not the logarithmic derivative of a k(t)-radical.
|
||
|
|
return None
|
||
|
|
|
||
|
|
Q, v = Qv
|
||
|
|
|
||
|
|
if Q.is_zero or v.is_zero:
|
||
|
|
return None
|
||
|
|
|
||
|
|
return (Q*N, Q*M, v)
|
||
|
|
|
||
|
|
|
||
|
|
def parametric_log_deriv(fa, fd, wa, wd, DE):
|
||
|
|
# TODO: Write the full algorithm using the structure theorems.
|
||
|
|
# try:
|
||
|
|
A = parametric_log_deriv_heu(fa, fd, wa, wd, DE)
|
||
|
|
# except NotImplementedError:
|
||
|
|
# Heuristic failed, we have to use the full method.
|
||
|
|
# TODO: This could be implemented more efficiently.
|
||
|
|
# It isn't too worrisome, because the heuristic handles most difficult
|
||
|
|
# cases.
|
||
|
|
return A
|
||
|
|
|
||
|
|
|
||
|
|
def is_deriv_k(fa, fd, DE):
|
||
|
|
r"""
|
||
|
|
Checks if Df/f is the derivative of an element of k(t).
|
||
|
|
|
||
|
|
Explanation
|
||
|
|
===========
|
||
|
|
|
||
|
|
a in k(t) is the derivative of an element of k(t) if there exists b in k(t)
|
||
|
|
such that a = Db. Either returns (ans, u), such that Df/f == Du, or None,
|
||
|
|
which means that Df/f is not the derivative of an element of k(t). ans is
|
||
|
|
a list of tuples such that Add(*[i*j for i, j in ans]) == u. This is useful
|
||
|
|
for seeing exactly which elements of k(t) produce u.
|
||
|
|
|
||
|
|
This function uses the structure theorem approach, which says that for any
|
||
|
|
f in K, Df/f is the derivative of a element of K if and only if there are ri
|
||
|
|
in QQ such that::
|
||
|
|
|
||
|
|
--- --- Dt
|
||
|
|
\ r * Dt + \ r * i Df
|
||
|
|
/ i i / i --- = --.
|
||
|
|
--- --- t f
|
||
|
|
i in L i in E i
|
||
|
|
K/C(x) K/C(x)
|
||
|
|
|
||
|
|
|
||
|
|
Where C = Const(K), L_K/C(x) = { i in {1, ..., n} such that t_i is
|
||
|
|
transcendental over C(x)(t_1, ..., t_i-1) and Dt_i = Da_i/a_i, for some a_i
|
||
|
|
in C(x)(t_1, ..., t_i-1)* } (i.e., the set of all indices of logarithmic
|
||
|
|
monomials of K over C(x)), and E_K/C(x) = { i in {1, ..., n} such that t_i
|
||
|
|
is transcendental over C(x)(t_1, ..., t_i-1) and Dt_i/t_i = Da_i, for some
|
||
|
|
a_i in C(x)(t_1, ..., t_i-1) } (i.e., the set of all indices of
|
||
|
|
hyperexponential monomials of K over C(x)). If K is an elementary extension
|
||
|
|
over C(x), then the cardinality of L_K/C(x) U E_K/C(x) is exactly the
|
||
|
|
transcendence degree of K over C(x). Furthermore, because Const_D(K) ==
|
||
|
|
Const_D(C(x)) == C, deg(Dt_i) == 1 when t_i is in E_K/C(x) and
|
||
|
|
deg(Dt_i) == 0 when t_i is in L_K/C(x), implying in particular that E_K/C(x)
|
||
|
|
and L_K/C(x) are disjoint.
|
||
|
|
|
||
|
|
The sets L_K/C(x) and E_K/C(x) must, by their nature, be computed
|
||
|
|
recursively using this same function. Therefore, it is required to pass
|
||
|
|
them as indices to D (or T). E_args are the arguments of the
|
||
|
|
hyperexponentials indexed by E_K (i.e., if i is in E_K, then T[i] ==
|
||
|
|
exp(E_args[i])). This is needed to compute the final answer u such that
|
||
|
|
Df/f == Du.
|
||
|
|
|
||
|
|
log(f) will be the same as u up to a additive constant. This is because
|
||
|
|
they will both behave the same as monomials. For example, both log(x) and
|
||
|
|
log(2*x) == log(x) + log(2) satisfy Dt == 1/x, because log(2) is constant.
|
||
|
|
Therefore, the term const is returned. const is such that
|
||
|
|
log(const) + f == u. This is calculated by dividing the arguments of one
|
||
|
|
logarithm from the other. Therefore, it is necessary to pass the arguments
|
||
|
|
of the logarithmic terms in L_args.
|
||
|
|
|
||
|
|
To handle the case where we are given Df/f, not f, use is_deriv_k_in_field().
|
||
|
|
|
||
|
|
See also
|
||
|
|
========
|
||
|
|
is_log_deriv_k_t_radical_in_field, is_log_deriv_k_t_radical
|
||
|
|
|
||
|
|
"""
|
||
|
|
# Compute Df/f
|
||
|
|
dfa, dfd = (fd*derivation(fa, DE) - fa*derivation(fd, DE)), fd*fa
|
||
|
|
dfa, dfd = dfa.cancel(dfd, include=True)
|
||
|
|
|
||
|
|
# Our assumption here is that each monomial is recursively transcendental
|
||
|
|
if len(DE.exts) != len(DE.D):
|
||
|
|
if [i for i in DE.cases if i == 'tan'] or \
|
||
|
|
({i for i in DE.cases if i == 'primitive'} -
|
||
|
|
set(DE.indices('log'))):
|
||
|
|
raise NotImplementedError("Real version of the structure "
|
||
|
|
"theorems with hypertangent support is not yet implemented.")
|
||
|
|
|
||
|
|
# TODO: What should really be done in this case?
|
||
|
|
raise NotImplementedError("Nonelementary extensions not supported "
|
||
|
|
"in the structure theorems.")
|
||
|
|
|
||
|
|
E_part = [DE.D[i].quo(Poly(DE.T[i], DE.T[i])).as_expr() for i in DE.indices('exp')]
|
||
|
|
L_part = [DE.D[i].as_expr() for i in DE.indices('log')]
|
||
|
|
|
||
|
|
# The expression dfa/dfd might not be polynomial in any of its symbols so we
|
||
|
|
# use a Dummy as the generator for PolyMatrix.
|
||
|
|
dum = Dummy()
|
||
|
|
lhs = Matrix([E_part + L_part], dum)
|
||
|
|
rhs = Matrix([dfa.as_expr()/dfd.as_expr()], dum)
|
||
|
|
|
||
|
|
A, u = constant_system(lhs, rhs, DE)
|
||
|
|
|
||
|
|
u = u.to_Matrix() # Poly to Expr
|
||
|
|
|
||
|
|
if not A or not all(derivation(i, DE, basic=True).is_zero for i in u):
|
||
|
|
# If the elements of u are not all constant
|
||
|
|
# Note: See comment in constant_system
|
||
|
|
|
||
|
|
# Also note: derivation(basic=True) calls cancel()
|
||
|
|
return None
|
||
|
|
else:
|
||
|
|
if not all(i.is_Rational for i in u):
|
||
|
|
raise NotImplementedError("Cannot work with non-rational "
|
||
|
|
"coefficients in this case.")
|
||
|
|
else:
|
||
|
|
terms = ([DE.extargs[i] for i in DE.indices('exp')] +
|
||
|
|
[DE.T[i] for i in DE.indices('log')])
|
||
|
|
ans = list(zip(terms, u))
|
||
|
|
result = Add(*[Mul(i, j) for i, j in ans])
|
||
|
|
argterms = ([DE.T[i] for i in DE.indices('exp')] +
|
||
|
|
[DE.extargs[i] for i in DE.indices('log')])
|
||
|
|
l = []
|
||
|
|
ld = []
|
||
|
|
for i, j in zip(argterms, u):
|
||
|
|
# We need to get around things like sqrt(x**2) != x
|
||
|
|
# and also sqrt(x**2 + 2*x + 1) != x + 1
|
||
|
|
# Issue 10798: i need not be a polynomial
|
||
|
|
i, d = i.as_numer_denom()
|
||
|
|
icoeff, iterms = sqf_list(i)
|
||
|
|
l.append(Mul(*([Pow(icoeff, j)] + [Pow(b, e*j) for b, e in iterms])))
|
||
|
|
dcoeff, dterms = sqf_list(d)
|
||
|
|
ld.append(Mul(*([Pow(dcoeff, j)] + [Pow(b, e*j) for b, e in dterms])))
|
||
|
|
const = cancel(fa.as_expr()/fd.as_expr()/Mul(*l)*Mul(*ld))
|
||
|
|
|
||
|
|
return (ans, result, const)
|
||
|
|
|
||
|
|
|
||
|
|
def is_log_deriv_k_t_radical(fa, fd, DE, Df=True):
|
||
|
|
r"""
|
||
|
|
Checks if Df is the logarithmic derivative of a k(t)-radical.
|
||
|
|
|
||
|
|
Explanation
|
||
|
|
===========
|
||
|
|
|
||
|
|
b 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*b == Du/u.
|
||
|
|
Either returns (ans, u, n, const) or None, which means that Df cannot be
|
||
|
|
written as the logarithmic derivative of a k(t)-radical. ans is a list of
|
||
|
|
tuples such that Mul(*[i**j for i, j in ans]) == u. This is useful for
|
||
|
|
seeing exactly what elements of k(t) produce u.
|
||
|
|
|
||
|
|
This function uses the structure theorem approach, which says that for any
|
||
|
|
f in K, Df is the logarithmic derivative of a K-radical if and only if there
|
||
|
|
are ri in QQ such that::
|
||
|
|
|
||
|
|
--- --- Dt
|
||
|
|
\ r * Dt + \ r * i
|
||
|
|
/ i i / i --- = Df.
|
||
|
|
--- --- t
|
||
|
|
i in L i in E i
|
||
|
|
K/C(x) K/C(x)
|
||
|
|
|
||
|
|
|
||
|
|
Where C = Const(K), L_K/C(x) = { i in {1, ..., n} such that t_i is
|
||
|
|
transcendental over C(x)(t_1, ..., t_i-1) and Dt_i = Da_i/a_i, for some a_i
|
||
|
|
in C(x)(t_1, ..., t_i-1)* } (i.e., the set of all indices of logarithmic
|
||
|
|
monomials of K over C(x)), and E_K/C(x) = { i in {1, ..., n} such that t_i
|
||
|
|
is transcendental over C(x)(t_1, ..., t_i-1) and Dt_i/t_i = Da_i, for some
|
||
|
|
a_i in C(x)(t_1, ..., t_i-1) } (i.e., the set of all indices of
|
||
|
|
hyperexponential monomials of K over C(x)). If K is an elementary extension
|
||
|
|
over C(x), then the cardinality of L_K/C(x) U E_K/C(x) is exactly the
|
||
|
|
transcendence degree of K over C(x). Furthermore, because Const_D(K) ==
|
||
|
|
Const_D(C(x)) == C, deg(Dt_i) == 1 when t_i is in E_K/C(x) and
|
||
|
|
deg(Dt_i) == 0 when t_i is in L_K/C(x), implying in particular that E_K/C(x)
|
||
|
|
and L_K/C(x) are disjoint.
|
||
|
|
|
||
|
|
The sets L_K/C(x) and E_K/C(x) must, by their nature, be computed
|
||
|
|
recursively using this same function. Therefore, it is required to pass
|
||
|
|
them as indices to D (or T). L_args are the arguments of the logarithms
|
||
|
|
indexed by L_K (i.e., if i is in L_K, then T[i] == log(L_args[i])). This is
|
||
|
|
needed to compute the final answer u such that n*f == Du/u.
|
||
|
|
|
||
|
|
exp(f) will be the same as u up to a multiplicative constant. This is
|
||
|
|
because they will both behave the same as monomials. For example, both
|
||
|
|
exp(x) and exp(x + 1) == E*exp(x) satisfy Dt == t. Therefore, the term const
|
||
|
|
is returned. const is such that exp(const)*f == u. This is calculated by
|
||
|
|
subtracting the arguments of one exponential from the other. Therefore, it
|
||
|
|
is necessary to pass the arguments of the exponential terms in E_args.
|
||
|
|
|
||
|
|
To handle the case where we are given Df, not f, use
|
||
|
|
is_log_deriv_k_t_radical_in_field().
|
||
|
|
|
||
|
|
See also
|
||
|
|
========
|
||
|
|
|
||
|
|
is_log_deriv_k_t_radical_in_field, is_deriv_k
|
||
|
|
|
||
|
|
"""
|
||
|
|
if Df:
|
||
|
|
dfa, dfd = (fd*derivation(fa, DE) - fa*derivation(fd, DE)).cancel(fd**2,
|
||
|
|
include=True)
|
||
|
|
else:
|
||
|
|
dfa, dfd = fa, fd
|
||
|
|
|
||
|
|
# Our assumption here is that each monomial is recursively transcendental
|
||
|
|
if len(DE.exts) != len(DE.D):
|
||
|
|
if [i for i in DE.cases if i == 'tan'] or \
|
||
|
|
({i for i in DE.cases if i == 'primitive'} -
|
||
|
|
set(DE.indices('log'))):
|
||
|
|
raise NotImplementedError("Real version of the structure "
|
||
|
|
"theorems with hypertangent support is not yet implemented.")
|
||
|
|
|
||
|
|
# TODO: What should really be done in this case?
|
||
|
|
raise NotImplementedError("Nonelementary extensions not supported "
|
||
|
|
"in the structure theorems.")
|
||
|
|
|
||
|
|
E_part = [DE.D[i].quo(Poly(DE.T[i], DE.T[i])).as_expr() for i in DE.indices('exp')]
|
||
|
|
L_part = [DE.D[i].as_expr() for i in DE.indices('log')]
|
||
|
|
|
||
|
|
# The expression dfa/dfd might not be polynomial in any of its symbols so we
|
||
|
|
# use a Dummy as the generator for PolyMatrix.
|
||
|
|
dum = Dummy()
|
||
|
|
lhs = Matrix([E_part + L_part], dum)
|
||
|
|
rhs = Matrix([dfa.as_expr()/dfd.as_expr()], dum)
|
||
|
|
|
||
|
|
A, u = constant_system(lhs, rhs, DE)
|
||
|
|
|
||
|
|
u = u.to_Matrix() # Poly to Expr
|
||
|
|
|
||
|
|
if not A or not all(derivation(i, DE, basic=True).is_zero for i in u):
|
||
|
|
# If the elements of u are not all constant
|
||
|
|
# Note: See comment in constant_system
|
||
|
|
|
||
|
|
# Also note: derivation(basic=True) calls cancel()
|
||
|
|
return None
|
||
|
|
else:
|
||
|
|
if not all(i.is_Rational for i in u):
|
||
|
|
# TODO: But maybe we can tell if they're not rational, like
|
||
|
|
# log(2)/log(3). Also, there should be an option to continue
|
||
|
|
# anyway, even if the result might potentially be wrong.
|
||
|
|
raise NotImplementedError("Cannot work with non-rational "
|
||
|
|
"coefficients in this case.")
|
||
|
|
else:
|
||
|
|
n = reduce(ilcm, [i.as_numer_denom()[1] for i in u])
|
||
|
|
u *= n
|
||
|
|
terms = ([DE.T[i] for i in DE.indices('exp')] +
|
||
|
|
[DE.extargs[i] for i in DE.indices('log')])
|
||
|
|
ans = list(zip(terms, u))
|
||
|
|
result = Mul(*[Pow(i, j) for i, j in ans])
|
||
|
|
|
||
|
|
# exp(f) will be the same as result up to a multiplicative
|
||
|
|
# constant. We now find the log of that constant.
|
||
|
|
argterms = ([DE.extargs[i] for i in DE.indices('exp')] +
|
||
|
|
[DE.T[i] for i in DE.indices('log')])
|
||
|
|
const = cancel(fa.as_expr()/fd.as_expr() -
|
||
|
|
Add(*[Mul(i, j/n) for i, j in zip(argterms, u)]))
|
||
|
|
|
||
|
|
return (ans, result, n, const)
|
||
|
|
|
||
|
|
|
||
|
|
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.
|
||
|
|
|
||
|
|
Explanation
|
||
|
|
===========
|
||
|
|
|
||
|
|
It differs from is_log_deriv_k_t_radical(fa, fd, DE, Df=False)
|
||
|
|
for any given fa, fd, DE in that it finds the solution in the
|
||
|
|
given field not in some (possibly unspecified extension) and
|
||
|
|
"in_field" with the function name is used to indicate that.
|
||
|
|
|
||
|
|
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 is 'auto',
|
||
|
|
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
|
||
|
|
|
||
|
|
# [(a, i), ...], where i*log(a) is a term in the log-part of the integral
|
||
|
|
# of f
|
||
|
|
respolys, residues = list(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 range(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
|
||
|
|
|
||
|
|
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.One)
|
||
|
|
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.One)
|
||
|
|
residueterms = [(i, j*common_denom) for i, j in residueterms]
|
||
|
|
m = common_denom//n
|
||
|
|
if common_denom != n*m: # Verify exact division
|
||
|
|
raise ValueError("Inexact division")
|
||
|
|
u = cancel(u**m*Mul(*[Pow(i, j) for i, j in residueterms]))
|
||
|
|
|
||
|
|
return (common_denom, u)
|