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/*
Copyright (C) 2015 Fredrik Johansson
This file is part of Arb.
Arb is free software: you can redistribute it and/or modify it under
the terms of the GNU Lesser General Public License (LGPL) as published
by the Free Software Foundation; either version 2.1 of the License, or
(at your option) any later version. See <http://www.gnu.org/licenses/>.
*/
#include "acb_hypgeom.h"
/*
[S(k+1)] = [ R(k) 0 ] [S(k)]
[T(k+1)] [ 1 1 ] [T(k)]
[S(k+1)] = [ P(k) / Q(k) 0 ] [S(k)]
[T(k+1)] [ 1 1 ] [T(k)]
1 [ P(k) ]
---- [ ]
Q(k) [ Q(k) Q(k) ]
[[A2 0] [B2 C2]] . [[A1 0] [B1 C1]] = [[A1 A2 0] [A1 B2 + B1 C2 C1 C2]
A1 B2 + B1 B2 = B2 (A1 + B1) -- use to save time?
*/
static void
factor(acb_t A, acb_t tmp, acb_srcptr a, slong p, const acb_t z, slong k, slong prec)
{
slong i;
if (p == 0)
{
if (z == NULL)
acb_one(A);
else
acb_set(A, z);
}
else
{
acb_add_ui(A, a, k, prec);
for (i = 1; i < p; i++)
{
acb_add_ui(tmp, a + i, k, prec);
acb_mul(A, A, tmp, prec);
}
if (z != NULL)
acb_mul(A, A, z, prec);
}
}
static void
bsplit(acb_t A1, acb_t B1, acb_t C1,
acb_srcptr a, slong p,
acb_srcptr b, slong q,
const acb_t z,
slong aa,
slong bb,
slong prec,
int invz)
{
if (bb - aa == 1)
{
factor(A1, B1, a, p, invz ? NULL : z, aa, prec);
factor(C1, B1, b, q, invz ? z : NULL, aa, prec);
/* acb_set(B1, C1); but we skip this */
}
else
{
slong m;
acb_t A2, B2, C2;
acb_init(A2);
acb_init(B2);
acb_init(C2);
m = aa + (bb - aa) / 2;
bsplit(A1, B1, C1, a, p, b, q, z, aa, m, prec, invz);
bsplit(A2, B2, C2, a, p, b, q, z, m, bb, prec, invz);
if (bb - m == 1) /* B2 = C2 */
{
if (m - aa == 1)
acb_add(B2, A1, C1, prec);
else
acb_add(B2, A1, B1, prec);
acb_mul(B1, B2, C2, prec);
}
else
{
if (m - aa == 1)
acb_mul(B1, C1, C2, prec);
else
acb_mul(B1, B1, C2, prec);
acb_addmul(B1, A1, B2, prec);
}
acb_mul(A1, A1, A2, prec);
acb_mul(C1, C1, C2, prec);
acb_clear(A2);
acb_clear(B2);
acb_clear(C2);
}
}
void
acb_hypgeom_pfq_sum_bs(acb_t s, acb_t t,
acb_srcptr a, slong p, acb_srcptr b, slong q, const acb_t z, slong n, slong prec)
{
acb_t u, v, w, tmp;
if (n < 4)
{
acb_hypgeom_pfq_sum_forward(s, t, a, p, b, q, z, n, prec);
return;
}
acb_init(u);
acb_init(v);
acb_init(w);
acb_init(tmp);
/* we compute to n-1 instead of n to avoid dividing by 0 in the
denominator when computing a hypergeometric polynomial
that terminates right before a pole */
bsplit(u, v, w, a, p, b, q, z, 0, n - 1, prec, 0);
acb_add(s, u, v, prec); /* s = s + t */
acb_div(s, s, w, prec);
/* split off last factor */
factor(t, tmp, a, p, z, n - 1, prec);
acb_mul(u, u, t, prec);
factor(t, tmp, b, q, NULL, n - 1, prec);
acb_mul(w, w, t, prec);
acb_div(t, u, w, prec);
acb_clear(u);
acb_clear(v);
acb_clear(w);
acb_clear(tmp);
}
void
acb_hypgeom_pfq_sum_bs_invz(acb_t s, acb_t t,
acb_srcptr a, slong p, acb_srcptr b, slong q, const acb_t z, slong n, slong prec)
{
acb_t u, v, w, tmp;
if (n < 4)
{
acb_init(u);
acb_inv(u, z, prec);
acb_hypgeom_pfq_sum_forward(s, t, a, p, b, q, u, n, prec);
acb_clear(u);
return;
}
acb_init(u);
acb_init(v);
acb_init(w);
acb_init(tmp);
/* we compute to n-1 instead of n to avoid dividing by 0 in the
denominator when computing a hypergeometric polynomial
that terminates right before a pole */
bsplit(u, v, w, a, p, b, q, z, 0, n - 1, prec, 1);
acb_add(s, u, v, prec); /* s = s + t */
acb_div(s, s, w, prec);
/* split off last factor */
factor(t, tmp, a, p, NULL, n - 1, prec);
acb_mul(u, u, t, prec);
factor(t, tmp, b, q, z, n - 1, prec);
acb_mul(w, w, t, prec);
acb_div(t, u, w, prec);
acb_clear(u);
acb_clear(v);
acb_clear(w);
acb_clear(tmp);
}
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