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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"
#include "arb_hypgeom.h"
void
acb_hypgeom_legendre_p(acb_t res, const acb_t n, const acb_t m,
const acb_t z, int type, slong prec)
{
acb_t a, b, c, w;
int flag;
if (!acb_is_finite(z))
{
acb_indeterminate(res);
return;
}
if (acb_is_real(z) && acb_is_zero(m) && acb_is_int(n) &&
arf_sgn(arb_midref(acb_realref(n))) >= 0 &&
arf_cmpabs_2exp_si(arb_midref(acb_realref(n)), FLINT_BITS - 1) < 0)
{
arb_hypgeom_legendre_p_ui(acb_realref(res), NULL,
arf_get_si(arb_midref(acb_realref(n)), ARF_RND_DOWN), acb_realref(z), prec);
arb_zero(acb_imagref(res));
return;
}
if (acb_is_int(n) && acb_is_int(m) && arb_is_nonnegative(acb_realref(n))
&& arb_is_nonnegative(acb_realref(m)) && type == 0)
{
arf_srcptr nn = arb_midref(acb_realref(n));
arf_srcptr mm = arb_midref(acb_realref(m));
if (arf_cmpabs(mm, nn) > 0)
{
acb_zero(res);
return;
}
if (arf_cmpabs_2exp_si(nn, FLINT_BITS - 1) < 0 &&
arf_cmpabs_2exp_si(mm, FLINT_BITS - 1) < 0)
{
slong nnn, mmm;
nnn = arf_get_si(nn, ARF_RND_DOWN);
mmm = arf_get_si(mm, ARF_RND_DOWN);
/* we will probably lose all accuracy... */
if (nnn - mmm > 2 * prec)
{
acb_indeterminate(res);
} /* hypergeometric series is faster at high precision, but avoid at 1 */
else if (prec < 500 + nnn * 10.0 || (nnn - mmm) < 10 ||
(arb_contains_si(acb_realref(z), 1) &&
arb_contains_zero(acb_imagref(z))))
{
if (mmm == 0)
{
acb_hypgeom_legendre_p_uiui_rec(res, nnn, mmm, z, prec);
}
else
{
acb_init(a);
acb_init(b);
acb_mul(a, z, z, prec);
acb_sub_ui(a, a, 1, prec);
acb_neg(a, a);
acb_mul_2exp_si(b, m, -1);
acb_pow(a, a, b, prec);
acb_hypgeom_legendre_p_uiui_rec(res, nnn, mmm, z, prec);
acb_mul(res, res, a, prec);
acb_clear(a);
acb_clear(b);
}
return;
}
}
}
acb_init(a);
acb_init(b);
acb_init(c);
acb_init(w);
acb_neg(a, n);
acb_add_ui(b, n, 1, prec);
acb_sub_ui(c, m, 1, prec);
acb_neg(c, c);
acb_sub_ui(w, z, 1, prec);
acb_neg(w, w);
acb_mul_2exp_si(w, w, -1);
/* a + b - c (which could be inexact) is integer iff c is integer */
flag = 1;
if (acb_is_int(c))
{
flag |= ACB_HYPGEOM_2F1_ABC;
}
acb_hypgeom_2f1(w, a, b, c, w, flag, prec);
if (!acb_is_zero(m))
{
acb_add_ui(a, z, 1, prec);
acb_sub_ui(b, z, 1, prec);
if (type == 0)
{
acb_neg(b, b);
}
else if (type != 1)
{
flint_printf("unsupported 'type' %d for legendre p\n", type);
flint_abort();
}
acb_mul_2exp_si(c, m, -1);
acb_pow(a, a, c, prec);
acb_neg(c, c);
acb_pow(b, b, c, prec);
acb_mul(w, w, a, prec);
acb_mul(w, w, b, prec);
}
acb_set(res, w);
acb_clear(a);
acb_clear(b);
acb_clear(c);
acb_clear(w);
}
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