mirror of
https://github.com/vale981/arb
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131 lines
3.3 KiB
C
131 lines
3.3 KiB
C
/*
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Copyright (C) 2013 Fredrik Johansson
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This file is part of Arb.
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Arb is free software: you can redistribute it and/or modify it under
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the terms of the GNU Lesser General Public License (LGPL) as published
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by the Free Software Foundation; either version 2.1 of the License, or
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(at your option) any later version. See <http://www.gnu.org/licenses/>.
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*/
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#include "bernoulli.h"
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#include "acb.h"
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#include "acb_hypgeom.h"
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void
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acb_gamma_stirling_choose_param(int * reflect, slong * r, slong * n,
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const acb_t z, int use_reflect, int digamma, slong prec);
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void acb_gamma_stirling_bound(mag_ptr err, const acb_t z, slong k0, slong knum, slong n);
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void arb_gamma_stirling_coeff(arb_t b, ulong k, int digamma, slong prec);
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void
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acb_gamma_stirling_eval(acb_t s, const acb_t z, slong nterms, int digamma, slong prec)
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{
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acb_t t, logz, zinv, zinv2;
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arb_t b;
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mag_t err;
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slong k, term_prec;
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double z_mag, term_mag;
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acb_init(t);
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acb_init(logz);
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acb_init(zinv);
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acb_init(zinv2);
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arb_init(b);
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acb_log(logz, z, prec);
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acb_inv(zinv, z, prec);
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nterms = FLINT_MAX(nterms, 1);
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acb_zero(s);
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if (nterms > 1)
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{
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acb_mul(zinv2, zinv, zinv, prec);
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z_mag = arf_get_d(arb_midref(acb_realref(logz)), ARF_RND_UP) * 1.44269504088896;
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for (k = nterms - 1; k >= 1; k--)
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{
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term_mag = bernoulli_bound_2exp_si(2 * k);
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term_mag -= (2 * k - 1) * z_mag;
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term_prec = prec + term_mag;
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term_prec = FLINT_MIN(term_prec, prec);
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term_prec = FLINT_MAX(term_prec, 10);
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arb_gamma_stirling_coeff(b, k, digamma, term_prec);
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if (prec > 2000)
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{
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acb_set_round(t, zinv2, term_prec);
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acb_mul(s, s, t, term_prec);
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}
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else
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acb_mul(s, s, zinv2, term_prec);
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arb_add(acb_realref(s), acb_realref(s), b, term_prec);
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}
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if (digamma)
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acb_mul(s, s, zinv2, prec);
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else
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acb_mul(s, s, zinv, prec);
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}
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/* remainder bound */
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mag_init(err);
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acb_gamma_stirling_bound(err, z, digamma ? 1 : 0, 1, nterms);
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mag_add(arb_radref(acb_realref(s)), arb_radref(acb_realref(s)), err);
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mag_add(arb_radref(acb_imagref(s)), arb_radref(acb_imagref(s)), err);
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mag_clear(err);
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if (digamma)
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{
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acb_neg(s, s);
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acb_mul_2exp_si(zinv, zinv, -1);
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acb_sub(s, s, zinv, prec);
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acb_add(s, s, logz, prec);
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}
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else
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{
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/* (z-0.5)*log(z) - z + log(2*pi)/2 */
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arb_one(b);
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arb_mul_2exp_si(b, b, -1);
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arb_set(acb_imagref(t), acb_imagref(z));
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arb_sub(acb_realref(t), acb_realref(z), b, prec);
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acb_mul(t, logz, t, prec);
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acb_add(s, s, t, prec);
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acb_sub(s, s, z, prec);
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arb_const_log_sqrt2pi(b, prec);
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arb_add(acb_realref(s), acb_realref(s), b, prec);
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}
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acb_clear(t);
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acb_clear(logz);
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acb_clear(zinv);
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acb_clear(zinv2);
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arb_clear(b);
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}
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void
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acb_gamma(acb_t y, const acb_t x, slong prec)
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{
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acb_hypgeom_gamma(y, x, prec);
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}
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void
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acb_rgamma(acb_t y, const acb_t x, slong prec)
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{
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acb_hypgeom_rgamma(y, x, prec);
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}
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void
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acb_lgamma(acb_t y, const acb_t x, slong prec)
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{
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acb_hypgeom_lgamma(y, x, prec);
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}
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