mirror of
https://github.com/vale981/arb
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119 lines
3.2 KiB
C
119 lines
3.2 KiB
C
/*=============================================================================
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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
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it under the terms of the GNU General Public License as published by
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the Free Software Foundation; either version 2 of the License, or
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(at your option) any later version.
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ARB is distributed in the hope that it will be useful,
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but WITHOUT ANY WARRANTY; without even the implied warranty of
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MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
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GNU General Public License for more details.
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You should have received a copy of the GNU General Public License
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along with ARB; if not, write to the Free Software
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Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA
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=============================================================================*/
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/******************************************************************************
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Copyright (C) 2012 Fredrik Johansson
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******************************************************************************/
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#include <math.h>
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#include "zeta.h"
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/*
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Let P(a,b) = prod_{a <= p <= b} (1 - p^(-s)).
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Then 1/zeta(s) = P(a,M) * P(M+1,inf).
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According to the analysis in S. Fillebrown,
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"Faster Computation of Bernoulli Numbers", Journal of Algorithms 13,
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431-445 (1992), it holds for all s >= 6 and M >= 1 that
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(1/P(M+1,inf) - 1) <= 2 * M^(1-s) / (s/2 - 1).
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Writing 1/zeta(s) = P(a,M) * (1 - eps) and solving for eps gives
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1/(1-eps) <= 1 + 2 * M^(1-s) / (s/2 - 1), so we have
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eps <= 2 * M^(1-s) / (s/2 - 1) = 4 * M^(1-s) / (s-2).
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Since 0 < P(a,M) <= 1, this bounds the absolute error of 1/zeta(s).
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*/
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static void
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add_error(fmprb_t z, ulong M, ulong s)
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{
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fmpr_t t;
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fmpr_init(t);
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fmpr_set_ui(t, M);
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fmpr_pow_sloppy_ui(t, t, s - 1, FMPRB_RAD_PREC, FMPR_RND_DOWN);
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fmpr_mul_ui(t, t, s - 2, FMPRB_RAD_PREC, FMPR_RND_DOWN);
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fmpr_ui_div(t, 4, t, FMPRB_RAD_PREC, FMPR_RND_UP);
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fmprb_add_error_fmpr(z, t);
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fmpr_clear(t);
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}
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void
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fmprb_zeta_inv_ui_euler_product(fmprb_t z, ulong s, long prec)
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{
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long wp, powprec;
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double powmag;
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fmprb_t t;
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ulong M;
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mp_limb_t p;
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if (s < 6)
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{
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printf("too small s!\n");
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abort();
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}
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/* heuristic */
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wp = prec + FLINT_BIT_COUNT(prec) + (prec/s) + 4;
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fmprb_init(t);
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/* z = 1 - 2^(-s) */
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fmprb_set_ui(z, 1UL);
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fmpr_set_ui_2exp_si(fmprb_midref(t), 1, -s);
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fmprb_sub(z, z, t, wp);
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M = 2;
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p = 3UL;
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while (1)
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{
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/* approximate magnitude of p^s */
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powmag = s * log(p) * 1.4426950408889634;
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powprec = FLINT_MAX(wp - powmag, 8);
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/* see error analysis */
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if ((powmag >= prec) &&
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((1.-s)*log(M-1.)) - log(s-2.) + 2 <= -(prec+1) * 0.69314718055995)
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break;
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M = p;
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fmprb_ui_pow_ui(t, p, s, powprec);
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fmprb_div(t, z, t, powprec);
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fmprb_sub(z, z, t, wp);
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p = n_nextprime(p, 0);
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}
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add_error(z, M, s);
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fmprb_clear(t);
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}
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void
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fmprb_zeta_ui_euler_product(fmprb_t z, ulong s, long prec)
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{
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fmprb_t one;
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fmprb_init(one);
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fmprb_set_ui(one, 1);
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fmprb_zeta_inv_ui_euler_product(z, s, prec);
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fmprb_div(z, one, z, prec);
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fmprb_clear(one);
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}
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