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https://github.com/vale981/arb
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139 lines
4.5 KiB
C
139 lines
4.5 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 "fmprb.h"
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/* With parameter n, the error is bounded by 3/(3+sqrt(8))^n */
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#define ERROR_A 1.5849625007211561815 /* log2(3) */
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#define ERROR_B 2.5431066063272239453 /* log2(3+sqrt(8)) */
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/*
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Computes zeta(s) for s = start + i*step, 0 <= i < num, writing the
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consecutive values to the array z. Uses Borwein's algorithm, here
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extended to support fast multi-evaluation (but also works well
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for a single s).
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Requires start >= 2. For efficiency, the largest s should be at most about as
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large as prec. Arguments approaching LONG_MAX will cause overflows.
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One should therefore only use this function for s up to about prec, and
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then switch to the Euler product.
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References:
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P. Borwein, "An Efficient Algorithm for the Riemann Zeta Function",
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Constructive experimental and nonlinear analysis,
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CMS Conference Proc. 27 (2000), 29–34
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http://www.cecm.sfu.ca/personal/pborwein/PAPERS/P155.pdf
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The MPFR team (2012), "MPFR Algorithms", http://www.mpfr.org/algo.html
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X. Gourdon and P. Sebah (2003),
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"Numerical evaluation of the Riemann Zeta-function"
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http://numbers.computation.free.fr/Constants/Miscellaneous/zetaevaluations.pdf
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The algorithm for single s is basically identical to the one used in MPFR
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(see the MPFR Algorithms paper for a detailed description).
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In particular, we evaluate the sum backwards to avoid temporary storage of
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the d_k coefficients, and use integer arithmetic throughout since it
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is convenient and the terms turn out to be slightly larger than 2^prec.
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The only numerical error in the main loop comes from the division by k^s,
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which adds less than 1 unit of error per term.
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For fast multi-evaluation, we perform repeated divisions by k^step.
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Each division decreases the input error and adds at most 1 unit of rounding
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error, so by induction, the error per term is always smaller than 2 units.
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*/
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void
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fmprb_zeta_ui_vec_borwein(fmprb_struct * z, ulong start, long num,
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ulong step, long prec)
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{
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long j, k, s, n, wp;
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fmpz_t c, d, t, u;
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fmpz * zeta;
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if (num < 1)
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return;
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wp = prec + FLINT_BIT_COUNT(prec);
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n = wp / 2.5431066063272239453 + 1;
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fmpz_init(c);
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fmpz_init(d);
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fmpz_init(t);
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fmpz_init(u);
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zeta = _fmpz_vec_init(num);
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fmpz_set_ui(c, 1);
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fmpz_mul_2exp(c, c, 2 * n - 1);
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fmpz_set(d, c);
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for (k = n; k > 0; k--)
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{
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/* divide by first k^s */
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fmpz_ui_pow_ui(u, k, start);
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fmpz_tdiv_q(t, d, u);
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if (k % 2 == 0)
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fmpz_neg(t, t);
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fmpz_add(zeta, zeta, t);
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/* remaining k^s */
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fmpz_ui_pow_ui(u, k, step);
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for (j = 1; j < num; j++)
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{
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fmpz_tdiv_q(t, t, u);
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fmpz_add(zeta + j, zeta + j, t);
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}
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/* hypergeometric recurrence */
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fmpz_mul2_uiui(c, c, k, 2 * k - 1);
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fmpz_divexact2_uiui(c, c, 2 * (n - k + 1), n + k - 1);
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fmpz_add(d, d, c);
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}
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for (k = 0; k < num; k++)
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{
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fmprb_struct * x = z + k;
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s = start + step * k;
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fmprb_set_fmpz(x, zeta + k);
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/* the error in each term in the main loop is < 2 */
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fmpr_set_ui(fmprb_radref(x), 2 * n);
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fmprb_div_fmpz(x, x, d, wp);
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/* mathematical error for eta(s), bounded by 3/(3+sqrt(8))^n */
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fmprb_add_error_2exp_si(x, (long) (ERROR_A - ERROR_B * n + 1));
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/* convert from eta(s) to zeta(s) */
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fmprb_div_2expm1_ui(x, x, s - 1, wp);
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fmprb_mul_2exp_si(x, x, s - 1);
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}
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fmpz_clear(c);
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fmpz_clear(d);
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fmpz_clear(t);
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fmpz_clear(u);
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_fmpz_vec_clear(zeta, num);
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}
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