mirror of
https://github.com/vale981/arb
synced 2025-03-05 09:21:38 -05:00
135 lines
3.3 KiB
C
135 lines
3.3 KiB
C
/*
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Copyright (C) 2017 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 "arb_hypgeom.h"
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#include "bernoulli.h"
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static const unsigned int central_bin_tab[] = {
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1, 2, 6, 20, 70, 252, 924, 3432, 12870, 48620, 184756, 705432, 2704156,
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10400600, 40116600, 155117520, 601080390, 2333606220U,
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};
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void arb_gamma_stirling_coeff(arb_t b, ulong k, int digamma, slong prec);
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/* See Richard P. Brent, "Asymptotic approximation of central binomial
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coefficients with rigorous error bounds". https://arxiv.org/abs/1608.04834 */
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static void
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arb_hypgeom_central_bin_ui_asymp(arb_t res, ulong n, slong prec)
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{
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arb_t s, t, u;
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fmpz_t n2;
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slong j, k, term_prec, wp;
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double term_mag, n2_mag;
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mag_t err, err2;
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arb_init(s);
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arb_init(t);
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arb_init(u);
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fmpz_init(n2);
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mag_init(err);
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mag_init(err2);
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wp = prec + 8;
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n2_mag = log(n) * 1.44269504088896;
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for (k = 1; k < prec; k++)
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{
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term_mag = bernoulli_bound_2exp_si(2 * k + 2) - (2 * k + 1) * n2_mag;
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term_mag -= (FLINT_BIT_COUNT((k + 1)*(2*k+1)) - 1);
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if (term_mag < -wp)
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break;
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}
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wp += 2 * FLINT_BIT_COUNT(k);
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BERNOULLI_ENSURE_CACHED(2*k)
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fmpz_set_ui(n2, n);
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fmpz_mul_ui(n2, n2, n);
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n2_mag *= 2;
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for (j = 0; j <= k - 1; j++)
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{
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term_mag = bernoulli_bound_2exp_si(2 * j + 2);
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term_mag -= j * n2_mag;
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term_prec = wp + term_mag;
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term_prec = FLINT_MIN(term_prec, wp);
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term_prec = FLINT_MAX(term_prec, 10);
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arb_gamma_stirling_coeff(t, j + 1, 0, term_prec);
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arb_mul_2exp_si(u, t, -2*j - 2);
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arb_sub(t, u, t, term_prec);
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arb_mul_2exp_si(t, t, 1);
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arb_addmul_fmpz(t, s, n2, wp);
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arb_swap(s, t);
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}
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arb_set_fmpz(t, n2);
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arb_pow_ui(t, t, k - 1, wp);
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arb_mul_ui(t, t, n, wp);
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arb_div(s, s, t, wp);
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/* error term: bernoulli(2k+2) / ((k+1)(2k+1)) / n^(2k+1) */
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mag_bernoulli_div_fac_ui(err, 2 * k + 2);
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mag_fac_ui(err2, 2 * k + 2);
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mag_mul(err, err, err2);
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mag_set_ui_lower(err2, n);
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mag_pow_ui_lower(err2, err2, 2 * k + 1);
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mag_mul_ui_lower(err2, err2, k + 1);
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mag_div(err, err, err2);
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arb_add_error_mag(s, err);
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arb_exp(s, s, wp);
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arb_const_pi(t, wp);
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arb_mul_ui(t, t, n, wp);
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arb_rsqrt(t, t, wp);
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arb_mul(res, s, t, prec);
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fmpz_set_ui(n2, n);
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fmpz_mul_2exp(n2, n2, 1);
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arb_mul_2exp_fmpz(res, res, n2);
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arb_clear(s);
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arb_clear(t);
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arb_clear(u);
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fmpz_clear(n2);
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mag_clear(err);
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mag_clear(err2);
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}
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void
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arb_hypgeom_central_bin_ui(arb_t res, ulong n, slong prec)
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{
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if (n <= 17)
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{
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arb_set_ui(res, central_bin_tab[n]);
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arb_set_round(res, res, prec);
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}
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else if (n < 6.0 * prec + 200.0)
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{
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fmpz_t t;
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fmpz_init(t);
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fmpz_bin_uiui(t, 2 * n, n);
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arb_set_round_fmpz(res, t, prec);
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fmpz_clear(t);
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}
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else
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{
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arb_hypgeom_central_bin_ui_asymp(res, n, prec);
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}
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}
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