mirror of
https://github.com/vale981/arb
synced 2025-03-06 01:41:39 -05:00
331 lines
7.2 KiB
C
331 lines
7.2 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 "double_extras.h"
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#include "hypgeom.h"
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void
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fmpr_gamma_ui_lbound(fmpr_t x, ulong n, long prec)
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{
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if (n == 0) abort();
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if (n < 250)
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{
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fmpz_t t;
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fmpz_init(t);
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fmpz_fac_ui(t, n - 1);
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fmpr_set_round_fmpz(x, t, prec, FMPR_RND_DOWN);
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fmpz_clear(t);
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}
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else
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{
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/* (2 pi/x)^(1/2) * (x/e)^x < Gamma(x) */
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fmpr_t t, u;
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fmpr_init(t);
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fmpr_init(u);
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/* lower bound for 2 pi */
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fmpr_set_ui_2exp_si(t, 843314855, -27);
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fmpr_div_ui(t, t, n, prec, FMPR_RND_DOWN);
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fmpr_sqrt(t, t, prec, FMPR_RND_DOWN);
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/* lower bound for 1/e */
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fmpr_set_ui_2exp_si(u, 197503771, -29);
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fmpr_mul_ui(u, u, n, prec, FMPR_RND_DOWN);
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fmpr_pow_sloppy_ui(u, u, n, prec, FMPR_RND_DOWN);
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fmpr_mul(x, t, u, prec, FMPR_RND_DOWN);
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fmpr_clear(t);
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fmpr_clear(u);
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}
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}
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void
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fmpr_gamma_ui_ubound(fmpr_t x, ulong n, long prec)
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{
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if (n == 0) abort();
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if (n < 250)
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{
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fmpz_t t;
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fmpz_init(t);
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fmpz_fac_ui(t, n - 1);
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fmpr_set_round_fmpz(x, t, prec, FMPR_RND_UP);
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fmpz_clear(t);
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}
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else
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{
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fmpr_t t, u;
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fmpr_init(t);
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/* Gamma(x) < e * (x / e)^x -- TODO: use a tighter bound */
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fmpr_init(t);
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fmpr_init(u);
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/* upper bound for 1/e */
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fmpr_set_ui_2exp_si(u, 197503773, -29);
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fmpr_mul_ui(u, u, n, prec, FMPR_RND_UP);
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fmpr_pow_sloppy_ui(u, u, n, prec, FMPR_RND_UP);
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/* upper bound for e */
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fmpr_set_ui_2exp_si(t, 364841613, -27);
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fmpr_mul(x, t, u, prec, FMPR_RND_UP);
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fmpr_clear(t);
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fmpr_clear(u);
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}
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}
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long
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hypgeom_root_bound(const mag_t z, int r)
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{
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if (r == 0)
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{
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return 0;
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}
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else
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{
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fmpr_t t;
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long v;
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fmpr_init(t);
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mag_get_fmpr(t, z);
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fmpr_root(t, t, r, MAG_BITS, FMPR_RND_UP);
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fmpr_add_ui(t, t, 1, MAG_BITS, FMPR_RND_UP);
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v = fmpr_get_si(t, FMPR_RND_UP);
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fmpr_clear(t);
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return v;
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}
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}
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/*
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Given T(K), compute bound for T(n) z^n.
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We need to multiply by
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z^n * 1/rf(K+1,m)^r * (rf(K+1,m)/rf(K+1-A,m)) * (rf(K+1-B,m)/rf(K+1-2B,m))
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where m = n - K. This is equal to
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z^n *
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(K+A)! (K-2B)! (K-B+m)!
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----------------------- * ((K+m)! / K!)^(1-r)
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(K-B)! (K-A+m)! (K-2B+m)!
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*/
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void
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hypgeom_term_bound(mag_t Tn, const mag_t TK, long K, long A, long B, int r, const mag_t z, long n)
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{
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mag_t t, u, num;
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long m;
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mag_init(t);
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mag_init(u);
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mag_init(num);
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m = n - K;
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if (m < 0)
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{
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printf("hypgeom term bound\n");
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abort();
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}
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/* TK * z^n */
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mag_pow_ui(t, z, n);
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mag_mul(num, TK, t);
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/* numerator: (K+A)! (K-2B)! (K-B+m)! */
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mag_fac_ui(t, K+A);
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mag_mul(num, num, t);
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mag_fac_ui(t, K-2*B);
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mag_mul(num, num, t);
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mag_fac_ui(t, K-B+m);
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mag_mul(num, num, t);
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/* denominator: (K-B)! (K-A+m)! (K-2B+m)! */
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mag_rfac_ui(t, K-B);
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mag_mul(num, num, t);
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mag_rfac_ui(t, K-A+m);
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mag_mul(num, num, t);
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mag_rfac_ui(t, K-2*B+m);
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mag_mul(num, num, t);
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/* ((K+m)! / K!)^(1-r) */
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if (r == 0)
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{
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mag_fac_ui(t, K+m);
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mag_mul(num, num, t);
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mag_rfac_ui(t, K);
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mag_mul(num, num, t);
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}
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else if (r != 1)
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{
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mag_fac_ui(t, K);
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mag_rfac_ui(u, K+m);
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mag_mul(t, t, u);
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mag_pow_ui(t, t, r-1);
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mag_mul(num, num, t);
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}
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mag_set(Tn, num);
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mag_clear(t);
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mag_clear(u);
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mag_clear(num);
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}
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/* z = max(x-y, 0), rounding down [lower bound] */
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void
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mag_sub_lower(mag_t z, const mag_t x, const mag_t y)
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{
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if (mag_is_special(x) || mag_is_special(y))
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{
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if (mag_is_zero(y))
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mag_set(z, x);
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else if (mag_is_zero(x) || mag_is_inf(y))
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mag_zero(z);
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else
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mag_inf(z);
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}
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else
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{
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fmpr_t t, u;
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fmpr_init(t);
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fmpr_init(u);
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mag_get_fmpr(t, x);
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mag_get_fmpr(u, y);
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fmpr_sub(t, t, u, MAG_BITS, FMPR_RND_DOWN);
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if (fmpr_sgn(t) <= 0)
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mag_zero(z);
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else /* XXX: exact? */
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mag_set_fmpz_2exp_fmpz_lower(z, fmpr_manref(t), fmpr_expref(t));
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fmpr_clear(t);
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fmpr_clear(u);
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}
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}
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long
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hypgeom_bound(mag_t error, int r,
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long A, long B, long K, const mag_t TK, const mag_t z, long tol_2exp)
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{
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mag_t Tn, t, u, one, tol, num, den;
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long n, m;
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mag_init(Tn);
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mag_init(t);
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mag_init(u);
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mag_init(one);
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mag_init(tol);
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mag_init(num);
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mag_init(den);
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mag_one(one);
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mag_set_ui_2exp_si(tol, 1UL, -tol_2exp);
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/* approximate number of needed terms */
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n = hypgeom_estimate_terms(z, r, tol_2exp);
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/* required for 1 + O(1/k) part to be decreasing */
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n = FLINT_MAX(n, K + 1);
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/* required for z^k / (k!)^r to be decreasing */
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m = hypgeom_root_bound(z, r);
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n = FLINT_MAX(n, m);
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/* We now have |R(k)| <= G(k) where G(k) is monotonically decreasing,
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and can bound the tail using a geometric series as soon
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as soon as G(k) < 1. */
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/* bound T(n-1) */
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hypgeom_term_bound(Tn, TK, K, A, B, r, z, n-1);
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while (1)
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{
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/* bound R(n) */
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mag_mul_ui(num, z, n);
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mag_mul_ui(num, num, n - B);
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mag_set_ui_lower(den, n - A);
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mag_mul_ui_lower(den, den, n - 2*B);
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if (r != 0)
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{
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mag_set_ui_lower(u, n);
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mag_pow_ui_lower(u, u, r);
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mag_mul_lower(den, den, u);
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}
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mag_div(t, num, den);
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/* multiply bound for T(n-1) by bound for R(n) to bound T(n) */
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mag_mul(Tn, Tn, t);
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/* geometric series termination check */
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/* u = max(1-t, 0), rounding down [lower bound] */
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mag_sub_lower(u, one, t);
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if (!mag_is_zero(u))
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{
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mag_div(u, Tn, u);
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if (mag_cmp(u, tol) < 0)
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{
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mag_set(error, u);
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break;
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}
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}
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/* move on to next term */
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n++;
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}
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mag_clear(Tn);
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mag_clear(t);
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mag_clear(u);
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mag_clear(one);
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mag_clear(tol);
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mag_clear(num);
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mag_clear(den);
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return n;
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}
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