mirror of
https://github.com/vale981/arb
synced 2025-03-06 18:01:39 -05:00
330 lines
7.8 KiB
C
330 lines
7.8 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 "fmprb.h"
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#include "double_extras.h"
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double fmpr_get_d(const fmpr_t x);
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#define LOG2 0.69314718055994530942
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#define EXP1 2.7182818284590452354
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double d_root(double x, int r)
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{
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if (r == 1)
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return x;
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if (r == 2)
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return sqrt(x);
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return pow(x, 1. / r);
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}
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/*
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Estimate the truncation point to obtain accuracy 2^(-prec) with the
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hypergeometric series |z|^k / (k!)^r.
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*/
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long
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estimate_nterms(double z, int r, long prec)
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{
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double y;
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z = fabs(z);
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if (z == 0)
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return 1;
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if (r == 0)
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{
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if (z >= 1)
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{
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printf("z must be smaller than 1\n");
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abort();
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}
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y = (log(1-z) - prec * LOG2) / log(z) + 1;
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}
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else
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{
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/* Solve k*log(z) - r*(k*log(k)-k) = -prec*log(2) */
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y = prec * LOG2 / (d_root(z, r) * EXP1 * r);
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y = prec * LOG2 / (r * d_lambertw(y)) + 1;
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}
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if (y >= LONG_MAX / 2)
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{
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printf("error: series will converge too slowly\n");
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abort();
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}
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return y;
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}
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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_fmpz(x, t);
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fmpr_set_round(x, x, 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_fmpz(x, t);
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fmpr_set_round(x, x, 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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/* FIXME: assumes no overflow when computing n + r */
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static void
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fmpr_rfac_uiui_ubound(fmpr_t x, ulong n, ulong r, long prec)
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{
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if (r == 0)
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{
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fmpr_one(x);
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}
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else if (r == 1)
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{
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fmpr_set_ui(x, n);
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}
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else
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{
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fmpr_t t;
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fmpr_init(t);
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fmpr_gamma_ui_ubound(x, n + r, prec);
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fmpr_gamma_ui_lbound(t, n, prec);
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fmpr_div(x, x, t, prec, FMPR_RND_UP);
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fmpr_clear(t);
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}
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}
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static void
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fmpr_rfac_uiui_lbound(fmpr_t x, ulong n, ulong r, long prec)
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{
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if (r == 0)
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{
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fmpr_one(x);
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}
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else if (r == 1)
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{
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fmpr_set_ui(x, n);
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}
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else
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{
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fmpr_t t;
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fmpr_init(t);
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fmpr_gamma_ui_lbound(x, n + r, prec);
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fmpr_gamma_ui_ubound(t, n, prec);
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fmpr_div(x, x, t, prec, FMPR_RND_DOWN);
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fmpr_clear(t);
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}
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}
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/*
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The general term T(k) is z^k / (k!)^r * R(k) where R(k) = 1 + O(1/k).
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We have precomputed integers A, B, K such that for all k > K,
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|R(k)| <= F(k) = k(k-B)/((k-A)(k-2B)) = (1+A/(k-A))(1+B/(k-2B)).
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*/
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long
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hypgeom_bound(fmpr_t error, int r,
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long A, long B, long K, const fmpr_t TK, const fmpr_t z, long prec)
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{
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fmpr_t Tn, t, u, one, tol;
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long wp = FMPRB_RAD_PREC;
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long n;
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double zd;
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fmpr_init(Tn);
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fmpr_init(t);
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fmpr_init(u);
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fmpr_init(one);
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fmpr_init(tol);
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fmpr_one(one);
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fmpr_set_ui_2exp_si(tol, 1UL, -prec);
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zd = fmpr_get_d(z);
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n = estimate_nterms(zd, r, prec);
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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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(TODO: don't use doubles for this) */
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if (r > 0)
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{
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long nbd = d_root(zd, r) + 2;
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n = FLINT_MAX(n, nbd);
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}
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/* We are now sure that |R(k)| is either decreasing or strictly
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smaller than 1 for k >= n, which means that we can bound the tail
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using a geometric series as soon as soon as |R(k)| < 1 */
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/* Compute an upper bound for T(n) */
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/* (z^n) / (n!)^r * TK * [(K+1)(K+2)...(n)] * [(K-B+1)(K-B+2)...(n-B)]
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---------------------------------------------
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[(K-A+1)(K-A+2)...(n)] * [(K-2B+1)(K-2B+2)...(n-2B)]
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*/
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/* z^n * TK */
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fmpr_pow_sloppy_ui(Tn, z, n, wp, FMPR_RND_UP);
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fmpr_mul(Tn, Tn, TK, wp, FMPR_RND_UP);
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/* divide by (n!)^r */
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if (r != 0)
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{
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fmpr_gamma_ui_lbound(t, n + 1, wp);
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fmpr_ui_div(t, 1UL, t, wp, FMPR_RND_UP);
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fmpr_pow_sloppy_ui(t, t, r, wp, FMPR_RND_UP);
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fmpr_mul(Tn, Tn, t, wp, FMPR_RND_UP);
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}
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fmpr_rfac_uiui_ubound(t, K+1, n-K, wp);
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fmpr_mul(Tn, Tn, t, wp, FMPR_RND_UP);
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fmpr_rfac_uiui_ubound(t, K-B+1, n-K, wp);
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fmpr_mul(Tn, Tn, t, wp, FMPR_RND_UP);
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fmpr_rfac_uiui_lbound(t, K-A+1, n-K, wp);
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fmpr_div(Tn, Tn, t, wp, FMPR_RND_UP);
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fmpr_rfac_uiui_lbound(t, K-2*B+1, n-K, wp);
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fmpr_div(Tn, Tn, t, wp, FMPR_RND_UP);
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while (1)
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{
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/* bound for term ratio: z * F(n) / n^r */
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/* F(n) <= n (n-B) / ((n-A) (n-2B)) */
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fmpr_set_ui(t, n);
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fmpr_mul_ui(t, t, n - B, wp, FMPR_RND_UP);
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fmpr_div_ui(t, t, n - A, wp, FMPR_RND_UP);
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fmpr_div_ui(t, t, n - 2*B, wp, FMPR_RND_UP);
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fmpr_mul(t, t, z, wp, FMPR_RND_UP);
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if (r != 0)
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{
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fmpr_div_ui(u, one, n, wp, FMPR_RND_UP);
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fmpr_pow_sloppy_ui(u, u, r, wp, FMPR_RND_UP);
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fmpr_mul(t, t, u, wp, FMPR_RND_UP);
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}
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/* bound by geometric series: Tn / (1 - t) */
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/* where the term ratio must be < 1 */
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fmpr_sub(u, one, t, wp, FMPR_RND_DOWN);
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if (fmpr_sgn(u) > 0)
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{
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fmpr_div(u, Tn, u, wp, FMPR_RND_UP);
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if (fmpr_cmp(u, tol) < 0)
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{
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fmpr_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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fmpr_mul(Tn, Tn, t, wp, FMPR_RND_UP);
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n++;
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}
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fmpr_clear(Tn);
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fmpr_clear(t);
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fmpr_clear(u);
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fmpr_clear(one);
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fmpr_clear(tol);
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return n;
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}
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