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bernoulli.h
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bernoulli.h
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/*=============================================================================
This file is part of ARB.
ARB is free software; you can redistribute it and/or modify
it under the terms of the GNU General Public License as published by
the Free Software Foundation; either version 2 of the License, or
(at your option) any later version.
ARB is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
GNU General Public License for more details.
You should have received a copy of the GNU General Public License
along with ARB; if not, write to the Free Software
Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA
=============================================================================*/
/******************************************************************************
Copyright (C) 2012 Fredrik Johansson
******************************************************************************/
#ifndef BERNOULLI_H
#define BERNOULLI_H
#include <math.h>
#include "flint.h"
#include "fmpz.h"
#include "fmpz_vec.h"
#include "fmpq.h"
#include "arith.h"
#include "fmprb.h"
#include "arb.h"
#ifdef __cplusplus
extern "C" {
#endif
extern long TLS_PREFIX bernoulli_cache_num;
extern TLS_PREFIX fmpq * bernoulli_cache;
void bernoulli_cache_compute(long n);
/*
Crude bound for the bits in d(n) = denom(B_n).
By von Staudt-Clausen, d(n) = prod_{p-1 | n} p
<= prod_{k | n} 2k
<= n^{sigma_0(n)}.
We get a more accurate estimate taking the square root of this.
Further, at least for sufficiently large n,
sigma_0(n) < exp(1.066 log(n) / log(log(n))).
*/
static __inline__ long bernoulli_denom_size(long n)
{
return 0.5 * 1.4427 * log(n) * pow(n, 1.066 / log(log(n)));
}
static __inline__ long bernoulli_zeta_terms(ulong s, long prec)
{
long N;
N = pow(2.0, (prec + 1.0) / (s - 1.0));
N += ((N % 2) == 0);
return N;
}
static __inline__ long bernoulli_power_prec(long i, ulong s1, long wp)
{
long p = wp - s1 * log(i) * 1.44269504088896341;
return FLINT_MAX(p, 10);
}
/* we should technically add O(log(n)) guard bits, but this is unnecessary
in practice since the denominator estimate is quite a bit larger
than the true denominators
*/
static __inline__ long bernoulli_global_prec(ulong nmax)
{
return arith_bernoulli_number_size(nmax) + bernoulli_denom_size(nmax);
}
/* avoid potential numerical problems for very small n */
#define BERNOULLI_REV_MIN 32
typedef struct
{
long alloc;
long prec;
long max_power;
fmpz * powers;
fmpz_t pow_error;
arb_t prefactor;
arb_t two_pi_squared;
ulong n;
}
bernoulli_rev_struct;
typedef bernoulli_rev_struct bernoulli_rev_t[1];
void bernoulli_rev_init(bernoulli_rev_t iter, ulong nmax);
void bernoulli_rev_next(fmpz_t numer, fmpz_t denom, bernoulli_rev_t iter);
void bernoulli_rev_clear(bernoulli_rev_t iter);
#define BERNOULLI_ENSURE_CACHED(n) \
do { \
long __n = (n); \
if (__n >= bernoulli_cache_num) \
bernoulli_cache_compute(__n + 1); \
} while (0); \
long bernoulli_bound_2exp_si(ulong n);
void bernoulli_fmprb_ui_zeta(fmprb_t b, ulong n, long prec);
void bernoulli_fmprb_ui(fmprb_t b, ulong n, long prec);
void _bernoulli_fmpq_ui_zeta(fmpz_t num, fmpz_t den, ulong n);
static __inline__ void
_bernoulli_fmpq_ui(fmpz_t num, fmpz_t den, ulong n)
{
if (n < (ulong) bernoulli_cache_num)
{
fmpz_set(num, fmpq_numref(bernoulli_cache + n));
fmpz_set(den, fmpq_denref(bernoulli_cache + n));
}
else
{
_bernoulli_fmpq_ui_zeta(num, den, n);
}
}
static __inline__ void
bernoulli_fmpq_ui(fmpq_t b, ulong n)
{
_bernoulli_fmpq_ui(fmpq_numref(b), fmpq_denref(b), n);
}
#ifdef __cplusplus
}
#endif
#endif