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arith.h
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arith.h
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/*============================================================================
This file is part of FLINT.
FLINT 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.
FLINT 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 FLINT; if not, write to the Free Software
Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA
===============================================================================*/
/******************************************************************************
Copyright (C) 2010-2012 Fredrik Johansson
******************************************************************************/
#ifndef ARITH_H
#define ARITH_H
#include <gmp.h>
#include <mpfr.h>
#include "flint.h"
#include "fmpz.h"
#include "fmpz_mat.h"
#include "fmpz_poly.h"
#include "fmpq_poly.h"
#include "fmpq.h"
#ifdef __cplusplus
extern "C" {
#endif
/* MPFR extras ***************************************************************/
void mpfr_zeta_inv_euler_product(mpfr_t res, ulong s, int char_4);
void mpfr_pi_chudnovsky(mpfr_t res, mpfr_rnd_t rnd);
/* Various arithmetic functions **********************************************/
void arith_primorial(fmpz_t res, slong n);
void _arith_harmonic_number(fmpz_t num, fmpz_t den, slong n);
void arith_harmonic_number(fmpq_t x, slong n);
void arith_ramanujan_tau(fmpz_t res, const fmpz_t n);
void arith_ramanujan_tau_series(fmpz_poly_t res, slong n);
void arith_divisors(fmpz_poly_t res, const fmpz_t n);
void arith_divisor_sigma(fmpz_t res, const fmpz_t n, ulong k);
int arith_moebius_mu(const fmpz_t n);
void arith_euler_phi(fmpz_t res, const fmpz_t n);
/* Stirling numbers **********************************************************/
void arith_stirling_number_1u(fmpz_t s, slong n, slong k);
void arith_stirling_number_1(fmpz_t s, slong n, slong k);
void arith_stirling_number_2(fmpz_t s, slong n, slong k);
void arith_stirling_number_1u_vec(fmpz * row, slong n, slong klen);
void arith_stirling_number_1_vec(fmpz * row, slong n, slong klen);
void arith_stirling_number_2_vec(fmpz * row, slong n, slong klen);
void arith_stirling_number_1u_vec_next(fmpz * row,
const fmpz * prev, slong n, slong klen);
void arith_stirling_number_1_vec_next(fmpz * row,
const fmpz * prev, slong n, slong klen);
void arith_stirling_number_2_vec_next(fmpz * row,
const fmpz * prev, slong n, slong klen);
void arith_stirling_matrix_1u(fmpz_mat_t mat);
void arith_stirling_matrix_1(fmpz_mat_t mat);
void arith_stirling_matrix_2(fmpz_mat_t mat);
/* Bell numbers **************************************************************/
#if FLINT64
#define BELL_NUMBER_TAB_SIZE 26
#else
#define BELL_NUMBER_TAB_SIZE 16
#endif
extern const mp_limb_t bell_number_tab[];
double arith_bell_number_size(ulong n);
void arith_bell_number(fmpz_t b, ulong n);
void arith_bell_number_bsplit(fmpz_t res, ulong n);
void arith_bell_number_multi_mod(fmpz_t res, ulong n);
void arith_bell_number_vec(fmpz * b, slong n);
void arith_bell_number_vec_recursive(fmpz * b, slong n);
void arith_bell_number_vec_multi_mod(fmpz * b, slong n);
mp_limb_t arith_bell_number_nmod(ulong n, nmod_t mod);
void arith_bell_number_nmod_vec(mp_ptr b, slong n, nmod_t mod);
void arith_bell_number_nmod_vec_recursive(mp_ptr b, slong n, nmod_t mod);
void arith_bell_number_nmod_vec_series(mp_ptr b, slong n, nmod_t mod);
/* Euler numbers *************************************************************/
#if FLINT64
#define SMALL_EULER_LIMIT 25
#else
#define SMALL_EULER_LIMIT 15
#endif
static const mp_limb_t euler_number_small[] = {
UWORD(1), UWORD(1), UWORD(5), UWORD(61), UWORD(1385), UWORD(50521), UWORD(2702765),
UWORD(199360981),
#if FLINT64
UWORD(19391512145), UWORD(2404879675441), UWORD(370371188237525),
UWORD(69348874393137901), UWORD(15514534163557086905)
#endif
};
double arith_euler_number_size(ulong n);
void arith_euler_number_vec(fmpz * res, slong n);
void _arith_euler_number_zeta(fmpz_t res, ulong n);
void arith_euler_number(fmpz_t res, ulong n);
void arith_euler_polynomial(fmpq_poly_t poly, ulong n);
/* Bernoulli numbers *********************************************************/
#if FLINT64
#define BERNOULLI_SMALL_NUMER_LIMIT 35
#else
#define BERNOULLI_SMALL_NUMER_LIMIT 27
#endif
static const slong _bernoulli_numer_small[] = {
WORD(1), WORD(1), WORD(-1), WORD(1), WORD(-1), WORD(5), WORD(-691), WORD(7), WORD(-3617), WORD(43867), WORD(-174611), WORD(854513),
WORD(-236364091), WORD(8553103),
#if FLINT64
WORD(-23749461029), WORD(8615841276005), WORD(-7709321041217), WORD(2577687858367)
#endif
};
void _arith_bernoulli_number(fmpz_t num, fmpz_t den, ulong n);
void arith_bernoulli_number(fmpq_t x, ulong n);
void _arith_bernoulli_number_vec(fmpz * num, fmpz * den, slong n);
void arith_bernoulli_number_vec(fmpq * num, slong n);
void arith_bernoulli_number_denom(fmpz_t den, ulong n);
double arith_bernoulli_number_size(ulong n);
void arith_bernoulli_polynomial(fmpq_poly_t poly, ulong n);
void _arith_bernoulli_number_zeta(fmpz_t num, fmpz_t den, ulong n);
void _arith_bernoulli_number_vec_multi_mod(fmpz * num, fmpz * den, slong n);
void _arith_bernoulli_number_vec_recursive(fmpz * num, fmpz * den, slong n);
void _arith_bernoulli_number_vec_zeta(fmpz * num, fmpz * den, slong n);
/* Cyclotomic polynomials ****************************************************/
void _arith_cyclotomic_polynomial(fmpz * a, ulong n, mp_ptr factors,
slong num_factors, ulong phi);
void arith_cyclotomic_polynomial(fmpz_poly_t poly, ulong n);
void _arith_cos_minpoly(fmpz * coeffs, slong d, ulong n);
void arith_cos_minpoly(fmpz_poly_t poly, ulong n);
/* Hypergeometric polynomials ************************************************/
void arith_legendre_polynomial(fmpq_poly_t poly, ulong n);
void arith_chebyshev_t_polynomial(fmpz_poly_t poly, ulong n);
void arith_chebyshev_u_polynomial(fmpz_poly_t poly, ulong n);
/* Swinnerton-Dyer polynomials ***********************************************/
void arith_swinnerton_dyer_polynomial(fmpz_poly_t poly, ulong n);
/* Landau function ***********************************************************/
void arith_landau_function_vec(fmpz * res, slong len);
/* Dedekind sums *************************************************************/
void arith_dedekind_sum_naive(fmpq_t s, const fmpz_t h, const fmpz_t k);
double arith_dedekind_sum_coprime_d(double h, double k);
void arith_dedekind_sum_coprime_large(fmpq_t s, const fmpz_t h, const fmpz_t k);
void arith_dedekind_sum_coprime(fmpq_t s, const fmpz_t h, const fmpz_t k);
void arith_dedekind_sum(fmpq_t s, const fmpz_t h, const fmpz_t k);
/* Exponential sums **********************************************************/
typedef struct
{
int n;
int prefactor;
mp_limb_t sqrt_p;
mp_limb_t sqrt_q;
mp_limb_signed_t cos_p[FLINT_BITS];
mp_limb_t cos_q[FLINT_BITS];
} trig_prod_struct;
typedef trig_prod_struct trig_prod_t[1];
static __inline__
void trig_prod_init(trig_prod_t sum)
{
sum->n = 0;
sum->prefactor = 1;
sum->sqrt_p = 1;
sum->sqrt_q = 1;
}
void arith_hrr_expsum_factored(trig_prod_t prod, mp_limb_t k, mp_limb_t n);
/* Number of partitions ******************************************************/
void arith_number_of_partitions_nmod_vec(mp_ptr res, slong len, nmod_t mod);
void arith_number_of_partitions_vec(fmpz * res, slong len);
void arith_number_of_partitions_mpfr(mpfr_t x, ulong n);
void arith_number_of_partitions(fmpz_t x, ulong n);
/* Number of sums of squares representations *********************************/
void arith_sum_of_squares(fmpz_t r, ulong k, const fmpz_t n);
void arith_sum_of_squares_vec(fmpz * r, ulong k, slong n);
#ifdef __cplusplus
}
#endif
#endif