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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) 2010-2011 Fredrik Johansson
******************************************************************************/
#ifndef ARITH_H
#define ARITH_H
#include <mpir.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"
void fmpz_primorial(fmpz_t res, long n);
void fmpz_poly_ramanujan_tau(fmpz_poly_t res, long n);
void fmpz_ramanujan_tau(fmpz_t res, const fmpz_t n);
void fmpz_divisors(fmpz_poly_t res, const fmpz_t n);
void fmpz_divisor_sigma(fmpz_t res, const fmpz_t n, ulong k);
int fmpz_moebius_mu(const fmpz_t n);
void fmpz_euler_phi(fmpz_t res, const fmpz_t n);
void _harmonic_number(fmpz_t num, fmpz_t den, long n);
void harmonic_number(fmpq_t x, long n);
/* Stirling numbers **********************************************************/
void stirling_number_1u(fmpz_t s, long n, long k);
void stirling_number_1(fmpz_t s, long n, long k);
void stirling_number_2(fmpz_t s, long n, long k);
void stirling_number_1u_vec(fmpz * row, long n, long klen);
void stirling_number_1_vec(fmpz * row, long n, long klen);
void stirling_number_2_vec(fmpz * row, long n, long klen);
void stirling_number_1u_vec_next(fmpz * row, fmpz * prev, long n, long klen);
void stirling_number_1_vec_next(fmpz * row, fmpz * prev, long n, long klen);
void stirling_number_2_vec_next(fmpz * row, fmpz * prev, long n, long klen);
void stirling_number_1u_mat(fmpz_mat_t mat);
void stirling_number_1_mat(fmpz_mat_t mat);
void stirling_number_2_mat(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 bell_number_size(ulong n);
void bell_number(fmpz_t b, ulong n);
void bell_number_bsplit(fmpz_t res, ulong n);
void bell_number_multi_mod(fmpz_t res, ulong n);
void bell_number_vec(fmpz * b, long n);
void bell_number_vec_recursive(fmpz * b, long n);
void bell_number_vec_multi_mod(fmpz * b, long n);
mp_limb_t bell_number_nmod(ulong n, nmod_t mod);
void bell_number_nmod_vec(mp_ptr b, long n, nmod_t mod);
void bell_number_nmod_vec_recursive(mp_ptr b, long n, nmod_t mod);
void bell_number_nmod_vec_series(mp_ptr b, long n, nmod_t mod);
/* Zeta Euler product ********************************************************/
void _zeta_inv_euler_product(mpfr_t res, ulong s, int char_4);
/* Euler numbers *************************************************************/
#if FLINT64
#define SMALL_EULER_LIMIT 25
#else
#define SMALL_EULER_LIMIT 15
#endif
static const mp_limb_t euler_number_small[] = {
1UL, 1UL, 5UL, 61UL, 1385UL, 50521UL, 2702765UL,
199360981UL,
#if FLINT64
19391512145UL, 2404879675441UL, 370371188237525UL,
69348874393137901UL, 15514534163557086905UL
#endif
};
double euler_number_size(ulong n);
void euler_number_vec(fmpz * res, long n);
void _euler_number_zeta(fmpz_t res, ulong n);
void euler_number(fmpz_t res, ulong n);
void 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 long _bernoulli_numer_small[] = {
1L, 1L, -1L, 1L, -1L, 5L, -691L, 7L, -3617L, 43867L, -174611L, 854513L,
-236364091L, 8553103L,
#if FLINT64
-23749461029L, 8615841276005L, -7709321041217L, 2577687858367L
#endif
};
void _bernoulli_number(fmpz_t num, fmpz_t den, ulong n);
void bernoulli_number(fmpq_t x, ulong n);
void _bernoulli_number_vec(fmpz * num, fmpz * den, long n);
void bernoulli_number_vec(fmpq * num, long n);
void bernoulli_number_denom(fmpz_t den, ulong n);
double bernoulli_number_size(ulong n);
void bernoulli_polynomial(fmpq_poly_t poly, ulong n);
void _bernoulli_number_zeta(fmpz_t num, fmpz_t den, ulong n);
void _bernoulli_number_vec_multi_mod(fmpz * num, fmpz * den, long n);
void _bernoulli_number_vec_recursive(fmpz * num, fmpz * den, long n);
void _bernoulli_number_vec_zeta(fmpz * num, fmpz * den, long n);
/* Cyclotomic polynomials ****************************************************/
void _cyclotomic_polynomial(fmpz * a, ulong n, mp_ptr factors,
long num_factors, ulong phi);
void cyclotomic_polynomial(fmpz_poly_t poly, ulong n);
void _cyclotomic_cos_polynomial(fmpz * coeffs, long d, ulong n);
void cyclotomic_cos_polynomial(fmpz_poly_t poly, ulong n);
/* Legendre polynomials ******************************************************/
void legendre_polynomial(fmpq_poly_t poly, ulong n);
/* Chebyshev polynomials *****************************************************/
void chebyshev_t_polynomial(fmpz_poly_t poly, ulong n);
void chebyshev_u_polynomial(fmpz_poly_t poly, ulong n);
/* Swinnerton-Dyer polynomials ***********************************************/
void swinnerton_dyer_polynomial(fmpz_poly_t poly, ulong n);
/* Landau function ***********************************************************/
void landau_function_vec(fmpz * res, long len);
/* Dedekind sums *************************************************************/
void dedekind_sum_naive(fmpq_t s, const fmpz_t h, const fmpz_t k);
double dedekind_sum_coprime_d(double h, double k);
void dedekind_sum_coprime_large(fmpq_t s, const fmpz_t h, const fmpz_t k);
void dedekind_sum_coprime(fmpq_t s, const fmpz_t h, const fmpz_t k);
void 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 dedekind_cosine_sum_factored(trig_prod_t prod, mp_limb_t k, mp_limb_t n);
/* Number of partitions ******************************************************/
void number_of_partitions_nmod_vec(mp_ptr res, long len, nmod_t mod);
void number_of_partitions_vec(fmpz * res, long len);
void number_of_partitions_mpfr(mpfr_t x, ulong n);
void number_of_partitions(fmpz_t x, ulong n);
void number_of_distinct_partitions_vec(fmpz * res, long len);
void number_of_distinct_partitions_nmod_vec(mp_ptr res, long len, nmod_t mod);
/* Number of sums of squares representations *********************************/
void sum_of_squares(fmpz_t r, ulong k, const fmpz_t n);
void sum_of_squares_vec(fmpz * r, ulong k, long n);
/* MPFR extras ***************************************************************/
void mpfr_pi_chudnovsky(mpfr_t res, mpfr_rnd_t rnd);
void mpfr_const_euler_brent_mcmillan(mpfr_t res, mpfr_rnd_t rnd);
void mpfr_zeta_ui_bsplit(mpfr_t x, ulong s, mpfr_rnd_t rnd);
#endif