/*============================================================================ 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 #include #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