forked from flintlib/flint
-
Notifications
You must be signed in to change notification settings - Fork 0
/
arith.h
244 lines (174 loc) · 8.57 KB
/
arith.h
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
/*============================================================================
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"
#include "double_extras.h"
#ifdef __cplusplus
extern "C" {
#endif
/* MPFR extras ***************************************************************/
FLINT_DLL void mpfr_zeta_inv_euler_product(mpfr_t res, ulong s, int char_4);
/* Various arithmetic functions **********************************************/
#define arith_primorial fmpz_primorial
FLINT_DLL void _arith_harmonic_number(fmpz_t num, fmpz_t den, slong n);
FLINT_DLL void arith_harmonic_number(fmpq_t x, slong n);
FLINT_DLL void arith_ramanujan_tau(fmpz_t res, const fmpz_t n);
FLINT_DLL void arith_ramanujan_tau_series(fmpz_poly_t res, slong n);
FLINT_DLL void arith_divisors(fmpz_poly_t res, const fmpz_t n);
#define arith_divisor_sigma fmpz_divisor_sigma
#define arith_moebius_mu fmpz_moebius_mu
#define arith_euler_phi fmpz_euler_phi
/* Stirling numbers **********************************************************/
FLINT_DLL void arith_stirling_number_1u(fmpz_t s, slong n, slong k);
FLINT_DLL void arith_stirling_number_1(fmpz_t s, slong n, slong k);
FLINT_DLL void arith_stirling_number_2(fmpz_t s, slong n, slong k);
FLINT_DLL void arith_stirling_number_1u_vec(fmpz * row, slong n, slong klen);
FLINT_DLL void arith_stirling_number_1_vec(fmpz * row, slong n, slong klen);
FLINT_DLL void arith_stirling_number_2_vec(fmpz * row, slong n, slong klen);
FLINT_DLL void arith_stirling_number_1u_vec_next(fmpz * row,
const fmpz * prev, slong n, slong klen);
FLINT_DLL void arith_stirling_number_1_vec_next(fmpz * row,
const fmpz * prev, slong n, slong klen);
FLINT_DLL void arith_stirling_number_2_vec_next(fmpz * row,
const fmpz * prev, slong n, slong klen);
FLINT_DLL void arith_stirling_matrix_1u(fmpz_mat_t mat);
FLINT_DLL void arith_stirling_matrix_1(fmpz_mat_t mat);
FLINT_DLL 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[];
FLINT_DLL double arith_bell_number_size(ulong n);
FLINT_DLL void arith_bell_number(fmpz_t b, ulong n);
FLINT_DLL void arith_bell_number_bsplit(fmpz_t res, ulong n);
FLINT_DLL void arith_bell_number_multi_mod(fmpz_t res, ulong n);
FLINT_DLL void arith_bell_number_vec(fmpz * b, slong n);
FLINT_DLL void arith_bell_number_vec_recursive(fmpz * b, slong n);
FLINT_DLL void arith_bell_number_vec_multi_mod(fmpz * b, slong n);
FLINT_DLL mp_limb_t arith_bell_number_nmod(ulong n, nmod_t mod);
FLINT_DLL void arith_bell_number_nmod_vec(mp_ptr b, slong n, nmod_t mod);
FLINT_DLL void arith_bell_number_nmod_vec_recursive(mp_ptr b, slong n, nmod_t mod);
FLINT_DLL 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
};
FLINT_DLL double arith_euler_number_size(ulong n);
FLINT_DLL void arith_euler_number_vec(fmpz * res, slong n);
FLINT_DLL void _arith_euler_number_zeta(fmpz_t res, ulong n);
FLINT_DLL void arith_euler_number(fmpz_t res, ulong n);
FLINT_DLL 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
};
FLINT_DLL void _arith_bernoulli_number(fmpz_t num, fmpz_t den, ulong n);
FLINT_DLL void arith_bernoulli_number(fmpq_t x, ulong n);
FLINT_DLL void _arith_bernoulli_number_vec(fmpz * num, fmpz * den, slong n);
FLINT_DLL void arith_bernoulli_number_vec(fmpq * num, slong n);
FLINT_DLL void arith_bernoulli_number_denom(fmpz_t den, ulong n);
FLINT_DLL double arith_bernoulli_number_size(ulong n);
FLINT_DLL void arith_bernoulli_polynomial(fmpq_poly_t poly, ulong n);
FLINT_DLL void _arith_bernoulli_number_zeta(fmpz_t num, fmpz_t den, ulong n);
FLINT_DLL void _arith_bernoulli_number_vec_multi_mod(fmpz * num, fmpz * den, slong n);
FLINT_DLL void _arith_bernoulli_number_vec_recursive(fmpz * num, fmpz * den, slong n);
FLINT_DLL void _arith_bernoulli_number_vec_zeta(fmpz * num, fmpz * den, slong n);
/* Cyclotomic polynomials ****************************************************/
#define _arith_cyclotomic_polynomial _fmpz_poly_cyclotomic
#define arith_cyclotomic_polynomial fmpz_poly_cyclotomic
FLINT_DLL void _arith_cos_minpoly(fmpz * coeffs, slong d, ulong n);
FLINT_DLL void arith_cos_minpoly(fmpz_poly_t poly, ulong n);
/* Hypergeometric polynomials ************************************************/
FLINT_DLL void arith_legendre_polynomial(fmpq_poly_t poly, ulong n);
#define _arith_chebyshev_t_polynomial _fmpz_poly_chebyshev_t
#define arith_chebyshev_t_polynomial fmpz_poly_chebyshev_t
#define _arith_chebyshev_u_polynomial _fmpz_poly_chebyshev_u
#define arith_chebyshev_u_polynomial fmpz_poly_chebyshev_u
/* Swinnerton-Dyer polynomials ***********************************************/
#define _arith_swinnerton_dyer_polynomial _fmpz_poly_swinnerton_dyer
#define arith_swinnerton_dyer_polynomial fmpz_poly_swinnerton_dyer
/* Landau function ***********************************************************/
FLINT_DLL void arith_landau_function_vec(fmpz * res, slong len);
/* Dedekind sums *************************************************************/
#define arith_dedekind_sum_naive fmpq_dedekind_sum_naive
#define arith_dedekind_sum_coprime_d d_dedekind_sum_coprime
#define arith_dedekind_sum_coprime_large fmpq_dedekind_sum_coprime_large
#define arith_dedekind_sum_coprime fmpq_dedekind_sum_coprime
#define arith_dedekind_sum fmpq_dedekind_sum
/* 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;
}
FLINT_DLL void arith_hrr_expsum_factored(trig_prod_t prod, mp_limb_t k, mp_limb_t n);
/* Number of partitions ******************************************************/
FLINT_DLL void arith_number_of_partitions_nmod_vec(mp_ptr res, slong len, nmod_t mod);
FLINT_DLL void arith_number_of_partitions_vec(fmpz * res, slong len);
FLINT_DLL void arith_number_of_partitions_mpfr(mpfr_t x, ulong n);
FLINT_DLL void arith_number_of_partitions(fmpz_t x, ulong n);
/* Number of sums of squares representations *********************************/
FLINT_DLL void arith_sum_of_squares(fmpz_t r, ulong k, const fmpz_t n);
FLINT_DLL void arith_sum_of_squares_vec(fmpz * r, ulong k, slong n);
#ifdef __cplusplus
}
#endif
#endif