1 /*
2 Copyright (C) 2012 Fredrik Johansson
3
4 This file is part of Arb.
5
6 Arb is free software: you can redistribute it and/or modify it under
7 the terms of the GNU Lesser General Public License (LGPL) as published
8 by the Free Software Foundation; either version 2.1 of the License, or
9 (at your option) any later version. See <http://www.gnu.org/licenses/>.
10 */
11
12 #include "arb.h"
13
14 /* With parameter n, the error is bounded by 3/(3+sqrt(8))^n */
15 #define ERROR_A 1.5849625007211561815 /* log2(3) */
16 #define ERROR_B 2.5431066063272239453 /* log2(3+sqrt(8)) */
17
18 void mag_borwein_error(mag_t err, slong n);
19
20 void
arb_zeta_ui_vec_borwein(arb_ptr z,ulong start,slong num,ulong step,slong prec)21 arb_zeta_ui_vec_borwein(arb_ptr z, ulong start, slong num, ulong step, slong prec)
22 {
23 slong j, k, s, n, wp;
24 fmpz_t c, d, t, u;
25 fmpz * zeta;
26 mag_t err;
27
28 if (num < 1)
29 return;
30
31 wp = prec + FLINT_BIT_COUNT(prec);
32 n = wp / 2.5431066063272239453 + 1;
33
34 fmpz_init(c);
35 fmpz_init(d);
36 fmpz_init(t);
37 fmpz_init(u);
38 zeta = _fmpz_vec_init(num);
39
40 fmpz_set_ui(c, 1);
41 fmpz_mul_2exp(c, c, 2 * n - 1);
42 fmpz_set(d, c);
43
44 for (k = n; k > 0; k--)
45 {
46 /* divide by first k^s */
47 fmpz_ui_pow_ui(u, k, start);
48 fmpz_tdiv_q(t, d, u);
49 if (k % 2 == 0)
50 fmpz_neg(t, t);
51 fmpz_add(zeta, zeta, t);
52
53 /* remaining k^s */
54 fmpz_ui_pow_ui(u, k, step);
55 for (j = 1; j < num; j++)
56 {
57 fmpz_tdiv_q(t, t, u);
58 fmpz_add(zeta + j, zeta + j, t);
59 }
60
61 /* hypergeometric recurrence */
62 fmpz_mul2_uiui(c, c, k, 2 * k - 1);
63 fmpz_divexact2_uiui(c, c, 2 * (n - k + 1), n + k - 1);
64 fmpz_add(d, d, c);
65 }
66
67 mag_init(err);
68 mag_borwein_error(err, n);
69
70 for (k = 0; k < num; k++)
71 {
72 arb_ptr x = z + k;
73 s = start + step * k;
74
75 arb_set_fmpz(x, zeta + k);
76 /* the error in each term in the main loop is < 2 */
77 mag_set_ui(arb_radref(x), 2 * n);
78 arb_div_fmpz(x, x, d, wp);
79
80 /* mathematical error for eta(s), bounded by 3/(3+sqrt(8))^n */
81 mag_add(arb_radref(x), arb_radref(x), err);
82
83 /* convert from eta(s) to zeta(s) */
84 arb_div_2expm1_ui(x, x, s - 1, wp);
85 arb_mul_2exp_si(x, x, s - 1);
86 }
87
88 mag_clear(err);
89
90 fmpz_clear(c);
91 fmpz_clear(d);
92 fmpz_clear(t);
93 fmpz_clear(u);
94 _fmpz_vec_clear(zeta, num);
95 }
96