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
|
/* mpfr_zeta_ui -- compute the Riemann Zeta function for integer argument.
Copyright 2005-2014 Free Software Foundation, Inc.
Contributed by the AriC and Caramel projects, INRIA.
This file is part of the GNU MPFR Library.
The GNU MPFR Library is free software; you can redistribute it and/or modify
it under the terms of the GNU Lesser General Public License as published by
the Free Software Foundation; either version 3 of the License, or (at your
option) any later version.
The GNU MPFR Library 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 Lesser General Public
License for more details.
You should have received a copy of the GNU Lesser General Public License
along with the GNU MPFR Library; see the file COPYING.LESSER. If not, see
http://www.gnu.org/licenses/ or write to the Free Software Foundation, Inc.,
51 Franklin St, Fifth Floor, Boston, MA 02110-1301, USA. */
#define MPFR_NEED_LONGLONG_H
#include "mpfr-impl.h"
int
mpfr_zeta_ui (mpfr_ptr z, unsigned long m, mpfr_rnd_t r)
{
MPFR_ZIV_DECL (loop);
if (m == 0)
{
mpfr_set_ui (z, 1, r);
mpfr_div_2ui (z, z, 1, r);
MPFR_CHANGE_SIGN (z);
MPFR_RET (0);
}
else if (m == 1)
{
MPFR_SET_INF (z);
MPFR_SET_POS (z);
MPFR_SET_DIVBY0 ();
return 0;
}
else /* m >= 2 */
{
mpfr_prec_t p = MPFR_PREC(z);
unsigned long n, k, err, kbits;
mpz_t d, t, s, q;
mpfr_t y;
int inex;
if (r == MPFR_RNDA)
r = MPFR_RNDU; /* since the result is always positive */
if (m >= p) /* 2^(-m) < ulp(1) = 2^(1-p). This means that
2^(-m) <= 1/2*ulp(1). We have 3^(-m)+4^(-m)+... < 2^(-m)
i.e. zeta(m) < 1+2*2^(-m) for m >= 3 */
{
if (m == 2) /* necessarily p=2 */
return mpfr_set_ui_2exp (z, 13, -3, r);
else if (r == MPFR_RNDZ || r == MPFR_RNDD || (r == MPFR_RNDN && m > p))
{
mpfr_set_ui (z, 1, r);
return -1;
}
else
{
mpfr_set_ui (z, 1, r);
mpfr_nextabove (z);
return 1;
}
}
/* now treat also the case where zeta(m) - (1+1/2^m) < 1/2*ulp(1),
and the result is either 1+2^(-m) or 1+2^(-m)+2^(1-p). */
mpfr_init2 (y, 31);
if (m >= p / 2) /* otherwise 4^(-m) > 2^(-p) */
{
/* the following is a lower bound for log(3)/log(2) */
mpfr_set_str_binary (y, "1.100101011100000000011010001110");
mpfr_mul_ui (y, y, m, MPFR_RNDZ); /* lower bound for log2(3^m) */
if (mpfr_cmp_ui (y, p + 2) >= 0)
{
mpfr_clear (y);
mpfr_set_ui (z, 1, MPFR_RNDZ);
mpfr_div_2ui (z, z, m, MPFR_RNDZ);
mpfr_add_ui (z, z, 1, MPFR_RNDZ);
if (r != MPFR_RNDU)
return -1;
mpfr_nextabove (z);
return 1;
}
}
mpz_init (s);
mpz_init (d);
mpz_init (t);
mpz_init (q);
p += MPFR_INT_CEIL_LOG2(p); /* account of the n term in the error */
p += MPFR_INT_CEIL_LOG2(p) + 15; /* initial value */
MPFR_ZIV_INIT (loop, p);
for(;;)
{
/* 0.39321985067869744 = log(2)/log(3+sqrt(8)) */
n = 1 + (unsigned long) (0.39321985067869744 * (double) p);
err = n + 4;
mpfr_set_prec (y, p);
/* computation of the d[k] */
mpz_set_ui (s, 0);
mpz_set_ui (t, 1);
mpz_mul_2exp (t, t, 2 * n - 1); /* t[n] */
mpz_set (d, t);
for (k = n; k > 0; k--)
{
count_leading_zeros (kbits, k);
kbits = GMP_NUMB_BITS - kbits;
/* if k^m is too large, use mpz_tdiv_q */
if (m * kbits > 2 * GMP_NUMB_BITS)
{
/* if we know in advance that k^m > d, then floor(d/k^m) will
be zero below, so there is no need to compute k^m */
kbits = (kbits - 1) * m + 1;
/* k^m has at least kbits bits */
if (kbits > mpz_sizeinbase (d, 2))
mpz_set_ui (q, 0);
else
{
mpz_ui_pow_ui (q, k, m);
mpz_tdiv_q (q, d, q);
}
}
else /* use several mpz_tdiv_q_ui calls */
{
unsigned long km = k, mm = m - 1;
while (mm > 0 && km < ULONG_MAX / k)
{
km *= k;
mm --;
}
mpz_tdiv_q_ui (q, d, km);
while (mm > 0)
{
km = k;
mm --;
while (mm > 0 && km < ULONG_MAX / k)
{
km *= k;
mm --;
}
mpz_tdiv_q_ui (q, q, km);
}
}
if (k % 2)
mpz_add (s, s, q);
else
mpz_sub (s, s, q);
/* we have d[k] = sum(t[i], i=k+1..n)
with t[i] = n*(n+i-1)!*4^i/(n-i)!/(2i)!
t[k-1]/t[k] = k*(2k-1)/(n-k+1)/(n+k-1)/2 */
#if (GMP_NUMB_BITS == 32)
#define KMAX 46341 /* max k such that k*(2k-1) < 2^32 */
#elif (GMP_NUMB_BITS == 64)
#define KMAX 3037000500
#endif
#ifdef KMAX
if (k <= KMAX)
mpz_mul_ui (t, t, k * (2 * k - 1));
else
#endif
{
mpz_mul_ui (t, t, k);
mpz_mul_ui (t, t, 2 * k - 1);
}
mpz_fdiv_q_2exp (t, t, 1);
/* Warning: the test below assumes that an unsigned long
has no padding bits. */
if (n < 1UL << ((sizeof(unsigned long) * CHAR_BIT) / 2))
/* (n - k + 1) * (n + k - 1) < n^2 */
mpz_divexact_ui (t, t, (n - k + 1) * (n + k - 1));
else
{
mpz_divexact_ui (t, t, n - k + 1);
mpz_divexact_ui (t, t, n + k - 1);
}
mpz_add (d, d, t);
}
/* multiply by 1/(1-2^(1-m)) = 1 + 2^(1-m) + 2^(2-m) + ... */
mpz_fdiv_q_2exp (t, s, m - 1);
do
{
err ++;
mpz_add (s, s, t);
mpz_fdiv_q_2exp (t, t, m - 1);
}
while (mpz_cmp_ui (t, 0) > 0);
/* divide by d[n] */
mpz_mul_2exp (s, s, p);
mpz_tdiv_q (s, s, d);
mpfr_set_z (y, s, MPFR_RNDN);
mpfr_div_2ui (y, y, p, MPFR_RNDN);
err = MPFR_INT_CEIL_LOG2 (err);
if (MPFR_LIKELY(MPFR_CAN_ROUND (y, p - err, MPFR_PREC(z), r)))
break;
MPFR_ZIV_NEXT (loop, p);
}
MPFR_ZIV_FREE (loop);
mpz_clear (d);
mpz_clear (t);
mpz_clear (q);
mpz_clear (s);
inex = mpfr_set (z, y, r);
mpfr_clear (y);
return inex;
}
}
|
