1/* mpfr_zeta_ui -- compute the Riemann Zeta function for integer argument.
2
3Copyright 2005, 2006, 2007, 2008, 2009, 2010, 2011 Free Software Foundation, Inc.
4Contributed by the Arenaire and Cacao projects, INRIA.
5
6This file is part of the GNU MPFR Library.
7
8The GNU MPFR Library is free software; you can redistribute it and/or modify
9it under the terms of the GNU Lesser General Public License as published by
10the Free Software Foundation; either version 3 of the License, or (at your
11option) any later version.
12
13The GNU MPFR Library is distributed in the hope that it will be useful, but
14WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
15or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU Lesser General Public
16License for more details.
17
18You should have received a copy of the GNU Lesser General Public License
19along with the GNU MPFR Library; see the file COPYING.LESSER.  If not, see
20http://www.gnu.org/licenses/ or write to the Free Software Foundation, Inc.,
2151 Franklin St, Fifth Floor, Boston, MA 02110-1301, USA. */
22
23#define MPFR_NEED_LONGLONG_H
24#include "mpfr-impl.h"
25
26int
27mpfr_zeta_ui (mpfr_ptr z, unsigned long m, mpfr_rnd_t r)
28{
29  MPFR_ZIV_DECL (loop);
30
31  if (m == 0)
32    {
33      mpfr_set_ui (z, 1, r);
34      mpfr_div_2ui (z, z, 1, r);
35      MPFR_CHANGE_SIGN (z);
36      MPFR_RET (0);
37    }
38  else if (m == 1)
39    {
40      MPFR_SET_INF (z);
41      MPFR_SET_POS (z);
42      return 0;
43    }
44  else /* m >= 2 */
45    {
46      mpfr_prec_t p = MPFR_PREC(z);
47      unsigned long n, k, err, kbits;
48      mpz_t d, t, s, q;
49      mpfr_t y;
50      int inex;
51
52      if (r == MPFR_RNDA)
53        r = MPFR_RNDU; /* since the result is always positive */
54
55      if (m >= p) /* 2^(-m) < ulp(1) = 2^(1-p). This means that
56                     2^(-m) <= 1/2*ulp(1). We have 3^(-m)+4^(-m)+... < 2^(-m)
57                     i.e. zeta(m) < 1+2*2^(-m) for m >= 3 */
58
59        {
60          if (m == 2) /* necessarily p=2 */
61            return mpfr_set_ui_2exp (z, 13, -3, r);
62          else if (r == MPFR_RNDZ || r == MPFR_RNDD || (r == MPFR_RNDN && m > p))
63            {
64              mpfr_set_ui (z, 1, r);
65              return -1;
66            }
67          else
68            {
69              mpfr_set_ui (z, 1, r);
70              mpfr_nextabove (z);
71              return 1;
72            }
73        }
74
75      /* now treat also the case where zeta(m) - (1+1/2^m) < 1/2*ulp(1),
76         and the result is either 1+2^(-m) or 1+2^(-m)+2^(1-p). */
77      mpfr_init2 (y, 31);
78
79      if (m >= p / 2) /* otherwise 4^(-m) > 2^(-p) */
80        {
81          /* the following is a lower bound for log(3)/log(2) */
82          mpfr_set_str_binary (y, "1.100101011100000000011010001110");
83          mpfr_mul_ui (y, y, m, MPFR_RNDZ); /* lower bound for log2(3^m) */
84          if (mpfr_cmp_ui (y, p + 2) >= 0)
85            {
86              mpfr_clear (y);
87              mpfr_set_ui (z, 1, MPFR_RNDZ);
88              mpfr_div_2ui (z, z, m, MPFR_RNDZ);
89              mpfr_add_ui (z, z, 1, MPFR_RNDZ);
90              if (r != MPFR_RNDU)
91                return -1;
92              mpfr_nextabove (z);
93              return 1;
94            }
95        }
96
97      mpz_init (s);
98      mpz_init (d);
99      mpz_init (t);
100      mpz_init (q);
101
102      p += MPFR_INT_CEIL_LOG2(p); /* account of the n term in the error */
103
104      p += MPFR_INT_CEIL_LOG2(p) + 15; /* initial value */
105
106      MPFR_ZIV_INIT (loop, p);
107      for(;;)
108        {
109          /* 0.39321985067869744 = log(2)/log(3+sqrt(8)) */
110          n = 1 + (unsigned long) (0.39321985067869744 * (double) p);
111          err = n + 4;
112
113          mpfr_set_prec (y, p);
114
115          /* computation of the d[k] */
116          mpz_set_ui (s, 0);
117          mpz_set_ui (t, 1);
118          mpz_mul_2exp (t, t, 2 * n - 1); /* t[n] */
119          mpz_set (d, t);
120          for (k = n; k > 0; k--)
121            {
122              count_leading_zeros (kbits, k);
123              kbits = GMP_NUMB_BITS - kbits;
124              /* if k^m is too large, use mpz_tdiv_q */
125              if (m * kbits > 2 * GMP_NUMB_BITS)
126                {
127                  /* if we know in advance that k^m > d, then floor(d/k^m) will
128                     be zero below, so there is no need to compute k^m */
129                  kbits = (kbits - 1) * m + 1;
130                  /* k^m has at least kbits bits */
131                  if (kbits > mpz_sizeinbase (d, 2))
132                    mpz_set_ui (q, 0);
133                  else
134                    {
135                      mpz_ui_pow_ui (q, k, m);
136                      mpz_tdiv_q (q, d, q);
137                    }
138                }
139              else /* use several mpz_tdiv_q_ui calls */
140                {
141                  unsigned long km = k, mm = m - 1;
142                  while (mm > 0 && km < ULONG_MAX / k)
143                    {
144                      km *= k;
145                      mm --;
146                    }
147                  mpz_tdiv_q_ui (q, d, km);
148                  while (mm > 0)
149                    {
150                      km = k;
151                      mm --;
152                      while (mm > 0 && km < ULONG_MAX / k)
153                        {
154                          km *= k;
155                          mm --;
156                        }
157                      mpz_tdiv_q_ui (q, q, km);
158                    }
159                }
160              if (k % 2)
161                mpz_add (s, s, q);
162              else
163                mpz_sub (s, s, q);
164
165              /* we have d[k] = sum(t[i], i=k+1..n)
166                 with t[i] = n*(n+i-1)!*4^i/(n-i)!/(2i)!
167                 t[k-1]/t[k] = k*(2k-1)/(n-k+1)/(n+k-1)/2 */
168#if (GMP_NUMB_BITS == 32)
169#define KMAX 46341 /* max k such that k*(2k-1) < 2^32 */
170#elif (GMP_NUMB_BITS == 64)
171#define KMAX 3037000500
172#endif
173#ifdef KMAX
174              if (k <= KMAX)
175                mpz_mul_ui (t, t, k * (2 * k - 1));
176              else
177#endif
178                {
179                  mpz_mul_ui (t, t, k);
180                  mpz_mul_ui (t, t, 2 * k - 1);
181                }
182              mpz_fdiv_q_2exp (t, t, 1);
183              /* Warning: the test below assumes that an unsigned long
184                 has no padding bits. */
185              if (n < 1UL << ((sizeof(unsigned long) * CHAR_BIT) / 2))
186                /* (n - k + 1) * (n + k - 1) < n^2 */
187                mpz_divexact_ui (t, t, (n - k + 1) * (n + k - 1));
188              else
189                {
190                  mpz_divexact_ui (t, t, n - k + 1);
191                  mpz_divexact_ui (t, t, n + k - 1);
192                }
193              mpz_add (d, d, t);
194            }
195
196          /* multiply by 1/(1-2^(1-m)) = 1 + 2^(1-m) + 2^(2-m) + ... */
197          mpz_fdiv_q_2exp (t, s, m - 1);
198          do
199            {
200              err ++;
201              mpz_add (s, s, t);
202              mpz_fdiv_q_2exp (t, t, m - 1);
203            }
204          while (mpz_cmp_ui (t, 0) > 0);
205
206          /* divide by d[n] */
207          mpz_mul_2exp (s, s, p);
208          mpz_tdiv_q (s, s, d);
209          mpfr_set_z (y, s, MPFR_RNDN);
210          mpfr_div_2ui (y, y, p, MPFR_RNDN);
211
212          err = MPFR_INT_CEIL_LOG2 (err);
213
214          if (MPFR_LIKELY(MPFR_CAN_ROUND (y, p - err, MPFR_PREC(z), r)))
215            break;
216
217          MPFR_ZIV_NEXT (loop, p);
218        }
219      MPFR_ZIV_FREE (loop);
220
221      mpz_clear (d);
222      mpz_clear (t);
223      mpz_clear (q);
224      mpz_clear (s);
225      inex = mpfr_set (z, y, r);
226      mpfr_clear (y);
227      return inex;
228    }
229}
230