1/* mpfr_zeta_ui -- compute the Riemann Zeta function for integer argument. 2 3Copyright 2005, 2006, 2007, 2008, 2009, 2010, 2011, 2012, 2013 Free Software Foundation, Inc. 4Contributed by the AriC and Caramel 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 mpfr_set_divby0 (); 43 return 0; 44 } 45 else /* m >= 2 */ 46 { 47 mpfr_prec_t p = MPFR_PREC(z); 48 unsigned long n, k, err, kbits; 49 mpz_t d, t, s, q; 50 mpfr_t y; 51 int inex; 52 53 if (r == MPFR_RNDA) 54 r = MPFR_RNDU; /* since the result is always positive */ 55 56 if (m >= p) /* 2^(-m) < ulp(1) = 2^(1-p). This means that 57 2^(-m) <= 1/2*ulp(1). We have 3^(-m)+4^(-m)+... < 2^(-m) 58 i.e. zeta(m) < 1+2*2^(-m) for m >= 3 */ 59 60 { 61 if (m == 2) /* necessarily p=2 */ 62 return mpfr_set_ui_2exp (z, 13, -3, r); 63 else if (r == MPFR_RNDZ || r == MPFR_RNDD || (r == MPFR_RNDN && m > p)) 64 { 65 mpfr_set_ui (z, 1, r); 66 return -1; 67 } 68 else 69 { 70 mpfr_set_ui (z, 1, r); 71 mpfr_nextabove (z); 72 return 1; 73 } 74 } 75 76 /* now treat also the case where zeta(m) - (1+1/2^m) < 1/2*ulp(1), 77 and the result is either 1+2^(-m) or 1+2^(-m)+2^(1-p). */ 78 mpfr_init2 (y, 31); 79 80 if (m >= p / 2) /* otherwise 4^(-m) > 2^(-p) */ 81 { 82 /* the following is a lower bound for log(3)/log(2) */ 83 mpfr_set_str_binary (y, "1.100101011100000000011010001110"); 84 mpfr_mul_ui (y, y, m, MPFR_RNDZ); /* lower bound for log2(3^m) */ 85 if (mpfr_cmp_ui (y, p + 2) >= 0) 86 { 87 mpfr_clear (y); 88 mpfr_set_ui (z, 1, MPFR_RNDZ); 89 mpfr_div_2ui (z, z, m, MPFR_RNDZ); 90 mpfr_add_ui (z, z, 1, MPFR_RNDZ); 91 if (r != MPFR_RNDU) 92 return -1; 93 mpfr_nextabove (z); 94 return 1; 95 } 96 } 97 98 mpz_init (s); 99 mpz_init (d); 100 mpz_init (t); 101 mpz_init (q); 102 103 p += MPFR_INT_CEIL_LOG2(p); /* account of the n term in the error */ 104 105 p += MPFR_INT_CEIL_LOG2(p) + 15; /* initial value */ 106 107 MPFR_ZIV_INIT (loop, p); 108 for(;;) 109 { 110 /* 0.39321985067869744 = log(2)/log(3+sqrt(8)) */ 111 n = 1 + (unsigned long) (0.39321985067869744 * (double) p); 112 err = n + 4; 113 114 mpfr_set_prec (y, p); 115 116 /* computation of the d[k] */ 117 mpz_set_ui (s, 0); 118 mpz_set_ui (t, 1); 119 mpz_mul_2exp (t, t, 2 * n - 1); /* t[n] */ 120 mpz_set (d, t); 121 for (k = n; k > 0; k--) 122 { 123 count_leading_zeros (kbits, k); 124 kbits = GMP_NUMB_BITS - kbits; 125 /* if k^m is too large, use mpz_tdiv_q */ 126 if (m * kbits > 2 * GMP_NUMB_BITS) 127 { 128 /* if we know in advance that k^m > d, then floor(d/k^m) will 129 be zero below, so there is no need to compute k^m */ 130 kbits = (kbits - 1) * m + 1; 131 /* k^m has at least kbits bits */ 132 if (kbits > mpz_sizeinbase (d, 2)) 133 mpz_set_ui (q, 0); 134 else 135 { 136 mpz_ui_pow_ui (q, k, m); 137 mpz_tdiv_q (q, d, q); 138 } 139 } 140 else /* use several mpz_tdiv_q_ui calls */ 141 { 142 unsigned long km = k, mm = m - 1; 143 while (mm > 0 && km < ULONG_MAX / k) 144 { 145 km *= k; 146 mm --; 147 } 148 mpz_tdiv_q_ui (q, d, km); 149 while (mm > 0) 150 { 151 km = k; 152 mm --; 153 while (mm > 0 && km < ULONG_MAX / k) 154 { 155 km *= k; 156 mm --; 157 } 158 mpz_tdiv_q_ui (q, q, km); 159 } 160 } 161 if (k % 2) 162 mpz_add (s, s, q); 163 else 164 mpz_sub (s, s, q); 165 166 /* we have d[k] = sum(t[i], i=k+1..n) 167 with t[i] = n*(n+i-1)!*4^i/(n-i)!/(2i)! 168 t[k-1]/t[k] = k*(2k-1)/(n-k+1)/(n+k-1)/2 */ 169#if (GMP_NUMB_BITS == 32) 170#define KMAX 46341 /* max k such that k*(2k-1) < 2^32 */ 171#elif (GMP_NUMB_BITS == 64) 172#define KMAX 3037000500 173#endif 174#ifdef KMAX 175 if (k <= KMAX) 176 mpz_mul_ui (t, t, k * (2 * k - 1)); 177 else 178#endif 179 { 180 mpz_mul_ui (t, t, k); 181 mpz_mul_ui (t, t, 2 * k - 1); 182 } 183 mpz_fdiv_q_2exp (t, t, 1); 184 /* Warning: the test below assumes that an unsigned long 185 has no padding bits. */ 186 if (n < 1UL << ((sizeof(unsigned long) * CHAR_BIT) / 2)) 187 /* (n - k + 1) * (n + k - 1) < n^2 */ 188 mpz_divexact_ui (t, t, (n - k + 1) * (n + k - 1)); 189 else 190 { 191 mpz_divexact_ui (t, t, n - k + 1); 192 mpz_divexact_ui (t, t, n + k - 1); 193 } 194 mpz_add (d, d, t); 195 } 196 197 /* multiply by 1/(1-2^(1-m)) = 1 + 2^(1-m) + 2^(2-m) + ... */ 198 mpz_fdiv_q_2exp (t, s, m - 1); 199 do 200 { 201 err ++; 202 mpz_add (s, s, t); 203 mpz_fdiv_q_2exp (t, t, m - 1); 204 } 205 while (mpz_cmp_ui (t, 0) > 0); 206 207 /* divide by d[n] */ 208 mpz_mul_2exp (s, s, p); 209 mpz_tdiv_q (s, s, d); 210 mpfr_set_z (y, s, MPFR_RNDN); 211 mpfr_div_2ui (y, y, p, MPFR_RNDN); 212 213 err = MPFR_INT_CEIL_LOG2 (err); 214 215 if (MPFR_LIKELY(MPFR_CAN_ROUND (y, p - err, MPFR_PREC(z), r))) 216 break; 217 218 MPFR_ZIV_NEXT (loop, p); 219 } 220 MPFR_ZIV_FREE (loop); 221 222 mpz_clear (d); 223 mpz_clear (t); 224 mpz_clear (q); 225 mpz_clear (s); 226 inex = mpfr_set (z, y, r); 227 mpfr_clear (y); 228 return inex; 229 } 230} 231