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