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Diffstat (limited to 'src/zeta_ui.c')
| -rw-r--r-- | src/zeta_ui.c | 229 |
1 files changed, 229 insertions, 0 deletions
diff --git a/src/zeta_ui.c b/src/zeta_ui.c new file mode 100644 index 000000000..eea589380 --- /dev/null +++ b/src/zeta_ui.c @@ -0,0 +1,229 @@ +/* mpfr_zeta_ui -- compute the Riemann Zeta function for integer argument. + +Copyright 2005, 2006, 2007, 2008, 2009, 2010 Free Software Foundation, Inc. +Contributed by the Arenaire 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); + 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; + } +} |