| /* mpfr_zeta_ui -- compute the Riemann Zeta function for integer argument. |
| |
| Copyright 2005, 2006, 2007, 2008, 2009 Free Software Foundation, Inc. |
| Contributed by the Arenaire and Cacao 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 2.1 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.LIB. If not, 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, mp_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 */ |
| { |
| mp_prec_t p = MPFR_PREC(z); |
| unsigned long n, k, err, kbits; |
| mpz_t d, t, s, q; |
| mpfr_t y; |
| int inex; |
| |
| 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 == GMP_RNDZ || r == GMP_RNDD || (r == GMP_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, GMP_RNDZ); /* lower bound for log2(3^m) */ |
| if (mpfr_cmp_ui (y, p + 2) >= 0) |
| { |
| mpfr_clear (y); |
| mpfr_set_ui (z, 1, GMP_RNDZ); |
| mpfr_div_2ui (z, z, m, GMP_RNDZ); |
| mpfr_add_ui (z, z, 1, GMP_RNDZ); |
| if (r != GMP_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 = BITS_PER_MP_LIMB - kbits; |
| /* if k^m is too large, use mpz_tdiv_q */ |
| if (m * kbits > 2 * BITS_PER_MP_LIMB) |
| { |
| /* 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 (BITS_PER_MP_LIMB == 32) |
| #define KMAX 46341 /* max k such that k*(2k-1) < 2^32 */ |
| #elif (BITS_PER_MP_LIMB == 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_div_2exp (t, t, 1); |
| if (n < 1UL << (BITS_PER_MP_LIMB / 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_div_2exp (t, s, m - 1); |
| do |
| { |
| err ++; |
| mpz_add (s, s, t); |
| mpz_div_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, GMP_RNDN); |
| mpfr_div_2ui (y, y, p, GMP_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; |
| } |
| } |
