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Diffstat (limited to 'Build/source/libs/mpfr/mpfr-3.1.2/src/zeta.c')
-rw-r--r-- | Build/source/libs/mpfr/mpfr-3.1.2/src/zeta.c | 466 |
1 files changed, 466 insertions, 0 deletions
diff --git a/Build/source/libs/mpfr/mpfr-3.1.2/src/zeta.c b/Build/source/libs/mpfr/mpfr-3.1.2/src/zeta.c new file mode 100644 index 00000000000..525e7af049f --- /dev/null +++ b/Build/source/libs/mpfr/mpfr-3.1.2/src/zeta.c @@ -0,0 +1,466 @@ +/* mpfr_zeta -- compute the Riemann Zeta function + +Copyright 2003, 2004, 2005, 2006, 2007, 2008, 2009, 2010, 2011, 2012, 2013 Free Software Foundation, Inc. +Contributed by the AriC 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" + +/* + Parameters: + s - the input floating-point number + n, p - parameters from the algorithm + tc - an array of p floating-point numbers tc[1]..tc[p] + Output: + b is the result, i.e. + sum(tc[i]*product((s+2j)*(s+2j-1)/n^2,j=1..i-1), i=1..p)*s*n^(-s-1) +*/ +static void +mpfr_zeta_part_b (mpfr_t b, mpfr_srcptr s, int n, int p, mpfr_t *tc) +{ + mpfr_t s1, d, u; + unsigned long n2; + int l, t; + MPFR_GROUP_DECL (group); + + if (p == 0) + { + MPFR_SET_ZERO (b); + MPFR_SET_POS (b); + return; + } + + n2 = n * n; + MPFR_GROUP_INIT_3 (group, MPFR_PREC (b), s1, d, u); + + /* t equals 2p-2, 2p-3, ... ; s1 equals s+t */ + t = 2 * p - 2; + mpfr_set (d, tc[p], MPFR_RNDN); + for (l = 1; l < p; l++) + { + mpfr_add_ui (s1, s, t, MPFR_RNDN); /* s + (2p-2l) */ + mpfr_mul (d, d, s1, MPFR_RNDN); + t = t - 1; + mpfr_add_ui (s1, s, t, MPFR_RNDN); /* s + (2p-2l-1) */ + mpfr_mul (d, d, s1, MPFR_RNDN); + t = t - 1; + mpfr_div_ui (d, d, n2, MPFR_RNDN); + mpfr_add (d, d, tc[p-l], MPFR_RNDN); + /* since s is positive and the tc[i] have alternate signs, + the following is unlikely */ + if (MPFR_UNLIKELY (mpfr_cmpabs (d, tc[p-l]) > 0)) + mpfr_set (d, tc[p-l], MPFR_RNDN); + } + mpfr_mul (d, d, s, MPFR_RNDN); + mpfr_add (s1, s, __gmpfr_one, MPFR_RNDN); + mpfr_neg (s1, s1, MPFR_RNDN); + mpfr_ui_pow (u, n, s1, MPFR_RNDN); + mpfr_mul (b, d, u, MPFR_RNDN); + + MPFR_GROUP_CLEAR (group); +} + +/* Input: p - an integer + Output: fills tc[1..p], tc[i] = bernoulli(2i)/(2i)! + tc[1]=1/12, tc[2]=-1/720, tc[3]=1/30240, ... +*/ +static void +mpfr_zeta_c (int p, mpfr_t *tc) +{ + mpfr_t d; + int k, l; + + if (p > 0) + { + mpfr_init2 (d, MPFR_PREC (tc[1])); + mpfr_div_ui (tc[1], __gmpfr_one, 12, MPFR_RNDN); + for (k = 2; k <= p; k++) + { + mpfr_set_ui (d, k-1, MPFR_RNDN); + mpfr_div_ui (d, d, 12*k+6, MPFR_RNDN); + for (l=2; l < k; l++) + { + mpfr_div_ui (d, d, 4*(2*k-2*l+3)*(2*k-2*l+2), MPFR_RNDN); + mpfr_add (d, d, tc[l], MPFR_RNDN); + } + mpfr_div_ui (tc[k], d, 24, MPFR_RNDN); + MPFR_CHANGE_SIGN (tc[k]); + } + mpfr_clear (d); + } +} + +/* Input: s - a floating-point number + n - an integer + Output: sum - a floating-point number approximating sum(1/i^s, i=1..n-1) */ +static void +mpfr_zeta_part_a (mpfr_t sum, mpfr_srcptr s, int n) +{ + mpfr_t u, s1; + int i; + MPFR_GROUP_DECL (group); + + MPFR_GROUP_INIT_2 (group, MPFR_PREC (sum), u, s1); + + mpfr_neg (s1, s, MPFR_RNDN); + mpfr_ui_pow (u, n, s1, MPFR_RNDN); + mpfr_div_2ui (u, u, 1, MPFR_RNDN); + mpfr_set (sum, u, MPFR_RNDN); + for (i=n-1; i>1; i--) + { + mpfr_ui_pow (u, i, s1, MPFR_RNDN); + mpfr_add (sum, sum, u, MPFR_RNDN); + } + mpfr_add (sum, sum, __gmpfr_one, MPFR_RNDN); + + MPFR_GROUP_CLEAR (group); +} + +/* Input: s - a floating-point number >= 1/2. + rnd_mode - a rounding mode. + Assumes s is neither NaN nor Infinite. + Output: z - Zeta(s) rounded to the precision of z with direction rnd_mode +*/ +static int +mpfr_zeta_pos (mpfr_t z, mpfr_srcptr s, mpfr_rnd_t rnd_mode) +{ + mpfr_t b, c, z_pre, f, s1; + double beta, sd, dnep; + mpfr_t *tc1; + mpfr_prec_t precz, precs, d, dint; + int p, n, l, add; + int inex; + MPFR_GROUP_DECL (group); + MPFR_ZIV_DECL (loop); + + MPFR_ASSERTD (MPFR_IS_POS (s) && MPFR_GET_EXP (s) >= 0); + + precz = MPFR_PREC (z); + precs = MPFR_PREC (s); + + /* Zeta(x) = 1+1/2^x+1/3^x+1/4^x+1/5^x+O(1/6^x) + so with 2^(EXP(x)-1) <= x < 2^EXP(x) + So for x > 2^3, k^x > k^8, so 2/k^x < 2/k^8 + Zeta(x) = 1 + 1/2^x*(1+(2/3)^x+(2/4)^x+...) + = 1 + 1/2^x*(1+sum((2/k)^x,k=3..infinity)) + <= 1 + 1/2^x*(1+sum((2/k)^8,k=3..infinity)) + And sum((2/k)^8,k=3..infinity) = -257+128*Pi^8/4725 ~= 0.0438035 + So Zeta(x) <= 1 + 1/2^x*2 for x >= 8 + The error is < 2^(-x+1) <= 2^(-2^(EXP(x)-1)+1) */ + if (MPFR_GET_EXP (s) > 3) + { + mpfr_exp_t err; + err = MPFR_GET_EXP (s) - 1; + if (err > (mpfr_exp_t) (sizeof (mpfr_exp_t)*CHAR_BIT-2)) + err = MPFR_EMAX_MAX; + else + err = ((mpfr_exp_t)1) << err; + err = 1 - (-err+1); /* GET_EXP(one) - (-err+1) = err :) */ + MPFR_FAST_COMPUTE_IF_SMALL_INPUT (z, __gmpfr_one, err, 0, 1, + rnd_mode, {}); + } + + d = precz + MPFR_INT_CEIL_LOG2(precz) + 10; + + /* we want that s1 = s-1 is exact, i.e. we should have PREC(s1) >= EXP(s) */ + dint = (mpfr_uexp_t) MPFR_GET_EXP (s); + mpfr_init2 (s1, MAX (precs, dint)); + inex = mpfr_sub (s1, s, __gmpfr_one, MPFR_RNDN); + MPFR_ASSERTD (inex == 0); + + /* case s=1 should have already been handled */ + MPFR_ASSERTD (!MPFR_IS_ZERO (s1)); + + MPFR_GROUP_INIT_4 (group, MPFR_PREC_MIN, b, c, z_pre, f); + + MPFR_ZIV_INIT (loop, d); + for (;;) + { + /* Principal loop: we compute, in z_pre, + an approximation of Zeta(s), that we send to can_round */ + if (MPFR_GET_EXP (s1) <= -(mpfr_exp_t) ((mpfr_prec_t) (d-3)/2)) + /* Branch 1: when s-1 is very small, one + uses the approximation Zeta(s)=1/(s-1)+gamma, + where gamma is Euler's constant */ + { + dint = MAX (d + 3, precs); + MPFR_TRACE (printf ("branch 1\ninternal precision=%lu\n", + (unsigned long) dint)); + MPFR_GROUP_REPREC_4 (group, dint, b, c, z_pre, f); + mpfr_div (z_pre, __gmpfr_one, s1, MPFR_RNDN); + mpfr_const_euler (f, MPFR_RNDN); + mpfr_add (z_pre, z_pre, f, MPFR_RNDN); + } + else /* Branch 2 */ + { + size_t size; + + MPFR_TRACE (printf ("branch 2\n")); + /* Computation of parameters n, p and working precision */ + dnep = (double) d * LOG2; + sd = mpfr_get_d (s, MPFR_RNDN); + /* beta = dnep + 0.61 + sd * log (6.2832 / sd); + but a larger value is ok */ +#define LOG6dot2832 1.83787940484160805532 + beta = dnep + 0.61 + sd * (LOG6dot2832 - LOG2 * + __gmpfr_floor_log2 (sd)); + if (beta <= 0.0) + { + p = 0; + /* n = 1 + (int) (exp ((dnep - LOG2) / sd)); */ + n = 1 + (int) __gmpfr_ceil_exp2 ((d - 1.0) / sd); + } + else + { + p = 1 + (int) beta / 2; + n = 1 + (int) ((sd + 2.0 * (double) p - 1.0) / 6.2832); + } + MPFR_TRACE (printf ("\nn=%d\np=%d\n",n,p)); + /* add = 4 + floor(1.5 * log(d) / log (2)). + We should have add >= 10, which is always fulfilled since + d = precz + 11 >= 12, thus ceil(log2(d)) >= 4 */ + add = 4 + (3 * MPFR_INT_CEIL_LOG2 (d)) / 2; + MPFR_ASSERTD(add >= 10); + dint = d + add; + if (dint < precs) + dint = precs; + + MPFR_TRACE (printf ("internal precision=%lu\n", + (unsigned long) dint)); + + size = (p + 1) * sizeof(mpfr_t); + tc1 = (mpfr_t*) (*__gmp_allocate_func) (size); + for (l=1; l<=p; l++) + mpfr_init2 (tc1[l], dint); + MPFR_GROUP_REPREC_4 (group, dint, b, c, z_pre, f); + + MPFR_TRACE (printf ("precision of z = %lu\n", + (unsigned long) precz)); + + /* Computation of the coefficients c_k */ + mpfr_zeta_c (p, tc1); + /* Computation of the 3 parts of the fonction Zeta. */ + mpfr_zeta_part_a (z_pre, s, n); + mpfr_zeta_part_b (b, s, n, p, tc1); + /* s1 = s-1 is already computed above */ + mpfr_div (c, __gmpfr_one, s1, MPFR_RNDN); + mpfr_ui_pow (f, n, s1, MPFR_RNDN); + mpfr_div (c, c, f, MPFR_RNDN); + MPFR_TRACE (MPFR_DUMP (c)); + mpfr_add (z_pre, z_pre, c, MPFR_RNDN); + mpfr_add (z_pre, z_pre, b, MPFR_RNDN); + for (l=1; l<=p; l++) + mpfr_clear (tc1[l]); + (*__gmp_free_func) (tc1, size); + /* End branch 2 */ + } + + MPFR_TRACE (MPFR_DUMP (z_pre)); + if (MPFR_LIKELY (MPFR_CAN_ROUND (z_pre, d-3, precz, rnd_mode))) + break; + MPFR_ZIV_NEXT (loop, d); + } + MPFR_ZIV_FREE (loop); + + inex = mpfr_set (z, z_pre, rnd_mode); + + MPFR_GROUP_CLEAR (group); + mpfr_clear (s1); + + return inex; +} + +int +mpfr_zeta (mpfr_t z, mpfr_srcptr s, mpfr_rnd_t rnd_mode) +{ + mpfr_t z_pre, s1, y, p; + double sd, eps, m1, c; + long add; + mpfr_prec_t precz, prec1, precs, precs1; + int inex; + MPFR_GROUP_DECL (group); + MPFR_ZIV_DECL (loop); + MPFR_SAVE_EXPO_DECL (expo); + + MPFR_LOG_FUNC ( + ("s[%Pu]=%.*Rg rnd=%d", mpfr_get_prec (s), mpfr_log_prec, s, rnd_mode), + ("z[%Pu]=%.*Rg inexact=%d", mpfr_get_prec (z), mpfr_log_prec, z, inex)); + + /* Zero, Nan or Inf ? */ + if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (s))) + { + if (MPFR_IS_NAN (s)) + { + MPFR_SET_NAN (z); + MPFR_RET_NAN; + } + else if (MPFR_IS_INF (s)) + { + if (MPFR_IS_POS (s)) + return mpfr_set_ui (z, 1, MPFR_RNDN); /* Zeta(+Inf) = 1 */ + MPFR_SET_NAN (z); /* Zeta(-Inf) = NaN */ + MPFR_RET_NAN; + } + else /* s iz zero */ + { + MPFR_ASSERTD (MPFR_IS_ZERO (s)); + return mpfr_set_si_2exp (z, -1, -1, rnd_mode); + } + } + + /* s is neither Nan, nor Inf, nor Zero */ + + /* check tiny s: we have zeta(s) = -1/2 - 1/2 log(2 Pi) s + ... around s=0, + and for |s| <= 0.074, we have |zeta(s) + 1/2| <= |s|. + Thus if |s| <= 1/4*ulp(1/2), we can deduce the correct rounding + (the 1/4 covers the case where |zeta(s)| < 1/2 and rounding to nearest). + A sufficient condition is that EXP(s) + 1 < -PREC(z). */ + if (MPFR_GET_EXP (s) + 1 < - (mpfr_exp_t) MPFR_PREC(z)) + { + int signs = MPFR_SIGN(s); + + MPFR_SAVE_EXPO_MARK (expo); + mpfr_set_si_2exp (z, -1, -1, rnd_mode); /* -1/2 */ + if (rnd_mode == MPFR_RNDA) + rnd_mode = MPFR_RNDD; /* the result is around -1/2, thus negative */ + if ((rnd_mode == MPFR_RNDU || rnd_mode == MPFR_RNDZ) && signs < 0) + { + mpfr_nextabove (z); /* z = -1/2 + epsilon */ + inex = 1; + } + else if (rnd_mode == MPFR_RNDD && signs > 0) + { + mpfr_nextbelow (z); /* z = -1/2 - epsilon */ + inex = -1; + } + else + { + if (rnd_mode == MPFR_RNDU) /* s > 0: z = -1/2 */ + inex = 1; + else if (rnd_mode == MPFR_RNDD) + inex = -1; /* s < 0: z = -1/2 */ + else /* (MPFR_RNDZ and s > 0) or MPFR_RNDN: z = -1/2 */ + inex = (signs > 0) ? 1 : -1; + } + MPFR_SAVE_EXPO_FREE (expo); + return mpfr_check_range (z, inex, rnd_mode); + } + + /* Check for case s= -2n */ + if (MPFR_IS_NEG (s)) + { + mpfr_t tmp; + tmp[0] = *s; + MPFR_EXP (tmp) = MPFR_GET_EXP (s) - 1; + if (mpfr_integer_p (tmp)) + { + MPFR_SET_ZERO (z); + MPFR_SET_POS (z); + MPFR_RET (0); + } + } + + /* Check for case s= 1 before changing the exponent range */ + if (mpfr_cmp (s, __gmpfr_one) ==0) + { + MPFR_SET_INF (z); + MPFR_SET_POS (z); + mpfr_set_divby0 (); + MPFR_RET (0); + } + + MPFR_SAVE_EXPO_MARK (expo); + + /* Compute Zeta */ + if (MPFR_IS_POS (s) && MPFR_GET_EXP (s) >= 0) /* Case s >= 1/2 */ + inex = mpfr_zeta_pos (z, s, rnd_mode); + else /* use reflection formula + zeta(s) = 2^s*Pi^(s-1)*sin(Pi*s/2)*gamma(1-s)*zeta(1-s) */ + { + int overflow = 0; + + precz = MPFR_PREC (z); + precs = MPFR_PREC (s); + + /* Precision precs1 needed to represent 1 - s, and s + 2, + without any truncation */ + precs1 = precs + 2 + MAX (0, - MPFR_GET_EXP (s)); + sd = mpfr_get_d (s, MPFR_RNDN) - 1.0; + if (sd < 0.0) + sd = -sd; /* now sd = abs(s-1.0) */ + /* Precision prec1 is the precision on elementary computations; + it ensures a final precision prec1 - add for zeta(s) */ + /* eps = pow (2.0, - (double) precz - 14.0); */ + eps = __gmpfr_ceil_exp2 (- (double) precz - 14.0); + m1 = 1.0 + MAX(1.0 / eps, 2.0 * sd) * (1.0 + eps); + c = (1.0 + eps) * (1.0 + eps * MAX(8.0, m1)); + /* add = 1 + floor(log(c*c*c*(13 + m1))/log(2)); */ + add = __gmpfr_ceil_log2 (c * c * c * (13.0 + m1)); + prec1 = precz + add; + prec1 = MAX (prec1, precs1) + 10; + + MPFR_GROUP_INIT_4 (group, prec1, z_pre, s1, y, p); + MPFR_ZIV_INIT (loop, prec1); + for (;;) + { + mpfr_sub (s1, __gmpfr_one, s, MPFR_RNDN);/* s1 = 1-s */ + mpfr_zeta_pos (z_pre, s1, MPFR_RNDN); /* zeta(1-s) */ + mpfr_gamma (y, s1, MPFR_RNDN); /* gamma(1-s) */ + if (MPFR_IS_INF (y)) /* Zeta(s) < 0 for -4k-2 < s < -4k, + Zeta(s) > 0 for -4k < s < -4k+2 */ + { + mpfr_div_2ui (s1, s, 2, MPFR_RNDN); /* s/4, exact */ + mpfr_frac (s1, s1, MPFR_RNDN); /* exact, -1 < s1 < 0 */ + overflow = (mpfr_cmp_si_2exp (s1, -1, -1) > 0) ? -1 : 1; + break; + } + mpfr_mul (z_pre, z_pre, y, MPFR_RNDN); /* gamma(1-s)*zeta(1-s) */ + mpfr_const_pi (p, MPFR_RNDD); + mpfr_mul (y, s, p, MPFR_RNDN); + mpfr_div_2ui (y, y, 1, MPFR_RNDN); /* s*Pi/2 */ + mpfr_sin (y, y, MPFR_RNDN); /* sin(Pi*s/2) */ + mpfr_mul (z_pre, z_pre, y, MPFR_RNDN); + mpfr_mul_2ui (y, p, 1, MPFR_RNDN); /* 2*Pi */ + mpfr_neg (s1, s1, MPFR_RNDN); /* s-1 */ + mpfr_pow (y, y, s1, MPFR_RNDN); /* (2*Pi)^(s-1) */ + mpfr_mul (z_pre, z_pre, y, MPFR_RNDN); + mpfr_mul_2ui (z_pre, z_pre, 1, MPFR_RNDN); + + if (MPFR_LIKELY (MPFR_CAN_ROUND (z_pre, prec1 - add, precz, + rnd_mode))) + break; + + MPFR_ZIV_NEXT (loop, prec1); + MPFR_GROUP_REPREC_4 (group, prec1, z_pre, s1, y, p); + } + MPFR_ZIV_FREE (loop); + if (overflow != 0) + { + inex = mpfr_overflow (z, rnd_mode, overflow); + MPFR_SAVE_EXPO_UPDATE_FLAGS (expo, MPFR_FLAGS_OVERFLOW); + } + else + inex = mpfr_set (z, z_pre, rnd_mode); + MPFR_GROUP_CLEAR (group); + } + + MPFR_SAVE_EXPO_FREE (expo); + return mpfr_check_range (z, inex, rnd_mode); +} |