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/* mpfr_zeta_ui -- compute the Riemann Zeta function for integer argument.
Copyright 2005, 2006 Free Software Foundation.
This file is part of the MPFR Library.
The 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 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 MPFR Library; see the file COPYING.LIB. If not, write to
the Free Software Foundation, Inc., 51 Franklin Place, 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)
{
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;
int st = 0;
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;
}
}
/* FIXME: treat also the case where zeta(m) - (1+1/2^m) < ulp(1).
For m >= 6, we have zeta(m) - (1+1/2^m) < 2^(-1.5*m),
thus if p <= 1.5*m, then zeta(m) - (1+1/2^m) < 2^(-p)
and the result is either 1 or 1+. */
mpz_init (s);
mpz_init (d);
mpz_init (t);
mpz_init (q);
mpfr_init2 (y, MPFR_PREC_MIN);
p += MPFR_INT_CEIL_LOG2(p); /* account of the n term in the error */
do
{
p += MPFR_INT_CEIL_LOG2(p) + 15;
/* 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);
st = 0;
for (k = n; k > 0; k--)
{
count_leading_zeros (kbits, k);
kbits = m * (BITS_PER_MP_LIMB - kbits);
/* if k^m is too large, use mpz_tdiv_q */
if (kbits > 2 * BITS_PER_MP_LIMB)
{
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] = (2k)*(2k-1)/(n-k+1)/(n+k-1)/4 */
#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_2exp (y, y, p, GMP_RNDN);
err = MPFR_INT_CEIL_LOG2 (err);
}
while (MPFR_CAN_ROUND (y, p - err, MPFR_PREC(z), r) == 0);
mpz_clear (d);
mpz_clear (t);
mpz_clear (q);
mpz_clear (s);
inex = mpfr_set (z, y, r);
mpfr_clear (y);
return inex;
}
}
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