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arb/fmprb/const_euler.c

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/*=============================================================================
This file is part of ARB.
ARB is free software; you can redistribute it and/or modify
it under the terms of the GNU General Public License as published by
the Free Software Foundation; either version 2 of the License, or
(at your option) any later version.
ARB 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 General Public License for more details.
You should have received a copy of the GNU General Public License
along with ARB; if not, write to the Free Software
Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA
=============================================================================*/
/******************************************************************************
Copyright (C) 2012, 2013 Fredrik Johansson
******************************************************************************/
#include "zeta.h"
#include "hypgeom.h"
typedef struct
{
fmprb_t P;
fmprb_t Q;
fmprb_t T;
fmprb_t C;
fmprb_t D;
fmprb_t V;
} euler_bsplit_struct;
typedef euler_bsplit_struct euler_bsplit_t[1];
static void euler_bsplit_init(euler_bsplit_t s)
{
fmprb_init(s->P);
fmprb_init(s->Q);
fmprb_init(s->T);
fmprb_init(s->C);
fmprb_init(s->D);
fmprb_init(s->V);
}
static void euler_bsplit_clear(euler_bsplit_t s)
{
fmprb_clear(s->P);
fmprb_clear(s->Q);
fmprb_clear(s->T);
fmprb_clear(s->C);
fmprb_clear(s->D);
fmprb_clear(s->V);
}
static void
euler_bsplit_1_merge(euler_bsplit_t s, euler_bsplit_t L, euler_bsplit_t R,
long wp, int cont)
{
fmprb_t t, u, v;
fmprb_init(t);
fmprb_init(u);
fmprb_init(v);
if (cont)
fmprb_mul(s->P, L->P, R->P, wp);
fmprb_mul(s->Q, L->Q, R->Q, wp);
fmprb_mul(s->D, L->D, R->D, wp);
/* T = LP RT + RQ LT*/
fmprb_mul(t, L->P, R->T, wp);
fmprb_mul(v, R->Q, L->T, wp);
fmprb_add(s->T, t, v, wp);
/* C = LC RD + RC LD */
if (cont)
{
fmprb_mul(s->C, L->C, R->D, wp);
fmprb_addmul(s->C, R->C, L->D, wp);
}
/* V = RD (RQ LV + LC LP RT) + LD LP RV */
fmprb_mul(u, L->P, R->V, wp);
fmprb_mul(u, u, L->D, wp);
fmprb_mul(v, R->Q, L->V, wp);
fmprb_addmul(v, t, L->C, wp);
fmprb_mul(v, v, R->D, wp);
fmprb_add(s->V, u, v, wp);
fmprb_clear(t);
fmprb_clear(u);
fmprb_clear(v);
}
void
euler_bsplit_1(euler_bsplit_t s, long n1, long n2, long N, long wp, int cont)
{
if (n2 - n1 == 1)
{
fmprb_set_si(s->P, N); /* p = N^2 */
fmprb_mul(s->P, s->P, s->P, wp);
fmprb_set_si(s->Q, n1 + 1); /* q = (k + 1)^2 */
fmprb_mul(s->Q, s->Q, s->Q, wp);
fmprb_set_si(s->C, 1);
fmprb_set_si(s->D, n1 + 1);
fmprb_set(s->T, s->P);
fmprb_set(s->V, s->P);
}
else
{
euler_bsplit_t L, R;
long m = (n1 + n2) / 2;
euler_bsplit_init(L);
euler_bsplit_init(R);
euler_bsplit_1(L, n1, m, N, wp, 1);
euler_bsplit_1(R, m, n2, N, wp, 1);
euler_bsplit_1_merge(s, L, R, wp, cont);
euler_bsplit_clear(L);
euler_bsplit_clear(R);
}
}
void
euler_bsplit_2(fmprb_t P, fmprb_t Q, fmprb_t T, long n1, long n2,
long N, long wp, int cont)
{
if (n2 - n1 == 1)
{
if (n1 == 0)
{
fmprb_set_si(P, 1);
fmprb_set_si(Q, 4 * N);
fmprb_set_si(T, 1);
}
else
{
fmprb_si_pow_ui(P, 1 - 2*n1, 3, wp);
fmprb_neg(P, P);
fmprb_set_si(Q, 32 * n1);
fmprb_mul_ui(Q, Q, N, wp);
fmprb_mul_ui(Q, Q, N, wp);
}
fmprb_set(T, P);
}
else
{
fmprb_t P2, Q2, T2;
long m = (n1 + n2) / 2;
fmprb_init(P2);
fmprb_init(Q2);
fmprb_init(T2);
euler_bsplit_2(P, Q, T, n1, m, N, wp, 1);
euler_bsplit_2(P2, Q2, T2, m, n2, N, wp, 1);
fmprb_mul(T, T, Q2, wp);
fmprb_mul(T2, T2, P, wp);
fmprb_add(T, T, T2, wp);
if (cont)
fmprb_mul(P, P, P2, wp);
fmprb_mul(Q, Q, Q2, wp);
fmprb_clear(P2);
fmprb_clear(Q2);
fmprb_clear(T2);
}
}
static void
atanh_bsplit(fmprb_t s, ulong c, long a, long prec)
{
fmprb_t t;
hypgeom_t series;
hypgeom_init(series);
fmprb_init(t);
fmpz_poly_set_ui(series->A, 1);
fmpz_poly_set_coeff_ui(series->B, 0, 1);
fmpz_poly_set_coeff_ui(series->B, 1, 2);
fmpz_poly_set_ui(series->P, 1);
fmpz_poly_set_ui(series->Q, c * c);
fmprb_hypgeom_infsum(s, t, series, prec, prec);
fmprb_mul_si(s, s, a, prec);
fmprb_mul_ui(t, t, c, prec);
fmprb_div(s, s, t, prec);
fmprb_clear(t);
hypgeom_clear(series);
}
static ulong
next_smooth(ulong n)
{
ulong t, k;
for (k = n; ; k++)
{
t = k;
while (t % 2 == 0) t /= 2;
while (t % 3 == 0) t /= 3;
while (t % 5 == 0) t /= 5;
if (t == 1)
return k;
}
}
void
fmprb_log_ui_smooth(fmprb_t s, ulong n, long prec)
{
ulong m, i, j, k;
fmprb_t t;
m = n;
i = j = k = 0;
while (m % 2 == 0) { m /= 2; i++; }
while (m % 3 == 0) { m /= 3; j++; }
while (m % 5 == 0) { m /= 5; k++; }
if (m != 1)
abort();
fmprb_init(t);
prec += FLINT_CLOG2(prec);
atanh_bsplit(s, 31, 14*i + 22*j + 32*k, prec);
atanh_bsplit(t, 49, 10*i + 16*j + 24*k, prec);
fmprb_add(s, s, t, prec);
atanh_bsplit(t, 161, 6*i + 10*j + 14*k, prec);
fmprb_add(s, s, t, prec);
fmprb_clear(t);
}
void fmpr_gamma_ui_lbound(fmpr_t x, ulong n, long prec);
void
fmprb_const_euler_eval(fmprb_t res, long prec)
{
euler_bsplit_t sum;
fmprb_t t, u, v, P2, T2, Q2;
long bits, wp, wp2, n, K, M;
bits = prec + 10;
n = 0.086643397569993163677 * bits + 1; /* log(2) / 8 */
/* round n to have many trailing zeros, speeding up arithmetic,
and make it smooth to allow computing the logarithm cheaply */
if (n > 256)
{
int b = FLINT_BIT_COUNT(n);
n = next_smooth((n >> (b-4)) + 1) << (b-4);
}
else
{
n = next_smooth(n);
}
K = 4.9706257595442318644 * n; /* 3/W(3/e) */
M = 2 * n;
wp = bits + 2 * FLINT_BIT_COUNT(n);
wp2 = bits/2 + 2 * FLINT_BIT_COUNT(n);
euler_bsplit_init(sum);
fmprb_init(P2);
fmprb_init(T2);
fmprb_init(Q2);
fmprb_init(t);
fmprb_init(u);
fmprb_init(v);
/* Compute S0 = V / (Q D) + eps1
I0 = 1 + T / Q + eps2 */
euler_bsplit_1(sum, 0, K, n, wp, 0);
/* I0 = T / Q + eps2 */
fmprb_add(sum->T, sum->T, sum->Q, wp);
/* Assuming K > 2 and K >= 4n, eps1 and eps2 are both bounded by
2 H_K / (K!)^2 * n^(2K) < 4 log(K) * n^(2K) / (K!)^2
*/
{
fmpr_t e, f;
fmpr_init(e);
fmpr_init(f);
fmpr_set_ui(e, n);
fmpr_pow_sloppy_ui(e, e, 2 * K, FMPRB_RAD_PREC, FMPR_RND_UP);
fmpr_set_ui(f, K);
fmpr_log(f, f, FMPRB_RAD_PREC, FMPR_RND_UP);
fmpr_mul(e, e, f, FMPRB_RAD_PREC, FMPR_RND_UP);
fmpr_mul_2exp_si(e, e, 2);
fmpr_gamma_ui_lbound(f, K + 1, FMPRB_RAD_PREC);
fmpr_mul(f, f, f, FMPRB_RAD_PREC, FMPR_RND_DOWN);
fmpr_div(e, e, f, FMPRB_RAD_PREC, FMPR_RND_UP);
/* T / Q + eps = (T + eps Q) / Q */
fmprb_get_abs_ubound_fmpr(f, sum->Q, FMPRB_RAD_PREC);
fmpr_mul(e, e, f, FMPRB_RAD_PREC, FMPR_RND_UP);
fmprb_add_error_fmpr(sum->T, e);
/* V / (Q D) + eps = (V + eps Q D) / (Q D) */
fmprb_get_abs_ubound_fmpr(f, sum->D, FMPRB_RAD_PREC);
fmpr_mul(e, e, f, FMPRB_RAD_PREC, FMPR_RND_UP);
fmprb_add_error_fmpr(sum->V, e);
fmpr_clear(e);
fmpr_clear(f);
}
/* Compute S0 / I0 = V / (D T) */
fmprb_mul(t, sum->T, sum->D, wp);
fmprb_div(res, sum->V, t, wp);
/* Compute K0 (actually I_0(2n) K_0(2n)) = T2 / Q2 + eps */
euler_bsplit_2(P2, Q2, T2, 0, M, n, wp2, 0);
/* assuming M = 2n, eps is bounded by 2 exp(-4n) / n, and
T2 / Q2 = s + eps <=> (T2 - Q2 eps) / Q2 = s */
{
fmpr_t e, f;
fmpr_init(e);
fmpr_init(f);
fmpr_set_si(f, -4*n);
fmpr_exp(f, f, FMPRB_RAD_PREC, FMPR_RND_UP);
fmpr_div_ui(f, f, n, FMPRB_RAD_PREC, FMPR_RND_UP);
fmpr_mul_2exp_si(f, f, 1);
fmprb_get_abs_ubound_fmpr(e, Q2, FMPRB_RAD_PREC);
fmpr_mul(e, e, f, FMPRB_RAD_PREC, FMPR_RND_UP);
fmprb_add_error_fmpr(T2, e);
fmpr_clear(e);
fmpr_clear(f);
}
/* Compute K0 / I^2 = Q^2 * T2 / (Q2 * T^2) */
fmprb_set_round(t, sum->Q, wp2);
fmprb_mul(t, t, t, wp2);
fmprb_mul(t, t, T2, wp2);
fmprb_set_round(u, sum->T, wp2);
fmprb_mul(u, u, u, wp2);
fmprb_mul(u, u, Q2, wp2);
fmprb_div(t, t, u, wp2);
fmprb_sub(res, res, t, wp);
/* subtract log(n) */
fmprb_log_ui_smooth(u, n, wp);
fmprb_sub(res, res, u, wp);
fmprb_clear(P2);
fmprb_clear(T2);
fmprb_clear(Q2);
fmprb_clear(t);
fmprb_clear(u);
fmprb_clear(v);
euler_bsplit_clear(sum);
}
DEF_CACHED_CONSTANT(fmprb_const_euler, fmprb_const_euler_eval)
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