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arb/partitions/hrr_sum_fmprb.c
Fredrik Johansson fe55e3c8a1 optionally allow use of doubles in partitions
2013-06-10 14:53:52 +02:00

415 lines
9.9 KiB
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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) 2013 Fredrik Johansson
******************************************************************************/
#include "partitions.h"
#include "arith.h"
#include "fmprb.h"
#include "math.h"
#define DOUBLE_CUTOFF 40
#define DOUBLE_ERR 1e-12
#define DOUBLE_PREC 53
#define MIN_PREC 20
#define PI 3.141592653589793238462643
#define INV_LOG2 (1.44269504088896340735992468 + 1e-12)
#define HRR_A (1.1143183348516376904 + 1e-12) /* 44*pi^2/(225*sqrt(3)) */
#define HRR_B (0.0592384391754448833 + 1e-12) /* pi*sqrt(2)/75 */
#define HRR_C (2.5650996603237281911 + 1e-12) /* pi*sqrt(2/3) */
#define HRR_D (1.2424533248940001551 + 1e-12) /* log(2) + log(3)/2 */
static double
partitions_remainder_bound(double n, double terms)
{
return HRR_A/sqrt(terms)
+ HRR_B*sqrt(terms/(n-1)) * sinh(HRR_C * sqrt(n)/terms);
}
/* Crude upper bound, sufficient to estimate the precision */
static double
log_sinh(double x)
{
if (x > 4)
return x;
else
return log(x) + x*x*(1/6.);
}
static double
partitions_remainder_bound_log2(double n, double N)
{
double t1, t2;
t1 = log(HRR_A) - 0.5*log(N);
t2 = log(HRR_B) + 0.5*(log(N) - log(n-1)) + log_sinh(HRR_C * sqrt(n)/N);
return (FLINT_MAX(t1, t2) + 1) * INV_LOG2;
}
static long
partitions_needed_terms(ulong n)
{
long N;
for (N = 1; partitions_remainder_bound_log2(n, N) > 10; N++);
for ( ; partitions_remainder_bound(n, N) > 0.4; N++);
return N;
}
static double
partitions_term_bound(double n, double k)
{
return ((PI*sqrt(24*n-1) / (6.0*k)) + HRR_D - log(24.0*n-1) + 0.5*log(k)) * INV_LOG2;
}
/* Bound number of prime factors in k */
static mp_limb_t primorial_tab[] = {
1, 2, 6, 30, 210, 2310, 30030, 510510, 9699690, 223092870,
#if FLINT64
6469693230UL, 200560490130UL, 7420738134810UL, 304250263527210UL,
13082761331670030UL, 614889782588491410UL
#endif
};
static __inline__ int
bound_primes(ulong k)
{
int i;
for (i = 0; i < sizeof(primorial_tab) / sizeof(mp_limb_t); i++)
if (k <= primorial_tab[i])
return i;
return i;
}
static __inline__ long
log2_ceil(double x)
{
/* ceil(log2(n)) = bitcount(n-1);
this is too large if x is a power of two */
return FLINT_BIT_COUNT((long) x);
}
static long
partitions_prec_bound(ulong n, long k, long N)
{
long prec;
prec = partitions_term_bound(n, k);
prec += log2_ceil(8 * N * (26 * (sqrt(n) / k) + 7 * bound_primes(k) + 22));
return prec;
}
static double
cos_pi_pq(mp_limb_signed_t p, mp_limb_signed_t q)
{
/* Force 0 <= p < q */
p = FLINT_ABS(p);
p %= (2 * q);
if (p >= q)
p = 2 * q - p;
if (4 * p <= q)
return cos(p * PI / q);
else if (4 * p < 3 * q)
return sin((q - 2*p) * PI / (2 * q));
else
return -cos((q - p) * PI / q);
}
static double
eval_trig_prod_d(trig_prod_t prod)
{
int i;
double s;
mp_limb_t v;
if (prod->prefactor == 0)
return 0.0;
s = prod->prefactor;
v = n_gcd(FLINT_MAX(prod->sqrt_p, prod->sqrt_q),
FLINT_MIN(prod->sqrt_p, prod->sqrt_q));
prod->sqrt_p /= v;
prod->sqrt_q /= v;
if (prod->sqrt_p != 1)
s *= sqrt(prod->sqrt_p);
if (prod->sqrt_q != 1)
s /= sqrt(prod->sqrt_q);
for (i = 0; i < prod->n; i++)
s *= cos_pi_pq(prod->cos_p[i], prod->cos_q[i]);
return s;
}
static void
eval_trig_prod(fmprb_t sum, trig_prod_t prod, long prec)
{
int i;
mp_limb_t v;
fmprb_t t;
if (prod->prefactor == 0)
{
fmprb_zero(sum);
return;
}
fmprb_init(t);
fmprb_set_si(sum, prod->prefactor);
v = n_gcd(FLINT_MAX(prod->sqrt_p, prod->sqrt_q),
FLINT_MIN(prod->sqrt_p, prod->sqrt_q));
prod->sqrt_p /= v;
prod->sqrt_q /= v;
if (prod->sqrt_p != 1)
{
fmprb_sqrt_ui(t, prod->sqrt_p, prec);
fmprb_mul(sum, sum, t, prec);
}
if (prod->sqrt_q != 1)
{
fmprb_rsqrt_ui(t, prod->sqrt_q, prec);
fmprb_mul(sum, sum, t, prec);
}
for (i = 0; i < prod->n; i++)
{
fmpq_t pq;
*fmpq_numref(pq) = prod->cos_p[i];
*fmpq_denref(pq) = prod->cos_q[i];
fmprb_cos_pi_fmpq(t, pq, prec);
fmprb_mul(sum, sum, t, prec);
}
fmprb_clear(t);
}
static void
sinh_cosh_divk_precomp(fmprb_t sh, fmprb_t ch, fmprb_t ex, long k, long prec)
{
fmprb_t t;
fmprb_init(t);
fmprb_set_round(t, ex, prec);
fmprb_root(ch, t, k, prec);
/* The second term doesn't need full precision,
but this doesn't affect performance that much... */
fmprb_ui_div(t, 1, ch, prec);
fmprb_sub(sh, ch, t, prec);
fmprb_add(ch, ch, t, prec);
fmprb_mul_2exp_si(ch, ch, -1);
fmprb_mul_2exp_si(sh, sh, -1);
fmprb_clear(t);
}
void
partitions_hrr_sum_fmprb(fmprb_t x, ulong n, long N0, long N, int use_doubles)
{
trig_prod_t prod;
fmprb_t acc, C, t1, t2, t3, t4, exp1;
fmpr_t bound;
fmpz_t n24;
long k, prec, res_prec, acc_prec, guard_bits;
double Cd;
if (n <= 2)
{
fmprb_set_ui(x, FLINT_MAX(1, n));
return;
}
/* compute initial precision */
guard_bits = 2 * FLINT_BIT_COUNT(N) + 32;
prec = partitions_remainder_bound_log2(n, N0) + guard_bits;
prec = FLINT_MAX(prec, DOUBLE_PREC);
res_prec = acc_prec = prec;
fmprb_init(acc);
fmprb_init(C);
fmprb_init(t1);
fmprb_init(t2);
fmprb_init(t3);
fmprb_init(t4);
fmprb_init(exp1);
fmpz_init(n24);
fmprb_zero(x);
/* n24 = 24n - 1 */
fmpz_set_ui(n24, n);
fmpz_mul_ui(n24, n24, 24);
fmpz_sub_ui(n24, n24, 1);
/* C = (pi/6) sqrt(24n-1) */
fmprb_const_pi(t1, prec);
fmprb_sqrt_fmpz(t2, n24, prec);
fmprb_mul(t1, t1, t2, prec);
fmprb_div_ui(C, t1, 6, prec);
/* exp1 = exp(C) */
fmprb_exp(exp1, C, prec);
Cd = PI * sqrt(24*n-1) / 6;
for (k = N0; k <= N; k++)
{
trig_prod_init(prod);
arith_hrr_expsum_factored(prod, k, n % k);
if (prod->prefactor != 0)
{
if (prec > MIN_PREC)
prec = partitions_prec_bound(n, k, N);
prod->prefactor *= 4;
prod->sqrt_p *= 3;
prod->sqrt_q *= k;
if (prec > DOUBLE_CUTOFF || !use_doubles)
{
/* Compute A_k(n) * sqrt(3/k) * 4 / (24*n-1) */
eval_trig_prod(t1, prod, prec);
fmprb_div_fmpz(t1, t1, n24, prec);
/* Multiply by (cosh(z) - sinh(z)/z) where z = C / k */
fmprb_set_round(t2, C, prec);
fmprb_div_ui(t2, t2, k, prec);
if (k < 35 && prec > 1000)
sinh_cosh_divk_precomp(t3, t4, exp1, k, prec);
else
fmprb_sinh_cosh(t3, t4, t2, prec);
fmprb_div(t3, t3, t2, prec);
fmprb_sub(t2, t4, t3, prec);
fmprb_mul(t1, t1, t2, prec);
}
else
{
double xx, zz, xxerr;
xx = eval_trig_prod_d(prod) / (24*n - 1);
zz = Cd / k;
xx = xx * (cosh(zz) - sinh(zz) / zz);
xxerr = fabs(xx) * DOUBLE_ERR + DOUBLE_ERR;
fmpr_set_d(fmprb_midref(t1), xx);
fmpr_set_d(fmprb_radref(t1), xxerr);
}
/* Add to accumulator */
fmprb_add(acc, acc, t1, acc_prec);
if (acc_prec > 2 * prec + 32)
{
fmprb_add(x, x, acc, res_prec);
fmprb_zero(acc);
acc_prec = prec + 32;
}
}
}
fmprb_add(x, x, acc, res_prec);
fmpr_init(bound);
partitions_rademacher_bound(bound, n, N);
/* printf("addery %lu %lu: ", n, N); fmpr_printd(bound, 20); printf("\n"); */
fmpr_add(fmprb_radref(x), fmprb_radref(x), bound, FMPRB_RAD_PREC, FMPR_RND_UP);
fmpr_clear(bound);
fmpz_clear(n24);
fmprb_clear(acc);
fmprb_clear(exp1);
fmprb_clear(C);
fmprb_clear(t1);
fmprb_clear(t2);
fmprb_clear(t3);
fmprb_clear(t4);
}
/* defined in flint*/
#define NUMBER_OF_SMALL_PARTITIONS 128
extern const unsigned int partitions_lookup[NUMBER_OF_SMALL_PARTITIONS];
void
partitions_fmpz_ui(fmpz_t p, ulong n)
{
fmprb_t x;
if (n < NUMBER_OF_SMALL_PARTITIONS)
{
fmpz_set_ui(p, partitions_lookup[n]);
return;
}
fmprb_init(x);
partitions_hrr_sum_fmprb(x, n, 1, partitions_needed_terms(n), 0);
if (!fmprb_get_unique_fmpz(p, x))
{
printf("not unique!\n");
fmprb_printd(x, 50);
printf("\n");
abort();
}
fmprb_clear(x);
}
void
partitions_fmpz_ui_using_doubles(fmpz_t p, ulong n)
{
fmprb_t x;
if (n < NUMBER_OF_SMALL_PARTITIONS)
{
fmpz_set_ui(p, partitions_lookup[n]);
return;
}
fmprb_init(x);
partitions_hrr_sum_fmprb(x, n, 1, partitions_needed_terms(n), 1);
if (!fmprb_get_unique_fmpz(p, x))
{
printf("not unique!\n");
fmprb_printd(x, 50);
printf("\n");
abort();
}
fmprb_clear(x);
}
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